literature review of single index model

Construction of Optimal Portfolio Using Sharpe’s Single Index Model and Markowitz Model: An Empirical Study on Nifty50 Stocks

14 Pages Posted: 25 Oct 2018

Ajay Kumar Patel

GL Bajaj Institute of Management and Research

GL Bajaj Institute of Management and Research, Students

Date Written: June 2, 2017

The main focus of this research is to construct an optimal portfolio in Indian Market with the help of Sharpe Single index model. The construction of an optimal portfolio has become increasing challenging in recent years, as investors expect to maximize return and minimize risk from their respective investment therefor a good combination of portfolio will give maximum return for a particular level of risk. An investor need to have proper knowledge of security analysis and portfolio theory for making corrective investment decision. In 1950, Markowitz who was considered the father of modern portfolio theory, mainly because he is the first person who gave a mathematical model for portfolio optimization and diversification.Modern portfolio theory (MPT) is a theory of finance that attempts to maximize portfolio expected return for a given amount of risk, or minimize the risk for a given level of expected return. Markowitz theory advise investors to invest in multiple securities rather than pulling all eggs in one basket. In order to overcome the Markowitz Model, William Sharpe, tried to simplify the process of data inputs and reaching a solution, by developing a simplified variant of the Markowitz model.In the Sharpe’s model, the desirability of any securities inclusion in the portfolio is directly related to its excess return-to-beta ratio. The proposed method formulates a unique cut off point (Cut off rate of return) and selects stocks having excess of their expected return over risk free rate of return surpassing this cut-off point. Then they are ranked from highest to lowest order and then the Percentage of investment in each of the selected security is then decided on the basis of respective weights assigned to each security. In this research,all 50 stocks of NSE NIFTY 50 Index are taken into consideration and &Weekly data of all these stock for the period of September 14,2016 to September 15.2017 have been considered which further converted into annually. Further the proportion of investment of each 50 stock in the optimal portfolio was also calculated and along with risk and return of the selected stock are also been calculated with the help of Markowitz Model This study gains more importance as stocks included in Nifty 50 represent majority of market capitalisation of NSE.The Nifty 50 hold about 62.9% of the market capitalization of the stocks listed on NSE.

Keywords: Optimal Portfolio, Beta, Investment, Stock, Market Capitalization

JEL Classification: G11, G12

Suggested Citation: Suggested Citation

Ajay Kumar Patel (Contact Author)

Gl bajaj institute of management and research ( email ).

Knowledge Park II Greater Noida Uttar Pradesh, 201307 India

Subhodeep Chakraborty

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Literature Review

• First Online: 18 October 2020

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• Elissaios Sarmas 22 ,
• Panos Xidonas 23 &
• Haris Doukas 22  

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 163))

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In the basis of the Markowitz mean–variance methodology [ 112 , 113 ], a series of new models have been developed, such as the single index models, the multi-index models, the average correlation models, the mixed models, and the utility models. Additionally, different criteria have been proposed such as geometric mean return, skewness, etc.

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Abdelaziz, F Ben, El Fayedh, Rimeh, and Rao, Ananth (2009). “A discrete stochastic goal program for portfolio selection: The case of United Arab Emirates equity market”. In: INFOR: Information Systems and Operational Research 47.1, pp. 5–13.

Google Scholar  

Abdelaziz, Fouad Ben and Masmoudi, Meryem (2014). “A multiple objective stochastic portfolio selection problem with random Beta”. In: International Transactions in Operational Research 21.6, pp. 919–933.

Ahmed, E and El-Alem, M (2005). “On multiobjective optimization in portfolio management”. In: Applied mathematics and computation 167.1, pp. 616–621.

Akbay, Mehmet Anil, Kalayci, Can B, and Polat, Olcay (2020). “A parallel variable neighborhood search algorithm with quadratic programming for cardinality constrained portfolio optimization”. In: Knowledge-Based Systems , p. 105944.

Albadvi, Amir, Chaharsooghi, S Kamal, and Esfahanipour, Akbar (2007). “Decision making in stock trading: An application of PROMETHEE”. In: European journal of operational research 177.2, pp. 673–683.

Alexander, Gordon J and Resnick, Bruce G (1985). “Using linear and goal programming to immunize bond portfolios”. In: Journal of Banking & Finance 9.1, pp. 35–54.

Amiri, Maghsoud, Zandieh, Mostafa, Vahdani, Behnam, Soltani, R, and Roshanaei, V (2010). “An integrated eigenvector–DEA–TOPSIS methodology for portfolio risk evaluation in the FOREX spot market”. In: Expert Systems with Applications 37.1, pp. 509–516.

Ammar, Elsaid El (2008). “On solutions of fuzzy random multiobjective quadratic programming with applications in portfolio problem”. In: Information Sciences 178.2, pp. 468–484.

Aouni, Belaid, Doumpos, Michael, Pérez-Gladish, Blanca, and Steuer, Ralph E (2018). “On the increasing importance of multiple criteria decision aid methods for portfolio selection”. In: Journal of the Operational Research Society 69.10, pp. 1525–1542.

Bagheri, Noushin (2019). “Deterministic goal programming approach for Islamic portfolio selection”. In: Operational Research , pp. 1–13.

Ballestero, E (1998). “Approximating the optimum portfolio for an investor with particular preferences”. In: Journal of the Operational Research Society 49.9, pp. 998–1000.

Ballestero, Enrique, Bravo, Mila, Pérez-Gladish, Blanca, Arenas-Parra, Mar, and Pla-Santamaria, David (2012). “Socially responsible investment: A multicriteria approach to portfolio selection combining ethical and financial objectives”. In: European Journal of Operational Research 216.2, pp. 487–494.

Ballestero, Enrique and Garcia-Bernabeu, Ana (2012). “Portfolio selection with multiple time horizons: A mean variance—Stochastic goal programming approach”. In: INFOR: Information Systems and Operational Research 50.3, pp. 106–116.

Ballestero, Enrique, Pérez-Gladish, Blanca, Arenas-Parra, Mar, and Bilbao- Terol, Amelia (2009). “Selecting portfolios given multiple Eurostoxx-based uncertainty scenarios: A stochastic goal programming approach from fuzzy betas”. In: INFOR: Information Systems and Operational Research 47.1, pp. 59–70.

Behzadian, Majid, Otaghsara, S Khanmohammadi, Yazdani, Morteza, and Ignatius, Joshua (2012). “A state-of the-art survey of TOPSIS applications”. In: Expert Systems with applications 39.17, pp. 13051–13069.

Bernard III, Taylor W and Keown, Arthur J (1978). “A goal programming application of capital project selection in the production area”. In: AIIE Transactions 10.1, pp. 52–57.

Bertsimas, Dimitris, Darnell, Christopher, and Soucy, Robert (1999). “Portfolio construction through mixed-integer programming at Grantham, Mayo, Van Otterloo and Company”. In: Interfaces 29.1, pp. 49–66.

Bilbao-Terol, Amelia, Arenas-Parra, Mar, and Cañal-Fernández, Verónica (2012). “Selection of socially responsible portfolios using goal programming and fuzzy technology”. In: Information Sciences 189, pp. 110–125.

Bilbao-Terol, Amelia, Arenas-Parra, Mar, Cañal-Fernández, Verónica, and Antomil-Ibias, José (2014). “Using TOPSIS for assessing the sustainability of government bond funds”. In: Omega 49, pp. 1–17.

Bilbao-Terol, Amelia, Arenas-Parra, Mar, Cañal-Fernández, Verónica, and Bilbao-Terol, Celia (2013). “Selection of socially responsible portfolios using hedonic prices”. In: Journal of business ethics 115.3, pp. 515–529.

Bilbao-Terol, Amelia, Arenas-Parra, Mar, Cañal-Fernández, Verónica, and Bilbao-Terol, Celia (2016a). “Multi-criteria decision making for choosing socially responsible investment within a behavioral portfolio theory framework: A new way of investing into a crisis environment”. In: Annals of Operations Research 247.2, pp. 549–580.

Bilbao-Terol, Amelia, Arenas-Parra, Mar, Cañal-Fernández, Verónica, and Jiménez, Mariano (2016b). “A sequential goal programming model with fuzzy hierarchies to sustainable and responsible portfolio selection problem”. In: Journal of the Operational Research Society 67.10, pp. 1259–1273.

Booth, G Geoffrey and Dash Jr, Gordon H (1977). “Bank Portfolio Management Using Nonlinear Two Stage Goal Programming”. In: Financial Review 12.1, pp. 59–69.

Bouri, Abdelfettah, Martel, Jean-Marc, and Chabchoub, Habib (2002). “A multi-criterion approach for selecting attractive portfolio”. In: Journal of Multi-Criteria Decision Analysis 11.4-5, pp. 269–277.

Branke, Jürgen, Scheckenbach, B, Stein, Michael, Deb, Kalyanmoy, and Schmeck, Hartmut (2009). “Portfolio optimization with an envelope-based multi-objective evolutionary algorithm”. In: European Journal of Operational Research 199.3, pp. 684–693.

Bravo, Mila, Pla-Santamaria, David, and Garcia-Bernabeu, Ana (2010). “Portfolio selection from multiple benchmarks: A goal programming approach to an actual case”. In: Journal of Multi-Criteria Decision Analysis 17.5-6, pp. 155–166.

Briec, Walter, Kerstens, Kristiaan, and Van de Woestyne, Ignace (2013). “Portfolio selection with skewness: A comparison of methods and a generalized one fund result”. In: European Journal of Operational Research 230.2, pp. 412–421.

Calvo, Clara, Ivorra, Carlos, and Liern, Vicente (2012). “On the computation of the efficient frontier of the portfolio selection problem”. In: Journal of Applied Mathematics 2012.

Calvo, Clara, Ivorra, Carlos, and Liern, Vicente (2015). “Finding socially responsible portfolios close to conventional ones”. In: International Review of Financial Analysis 40, pp. 52–63.

Calvo, Clara, Ivorra, Carlos, and Liern, Vicente (2016). “Fuzzy portfolio selection with non-financial goals: exploring the efficient frontier”. In: Annals of Operations Research 245.1–2, pp. 31–46.

Calvo, Clara, Ivorra, Carlos, and Liern, Vicente (2018). “Controlling risk through diversification in portfolio selection with non-historical information”. In: Journal of the Operational Research Society 69.10, pp. 1543–1548.

Caplin, DA and Kornbluth, JSH (1975). “Multiobjective investment planning under uncertainty”. In: Omega 3.4, pp. 423–441.

Chen, Chen-Tung and Hung, Wei-Zhan (2009). “A new decision-making method for stock portfolio selection based on computing with linguistic assessment”. In: Advances in Decision Sciences 2009.

Chen, S (1995). “Multiobjective decision making on the portfolio selection”. In: Proceeding of the 6th National Conference on Multiple Criteria Decision Making, Beijing, SCI-TECH Information Services, Windsor, England , pp. 169–172.

Cheung, Michael Tow and Liao, Ziqi (2009). “Investing in real-world equity markets with an AHP-based decision framework”. In: Journal of decision systems 18.2, pp. 149–163.

Chunhachinda, Pornchai, Dandapani, Krishnan, Hamid, Shahid, and Prakash, Arun J (1997). “Portfolio selection and skewness: Evidence from international stock markets”. In: Journal of Banking & Finance 21.2, pp. 143–167.

Coffin, Mark A and Taylor III, Bernard W (1996). “Multiple criteria R&D project selection and scheduling using fuzzy logic”. In: Computers & Operations Research 23.3, pp. 207–220.

Colson, Gerard and De Bruyn, Christian (1989). “An integrated multiobjective portfolio management system”. In: Models and Methods in Multiple Criteria Decision Making . Elsevier, pp. 1359–1381.

Cooper, WW, Lelas, V, and Sueyoshi, T (1997). “Goal programming models and their duality relations for use in evaluating security portfolio and regression relations”. In: European Journal of Operational Research 98.2, pp. 431–443.

Davies, Ryan J, Kat, Harry M, and Lu, Sa (2016). “Fund of hedge funds portfolio selection: A multiple-objective approach”. In: Derivatives and Hedge Funds . Springer, pp. 45–71.

Dominiak, Cezary (1997). “An application of interactive multiple goal programming on the Warsaw stock exchange”. In: Advances in multiple objective and goal programming . Springer, pp. 66–74.

Doumpos, Michael, Spanos, M, and Zopounidis, Constantin (1999). “On the use of goal programming techniques in the assessment of financial risks”. In: Journal of Euro-Asian Management 5.1, pp. 83–100.

Doumpos, Michael and Zopounidis, Constantin (2002). Multicriteria decision aid classification methods . Vol. 73. Springer Science & Business Media.

Ehrgott, Matthias, Klamroth, Kathrin, and Schwehm, Christian (2004). “An MCDM approach to portfolio optimization”. In: European Journal of Operational Research 155.3, pp. 752–770.

Fauzi, Nur Aqilah Mohammed, Ismail, Munira, Jaaman, Saiful Hafizah, and Kamaruddin, Siti Noor Diana Mohd (2019). “Applicability of TOPSIS Model and Markowitz Model”. In: Journal of Physics: Conference Series . Vol. 1212. 1. IOP Publishing, p. 012032.

Garcia-Melón, Mónica, Pérez-Gladish, Blanca, Gómez-Navarro, Tomás, and Mendez-Rodriguez, Paz (2016). “Assessing mutual funds’ corporate social responsibility: a multistakeholder-AHP based methodology”. In: Annals of Operations Research 244.2, pp. 475–503.

Garcia, Fernando, Guijarro, Francisco, and Moya, Ismael (2013). “A multiobjective model for passive portfolio management: An application on the S&P 100 index”. In: Journal of Business Economics and Management 14.4, pp. 758–775.

Ghahtarani, Alireza and Najafi, Amir Abbas (2013). “Robust goal programming for multi-objective portfolio selection problem”. In: Economic Modelling 33, pp. 588–592.

Gladish, B Perez, Jones, DF, Tamiz, Merdah, and Terol, A Bilbao (2007). “An interactive three-stage model for mutual funds portfolio selection”. In: Omega 35.1, pp. 75–88.

Gupta, Pankaj, Mehlawat, Mukesh Kumar, Yadav, Sanjay, and Kumar, Arun (2019). “A polynomial goal programming approach for intuitionistic fuzzy portfolio optimization using entropy and higher moments”. In: Applied Soft Computing 85, p. 105781.

Gupta, Pankaj, Mittal, Garima, and Mehlawat, Mukesh Kumar (2013). “Multiobjective expected value model for portfolio selection in fuzzy environment”. In: Optimization Letters 7.8, pp. 1765–1791.

Han, Bing, Zhang, Xiao-Xiong, and Yi, Yujiang (2019). “Multi-criteria project selection using fuzzy preference relations based AHP and TOPSIS”. In: 2019 Chinese Control And Decision Conference (CCDC) . IEEE, pp. 3809–3816.

Harrington, Thomas C and Fischer, William A (1980). “Portfolio modeling in multiple-criteria situations under uncertainty: Comment”. In: Decision Sciences 11.1, pp. 171–177.

Hasuike, Takashi and Katagiri, Hideki (2014). “Risk-controlled multiobjective portfolio selection problem using a principle of compromise”. In: Mathematical Problems in Engineering 2014.

Hurson, Ch and Zopounidis, Constantin (1997). “On the use of multicriteria decision aid methods to portfolio selection”. In: Multicriteria analysis . Springer, pp. 496–507.

Hurson, Christian and Zopounidis, Constantin (1996). Gestion de portefeuille et analyse multicritere . FeniXX.

Jayasekara, Pubudu LW, Adelgren, Nathan, and Wiecek, Margaret M (n.d.). “On convex multiobjective programs with application to portfolio optimization”. In: Journal of Multi-Criteria Decision Analysis .

Khan, Ameer Hamza, Cao, Xinwei, Katsikis, Vasilios N, Stanimirović, Predrag, Brajević, Ivona, Li, Shuai, Kadry, Seifedine, and Nam, Yunyoung (2020). “Optimal Portfolio Management for Engineering Problems Using Nonconvex Cardinality Constraint: A Computing Perspective”. In: IEEE Access 8, pp. 57437–57450.

Khorramshahgol, Reza and Okoruwa, A Ason (1994). “A goal programming approach to investment decisions: A case study of fund allocation among different shopping malls”. In: European Journal of Operational Research 73.1, pp. 17–22.

Khoury, Nabil, Martel, Jean-Marc, and Veilleux, Marc (1993). “Méthode multicritère de sélection de portefeuilles indiciels internationaux”. In: L’Actualité économique 69.1, pp. 171–190.

K100r100ş, Şafak and Ustun, Ozden (2012). “An integrated approach for stock evaluation and portfolio optimization”. In: Optimization 61.4, pp. 423–441.

Kobayashi, Kiyoshi, Tanino, Tetuzo, Wierzbicki, Andrzej P, and Okumura, Makoto (1987). “Multifactor decision analysis for regional investment allocation”. In: Toward Interactive and Intelligent Decision Support Systems . Springer, pp. 422–431.

Kocadağl100, Ozan and Keskin, R100dvan (2015). “A novel portfolio selection model based on fuzzy goal programming with different importance and priorities”. In: Expert Systems with Applications 42.20, pp. 6898–6912.

Konno, Hiroshi, Shirakawa, Hiroshi, and Yamazaki, Hiroaki (1993). “A mean-absolute deviation-skewness portfolio optimization model”. In: Annals of Operations Research 45.1, pp. 205–220.

Konno, Hiroshi and Suzuki, Ken-ichi (1995). “A mean-variance-skewness portfolio optimization model”. In: Journal of the Operations Research Society of Japan 38.2, pp. 173–187.

Kumar, PC and Philippatos, GC (1979). “Conflict resolution in investment decisions: Implementation of goal programming methodology for dual- purpose funds”. In: Decision Sciences 10.4, pp. 562–576.

Kumar, PC, Philippatos, George C, and Ezzell, John R (1978). “Goal Programming and the Selection of Portfolios by Dual-Purpose Funds”. In: The Journal of Finance 33.1, pp. 303–310.

L’Hoir, H and Teghem, J (1995). “Portfolio selection by MOLP using an interactive branch and bound”. In: Foundations of Computing and Decision Sciences 20.3, pp. 175–185.

Lamata, Maria Teresa, Liern, Vicente, and Pérez-Gladish, Blanca (2018). “Doing good by doing well: A MCDM framework for evaluating corporate social responsibility attractiveness”. In: Annals of Operations Research 267.1–2, pp. 249–266.

Leão, Heloise Acco Tives, Canedo, Edna Dias, Costa, Pedro Henrique Teixeira, Okimoto, Márcio Vinicius, and Santos, Giovanni Almeida (2019). “Use of AHP and Promethee for Research Project Portfolio Selection”. In: International Conference on Computational Science and Its Applications . Springer, pp. 504–517.

Lee, Sang M and Chesser, Delton L (1980). “Goal programming for portfolio selection”. In: The Journal of Portfolio Management 6.3, pp. 22–26.

Lee, Sang M and Lerro, AJ (1973). “Optimizing the portfolio selection for mutual funds”. In: The Journal of Finance 28.5, pp. 1087–1101.

Leung, Mark T, Daouk, Hazem, and Chen, An-Sing (2001). “Using investment portfolio return to combine forecasts: a multiobjective approach”. In: European Journal of Operational Research 134.1, pp. 84–102.

Liu, Yong-Jun, Zhang, Wei-Guo, and Xu, Wei-Jun (2012). “Fuzzy multi-period portfolio selection optimization models using multiple criteria”. In: Automatica 48.12, pp. 3042–3053.

Mansini, Renata and Speranza, Maria Grazia (1999). “Heuristic algorithms for the portfolio selection problem with minimum transaction lots”. In: European Journal of Operational Research 114.2, pp. 219–233.

Marasović, Branka and Babić, Zoran (2011). “Two-step multi-criteria model for selecting optimal portfolio”. In: International Journal of Production Economics 134.1, pp. 58–66.

Markowitz, Harry (1952). “Portfolio Selection”. In: The Journal of Finance 7.1, pp. 77–91.

Markowitz, Harry (1959). Portfolio selection: Efficient diversification of investments . Vol. 16. John Wiley New York.

Marqués, Ana I, Garc100éa, Vicente, and Sánchez, J Salvador (n.d.). “Ranking-based MCDM models in financial management applications: analysis and emerging challenges”. In:.

Martel, J-M, Khoury, NT, and Bergeron, M (1988). “An application of a multicriteria approach to portfolio comparisons”. In: Journal of the Operational Research Society 39.7, pp. 617–628.

Martel, Jean-Marc, Khoury, Nabil, and M’Zali, Bouchra (1991). “Comparaison performance-taille des fonds mutuels par une analyse multicritère”. In: L’Actualité économique 67.3, pp. 306–324.

Masmoudi, Meryem and Abdelaziz, Fouad Ben (2017). “A chance constrained recourse approach for the portfolio selection problem”. In: Annals of Operations Research 251.1-2, pp. 243–254.

Masri, Hatem (2018). “A Shariah-compliant portfolio selection model”. In: Journal of the Operational Research Society 69.10, pp. 1568–1575.

Mavrotas, George, aas, Panos, and Psarras, John (2008). “An integrated multiple criteria methodology for supporting common stock portfolio selection decisions”. In: Proceedings of the 67th meeting of the European working group on multiple criteria decision aiding. Rovaniemi, Finland , pp. 56–71.

Mehlawat, Mukesh Kumar (2016). “Credibilistic mean-entropy models for multi-period portfolio selection with multi-choice aspiration levels”. In: Information Sciences 345, pp. 9–26.

Messaoudi, Laila, Aouni, Belaid, and Rebai, Abdelwaheb (2017). “Fuzzy chance-constrained goal programming model for multi-attribute financial portfolio selection”. In: Annals of Operations Research 251.1–2, pp. 193–204.

Moon, Yongma and Yao, Tao (2011). “A robust mean absolute deviation model for portfolio optimization”. In: Computers & Operations Research 38.9, pp. 1251–1258.

Muhlemann, Alan Paul and Lockett, Alan Geoffrey (1980). “Portfolio modeling in multiple-criteria situations under uncertainty: rejoinder”. In: Decision Sciences 11.1, pp. 178–180.

Muhlemann, Alan Paul, Lockett, Alan Geoffrey, and Gear, Anthony Edward (1978). “Portfolio modeling in multiple-criteria situations under uncertainty”. In: Decision Sciences 9.4, pp. 612–626.

Nakayama, H, Takeguchi, T, and Sano, M (1983). “Interactive graphics for portfolio selection”. In: Essays and Surveys on Multiple Criteria Decision Making . Springer, pp. 280–289.

Nguyen, Thi Thanh and Gordon-Brown, Lee (2011). “Constrained fuzzy hierarchical analysis for portfolio selection under higher moments”. In: IEEE Transactions on Fuzzy Systems 20.4, pp. 666–682.

Ogryczak, Wlodzimierz (2002). “Multiple criteria optimization and decisions under risk”. In: Control and Cybernetics 31.4, pp. 975–1003.

Ogryczak, Wlodzimierz (2000). “Multiple criteria linear programming model for portfolio selection”. In: Annals of Operations Research 97.1-4, pp. 143–162.

