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1、 CHAPTER8279Autoregressive Conditional Heteroskedastic Modelsn linear regression analysis, a standard assumption is that the variance of all squared error terms is the same. This assumption is called homoske- dasticity (constant variance). However, many time series data exhibit het- eroskedasticity,
2、 where the variances of the error terms are not equal, and in which the error terms may be expected to be larger for some observations or periods of the data than for others. The issue is then how to constructmodels that accommodate heteroskedasticity so that valid coefficient esti- mates and models
3、 are obtained for the variance of the error terms. Autore- gressive conditional heteroskedasticity(ARCH) models are the topic ofthis chapter. They have proven to be very useful in finance to model returnvariance or volatility of major asset classes including equity, fixed income, and foreign exchang
4、e. Understanding the behavior of the variance of the return process is important for forecasting as well as pricing option-type derivative instruments since the variance is a proxy for risk. Although asset returns, such as stock and exchange rate returns, appear to follow a martingale difference seq
5、uence, observation of the daily return plots shows that the amplitude of the returns varies across time. Awidely observed phenomenon in finance confirming this fact is the so- called volatility clustering. This refers to the tendency of large changes in asset prices (either positive or negative) to
6、be followed by large changes and small changes to be followed by small changes. Hence, there is tempo- ral dependence in asset returns. Typically, they are not even close to being independently and identically distributed (IID). This pattern in the volatil-ity of asset returns was first reported by
7、Mandelbrot.1Time-varying vola-1 Benoit B. Mandelbrot, “The Variation of Certain Speculative Prices,” Journal of Business36 (1963), pp. 394419.Ic08-ARCHModels Page 279 Thursday, October 26, 2006 2:07 PM280FINANCIAL ECONOMETRICStility and heavy tails found in daily asset returns data are two of the ty
8、picalstylized factsassociated with financial return series.2The ARCH model and its generalization, the generalizedautoregres- sive conditional heteroskedasticity(GARCH) model, provide a conve- nient framework to study the problem of modeling volatility clustering. While these models do not answer th
9、e question of what causes this phe- nomenon, they model the underlying time-varying behavior so that fore- casts models can be developed. As it turns out, ARCH/GARCH models allow for both volatility clustering and unconditional heavy tails. TheARCH model is one of the pivotal developments in the fin
10、ancial econo-metrics field and seems to be purposely built for applications in finance.3It was introduced in the initial paper by Engle to model inflation rates. Since the seminal papers of Engle4in 1982 and Bollerslev5in 1987, a large number of variants of the initial ARCH and GARCH models have bee
11、n developed. They all have the common goal of modeling time-vary- ing volatility, but at the same time they allow extensions to capture moredetailed features of financial time series. In addition to ARCH/GARCH models, there are other models of time-varying volatility, such as sto- chastic-volatility
12、 models, which are beyond our objectives here. In this chapter, we will focus on basic ARCH and GARCH models, discuss their structural properties, their estimation, and how they can be used in forecasting. Additionally, we will discuss important variants of these models along with the relevance for
13、practical use.ARCH PROCESSThe ARCH process describes a process in which volatility changes in a particular way. Consider an ARCH(q) model for yt2 The term stylized facts is used to describe well-known characteristics or empirical regularities of financial return series. For example, daily stock inde
14、x returns display volatility clustering, fat tails, and almost no autocorrelation. These three major styl- ized facts can be explained by the ARCH family of models. Additional stylized facts include leverage effect and long memory effect described later in the chapter.3 Tim Bollerslev, “The Financia
15、l Econometrics: Past Developments and Future Chal- lenges,” Journal of Econometrics100 (2001), pp. 4151.4 Robert F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of U.K. Inflation,” Econometrica50 (1982), pp. 9871008. 5 Tim Bollerslev, “A Conditionally Heterosc
16、edastic Time-Series Models for Security Prices and Rates of Return Data,” Review of Econometrics and Statistics69 (1987), pp. 542547.c08-ARCHModels Page 280 Thursday, October 26, 2006 2:07 PMAutoregressive Conditional Heteroskedastic Models281yt= t(8.1), (8.2)(8.3)where htis the variance of tconditional on the information available at time t. htis called the conditional variance oft. The sequence t is the error process to be modeled. Expression (8.1) is typically extended to t
