Bootstrapping forecast intervals in ARCH

In this paper we develop a bootstrap method for the construction of prediction intervals for an ARMA model when its innovations are an autoregressive conditional heteroscedastic process. VCe give a proof of the validity of the proposed bootstrap for this process. For this purpose we prove the convergence to zero in probability of the Mallows metric between the empirical distribution function and the theoretical distribution function of the residuals. The potential of the proposed method is assessed through a simulation study. K e y W o r d s : ARCH models, bootstrap method, prediction intervals. A M S subject classification: 62M20, 62G09, 62E25 1 I n t r o d u c t i o n A m o n g s t the different ob jec t ives of app l i ed s t a t i s t i ca l ana lys is , one of the m o s t in te res t ing w h e n s t u d y i n g t ime series is forecas t ing. In th is contex t , the m e t h o d o l o g y p re sen t ed here is an app l i ca t i on of t he b o o t s t r a p proced u r e to the p r o b l e m of in te rva l fo recas t ing for A R M A ( p , q ) A R C H ( r ) t i m e series. T h e A R C H class of mode l s was or ig inal ly i n t r o d u c e d by Engle (1982) to r ep resen t empi r i ca l ev idence in those series in which u n c e r t a i n t y plays an essent ia l role, as a conven ien t way of mode l l i ng t i m e d e p e n d e n t cond i t iona l he te roscedas t ic i ty . Engle and Kra f t (1983) were the first to consider t he effect of A R C H on forecas t ing . T h e y der ived express ions for the m u l t i s t e p p red ic t ion e r ror "'Correspondence to: Jesfis A. Miguel, Departalnento de M6todos Estadfsticos, Facultad de Ciencias Econdmicas y Empresariales, Universidad de Zaragoza, Zaragoza, Spain. E-mail: jamiguel~(t posta.unizar.es This research was partially supported by Grant UZ-228-26 from the Spanish Ministry of Education and Grant UZ-228-25 from University of Zaragoza. Received: June 1997; Accepted: November 1998 346 J.A. Miguel and P. Olave variance in ARMA models with ARCH errors, but did not discuss tile characteristics of tile prediction error distribution. Baillie and Bollerslev (1992) found ex ante prediction confidence intervals by using parametric hypotheses on tile conditional error distribution. However, these conditions are usually not satisfied, and forecast intervals typically depend upon an assmnption on the conditional error distribution. In addition, tile standardized prediction error distr ibution depends nontrivially on tile information set at t ime t. This complicates the expressions for tile higher-order conditional moments, and consequently tile construction of tile forecast intervals. Whilst the major i ty of empirical studies using ARCH models tend to rely on parametric specifications, we propose a boots t rap procedure when tile assmnptions on tile conditional error distr ibution may not be justified. Efi'on's (1979) boots t rap provides a non-parametric method of estimating distr ibutions with small sample sizes, when derivations of the sampling dist r ibut ion are analytically intractable. In particular, we develop a boots t rap method for estimating the conditional distr ibution of t + s, given tile observations up to time t. Tile s tudy of the boots t rap for t ime series and dynamic regression models was begun by Freedman (1981). Thombs and Schucany (1990) give boots t rap prediction intervals for AR models, while McCullough (1994) implements tile bias-correction boots t rap to real data. Kreiss and Franke (1992) prove the asymptot ic validity of the boots t rap applied to M-estimators of s tat ionary ARMA models. In this paper we use a recent boots t rap procedure for constructing prediction confidence intervals, which does not require any additional hypotheses on tile conditional error distribution. It is based on tile works of Febrero et al. (1995) and Cao et al. (1995), and is designed to nfinfic the distribution of the forward values, conditional on all previous values of the series. We will only resample ahead conditional on tile estimates of tile model parameters, since the ARCH models are "ad hoc" methods for measuring shifts in the variance over time. The proposed method does not require a knowledge of the backward residuals which are difficult to estimate in dynamic models with timedependent conditional variances, as several parametric specifications of tile conditional variance function can be evaluated. Consequently, no single parametr ic specification of tile conditional density appears to be suitable for all conditionally heteroscedastic data. Moreover, it is computat ional ly faster than the method proposed by Thombs and Schucany (1990) and ex-