Orne, Daniel L, Rao, Ashok, and Wallace, William A (1975). “Profit maximization with the aid of goal programming for speculative housing estate developers”. In: Journal of the Operational Research Society 26.4, pp. 813–826.

Pardalos, Panos M, Sandström, Mattias, and Zopounidis, Costas (1994). “On the use of optimization models for portfolio selection: A review and some computational results”. In: Computational Economics 7.4, pp. 227–244.

Parra, M Arenas, Terol, A Bilbao, and Ur100a, MV Rodr100guez (2001). “A fuzzy goal programming approach to portfolio selection”. In: European Journal of Operational Research 133.2, pp. 287–297.

Pätäri, Eero, Karell, Ville, Luukka, Pasi, and Yeomans, Julian S (2018). “Comparison of the multicriteria decision-making methods for equity portfolio selection: The US evidence”. In: European Journal of Operational Research 265.2, pp. 655–672.

Pendaraki, Konstantina, Zopounidis, Constantin, and Doumpos, Michael (2005). “On the construction of mutual fund portfolios: A multicriteria methodology and an application to the Greek market of equity mutual funds”. In: European Journal of Operational Research 163.2, pp. 462–481.

Perez-Gladish, Blanca and M’Zali, Bouchra (2010). “An AHP-based approach to mutual funds’ social performance measurement”. In: International Journal of Multicriteria Decision Making 1.1, pp. 103–127.

Petrillo, Antonella, De Felice, Fabio, Garc100éa-Melón, Mónica, and Pérez-Gladish, Blanca (2016). “Investing in socially responsible mutual funds: Proposal of non-financial ranking in Italian market”. In: Research in International Business and Finance 37, pp. 541–555.

Powell, John G and Premachandra, IM (1998). “Accommodating diverse institutional investment objectives and constraints using non-linear goal programming”. In: European Journal of Operational Research 105.3, pp. 447–456.

Prakash, Arun J, Chang, Chun-Hao, and Pactwa, Therese E (2003). “Selecting a portfolio with skewness: Recent evidence from US, European, and Latin American equity markets”. In: Journal of Banking & Finance 27.7, pp. 1375–1390.

Qi, Yue (2019). “Parametrically computing efficient frontiers of portfolio selection and reporting and utilizing the piecewise-segment structure”. In: Journal of the Operational Research Society , pp. 1–16.

R100éos-Garc1̌00a, S and Rios-Insua, S (1983). “The portfolio selection problem with multiattributes and multiple criteria”. In: Essays and Surveys on Multiple Criteria Decision Making . Springer, pp. 317–325.

Roman, Diana, Darby-Dowman, Kenneth, and Mitra, Gautam (2007). “Mean-risk models using two risk measures: a multi-objective approach”. In: Quantitative Finance 7.4, pp. 443–458.

Rustem, Berc and Rustem, B (1998). Algorithms for nonlinear programming and multiple-objective decisions . Wiley Chichester.

Rys, Tomasz and Ziemba, W (1991). “Multiobjective investment scheduling problem”. In: Lecture Notes in Economics and Mathematical Systems 351, pp. 245–253.

Schlottmann, Frank and Seese, Detlef (2004). “A hybrid heuristic approach to discrete multi-objective optimization of credit portfolios”. In: Computational statistics & data analysis 47.2, pp. 373–399.

Schniederjans, AM and Schniederjans, MJ (2000). “Using the analytic hierarchy process and multi-objective programming for internet portfolio analysis”. In: Multi-Criteria Applications, JAI/Elsevier, Amsterdam , pp. 103–137.

Schniederjans, Marc J (1984). Linear goal programming . Petrocelli Books.

Schniederjans, Marc J, Zorn, Thomas S, and Johnson, Robert R (1993). “Allocating total wealth: A goal programming approach”. In: Computers & operations research 20.7, pp. 679–685.

Sealey Jr, CW (1977). “Commercial bank portfolio management with multiple objectives”. In: Journal of Commercial Bank Lending 59.6, pp. 39–48.

Sevastjanov, Pavel and Dymova, Ludmila (2009). “Stock screening with use of multiple criteria decision making and optimization”. In: Omega 37.3, pp. 659–671.

Shapiro, Jeremy F (1976). “Multiple criteria public investment decision making by mixed integer programming”. In: Multiple Criteria Decision Making . Springer, pp. 170–182.

Sheshai, KM El, Harwood, GB, and Hermanson, RH (1977). “Cost volume profit analysis with integer goal programming”. In: Management Accounting 59.4, pp. 43–47.

Shing, Chue and Nagasawa, Hiroyuki (1999). “Interactive decision system in stochastic multiobjective portfolio selection”. In: International Journal of Production Economics 60, pp. 187–193.

Skocz, Maciej, Zebrowski, Maciej, and Ziembla, Wieslaw (1989). “Spatial allocation and investment scheduling in the development programming”. In: Aspiration Based Decision Support Systems . Springer, pp. 322–338.

Skulimowski, Andrzej MJ (1996). “Dynamic portfolio optimization based on reference trajectories”. In: System Modelling and Optimization . Springer, pp. 171–178.

Speranza, M Grazia (1993). Linear programming models for portfolio optimization .

Speranza, M Grazia (1996). “A heuristic algorithm for a portfolio optimization model applied to the Milan stock market”. In: Computers & Operations Research 23.5, pp. 433–441.

Speranza, Maria Grazia (1994). “Linear models for portfolio selection and their application to the Milano stock market”. In: Financial modelling . Springer, pp. 320–333.

Spronk, Jaap, Steuer, Ralph E, and Zopounidis, Constantin (2005). “Multicriteria decision aid/analysis in finance”. In: Multiple criteria decision analysis: State of the art surveys . Springer, pp. 799–848.

Spronk, Jaap and Veeneklaas, Frank (1983). “A feasibility study of economic and environmental scenarios by means of interactive multiple goal programming”. In: Regional Science and Urban Economics 13.1, pp. 141–160.

Spronk, Jaap and Zambruno, G (1985). “Interactive multiple goal programming for bank portfolio selection”. In: Multiple Criteria Decision Methods and Applications . Springer, pp. 289–306.

Spronk, Jaap and Zambruno, Giovanni M (1981). A multiple-criteria approach to portfolio selection .

Steuer, Ralph E and Na, Paul (2003). “Multiple criteria decision making combined with finance: A categorized bibliographic study”. In: European Journal of operational research 150.3, pp. 496–515.

Steuer, Ralph E, Qi, Yue, and Hirschberger, Markus (2005). “Multiple objectives in portfolio selection”. In: Journal of Financial Decision Making 1.1, pp. 5–20.

Steuer, Ralph E, Qi, Yue, and Hirschberger, Markus (2006). “Developments in multi-attribute portfolio selection”. In: Multiple criteria decision making 5, pp. 251–262.

Steuer, Ralph E, Qi, Yue, and Hirschberger, Markus (2007). “Suitable-portfolio investors, nondominated frontier sensitivity, and the effect of multiple objectives on standard portfolio selection”. In: Annals of Operations Research 152.1, pp. 297–317.

Steuer, Ralph E, Qi, Yue, and Hirschberger, Markus (2008a). “Portfolio selection in the presence of multiple criteria”. In: Handbook of financial engineering . Springer, pp. 3–24.

Steuer, Ralph E, Qi, Yue, and Hirschberger, Markus (2008b). “Portfolio selection in the presence of multiple criteria”. In: Handbook of financial engineering . Springer, pp. 3–24.

Steuer, RE (1974). “Interval criterion weights programming: A portfolio selection example, gradient cone modification, and computational experience”. In: Proceedings: Tenth annual southeastern regional TIMS meeting , pp. 246–255.

Stone, Bernell K (1973). “A linear programming formulation of the general portfolio selection problem”. In: Journal of Financial and Quantitative Analysis 8.4, pp. 621–636.

Stone, Bernell K and Reback, Robert (1975). “Constructing a model for managing portfolio revisions”. In: Journal of Bank Research 6.1, pp. 48–60.

Sugrue, Paul K, Mehrotra, Anuj, and Orehovec, Paul M (2006). “Financial aid management: an optimisation approach”. In: International Journal of Operational Research 1.3, pp. 267–282.

Tamiz, Mehrdad and Azmi, Rania A (2019). “Goal programming with extended factors for portfolio selection”. In: International Transactions in Operational Research 26.6, pp. 2324–2336.

Tamiz, Mehrdad, Azmi, Rania A, and Jones, Dylan F (2013). “On selecting portfolio of international mutual funds using goal programming with extended factors”. In: European Journal of Operational Research 226.3, pp. 560–576.

Tamiz, Mehrdad, Hasham, R, and Jones, DF (1997). “A comparison between goal programming and regression analysis for portfolio selection”. In: Multiple Criteria Decision Making . Springer, pp. 421–432.

Tamiz, Mehrdad, Hasham, R, Jones, DF, Hesni, B, and Fargher, EK (1996). “A two staged goal programming model for portfolio selection”. In: Multi-Objective Programming and Goal Programming . Springer, pp. 286–299.

Tiryaki, Fatma and Ahlatcioglu, Beyza (2009). “Fuzzy portfolio selection using fuzzy analytic hierarchy process”. In: Information Sciences 179.1–2, pp. 53–69.

Tiryaki, Fatma and Ahlatcioglu, Mehmet (2005). “Fuzzy stock selection using a new fuzzy ranking and weighting algorithm”. In: Applied Mathematics and Computation 170.1, pp. 144–157.

Trenado, Manuel, Romero, Mar100éa, Cuadrado, Mar100éa L, and Romero, Carlos (2014). “Corporate social responsibility in portfolio selection: A “goal games” against nature approach”. In: Computers & Industrial Engineering 75, pp. 260–265.

Vermeulen, Erik M, Spronk, Jaap, and Wijst, Nico van der (1993). “A new approach to firm evaluation”. In: Annals of Operations Research 45.1, pp. 387–403.

Vezmelai, A, Lashgari, Z, and Keyghobadi, A (2015). “Portfolio selection using ELECTRE III: Evidence from Tehran stock exchange”. In: Decision Science Letters 4.2, pp. 227–236.

Weber, Charles A and Current, John R (1993). “A multiobjective approach to vendor selection”. In: European journal of operational research 68.2, pp. 173–184.

Wei, Hechuan, Niu, Caiyun, Xia, Boyuan, Dou, Yajie, and Hu, Xuejun (2020). “A refined selection method for project portfolio optimization considering project interactions”. In: Expert Systems with Applications 142, p. 112952.

Wilhelm, Jochen (1980). “Multiperiod portfolio selection and capital asset pricing”. In: Multiple Criteria Decision Making Theory and Application . Springer, pp. 487–510.

Wu, Liang-Chuan, Chou, Seng-Cho, Yang, Chau-Chen, and Ong, Chorng- Shyong (2007). “Enhanced index investing based on goal programming”. In: The Journal of Portfolio Management 33.3, pp. 49–56.

Xidonas, Panos, Askounis, Dimitrios, and Psarras, John (2009). “Common stock portfolio selection: a multiple criteria decision making methodology and an application to the Athens Stock Exchange”. In: Operational Research 9.1, p. 55.

Xidonas, Panos, Askounis, Dimitrios, Psarras, John, and Mavrotas, George (2009). “Portfolio engineering using the IPSSIS multiobjective optimisation decision support system”. In: International journal of decision sciences, risk and management 1.1–2, pp. 36–53.

Xidonas, Panos and Mavrotas, George (2014). “Multiobjective portfolio optimization with non-convex policy constraints: Evidence from the Eurostoxx 50”. In: The European Journal of Finance 20.11, pp. 957–977.

Xidonas, Panos, Mavrotas, George, Askounis, Dimitrios, and Psarras, John (2008). “Portfolio selection in the presence of multiple criteria”. In: pp.631–648.

Xidonas, Panos, Mavrotas, George, Krintas, Theodore, Psarras, John, and Zopounidis, Constantin (2012). Multicriteria portfolio management . Springer, New York.

Book   Google Scholar  

Xidonas, Panos, Mavrotas, George, and Psarras, John (2009). “A multicriteria methodology for equity selection using financial analysis”. In: Computers & operations research 36.12, pp. 3187–3203.

Xidonas, Panos, Mavrotas, George, and Psarras, John (2010). “Portfolio management within the frame of multiobjective mathematical programming: a categorised bibliographic study”. In: International Journal of Operational Research 8.1, pp. 21–41.

Xidonas, Panos, Mavrotas, George, Zopounidis, Constantin, and Psarras, John (2011). “IPSSIS: An integrated multicriteria decision support system for equity portfolio construction and selection”. In: European Journal of Operational Research 210.2, pp. 398–409.

Xidonas, Panos and Psarras, John (2009). “Equity portfolio management within the MCDM frame: a literature review”. In: International Journal of Banking, Accounting and Finance 1.3, pp. 285–309.

Yodmun, Satit and Witayakiattilerd, Wichai (2016). “Stock selection into portfolio by fuzzy quantitative analysis and fuzzy multicriteria decision making”. In: Advances in Operations Research , pp.1–14.

Zopounidis, Constantin (1999). “Multicriteria decision aid in financial management”. In: European Journal of Operational Research 119.2, pp. 404–415.

Zopounidis, Constantin, Despotis, DK, and Kamaratou, I (1998). “Portfolio Selection Using the ADELAIS Multiobjective Linear Programming System”. In: Computational Economics 11.3, pp. 189–204.

Zopounidis, Constantin and Doumpos, Michael (2000b). “INVESTOR: A decision support system based on multiple criteria for portfolio selection and composition”. In: A-MCD-A Aide Multi Critere a la Decision-Multiple Criteria Decision Aiding , pp. 371–381.

Zopounidis, Constantin and Doumpos, Michael (2002). “Multi-criteria decision aid in financial decision making: methodologies and literature review”. In: Journal of Multi-Criteria Decision Analysis 11.4–5, pp. 167–186.

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Sarmas, E., Xidonas, P., Doukas, H. (2020). Literature Review. In: Multicriteria Portfolio Construction with Python. Springer Optimization and Its Applications, vol 163. Springer, Cham. https://doi.org/10.1007/978-3-030-53743-2_4

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Introduction to Computational Finance and Financial Econometrics with R

16 single index model.

Updated: October 16, 2020

Copyright © Eric Zivot 2015

• always pd, reduces number of estimated components in matrix.
• Compare results with sample covariance matrix
• Risk analysis of asset and portfolios: factor risk reports
• Standard errors
• Asymptotic distributions and confidence intervals
• Tests for coefficients
• Tests for model assumptions: check to see if residual correlation matrix is diagonal!
• Diagnostics for covariance stationarity
• Rolling estimates from the single index model

16.1 Motivation

The CER model for monthly asset returns assumes that returns are jointly normally distributed with a constant mean vector and covariance matrix. This model captures many of the stylized facts for monthly asset returns. While the CER model captures the covariances and correlations between different asset returns it does not explain where these covariances and correlations come from. The single index model is an extension of the CER model that explains the covariance and correlation structure among asset returns as resulting from common exposures to an underlying market index. The intuition behind the single index model can be illustrated by examining some example monthly returns.

For the single index model examples in this chapter, consider the monthly adjusted closing prices on four Northwest stocks (with tickers in parentheses): Boeing (BA), Nordstrom (JWN), Microsoft (MSFT), Starbucks (SBUX). We will also use the monthly adjusted closing prices on the S&P 500 index (^GSPC) as the proxy for the market index. These prices are extracted from the IntroCompFinR package as follows:

The data sample is from January, 1998 through May, 2012 and returns are simple returns:

Prices are shown in Figure 16.1 , created using:

Figure 16.1: Monthly closing prices on four Northwest stocks and the S&P 500 index.

The time plots of prices show some common movements among the stocks that are similar to movements of the S&P 500 index . During the dot-com boom-bust at the beginning of the sample, except for Starbucks, prices rise during the boom and fall during the bust. For all assets, prices rise during the five year boom period prior to the financial crisis, fall sharply after 2008 during the bust, and then rise afterward.

Figure 16.2 shows time plots of the monthly returns on each stock together with the return on the S&P 500 index, created using

Figure 16.2: Monthly returns on four Northwest stocks. The orange line in each panel is the monthly return on the stock, and the blue line is the monthly return on the S&P 500 index.

Figure 16.2 shows that the individual stock returns are more volatile than the S&P 500 returns, and that the movement in stock returns (orange lines) tends to follow the movements in the S&P 500 returns indicating positive covariances and correlations.

The sample return covariance and correlation matrices are computed using:

The sample correlation matrix is visualized in Figure 16.3 using the corrplot function corrplot.mixed() :

Figure 16.3: Sample correlation matrix of monthly returns.

All returns are positively correlated and each stock return has the highest positive correlation with the S&P 500 index.

The positive covariance and correlation of each stock return with the market return can also be visualized with scatterplots as illustrated in Figure 16.4 , created with:

Figure 16.4: Monthly return scatterplots of each stock vs. the S&P 500 index.

The scatterplots show that as the market return increases, the returns on each stock increase in a linear way.

\(\blacksquare\)

16.2 William Sharpe’s SI Model

• Provides an explanation as to why assets are correlated with each other: share an exposure to a common source.
• Provides a simplification of the covariance matrix of a large number of asset returns.
• Provides additional intuition about risk reduction through diversification.
• Example of a factor model for asset returns. Factor models are used heavily in academic theories of asset pricing and in industry for explaining asset returns, portfolio construction and risk analysis. Sharpe’s SI model is the most widely used.
• Motivate by showing positive correlation between individual asset returns and market index (sp500)
• Mostly used for simple returns; but can be used for cc returns for risk analysis.

Let \(R_{it}\) denote the simple return on asset \(i\) over the investment horizon between times \(t-1\) and \(t\) . Let \(R_{Mt}\) denote the simple return on a well diversified market index portfolio, such as the S&P 500 index. The single index (SI) model for \(R_{it}\) has the form: 96 \[\begin{eqnarray} R_{it} & = & \alpha_{i}+\beta_{i}R_{Mt}+\varepsilon_{it},\tag{16.1}\\ R_{Mt} & \sim & iid\,N(0,\sigma_{M}^{2}),\tag{16.2}\\ \epsilon_{it} & \sim & \mathrm{GWN}\,(0,\sigma_{\epsilon,i}^{2}),\tag{16.3}\\ \mathrm{cov}(R_{Mt},\epsilon_{is}) & = & 0,\,\mathrm{f}\mathrm{or\,all}\,t\,\mathrm{and}\,s,\tag{16.4}\\ \mathrm{cov}(\epsilon_{it},\epsilon_{js}) & = & 0,\,\mathrm{for\,all}\,i\neq j\,\mathrm{and\,all}\,t\,\mathrm{and}\,s.\tag{16.5} \end{eqnarray}\] The SI model (16.1) - (16.5) assumes that all individual asset returns, \(R_{it},\) are covariance stationary and are a linear function of the market index return, \(R_{Mt},\) and an independent error term, \(\epsilon_{it}.\) The SI model is an important extension of the regression form of the CER model. In the SI model individual asset returns are explained by two distinct sources: (1) a common (to all assets) market-wide source \(R_{Mt}\) ; (2) and asset specific source \(\epsilon_{it}\) .

16.2.1 Economic interpretation of the SI model

16.2.1.1 interpretation of \(\beta_{i}\).

First, consider the interpretation of \(\beta_{i}\) in (16.1) . The coefficient \(\beta_{i}\) is called the asset’s market exposure or market “beta” {Market beta}. It represents the slope coefficient in the linear relationship between \(R_{it}\) and \(R_{Mt}\) . Because it is assumed that \(R_{Mt}\) and \(\epsilon_{it}\) are independent, \(\beta_{i}\) can be interpreted as the partial derivative \[ \frac{\partial R_{it}}{\partial R_{Mt}}=\frac{\partial}{\partial R_{Mt}}(\alpha_{i}+\beta_{i}R_{Mt}+\varepsilon_{it})=\beta_{i}. \] Then, for small changes in \(R_{Mt}\) denoted \(\Delta R_{Mt}\) and holding \(\varepsilon_{it}\) fixed, we have the approximation \[ \frac{\Delta R_{it}}{\Delta R_{Mt}}\approx\beta_{i}\Rightarrow\Delta R_{it}\approx\beta_{i}\times\Delta R_{Mt}, \] Hence, for a given change in the market return \(\beta_{i}\) determines the magnitude of the response of asset \(i's\) return. The larger (smaller) is \(\beta_{i}\) the larger (smaller) is the response of asset \(i\) to the movement in the market return.

The coefficient \(\beta_{i}\) in (16.1) has another interpretation that is directly related to portfolio risk budgeting. In particular, from chapter 14 \[\begin{equation} \beta_{i}=\frac{\mathrm{cov}(R_{it},R_{Mt})}{\mathrm{var}(R_{Mt})}=\frac{\sigma_{iM}}{\sigma_{M}^{2}},\tag{16.6} \end{equation}\] and \[ \mathrm{MC}\mathrm{R}_{i}^{\sigma_{M}}=\beta_{i}\sigma_{M}. \] Here, we see that \(\beta_{i}\) is the “beta” of asset \(i\) with respect to the market portfolio and so it is proportional to the marginal contribution of asset \(i\) to the volatility of the market portfolio. Therefore, assets with large (small) values of \(\beta_{i}\) have large (small) contributions to the volatility of the market portfolio. In this regard, \(\beta_{i}\) can be thought of as a measure of portfolio risk. Increasing allocations to assets with high (low) \(\beta_{i}\) values will increase (decrease) portfolio risk (as measured by portfolio volatility). In particular, when \(\beta_{i}=1\) asset \(i's\) percent contribution to the volatility of the market portfolio is its allocation weight. When \(\beta_{i}>1\) asset \(i's\) percent contribution to the volatility of the market portfolio is greater than its allocation weight, and when \(\beta_{i}<1\) asset \(i's\) percent contribution to the volatility of the market portfolio is less than its allocation weight.

The derivation of (16.6) is straightforward. Using (16.1) we can write \[\begin{align*} \mathrm{cov}(R_{it},R_{Mt}) & =\mathrm{cov}(\alpha_{i}+\beta_{i}R_{Mt}+\varepsilon_{it},R_{Mt})\\ & =\mathrm{cov}(\beta_{i}R_{Mt},R_{Mt})+\mathrm{cov}(\varepsilon_{it},R_{Mt})\\ & =\beta_{i}\mathrm{var}(R_{Mt})\text{ }\ (\text{since }\mathrm{cov}(\varepsilon_{it},R_{Mt})=0)\\ & \Rightarrow\beta_{i}=\frac{\mathrm{cov}(R_{it},R_{Mt})}{\mathrm{var}(R_{Mt})}. \end{align*}\]

16.2.1.2 Interpretation of \(R_{Mt}\) and \(\epsilon_{it}\)

To aid in the interpretation of \(R_{Mt}\) and \(\epsilon_{it}\) in (16.1) , re-write (16.3) as \[\begin{equation} \varepsilon_{it}=R_{it}-\alpha_{i}-\beta_{i}R_{Mt}.\tag{16.7} \end{equation}\] In (16.3) , we see that \(\epsilon_{it}\) is the difference between asset \(i's\) return, \(R_{it},\) and the portion of asset \(i's\) return that is explained by the market return, \(\beta_{i}R_{Mt}\) , and the intercept, \(\alpha_{i}.\) We can think of \(R_{Mt}\) as capturing “market-wide” news at time \(t\) that is common to all assets, and \(\beta_{i}\) captures the sensitivity or exposure of asset \(i\) to this market-wide news. An example of market-wide news is the release of information by the government about the national unemployment rate. If the news is good then we might expect \(R_{Mt}\) to increase because general business conditions are good. Different assets will respond differently to this news and this differential impact is captured by \(\beta_{i}.\) Assets with positive values of \(\beta_{i}\) will see their returns increase because of good market-wide news, and assets with negative values of \(\beta_{i}\) will see their returns decrease because of this good news. Hence, the magnitude and direction of correlation between asset returns can be partially explained by their exposures to common market-wide news. Because \(\epsilon_{it}\) is assumed to be independent of \(R_{Mt}\) and of \(\epsilon_{jt}\) , we can think of \(\epsilon_{it}\) as capturing specific news to asset \(i\) that is unrelated to market news or to specific news to any other asset \(j\) . Examples of asset-specific news are company earnings reports and reports about corporate restructuring (e.g., CEO resigning). Hence, specific news for asset \(i\) only effects the return on asset \(i\) and not the return on any other asset \(j\) .

