Long memory and nonlinearities in realized volatility: A Markov switching approach

Abstract

Realized volatility is studied using nonlinear and highly persistent dynamics. In particular, a model is proposed that simultaneously captures long memory and nonlinearities in which level and persistence shift through a Markov switching dynamics. Inference is based on an efficient Markov chain Monte Carlo (MCMC) algorithm that is used to estimate parameters, latent process and predictive densities. The in-sample results show that both long memory and nonlinearities are significant and improve the description of the data. The out-sample results at several forecast horizons show that introducing these nonlinearities produces superior forecasts over those obtained using nested models.

Introduction

It is well-known that accurately measuring and forecasting financial volatility plays a central role in many pricing and risk management problems. With high frequency intra-daily data sets becoming widely available, more accurate estimates of volatility can be obtained. Realized volatility (RV), i.e. the sum of intra-day squared returns, reduces the noise in the volatility estimate considerably when compared to other volatility measures, such as squared or absolute daily returns. Thus, volatility becomes observable and can be modeled directly, rather than being treated as a latent process both in a GARCH or in a stochastic volatility setup.

Many empirical regularities on RV have been well-documented in recent literature and a detailed review has been provided by McAleer and Medeiros (2008b). One of the most relevant empirical regularities is that RV exhibits high persistence, as is evidenced in the research of Andersen et al. (2003), Koopman et al. (2005) and Corsi (2009). For this reason, linear fractionally integrated models (ARFIMA) are generally used to capture this feature. A flexible strategy to model serial dependencies for RV has been proposed in Barndorff-Nielsen and Shephard (2002) through a superposition of ARMA(1, 1) processes, whereas Corsi (2009) has introduced the Heterogeneous Autoregressive model (HAR) given by a combination of volatilities measured over different time horizons. However, Granger and Ding (1996) found out that persistence in volatility tends to be non-constant over time and Longin (1997) provided evidence of an usually high level of persistence when volatility is low, thus suggesting the presence of nonlinearities.

On the other hand, it is well-known that a long memory can be overestimated when a regime shifts or structural breaks are not taken into account. In fact, these are confounding factors and distinguishing between them can be rather troublesome (Diebold and Inoue, 2001). However, recent statistical tests aimed at disentangling the effects of long memory and level shifts have been proposed in Baillie and Kapetanios (2007) and in Ohanissian et al. (2008). Baillie and Kapetanios (2007) found the presence of nonlinearity together with a long memory in realized volatilities for currencies, whereas Ohanissian et al. (2008), in their empirical application, provided evidence that realized volatility of exchange rates, such as DM/$ and Yen/$ is properly described by a true long memory process. This latter result seems in contrast with Perron and Qu (2010), who claim that short memory models with level shifts are appropriate to describe volatility. Similarly, Carvalho and Lopes (2007) model volatility with a short memory switching regime dynamics. Chen et al. (2008) propose a range-based threshold heteroskedastic model, whereas He and Maheu (2010) propose a GARCH model subject to an unknown number of structural breaks.

The question of whether long memory is spurious is still debateable. It thus appears of obvious interest to join both models’ features into a single time series model. In this way, it is also possible to check whether the benefits of combining long memory and nonlinearities represent an improvement in forecasting accuracy. Recently, some time series models have suggested combining long memory and nonlinearities to describe conditional variances. Contributions in this direction are, respectively, Martens et al. (2004) and Hillebrand and Medeiros (2008) who built from an ARFIMA model by allowing for smooth level shifts, day of the week effects and leverage. McAleer and Medeiros (2008a) propose a different strategy by introducing a multiple regime smooth transition extension of HAR. Finally, Lux and Morales-Arias (2010) introduce a regime-switching multifractal model.

In this paper, we consider a different strategy to model abrupt changes in the conditional mean and time varying long range dependency. We base our analysis on a Markov switching model that is in line with the seminal work of Hamilton (1989) in which level shifts are modeled through a binary non-observable Markov process and in which the parameters, including the degree of persistence, are state dependent. Persistence is introduced through a standard ARFIMA model.

Furthermore, we think it is important to include exogenous regressors in the model’s specification. Following Bandi and Perron (2006), we consider the implied volatility as a predictor, since it is proven to be an unbiased long run forecast of future RV, once controlling for a fractionally cointegration relation. An interesting study on the relation between realized volatility, long range persistence and exogenous regressors has recently been completed by Corsi and Renò (2010).

We use Bayesian estimation techniques and goodness-of-fit indicators to assess the in-sample performance of our model. We also pay particular attention to the forecasting ability of the proposed models.

We base our empirical analysis on the 5 min intra-daily series of Standard & Poor’s 500 (S&P500) stock index over the period 1 January 2000 to 28 February 2005. Our results showed that implied volatility is important for predicting RV and also that, in the short run, long memory, together with nonlinearities improve forecasting performance. In the long run, the ARFIMA effect seems to be dominant with respect to the switching regime mechanism.

The remainder of the paper is organized as follows. Our data set is described in Section 2. Section 3 contains the definitions of the Markov switching models that are considered in this research. Our inferential solution is outlined in Section 4. The forecasting methodology is explained in Section 5 and empirical results based on simulated and real data are illustrated in Section 6 and in Section 7, respectively.