Estimating value-at-risk via Markov switching ARCH models – an empirical study on stock index returns

Abstract

This paper estimates the Value-at-Risk (VaR) on returns of stock market indexes including Dow Jones, Nikkei, Frankfurt Commerzbank index, and FTSE via Markov Switching ARCH (SWARCH) models. It is conjectured that structural changes contribute to non-normality in stock return distributions. SWARCH models, which admit parameters based on various states to control structural changes in the estimating periods, may thus help mitigate kurtosis, tail-fatness and skewness problems in estimating VaR. Significant kurtosis and skewness in return distributions of Dow Jones, Nikkei, FCI and FTSE and significant tail-fatness (tail-thinness) in the 1% (5%) region critical probability are documented. Moreover, it is shown that the more generalized SWARCH outshines both ARCH and GARCH in capturing non-normalities with respect to both in- and out-sample VaR violation rate tests.

Acknowledgements

The Authors are grateful to Darrel Duffie and James D. Hamilton for their invaluable comments and suggestions.

Notes

 The G10 group includes Belgium, Canada, France, Germany, Italy, Japan, Netherlands, Sweden, Switzerland, UK, USA and Luxembourg.

 This statement is especially descriptive for banks with large trading accounts.

 One of the most frequently adopted VaR models is the RiskMetrics of J. P. Morgan Financial Service Co. Incorporated. RiskMetrics, which is publicly available on http:\\www.jpmorgan.com, is a representative parametric model.

 For instance, Hendricks (1994), Beder (1995), Simons (1996), Fong and Vasicek (1997) investigate the strengths and weaknesses of various VaR measures including parametric, historical simulation and Monte Carlo simulation methods. Jorion (1997) employs the student t-distribution in estimating VaR. Venkataraman (1997) adopts the mixture of normal distributions and the quasi-Bayesian estimation techniques for measuring the VaRs for a sample of eight exchange rates.

 This study is also in contrast with Venkataraman (1997), who uses the binominal probability distributions for jumps from one distribution to another. Both mixing normal and MS incorporate two or more distributions to form a new distribution. Nevertheless, if the perceived non-normality is due to the above-mentioned structural change events, MS models should outperform mixing-normal models. Specifically, if there exists persistence and if period t returns are drawn from the first distribution, then the probability that the period t + 1 return is drawn from the same distribution should be greater than p.

 SWARCH incorporates MS and ARCH models, with the former setting filtering out return volatility, and ARCH controlling residual return volatilities. Specifically, MS models incorporate the discrete state variables to control the structural changes in the test period and mitigate persistence problems for estimating volatility in ARCH and GARCH. MS models therefore help capture kurtosis, skewness and tail-fatness in VaR estimation.

 Note that, in contrast with Hamilton and Susmel (1994), who specify three regimes for return volatility, this study selects a more simplified two-regime setting. With limited number of prior observations, two- instead of three-regime settings may be sufficient.

 Specifically, when the information set includes signals dated up to time t, the regime probability is referred to as a filtering probability, p(s_t |y_t , y_{t −1},…). On the other hand, when the overall sample period information set is used to estimate the state at t, the probability may be referred to a smoothing probability, p(s_t |y_T , y_{T −1},…). In contrast, a predicting probability, p(s_t |y_{t −1}, y_{t −2},…), is the regime probability for ex ante estimation, with the information set including signals dated up to period t − 1.

 See Simons (1996) and Ho and Lin (1999).

 The parameter estimates for both specifications of higher-ordered prior-period conditional variance and specifications of higher-ordered error sum of squares are insignificantly different from zero.

 OPTIMUM, a GAUSS package program, and the built-in Boyden, Fletcher, Goldfarb, and Shanno (BFGS) algebra are used to compute the negative minimum likelihood function. BFGS algebra is effective for deriving the maximum value of non-linear likelihood functions. See Luenberger (1984).

 Refer to Schwarz (1978) for Schwarz value and Akaike (1976) for AIC.

 For example, the estimate of g ₂ for Dow Jones is 5.292, indicating that the standard deviation of regime 2 returns is 5.292 times the standard deviation of regime 1 returns.

 The Basle Committee proposes that the learning window should at least include 250 pre-VaR trading days. In the rolling estimation process used, one more observation are introduced for each time point. To facilitate convergence in the process, the estimate for each period is used as the initial value in the non-linear estimation for the immediately subsequent period.

 It is a common practice of estimating one-holding-day VaRs.

 In contrast, there exist mixed results as to the order of sample means with respect to regimes 1 and 2.

 Lopez and Walter (2001) and Engle and Manginelli (1999) may be referred to for other statistical tests for VaR estimates.

 Refer to Venkataraman (1997), who assumes symmetry of the two ends and aggregates the tailed region measures.

 Similar results are found for the other five portfolios.

 VaRs with 1000-prior-trading-day windows are estimated.

 According to the above-mentioned results, the linear, ARCH and GARCH models are inaccurate in estimating the 1% VaRs.

 See Bollerslev et al. (1994) for the details regarding these approaches.