The CER model does not distinguish between overall market and asset specific news and so allows the unexpected news shocks \(\epsilon_{it}\) to be correlated across assets. Some news is common to all assets but some is specific to a given asset and the CER model error includes both types of news. In this regard, the CER model for an asset’s return is not a special case of the SI model when \(\beta_{i}=0\) because the SI model assumes that \(\epsilon_{it}\) is uncorrelated across assets.

16.2.2 Statistical properties of returns in the SI model

In this sub-section we present and derive the statistical properties of returns in the SI model (16.1) - (16.5) . We will derive two types of statistical properties: unconditional and conditional. The unconditional properties are based on unconditional or marginal distribution of returns. The conditional properties are based on the distribution of returns conditional on the value of the market return.

16.2.2.1 Unconditional properties

The unconditional properties of returns in the SI model (16.1) - (16.5) are: \[\begin{eqnarray} E[R_{it}] & = & \mu_{i}=\alpha_{i}+\beta_{i}\mu_{M},\tag{16.8}\\ \mathrm{var}(R_{it}) & = & \sigma_{i}^{2}=\beta_{i}^{2}\sigma_{M}^{2}+\sigma_{\epsilon,i}^{2},\tag{16.9}\\ \mathrm{cov}(R_{it},R_{jt}) & = & \sigma_{ij}=\beta_{i}\beta_{j}\sigma_{M}^{2},\tag{16.10}\\ \mathrm{cor}(R_{it},R_{jt}) & = & \rho_{ij}=\frac{\beta_{i}\beta_{j}\sigma_{M}^{2}}{\sqrt{(\beta_{i}^{2}\sigma_{M}^{2}+\sigma_{\epsilon,i}^{2})(\beta_{j}^{2}\sigma_{M}^{2}+\sigma_{\epsilon,j}^{2})}},\tag{16.11}\\ R_{it} & \sim & iid\,N(\mu_{i},\sigma_{i}^{2})=N(\alpha_{i}+\beta_{i}\mu_{M},\beta_{i}^{2}\sigma_{M}^{2}+\sigma_{\epsilon,i}^{2})\tag{16.12} \end{eqnarray}\] The derivations of these properties are straightforward and are left as end-of-chapter exercises.

Property (16.8) shows that \[\begin{equation} \alpha_{i}=\mu_{i}-\beta_{i}\mu_{M}.\tag{16.13} \end{equation}\] Hence, the intercept term \(\alpha_{i}\) can be interpreted as the average return on asset \(i\) that is in excess of the average return due to the market.

From property (16.9) , an asset’s return variance is additively decomposed into two independent components: \[\begin{eqnarray} \mathrm{var}(R_{it}) & = & \sigma_{i}^{2}=\beta_{i}^{2}\sigma_{M}^{2}+\sigma_{\epsilon,i}^{2}\tag{16.14}\\ \mathrm{(total\,asset\,}i\,\mathrm{variance} & = & \mathrm{market\,variance}+\mathrm{asset\,specific\,variance)}\nonumber \end{eqnarray}\] Here, \(\beta_{i}^{2}\sigma_{M}^{2}\) is the contribution of the market index return \(R_{Mt}\) to the total variance of asset \(i\) , and \(\sigma_{\epsilon,i}^{2}\) is the contribution of the asset specific component \(\epsilon_{it}\) to the total return variance, respectively. If we divide both sides of (16.14) by \(\sigma_{i}^{2}\) we get \[\begin{eqnarray*} 1 & = & \frac{\beta_{i}^{2}\sigma_{M}^{2}}{\sigma_{i}^{2}}+\frac{\sigma_{\epsilon,i}^{2}}{\sigma_{i}^{2}}\\ & = & R+(1-R^{2}), \end{eqnarray*}\] where \[\begin{equation} R^{2}=\frac{\beta_{i}^{2}\sigma_{M}^{2}}{\sigma_{i}^{2}}\tag{16.15} \end{equation}\] is the proportion of asset \(i's\) variance that is explained by the variability of the market return \(R_{Mt}\) and \[\begin{equation} 1-R^{2}=\frac{\sigma_{\epsilon,i}^{2}}{\sigma_{i}^{2}}\tag{16.16} \end{equation}\] is the proportion of asset \(i's\) variance that is explained by the variability of the asset specific component \(\epsilon_{it}.\)

• Sharpe’s rule of thumb. A typical stock has \(R^{2}=0.30\) . That is, 30% of an asset’s variability is explained by the market movements. - \(R^{2}\) can be interpreted as the fraction of risk that is non-diversifiable, \(1-R^{2}\) gives the fraction of risk that is diversifiable. Come back to this point after discussing the SI model and portfolios below.
• This is an example of factor risk budgeting

Properties (16.10) and (16.11) show how assets are correlated in the SI model. In particular,

• \(\sigma_{ij}=0\) if \(\beta_{i}=0\) or \(\beta_{j}=0\) or both. Assets \(i\) and \(j\) are uncorrelated if asset \(i\) or asset \(j\) or both do not respond to market news.
• \(\sigma_{ij}>0\) if \(\beta_{i},\beta_{j}>0\) or \(\beta_{i},\beta_{j}<0\) . Assets \(i\) and \(j\) are positively correlated if both assets respond to market news in the same direction.
• \(\sigma_{ij}<0\) if \(\beta_{i}>0\) and \(\beta_{j}<0\) or if \(\beta_{i}<0\) and \(\beta_{j}>0\) . Assets \(i\) and \(j\) are negatively correlated if they respond to market news in opposite directions.

From (16.11) , assets \(i\) and \(j\) are perfectly correlated ( \(\rho_{ij}=\pm1)\) only if \(\sigma_{\epsilon,i}=\sigma_{\epsilon,j}=0\) .

Property (16.12) shows that the distribution of asset returns in the SI model is normal with mean and variance given by () and (), respectively.

In summary, the unconditional properties of returns in the SI model are similar to the properties of returns in the CER model: Returns are covariance stationary with constant means, variances, and covariances. Returns on different assets can be contemporaneously correlated and all asset returns uncorrelated over time. The SI model puts more structure on the expected returns, variances and covariances than the CER model and this allows for a deeper understanding of the behavior of asset returns.

16.2.2.2 Conditional properties

The properties of returns in the SI model (16.1) - (16.5) conditional on \(R_{Mt}=r_{Mt}\) are: \[\begin{eqnarray} E[R_{it}|R_{Mt} & = & r_{Mt}]=\alpha_{i}+\beta_{i}r_{Mt},\\ \mathrm{var}(R_{it}|R_{Mt} & = & r_{Mt}]=\sigma_{\epsilon,i}^{2},\\ \mathrm{cov}(R_{it},R_{jt}|R_{Mt} & = & r_{Mt}]=0,\\ \mathrm{cor}(R_{it},R_{jt}|R_{Mt} & = & r_{Mt}]=0,\\ R_{it}|R_{Mt} & \sim & iid\,N(\alpha_{i}+\beta_{i}r_{Mt},\sigma_{\epsilon,i}^{2}). \end{eqnarray}\]

Recall, conditioning on a random variable means we observe its value. In the SI model, once we observe the market return two important things happen: (1) an asset’s return variance reduces to its asset specific variance; and (2) asset returns become uncorrelated.

16.2.3 SI model and portfolios

A nice feature of the SI model for asset returns is that it also holds for a portfolio of asset returns. This property follows because asset returns are a linear function of the market return. To illustrate, consider a two asset portfolio with investment weights \(x_{1}\) and \(x_{2}\) where each asset return is explained by the SI model: \[\begin{eqnarray*} R_{1t} & = & \alpha_{1}+\beta_{1}R_{Mt}+\epsilon_{1t},\\ R_{2t} & = & \alpha_{2}+\beta_{2}R_{Mt}+\epsilon_{2t}. \end{eqnarray*}\] Then the portfolio return is \[\begin{align*} R_{p,t} & =x_{1}R_{1t}+x_{2}R_{2t}\\ & =x_{1}(\alpha_{1}+\beta_{1}R_{Mt}+\varepsilon_{1t})+x_{2}(\alpha_{2}+\beta_{2}R_{Mt}+\varepsilon_{2t})\\ & =\left(x_{1}\alpha_{1}+x_{2}\alpha_{2}\right)+\left(x_{1}\beta_{1}+x_{2}\beta_{2}\right)R_{Mt}+\left(x_{1}\varepsilon_{1t}+x_{2}\varepsilon_{2t}\right)\\ & =\alpha_{p}+\beta_{p}R_{Mt}+\varepsilon_{p,t} \end{align*}\] where \(\alpha_{p}=x_{1}\alpha_{1}+x_{2}\alpha_{2}\) , \(\beta_{p}=x_{1}\beta_{1}+x_{2}\beta_{2}\) , and \(\varepsilon_{p,t}=x_{1}\varepsilon_{1t}+x_{2}\varepsilon_{2t}\) .

16.2.3.1 SI model and large portfolios

Consider an equally weighted portfolio of \(N\) assets, where \(N\) is a large number (e.g.  \(N=500\) ) whose returns are described by the SI model. Here, \(x_{i}=1/N\) for \(i=1,\ldots,N\) . Then the portfolio return is \[\begin{align*} R_{p,t} & =\sum_{i=1}^{N}x_{i}R_{it}\\ & =\sum_{i=1}^{N}x_{i}\left(\alpha_{i}+\beta_{i}R_{Mt}+\varepsilon_{it}\right)\\ & =\sum_{i=1}^{N}x_{i}\alpha_{i}+\left(\sum_{i=1}^{N}x_{i}\beta_{i}\right)R_{Mt}+\sum_{i=1}^{N}x_{i}\varepsilon_{it}\\ & =\frac{1}{N}\sum_{i=1}^{N}\alpha_{i}+\left(\frac{1}{N}\sum_{i=1}^{N}\beta_{i}\right)R_{Mt}+\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{it}\\ & =\bar{\alpha}+\bar{\beta}R_{Mt}+\bar{\varepsilon}_{t}, \end{align*}\] where \(\bar{\alpha}=\frac{1}{N}\sum_{i=1}^{N}\alpha_{i}\) , \(\bar{\beta}=\frac{1}{N}\sum_{i=1}^{N}\beta_{i}\) and \(\bar{\varepsilon}_{t}=\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{it}\) . Now, \[ \mathrm{var}(\bar{\varepsilon}_{t})=\mathrm{var}\left(\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{it}\right)=\frac{1}{N^{2}}\sum_{i=1}^{N}\mathrm{var}(\varepsilon_{it})=\frac{1}{N}\left(\frac{1}{N}\sum_{i=1}^{N}\sigma_{\epsilon,i}^{2}\right)=\frac{1}{N}\bar{\sigma}^{2} \] where \(\bar{\sigma}^{2}=\frac{1}{N}\sum_{i=1}^{N}\sigma_{\epsilon,i}^{2}\) is the average of the asset specific variances. For large \(N\) , \(\frac{1}{N}\bar{\sigma}^{2}\approx0\) and we have the Law of Large Numbers result \[ \bar{\varepsilon}_{t}=\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{it}\approx E[\varepsilon_{it}]=0. \] As a result, in a large equally weighted portfolio we have the following:

• \(R_{p,t}\approx\bar{\alpha}+\bar{\beta}R_{Mt}:\) all non-market asset-specific risk is diversified away and only market risk remains.
• \(\mathrm{var}(R_{p,t})=\bar{\beta}^{2}\mathrm{var}(R_{Mt})\Rightarrow\mathrm{SD}(R_{p,t})=|\bar{\beta}|\times\mathrm{SD}(R_{Mt}):\) portfolio volatility is proportional to market volatility where the factor of proportionality is the absolute value of portfolio beta.
• \(R^{2}\approx1:\) Approximately 100% of portfolio variance is due to market variance.
• \(\bar{\beta}\approx1\) . A large equally weighted portfolio resembles the market portfolio (e.g., as proxied by the S&P 500 index) and so the beta of a well diversified portfolio will be close to the beta of the market portfolio which is one by definition. 97

These results help us to understand the type of risk that gets diversified away and the type of risk that remains when we form diversified portfolios. Asset specific risk, which is uncorrelated across assets, gets diversified away whereas market risk, which is common to all assets, does not get diversified away.

• (Relate to average covariance calculation from portfolio theory chapter).
• Related to asset R2 discussed earlier. R2 of an asset shows the portion of risk that cannot be diversified away when forming portfolios.

16.2.4 The SI model in matrix notation

• Need to emphasize that the SI model covariance matrix is always positive definite. This is an important result because it allows for the mean-variance analysis of very large portfolios.

For \(i=1,\ldots,N\) assets, stacking (16.1) gives the SI model in matrix notation \[ \left(\begin{array}{c} R_{1t}\\ \vdots\\ R_{Nt} \end{array}\right)=\left(\begin{array}{c} \alpha_{1}\\ \vdots\\ \alpha_{N} \end{array}\right)+\left(\begin{array}{c} \beta_{1}\\ \vdots\\ \beta_{N} \end{array}\right)R_{Mt}+\left(\begin{array}{c} \epsilon_{1t}\\ \vdots\\ \epsilon_{Nt} \end{array}\right), \] or \[\begin{equation} \mathbf{R}_{t}=\alpha+\beta R_{Mt}+\epsilon_{t}.\tag{16.17} \end{equation}\] The unconditional statistical properties of returns (16.8) , (16.9) , (16.10) and (16.12) can be re-expressed using matrix notation as follows: \[\begin{eqnarray} E[\mathbf{R}_{t}] & = & \mu=\alpha+\beta\mu_{M},\tag{16.18}\\ \mathrm{var}(\mathbf{R}_{t}) & = & \Sigma=\sigma_{M}^{2}\beta \beta ^{\prime}+\mathbf{D},\tag{16.19}\\ \mathbf{R}_{t} & \sim & iid\,N(\mu,\Sigma)=N(\alpha+\beta\mu_{M},\sigma_{M}^{2}\beta \beta ^{\prime}+\mathbf{D}),\tag{16.20} \end{eqnarray}\] where \[\begin{eqnarray} \mathbf{D} & = & \mathrm{var}(\epsilon_{t})=\left(\begin{array}{cccc} \sigma_{\epsilon,1}^{2} & 0 & \cdots & 0\\ 0 & \sigma_{\epsilon,2}^{2} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & \sigma_{\epsilon,N}^{2} \end{array}\right)\\ & = & \mathrm{diag}(\sigma_{\epsilon,1}^{2},\sigma_{\epsilon,1}^{2},\ldots,\sigma_{\epsilon,N}^{2}).\nonumber \end{eqnarray}\]

The derivation of the SI model covariance matrix (16.19) is \[\begin{eqnarray*} \mathrm{var}(\mathbf{R}_{t}) & = & \Sigma=\beta \mathrm{var}(R_{Mt})\beta ^{\prime}+\mathrm{var}(\epsilon_{t})\\ & = & \sigma_{M}^{2}\beta \beta ^{\prime}+\mathbf{D}, \end{eqnarray*}\] which uses the assumption that the market return \(R_{Mt}\) is uncorrelated will all asset specific error terms in \(\epsilon_{t}\) .

It is useful to examine the SI covariance matrix (16.19) for a three asset portfolio. In this case, we have \[\begin{align*} R_{it} & =\alpha_{i}+\beta_{i}R_{Mt}+\varepsilon_{it},\text{ }i=1,2,3\\ \sigma_{i}^{2} & =\mathrm{var}(R_{it})=\beta_{i}^{2}\sigma_{M}^{2}+\sigma_{\varepsilon,i}^{2}\\ \sigma_{ij} & =\mathrm{cov}(R_{it},R_{jt})=\sigma_{M}^{2}\beta_{i}\beta_{j} \end{align*}\] The \(3\times3\) covariance matrix is \[\begin{align*} \Sigma & =\left(\begin{array}{ccc} \sigma_{1}^{2} & \sigma_{12} & \sigma_{13}\\ \sigma_{12} & \sigma_{2}^{2} & \sigma_{23}\\ \sigma_{13} & \sigma_{23} & \sigma_{3}^{2} \end{array}\right)\\ & =\left(\begin{array}{ccc} \beta_{1}^{2}\sigma_{M}^{2}+\sigma_{\varepsilon,1}^{2} & \sigma_{M}^{2}\beta_{1}\beta_{2} & \sigma_{M}^{2}\beta_{1}\beta_{3}\\ \sigma_{M}^{2}\beta_{1}\beta_{2} & \beta_{2}^{2}\sigma_{M}^{2}+\sigma_{\varepsilon,2}^{2} & \sigma_{M}^{2}\beta_{2}\beta_{3}\\ \sigma_{M}^{2}\beta_{1}\beta_{3} & \sigma_{M}^{2}\beta_{2}\beta_{3} & \beta_{3}^{2}\sigma_{M}^{2}+\sigma_{\varepsilon,3}^{2} \end{array}\right)\\ & =\sigma_{M}^{2}\left(\begin{array}{ccc} \beta_{1}^{2} & \beta_{1}\beta_{2} & \beta_{1}\beta_{3}\\ \beta_{1}\beta_{2} & \beta_{2}^{2} & \beta_{2}\beta_{3}\\ \beta_{1}\beta_{3} & \beta_{2}\beta_{3} & \beta_{3}^{2} \end{array}\right)+\left(\begin{array}{ccc} \sigma_{\varepsilon,1}^{2} & 0 & 0\\ 0 & \sigma_{\varepsilon,2}^{2} & 0\\ 0 & 0 & \sigma_{\varepsilon,3}^{2} \end{array}\right). \end{align*}\] The first matrix shows the return variance and covariance contributions due to the market returns, and the second matrix shows the contributions due to the asset specific errors. Define \(\beta =(\beta_{1},\beta_{2},\beta_{3})^{\prime}.\) Then \[\begin{eqnarray*} \sigma_{M}^{2}\beta \beta ^{\prime} & = & \sigma_{M}^{2}\left(\begin{array}{c} \beta_{1}\\ \beta_{2}\\ \beta_{3} \end{array}\right)\left(\begin{array}{ccc} \beta_{1} & \beta_{2} & \beta_{3}\end{array}\right)=\sigma_{M}^{2}\left(\begin{array}{ccc} \beta_{1}^{2} & \beta_{1}\beta_{2} & \beta_{1}\beta_{3}\\ \beta_{1}\beta_{2} & \beta_{2}^{2} & \beta_{2}\beta_{3}\\ \beta_{1}\beta_{3} & \beta_{2}\beta_{3} & \beta_{3}^{2} \end{array}\right),\\ \mathbf{D} & = & \mathrm{diag}(\sigma_{\varepsilon,1}^{2},\sigma_{\varepsilon,2}^{2},\sigma_{\varepsilon,3}^{2})=\left(\begin{array}{ccc} \sigma_{\varepsilon,1}^{2} & 0 & 0\\ 0 & \sigma_{\varepsilon,2}^{2} & 0\\ 0 & 0 & \sigma_{\varepsilon,3}^{2} \end{array}\right), \end{eqnarray*}\] and so \[ \Sigma=\sigma_{M}^{2}\beta \beta ^{\prime}+\mathbf{D}. \]

The matrix form of the SI model (16.18) - (16.20) is useful for portfolio analysis. For example, consider a portfolio with \(N\times1\) weight vector \(\mathbf{x}=(x_{1},\ldots,x_{N})^{\prime}.\) Using (16.17) , the SI model for the portfolio return \(R_{p,t}=\mathbf{x}^{\prime}R_{t}\) is \[\begin{eqnarray*} R_{p,t} & = & \mathbf{x}^{\prime}(\alpha+\beta R_{Mt}+\epsilon_{t})\\ & = & \mathbf{x}^{\prime}\alpha+\mathbf{x}^{\prime}\beta R_{Mt}+\mathbf{x}^{\prime}\epsilon_{t}\\ & = & \alpha_{p}+\beta_{p}R_{Mt}+\epsilon_{p,t}, \end{eqnarray*}\] where \(\alpha_{p}=\mathbf{x}^{\prime}\alpha\) , \(\beta_{p}=\mathbf{x}^{\prime}\beta\) and \(\epsilon_{p,t}=\mathbf{x}^{\prime}\epsilon_{t}\) .

16.3 Monte Carlo Simulation of the SI Model

To give a first step reality check for the SI model, consider simulating data from the SI model for a single asset. The steps to create a Monte Carlo simulation from the SI model are:

• Fix values for the SI model parameters \(\alpha\) , \(\beta\) , \(\sigma_{\epsilon}\) , \(\mu_{M}\) and \(\sigma_{M}\)
• Determine the number of simulated returns, \(T\) , to create.
• Use a computer random number generator to simulate \(T\) iid values of \(R_{Mt}\sim N(\mu_{M},\sigma_{M}^{2})\) and \(\epsilon_{t}\sim N(0,\sigma_{\epsilon}^{2})\) , where \(R_{Mt}\) is simulated independently from \(\epsilon_{t}\) .
• Create the simulated asset returns \(\tilde{R}_{t}=\alpha+\beta\tilde{R}_{M}\) + \(\tilde{\epsilon}_{t}\) for \(t=1,\ldots,T\)

The following example illustrates simulating SI model returns for Boeing.

Consider simulating monthly returns on Boeing from the SI model where the S&P 500 index is used as the market index. The SI model parameters are calibrated from the actual monthly returns on Boeing and the S&P 500 index using sample statistics, as discussed in the next section, and are given by: \[ \alpha_{BA}=0.005,\,\beta_{BA}=0.98,\,\sigma_{\epsilon,BA}=0.08,\,\mu_{M}=0.003,\,\sigma_{M}=0.048. \] These values are set in R using:

The simulated returns for the S&P 500 index and Boeing are created using:

Figure 16.5: Simulated SI model monthly returns on the S&P 500 index and Boeing.

