Year 2025,
Volume: 25 Issue: 3, 609 - 626, 06.08.2025
Musa Ovalı
,
Koray Kayalıdere
References
-
Abdymomunov, A., & Morley, J. (2011). Time variation of CAPM betas across market volatility regimes. Applied Financial Economics(21), 1463-1478.
-
Agrrawal, P., Gilbert, F., & Harkins, J. (2022). Time Dependence of CAPM Betas on the Choice of Interval Frequency and Return Timeframes: Is There an Optimum? Journal of Risk and Financial Management, 15(11).
-
Akyatan, A., & Çetin, M. K. (2020). Return prediction with time varying betas: a research in BIST. International Journal of Accounting and Finance, 10(1), 64-86.
-
Alexander, G. J., & Chervany, N. L. (1980). On the estimation and stability of beta. Journal of Financial and Quantitative Analysis, 15(1), 123-137.
-
Baesel, J. B. (1974). On the Assessment of Risk: Some Further Considerations. Journal of Finance(29), 1491-1494.
-
Bartholdy, J., & Peare, P. (2001, 03 16). The relative efficiency of beta estimates. Aarhus, Denmark. Retrived on 04 04, 2023 from https://papers.ssrn.com/sol3/papers.cfm?abstract_id=263745.
-
Blasques, F., Francq, C., & Laurent, S. (2024). Autoregressive conditional betas. Journal of Econometrics, 238(2), 1-22.
-
Blume, M. E. (1975). Betas and their regression tendencies. The Journal of Finance, 30(3), 785-795.
Bollerslev, T., Engle, R. F., & Wooldridge, J. M. (1988). A capital asset pricing model with time-varying covariances. Journal of political Economy, 96(1), 116-131.
-
Bos, T., & Newbold, P. (1984). An empirical investigation of the possibility of stochastic systematic risk in the market model. Journal of Business, 35-41.
-
Brenner, M., & Smidt, S. (1977). A simple model of non-stationarity of systematic risk. The Journal of Finance, 32(4), 1081-1092.
-
Brooks, R. D., Faff, R. W., & Lee, J. H. (1992). The form of time variation of systematic risk: Some Australian evidence. Applied Financial Economics, 2(4), 191-198.
-
Brzeszczynski, J., Gajdka, J., & Schabek, T. (2011). The role of stock size and trading intensity in the magnitude of the" interval effect" in beta estimation: Empirical evidence from the Polish capital market. Emerging Markets Finance and Trade, 47(1), 28-49.
-
Cai, Z., & Juhl, T. (2023). The distribution of rolling regression estimators. Journal of Econometrics, 235(2), 1447-1463.
-
Chakrabarti, G., & Das, R. (2021). Time-varying beta, market volatility and stress: A comparison between the United States and India. IIMB Management Review, 33(1), 50-63.
-
Choudhry, T., & Wu, H. (2008). Forecasting Ability of GARCH vs Kalman Filter Method: Evidence from Daily UK Time-Varying Beta. Journal of Forecasting, 27(8), 670-689.
-
Choudhry, T., & Wu, H. (2009). Forecasting the weekly time-varying beta of UK firms: GARCH models vs. Kalman filter method. The European Journal of Finance, 15(4), 437-444.
-
Ciner, C. (2015). Time variation in systematic risk, returns and trading volume: Evidence from precious metals mining stocks. International Review of Financial Analysis(41), 277-283.
-
Collins, D. W., Ledolter, J., & Rayburn, J. (1987). Some Further Evidence on the Stochastic Properties of Systematic Risk. Journal of Business, 60(3), 425-448.
-
Corhay, A., & Rad, A. T. (1993). Return interval, firm size and systematic risk on the Dutch stock market. Review of Financial Economics, 2(2), s. 19-28.
-
Çayırlı, Ö., Kayalıdere, K., & Aktaş, H. (2022). Toplam Getiri Yerine Fiyat Getirisi Kullanılmasının Varlık Fiyatlama Modelleri ve Portföy Seçimi Üzerine Yapılan Çalışma Sonuçlarına Etkisi. Muhasebe ve Finansman Dergisi, 75-92.
