Double threshold autoregressive conditionally heteroscedastic model building by genetic algorithms

Abstract

Asymmetric behaviour in both mean and variance is often observed in real time series. The approach we adopt is based on double threshold autoregressive conditionally heteroscedastic (DTARCH) model with normal innovations. This model allows threshold nonlinearity in mean and volatility to be modelled as a result of the impact of lagged changes in assets and squared shocks, respectively. A methodology for building DTARCH models is proposed based on genetic algorithms (GAs). The most important structural parameters, that is regimes and thresholds, are searched for by GAs, while the remaining structural parameters, that is the delay parameters and models orders, vary in some pre-specified intervals and are determined using exhaustive search and an Asymptotic Information Criterion (AIC) like criterion. For each structural parameters trial set, a DTARCH model is fitted that maximizes the (penalized) likelihood (AIC criterion). For this purpose the iteratively weighted least squares algorithm is used. Then the best model according to the AIC criterion is chosen. Extension to the double threshold generalized ARCH (DTGARCH) model is also considered. The proposed methodology is checked using both simulated and market index data. Our findings show that our GAs-based procedure yields results that comparable to that reported in the literature and concerned with real time series. As far as artificial time series are considered, the proposed procedure seems to be able to fit the data quite well. In particular, a comparison is performed between the present procedure and the method proposed by Tsay [Tsay, R.S., 1989, Testing and modeling threshold autoregressive processes. Journal of the American Statistical Association, Theory and Methods, 84, 231-240.] for estimating the delay parameter. The former almost always yields better results than the latter. However, adopting Tsay's procedure as a preliminary stage for finding the appropriate delay parameter may save computational time specially if the delay parameter may vary in a large interval

Compstat 2006 - Proceedings in Computational Statistics

“…For these series the delay 1 was suggested by [LL96]. As a consequence the number of observations in each of the two regimes differ from that found in previous studies (see, for instance, [BC05], whose results are closer to those provided by [LL96]). Tsay's test seems to support the GA choice for years 1988 and 1991 only, whilst the suggested delays are 2 for year 1987, 3 for year 1989 and no threshold structure is found for year 1990.…”

“…Such a coding (adopted for example by [BC05]), in spite of its simplicity, has the disadvantage of generating chromosomes that may violate the two above requirements (no more than K regimes, and at least M observations in each regime), therefore it needs a "legalization" stage, in which each chromosome is analyzed, and those violating the requirements are discarded. We first choose a maximum number of regimes, K, and a minimum number of observations for each regime, M .…”

“…Genetic algorithms for double threshold ARCH models are discussed by [BC05]; in the present paper we extend to the generalized specification GARCH for the conditional variance, and use a different more convenient coding for the genetic algorithm.…”

Unit Root Tests in Time Series Volume 2

“…For example, see Maki (2009) for a review and assessment of unit root tests in a three regime model, and Battaglia and Protopapas (2010), who considered both nonlinearity and nonstationarity and allow the regime switching to be self-exciting, smooth transition or piecewise linear. Threshold models can be extended to the variance as in TAR versions of ARCH (TARCH) models (see, for example, Baragona and Cucina, 2008). Kejriwal and Perron (2010) extend the work of Bai (1997) and Bai and Perron (1998), considering a sequential procedure to determine the number and timing of structural breaks.…”