Figure 16.6: Simulated SI model monthly returns on the S&P 500 index and Boeing. The orange line is the equation \(R_{BA,t}=\alpha_{BA}+\beta_{BA}R_{Mt}=0.005+0.98\times R_{Mt}.\)

The simulate returns are illustrated in Figures 16.5 and 16.6 . The time plot and scatterplot of the simulated returns on the S&P 500 index look much like the corresponding plots of the actual returns shows in the top left panels of Figures 16.2 and 16.4 , respectively. Hence, the SI model for the S&P 500 index and Boeing passes the first step reality check. In Figure 16.6 , the line \(\alpha_{i}+\beta_{i}R_{Mt}=0.005+0.98R_{Mt}\) is shown in orange. This line represents the predictions of Boeing’s return given the S&P 500 return. For example, if \(R_{Mt}=0.05\) then the predicted return for Boeing is \(R_{BA}=0.005+0.98\times(0.05)=0.054\) . The differences between the observed Boeing returns and (blue dots) and the predicted returns (orange line) are the asset specific error terms. These are the random asset specific news components. The standard deviation of these components, \(\sigma_{\epsilon,BA}=0.08\) , represents the typical magnitude of these components. Here, we would not be too surprised if the observed Boeing return is \(0.08\) above or below the orange line. How close the Boeing returns are to the predicted returns is determined by the \(R^{2}\) value given by (16.15) . Here, \(\sigma_{BA}^{2}=\beta_{BA}^{2}\sigma_{M}^{2}+\sigma_{\epsilon,BA}^{2}=(0.98)^{2}(0.048)^{2}+(0.08)^{2}=0.00861\) and so \(R^{2}=(0.98)^{2}(0.048)^{2}/0.00861=0.257\) . Hence, 25.7% of the variability (risk) of Boeing returns are explained by the variability of the S&P 500 returns. As a result, 74.3% \((1-R^{2}=0.743)\) of the variability of Boeing returns are due to the random asset specific terms.

16.4 Estimation of SI Model

Consider the SI model (16.1) - (16.5) . The asset specific parameters to be estimated are \(\alpha_{i}\) , \(\beta_{i}\) and \(\sigma_{\epsilon,i}^{2}\) , \((i=1,\ldots,N)\) , and the market parameters to be estimated are \(\mu_{M}\) and \(\sigma_{M}^{2}\) . These parameters can be estimated using the plug-in principle, linear regression, and maximum likelihood. All methods give essentially the same estimators for the SI model parameters.

16.4.1 Plug-in principle estimates

Let \(\{(r_{it},r_{Mt})\}_{t=1}^{T}\) denote a sample of size \(T\) of observed returns on asset \(i\) and the market return which are assumed to be generated from the SI model. (16.1) - (16.12) . Recall, the plug-in principle says to estimate population model parameters using appropriate sample statistics. For the market parameters, the plug-in principle estimates are the same as the CER model estimates \[\begin{eqnarray*} \hat{\mu}_{M} & = & \frac{1}{T}\sum_{t=1}^{T}r_{Mt},\\ \hat{\sigma}_{M}^{2} & = & \frac{1}{T-1}\sum_{t=1}^{T}(r_{Mt}-\hat{\mu}_{M})^{2}. \end{eqnarray*}\] From (16.13) - (16.6) we see that \(\alpha_{i}\) and \(\beta_{i}\) are functions of population parameters \[\begin{eqnarray*} \alpha_{i} & = & \mu_{i}-\beta_{i}\mu_{M},\\ \beta_{i} & = & \frac{\mathrm{cov}(R_{it},R_{Mt})}{\mathrm{var}(R_{Mt})}=\frac{\sigma_{iM}}{\sigma_{M}^{2}}. \end{eqnarray*}\] The corresponding plug-in principle estimates are then: \[\begin{eqnarray} \hat{\alpha}_{i} & = & \hat{\mu}_{i}-\hat{\beta}_{i}\hat{\mu}_{M},\tag{16.21}\\ \hat{\beta}_{i} & = & \frac{\hat{\sigma}_{iM}}{\hat{\sigma}_{M}^{2}},\tag{16.22} \end{eqnarray}\] where \[\begin{eqnarray*} \hat{\mu}_{i} & = & \frac{1}{T}\sum_{t=1}^{T}r_{it},\\ \hat{\sigma}_{iM} & = & \frac{1}{T-1}\sum_{t=1}^{T}(r_{it}-\hat{\mu}_{i})(r_{Mt}-\hat{\mu}_{M}). \end{eqnarray*}\]

Given the plug-in principle estimates \(\hat{\alpha}_{i}\) and \(\hat{\beta}_{i}\) , the plug-in principle estimate of \(\epsilon_{it}\) is \[\begin{equation} \hat{\varepsilon}_{it}=r_{it}-\hat{\alpha}_{i}-\hat{\beta}_{i}r_{Mt},\,t=1,\ldots,T.\tag{16.23} \end{equation}\] Using (16.23) , the plug-in principle estimate of \(\sigma_{\epsilon,i}^{2}\) is the sample variance of \(\{\hat{\epsilon}_{it}\}_{t=1}^{T}\) (adjusted for the number of degrees of freedom): \[\begin{align} \hat{\sigma}_{\varepsilon,i}^{2} & =\frac{1}{T-2}\sum_{t=1}^{T}\hat{\varepsilon}_{t}^{2}=\frac{1}{T-2}\sum_{t=1}^{T}\left(r_{it}-\hat{\alpha}_{i}-\hat{\beta}_{i}r_{Mt}\right)^{2}.\tag{16.24} \end{align}\]

Plug-in principle estimates of \(R^{2}\) based on (16.15) can be computed using \[\begin{align} \hat{R}^{2} & =\frac{\hat{\beta}_{i}^{2}\hat{\sigma}_{M}^{2}}{\hat{\sigma}_{i}^{2}}=1-\frac{\hat{\sigma}_{\varepsilon,i}^{2}}{\hat{\sigma}_{i}^{2}}.\tag{16.25} \end{align}\]

Consider computing the plug-in principle estimates for \(\alpha_{i},\) \(\beta_{i}\) and \(\sigma_{\epsilon,i}^{2}\) from the example data using the formulas (16.21) , (16.22) and (16.24) , respectively. First, extract the sample statistics \(\hat{\mu}_{i}\) , \(\hat{\sigma}_{iM}\) , \(\hat{\mu}_{M}\) , and \(\hat{\sigma}_{M}^{2}\) :

Next, estimate \(\hat{\beta}_{i}\) using

Here, we see that \(\hat{\beta}_{BA}\) and \(\hat{\beta}_{SBUX}\) are very close to one and that \(\hat{\beta}_{JWN}\) and \(\hat{\beta}_{MSFT}\) are slightly bigger than one. Using the estimates of \(\hat{\beta}_{i}\) and the sample statistics \(\hat{\mu}_{i}\) and \(\hat{\mu}_{M}\) the estimates for \(\hat{\alpha}_{i}\) are

All of the estimates of \(\hat{\alpha}_{i}\) are close to zero. The estimates of \(\sigma_{\epsilon,i}^{2}\) can be computed using:

Lastly, the estimates of \(R^{2}\) can be computed using

16.4.2 Least squares estimates

The SI model representation (16.1) shows that returns are a linear function of the market return and an asset specific error term \[ R_{it}=\alpha_{i}+\beta_{i}R_{Mt}+\epsilon_{it}, \] here \(\alpha_{i}\) is the intercept and \(\beta_{i}\) is the slope. Least squares regression is a method for estimating \(\alpha_{i}\) and \(\beta_{i}\) by finding the “best fitting” line to the scatterplot of returns where \(R_{it}\) is on the vertical axis and \(R_{Mt}\) is on the horizontal axis.

{[}Insert Figure here{]} To be completed…

To see how the method of least squares determines the “best fitting” line, consider the scatterplot of the sample returns on Boeing and the S&P 500 index illustrated in Figure xxx. In the figure, the black line is a fitted line with initial guess \(\hat{\alpha}_{BA}=0\) and \(\hat{\beta}_{BA}=0.5.\) The differences between the observed returns (blue dots) and the values on the fitted line are the estimated errors \(\hat{\epsilon}_{BA,t}=r_{BA,t}-\hat{\alpha}_{BA}-\hat{\beta}_{BA}r_{Mt}=r_{BA,t}-0-0.5\times R_{Mt}.\) Some estimated errors are big and some are small. The overall fit of the line can be measured using a statistic based on all \(t=1,\ldots,T\) of the estimated errors. A natural choice is the sum of the errors \(\sum_{t=1}^{T}\hat{\epsilon}_{t}\) . However, this choice can be misleading due to the canceling out of large positive and negative errors. To avoid this problem, it is better to measure the overall fit using \(\sum_{t=1}^{T}\hat{|\epsilon}_{t}|\) or \(\sum_{t=1}^{T}\hat{\epsilon}_{t}^{2}.\) Then the best fitting line can be determined by finding the intercept and slope values that minimize \(\sum_{t=1}^{T}\hat{|\epsilon}_{t}|\) or \(\sum_{t=1}^{T}\hat{\epsilon}_{t}^{2}.\)

The method of least squares regression defines the “best fitting” line by finding the intercept and slope values that minimize the sum of squared errors \[\begin{equation} \mathrm{SSE}(\hat{\alpha}_{i},\hat{\beta}_{i})=\sum_{t=1}^{T}\hat{\epsilon}_{it}^{2}=\sum_{t=1}^{T}\left(r_{it}-\hat{\alpha}_{i}-\hat{\beta}_{i}r_{Mt}\right)^{2}.\tag{16.26} \end{equation}\] Because \(\mathrm{SSE}(\hat{\alpha},\hat{\beta})\) is a continuous and differential function of \(\hat{\alpha}_{i}\) and \(\hat{\beta}_{i}\) , the minimizing values of \(\hat{\alpha}_{i}\) and \(\hat{\beta}_{i}\) can be determined using simple calculus. The first order conditions for a minimum are: \[\begin{align} 0 & =\frac{\partial\mathrm{SSE}(\hat{\alpha}_{i},\hat{\beta}_{i})}{\partial\hat{\alpha}_{i}}=-2\sum_{t=1}^{T}(r_{it}-\hat{\alpha}_{i}-\hat{\beta}_{i}r_{Mt})=-2\sum_{t=1}^{T}\hat{\varepsilon}_{it},\tag{16.27}\\ 0 & =\frac{\partial\mathrm{SSE}(\hat{\alpha}_{i},\hat{\beta}_{i})}{\partial\hat{\beta}_{i}}=-2\sum_{t=1}^{T}(r_{it}-\hat{\alpha}_{i}-\hat{\beta}_{i}r_{Mt})r_{Mt}=-2\sum_{t=1}^{T}\hat{\varepsilon}_{it}r_{Mt}.\tag{16.28} \end{align}\] These are two linear equations in two unknowns which can be re-expressed as \[\begin{eqnarray*} \hat{\alpha}_{i}T+\hat{\beta}_{i}\sum_{t=1}^{T}r_{Mt} & = & \sum_{t=1}^{T}r_{it},\\ \hat{\alpha}_{i}\sum_{t=1}^{T}r_{Mt}+\hat{\beta}_{i}\sum_{t=1}^{T}r_{Mt}^{2} & = & \sum_{t=1}^{T}r_{it}r_{Mt}. \end{eqnarray*}\] Using matrix algebra, we can write these equations as: \[\begin{equation} \left(\begin{array}{cc} T & \sum_{t=1}^{T}r_{Mt}\\ \sum_{t=1}^{T}r_{Mt} & \sum_{t=1}^{T}r_{Mt}^{2} \end{array}\right)\left(\begin{array}{c} \hat{\alpha}_{i}\\ \hat{\beta}_{i} \end{array}\right)=\left(\begin{array}{c} \sum_{t=1}^{T}r_{it}\\ \sum_{t=1}^{T}r_{it}r_{Mt} \end{array}\right),\tag{16.29} \end{equation}\] which is of the form \(\mathbf{Ax}=\mathbf{b}\) with \[ \mathbf{A}=\left(\begin{array}{cc} T & \sum_{t=1}^{T}r_{Mt}\\ \sum_{t=1}^{T}r_{Mt} & \sum_{t=1}^{T}r_{Mt}^{2} \end{array}\right),\,\mathbf{x}=\left(\begin{array}{c} \hat{\alpha}_{i}\\ \hat{\beta}_{i} \end{array}\right),\,\mathbf{b}=\left(\begin{array}{c} \sum_{t=1}^{T}r_{it}\\ \sum_{t=1}^{T}r_{it}r_{Mt} \end{array}\right). \] Hence, we can determine \(\hat{\alpha}_{i}\) and \(\hat{\beta}_{i}\) by solving \(\mathbf{x}=\mathbf{A}^{-1}\mathbf{b}.\) Now, 98 \[\begin{eqnarray*} \mathbf{A}^{-1} & = & \frac{1}{\mathrm{det}(\mathbf{A})}\left(\begin{array}{cc} \sum_{t=1}^{T}r_{Mt}^{2} & -\sum_{t=1}^{T}r_{Mt}\\ -\sum_{t=1}^{T}r_{Mt} & T \end{array}\right),\\ \mathrm{det}(\mathbf{A}) & = & T\sum_{t=1}^{T}r_{Mt}^{2}-\left(\sum_{t=1}^{T}r_{Mt}\right)^{2}=T\sum_{t=1}^{T}\left(r_{Mt}-\hat{\mu}_{M}\right)^{2},\\ \hat{\mu}_{M} & = & \frac{1}{T}\sum_{t=1}^{T}r_{Mt}. \end{eqnarray*}\] Consequently, \[\begin{equation} \left(\begin{array}{c} \hat{\alpha}_{i}\\ \hat{\beta}_{i} \end{array}\right)=\frac{1}{T\sum_{t=1}^{T}\left(r_{Mt}-\hat{\mu}_{M}\right)^{2}}\left(\begin{array}{cc} \sum_{t=1}^{T}r_{Mt}^{2} & -\sum_{t=1}^{T}r_{Mt}\\ -\sum_{t=1}^{T}r_{Mt} & T \end{array}\right)\left(\begin{array}{c} \sum_{t=1}^{T}r_{it}\\ \sum_{t=1}^{T}r_{it}r_{Mt} \end{array}\right)\tag{16.30} \end{equation}\] and so \[\begin{eqnarray} \hat{\alpha}_{i} & = & \frac{\sum_{t=1}^{T}r_{Mt}^{2}\sum_{t=1}^{T}r_{it}-\sum_{t=1}^{T}r_{Mt}^{2}\sum_{t=1}^{T}r_{it}r_{Mt}}{T\sum_{t=1}^{T}\left(r_{Mt}-\hat{\mu}_{M}\right)^{2}},\tag{16.31}\\ \hat{\beta}_{i} & = & \frac{T\sum_{t=1}^{T}r_{it}r_{Mt}-\sum_{t=1}^{T}r_{Mt}\sum_{t=1}^{T}r_{it}}{T\sum_{t=1}^{T}\left(r_{Mt}-\hat{\mu}_{M}\right)^{2}}.\tag{16.32} \end{eqnarray}\] After a little bit of algebra (see end-of-chapter exercises) it can be shown that \[\begin{eqnarray*} \hat{\alpha}_{i} & = & \hat{\mu}_{i}-\hat{\beta}_{i}\hat{\mu}_{M},\\ \hat{\beta}_{i} & = & \frac{\hat{\sigma}_{iM}}{\hat{\sigma}_{M}^{2}}, \end{eqnarray*}\] which are plug-in estimates for \(\hat{\alpha}_{i}\) and \(\hat{\beta}_{i}\) determined earlier. Hence, the least squares estimates of \(\hat{\alpha}_{i}\) and \(\hat{\beta}_{i}\) are identical to the plug-in estimates.

The solution for the least squares estimates in (16.30) has an elegant representation using matrix algebra. To see this, define the \(T\times1\) vectors \(\mathbf{r}_{i}=(r_{i1},\ldots,r_{iT})^{\prime}\) , \(\mathbf{r}_{M}=(r_{M1},\ldots,r_{MT})^{\prime}\) and \(\mathbf{1}=(1,\ldots,1)^{\prime}\) . Then we can re-write (16.29) as \[ \left(\begin{array}{cc} \mathbf{1}^{\prime}\mathbf{1} & \mathbf{1}^{\prime}\mathbf{r}_{M}\\ \mathbf{1}^{\prime}\mathbf{r}_{M} & \mathbf{r}_{M}^{\prime}\mathbf{r}_{M} \end{array}\right)\left(\begin{array}{c} \hat{\alpha}_{i}\\ \hat{\beta}_{i} \end{array}\right)=\left(\begin{array}{c} \mathbf{1}^{\prime}\mathbf{r}_{i}\\ \mathbf{r}_{M}^{\prime}\mathbf{r}_{i} \end{array}\right) \] or \[\begin{equation} \mathbf{X}^{\prime}\mathbf{X}\hat{\gamma}_{i}=\mathbf{X}^{\prime}\mathbf{r}_{i}\tag{16.33} \end{equation}\] where \(\mathbf{X}=(\begin{array}{cc} \mathbf{1} & \mathbf{r}_{M})\end{array}\) is a \(T\times2\) matrix and \(\hat{\gamma}=(\hat{\alpha}_{i},\hat{\beta}_{i})^{\prime}\) . Provided \(\mathbf{X}^{\prime}\mathbf{X}\) is invertible, solving (16.33) for \(\hat{\gamma}_{i}\) gives the least squares estimates in matrix form: \[\begin{equation} \hat{\gamma}_{i}=\left(\mathbf{X}^{\prime}\mathbf{X}\right)^{-1}\mathbf{X}^{\prime}\mathbf{r}_{i}.\tag{16.34} \end{equation}\] The matrix form solution (16.34) is especially convenient for computation in R.

The least squares estimates of \(\epsilon_{t}\) , \(\sigma_{\epsilon,i}^{2}\) and \(R^{2}\) are the same as the plug-in estimators (16.23) , (16.24) and (16.25) , respectively. In the context of least squares estimation, the estimate \(\hat{\sigma}_{\epsilon,i}=\sqrt{\hat{\sigma}_{\epsilon,i}^{2}}\) is called the standard error of the regression and measures the typical magnitude of \(\hat{\epsilon}_{t}\) (difference between observed return and fitted regression line).

16.4.3 Simple linear regression in R

• don’t do regression examples until statistical theory is discussed
• computing least squares estimates using matrix algebra formulas
• See discussion from my regression chapter in MFTSR
• describe structure of lm() function, extractor and method functions
• Do analysis of example data To be completed…

16.4.4 Maximum likelihood estimates

The SI model parameters can also be estimated using the method of maximum likelihood, which was introduced in chapter (GARCH estimation chapter). To construct the likelihood function, we use property () of the SI model that conditional on \(R_{Mt}=r_{Mt}\) the distribution of \(R_{it}\) is normal with mean \(\alpha_{i}+\beta_{i}r_{Mt}\) and variance \(\sigma_{\epsilon,i}^{2}\) . The pdf of \(R_{it}|R_{Mt}=r_{mt}\) is then \[ f(r_{it}|r_{mt},\theta_{i})=(2\pi\sigma_{\varepsilon,i}^{2})^{-1/2}\exp\left(\frac{-1}{2\sigma_{\varepsilon,i}^{2}}\left(r_{it}-\alpha_{i}+\beta_{i}r_{Mt}\right)^{2}\right),\,t=1,\ldots,T, \] where \(\theta_{i}=(\alpha_{i},\beta_{i},\sigma_{\epsilon,i}^{2})^{\prime}\) . Given a sample \(\{(r_{it},r_{Mt})\}_{t=1}^{T}=\{\mathbf{r}_{i},\mathbf{r}_{M}\}\) of observed returns on asset \(i\) and the market return, which are assumed to be generated from the SI model, the joint density of asset returns given the market returns is \[\begin{eqnarray*} f(\mathbf{r}_{i}|\mathbf{r}_{m}) & = & \prod_{t=1}^{T}(2\pi\sigma_{\varepsilon,i}^{2})^{-1/2}\exp\left(\frac{-1}{2\sigma_{\varepsilon,i}^{2}}\left(r_{it}-\alpha_{i}+\beta_{i}r_{Mt}\right)^{2}\right)\\ & = & (2\pi\sigma_{\varepsilon,i}^{2})^{-T/2}\exp\left(\frac{-1}{2\sigma_{\varepsilon,i}^{2}}\sum_{t=1}^{T}\left(r_{it}-\alpha_{i}+\beta_{i}r_{Mt}\right)^{2}\right)\\ & = & (2\pi\sigma_{\varepsilon,i}^{2})^{-T/2}\exp\left(\frac{-1}{2\sigma_{\varepsilon,i}^{2}}\mathrm{SSE}(\alpha_{i},\beta_{i})\right). \end{eqnarray*}\] where \(\mathrm{SSE}(\alpha_{i},\beta_{i})\) is the sum of squared residuals (16.26) used to determine the least squares estimates. The log-likelihood function for \(\theta_{i}\) is then \[\begin{eqnarray} lnL(\theta_{i}|\mathbf{r}_{i},\mathbf{r}_{M}) & = & \frac{-T}{2}\ln(2\pi)-\frac{T}{2}\ln(\sigma_{\varepsilon,i}^{2})-\frac{1}{2\sigma_{\varepsilon,i}^{2}}\mathrm{SSE}(\alpha_{i},\beta_{i}).\tag{16.35} \end{eqnarray}\] From (16.35) , it can be seen that the values of \(\alpha_{i}\) and \(\beta_{i}\) that maximize the log-likelihood are the values that minimize \(\mathrm{SSE}(\alpha_{i},\beta_{i})\) . Hence, the ML estimates of \(\alpha_{i}\) and \(\beta_{i}\) are the least squares estimates.

To find the ML estimate for \(\sigma_{\epsilon,i}^{2}\) , plug the ML estimates of \(\alpha_{i}\) and \(\beta_{i}\) into (16.35) giving \[ lnL(\hat{\alpha}_{i},\hat{\beta}_{i},\sigma_{\epsilon,i}^{2}|\mathbf{r}_{i},\mathbf{r}_{M})=\frac{-T}{2}\ln(2\pi)-\frac{T}{2}\ln(\sigma_{\varepsilon,i}^{2})-\frac{1}{2\sigma_{\varepsilon,i}^{2}}\mathrm{SSE}(\hat{\alpha}_{i},\hat{\beta}_{i}). \] Maximization with respect to \(\sigma_{\epsilon,i}^{2}\) gives the first order condition \[ \frac{\partial lnL(\hat{\alpha}_{i},\hat{\beta}_{i},\sigma_{\epsilon,i}^{2}|\mathbf{r}_{i},\mathbf{r}_{M})}{\partial\sigma_{\epsilon,i}^{2}}=-\frac{T}{2\hat{\sigma}_{\epsilon,i}^{2}}+\frac{1}{2\left(\hat{\sigma}_{\epsilon,i}^{2}\right)^{2}}\mathrm{SSE}(\hat{\alpha}_{i},\hat{\beta}_{i})=0. \] Solving for \(\hat{\sigma}_{\epsilon,i}^{2}\) gives the ML estimate for \(\sigma_{\epsilon,i}^{2}\) : \[ \hat{\sigma}_{\epsilon,i}^{2}=\frac{\mathrm{SSE}(\hat{\alpha}_{i},\hat{\beta}_{i})}{T}=\frac{1}{T}\sum_{t=1}^{T}\hat{\epsilon}_{t}^{2}, \] which is plug-in principle estimate (16.24) not adjusted for degrees-of-freedom.