-
Çelik, S. (2013). Testing the Stability of Beta: A Sectoral Analysis in Turkish Stock Market. Journal of Economics and Behavioral Studies, 5(1), 18-23.
-
Çenesizoğlu, T., Liu, Q., Reeves, J., & Wu, H. (2016). Monthly Beta Forecasting with Low-, Medium- and High-Frequency Stock Returns. Journal of Forecasting, 35(6), 528-541.
-
Damodaran, A. (1999). Estimating Risk Parameters. New York, USA. Retrived on 04 04, 2023 from https://archive.nyu.edu/bitstream/2451/26906/2/wpa99019.pdf.
-
Damodaran, A. (2013, Mart). Equity Risk Premiums (ERP): Determinants, Estimation and Implications- The 2013 Edition. (6). http://ssrn.com/abstract=2238064.
-
Das, A., & Ghoshal, T. K. (2010). Market risk beta estimation using adaptive Kalman Filter. International Journal of Engineering Science and Technology, 2(6), 1923-1934.
-
Das, S., & Barai, P. (2015). Time-varying industry beta in Indian stock market and forecasting errors. International Journal of Emerging Markets, 10(3), 521-534.
-
Dasgupta, S., Gan, J., & Gao, N. (2010). Transparency, price informativeness, and stock return synchronicity: Theory and evidence. Journal of Financial and Quantitative Analysis, 45(5), 1189-1220.
-
Daves, P. R., Ehrhardt, M. C., & Kunkel, R. A. (2000). Estimating systematic risk: the choice of return interval and estimation period. Journal of Financial and Strategic Decisions, 13(1), 7-13.
-
Domenech, N., Orbe Mandaluniz, S., & Zarraga, A. A. (2011). Time-varying beta estimators in the Mexican emerging market.
-
Drobetz, W., Hollstein, F., Otto, T., & Prokopczuk, M. (2023, 12 14). Estimating Stock Market Betas via Machine Learning. Retrived on 01 02, 2024 from https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3933048.
-
Estrada, J. (2000). The temporal dimension of risk. The Quarterly Review of Economics and Finance, 40(2), 189-204.
-
Fabozzi, F. J., & Francis, J. C. (1978). Beta as a Random Coefficient. Journal of Financial and Quantitative Analysis, 13(1), 101-116.
-
Faff, R. W., Hillier, D., & Hillier, J. (2000). Time Varying Beta Risk: An Analysisof Alternative ModellingTechniques. Journal of Business Finance & Accounting, 27(5), 523-554.
-
Faff, R. W., Lee, J. h., & Fry, T. R. (1992). Time stationarity of systematic risk: Some Australian evidence. Journal of Business Finance & Accounting, 19(2), 253-270.
-
Fama, E. F., & Macbeth, J. D. (1973). Risk, Return, and Equilibrium: Empirical Tests. The University of Chicago Press Journals, 81(3), s. 607-636.
-
Fama, E., & French, K. (1997). Industry Costs of Equity. Journal of Financial Economics(43), 153-193.
Frankfurter, G. M., Leung, W. K., & Brockman, P. D. (1994). Compounding period length and the market model. Journal of Economics and Business, 46(3), 179-193.
-
Gong, S. X., Firth, M., & Cullinane, K. (2006). Beta estimation and stability in the US-listed international transportation industry. Review of Pacific Basin Financial Markets and Policies, 9(3), 463-490.
-
Groenewold, N., & Fraser, P. (1999). Time-varying estimates of CAPM betas. Mathematics and Computers in Simulation, 48, 531-539.
-
Groenewold, N., & Fraser, P. (2000). Forecasting Beta: How Well Doesthe `Five-Year Rule of Thumb' Do? Journal of Business Finance & Accounting, 27(7), 953-982.
