پویایی بازار سهام در یک مدل نامتقارن قدرت GARCH سوئیچینگ نظام

This paper analyzes the dynamics of Asian stock index returns through a Regime-Switching Asymmetric Power GARCH model (RS-APGARCH). The model confirms some stylized facts already discussed in former studies but also highlights interesting new characteristics of stock market returns and volatilities. Mainly, it improves the traditional regime-switching GARCH models by including an asymmetric response to news and, above all, by allowing the power transformations of the heteroskedasticity equations to be estimated directly from the data. Several mixture models are compared where a first-order Markov process governs the switching between regimes.

Since the availability of high frequency financial data, a growing body of empirical studies, starting with Fama and French (1989), has investigated the predictability of mean and volatility of asset returns. Volatility of financial returns is indeed a central parameter for many financial decisions including the pricing and hedging of derivative products and risk management. Most of the volatility models presented in the empirical literature are based on the observation that volatility is time-varying and that periods of high volatility tend to cluster. The autoregressive conditional heteroskedasticity (ARCH) models, as introduced by Engle (1982) and extended to Generalized ARCH (GARCH) in Bollerslev (1986), have proven to be useful means for empirically capturing these stylised facts. Although such approaches provide an improvement in fit compared with constant variance models, recent evidence from financial market data seems to suggest that persistence in variance, as measured by GARCH models, is so substantial that it sometimes implies an explosive conditional variance. To account for this apparent empirical regularity, Engle and Bollerslev (1986) introduce the Integrated-GARCH (I-GARCH) process, in which shocks to the variance do not decay over time. However, Lamoureux and Lastrapes (1990) show that one potential source of misspecification of ARCH/GARCH models is that the structural form of conditional means and variances is relatively inflexible and is held fixed throughout the entire sample period. As explained in Timmermann (2000) if the variance is high but constant for some time and low but constant otherwise, the persistence of such high- and low- volatility homoskedastic periods already results in volatility persistence. GARCH models, that cannot capture the persistence of such periods, put all the volatility persistence in the persistence of individual shocks, biasing thus upward our assessment of the degree to which conditional variance is persistent. Although the ad hoc introduction of deterministic shifts into the variance process represents one possibility to allow for periods with different unconditional variances, the most promising approach to modelling these nonlinearities is by endogenizing changes in the data generating process through a Markov regime-switching model as introduced in Hamilton (1989). The model relies on different coefficients for each regime to account for the possibility that the economic mechanism generating the asset returns may undergo a finite number of changes over the sample period. In order not to rule out within-regime heteroskedasticity, Gray (1996) extends Hamilton's (1989) model to accommodate within-regime GARCH effects with a so-called Regime-Switching GARCH model (RS-GARCH). RS-GARCH models have the attractive feature of incorporating significant nonlinearities, while remaining tractable and easy to estimate. Although they represent a suitable framework to investigate how the volatility dynamics is affected by the states of the economy, surprisingly few improvements of the single-regime ARCH/GARCH literature have been adapted and tested in their regime-switching counterparts. In particular, under classical GARCH models, shocks to the variance persist according to an autoregressive moving average (ARMA) structure of the squared residuals of the process. However, it is not necessary to impose a squared power term in the second moment equation as in Bollerslev (1986). Taylor's (1986) and Schwert's (1989) class of GARCH models, for instance, relate the conditional standard deviation of a series to lagged absolute residuals and past standard deviations. More recently, Ding, Granger, and Engle (1993) suggest an extension of the GARCH family models that analyses a wider class of power transformations than simply taking the absolute value or squaring the data as in the traditional heteroskedastic models. Known as the Power GARCH (PGARCH) models, this addition to the GARCH family has been shown to be superior in fit to its less sophisticated counterparts (see Brooks, Faff, McKenzie, and Mitchell (2000) for an empirical investigation in a single-regime framework). Nesting the major two classes of GARCH models (namely, Bollerslev's and Taylor–Schwert's) the PGARCH specification also provides an encompassing framework which facilitates comparison. An important contribution of the current paper is to highlight whether and to what extent these more flexible models improve both the fit and our understanding of asset returns dynamics when the assumption of a single regime is relaxed in favor of a regime-switching model. To this end, we introduce a new Regime-Switching Asymmetric Power GARCH (RS-APGARCH) model to analyze empirically Asian stock index returns. Our findings shed light on several interesting stylized facts about the relationships between both the dynamics of the conditional mean and variance and the state of the economy. It is shown that the RS-APGARCH model proposed in this paper is able to match some empirical regularities of stock index returns that could not be captured with the traditional regime-switching models already introduced in the literature, let alone using a single-regime GARCH model. Another important novelty of our approach compared to the classical literature on regime-switching processes regards the choice of the underlying conditional distributions. Indeed, a regime-switching model relies on a mixture of conditional distributions where the parameters are either held constant – Hamilton (1989) – or rendered time-varying – Gray (1996) –. Following the traditional literature on mixture of distributions (see Kon (1984) or Ané and Labidi (2001)) most Markov regime-switching models adopt conditional Gaussian mixtures. Since our analysis focuses on recent years where stock markets have undergone important shocks (both economical and political), financial assets have experienced periods of extreme volatility. In order to capture a higher degree of kurtosis in asset returns, we follow Perez-Quiros and Timmermann (2001) and introduce in our RS-APGARCH model a mixture of a Gaussian distribution and a Student-t density. In such a mixture, outliers or extreme returns will be modeled as drawn from a fat-tailed t-distribution with few degrees of freedom whereas the moderate returns will be generated by the conditional Gaussian density. With this additional characteristic our model enables us to differentiate the effect of the states of the economy on the dynamics of asset returns far beyond the mere difference of parameter values and/or conditional mean and variance structure: it allows for higher order conditional differences through conditional densities that exhibit very different probabilistic structures. We then test the necessity of introducing two leptokurtic densities in the model. Finally, another contribution of this paper arises from the APGARCH structure used on the volatility of each regime. Ding and Granger (1996) show that the power term transformation of this model can be related to the long run temporal dependency in the volatility also called the long memory property of the volatility. Using APGARCH models in a regime-switching framework we are thus able to investigate whether the degree of temporal dependency changes with the states of the economy. The remainder of the paper is organized as follows. Section 2 describes the new Regime-Switching Asymmetric Power GARCH model introduced in this paper. The data and a preliminary empirical investigation motivating the use of our model are presented in Section 3. Section 4 contains the main empirical findings and the goodness-of-fit tests while Section 5 concludes.

