Forecasting exchange rate volatility using conditional variance models selected by information criteria

Abstract

This paper uses appropriately modified information criteria to select models from the GARCH family, which are subsequently used for predicting US dollar exchange rate return volatility. The out of sample forecast accuracy of models chosen in this manner compares favourably on mean absolute error grounds, although less favourably on mean squared error grounds, with those generated by the commonly used GARCH(1, 1) model. An examination of the orders of models selected by the criteria reveals that (1, 1) models are typically selected less than 20% of the time.

Introduction

The use of GARCH models (a class first proposed by Bollerslev, 1986, and first applied to exchange rates by Hsieh, 1988), for modelling and predicting volatility is now very common in finance (see, for example, Akgiray, 1989, or Day and Lewis, 1992). A typical finding is that these models provide superior forecasts of volatility than those which simply use historical means of squared returns assuming homoscedasticity. However, the vast majority of extant studies, restrict the conditionally heteroscedastic model to be GARCH(1, 1) (see Bollerslev et al., 1992, for a comprehensive survey of such papers). This approach seems arbitrary, and is certainly not grounded in financial or economic theory. Until recently, the only method of determining the appropriate orders for GARCH(r, m) models was by starting with a “large” model and testing down using a series of likelihood ratio-type restrictions (this procedure is used, for example, by Akgiray, 1989and Cao and Tsay, 1992). However, Brooks and Burke (1997)have recently proposed a set of information criteria which allow the researcher to select an “optimal” in-sample model from the AR(p)-GARCH(r, m) class, where the maximum permitted orders of p, r, and m are specified in advance. The criteria are based upon estimation of the Kullback–Leibler discrepancy (see Sin and White, 1996). Models from this family can be expressed as

$$\text{y}{}_{\mathrm{t}}\text{=μ+}\text{∑}\text{i=1}\text{p}\text{φ}{}_{\mathrm{i}}\text{y}{}_{\mathrm{t-i}}\text{+u}{}_{\mathrm{t}}$$

$$\text{u}{}_{\mathrm{t}}\text{∼}\text{NIID}\text{(0,}\text{σ}{}^2{}_{\mathrm{t}}\text{)}$$

$$\text{σ}{}^2{}_{\mathrm{t}}\text{=α}{}_0\text{+}\text{∑}\text{j=1}\text{m}\text{α}{}_{\mathrm{j}}\text{u}{}^2{}_{\mathrm{t-j}}\text{+}\text{∑}\text{k=1}\text{r}\text{β}{}_{\mathrm{k}}\text{σ}{}^2{}_{\mathrm{t-k}}$$

The purpose of this paper is to determine whether the new information criteria lead to the selection of models which give improved out of sample forecasting performance compared with GARCH(1, 1) models. To this end, we use exactly the same data as West and Cho (1995), henceforth WC, who found that GARCH models gave slightly more accurate forecasts than homoscedastic, IGARCH, autoregressive volatility, or nonparametric models.