-
Gümrah, Ü., & Konuk, S. (2018). Zamanla Değişen Beta: Borsa İstanbul Bankacılık Sektörü Uygulaması. Ekonomik ve Sosyal Araştırmalar Dergisi, 14(1), 51-66.
-
Hasnaoui, H., & Fatnassi, I. (2014). Time-Varying Beta And The Subprime Financial Crisis: Evidence From U.S. Industrial Sectors. The Journal of Applied Business Research, 30(5), 1465-1476.
-
Hawawini, G. (1983). Why beta shifts as the return interval changes. Financial Analysts Journal, 39(3), 73-77.
He, L. T. (2005). Instability and predictability of factor betas of industrial stocks: The Flexible Least Squares solutions. The Quarterly Review of Economics and Finance(45), 619-640.
-
Hollstein, F., Prokopczuk, M., & Simen, C. (2019). Estimating beta: Forecast adjustments and the impact of stock characteristics for a broad cross-section. Journal of Financial Markets(44), 91-118.
-
Iqbal, J., & Brooks, R. (2007). Alternative beta risk estimators and asset pricing tests in emerging markets: The case of Pakistan. Journal of Multinational Financial Management, 17(1), 75-93.
-
Kalnina, I. (2022). Inference for nonparametric high-frequency estimators with an application to time variation in betas. Journal of Business & Economic Statistics, 1-12.
-
Kaminsky, G. L., & Kumar, M. S. (1990, 12 01). Time Varying Risk Premia in Futures Markets. Retrived on 06.07.2023 from https://www.elibrary.imf.org/view/journals/001/1990/116/article-A001-en.xml#A01equ13
-
Kurach, R., & Stelmach, J. (2014). Time-varying behaviour of sector beta risk–the case of Poland. Romanian journal of economic forecasting, 17(1), 139-159.
-
Levy, H., Guttman, I., & Tkatch, I. (2001). Regression, correlation, and the time interval: Additive-multiplicative framework. Management Science, 47(8), s. 1150-1159.
-
Lintner, J. (1965). Security Prices, Risk, And Maximal Gains From Diversification. The Journal of Finance, 20(4), 587-615.
-
Lopez Herrera, F. G., Jimenez, J., & Reyes Santiago, A. (2022). Forecasting performance of different betas: Mexican stocks before and during the covid-19 pandemic. Emerging Markets Finance and Trade, 58(13), 3868-3880.
-
Mantsios, G., & Xanthopoulos, S. (2016). The Beta intervalling effect during a deep economic crisis-evidence from Greece. International Journal of Business and Economic Sciences Applied Research, 9(1), s. 19-26.
Markowitz, H. (1952). Portfolio Selection. Journal of Finance(7), 77-91.
-
Marti, D. (2006). European Financial Management Association Meeting. The accuracy of time-varying betas and the cross-section of stock returns.
-
Mergner, S., & Bulla, J. (2008). Time-varying beta risk of Pan-European industry portfolios: A comparisonof alternative modeling techniques. The European Journal of Finance, 14(8), 771-802.
-
Messis, P., & Zapranis, A. (2016). Forecasting time-varying daily betas: A new nonlinear approach. Managerial Finance, 42(2), 54-73.
-
Meyers, S. L. (1973). The stationarity problem in the use of the market model of security price behavior. The Accounting Review, 48(2), 318-322.
-
Moonis, S. A., & Shah, A. (2003). Testing for time-variation in beta in India. Journal of Emerging Market Finance, 2(2), 163-180.
-
Mossin, J. (1966). Equilibrium in A Capital Asset Market. Econometrica: Journal of The Econometric Society, 768-783.
-
Nieto, B., Orbe, S., & Zarraga, A. (2014). Time-varying market beta: does the estimation methodology matter? Sort, 38(1), 13-42.
-
Patton, A. J., & Verardo, M. (2012). Does beta move with news? Firm-specific information flows and learning about profitability. The Review of Financial Studies, 25(9), 2789-2839.
