A dynamic analysis of stock markets using a hidden Markov model

Abstract

This paper proposes a framework to detect financial crises, pinpoint the end of a crisis in stock markets and support investment decision-making processes. This proposal is based on a hidden Markov model (HMM) and allows for a specific focus on conditional mean returns. By analysing weekly changes in the US stock market indexes over a period of 20 years, this study obtains an accurate detection of stable and turmoil periods and a probabilistic measure of switching between different stock market conditions. The results contribute to the discussion of the capabilities of Markov-switching models of analysing stock market behaviour. In particular, we find evidence that HMM outperforms threshold GARCH model with Student-t innovations both in-sample and out-of-sample, giving financial operators some appealing investment strategies.

Keywords:

• stock market pattern analysis
• regime-switching
• hidden Markov model
• financial crises
• market stability periods

Acknowledgements

Both authors express their gratitude to the editor of Journal of Applied Statistics and to the two anonymous reviewers for their useful suggestions. De Angelis gratefully acknowledges funding from the Italian Ministry of Education, University and Research (MIUR) through PRIN project ‘Multivariate statistical models for risk assessment’.

Notes

In our HMM approach, we do not directly estimate conditional variances which are restricted to be equal across states. Instead, we obtain measures of the variability within each latent state. We are thus able to achieve two goals: (i) avoiding the estimation of conditional variances allows the model to cluster observations into latent states on the basis of the conditional means and first-order correlation structure of the data, thus attaining a reliable and easily interpretable classification of the stock market return distribution; (ii) restricting the conditional variances enables us to decrease the number of parameters to be estimated. It must be noted that the model we adopt allows us to easily discriminate between periods characterized by low and high-volatility (see Section 4.2).

Markov chain order identification in HMM remains an unresolved issue (see [[Chapter 15]18 for a recent discussion]), and there are several concerns about the robustness and reliability of information criteria. We also agree with the concerns about a uncritical use of these indicators. However, we believe that they can contribute to current procedures, for which the choice of latent states is somewhat arbitrary.

The posterior probabilities are estimated using the properties of the forward and backwards probabilities. In particular, the posterior probability for the latent state j is given by f(y _t =j| z)>f(z, y _t =j)/f(z) (see 34 for further details).

For weekly frequency of the data, the simple net returns z _t we use in this paper are approximately equal to the (continuously compounded) log-returns obtained as z _t =log(p _t )−log(p _t−1) and which are sometimes used in financial studies. Therefore, the results are not affected by this choice.

Source: http://www.nber.org/cycles.html.

The Wald statistic tests the significance of sets of parameters. The general expression for the Wald test statistic is given by W=(C′ϕ)′(C′Σ C)⁻¹(C′ϕ), where the tested set of linear constraints is C′ϕ=0 and Σ is the estimated variance–covariance matrix. This test is distributed as a chi-square where the number of degrees of freedom equals the number of contraints. In our analysis, we test the equality of conditional means between states and, thus, we constrain the means of six latent states to be equal to the one of the remaining latent state. Hence, the number of degrees of freedom of both Wald and ANOVA tests reported in the text is 6. The (one-way) ANOVA test for comparing conditional means is performed using latent state membership as factor.

However, p ₅₄ is the highest transition probability with respect to the other transition probabilities p _j4 for j≠4 (see Table 4), and latent state 5 is the last visited regime before the switch to latent state 4 in both cases in the analysed data set.

The results obtained for the expected (monetary) value criterion are equivalent to the ones achieved using another traditional criterion in decision theory, namely the expected opportunity loss which suggests the strategy associated to the lowest weighted expected value for the regret.

The criterion adopted by Investor B provides the best outcomes using the forecasts given by the estimated T-GARCH model.