Home | Business News | Browse by Publication | I | International Advances in Economic Research

Forecasting irregularly spaced UHF financial data: realized volatility vs UHF-GARCH models.

Article Excerpt
Introduction

The recent implementation of electronic order-matching systems on financial markets has entailed increasing numbers and frequencies of trades. While data on prices and volumes were registered daily two decades ago, transactions (and especially those due to electronic systems) are now recorded instantaneously with an accuracy of a fraction of second. The growing interest devoted to intra-daily models in the financial literature is a direct consequence of the availability of higher frequency measurements. This phenomenon, stylized by increasing frequencies of observations, is at the origin of the concept of ultra-high frequency. In this context, the development of econometric methods for the analysis of ultra-high frequency data seems to be promising. However, the other side of the coin is the problem induced by the irregularity at which the observations arrive. For example, when we estimate a simple GARCH process on the S & P500, we usually use the returns observed every day or every week. In this case, the interval between each observation is fixed: 1 day or 1 week. But when analyzing intra-day observations, the information arrives sometimes in clusters and at different time intervals. This problem is called time deformation because measure is not the same as calendar time. The fact that the arrival of information is irregularly spaced is a salient feature of ultra-high frequency data. Aggregating this data up to fixed intervals of time gives way to an important loss of information. To avoid this loss, Engle and Russell (1998) and Engle (2000) have recently developed methods that are directly tailored to irregular spacing of the data. The basic model is the autoregressive conditional duration (ACD) model which is in the family of dependent Poisson processes. The ACD model applied to IBM transactions arrival times by Engle (2000) in a GARCH framework has produced ultra-high frequency measures of volatility. The results observed by Engle (2000) are interesting and indicate that this theoretical specification seems to be accurate to estimate models for ultra-high frequency data or transaction data. The ACD-GARCH volatility model and its extended version proposed by Engle (2000) lead indeed to large gains in forecast error accuracy from a theoretical perspective. Considering this result, it would be very interesting to apply these models to ultra-high frequency data.

This paper proposes two main contributions. Our first aim in this paper is to develop an empirical application of ACD-GARCH models in forecasting future volatilities. Our second contribution consists in comparing the performance of ACD-GARCH models to a new and simple way of modeling financial market volatility using high-frequency data recently developed by Bollerslev and Wright (2001): the integrated volatility or recently called, the realized volatility (Barndorff-Nielsen and Shephard 2002a; Andersen et al. 2001). According to Bollerslev and Wright (2001), volatility dynamics may be modeled by fitting a long autoregressive (AR) representation to ultra-high frequency data. The main problem with their approach is that they ignore the fact that data arrive at irregular intervals. Thus, we have to make an adjustment to take into account these fundamental features of ultra-high frequency data.

The plan of this paper is as follows. First, we present the models and their necessary adjustments. We also show how to use volatility forecasts to compute daily Value at Risk (VaR) with and without the historical simulation approach. Then, a section is devoted to the discussion of the ultra-high frequency data and the adjustment procedures employed. A further section gives details of the volatility calculations and volatility forecasts, followed by a comparison of the results and a short discussion. Finally, we conclude with some suggestions for further research.

Ultra-High-Frequency Variance Models

UHF-GARCH(1,1) Model

Ultra-high-frequency GARCH(1,1) model allows taking into account the irregular character of market transactions even if the current durations of these transactions are not explicitly considered as additional sources of information. In this sense, it is the simplest model of conditional variance defined at ultra-high-frequency. This model may be written as follows:

[[sigma].sub.i.sup.2] = [omega] + [alpha][[epsilon].sub.i-1.sup.2] + [beta][[sigma].sub.i-1.sup.2] (1)

with [[sigma].sub.i.sup.2], the conditional variance and [[epsilon].sub.i], the innovation.

The ACD-GARCH Model

The autoregressive conditional duration (ACD) model was firstly developed by Engle and Russel (1998). Later, Gourieroux et al. (1999) and Engle (2000) refined and applied the model in a similar context.

The basic formulation of the ACD is specified in terms of the conditional density of the durations. The duration is the interval between two arrival times denoted by [x.sub.i] = [t.sub.i] - [t.sub.i-1]. The expectation of the ith duration is given by the following function:

E([x.sub.i]|[x.sub.i-1],..., [x.sub.1]) = [theta]([x.sub.i-1],..., [x.sub.1]; [psi]) [equivalent to] [[theta].sub.i] (2)

under the assumption that:

[x.sub.i] = [[theta].sub.i][e.sub.i] (3)

where {[e.sub.i]}~ i.i.d., and [theta] is a set of parameters to be estimated. By definition,...