-
Rizvi, S. A., & Arshad, S. (2018). Understanding time-varying systematic risks in Islamic and conventional sectoral indices. Economic Modelling(70), 561-570.
-
Rossi, M. (2016). The capital asset pricing model: a critical literature review. Global Business and Economics Review, 18(5), s. 604-617.
-
Sharpe, W. (1964). Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions of Risk. The Journal of Finance, 19(3), 425-442.
-
Sunder, S. (1980). Stationarity of Market Risks: Random Coeficient Tests for Individual Stock. Journal of Finance, 35(4), 883-896.
-
Theobald, M. (1981). Beta stationarity and estimation period: Some analytical results. Journal of Financial and Quantitative Analysis, 16(5), 747-757.
-
Wijethunga, C., & Dayaratne, A. I. (2015). Time Varying Estimate of Beta (Systemic Risk): Evidence from Colombo Stock Exchange. International Journal of Accounting & Business Finance(1), 64-74.
-
Yeo, J. (2001). Modelling Time-Varying Systematic Risk in Australia. International Congress on Modelling & Simulation (s. 1565-1570). Modelling and Simulation Society of Australia and New Zealand Inc.
-
Zhang, X., & Zhang, S. (2021). Optimal time-varying tail risk network with a rolling window approach. Physica A, 1-15.
-
Zhou, J. (2013). Conditional Market Beta for REITs: A Comparison of Modeling Techniques. Economic Modelling(30), 196-204.
Time-Varying Betas and Effects of Data Frequency and Estimation Window Preferences: Case of Istanbul Stock Exchange
Year 2025,
Volume: 25 Issue: 3, 609 - 626, 06.08.2025
Musa Ovalı
,
Koray Kayalıdere
Abstract
In this study, we analyzed the changes in Beta over time for the leading indexes of Borsa Istanbul (XU100, XUHIZ, XUMAL, XUSIN, XUTEK) across 5-10 year and 15-year intervals from 2008 to 2023. We utilized Rolling regression and Recursive regression methods to estimate the fluctuations in Beta over time and compared the performance of these estimation techniques. To evaluate the effect of the estimation window length on Beta, we incorporated daily and weekly estimation windows of various lengths: 252 days, 126 days, 52 weeks, and 26 weeks. Additionally, we examined how data frequency affects Beta estimation using daily and weekly datasets. Our analysis showed that the Rolling regression method consistently outperformed the recursive method. Moreover, we found that employing daily datasets, instead of monthly datasets, significantly enhanced Beta forecast performance. We also found that a 126-day window is the most effective length for the estimation window.
Ethical Statement
It has been declared that relevant study complies with the ethical rules.
Thanks
We would like to express our gratitude to Dr. Ömer ÇAYIRLI for his valuable insights and suggestions during the preparation of this study.
References
-
Abdymomunov, A., & Morley, J. (2011). Time variation of CAPM betas across market volatility regimes. Applied Financial Economics(21), 1463-1478.
-
Agrrawal, P., Gilbert, F., & Harkins, J. (2022). Time Dependence of CAPM Betas on the Choice of Interval Frequency and Return Timeframes: Is There an Optimum? Journal of Risk and Financial Management, 15(11).
-
Akyatan, A., & Çetin, M. K. (2020). Return prediction with time varying betas: a research in BIST. International Journal of Accounting and Finance, 10(1), 64-86.
-
Alexander, G. J., & Chervany, N. L. (1980). On the estimation and stability of beta. Journal of Financial and Quantitative Analysis, 15(1), 123-137.
-
Baesel, J. B. (1974). On the Assessment of Risk: Some Further Considerations. Journal of Finance(29), 1491-1494.
-
Bartholdy, J., & Peare, P. (2001, 03 16). The relative efficiency of beta estimates. Aarhus, Denmark. Retrived on 04 04, 2023 from https://papers.ssrn.com/sol3/papers.cfm?abstract_id=263745.
-
Blasques, F., Francq, C., & Laurent, S. (2024). Autoregressive conditional betas. Journal of Econometrics, 238(2), 1-22.