This paper develops a new general class of regime-switching models called Regime-Switching Asymmetric Power GARCH model. It allows a free power term for the GARCH specification of each regime rather than assuming an absolute or squared term like most of the classical models. Since this type of APGARCH model has not yet been considered in a regime switching context, an important contribution of the current paper is to augment our understanding of whether and to what extent these types of more flexible models are statistically superior to their less sophisticated counterparts. The empirical investigation uses four Asian stock market indices corresponding to various market situations and provides interesting conclusions about how to understand time variations in stock index returns. Most obviously, it seems that commonly used single-state specifications for stock index returns that adopt the same model for recessions and expansions are misspecified and can be strongly rejected against a two-state model. We also find that the APGARCH structure provides a considerable improvement over the classical GARCH structure in both regimes. The empirical results do indicate that all generalizations brought by our model are statistically and economically significant. More specifically, a variety of new stylized facts about the dynamics of stock index returns has emerged from this RS-APGARCH model. We first recover, but with a higher level of significance however, a now classical result of the regime-switching literature: one regime could be regarded as modeling expansion periods while the second regime is clearly identified to recessions. Expansions are characterized by a positive conditional mean and a low-volatility regime while recessions exhibit a negative conditional mean and are always synonym of a much higher volatility level. Even if a basic regime-switching model with constant parameters would result in a leptokurtic process, we tried several conditional distributions for each regime to investigate the within-regime level of leptokurtosicity. The classical conditional Gaussian densities are shown not to be sufficient, even in a two-state framework, to incorporate all the kurtosis of the underlying series. We find that, whatever the investigated stock market index, both regimes are best modeled with a conditional Student-t distribution. However, we do recover, to some extent, the interesting result obtained by Perez-Quiros and Timmermann (2001) on U.S. stock returns using a mixture of Gaussian and Student-t densities. Indeed, it seems that for developed markets, the estimated degree of freedom of one Student-t distribution is large enough to statistically accept the convergence to conditional normality in this regime. The second regime, however, exhibits a strong leptokurtosicity and captures all extreme returns. Presenting both developed and emerging markets in this study, we are able to refine the result. Indeed, the “convergence” of one regime to normality is not obtained for emerging markets where the level of leptokurtosicity remains very strong and even comparable for recession and expansion cycles. This result should of course be tested on a larger sample of stock market indices. If it were confirmed, this would indicate that the degree of leptokurtosicity of the regime representing the good market condition (expansions) could be used to assess the level of development of a financial market. The RS-APGARCH model also introduces the possibility of within-regime asymmetric response to news. It is found that the introduction of such parameters is strongly significant and that asymmetries are important in both states of the economy. Although the classical leverage effect of stock market returns is obtained for both regimes, the asymmetric response to news is consistently stronger in the low-volatility regime. Market participants thus seem to differentiate less between good and bad news during extremely volatile periods. The unusually high level of volatility in the latter periods could bias the market participants perception of news and reduce their ability of differentiating between good and bad news. When markets evolve more smoothly, however, the information process may be less noisy and market participants may recover their ability to assess the content of new information and to react accordingly. If confirmed by other empirical studies, this result should open a challenging avenue of research for microstructure models of agent behavior and price formation in financial markets. Moreover, the improvements of the RS-APGARCH model are mainly due to the endogenous determination of the power transformation term used in the GARCH structure of each regime. Using Gray's RS-GARCH model, we find that the within-regime volatility persistence is consequently reduced relative to the single-state GARCH model. Such result gives credit to Lamoureux and Lastrapes (1990) thesis that structural breaks account for most of the volatility persistence observed with a single-state model through the regime persistence (very high P and Q). When the power GARCH term is introduced, however, we observe that not only the ARCH and GARCH parameters become statistically much more significant, but also that the within-regime heteroskedasticity increases strongly compared to the RS-GARCH level. This seems to indicate that the squared terms arbitrarily used in the traditional RS-GARCH models are sub-optimal and do not allow to fully capture the within-regime clustering effects. Lastly, as explained in Ding and Granger (1996) the APGARCH class of model we use on both regimes has no memory in return themselves, but long memory in absolute returns and their power transformations. The estimated power terms δst are significant and different in both regimes, implying the existence of long memory in both states of the economy. Again, this shows that the existing long memory in stock returns does not only result from structural breaks as it has often been argued in the single-state literature: we do find the existence of within-regime long memory. Nevertheless, the values of δst obtained for all stock indices are too close from one regime to the other to conclude that one regime predominantly captures short-run dependencies while the other regime exhibits long memory.