Asymmetric and Heavy-Tailed GARCH Models

Why Model Asymmetry and Heavy Tails?

A symmetric GARCH model gives shocks of equal magnitude the same immediate effect on conditional variance, whether the shock is positive or negative. Financial and other economic series may instead show a larger volatility response after negative shocks. Separately, unusually large standardized shocks may occur more often than a Gaussian innovation model predicts. These are distinct features: asymmetry concerns how volatility responds to the sign of a shock, while heavy tails concern the distribution of shocks after scaling by conditional volatility.

This tutorial develops two common asymmetric specifications, GJR-GARCH and EGARCH, and then replaces Gaussian innovations with standardized Student-t innovations. Each change modifies a different part of the conditional model. The resulting estimates and prediction intervals should therefore be interpreted in light of both the variance recursion and the assumed innovation distribution.

Symmetric GARCH as a Reference

For comparison, a GARCH(1,1) variance recursion is

A standard sufficient set of restrictions for positive conditional variance is \(\omega>0\), \(\alpha\geq 0\), and \(\beta\geq 0\), with a positive initial variance. Because the shock enters as its square, replacing \(\epsilon_{t-1}\) by \(-\epsilon_{t-1}\) leaves the next variance unchanged. When the standardized shocks are symmetric, a finite unconditional variance additionally requires \(\alpha+\beta<1\), in which case it is \(\omega/(1-\alpha-\beta)\). These baseline properties make the two asymmetric extensions easy to compare.

GJR-GARCH: A Sign-Dependent Shock Effect

GJR-GARCH adds an indicator for a negative innovation. Under the convention used here, the model is

where \(I(\epsilon_{t-1}<0)\) equals 1 when the previous innovation is negative and 0 otherwise. With \(\gamma>0\), negative shocks have a larger effect on next-period variance than positive shocks of equal magnitude. A common sufficient restriction for positive variance is \(\omega>0\), \(\alpha\geq0\), \(\beta\geq0\), and \(\alpha+\gamma\geq0\). The often-used leverage interpretation takes \(\gamma\geq0\), so the last condition follows from \(\alpha\geq0\).

If standardized innovations are symmetric with unit variance, then \(E[I(z_t<0)z_t^2]=1/2\). Thus a finite unconditional variance requires \(\alpha+\beta+\gamma/2<1\), and under this condition the unconditional variance is \(\omega/(1-\alpha-\beta-\gamma/2)\). This persistence calculation relies on symmetry; it is not a general formula for an asymmetric innovation distribution.

EGARCH: Modeling Log Variance

EGARCH models the logarithm of the variance and uses a standardized innovation to separate shock size from the current volatility scale. A common EGARCH(1,1) form is

The centered absolute-value term captures the effect of shock magnitude, while \(\gamma z_{t-1}\) captures sign asymmetry. For shocks \(+a\) and \(-a\), where \(a>0\), the log-variance difference is

Consequently, under this sign convention, \(\gamma<0\) means that the negative shock produces the higher next-period log variance. The same comparison also gives a variance ratio of \(\exp(-2\gamma a)\). Since \(h_t=\exp(\log h_t)\), the conditional variance is positive whenever the log variance is finite; EGARCH does not require nonnegative coefficients on the magnitude and asymmetry terms to keep \(h_t\) positive. A common stability condition for the log-variance recursion is \(|\beta|<1\), though full stationarity and moment properties depend on the innovation distribution and the model parameters.

Student-t Innovations and Heavy Tails

A Student-t innovation distribution allows more probability in the tails than a Gaussian distribution. To preserve the unit-variance convention for \(z_t\), let \(T\) have a Student-t distribution with \(\nu\) degrees of freedom and define \(z=\sqrt{(\nu-2)/\nu}\,T\). This scaling is valid for \(\nu>2\), because the unscaled \(T\) has variance \(\nu/(\nu-2)\). The resulting standardized innovation has mean zero and variance one. Its density is

In a conditional likelihood, the density for \(\epsilon_t\) is \(f(\epsilon_t/\sqrt{h_t};\nu)/\sqrt{h_t}\). Thus the innovation distribution and the variance recursion jointly determine each likelihood contribution. Estimating a t-GARCH model changes the fit because large standardized residuals receive different likelihood weights than under a Gaussian model; the estimated variance parameters may consequently change as well. A heavy-tailed distribution does not by itself make the variance response asymmetric.

How Tail Assumptions Change Prediction Intervals

For a one-step forecast, suppose the conditional mean is \(\mu_{t+1|t}\) and the forecast variance is \(h_{t+1|t}\). Under Gaussian innovations, a central 95% conditional prediction interval is \(\mu_{t+1|t}\pm1.96\sqrt{h_{t+1|t}}\). Under standardized t innovations with degrees of freedom \(\nu\), it is \(\mu_{t+1|t}\pm q_{0.975,\nu}^{*}\sqrt{h_{t+1|t}}\), where \(q_{0.975,\nu}^{*}\) is the 97.5th percentile of the unit-variance standardized t distribution. The interval accounts for the assumed one-step conditional distribution, not parameter-estimation uncertainty.

Practical Model Choice and Common Pitfall

GJR-GARCH and EGARCH describe sign-dependent volatility responses in different ways: GJR-GARCH adds a negative-shock term to the variance recursion, while EGARCH adds a signed term to log variance. Student-t innovations instead change the conditional distribution of standardized shocks. These features can be combined, for example in an asymmetric GARCH model with t innovations, but each added component should be motivated and checked rather than treated as interchangeable.

A frequent mistake is to see a large negative residual and conclude that a t distribution alone captures the negative-shock volatility response. It does not: the t distribution raises tail probability for large standardized residuals of either sign. Conversely, an asymmetry parameter does not by itself account for unusually heavy tails. A sound analysis checks whether the variance specification captures sign-related persistence and whether the chosen innovation distribution gives an adequate description of standardized residuals. It also verifies parameter restrictions and reports the distributional assumptions behind prediction intervals.