-
Blume, M. E. (1975). Betas and their regression tendencies. The Journal of Finance, 30(3), 785-795.
Bollerslev, T., Engle, R. F., & Wooldridge, J. M. (1988). A capital asset pricing model with time-varying covariances. Journal of political Economy, 96(1), 116-131.
-
Bos, T., & Newbold, P. (1984). An empirical investigation of the possibility of stochastic systematic risk in the market model. Journal of Business, 35-41.
-
Brenner, M., & Smidt, S. (1977). A simple model of non-stationarity of systematic risk. The Journal of Finance, 32(4), 1081-1092.
-
Brooks, R. D., Faff, R. W., & Lee, J. H. (1992). The form of time variation of systematic risk: Some Australian evidence. Applied Financial Economics, 2(4), 191-198.
-
Brzeszczynski, J., Gajdka, J., & Schabek, T. (2011). The role of stock size and trading intensity in the magnitude of the" interval effect" in beta estimation: Empirical evidence from the Polish capital market. Emerging Markets Finance and Trade, 47(1), 28-49.
-
Cai, Z., & Juhl, T. (2023). The distribution of rolling regression estimators. Journal of Econometrics, 235(2), 1447-1463.
-
Chakrabarti, G., & Das, R. (2021). Time-varying beta, market volatility and stress: A comparison between the United States and India. IIMB Management Review, 33(1), 50-63.
-
Choudhry, T., & Wu, H. (2008). Forecasting Ability of GARCH vs Kalman Filter Method: Evidence from Daily UK Time-Varying Beta. Journal of Forecasting, 27(8), 670-689.
-
Choudhry, T., & Wu, H. (2009). Forecasting the weekly time-varying beta of UK firms: GARCH models vs. Kalman filter method. The European Journal of Finance, 15(4), 437-444.
-
Ciner, C. (2015). Time variation in systematic risk, returns and trading volume: Evidence from precious metals mining stocks. International Review of Financial Analysis(41), 277-283.
-
Collins, D. W., Ledolter, J., & Rayburn, J. (1987). Some Further Evidence on the Stochastic Properties of Systematic Risk. Journal of Business, 60(3), 425-448.
-
Corhay, A., & Rad, A. T. (1993). Return interval, firm size and systematic risk on the Dutch stock market. Review of Financial Economics, 2(2), s. 19-28.
-
Çayırlı, Ö., Kayalıdere, K., & Aktaş, H. (2022). Toplam Getiri Yerine Fiyat Getirisi Kullanılmasının Varlık Fiyatlama Modelleri ve Portföy Seçimi Üzerine Yapılan Çalışma Sonuçlarına Etkisi. Muhasebe ve Finansman Dergisi, 75-92.
-
Çelik, S. (2013). Testing the Stability of Beta: A Sectoral Analysis in Turkish Stock Market. Journal of Economics and Behavioral Studies, 5(1), 18-23.
-
Çenesizoğlu, T., Liu, Q., Reeves, J., & Wu, H. (2016). Monthly Beta Forecasting with Low-, Medium- and High-Frequency Stock Returns. Journal of Forecasting, 35(6), 528-541.
-
Damodaran, A. (1999). Estimating Risk Parameters. New York, USA. Retrived on 04 04, 2023 from https://archive.nyu.edu/bitstream/2451/26906/2/wpa99019.pdf.
-
Damodaran, A. (2013, Mart). Equity Risk Premiums (ERP): Determinants, Estimation and Implications- The 2013 Edition. (6). http://ssrn.com/abstract=2238064.
-
Das, A., & Ghoshal, T. K. (2010). Market risk beta estimation using adaptive Kalman Filter. International Journal of Engineering Science and Technology, 2(6), 1923-1934.
-
Das, S., & Barai, P. (2015). Time-varying industry beta in Indian stock market and forecasting errors. International Journal of Emerging Markets, 10(3), 521-534.
