by Josef A

Created on: June 14, 2009   Last Updated: June 20, 2009

When investigating empirical issues, it is in most cases sufficient to analyze the first moment (the mean) of an econometric model. A stylized fact in econometrics is heteroskedasticity meaning the variance is not constant over time. There are several methods to adjust for heteroskedasticity in order to get unbiased test statistics and enable inference of the obtained estimates. However, the second moment of an econometric model, the variance, may also be of prior interest and can be modeled as a separate process. Such applications are quite common in many fields of finance as volatility itself is an essential part of analyzing financial markets. The basic empirical model to capture volatility dynamics is referred to as the Autoregressive Conditional Heteroskedasticity model or short ARCH. The main characteristics of such a model are described in the following.

To model the variance there must be some kind of model for the first moment, the mean, which can be any econometric model, rather simple or very complex. In any case, estimating the model will yield a series of error terms. Those error terms are the determining variables for the ARCH process. The conditional volatility can be modeled taking the standard deviation or the variance as the dependent variable. Most commonly, the conditional variance is taken and we also use this specification.

The conditional variance is explained by a constant and a number of lagged squared residuals (errors) from the estimation of the mean. In general, an ARCH(q) model describes a model including q lagged squared residuals. In order to test for the lag-length, a large number of lags is included and the lag-length is determined by the last significant estimate for a specific lagged residual. There is some more maths involved to derive the ARCH model, but the procedure is in principal quite straightforward. ARCH models have to be estimated using e.g. Maximum Likelihood (ML) as Ordinary Least Squares (OLS) does not work.

Although the ARCH model is the basic specification when investigating on conditional volatility, there are some problems that make the model difficult to apply in practice. The issue of using Maximum Likelihood or an alternative method to estimate the model is not very significant anymore, as current software packages handle such estimations easily. The more relevant problem is that it is difficult to determine the exact number of lags and a quite large number is often required. As a result, more advanced models are often applied that are based on the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) approach.

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