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Modeling time-varying beta in a sustainable stock market with a three-regime threshold GARCH model

  • S.I.: Risk in Financial Economics
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Abstract

This study revisits an important issue in financial theory: the instability of market beta. To this end, we demonstrate that the linear constant risk model is misleading and does not reproduce changes in beta correctly. We develop a new nonlinear market model to capture beta instability over time for three main states: bear, normal, and bull markets. Our model endogenously identifies these states and their thresholds. We then apply this econometric specification to four major sustainable stock indexes in the US, Europe, Asia, and the World for 2004–2015. The results provide three main findings. First, the market beta is time-varying and changes asymmetrically and nonlinearly, suggesting that the systematic risk statistically differs between market regimes for the US, Europe, and the World. Second, the positive sign of beta in the bull market for these three regions suggests that systematic risk increases as economic conditions improve. Third, the lowest level of beta in the bear market indicates the usefulness of the sustainable stock index to hedge and cover investors’ portfolios against risk, particularly in a bear market.

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Notes

  1. Many related studies focused on beta for different financial assets: individual securities (Fabozzi and Francis 1977; Kim and Zumwalt 1979), mutual funds (Fabozzi and Francis 1979), size based portfolios (Wiggins 1992; Howton and Peterson 1998), risk based portfolios (Wiggins 1992), exchange rates, and so on.

  2. Chen (1982) and Wiggins (1992) used the excess market return to zero for the threshold. Bhardwaj and Brooks (1993) used the median return for the threshold. Woodward and Marisetty (2005) used different proxies to identify bear and bull markets (6-month rolling moving average, coincident economic indicator, 4-month lagged yield spread, and excess market return). Ang et al. (2006) showed that market return does not help to identify market states and prefer the trend base market indicator, which provides a smoother indicator. Woodward and Anderson (2009) used a moving average specification of the market return, which capturers the cyclical upturn and downturn, and therefore bull and bear markets. Chiarella et al. (2011) used a market trend to capture the change in investors' behavior.

  3. According to Tsay (1986, 1989) and Hansen (1996), the optimal transition variable is that from which linearity is the most strongly rejected.

  4. The parameter d denotes the delay parameter when the transition variable is a lagged variable.

  5. We omit the unit root test results to save space, but they are available upon request.

  6. The Teräsvirta and Lukkonen (1988) tests use same hypotheses H \( _{01} \), H \( _{02} \), H \( _{03} \), and H \( _{12} \). While the first test tests linearity against a smooth transition autoregressive model, the second tests linearity against an STR (Smooth Transition Regression) model; the threshold variable might be an exogenous explanatory variable.

  7. Using these two classes of tests enables a check of the appropriate transition speed between market states and resolves the limitation in Woodward and Anderson (2009) regarding the transition function choice.

  8. The difference in market model specification for the World and the US data might be justified by the fact that sustainable funds are not well represented when considering the World data.

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Authors and Affiliations

  1. University of Evry, Evry, France

    Fredj Jawadi

  2. ESSCA School of Management, Angers, France

    Wael Louhichi

  3. EDC Paris Business School, Courbevoie, France

    Abdoulkarim Idi Cheffou

  4. INSEEC Business School, Paris, France

    Hachmi Ben Ameur

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  1. Fredj Jawadi
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  2. Wael Louhichi
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  3. Abdoulkarim Idi Cheffou
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  4. Hachmi Ben Ameur
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Correspondence to Hachmi Ben Ameur.

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Jawadi, F., Louhichi, W., Cheffou, A.I. et al. Modeling time-varying beta in a sustainable stock market with a three-regime threshold GARCH model. Ann Oper Res 281, 275–295 (2019). https://doi.org/10.1007/s10479-018-2793-3

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