The GJR model is a GARCH variant that includes leverage terms for modeling asymmetric volatility clustering. In the GJR formulation, large negative changes are more likely to be clustered than positive changes. The GJR model is named for Glosten, Jagannathan, and Runkle [1]. Close similarities exist between the GJR model and the threshold GARCH (TGARCH) model—a GJR model is a recursive equation for the variance process, and a TGARCH is the same recursion applied to the standard deviation process.
The GJR(P,Q) model has P GARCH coefficients associated with lagged variances, Q ARCH coefficients associated with lagged squared innovations, and Q leverage coefficients associated with the square of negative lagged innovations. The form of the GJR(P,Q) model in Econometrics Toolbox™ is
$${y}_{t}=\mu +{\epsilon}_{t},$$
where$${\epsilon}_{t}={\sigma}_{t}{z}_{t}$$ and
$${\sigma}_{t}^{2}=\kappa +{\displaystyle \sum _{i=1}^{P}{\gamma}_{i}{\sigma}_{t-i}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\alpha}_{j}}{\epsilon}_{t-j}^{2}+{\displaystyle \sum _{j=1}^{Q}{\xi}_{j}}I\left[{\epsilon}_{t-j}<0\right]{\epsilon}_{t-j}^{2}.$$
The indicator function $$I\left[{\epsilon}_{t-j}<0\right]$$ equals 1 if $${\epsilon}_{t-j}<0$$, and 0 otherwise. Thus, the leverage coefficients are applied to negative innovations, giving negative changes additional weight.
Note:
The |
For stationarity and positivity, the GJR model has the following constraints:
$$\kappa >0$$
$${\gamma}_{i}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha}_{j}\ge 0$$
$${\alpha}_{j}+{\xi}_{j}\ge 0$$
$${\sum}_{i=1}^{P}{\gamma}_{i}}+{\displaystyle {\sum}_{j=1}^{Q}{\alpha}_{j}+\frac{1}{2}}{\displaystyle {\sum}_{j=1}^{Q}{\xi}_{j}}<1$$
The GARCH model is nested in the GJR model. If all leverage coefficients are zero, then the GJR model reduces to the GARCH model. This means you can test a GARCH model against a GJR model using the likelihood ratio test.
[1] Glosten, L. R., R. Jagannathan, and D. E. Runkle. "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks." The Journal of Finance. Vol. 48, No. 5, 1993, pp. 1779–1801.