Exponential General Autoregressive Conditional Heteroskedasticity-EGARCH

### What is an EGARCH model?

*EGARCH* means *Exponential General Autoregressive Conditional Heteroskedasticity* and is an empirical approach mostly used in financial modeling. It is often not sufficient to model the first moment (the mean) of a financial time series. The reason is that the second moment (volatility) is not always constant. In fact, in most financial time series it is changing over time. This empirical phenomenon is called heteroskedasticity. In many other areas of research it is sufficient to correct standard errors in case of heteroscadisticity as researchers are primarily interested in the first moment and need correct standard errors for inference. In financial economics the development of the second moment itself is often of interest and is modeled more in detail. Volatility is a common measure for risk and the level of volatility is an important factor for many business decisions, not only in finance. The standard approaches therefor are ARCH (Autoregressive Conditional Heteroskedasticity) and GARCH (General Autoregressive Conditional Heteroskedasticity) models.

What are now the particularities and the main implications of the *EGARCH* model? One common stylized fact in financial economics is the leverage effect. It describes the observation that negative shocks often increase volatility to a greater extent than positive shocks. Including additional variables for measuring such and effect into a regular GARCH framework would imply the possibility of negative conditional volatility. This is not possible and the *EGARCH* model is often applied to model the leverage effect. The mathematical procedures are straightforward and sketched only briefly. First the logarithm is taken on both sides of the equation and then solved for the conditional volatility (see Wikipedia for equations). The left hand side of the equation is now an exponential function (therefore the name *exponential *GARCH model). It is well-known that the exponential function is positive at all times and the problem of non-negativity of the variance is solved with this simple model.

Although the mentioned leverage effect is a common application of the *EGARCH*model, there are many other situations where this approach can come in handy. Whenever other variables that can take on negative values are included in the specification of the conditional variance, it is useful to apply this approach. Standard software packages make an estimation of the model easy and computation is usually fast. It should be noted that neither OLS (Ordinary Least Squares) nor ML (Maximum Likelihood) can be used for estimating an *EGARCH*model. Common methods are QML (Quasi Maximum Likelihood) and GMM (Generalized Method of Moments).

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