GARCH model analysis using python

Question

I have an AR(3)-GJR-GARCH(2,2,2) model. How can I test the presence of ‘leverage effects’ (i.e. asymmetric responses of the conditional variance to the positive and negative shocks) with 5% significance level?

Below is my code for the model:

startdate = '2009-01-01'
enddate = '2021-12-31'
data = yf.download('GD', start = startdate, end = enddate)
data.rename(columns={"Adj Close": "price"}, inplace = True)
log_returns = np.log(data['price']/data['price'].shift(1))*100 # Log return in %
log_returns.dropna(inplace = True)
startdate = '2010-01-01'
enddate = '2018-12-31'
in_sample_return = log_returns.loc[startdate:enddate]
gjr_garch = arch_model(in_sample_return,mean='AR',lags=3,vol='GARCH',p=2,o=2,q=2,dist='t').fit(update_freq=5)

What do I do next to check if ‘leverage effects’ is present at 5% significance level?

As I know the gamma parameter is the leverage and when gamma is non-zero it means that the model has leverage effect, but the problem is here in this model I have two gamma parameters.

I thought checking gamma coefficient is enough but as it mentioned "5% significance level", I believe the p-value needs to be calculated and I'm not sure how do I do it.

Answer 1

You need to assess the joint significance of the two gamma coefficients. For that you will need to extract the estimated covariance matrix of the parameters from the fitted model object and do an $F$-test (or a $\chi^2$ test if the sample is large enough). I am not sure if there is a function that does the $F$-test given just the fitted ARMA-GJR-GARCH model object or the covariance matrix and the two point estimates. However, you could always do this by hand. Conceptually, it is no more difficult than doing an $F$-test in a multiple regression model by hand. Finding worked out examples for the latter is probably quite easy.