-
Dasgupta, S., Gan, J., & Gao, N. (2010). Transparency, price informativeness, and stock return synchronicity: Theory and evidence. Journal of Financial and Quantitative Analysis, 45(5), 1189-1220.
-
Daves, P. R., Ehrhardt, M. C., & Kunkel, R. A. (2000). Estimating systematic risk: the choice of return interval and estimation period. Journal of Financial and Strategic Decisions, 13(1), 7-13.
-
Domenech, N., Orbe Mandaluniz, S., & Zarraga, A. A. (2011). Time-varying beta estimators in the Mexican emerging market.
-
Drobetz, W., Hollstein, F., Otto, T., & Prokopczuk, M. (2023, 12 14). Estimating Stock Market Betas via Machine Learning. Retrived on 01 02, 2024 from https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3933048.
-
Estrada, J. (2000). The temporal dimension of risk. The Quarterly Review of Economics and Finance, 40(2), 189-204.
-
Fabozzi, F. J., & Francis, J. C. (1978). Beta as a Random Coefficient. Journal of Financial and Quantitative Analysis, 13(1), 101-116.
-
Faff, R. W., Hillier, D., & Hillier, J. (2000). Time Varying Beta Risk: An Analysisof Alternative ModellingTechniques. Journal of Business Finance & Accounting, 27(5), 523-554.
-
Faff, R. W., Lee, J. h., & Fry, T. R. (1992). Time stationarity of systematic risk: Some Australian evidence. Journal of Business Finance & Accounting, 19(2), 253-270.
-
Fama, E. F., & Macbeth, J. D. (1973). Risk, Return, and Equilibrium: Empirical Tests. The University of Chicago Press Journals, 81(3), s. 607-636.
-
Fama, E., & French, K. (1997). Industry Costs of Equity. Journal of Financial Economics(43), 153-193.
Frankfurter, G. M., Leung, W. K., & Brockman, P. D. (1994). Compounding period length and the market model. Journal of Economics and Business, 46(3), 179-193.
-
Gong, S. X., Firth, M., & Cullinane, K. (2006). Beta estimation and stability in the US-listed international transportation industry. Review of Pacific Basin Financial Markets and Policies, 9(3), 463-490.
-
Groenewold, N., & Fraser, P. (1999). Time-varying estimates of CAPM betas. Mathematics and Computers in Simulation, 48, 531-539.
-
Groenewold, N., & Fraser, P. (2000). Forecasting Beta: How Well Doesthe `Five-Year Rule of Thumb' Do? Journal of Business Finance & Accounting, 27(7), 953-982.
-
Gümrah, Ü., & Konuk, S. (2018). Zamanla Değişen Beta: Borsa İstanbul Bankacılık Sektörü Uygulaması. Ekonomik ve Sosyal Araştırmalar Dergisi, 14(1), 51-66.
-
Hasnaoui, H., & Fatnassi, I. (2014). Time-Varying Beta And The Subprime Financial Crisis: Evidence From U.S. Industrial Sectors. The Journal of Applied Business Research, 30(5), 1465-1476.
-
Hawawini, G. (1983). Why beta shifts as the return interval changes. Financial Analysts Journal, 39(3), 73-77.
He, L. T. (2005). Instability and predictability of factor betas of industrial stocks: The Flexible Least Squares solutions. The Quarterly Review of Economics and Finance(45), 619-640.
-
Hollstein, F., Prokopczuk, M., & Simen, C. (2019). Estimating beta: Forecast adjustments and the impact of stock characteristics for a broad cross-section. Journal of Financial Markets(44), 91-118.
-
Iqbal, J., & Brooks, R. (2007). Alternative beta risk estimators and asset pricing tests in emerging markets: The case of Pakistan. Journal of Multinational Financial Management, 17(1), 75-93.
-
Kalnina, I. (2022). Inference for nonparametric high-frequency estimators with an application to time variation in betas. Journal of Business & Economic Statistics, 1-12.
