Leverage effect
1. Leverage effect
The standard GARCH model uses the squared prediction error to forecast the return variance using one single equation. It does not distinguish between positive and negative prediction errors. In reality, the sign of the prediction error matters. Variance shoots up more after a large negative unexpected return than after a large positive unexpected return.2. Negative returns induce higher leverage
The rationale for a higher volatility after a negative return is that, since the market value of the company has dropped, the ratio between debt and market value of the firm has increased. This means that the firm has a higher leverage and is therefore more risky.3. Two equations
We thus need two equations: one equation explaining the GARCH variance following a negative unexpected return. And one equation describing the variance reaction after a positive surprise in the returns.4. In case of a positive surprise
In case of a positive prediction error, and thus a positive surprise in returns, it is common to take the usual GARCH one-one equation.5. In case of a negative surprise
However, in case of a negative surprise, the predicted variance should be higher than after a positive surprise. This means that we need to apply a larger multiplier on the squared prediction error. Formally, this means using as a coefficient $\alpha+\gamma$ instead of $\alpha$ with the additional gamma parameter being positive. These two equations together define the GJR GARCH model. The name GJR refers to the inventors of the model, namely Glosten, Jagannathan and Runkle.6. How?
This is simple to do in R. It suffices to change the argument variance.model in ugarchspec from sGARCH to gjrGARCH.7. Illustration on MSFT returns
We then use this model to analyze the GARCH dynamics for the daily Microsoft returns. We find that indeed, the estimated gamma is positive. It equals 0-point-055. The impact of the squared error on the variance prediction after a negative error is 0-point-09, which is the sum of alpha1 and gamma 1, namely 0-point-034 plus 0-point-055. Comparing the 0-point-03 in the second equation with the 0-point-09 in the first equation, you see that the response of the variance after a negative surprise in return is almost three times as high than following a positive surprise in return.8. Visualize volatility response using newsimpact()
You can use a news impact curve to visualize the impact of the prediction error on the variance. Using the newsimpact method in rugarch we obtain for Microsoft the asymmetric news impact shown in the slide. The quadratic curvature is because the gjrmodel predicts variances using squared prediction errors. Note that, when the prediction error is negative, the variance reacts with a multiple of 0-point-09 to the squared prediction error, while for positive prediction errors, it only reacts with a multiple of 0-point-03.9. Let's estimate a GJR GARCH model.
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