16.5 Statistical Properties of SI Model Estimates

To determine the statistical properties of the plug-in principle/least squares/ML estimators \(\hat{\alpha}_{i}\) , \(\hat{\beta}_{i}\) and \(\hat{\sigma}_{\epsilon,i}^{2}\) in the SI model, we treat them as functions of the random variables \(\{(R_{i,t},R_{Mt})\}_{t=1}^{T}\) where \(R_{t}\) and \(R_{Mt}\) are assumed to be generated by the SI model (16.1) - (16.5) .

16.5.1 Bias

In the SI model, the estimators \(\hat{\alpha}_{i}\) , \(\hat{\beta}_{i}\) and \(\hat{\sigma}_{\epsilon,i}^{2}\) (with degrees-of-freedom adjustment) are unbiased: \[\begin{eqnarray*} E[\hat{\alpha}_{i}] & = & \alpha_{i},\\ E[\hat{\beta}_{i}] & = & \beta_{i},\\ E[\hat{\sigma}_{\epsilon,i}^{2}] & = & \sigma_{\epsilon,i}^{2}. \end{eqnarray*}\] To shows that \(\hat{\alpha}_{i}\) and \(\hat{\beta}_{i}\) are unbiased, it is useful to consider the SI model for asset \(i\) in matrix form for \(t=1,\ldots,T\) : \[ \mathbf{R}_{i}=\alpha_{i}\mathbf{1}+\beta_{i}\mathbf{R}_{M}+\epsilon_{i}=\mathbf{X}\gamma_{i}+\epsilon_{i}, \] where \(\mathbf{R}_{i}=(R_{i1},\ldots,R_{iT})^{\prime},\) \(\mathbf{R}_{M}=(R_{M1},\ldots,R_{MT})^{\prime}\) , \(\epsilon_{i}=(\epsilon_{i1},\ldots,\epsilon_{iT})^{\prime}\) , \(\mathbf{1}=(1,\ldots,1)^{\prime}\) , \(\mathbf{X}=(\begin{array}{cc} \mathbf{1} & \mathbf{R}_{M})\end{array},\) and \(\gamma_{i}=(\alpha_{i},\beta_{i})^{\prime}.\) The estimator for \(\gamma_{i}\) is \[\begin{eqnarray*} \hat{\gamma}_{i} & = & (\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\mathbf{R}_{i}. \end{eqnarray*}\] Pluging in \(\mathbf{R}_{i}=\mathbf{X}\gamma_{i}+\epsilon_{i}\) gives \[\begin{eqnarray*} \hat{\gamma}_{i} & = & (\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\left(\mathbf{X}\gamma_{i}+\epsilon_{i}\right)=(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\mathbf{X}\gamma_{i}+(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\epsilon_{i}\\ & = & \gamma_{i}+(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\epsilon_{i}. \end{eqnarray*}\] Then \[\begin{eqnarray*} E[\hat{\gamma}_{i}] & = & \gamma_{i}+E\left[(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\epsilon_{i}\right]\\ & = & \gamma_{i}+E\left[(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\right]E[\epsilon_{i}]\,(\mathrm{because}\,\epsilon_{it}\,\mathrm{is}\,\mathrm{independent}\,\mathrm{of}\,R_{Mt})\\ & = & \gamma_{i}\,(\mathrm{because}\,E[\epsilon_{i}]=\mathbf{0}). \end{eqnarray*}\] The derivation of \(E[\hat{\sigma}_{\epsilon,i}^{2}]=\sigma_{\epsilon,i}^{2}\) is beyond the scope of this book and can be found in graduate econometrics textbooks such as Hayashi (1980).

16.5.2 Precision

Under the assumptions of the SI model, analytic formulas for estimates of the standard errors for \(\hat{\alpha}_{i}\) and \(\hat{\beta}_{i}\) are given by: \[\begin{align} \widehat{\mathrm{se}}(\hat{\alpha}_{i}) & \approx\frac{\hat{\sigma}_{\varepsilon,i}}{\sqrt{T\cdot\hat{\sigma}_{M}^{2}}}\cdot\sqrt{\frac{1}{T}\sum_{t=1}^{T}r_{Mt}^{2}},\tag{16.36}\\ \widehat{\mathrm{se}}(\hat{\beta}_{i}) & \approx\frac{\hat{\sigma}_{\varepsilon,i}}{\sqrt{T\cdot\hat{\sigma}_{M}^{2}}},\tag{16.37} \end{align}\] where “ \(\approx\) ” denotes an approximation based on the CLT that gets more accurate the larger the sample size. Remarks:

• \(\widehat{\mathrm{se}}(\hat{\alpha}_{i})\) and \(\widehat{\mathrm{se}}(\hat{\beta}_{i})\) are smaller the smaller is \(\hat{\sigma}_{\varepsilon,i}\) . That is, the closer are returns to the fitted regression line the smaller are the estimation errors in \(\hat{\alpha}_{i}\) and \(\hat{\beta}_{i}\) .
• \(\widehat{\mathrm{se}}(\hat{\beta}_{i})\) is smaller the larger is \(\hat{\sigma}_{M}^{2}\) . That is, the greater the variability in the market return \(R_{Mt}\) the smaller is the estimation error in the estimated slope coefficient \(\hat{\beta}_{i}\) . This is illustrated in Figure xxx. The left panel shows a data sample from the SI model with a small value of \(\hat{\sigma}_{M}^{2}\) and the right panel shows a sample with a large value of \(\hat{\sigma}_{M}^{2}\) . The right panel shows that the high variability in \(R_{Mt}\) makes it easier to identify the slope of the line.
• Both \(\widehat{\mathrm{se}}(\hat{\alpha}_{i})\) and \(\widehat{\mathrm{se}}(\hat{\beta}_{i})\) go to zero as the sample size, \(T\) , gets large. Since \(\hat{\alpha}_{i}\) and \(\hat{\beta}_{i}\) are unbiased estimators, this implies that they are also consistent estimators. That is, they converge to the true values \(\alpha_{i}\) and \(\beta_{i}\) , respectively, as \(T\rightarrow\infty\) .
• In R, the standard error values (16.36) and (16.37) are computed using the summary() function on an “ lm ” object.

There are no easy formulas for the estimated standard errors for \(\hat{\sigma}_{\epsilon,i}^{2}\) , \(\hat{\sigma}_{\varepsilon,i}\) and \(\hat{R}^{2}\) . Estimated standard errors for these estimators, however, can be easily computed using the bootstrap.

to be completed

16.5.3 Sampling distribution and confidence intervals.

Using arguments based on the CLT, it can be shown that for large enough \(T\) the estimators \(\hat{\alpha}_{i}\) and \(\hat{\beta}_{i}\) are approximately normally distributed: \[\begin{eqnarray*} \hat{\alpha}_{i} & \sim & N(\alpha_{i},\widehat{\mathrm{se}}(\hat{\alpha}_{i})^{2}),\\ \hat{\beta}_{i} & \sim & N(\beta_{i},\widehat{\mathrm{se}}(\hat{\beta}_{i})^{2}), \end{eqnarray*}\] where \(\widehat{\mathrm{se}}(\hat{\alpha}_{i})\) and \(\widehat{\mathrm{se}}(\hat{\beta}_{i})\) are given by (16.36) and (16.37) , respectively.

16.6 Further Reading: Single Index Model

To be completed …

16.7 Problems: Single Index Model

16.8 Solutions to Selected Problems

{Ruppert} Ruppert, D.A.. , Springer-Verlag.

{MartinSchererYollin}Martin, R.D., Scherer, B., and Yollin, G. (2016).

The single index model is also called the* market model* or the single factor model . ↩︎

The S&P 500 index is a value weighted index and so is not equal to an equally weighted portfolio of the stocks in the S&P 500 index. However, since most stocks in the S&P 500 index have large market capitalizations the values weights are not too different from equal weights. ↩︎

The matrix \(\mathbf{A}\) is invertible provided \(\mathrm{det}(A)\neq0.\) This requires the sample variance of \(R_{Mt}\) to be non-zero. ↩︎

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• v.9(7); 2013 Jul

Ten Simple Rules for Writing a Literature Review

Marco pautasso.

1 Centre for Functional and Evolutionary Ecology (CEFE), CNRS, Montpellier, France

2 Centre for Biodiversity Synthesis and Analysis (CESAB), FRB, Aix-en-Provence, France

Literature reviews are in great demand in most scientific fields. Their need stems from the ever-increasing output of scientific publications [1] . For example, compared to 1991, in 2008 three, eight, and forty times more papers were indexed in Web of Science on malaria, obesity, and biodiversity, respectively [2] . Given such mountains of papers, scientists cannot be expected to examine in detail every single new paper relevant to their interests [3] . Thus, it is both advantageous and necessary to rely on regular summaries of the recent literature. Although recognition for scientists mainly comes from primary research, timely literature reviews can lead to new synthetic insights and are often widely read [4] . For such summaries to be useful, however, they need to be compiled in a professional way [5] .

When starting from scratch, reviewing the literature can require a titanic amount of work. That is why researchers who have spent their career working on a certain research issue are in a perfect position to review that literature. Some graduate schools are now offering courses in reviewing the literature, given that most research students start their project by producing an overview of what has already been done on their research issue [6] . However, it is likely that most scientists have not thought in detail about how to approach and carry out a literature review.

Reviewing the literature requires the ability to juggle multiple tasks, from finding and evaluating relevant material to synthesising information from various sources, from critical thinking to paraphrasing, evaluating, and citation skills [7] . In this contribution, I share ten simple rules I learned working on about 25 literature reviews as a PhD and postdoctoral student. Ideas and insights also come from discussions with coauthors and colleagues, as well as feedback from reviewers and editors.

Rule 1: Define a Topic and Audience

How to choose which topic to review? There are so many issues in contemporary science that you could spend a lifetime of attending conferences and reading the literature just pondering what to review. On the one hand, if you take several years to choose, several other people may have had the same idea in the meantime. On the other hand, only a well-considered topic is likely to lead to a brilliant literature review [8] . The topic must at least be:

• interesting to you (ideally, you should have come across a series of recent papers related to your line of work that call for a critical summary),
• an important aspect of the field (so that many readers will be interested in the review and there will be enough material to write it), and
• a well-defined issue (otherwise you could potentially include thousands of publications, which would make the review unhelpful).

Ideas for potential reviews may come from papers providing lists of key research questions to be answered [9] , but also from serendipitous moments during desultory reading and discussions. In addition to choosing your topic, you should also select a target audience. In many cases, the topic (e.g., web services in computational biology) will automatically define an audience (e.g., computational biologists), but that same topic may also be of interest to neighbouring fields (e.g., computer science, biology, etc.).

Rule 2: Search and Re-search the Literature

After having chosen your topic and audience, start by checking the literature and downloading relevant papers. Five pieces of advice here:

• keep track of the search items you use (so that your search can be replicated [10] ),
• keep a list of papers whose pdfs you cannot access immediately (so as to retrieve them later with alternative strategies),
• use a paper management system (e.g., Mendeley, Papers, Qiqqa, Sente),
• define early in the process some criteria for exclusion of irrelevant papers (these criteria can then be described in the review to help define its scope), and
• do not just look for research papers in the area you wish to review, but also seek previous reviews.

The chances are high that someone will already have published a literature review ( Figure 1 ), if not exactly on the issue you are planning to tackle, at least on a related topic. If there are already a few or several reviews of the literature on your issue, my advice is not to give up, but to carry on with your own literature review,

The bottom-right situation (many literature reviews but few research papers) is not just a theoretical situation; it applies, for example, to the study of the impacts of climate change on plant diseases, where there appear to be more literature reviews than research studies [33] .

• discussing in your review the approaches, limitations, and conclusions of past reviews,
• trying to find a new angle that has not been covered adequately in the previous reviews, and
• incorporating new material that has inevitably accumulated since their appearance.

When searching the literature for pertinent papers and reviews, the usual rules apply:

• be thorough,
• use different keywords and database sources (e.g., DBLP, Google Scholar, ISI Proceedings, JSTOR Search, Medline, Scopus, Web of Science), and
• look at who has cited past relevant papers and book chapters.

Rule 3: Take Notes While Reading

If you read the papers first, and only afterwards start writing the review, you will need a very good memory to remember who wrote what, and what your impressions and associations were while reading each single paper. My advice is, while reading, to start writing down interesting pieces of information, insights about how to organize the review, and thoughts on what to write. This way, by the time you have read the literature you selected, you will already have a rough draft of the review.

Of course, this draft will still need much rewriting, restructuring, and rethinking to obtain a text with a coherent argument [11] , but you will have avoided the danger posed by staring at a blank document. Be careful when taking notes to use quotation marks if you are provisionally copying verbatim from the literature. It is advisable then to reformulate such quotes with your own words in the final draft. It is important to be careful in noting the references already at this stage, so as to avoid misattributions. Using referencing software from the very beginning of your endeavour will save you time.

Rule 4: Choose the Type of Review You Wish to Write

After having taken notes while reading the literature, you will have a rough idea of the amount of material available for the review. This is probably a good time to decide whether to go for a mini- or a full review. Some journals are now favouring the publication of rather short reviews focusing on the last few years, with a limit on the number of words and citations. A mini-review is not necessarily a minor review: it may well attract more attention from busy readers, although it will inevitably simplify some issues and leave out some relevant material due to space limitations. A full review will have the advantage of more freedom to cover in detail the complexities of a particular scientific development, but may then be left in the pile of the very important papers “to be read” by readers with little time to spare for major monographs.

There is probably a continuum between mini- and full reviews. The same point applies to the dichotomy of descriptive vs. integrative reviews. While descriptive reviews focus on the methodology, findings, and interpretation of each reviewed study, integrative reviews attempt to find common ideas and concepts from the reviewed material [12] . A similar distinction exists between narrative and systematic reviews: while narrative reviews are qualitative, systematic reviews attempt to test a hypothesis based on the published evidence, which is gathered using a predefined protocol to reduce bias [13] , [14] . When systematic reviews analyse quantitative results in a quantitative way, they become meta-analyses. The choice between different review types will have to be made on a case-by-case basis, depending not just on the nature of the material found and the preferences of the target journal(s), but also on the time available to write the review and the number of coauthors [15] .

Rule 5: Keep the Review Focused, but Make It of Broad Interest

Whether your plan is to write a mini- or a full review, it is good advice to keep it focused 16 , 17 . Including material just for the sake of it can easily lead to reviews that are trying to do too many things at once. The need to keep a review focused can be problematic for interdisciplinary reviews, where the aim is to bridge the gap between fields [18] . If you are writing a review on, for example, how epidemiological approaches are used in modelling the spread of ideas, you may be inclined to include material from both parent fields, epidemiology and the study of cultural diffusion. This may be necessary to some extent, but in this case a focused review would only deal in detail with those studies at the interface between epidemiology and the spread of ideas.

While focus is an important feature of a successful review, this requirement has to be balanced with the need to make the review relevant to a broad audience. This square may be circled by discussing the wider implications of the reviewed topic for other disciplines.

Rule 6: Be Critical and Consistent

Reviewing the literature is not stamp collecting. A good review does not just summarize the literature, but discusses it critically, identifies methodological problems, and points out research gaps [19] . After having read a review of the literature, a reader should have a rough idea of:

• the major achievements in the reviewed field,
• the main areas of debate, and
• the outstanding research questions.

It is challenging to achieve a successful review on all these fronts. A solution can be to involve a set of complementary coauthors: some people are excellent at mapping what has been achieved, some others are very good at identifying dark clouds on the horizon, and some have instead a knack at predicting where solutions are going to come from. If your journal club has exactly this sort of team, then you should definitely write a review of the literature! In addition to critical thinking, a literature review needs consistency, for example in the choice of passive vs. active voice and present vs. past tense.

Rule 7: Find a Logical Structure

Like a well-baked cake, a good review has a number of telling features: it is worth the reader's time, timely, systematic, well written, focused, and critical. It also needs a good structure. With reviews, the usual subdivision of research papers into introduction, methods, results, and discussion does not work or is rarely used. However, a general introduction of the context and, toward the end, a recapitulation of the main points covered and take-home messages make sense also in the case of reviews. For systematic reviews, there is a trend towards including information about how the literature was searched (database, keywords, time limits) [20] .

How can you organize the flow of the main body of the review so that the reader will be drawn into and guided through it? It is generally helpful to draw a conceptual scheme of the review, e.g., with mind-mapping techniques. Such diagrams can help recognize a logical way to order and link the various sections of a review [21] . This is the case not just at the writing stage, but also for readers if the diagram is included in the review as a figure. A careful selection of diagrams and figures relevant to the reviewed topic can be very helpful to structure the text too [22] .

Rule 8: Make Use of Feedback

Reviews of the literature are normally peer-reviewed in the same way as research papers, and rightly so [23] . As a rule, incorporating feedback from reviewers greatly helps improve a review draft. Having read the review with a fresh mind, reviewers may spot inaccuracies, inconsistencies, and ambiguities that had not been noticed by the writers due to rereading the typescript too many times. It is however advisable to reread the draft one more time before submission, as a last-minute correction of typos, leaps, and muddled sentences may enable the reviewers to focus on providing advice on the content rather than the form.

Feedback is vital to writing a good review, and should be sought from a variety of colleagues, so as to obtain a diversity of views on the draft. This may lead in some cases to conflicting views on the merits of the paper, and on how to improve it, but such a situation is better than the absence of feedback. A diversity of feedback perspectives on a literature review can help identify where the consensus view stands in the landscape of the current scientific understanding of an issue [24] .

Rule 9: Include Your Own Relevant Research, but Be Objective

In many cases, reviewers of the literature will have published studies relevant to the review they are writing. This could create a conflict of interest: how can reviewers report objectively on their own work [25] ? Some scientists may be overly enthusiastic about what they have published, and thus risk giving too much importance to their own findings in the review. However, bias could also occur in the other direction: some scientists may be unduly dismissive of their own achievements, so that they will tend to downplay their contribution (if any) to a field when reviewing it.

In general, a review of the literature should neither be a public relations brochure nor an exercise in competitive self-denial. If a reviewer is up to the job of producing a well-organized and methodical review, which flows well and provides a service to the readership, then it should be possible to be objective in reviewing one's own relevant findings. In reviews written by multiple authors, this may be achieved by assigning the review of the results of a coauthor to different coauthors.

Rule 10: Be Up-to-Date, but Do Not Forget Older Studies

Given the progressive acceleration in the publication of scientific papers, today's reviews of the literature need awareness not just of the overall direction and achievements of a field of inquiry, but also of the latest studies, so as not to become out-of-date before they have been published. Ideally, a literature review should not identify as a major research gap an issue that has just been addressed in a series of papers in press (the same applies, of course, to older, overlooked studies (“sleeping beauties” [26] )). This implies that literature reviewers would do well to keep an eye on electronic lists of papers in press, given that it can take months before these appear in scientific databases. Some reviews declare that they have scanned the literature up to a certain point in time, but given that peer review can be a rather lengthy process, a full search for newly appeared literature at the revision stage may be worthwhile. Assessing the contribution of papers that have just appeared is particularly challenging, because there is little perspective with which to gauge their significance and impact on further research and society.

Inevitably, new papers on the reviewed topic (including independently written literature reviews) will appear from all quarters after the review has been published, so that there may soon be the need for an updated review. But this is the nature of science [27] – [32] . I wish everybody good luck with writing a review of the literature.

Acknowledgments

Many thanks to M. Barbosa, K. Dehnen-Schmutz, T. Döring, D. Fontaneto, M. Garbelotto, O. Holdenrieder, M. Jeger, D. Lonsdale, A. MacLeod, P. Mills, M. Moslonka-Lefebvre, G. Stancanelli, P. Weisberg, and X. Xu for insights and discussions, and to P. Bourne, T. Matoni, and D. Smith for helpful comments on a previous draft.

Funding Statement

This work was funded by the French Foundation for Research on Biodiversity (FRB) through its Centre for Synthesis and Analysis of Biodiversity data (CESAB), as part of the NETSEED research project. The funders had no role in the preparation of the manuscript.

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• Published: 26 August 2024

Unveiling the nexus: exploring the influence of terrorism on energy trade in China and the Belt and Road countries

• Wenlong Yang 1 ,
• Wentian Shi   ORCID: orcid.org/0000-0003-1905-0465 2 &
• Dongcheng Chen 3  

Humanities and Social Sciences Communications volume  11 , Article number:  1088 ( 2024 ) Cite this article

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• Politics and international relations

An examination of the scale of energy trade between China and the countries along the Belt and Road, as well as the terrorist activities in these countries from 2008 to 2019, has been conducted in this study. The spatial and temporal evolution law is revealed through the use of exploratory spatial data analysis. The spatial Durbin model is employed to investigate the impacts and spatial spillover effects of terrorist activities along the Belt and Road on China’s energy trade with these countries. The findings of the study are as follows: Spatial clustering is exhibited by the energy scale of China’s imports and exports with countries along the routes and terrorist activities along the routes, with energy imports concentrated in West Asia and North Africa, energy exports concentrated in Southeast Asia, and terrorist activities concentrated in West Asia and North Africa, South Asia, and Central Asia. A negative effect on the scale of China’s energy imports from the countries along the route is exerted by the terrorism index, and a positive spatial spillover effect is present. The number of terrorist activities and deaths have a negative effect on the scale of China’s energy imports from the countries along the route, but there is no spatial spillover effect. The terrorism index does not affect the scale of energy exports from China to countries along the route, but there is a negative spatial spillover effect. There are differences in the spatial spillover effects of the number of terrorist activities and deaths on the scale of China’s energy exports. The level of economic development, liner transportation convenience, and energy exports dependence of the countries along the route also significantly influence China’s energy trade with these countries.

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Geopolitics and energy security: a comprehensive exploration of evolution, collaborations, and future directions

Introduction.

Throughout the transition from the coal-dominated era to that of oil and gas, energy has steadily emerged as a pivotal force sculpting the worldwide geopolitical map, wielding ever greater influence within the discourse of international relations.(Costa-Campi et al., 2017 ; Schuetze and Hussein, 2023 ; Vakulchuk et al., 2020 ; Yang et al., 2023 ; Yang et al., 2020 ). Over the years, the close relationship between energy and geopolitical competition, geopolitical conflicts, and even interstate wars has been demonstrated by events such as the Gulf War, the Iraq War, the Libyan War, the Syrian War, and the attacks on Saudi oil fields (Bradshaw, 2010 ; Firat, 2023 ; Painter, 2014 ). The conflict between Russia and Ukraine, which is set to commence in 2022, is anticipated to be a significant energy game involving the United States, Russia, and the European Union (Zhang et al., 2024 ). Daniel Yergin astutely observes that politics is an innate characteristic of energy, contending that in the case of oil, its composition is 90% political and merely 10% economic (Yergin, 2011 ). The Belt and Road route encompasses the “heartland” of the global energy supply. By 2019, it was reported that the proven oil and natural gas reserves along the route accounted for 57 and 78% of the world’s total, respectively. The recoverable resources yet to be discovered were found to account for 32 and 47% of the world’s total, respectively (Dudley, 2018 ). The maritime channels along the route, such as the North Indian Ocean, the Strait of Malacca, and the South China Sea, are important lifelines for global energy transportation. Overall, the security and stability of energy production, supply, and transportation in the countries along the Belt and Road are of utmost importance, which is not merely tied to the trajectory of the international energy market, but is also deeply interconnected with the broader geopolitical climate and the worldwide economic framework (Burke and Stephens, 2018 ; Palle, 2021 ; Pascual and Zambetakis, 2010 ; Rhodes, 2018 ; Zhang et al., 2015 ).

