GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics
he great workhorse of applied econometrics is the least squares model. This is a natural choice, because applied econometricians are typically called upon to determine how much one variable will change in response to a change insome other variable. Increasingly however, econometricians are being asked to forecast and analyze the size of the errors of the model. In this case, the questions are about volatility, and the standard tools have become the ARCH/ GARCH models. The basic version of the least squares model assumes that the expected value of all error terms, when squared, is the same at any given point. This assumptionis called homoskedasticity, and it is this assumption that is the focus of ARCH/ GARCH models. Data in which the variances of the error terms are not equal, in which the error terms may reasonably be expected to be larger for some points or ranges of the data than for others, are said to suffer from heteroskedasticity. The standard warning is that in the presence of heteroskedasticity, theregression coefﬁcients for an ordinary least squares regression are still unbiased, but the standard errors and conﬁdence intervals estimated by conventional procedures will be too narrow, giving a false sense of precision. Instead of considering this as a problem to be corrected, ARCH and GARCH models treat heteroskedasticity as a variance to be modeled. As a result, not only are the deﬁciencies ofleast squares corrected, but a prediction is computed for the variance of each error term. This prediction turns out often to be of interest, particularly in applications in ﬁnance. The warnings about heteroskedasticity have usually been applied only to cross-section models, not to time series models. For example, if one looked at the
y Robert Engle is the Michael Armellino Professor of Finance,Stern School of Business, New
York University, New York, New York, and Chancellor’s Associates Professor of Economics, University of California at San Diego, La Jolla, California.
Journal of Economic Perspectives
cross-section relationship between income and consumption in household data, one might expect to ﬁnd that the consumption of low-income households is more closely tied toincome than that of high-income households, because the dollars of savings or deﬁcit by poor households are likely to be much smaller in absolute value than high income households. In a cross-section regression of household consumption on income, the error terms seem likely to be systematically larger in absolute value for high-income than for low-income households, and the assumption ofhomoskedasticity seems implausible. In contrast, if one looked at an aggregate time series consumption function, comparing national income to consumption, it seems more plausible to assume that the variance of the error terms doesn’t change much over time. A recent development in estimation of standard errors, known as “robust standard errors,” has also reduced the concern over heteroskedasticity. If thesample size is large, then robust standard errors give quite a good estimate of standard errors even with heteroskedasticity. If the sample is small, the need for a heteroskedasticity correction that does not affect the coefﬁcients, and only asymptotically corrects the standard errors, can be debated. However, sometimes the natural question facing the applied econometrician is the accuracy of thepredictions of the model. In this case, the key issue is the variance of the error terms and what makes them large. This question often arises in ﬁnancial applications where the dependent variable is the return on an asset or portfolio and the variance of the return represents the risk level of those returns. These are time series applications, but it is nonetheless likely that heteroskedasticity...