-
Kaminsky, G. L., & Kumar, M. S. (1990, 12 01). Time Varying Risk Premia in Futures Markets. Retrived on 06.07.2023 from https://www.elibrary.imf.org/view/journals/001/1990/116/article-A001-en.xml#A01equ13
-
Kurach, R., & Stelmach, J. (2014). Time-varying behaviour of sector beta risk–the case of Poland. Romanian journal of economic forecasting, 17(1), 139-159.
-
Levy, H., Guttman, I., & Tkatch, I. (2001). Regression, correlation, and the time interval: Additive-multiplicative framework. Management Science, 47(8), s. 1150-1159.
-
Lintner, J. (1965). Security Prices, Risk, And Maximal Gains From Diversification. The Journal of Finance, 20(4), 587-615.
-
Lopez Herrera, F. G., Jimenez, J., & Reyes Santiago, A. (2022). Forecasting performance of different betas: Mexican stocks before and during the covid-19 pandemic. Emerging Markets Finance and Trade, 58(13), 3868-3880.
-
Mantsios, G., & Xanthopoulos, S. (2016). The Beta intervalling effect during a deep economic crisis-evidence from Greece. International Journal of Business and Economic Sciences Applied Research, 9(1), s. 19-26.
Markowitz, H. (1952). Portfolio Selection. Journal of Finance(7), 77-91.
-
Marti, D. (2006). European Financial Management Association Meeting. The accuracy of time-varying betas and the cross-section of stock returns.
-
Mergner, S., & Bulla, J. (2008). Time-varying beta risk of Pan-European industry portfolios: A comparisonof alternative modeling techniques. The European Journal of Finance, 14(8), 771-802.
-
Messis, P., & Zapranis, A. (2016). Forecasting time-varying daily betas: A new nonlinear approach. Managerial Finance, 42(2), 54-73.
-
Meyers, S. L. (1973). The stationarity problem in the use of the market model of security price behavior. The Accounting Review, 48(2), 318-322.
-
Moonis, S. A., & Shah, A. (2003). Testing for time-variation in beta in India. Journal of Emerging Market Finance, 2(2), 163-180.
-
Mossin, J. (1966). Equilibrium in A Capital Asset Market. Econometrica: Journal of The Econometric Society, 768-783.
-
Nieto, B., Orbe, S., & Zarraga, A. (2014). Time-varying market beta: does the estimation methodology matter? Sort, 38(1), 13-42.
-
Patton, A. J., & Verardo, M. (2012). Does beta move with news? Firm-specific information flows and learning about profitability. The Review of Financial Studies, 25(9), 2789-2839.
-
Rizvi, S. A., & Arshad, S. (2018). Understanding time-varying systematic risks in Islamic and conventional sectoral indices. Economic Modelling(70), 561-570.
-
Rossi, M. (2016). The capital asset pricing model: a critical literature review. Global Business and Economics Review, 18(5), s. 604-617.
-
Sharpe, W. (1964). Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions of Risk. The Journal of Finance, 19(3), 425-442.
-
Sunder, S. (1980). Stationarity of Market Risks: Random Coeficient Tests for Individual Stock. Journal of Finance, 35(4), 883-896.
-
Theobald, M. (1981). Beta stationarity and estimation period: Some analytical results. Journal of Financial and Quantitative Analysis, 16(5), 747-757.
-
Wijethunga, C., & Dayaratne, A. I. (2015). Time Varying Estimate of Beta (Systemic Risk): Evidence from Colombo Stock Exchange. International Journal of Accounting & Business Finance(1), 64-74.
-
Yeo, J. (2001). Modelling Time-Varying Systematic Risk in Australia. International Congress on Modelling & Simulation (s. 1565-1570). Modelling and Simulation Society of Australia and New Zealand Inc.
-
Zhang, X., & Zhang, S. (2021). Optimal time-varying tail risk network with a rolling window approach. Physica A, 1-15.
-
Zhou, J. (2013). Conditional Market Beta for REITs: A Comparison of Modeling Techniques. Economic Modelling(30), 196-204.