However, the frictions and conflicts between religious religions, nationalities, and races in countries along the Belt and Road are escalating, extremist forces are intertwined, and terrorist activities are showing a trend of proliferation and rampant. According to statistics from the Study of Terrorism and Responses to Terrorism, nearly 85% of global terrorist attacks have erupted in regions along the route since 2003, resulting in 81.6% of global casualties. Among them, there were over 10 thousand global terrorist attacks in 2013, with over 9 thousand occurring in the region. Terrorist activities pose a substantial threat to the energy production, supply, and transportation along the route, thereby exposing the international energy market to significant vulnerabilities and risks (Klinger, 2020 ; Minyan et al., 2016 ). For instance, in 2012, terrorist attacks on Syria’s gas pipelines led to a severe strain on international gas supplies and a surge in prices. In 2019, drone strikes on two Saudi oil facilities caused the most significant disruption to energy supply in history, slashing global crude oil production by nearly 5.7 million barrels per day and reducing ethane and liquefied natural gas supplies by around 50%. Obviously, terrorist activities play a crucial role in the international energy market, with its impacts notably affecting energy prices and causing substantial fluctuations (Li et al., 2023 ; Ramiah et al., 2019 ; Song et al., 2022 ). Meanwhile, extensive research has confirmed that terrorist activities have a strong spatial spillover effect and exhibit obvious diffusion characteristics (Gong et al., 2023 ; Jin et al., 2023 ). The question of whether these spatial spillover effects amplify risks across a broader scope of energy transactions, and how such effects propagate in space, has emerged as a crucial topic requiring meticulous examination. Delving into these issues is of immeasurable strategic value for countries along the Belt and Road to accurately assess security risks in energy trade, and to enhance the effectiveness and resilience of their external energy cooperation endeavors.

This paper attempts to make marginal contributions in the following three aspects: Firstly, it provides a comprehensive measurement of the spatial characteristics of energy imports and exports trade and terrorist activities at the national level. It discusses the spatiotemporal impact relationship between energy imports and exports trade and terrorist activities using scientific and quantitative research methods. Secondly, by integrating spatial interdependence theory and employing spatial econometric models, the paper separately explores the pathways through which terrorist activities influence energy imports and exports. It delves into whether these pathways exhibit spatial effects and comprehensively evaluates the impact of terrorist activities on energy imports and exports. Thirdly, the paper utilizes disaggregated indicators of terrorist activity to more accurately gauge the differential impacts of terrorist activities of varying severity on energy imports and exports.

The rest of this paper proceeds as follows: “Literature review” reviews the available literature on the spatial measurement and driving factors of energy trade, and the impact of terrorist activities on energy trade. “Theoretical analysis and hypotheses” is theoretical analysis and hypotheses, which provide theoretical support for exploring the impact of terrorist activities on energy imports and exports trade and the spatial spillover effects of terrorist activities on energy imports and exports trade. “Data and methodology” introduces the study area, the data, and the methods used in the study. “Results” presents the results, including the temporal and spatial evolution of energy imports and exports and various indicators of terrorist activities between China and countries along the Belt and Road, as well as the analysis of the mechanism of terrorist activities on energy imports and exports trade. The conclusion and implication are provided in “Conclusions and implications”.

Literature review

It has been observed that spatial dislocations in the supply and demand of energy necessitate the trade of energy as a pivotal conduit linking centers of energy production with those of consumption (Balta-Ozkan et al., 2015 ). The scholarly examination of the spatial dimensions of energy trade and the corridors of energy transportation is an area that has seen a burgeoning of interest. The genesis of spatial analysis in energy trade can be traced to the year 1948, wherein Barber provided a comprehensive delineation of the situation of oil-producing countries in the Middle East, and the spatial configuration of energy reserves, production, and consumption was analyzed on the basis of geographic units (Barber, 1948 ). Subsequently, Wagstaff offered a refinement and an update to the pattern of global energy distribution and its developmental trajectory, engaging in a more granular discussion on the distribution and consumption of energy, among other aspects (Wagstaff, 1974 ). The advent of the U.S. shale technology breakthrough has positioned the United States as a significant exporter within the global energy market, prompting notable shifts in the patterns of global energy production and consumption. Scholars have identified two principal shifts in the epicenter of global energy production: the initial shift transpired around the period of the Second World War, transitioning from the Gulf of Mexico region to the Gulf region; the subsequent shift was witnessed in the 1990s, moving from the Gulf region to the Middle East, Russia, and the Central Asia-Caspian Sea region, with the Americas adopting a predominantly multicentric pattern. The pattern of energy consumption is increasingly characterized by a trend towards multipolar decentralization, with Asia emerging as the region with the most rapid growth in oil and gas consumption (Bridge and Dodge, 2022 ). The formation of two energy export belts, encompassing Russia, the Middle East, Central Asia, and North America, and three energy imports centers in Asia, Europe, and North America, is indicative of an evolving pattern of energy trade (Du et al., 2016 ; Hao et al., 2016 ; Wang and Chen, 2016 ). In the year 2013, with the advancement of the Belt and Road, the study of energy imports and exports trade between China and the countries along the route has emerged as a focal point of research (Ahakwa, 2023 ; Xiang et al., 2022 ; Yu, 2018 ; Zhao et al., 2019 ). It has been demonstrated by certain studies that a spatial correlation exists among the energy production activities within the countries aligned with the Belt and Road, and a spatial spillover effect of energy trade is observed amongst these countries(Lin et al., 2020 ; Zhang et al., 2019 ). To illustrate, countries that produce energy are predominantly located in the Middle East and Central Asia, while the hubs for energy processing are generally found in proximity to the energy-producing countries (Thomson and Horii, 2009 ; Zhao, 2008 ). While the spatial agglomeration of energy trade has been depicted extensively in the existing body of literature, the spatial mechanisms governing the energy trade warrant further investigation.

Energy trade is influenced by a multitude of factors, including the resource endowment, pricing, transportation, and the trade policies of the exporting countries, as well as the geographical positioning, associated risks, and bilateral trade agreements. Spatial distance is one of the crucial factors affecting the cost of international trade, with a particularly significant impact on energy trade. Research conducted by Kitamura and Managi, utilizing the gravity model to analyze the global oil trade network, has indicated that spatial distance acts as a barrier in the selection of trade partners, compelling countries to engage in energy trade with geographically proximate oil-exporting countries due to transportation costs (Kitamura and Managi, 2017 ). Similarly, Leung has highlighted that, under identical conditions of resource endowment and other variables, transportation costs are a critical factor in the energy trade, leading to a preference for geographically neighboring countries for energy imports (Leung, 2011 ). The economic condition of energy-exporting countries has a direct influence on their capacity to export energy. It has been noted that economic expansion, population growth, and urbanization within these countries generate substantial domestic energy demands, thereby diminishing the reliance on energy exports. A case in point is Indonesia, which, during the early 1990s, was a major exporter of crude oil to China. However, with the acceleration of Indonesia’s economic development, a marked decrease in energy exports to China was observed (Shao et al., 2017 ). Trade and investment agreements between countries, political relations, and market access conditions all have profound effects on energy trade. Established trading partnerships and bilateral agreements serve as structural links in the international energy market, with stable relations mitigating trade risks and fostering future energy transactions between the involved entities (Du et al., 2017 ). It has been demonstrated that overseas energy investments constitute a pivotal method by which countries can fulfill their escalating energy requirements (Shao et al., 2017 ). Furthermore, the influence of geopolitical risks upon energy commerce warrants considerable scrutiny, encompassing fluctuations in oil prices, internal political strife, piracy, terrorism, and additional hazards (Sun et al., 2017 ). For instance, when engaging in trade with the Middle East and Africa, regions that are hubs of global energy resources, the potential perils present within the supplier’s locale must be taken into account. Research by Mohamed and Jebli suggests the existence of a causal relationship between energy dynamics and terrorist activities (Mohamed et al., 2019 ). Terrorist actions exert a profound effect on the trade of energy, particularly given the heightened susceptibility of energy trading networks, where an isolated incident of energy export disruption due to terrorism can result in the crippling of the entire network (Ji et al., 2014 ).

Following the September 11 Attacks, terrorist activities being the first of the five major non-traditional security issues along the route, have been identified as the largest uncertainty factor for energy imports and exports trade (Bandyopadhyay and Sandler, 2014 ; Bandyopadhyay et al., 2018 ). The impact of these activities on energy trade has been extensively studied. It is believed by some scholars that the potential for energy trade between China and countries along the route has been largely inhibited by terrorist activities (Bouri et al., 2019 ; Cheng and Chiu, 2018 ; Stokes, 2015 ). For instance, a significant reduction in the production and supply of energy, an increase in China’s energy transaction costs (Bandyopadhyay et al., 2014 ; Seung-Whan, 2015 ), threats to sea lanes such as the Malacca Straits and land lanes such as the China-Myanmar Pipeline, and an increase in China’s energy transportation risks have all been attributed to terrorist activities along the routes. The Middle East, where terrorist activities and great power games are intertwined and agitated, has been identified as the main geopolitical risk zone for China’s energy imports (Kelemen and Fergnani, 2020 ; Smith, 2009 ). Contrarily, other scholars hold the belief that the impact of terrorist activities along the route on China’s energy imports and exports trade has regional characteristics. For example, it has been observed that despite the frequent occurrence of terrorist activities along the route, the trade of imports and exports between China and the countries along the route has not decreased but increased (Bove et al., 2014 ; Martin et al., 2012 ). This is primarily attributed to the entrepreneurial and adventurous spirit of China’s economic and trade personnel, the “five principles of peaceful coexistence” foreign policy that can escort China’s economic and trade, and the comparative advantage of immunity to terrorist activities that China’s energy imports and exports trade possesses (Blustein, 2009 ). In addition, the existence of a spatial spillover effect of terrorist activities along the border has been discovered by other scholars(Kelejian and Mukerji, 2022 ; Pham and Doucouliagos, 2017 ). For instance, De Sousa has concluded that neighbors adjacent to terror, even when they do not source it, have trade reduced through enhanced security measures (De Sousa et al., 2018 ). Minyan has pointed out that the distribution of terrorist activities along the route takes the form of an arc-shaped shock zone (Minyan et al., 2016 ). Neumayer and Plümper have concluded that there is a spatial spillover effect of terrorist activities in China, Mongolia, Russia, Central Asia, Southeast Asia, South Asia, and West Asia (Neumayer and Plümper, 2016 ). However, little research exists in the current literature on whether the spatial spillover effect of terrorist activities along the routes affects the energy imports and exports trade between China and the countries along the routes.

Reviewing the existing literature, there is a scarcity of studies that explore the spatial association between energy imports and exports trade and terrorist activities, and even fewer that investigate the spatial spillover effects of terrorism on energy trade. In light of this gap, this paper takes China’s energy imports and exports with countries along the Belt and Road as a case study. Utilizing exploratory spatial data analysis methods, we aim to objectively reveal the spatial patterns and associations of energy trade between China and these countries, alongside the spatial distribution of terrorist activities in the region. We construct a Spatial Durbin Model to further analyze how terrorist activities and their spatial spillover effects impact China’s energy trade with the Belt and Road countries. This research addresses questions such as whether terrorist incidents in one country affect the energy trade of neighboring countries and to what extent. The study endeavors to furnish more cogent perspectives on the energy security concerns of the Belt and Road countries and to lay a scientific foundation for the formulation of counter-terrorism policies.

Theoretical analysis and hypotheses

Terrorism poses a significant threat to national economies and acts as a major impediment to international trade. Numerous studies have demonstrated that terrorism has a pronounced negative impact on imports and exports trade (Shang, 2020 ). Research by Nitsch and Schumacher indicates that a doubling of terrorist attacks in a trading partner country can lead to a reduction of approximately 4% in bilateral trade, with this decrease largely attributable to the effects of transnational terrorism (Nitsch and Schumacher, 2004 ). Moreover, the mechanisms through which terrorism affects imports and exports trade exhibit heterogeneity. Firstly, different types of terrorist activities have varying impacts. According to Mirza and Verdier, for every 1% increase in the frequency of terrorist attacks, the total value of goods imported by the United States from countries experiencing such attacks decreases by about 0.01% (Mirza and Verdier, 2014 ). Gulazat and Adiljan empirically tested the relationship between trade and terrorism risk between China and India, Pakistan, and Afghanistan, finding that higher casualties from terrorist attacks correlate with larger reductions in China’s exports to these three countries (Gulazat and Adiljan, 2018 ). Secondly, terrorism impacts different categories of traded goods differently. Bandyopadhyay et al., using country-pair data from 1995 to 2012, applied a gravity model to assess the effect of terrorism on bilateral trade among 151 countries worldwide. They discovered that terrorism does not significantly affect primary product trade but does notably suppress trade in manufactured goods (Bandyopadhyay et al., 2014 ). Thirdly, the influence of terrorism shows temporal variations. Egger and Gassebner, utilizing monthly data on terrorism and bilateral trade, revealed that transnational terrorist attacks only have negative effects on U.S. bilateral and multilateral trade in the medium to long term, with negligible short-term impacts (Egger and Gassebner, 2015 ). Additionally, some scholars argue that terrorism has differential impacts on imports versus exports, suggesting that each should be studied separately (Li and Yan, 2018 ).

The impact of terrorism on energy imports and exports trade can be understood through three primary pathways. Pathway 1: The negative effects of terrorist activities spill over into a series of economic activities including domestic production, distribution, exchange, and consumption, thereby affecting imports and exports trade (Sandler, 2015 ). Pathway 2: Terrorism alters government fiscal spending within the country, increasing expenditure on counter-terrorism measures, which can crowd out investment and production, further impacting trade (Blomberg et al., 2004 ). Pathway 3: Terrorism has a significant adverse effect on transportation, particularly more pronounced for short-distance transport, consequently affecting trade flows (Mitra et al., 2018 ). Thus, the influence of terrorism on energy trade can be summarized as direct and indirect impacts. Direct impacts include large-scale shutdowns or withdrawals by energy companies due to terrorism, leading to reduced energy production and supply. Illustratively, subsequent to the Syrian crisis’ inception, assaults on Syrian oil and natural gas installations by terrorists became commonplace, resulting in a precipitous decline in crude oil production from 400,000 barrels daily in 2011 to a mere 24,000 in 2019, alongside a reduction in natural gas output from 7.4 billion cubic meters to 3.6 billion over the same period. Concurrently, the protracted conflict in Yemen precipitated the near collapse of its oil market, with the volume of Chinese energy imports from Yemen diminishing by over fourfold. Indirect impacts involve threats to the security of energy transportation, increased transaction costs for energy, and instigating volatility in global energy prices. An illustration of this is the 2022 terrorist attack on the Nord Stream pipeline, which led to a nearly 200% increase in European gas prices. Similarly, every significant rise in international oil prices prompts China to tap into its strategic reserves to stabilize domestic inflation, thereby reducing its participation in international energy markets. Based on these insights, the following hypotheses are formulated:

Hypothesis 1: Terrorist activities inhibit energy exports, which having a negative impact on China’s energy imports from countries along the Belt and Road.

Hypothesis 2: Terrorist activities inhibit energy imports, which having a negative impact on China’s energy exports to countries along the Belt and Road.

The impact of terrorism often transcends geographical boundaries, spreading from one country to another, with the spillover effect initially manifesting in neighboring countries of where the terrorist acts occur. Doucouliagos used the gravity model of trade to examine the impact of terrorism in neighboring countries on trade across 160 nations globally from 1976 to 2014. He found that a 1% increase in terrorist attacks in a neighboring country reduces trade by approximately 6.4 million in the home country, and this negative impact is present even when the terrorist events in the neighbor result in zero casualties (Doucouliagos, 2017 ). De Sousa explored the causal chain between terrorism and imports and exports trade through geographic proximity, discovering that the closer a country is to the source of terrorism, the greater the negative spillover effect on its imports and exports trade. Regions adjacent to terrorist activities are more susceptible to disruptions and damage (De Sousa et al., 2009 ). Terrorism leads to heightened security measures, induces insecurity among trading partners, disrupts trade agreements, and results in constrained economic development in neighboring countries, all of which hinder their imports and exports trade (Anderson and Marcouiller, 2002 ; Czinkota et al., 2010 ). Terrorist activities can undermine bilateral or multilateral trade agreements, leading to higher trade barriers and costs. Countries that are geographically close often fall under the same trade agreements, so these impacts are not confined to the country where the terrorism occurs but also frequently affect neighboring countries (Doucouliagos, 2017 ). Terrorism increases insecurity among trading partners regarding the affected country, and this insecurity adds to the implicit costs of trade (Anderson and Marcouiller, 2002 ). Increased trade insecurity and risks in the supply chain can alter the trade preferences of partner countries, compelling them to seek alternative partners in the international market. For ordinary goods, there is a broader scope for international alternatives, and trading partners will opt for transactions with countries farther away from terrorist activities. However, energy as a special commodity in the international market, exhibits regional distribution, immobility of production, and concentration of supply, especially in regions like the Middle East and Central Asia, making it difficult to find international alternatives. Contrarily, terrorist activities can have a stimulating effect on energy exports from countries in close proximity to the area of occurrence. For instance, when Yemen’s energy exports were obstructed, other countries increased their imports from Saudi Arabia and Iraq. This demonstrates that terrorism has spatial spillover effects on trade, but these effects differ significantly between energy imports and exports.

The impact of terrorism on the energy exports of countries neighboring the affected country can be explained through two primary pathways. Pathway 1 involves the substitution mechanism in energy trade. When terrorism becomes prevalent, trading partners tend to prefer dealing with countries that offer political and social stability along with abundant energy resources, given that energy resources are geographically concentrated. Pathway 2 concerns fluctuations in international energy prices. Terrorism can trigger significant increases in global energy prices. Neighboring countries endowed with substantial energy resources might seize the opportunity presented by higher prices to boost their energy exports, aiming to maximize their economic returns. Based on these insights, the following hypothesis is formulated:

Hypothesis 3: Terrorism promotes the exports trade of countries neighboring the affected nation, resulting in a positive spatial spillover effect on China’s imports of energy from countries along the Belt and Road.

The impact of terrorism on energy imports in regions neighboring the affected countries can occur through three primary pathways. Pathway 1 involves enhanced border security measures by neighboring countries. Following terrorist incidents, countries in close proximity are likely to implement stricter border controls, amplifying spatial spillover effects. After the Paris attacks in 2015, European nations announced new measures to combat terrorism, which included tightened controls on goods and people crossing borders. These additional security costs are often borne by traders in the form of higher trade expenses, which can subsequently affect import volumes (Walkenhorst and Dihel, 2006 ). Pathway 2 relates to the spatial displacement of terrorist organizations. After terrorist acts, as counter-terrorism efforts intensify domestically, terrorists may flee to neighboring countries, posing a risk of renewed attacks, which in turn can impact imports (Peiyuan and Yafang, 2020 ). Pathway 3 involves the fear and panic induced by terrorist attacks in both the target country and its neighbors. This can dampen consumer demand, increase uncertainty about income, and lead to delays or interruptions in the purchase of imported goods (Czinkota et al., 2010 ). Based on these insights, the following hypothesis is formulated:

Hypothesis 4: Terrorism suppresses the imports trade of countries neighboring the affected nation, leading to a negative spatial spillover effect on China’s energy exports to countries along the Belt and Road.

Existing research indicates that both energy imports and exports trade and terrorist activities exhibit strong spatial correlations. Ignoring the inherent spatial spillover effects can lead to biased empirical results. Therefore, this study considers employing a spatial panel model for empirical analysis, incorporating the spatial lag terms of both variables into the model in order to control for their spatial correlations. Furthermore, spatial correlation can originate not only from the dependent variable itself but also from explanatory variables and error terms. The Spatial Durbin Model can effectively reflect spatial correlations from various sources (Elhorst, 2014 ). Consequently, integrating the theoretical analysis and research hypotheses outlined above, we opt to use the Spatial Durbin Model for our investigation.

Data and methodology

In alignment with the vision and actions to foster the construction of the “One Belt, One Road” initiative and the existing literature, this paper identifies Mongolia in Northeast Asia, 11 countries in Southeast Asia (Timor-Leste, the Philippines, Cambodia, Laos, Malaysia, Myanmar, Thailand, Brunei, Singapore, Indonesia, and Vietnam), 8 countries in South Asia (Afghanistan, Pakistan, Bhutan, Maldives, Bangladesh, Nepal, Sri Lanka, India), 19 countries in Western Asia and North Africa (UAE, Oman, Azerbaijan, Egypt, Palestine, Bahrain, Georgia, Qatar, Kuwait, Lebanon, Saudi Arabia, Turkey, Syria, Armenia, Yemen, Iraq, Iran, Israel, Jordan), 20 countries in Central and Eastern Europe (Albania, Estonia, Belarus, Bulgaria, North Macedonia, Bosnia and Herzegovina, Poland, Russia, Montenegro, Czech Republic, Croatia, Latvia, Lithuania, Romania, Moldova, Serbia, Slovakia, Slovenia, Ukraine, Hungary), and five countries in Central Asia (Kazakhstan, Kyrgyzstan, Tajikistan, Turkmenistan, and Uzbekistan), along with 64 other countries as the subjects of study (Fig. 1 ).

Notes: This study involves countries along the Belt and Road, including China and six regional areas: Northeast Asia (red legend), Southeast Asia (light green legend), South Asia (yellow legend), Central Asia (dark blue legend), Western Asia and North Africa (orange legend), Central and Eastern Europe (light blue legend).

Research methodologies

Exploratory spatial data analysis.

Global spatial autocorrelation mainly studies the spatial distribution characteristics of energy imports and exports between China and countries along the Belt and Road. The formula for global spatial autocorrelation (Moran’s I ) is given as follows:

where GMI represents the global spatial autocorrelation index; n is the number of countries; x i and x j are the values of country i and country j related to China’s energy imports scale or export scale or terrorist activity related indicators; \(\bar{x}\) is the mean value of x , and w ij are the elements in the spatial weight matrix W . In this study, the border bordering carve-out is chosen. When there is a territorial bordering relationship between countries, w ij  = 1, otherwise w ij  = 0. For an island country, the country and the two nearest continental countries have matrix element w ij  = 1.

Local spatial autocorrelation reveals spatial agglomeration as well as spatial interactions using LISA diagrams (Yang et al., 2021 ), which is calculated using the following formula:

Where LMI i is the local spatial autocorrelation index; Z i and Z j are the normalized values of country i and country j in relation to China’s energy imports scale or export scale or terrorist activity indicators; x i is the value of country i in relation to China’s energy imports scale or export scale or terrorist activity indicators; \(\bar{x}\) is the mean value of x , and w ij is the element in the spatial weighting matrix W as Eq. ( 1 ).

