Sep 28, 2007 - In his autoregressive conditional heteroskedasticity (ARCH) model, the .... evolves as an autoregressive moving average (ARMA) process:.
Imposing Necessary Conditions for Stationarity on ARCH and GARCH Models Christopher J. O’Donnell
Vanessa Rayner
University of Queensland
University of Queensland
September 28, 2007
Abstract In their seminal papers on ARCH and GARCH models, Engle (1982) and Bollerslev (1986) speci…ed parametric constraints that were su¢ cient but not necessary for positivity and stationarity of the estimated conditional variance function. Incorporating these constraints into the estimation process yields parameter estimates that are theoretically over-constrained. This paper uses Bayesian methodology to impose less restrictive necessary and su¢ cient constraints on the parameters. change.
The application is to data from the London Metals Ex-
Uncertainty surrounding the selection of an ARCH(3) or GARCH(1,1) model is
resolved using Bayesian model averaging. Results include estimated posterior pdfs for onestep-ahead conditional variance forecasts.
1
Introduction
Many economic and …nancial time series exhibit periods of high volatility followed by periods of relative tranquility. There is considerable interest in forecasting this special type of intertemporal variability. Forecasts of exchange rate volatility, for example, are used in international portfolio management and macroeconomic policy-making (Hsieh, 1988; West and Cho, 1995; Henry and Summers, 1999), while forecasts of volatility in asset returns are used to design dynamic hedging strategies and to price derivative securities (Kearns and Pagan, 1993; Vrontos et al 2000; Brooks and Persand, 2003; Geweke and McCausland, 2001). Engle (1982) was the …rst to design an econometric model for estimating and forecasting time-varying levels of volatility. In his autoregressive conditional heteroskedasticity (ARCH) model, the unconditional error variance is assumed constant, but the conditional variance is assumed to depend on past realizations of the error process. The model was partly motivated by the work Lucas (1972), whose rational expectations hypothesis implies that the conditional distribution of a series is paramount when it comes to forecasting. The ability of the ARCH model to explain variations in the conditional moments of a series is compatible with the Lucas hypothesis and ensures it plays a key role in the analysis of economic behaviour in the presence of uncertainty. The basic ARCH model has been extended in several ways. The most in‡uential and widely-used extension is Bollerslev’s (1986) generalized autoregressive conditional heteroskedasticity (GARCH) model, which assumes the conditional variance is also a function of its own history. The asymmetric GARCH models of Nelson (1991) and Glosten, Jagannathan and Runkle (1993) are also widely used, particularly in equity markets where bad news tends to have a more pronounced e¤ect on volatility than good news.
In their paper on posterior
integration in GARCH models, Müller and Pole (1998) identi…ed more than three hundred papers in mainstream economics and statistics journals discussing the theoretical properties and applications of various extensions to the ARCH model. ARCH and GARCH models are commonly estimated using unconstrained maximum likelihood (ML) methods. This empirical practice takes no account of the fact that a conditional variance process must be positive and stationary. The result is that point and interval es-
1
timates of parameters and variances may be statistically or economically implausible. One way forward is to impose parametric constraints that are su¢ cient but not necessary for positivity and stationarity of the conditional variances. Of course, this approach over-constrains the parameter space and, consequently, yields parameter estimates and conditional variance forecasts that are potentially biased. This paper illustrates Bayesian methodology for imposing a less restrictive set of su¢ cient and necessary regularity constraints on the estimated parameters. As it happens, in an application to data from the London Metals Exchange (LME), we …nd that estimates of the parameters of two models are largely una¤ected by the restrictiveness of these positivity and stationarity constraints. In our empirical application, and in most other practical settings, there is considerable uncertainty concerning the exact form of the conditional variance process. Sampling theorists can sometimes resolve this uncertainty using common model selection criteria and/or hypothesis tests. One well-known and undesirable consequence of this strategy is that it can lead to a false picture of the reliability of estimates of economic quantities of interest (e.g., Danilov and Magnus, 2004).
The Bayesian solution involves computing posterior model
probabilities and estimating economic quantities of interest using model averaging. In the case of inequality-constrained ARCH and GARCH models, a method proposed by Gelfand and Dey (1994) for computing posterior model probabilities is not immediately available due to the fact that inequality-restricted prior densities are only known up to a multiplicative constant.
This paper uses importance sampling to estimate these normalising constants
and, subsequently, posterior model probabilities. The paper is organized as follows. In Section 2, we present the standard linear regression model with GARCH errors, and we identify two sets of constraints for ensuring the unconditional variance is positive and the conditional variance process is both positive and stationary. In Section 3, we discuss sampling theory methods for selecting plausible ARCH and GARCH models for estimation. In Section 4, we discuss maximum likelihood estimation of GARCH models under the assumption the errors are normally distributed. In Section 5, we decribe Bayesian methods for incorporating positivity and stationarity constraints into the estimation process. In Section 6, we use sampling theory methods to select and estimate an ARCH(3) and a GARCH(1,1) model of an LME time series. Bayesian methods are also 2
used to estimate the two models, and to resolve model uncertainty through computation of posterior model probabilities and model averaging. In Section 7 we o¤er some concluding remarks.
2
ARCH and GARCH Models
Consider the standard linear regression model, (1)
y t = x t + "t for t = 1; :::; T , where xt is an N
1 vector and "t is a white noise error with V ar("t ) =
2
.
Volatility models are underpinned by an additional assumption concerning the conditional error variance, namely V ar("t j "t 1 ; "t 2 ; :::) = ht > 0: Error processes with this property are said to be conditionally heteroskesdastic. Di¤erent assumptions concerning the evolution of ht give rise to di¤erent models, including the ARCH and GARCH models developed by Engle (1982) and Bollerslev (1986). The ARCH(q) model is obtained under the speci…c assumption that the conditional variance evolves as a moving average process: ht = where L is the lag operator and (L) =
0
+ (L)"2t 2
(2) q
. To see that this assumption p is compatible with V ar("t ) = 2 , it is convenient to write "t = ut ht where ut is a white 1L +
2L
+ ::: +
qL
noise error with unit variance. Then E("2t ) = E(ht ) = E(
0
+ (L)"2t ) =
0
+ (1)E("2t ) =
0
1
(1)
=
2
:
(3)
The GARCH(p; q) model is obtained under the slightly more general assumption that ht evolves as an autoregressive moving average (ARMA) process: ht = where (L) =
1L
+
2L
2
+ ::: +
pL 2
0 p
+ (L)ht + (L)"2t
(4)
. For this model, the unconditional error variance is
=
0
1
(1) 3
(1)
:
(5)
Equations (2) to (5) are of particular interest because they suggest the following su¢ cient but not necessary conditions for positive, …nite and stationary variances (Engle, 1982; Bollerslev, 1986): S.1
2
will be positive and …nite if 0
0 for all j and
S.3 ht will be stationary if j
1j
+j
2j
k
+ ::: +
p
> 0 for all k; and < 1.
Because they are not necessary conditions, imposition of S.1 to S.3 over-constrains the parameter space. The following is an alternative set of less-restrictive necessary and su¢ cient conditions: NS.1
2
will will be positive and …nite if and only if 0