Spatial Durbin model

To circumvent the issue of multicollinearity in the model, this paper incorporates the terrorism index and three indicators of the number of terrorist activities, the number of deaths, and the number of injuries into the spatial Durbin model. Due to the strong correlation between the terrorism index and the number of occurrences, deaths, and injuries of terrorist activities, this article separates the terrorism index from these three variables and regresses them to form two independent models. Meanwhile, VIF test was conducted on the number of occurrences, the number of deaths, and the number of injuries, and the results showed that there was no multicollinearity with these three variables. This is done to explore the impact of terrorist activities on the scale of China’s energy imports and exports from countries along the route, and four models are established (Sterne, 2009 ). The energy imports model formula is as follows:

Where I it is the energy scale of China’s importing country I in year t ; terrorismI it , terrorismN it , terrorismD it , terrorismW it are the terrorism index, as well as the number of occurrences, deaths, and injuries of terrorist activities in country I in year t , respectively; GDP it , LSCI it , REP it are the GDP, liner shipping connectivity index, and energy exports dependence of country I in year t , respectively; W is the spatial weight matrix, as illustrated in Eq. ( 1 ); α is the constant term, l is the n × 1 order unit matrix; β (0−5) and θ (0−5) are the regression coefficients; ρ is the spatial autocorrelation coefficient and ε it is the error term. The energy exports model equation is as follows:

Where E jt is the energy scale of China’s exporting country j in year t ; other variables and coefficients are as explained in the previous Eq. ( 4 ) And (5). The variables in the model are spatial panel data, and the model regression is realized with the help of MATLAB software.

The study period for this paper is selected as 2008–2019, the dependent variable of this study is the energy scale of China’s imports and exports countries along the Belt and Road, the kernel variable is the terrorism index and its sub-indicators, and the control variable is gross domestic product, liner shipping connectivity index, and energy exports dependence (Table 1 ). Data on the scale of energy imports and exports between China and the countries along the Belt and Road are obtained from the United Nations Commodity Trade Statistics Database (UN Comtrade). The types of energy include coal, natural gas, Oil, and other fossil fuels. Countries along the route that have import and export relations with China have missing data in individual years. There are 17 instances of missing import data and 6 instances of missing export data, accounting for 2.54% and 0.90% respectively. These missing data are interpolated and supplemented by calculating the average values of the two years before and after. The data for the terrorism index and its sub-indicators (number of occurrences of terrorist activities, number of deaths, number of injuries) are sourced from the Global terrorism index Report and the Global Terrorism Database. The terrorism index is calculated by weighting the indicators related to terrorist activities, with a weighted value of 1 for the number of occurrences, 3 for the number of deaths, 0.5 for the number of injuries, and 0.5 for the total amount of property damage. The final dimensionless processing results in a value in the range of 0–10. The data of GDP, liner shipping connectivity index, and energy exports dependence were obtained from the World Bank Database. Liner shipping connectivity index mainly reflects the degree of connectivity between countries and the global maritime transportation network. Energy exports dependence is the share of energy exports in a country’s trade. The statistical characteristics of the variables are shown in Table 2 . In order to eliminate the dimensional influence between different indicators, all independent variables are transformed by ln(x + 0.1) in this paper.

Temporal evolution characteristics

A significant volatility is observed in the scale of China’s energy imports into countries along the route and the development of terrorist activities along the route (Fig. 2 ). It has been recorded that the value of energy imports by China from these countries experienced an increase from 98.8 billion U.S. dollars in 2008 to 207.3 billion U.S. dollars in 2014, followed by a decline to 107.4 billion U.S. dollars in 2016, and a subsequent rise to 211.3 billion U.S. dollars in 2019. Conversely, the scale of energy exports from China to these countries has been noted to exhibit a consistent year-on-year growth, albeit at a lower magnitude when juxtaposed with the imports, escalating from 6.6 billion U.S. dollars in 2008 to 21.2 billion U.S. dollars in 2019. The implementation of strategies for energy diversification and clean energy by China has been inferred to mitigate the dependency on energy imports from the aforementioned countries to a certain degree.

Notes: This figure shows the trends of various indicators over time for China and countries along the Belt and Road. The x -axis represents the years, the left y -axis represents the trade volume, and the right y -axis represents the number of incidents. The symbols used include: blue bars represent energy imports, orange bars represent energy exports, gray line represents the number of occurrences of terrorist activities, yellow line represents the number of deaths from terrorist activities, blue line represents the number of injuries from terrorist activities.

Regarding terrorism, a pattern of initial escalation followed by a reduction has been discerned in the frequency of terrorist incidents, deaths, and injuries. The number of occurrences of terrorist activities augmented from 3870 in 2009 to 12,584 in 2014, then diminished to 5432 in 2019. The number of deaths rose from 7175 in 2009 to 29,471 in 2014, then fell to 13,423 in 2019. The number of injuries escalated from 15,493 in 2009 to 34,559 in 2015, then decreased to 14,944 in 2019. The underlying causes have been attributed to the financial crisis precipitating substantial shifts in the geopolitical contours of the Middle East and Asia-Pacific regions, coupled with an intensification of terrorist activities. The interplay of influence exerted by the U.S. and Western countries has been linked to a concentrated surge in conflicts across the Belt and Road countries, while domestic policies, such as the curtailment of welfare expenditures, have incited societal unrest and a resurgence in ideologically driven terrorist acts. A series of counter-terrorism agreements and accords have been signed by the countries along the route, and a fortification of counter-terrorism cooperation has been observed, which to a certain extent has mitigated the prevalence of terrorist activities. For instance, China and Afghanistan have bolstered security and law enforcement operations and mobilized their cooperation with Tajikistan and Pakistan to hold the Senior Leaders’ Meeting on Counter-Terrorism Cooperation among the Military Forces of the Four Countries of Afghanistan, China, Pakistan, and Thailand. China and Vietnam have conducted the “Thien Thanh-2016” joint counter-terrorism exercise, and the border guard forces of the two sides have enhanced cooperation on terrorist activities.

Spatial evolution characteristics

Characteristics of global spatial autocorrelation.

Overall, the scale of China’s energy imports and exports from the countries along the route and the terrorism index of the countries along the route exhibit spatial clustering (Table 3 ). Except for the Moran’s I of energy imports in 2009 and energy exports in 2008, which are not significant, all other years are significant and have positive coefficients. This indicates that countries with a high scale of China’s energy imports and exports are geographically close to each other, as are countries with a high terrorism index.

The spatial pattern of China’s energy imports exhibits a tendency towards polarization. The Moran’s I 2008 to 2011 were modest, all falling below 0.20, whereas those from 2012 to 2019 ranged between 0.23 and 0.31, peaking at 0.304 in 2014 and reaching a nadir of 0.235 in 2017. Conversely, the spatial pattern of China’s energy exports appears more decentralized, with only the 2017 Moran’s I attaining 0.22, while all other years remained below this threshold. Lastly, the spatial pattern of the terrorism index along the route initially demonstrated a trend towards dispersion, followed by subsequent polarization. The Moran’s I declined from 0.261 in 2008 to 0.206 in 2012, suggesting a diffusion of terrorist activities. However, from 2013 to 2019, the index rose from 0.230 to 0.360, indicating a contraction in the spatial extent of terrorist activities in the countries along the route.

Characteristics of local spatial autocorrelation

The LISA agglomeration map is categorized into four spatial types (Fig. 3 ). High-High Agglomeration Area (H-H): Countries with a high scale of energy imports and exports with China or a high terrorism index are neighbored by countries that also have a high scale of energy imports and exports with China or a high terrorism index. High-Low Agglomeration Area (H-L): Countries with high energy imports and exports to and from China or a high terrorism index are neighbored by countries with low energy imports and exports to and from China or a low terrorism index. Low-High Agglomeration Area (L-H): Countries with low energy imports and exports with China or a low terrorism index are neighbored by countries with high energy imports and exports with China or a high terrorism index. Low-Low Agglomeration Area (L-L): Countries with a low scale of energy imports and exports with China or a low terrorism index are neighbored by countries that also have a low scale of energy imports and exports with China or a low terrorism index.

a energy imports in 2008, b energy imports in 2019, c energy exports in 2009, d energy exports in 2019, e terrorism index in 2008, f terrorism index in 2019.

High-high agglomeration area (H-H)

China’s energy imports are concentrated in West Asia and North Africa. Kuwait, Oman, Saudi Arabia, and the United Arab Emirates in 2008 and 2019 were all of this type. This type added Iraq, Mongolia, and Qatar in 2019 compared to 2008, and removed Yemen. These countries, alongside their neighboring territories, serve as pivotal growth poles for China’s energy importation endeavors. The aforementioned regions of West Asia and North Africa are strategically situated within the ‘energy heartland’, a locale reputed for harboring approximately half of the global reserves of oil and natural gas, thereby constituting a critical zone for China’s energy importation. Conversely, China’s energy exportation is notably concentrated in Southeast Asia, with Singapore and the Philippines being the sole recipients in 2009 and 2019, respectively. The rapid industrialization phase currently experienced by Southeast Asian countries necessitates the importation of substantial coal quantities due to inherent energy structure limitations, hence the focal concentration of China’s energy exports in this region. Furthermore, the areas registering high values on the terrorism index—South Asia, Central Asia, and West Asia North Africa—demonstrate a spatial coupling with China’s energy importation patterns. The years 2008 and 2019 have seen Jordan, Iran, Pakistan, and Syria categorized under this typology, with an increase in Iraq, Bangladesh, Israel, and Palestine, and a decrease in India and Bhutan for the latter year. The regions of West Asia and North Africa have been historically entangled in political and economic challenges, compounded by issues related to terrorist activities, which have escalated in their global proportion. South Asia, with its many religious sects and a large number of terrorist organizations, is the region of the contemporary world with the most intense terrorist activity outside of the Middle East. Religious extremist organizations, such as the Taliban, are frequently responsible for terrorist activities in eastern Tajikistan, southern Kyrgyzstan, and the Ferghana Valley of Uzbekistan, and have become a major problem for regional security in Central Asia.

High-low agglomeration area (H-L)

The number of countries in this category is limited, with only terrorism index of Russia appearing in this type of region in the 2019. After the dissolution of the Soviet Union, Russia became one of the major countries in the world that suffered the most terrorist attacks, with a much higher frequency of terrorist activities than neighboring Eastern European and East Asian countries. Russia inherited a significant number of the erstwhile superpower’s challenges, including ethnic separatism and regional conflicts. These issues have been particularly pronounced in areas like Chechnya, where incidents of terrorism are frequently instigated by ethnic separatist factions. Additionally, religious extremism has been identified as a contributing factor, with certain organizations exploiting religious doctrines to foment and facilitate acts of terror, thereby undermining the societal stability and security of the Russian Federation. The Islamic State, for instance, has been implicated in sowing discord and devastation within the country. Economic disparities, such as the pronounced divide between affluent and impoverished demographics, alongside economic instability, unemployment, and poverty, have been recognized as pivotal elements influencing the prevalence of terrorism in Russia. Between the years 2008 and 2019, Russia experienced 1245 terrorist incidents, resulting in 3477 casualties. Despite a reduction in terrorist activities in 2019, the frequency of such events in Russia remains elevated when compared to neighboring countries, including Estonia, Latvia, and Belarus.

Low-high agglomeration area (L-H)

The low-high agglomeration area of China’s energy imports is distributed in Central and Eastern Europe and West Asia. In 2008, this included Azerbaijan, Mongolia, Qatar, Bahrain, Iraq, and Timor-Leste, and in 2019, it included Yemen, Bahrain, Jordan, Estonia, and Kazakhstan. China imports less energy from these countries than from its neighbors. China’s import of energy from Yemen and Iraq have changed considerably. The conflict in Yemen, triggered by the struggle between the Houthis and the remaining forces of the Hadi government in 2015, led to a sharp decline in China’s import of energy from Yemen, from 3.12 billion dollars to 0.63 million dollars. China’s import of energy from Iraq were hampered by the U.S. war in Iraq at the beginning of the twenty-first century. Following the end of the war in Iraq in 2010, China’s import of Iraqi energy rose sharply, from 1.32 billion dollars to 13.81 billion dollars. China’s energy cooperation with Central and Eastern Europe focuses on renewable energy, and the traditional energy imports and exports scale is low. The low-high agglomeration area of China’s energy exports is distributed in Southeast Asia, specifically in Cambodia and Brunei in 2008 and 2019. Cambodia and Brunei have low energy demand, and the energy exported by China to these two countries differs greatly from its neighboring countries. The low-high agglomeration area for terrorism index includes Kuwait and the Maldives. Kuwait has a low frequency of terrorist activity compared to Saudi Arabia and Iraq. The Maldives, being far away from Eurasia, is thus protected from the volatility of South Asia.

Low-low agglomeration area (L-L)

The low-low agglomeration areas for China’s energy imports and exports from countries along the route, as well as for terrorist activities in countries along the route, are all located in Central and Eastern Europe. China’s energy imports Low-Low agglomeration area included India, Serbia, Croatia, Bosnia and Herzegovina, and Montenegro in 2008. Apart from India, all other countries belong to Central and Eastern Europe. In 2019, there were Central and Eastern European countries such as Serbia, Croatia, Bosnia and Herzegovina, Romania, Montenegro, and Albania. China’s energy exports low-low agglomeration area included Serbia, Croatia, and Romania in 2009. In 2019, it included Serbia, Croatia, Romania, Hungary, and North Macedonia, all of which belong to Central and Eastern Europe. The development of oil and gas resources in Central and Eastern European countries is limited, and these countries are highly dependent on Russia. Consequently, the scale of China’s energy imports and exports to Central and Eastern European countries is low. Terrorism index low-low agglomeration area included Serbia, Hungary, and the Czech Republic in 2008. In 2019, it included Serbia, Croatia, Hungary, North Macedonia, and Bosnia and Herzegovina. These countries all belong to Central and Eastern Europe. After the September 11 Attacks, European counter-terrorism cooperation has been strengthened. It has evolved from initial passive emergency response measures into a multi-dimensional counter-terrorism policy system with medium- and long-term strategic planning, which has played a positive role in European security.

Spatial model analysis results

Baseline regression analysis.

The presence of spatial autocorrelation in the indicators related to energy imports and exports, as well as terrorist activities between China and the Belt and Road affiliated countries, was identified, leading to the selection of the spatial Durbin model for analysis. It was observed in Table 4 that the original hypothesis was rejected by both the LR test and the Wald test at the 1% significance level, which substantiated the appropriateness of the spatial Durbin model’s application. Furthermore, the rejection of the original hypothesis by the Hausman statistic at the 1% significance level, as indicated in Table 4 , necessitated the adoption of the fixed effects model in this research. Significance at the 1% level was confirmed for both individual and temporal fixed effects across Models 1 to 4.

Subsequently, the Variance Inflation Factor (VIF) test was administered to all variables within the model to ascertain the extent of covariance issues. The findings, as presented in Table 5 , revealed that the VIF values for the variables did not exceed 8.5, and the aggregate VIF for each model did not surpass 2.21, a figure considerably below the threshold of 10, thereby eliminating concerns of multicollinearity within the regressions. Given that the spatial Durbin model incorporates a lagged dependent variable, its estimation was conducted utilizing the maximum likelihood (ML) method.

In Table 4 , the regression outcomes of the spatial Durbin model are presented, having incorporated fixed effects, and it is observed that the models exhibit a commendable fit, evidenced by R 2 values surpassing the 0.85 threshold, thereby indicating a high level of regression reliability. The import model’s ρ (Model 1 and 2) is validated at the 1% significance level, as is the export model’s ρ (Model 3 and 4), with both coefficients being positive, denoting the presence of spatial spillover effects in China’s energy trade, both imports and exports, with the Belt and Road countries, aligning with the positive Moran’s I value. TerrorismI is found to be significant at the 1% level with a negative coefficient in the import model, yet it does not hold significance in the export model. This differential suggests that the terrorism index variably impacts the magnitude of China’s energy trade with these countries, with terrorist activities ostensibly impeding China’s energy imports from the countries along the route. This outcome is consistent with the expectations set by Hypothesis 1, but it does not conform to what was anticipated by Hypothesis 2. This impediment is attributed to the concentration of China’s energy imports from the West Asia and North Africa region, which is notably afflicted by terrorism, with such activities curtailing energy production and supply in the region and escalating the costs associated with energy transactions. In the realm of economic geography, it has been observed that the exportation of energy by China to countries within Southeast Asia correlates with a diminution in terrorist attacks, attributed to the enhanced counter-terrorism measures undertaken by these countries. A divergence is noted when the influence of terrorist activities on merchandise trade is considered. Some studies have elucidated that terrorist activities exert a substantial detrimental effect on the importation and exportation of commodities by China from countries aligned with the Belt and Road. The magnitude of the impact wrought by terrorist attacks on trade is comparable to an escalation in tariffs by 30% (Bing and Xiaochen, 2018 ). In a similar vein, Shang has identified that terrorist activities adversely affect China’s bilateral trade with Central Asian countries, albeit the extent of this impact is relatively marginal (Shang, 2020 ). Furthermore, Bandyopadhyay have accentuated that terrorist activities predominantly curtail trade in manufactured goods, whilst concurrently presenting the potential for an augmentation in the trade of primary commodities (Bandyopadhyay et al., 2018 ).

Spatial effects decomposition

In the decomposition of spatial effects, it has been elucidated by Dubé and Legros that the bias in spatial regression estimations can be circumvented through the segregation of spatial effects into constituent direct and indirect influences (Dubé and Legros, 2014 ). Pursuant to this methodology, the decomposition of spatial effects was executed in the present analysis, yielding the subsequent findings.

The regression outcomes pertaining to direct effects, as delineated in Table 6 , elucidate the repercussions of terrorist activities within a nation upon the energy trade between China and the said nation. It is discerned from Model 1 that TerrorismI inversely correlates with the magnitude of China’s energy imports from the nation in question, as evidenced by a coefficient of −0.339 at a 1% significance threshold. Model 2 further reveals that both TerrorismN and TerrorismD are inversely proportional to Chinese energy imports, with coefficients of −0.359 and −0.719 respectively, surpassing the 10% significance criterion, which means that the frequency of terrorist activities and the number of injured individuals will suppress China’s energy imports. These findings are in alignment with the research by Shang, which posits a negative and significant correlation between the frequency of terrorist attacks and a nation’s export volume of goods, while the impact of injuries and fatalities resulting from such attacks on export volumes is negative yet not statistically significant (Shang, 2020 ). In the conducted analyses, it has been demonstrated through Models 1 and 2 that the control variables of GDP and REP are positively correlated with energy exports to China, as evidenced by their passage of the 1% significance level test. This suggests that a larger economic size and a higher reliance on energy exports are associated with increased energy exports to China. Conversely, the TerrorismI was not found to be significant in Model 3, indicating a negligible impact on China’s energy exports to countries along the Belt and Road. However, Model 4 reveals a different aspect, where TerrorismN is positively correlated with the scale of China’s energy exports, as it passes the 5% significance level with a coefficient of 0.125, which means that the frequency of terrorist activities has a positive impact on the energy scale of countries along China’s export routes. This finding aligns with the previous studies, which observed that frequent terrorist activities in a country lead to a substantial decrease in energy imports from neighboring countries, thereby triggering a substitution mechanism in energy trade that indirectly boosts China’s energy exports (Bing and Xiaochen, 2018 ). Furthermore, De Sousa have identified a positive influence of terrorist activities on the enhancement of trade security measures in proximate countries, which in turn benefits the export trade of more distant countries (De Sousa et al., 2018 ). Lastly, the analyses of Models 3 and 4 also confirm that countries with a larger economic scale and a higher Liner Shipping Bilateral Connectivity Index are likely to import more from China, as indicated by the positive coefficients and passage of the 1% significance level test for these control variables.

In the regression analysis of indirect effects presented in Table 6 , it is demonstrated that the terrorist activities within a nation’s adjacent countries exert an influence on the energy imports and exports of China from said nation. It is indicated by Model 1, wherein TerrorismI attains a coefficient of 0.207, surpassing the 10% significance threshold, that a positive spatial spillover effect is exerted by the terrorism index on the energy imports by China from countries situated along the route. This effect is characterized by an augmentation in the magnitude of China’s energy imports commensurate with the escalation of the terrorism index in a neighboring country. This outcome is consistent with the expectations set by Hypothesis 3, and it presents a divergence from studies on general merchandise trade. For example, Nitsch and Rabaud found that in the six months after the attack on France, there was a decrease of over 6 billion euros in commodity trade, and the impact on commodity exports from partner countries with lower border barriers to France was stronger (Nitsch and Rabaud, 2022 ). De Sousa have posited that the implementation of more stringent border security measures by countries in proximity to terrorist activities culminates in escalated trade costs and a contraction in the volume of merchandise exports (De Sousa et al., 2018 ). The disparity between the outcomes of this investigation and those pertaining to other commodity trades is principally attributable to the unique nature of energy as a commodity, characterized by geographical concentration and agglomeration of production. While general commodity trade is susceptible to international substitution, prompting the long-distance relocation of substitutable commodity trade in response to terrorist activities, the energy production and supply along the Belt and Road is predominantly centralized in the Middle East. Consequently, in the event of the emergence of terrorist activities in a particular country, China is compelled to engage in energy trade with neighboring countries that are endowed with abundant energy resources. In the analysis presented within Model 2, it is observed that the variables TerrorismN, TerrorismD, and TerrorismW hold no significant weight. Similarly, the control variables GDP, LSCI, and REP are deemed to be of no consequence in both Models 1 and 2, from which it can be inferred that the economic growth, maritime commerce, and reliance on energy exports of the adjacent countries exert no notable influence on the magnitude of energy imports by China from said countries. Within the confines of Model 3, TerrorismI is found to surpass the threshold of significance at the 10% level, with a coefficient of −0.774, suggesting the presence of a negative spatial spillover effect stemming from the terrorism index on the exportation of energy by China from countries in route, whereby an elevated terrorism index in a neighboring country correlates with a diminished scale of energy exports from China to that country. This outcome is consistent with the expectations of Hypothesis 4, primarily influenced by factors such as the displacement of terrorist groups and the reinforcement of border controls by neighboring countries. In Model 4, TerrorismN is significant at the 5% level, and TerrorismD at the 10% level, with coefficients of 0.335 and −0.398, respectively, indicating that while recurrent terrorist activities in a neighboring country amplify the scale of China’s energy exports to that nation, an excessive tally of fatalities due to terrorism inversely affects the scale of these exports. This phenomenon corroborates the assertion that terrorist activities incite shifts in international trade over extended distances (De Sousa et al., 2018 ). GDP attains significance at the 10% level in both Models 3 and 4, with coefficients of 2.096 and 0.540, respectively, denoting a spatial spillover effect of the economic magnitude of a country on the scale of China’s energy exports, with the implication being that a higher economic stature of a neighboring country enhances the scale of China’s energy exports thereto.

Spatial heterogeneity analysis

In the context of the Belt and Road, significant regional disparities have been observed in the energy trade between China and the participating countries, as well as in the prevalence of terrorist activities. An analysis was conducted on the regional heterogeneity of the impacts that terrorist activities have on the eastern, central, and western regions along the Belt and Road (Table 7 ). The division of regions was based on the following criteria: Northeast Asia and Southeast Asia were classified as the eastern region; South Asia, Central Asia, West Asia, and North Africa as the central region; and Central and Eastern Europe as the western region. It was found through regression analysis that the terrorism index exerts a significant negative influence on the energy imports of China from all regions, with the central and western regions being notably affected. The magnitude of the coefficients indicates that the central region’s terrorist activities have the most substantial effect on China’s energy imports, surpassing those of the eastern region, and the western region’s impact being the least. This phenomenon is primarily attributed to the fact that South Asia, Central Asia, West Asia, and North Africa constitute the Eurasian continent’s peripheral zone, which is a hotspot for terrorist activities, thus posing the greatest geopolitical risk to the energy trade between China and the countries along the Belt and Road. Within the framework of the Terrorism Index’s sub-indicators, it has been observed that the incidence of injuries resulting from terrorist acts in the eastern and central regions exerts a negative influence on the energy trade between China and the countries within these areas. Notably, the central region’s impact magnitude surpasses that of the eastern region. Conversely, the occurrence of deaths due to terrorism in the central region is associated with a positive effect on the energy imports by China from the affected countries. This phenomenon is largely attributed to the nature of high-death terrorist incidents, which are typically political, radical, and ephemeral, leading to a substantial escalation in international energy prices. The volatility in energy prices, particularly in the short term, necessitates a risk premium that amplifies China’s energy imports volume from South Asia, Central Asia, as well as West Asian and North African countries. Regarding control variables, the central region’s economic development level is positively correlated with the energy trade between China and the regional countries. Furthermore, the liner shipping indices pertinent to the eastern and western regions are found to positively influence the energy trade. Lastly, the central region’s reliance on energy exports is seen to positively impact the scale of energy imports by China from the region, while simultaneously exerting a negative effect on the export scale.

Robustness test

To ensure the reliability of the empirical results, this paper employs three methods to test the robustness of the models: substituting the spatial weight matrix, lagging the explanatory variables and control variables by one period, and utilizing a dynamic spatial Durbin model. The outcomes indicate that, overall, there is little change in the sign and significance of the coefficients for the primary observed variables, suggesting that the conclusions drawn in this study exhibit good robustness.

Following the approach by Shao (Shao et al., 2022 ), this paper employs the geographic distance between countries to construct the spatial weight matrix for the robustness test. The outcomes, as depicted in Table 8 , reveal that the variable coefficients’ directionalities are in alignment with the empirical observations documented in Table 6 . Furthermore, it is observed that the absolute magnitudes of the coefficients associated with terrorist activities surpass those in Table 6 . This phenomenon suggests an expanded spatial influence of terrorist activities on the energy trade dynamics, specifically the imports and exports transactions between China and the countries situated along the Belt and Road.

Drawing on Elhorst’s method (Elhorst, 2014 ), the explanatory and control variables are lagged by one period and replaced with the original variables for the robustness test. The results in Table 9 demonstrate that apart from the variable TerrorismD changing from insignificant to significant at the 10% level, the other regression outcomes remain consistent with those in Table 4 , indicating the robustness of the empirical results.

The dynamic spatial Durbin Model offers improvements over the Static Spatial Durbin Model by addressing endogeneity issues to a greater extent. It does so by accounting for the spatial and temporal lags of the dependent variable, which helps mitigate potential biases arising from omitted variable problems and the autocorrelation of errors. The dynamic spatial Durbin model takes into account lagged dependent variables, spatial factors, and endogeneity issues, which can better explain the relationship between terrorist activities and energy imports and exports trade. Following the method by Liu (Liu et al., 2024 ), our re-estimation using the dynamic spatial Durbin Model (Table 10 ) yields stable results.

Conclusions and implications

Conclusions.

Firstly, this paper analyzes the temporal dynamics and spatial heterogeneity of China’s energy imports and exports trade with countries along the Belt and Road, as well as the terrorism activities in these countries. Through the analysis, we find that all three phenomena exhibit characteristics of spatial agglomeration. In terms of time, there is significant fluctuation in China’s energy imports and terrorist activities in countries along the Belt and Road, roughly showing a “falling-rising-falling” trajectory, while China’s energy exports are relatively low and stable. Geographically, a clustering pattern is discernible, with energy imports and exports, as well as the incidence of terrorist activities, being regionally concentrated. Predominantly, the imports are centralized in West Asia and North Africa, with countries such as Saudi Arabia, Russia, the United Arab Emirates, Kuwait, and Iran being the primary contributors. In contrast, the exports from China are primarily directed towards Southeast Asian countries, including Singapore, Cambodia, and Brunei. The prevalence of terrorist activities is notably higher in regions where China’s energy importation is substantial, particularly in West Asia, North Africa, South Asia, and Central Asia. This correlation indicates that the countries with significant energy ties to China also experience a higher frequency of terrorist incidents.

Subsequently, this paper employed the spatial Durbin model to investigate the impact of terrorist activities on energy imports and exports trade between China and countries along the Belt and Road. Interestingly, we have found that terrorist activities have significant spatial spillover effects on energy imports and exports trade between China and countries along the Belt and Road. Terrorist activities have spatial spillover effects on energy trade, which can reduce the scale of energy trade with neighboring countries. This study distinguished the impact of terrorist activities on energy imports and exports.

Regarding the interrelation between terrorist activities and the importation of energy by China, it is observed that a reduction in energy production and market supply is caused by such activities. Consequently, an escalation in energy prices and transportation risks is witnessed, thereby hindering the importation of energy into China. It is discerned that the central region, encompassing South Asia, Central Asia, West Asia, and North Africa, experiences the most pronounced adverse effects of terrorist activities. This is followed by the eastern region, which includes Northeast Asia and Southeast Asia, while the western region, comprising Central and Eastern Europe, is affected to a lesser extent. The detrimental effects on China’s energy imports from countries situated along the routes are more significant when high-death occurrences are considered, as opposed to high-frequency events. Moreover, a positive spatial spillover effect is identified, wherein the impact of terrorist activities on China’s energy imports is augmented. The clustering of energy’s spatial distribution, coupled with the substitution mechanism inherent in international trade, results in the terrorism index of a nation’s neighboring countries exerting a positive influence on China’s importation of that nation’s energy. Additionally, the national economic development and the dependency on energy exports are factors that positively influence the importation of energy by China.

In the examination of the interplay between terrorist activities and the energy exportation endeavors of China, it has been observed that the terrorism index exerts an insubstantial influence on the energy exports directed towards the countries along the Belt and Road. Conversely, a discernible negative effect has been identified on the energy exports to the central and western regions, with a more pronounced impact in the central region compared to the western region. It is posited that an increase in the frequency of terrorist activities serves as a catalyst for the augmentation of China’s energy exports. This phenomenon can be attributed to the fact that escalated frequencies of terrorist incidents amplify the revenues derived from energy exports, thereby providing an impetus for Chinese trade and economic agents to intensify efforts in expanding energy exportation. Furthermore, such frequent terrorist occurrences have been found to diminish the accessibility of energy in adjacent regions, consequently fostering the importation of energy by China from countries along the route. Nevertheless, terrorist activities of a high-death nature are observed to curtail the magnitude of China’s energy exports to proximate countries. This is primarily due to the substantial geopolitical risks incited by high-death terrorist events, culminating in the closure of numerous international trade corridors. It is also noted that the progression of national economic development and the facilitation of liner shipping are factors that positively modulate the energy exports of China.

Implications

Considering the concentration of China’s energy imports and the adverse effects posed by terrorist activities, it is proposed that measures be adopted to enhance the security of China’s energy trade and the energy supply stability in the countries along the route. It is recommended that China’s energy reserve capabilities be augmented. Efforts to import energy from geopolitically stable countries, such as Saudi Arabia and Kuwait, should be intensified, and the strategic energy reserves expanded. The acceleration of reserve base construction near key coastal ports and petrochemical industry hubs is advocated, alongside the innovation of the oil and gas reserve model through the promotion of energy infrastructure development. The establishment of a multifaceted reserve system, integrating state, commercial, and societal reserves, is envisaged through the mobilization of the general populace and the engagement of private capital in oil reserves. Furthermore, the exploration of China’s indigenous energy resources should be escalated. A continuous increase in energy prospecting, development, and technological advancement is advised, with an emphasis on maritime energy expansion beyond terrestrial endeavors. Prioritization should be given to the enhancement of marine oil and gas resource exploitation and the augmentation of offshore oil drilling platform construction, leveraging progressively sophisticated scientific and technological methods and manufacturing capacities. The procurement of sea-based resources is anticipated to constitute a significant avenue for the realization of China’s energy diversification strategy. Thirdly, the development of international energy space ought to be further pursued by China. Energy cooperation with Russia is to be strengthened, and the expansion of energy imports from countries such as Kazakhstan, Turkmenistan, and Myanmar, which are geographically proximate and have a solid foundation for political and economic cooperation, is to be broadened. For instance, emphasis should be placed on the construction of the China-Russia crude oil pipeline, the acceleration of the western energy corridor’s construction linking Kazakhstan should be hastened, and the advancement of the oil corridor’s construction with Pakistan’s Gwadar Port is to be continuously promoted. Fourthly, the establishment of a coordinated counter-terrorism mechanism with pivotal countries along the route is imperative for China. The creation of terrorism databases and intelligence-sharing platforms in countries situated along the routes should be undertaken, the systematic assessment of the terrorism situation in these countries should be fortified, and the summarization of terrorism’s impact on the energy imports and exports trade between China and the countries along the Belt and Road should be compiled. Consequently, high-, medium-, and low-risk zones for terrorism are to be delineated, enabling the execution of more targeted counter-terrorism cooperation. Upon this foundation, it is proposed that an alliance for counter-terrorism cooperation be established. The deepening of policy and legal cooperation in counterterrorism with countries situated along the designated routes is to be pursued, and the formulation of a unified international counter-terrorism strategy at the highest echelon is necessitated. Furthermore, the establishment of a pragmatic mechanism for international criminal cooperation in counterterrorism is imperative. The enhancement of coordinated counter-terrorism capabilities through the strengthening of multi-party counter-terrorism joint exercises with countries in South Asia, Central Asia, West Asia, and North Africa is advocated. Additionally, it is incumbent upon China’s foreign energy enterprises to disseminate counter-terrorism security knowledge and devise contingency plans for terrorist incidents, thereby augmenting the counter-terrorism security consciousness and proficiency of their personnel.

Data availability

All data generated or analysed during this study are included in this published article.

Ahakwa I (2023) The role of economic production, energy consumption, and trade openness in urbanization-environment nexus: a heterogeneous analysis on developing economies along the Belt and Road route Environ Sci Pollut R 30(17):49798–49816

Article   Google Scholar  

Anderson JE, Marcouiller D (2002) Insecurity and the pattern of trade: an empirical investigation. Rev Econ Stat 84(2):342–352

Balta-Ozkan N, Watson T, Mocca E (2015) Spatially uneven development and low carbon transitions: insights from urban and regional planning. Energy Policy 85:500–510

Bandyopadhyay S, Sandler T (2014) The effects of terrorism on trade: a factor supply approach. Fed Reserve Bank St Louis Rev 96(3):229–241

Google Scholar  

Bandyopadhyay S, Sandler T, Younas J (2014) Foreign direct investment, aid, and terrorism. Oxf Econ Pap 66(1):25–50

Bandyopadhyay S, Sandler T, Younas J (2018) Trade and terrorism: a disaggregated approach. J Peace Res 55(5):656–670

Barber CT (1948) Review of Middle East oil. Petroleum 10:48–62

Bing L, Xiaochen Y (2018) China’s new comparative advantage of trading with Belt and Road countries: public security perspectives. Econ Res J 53(1):183–197

Blomberg SB, Hess GD, Orphanides A (2004) The macroeconomic consequences of terrorism. J Monet Econ 51(5):1007–1032

Blustein P (2009) Misadventures of the most favored nations: Clashing egos, inflated ambitions, and the great shambles of the world trade system. Public Affairs, New York

Bouri E, Demirer R, Gupta R, Marfatia HA (2019) Geopolitical risks and movements in Islamic bond and equity markets: a note. Def Peace Econ 30(3):367–379

Bove V, Elia L, Sekeris PG (2014) US security strategy and the gains from bilateral trade. Rev Int Econ 22(5):863–885

Bradshaw MJ (2010) Global energy dilemmas: a geographical perspective. Geogr J 176(4):275–290

Bridge G, Dodge A (2022) Regional assets and network switching: shifting geographies of ownership, control and capital in UK offshore oil. Camb J Reg, Econ Soc 15(2):367–388

Burke MJ, Stephens JC (2018) Political power and renewable energy futures: a critical review. Energy Res Soc Sci 35:78–93

Cheng CHJ, Chiu CJ (2018) How important are global geopolitical risks to emerging countries? Int Econ 156:305–325

Costa-Campi MT, Del Rio P, Trujillo-Baute E (2017) Trade-offs in energy and environmental policy. Energy Policy 104:415–418

Czinkota MR, Knight G, Liesch PW, Steen J (2010) Terrorism and international business: a research agenda. J Int Bus Stud 41:826–843

De Sousa J, Mirza D, Verdier T (2009) Trade and the spillovers of transnational terrorism. Swiss J Econ Stat 145:453–461

De Sousa J, Mirza D, Verdier T (2018) Terror networks and trade: does the neighbor hurt? Eur Econ Rev 107:27–56

Doucouliagos H (2017) An injury to one is an injury to all: terrorism’s spillover effects on bilateral trade, IZA Discussion Papers, No: 10859. Bonn, Institute of Labor Economics

Du R, Dong G, Tian L et al. (2016) A complex network perspective on features and evolution of world crude oil trade. Energy Procedia 104:221–226

Du R, Wang Y, Dong G et al. (2017) A complex network perspective on interrelations and evolution features of international oil trade, 2002–2013. Appl Energ 196:142–151

Article   ADS   Google Scholar  

Dubé J, Legros D (2014) Spatial econometrics using microdata. John Wiley & Sons, Hoboken, USA

Book   Google Scholar  

Dudley B (2018) BP statistical review of world energy. London, British Petroleum

Egger P, Gassebner M (2015) International terrorism as a trade impediment? Oxf Econ Pap 67(1):42–62

Elhorst JP (2014) Spatial panel data models. Spat Econ: cross-Sect data Spat panels 10:37–93

Firat B (2023) Geopolitics, infrastructure and scale-making in the southern gas corridor. Ethnos 0(0):1–22

Gong X, Feng Y, Liu J, Xiong X (2023) Study on international energy market and geopolitical risk contagion based on complex network. Resour Policy 82:103495

Gulazat T, Adiljan A (2018) Terrorism in three countries in south section of the Belt and Road Initiative: understanding its impact to Chinese export. South Asian Stud Q 1:101–108

Hao X, An H, Qi H, Gao X (2016) Evolution of the exergy flow network embodied in the global fossil energy trade: based on complex network. Appl Energ 162:1515–1522

Ji Q, Zhang H, Fan Y (2014) Identification of global oil trade patterns: an empirical research based on complex network theory. Energy Convers Manag 85:856–865

Jin Y, Zhao H, Bu L, Zhang D (2023) Geopolitical risk, climate risk and energy markets: a dynamic spillover analysis. Int Rev Financ Anal 87:102597

Kelejian HH, Mukerji P (2022) Causal factors of terrorist attacks on countries, and corresponding spill-overs between them. Eur J Polit Econ 72:102092

Kelemen B, Fergnani A (2020) The futures of terrorism against China in the Greater Middle East. Futures 124:102643

Kitamura T, Managi S (2017) Driving force and resistance: network feature in oil trade. Appl Energ 208:361–375

Klinger JM (2020) Environment, development, and security politics in the production of Belt and Road spaces. Territ Polit Gov 8(5):657–675

Leung GC (2011) China’s energy security: perception and reality. Energy Policy 39(3):1330–1337

Li B, Yan XC (2018) China’s new comparative advantage of trading with Belt and Road countries: public security perspectives. Econ Res J 53(1):183–197

Li S, Zhang W, Zhang W (2023) Dynamic time-frequency connectedness and risk spillover between geopolitical risks and natural resources. Resour Policy 82:103554

Lin B, Bae N, Bega F (2020) China’s Belt & Road Initiative nuclear export: implications for energy cooperation. Energy Policy 142:111519

Liu SJ, Li J, Wu D et al. (2024) Risk communication in multistakeholder engagement: a novel spatial econometric model. Risk Anal 44(1):87–107

Article   PubMed   Google Scholar  

Martin P, Mayer T, Thoenig M (2012) The geography of conflicts and regional trade agreements. Am Econ J: Macroecon 4(4):1–35

Minyan Z, Suocheng D, Zhe W et al. (2016) Assessment of countries’ security situation along the belt and road and countermeasures. Bull Chin Acad Sci 6:689–696

Mirza D, Verdier T (2014) Are lives a substitute for livelihoods? Terrorism, security, and US bilateral imports. J Confl Resolut 58(6):943–975

Mitra D, Pham CS, Bandyopadhyay S (2018) Terrorism and international air travel: a gravity approach. World Econ 41(11):2852–2882

Mohamed H, Jebli MB, Youssef SB (2019) Renewable and fossil energy, terrorism, economic growth, and trade: evidence from France. Renew Energ 139:459–467

Neumayer E, Plümper T (2016) Spatial spill-overs from terrorism on tourism: Western victims in Islamic destination countries. Public Choice 169(3):195–206

Nitsch V, Rabaud I (2022) Under attack: terrorism and international trade in France, 2014–2016. Oxf Econ Pap 74(4):976–998

Nitsch V, Schumacher D (2004) Terrorism and international trade: an empirical investigation. Eur J Polit Econ 20(2):423–433

Painter DS (2014) Oil and geopolitics: the oil crises of the 1970s and the Cold War. Hist Soc Res 4(39):186–208

Palle A (2021) Bringing geopolitics to energy transition research. Energy Res Soc Sci 81:102233

Pascual C, Zambetakis E (2010) Energy security: economics, politics, strategies, and implications. Brookings Institution Press, Washington, D. C

Peiyuan X, Yafang L (2020) The impact of terrorist activities on tourism development in countries along the Belt and Road. Econ Geogr 40(3):216–224

Pham CS, Doucouliagos C (2017) An Injury to one is an injury to all: Terrorism’s Spillover Effects on bilateral Trade. IZA Discussion Papers, Bonn

Ramiah V, Wallace D, Veron JF et al. (2019) The effects of recent terrorist attacks on risk and return in commodity markets. Energy Econ 77:13–22

Rhodes R (2018) Energy: A human history. Simon and Schuster, New York

Sandler T (2015) Terrorism and counterterrorism: an overview. Oxf Econ Pap 67(1):1–20

Schuetze B, Hussein H (2023) The geopolitical economy of an undermined energy transition: the case of Jordan. Energy Policy 180:113655

Seung-Whan C (2015) Economic growth and terrorism: domestic, international, and suicide. Oxf Econ Pap 67(1):157–181

Shang B (2020) Impact of terrorism on bilateral trade between china and five central asian countries: based on an expended gravity model. Transform Bus Econ 19(3 C):431–447

Shao S, Fan MT, Yang LL (2022) Economic restructuring, green technical progress, and low-carbon transition development in China: an empirical investigation based on the overall technology frontier and spatial spillover effect. J Manag World 38(2):46–69

Shao Y, Qiao H, Wang S (2017) What determines China’s crude oil importing trade patterns? Empirical evidence from 55 countries between 1992 and 2015. Energy Policy 109:854–862

Smith PJ (2009) China’s economic and political rise: Implications for global terrorism and US-China cooperation. Stud Confl Terror 32(7):627–645

Song Y, Chen B, Hou N, Yang Y (2022) Terrorist attacks and oil prices: a time-varying causal relationship analysis. Energy 246:123340

Sterne JA (2009) Meta-analysis in Stata: an updated collection from the Stata Journal. College Station, TX, Stata Press

Stokes J (2015) China’s road rules: Beijing looks west toward Eurasian integration. Foreign Aff 19:1–17

Sun X, Liu C, Chen X, Li J (2017) Modeling systemic risk of crude oil imports: Case of China’s global oil supply chain. Energy 121:449–465

Thomson E, Horii N (2009) China’s energy security: challenges and priorities. Eurasia Geogr Econ 50(6):643–664

Vakulchuk R, Overland I, Scholten D (2020) Renewable energy and geopolitics: a review. Renew Sustain Energy Rev 122:109547

Wagstaff HR (1974) A geography of energy. Texas, United States, W. C. Brown Co

Walkenhorst P, Dihel N (2006) Trade impacts of increased border security concerns. Int Trade J 20(1):1–31

Wang S, Chen B (2016) Hybrid ecological network and flow-distance analysis for international oil trade. Energy Procedia 104:209–214

Xiang Y, Zhang Q, Wang D, Wu S (2022) Mining investment risk assessment for Nations along the Belt and Road Initiative. Land 11(8):1287

Yang W, Shi W, Du D (2021) The spatial model and spatial mechanism of the global geo-economic co-operation: an empirical research based on the scale of commodity trade. Sci Geogr Sin 41(11):1875–1883

Yang Y, Xia S, Qian X (2023) Geopolitics of the energy transition. J Geogr Sci 33(4):683–704

Yang Y, Yu HY, Lu G et al. (2020) Interview on the unprecedented changes of energy geopolitics and national energy security. J Nat Resour 35(11):2803–2820

Yergin D (2011) The prize: The epic quest for oil, money & power. Simon and Schuster, New York

Yu K (2018) Energy cooperation in the Belt and Road Initiative: EU experience of the trans-european networks for energy. Asia Eur J 16:251–265

Zhang C, Fu J, Pu Z (2019) A study of the petroleum trade network of countries along “The Belt and Road Initiative”. J Clean Prod 222:593–605

Zhang H, Ji Q, Fan Y (2015) What drives the formation of global oil trade patterns? Energy Econ 49:639–648

Zhang Q, Hu Y, Jiao J, Wang S (2024) The impact of Russia–Ukraine war on crude oil prices: an EMC framework. Hum Soc Sci Commun 11(1):1–12

Zhao S (2008) China’s global search for energy security: cooperation and competition in Asia–Pacific. J Contemp China 17(55):207–227

Zhao Y, Liu X, Wang S, Ge Y (2019) Energy relations between China and the countries along the Belt and Road: an analysis of the distribution of energy resources and interdependence relationships. Renew Sustain Energy Rev 107:133–144

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 420001135).

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Institute of World Economy, Shanghai Academy of Social Sciences, Shanghai, China

Wenlong Yang

Department of Hospitality Management, Shanghai Business School, Shanghai, China

Wentian Shi

School of Economics, Hefei University of Technology, Hefei, China

Dongcheng Chen

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Conceptualization: Wenlong Yang, Wentian Shi; Data curation: Wenlong Yang, Wentian Shi; Formal analysis: Wenlong Yang, Wentian Shi; Methodology: Wenlong Yang, Dongcheng Chen; Resources: Wenlong Yang, Wentian Shi; Software: Wenlong Yang, Dongcheng Chen; Writing – original draft: Wenlong Yang, Wentian Shi; Writing – review & editing: Wenlong Yang, Wentian Shi, Dongcheng Chen.

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Yang, W., Shi, W. & Chen, D. Unveiling the nexus: exploring the influence of terrorism on energy trade in China and the Belt and Road countries. Humanit Soc Sci Commun 11 , 1088 (2024). https://doi.org/10.1057/s41599-024-03619-9

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Received : 26 January 2024

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DOI : https://doi.org/10.1057/s41599-024-03619-9

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