### GJR-GARCH-M Estimation for Stock Market Volatility

Posted:

**Sat Mar 27, 2021 9:57 am**Hi there!

I am doing a research project where I am examining the effect that Covid-19 has had on the returns and volatility of the financial markets. I have opted to use a GJR-GARCH-M (1,1) model to properly account for the leverage effect. I have purposely avoided using EGARCH and other asymmetric GARCH models because GJR is less explored in this context.

I am accounting for stock market volatility using 10 market indices for different countries, then I am taking my "volatility" to be ln(P_t/P_(t-1)), where P_t is the closing price of a particular index on day t, as specified in a number of different academic papers and text books (e.g. Brooks). Other independent variables I have currently collected are daily Covid-19 case data (cases and deaths) on a country-specific and global level. The Government Response Index from Oxford Covid Government Response Tracker which gives a measure of how stringent a particular government's covid measures have been (e.g. school closures, social distancing restrictions etc). Data has all been pre-formatted in Excel so that the dates all properly align and missing values (e.g. stock market holidays) were filled in using previous available value to ensure comparability cross-country.

As suggested in Brooks (2019) I ran an OLS regression of my "volatility" on all of my independent variables and then checked the correlogram of squared residuals which indicated presence of ARCH effects (All P-values zero up to 36 lags).

My current settings within EViews for GJR-GARCH-M are all "standard" apart from I selected 'EViews Legacy' - which was a recommendation I picked up at some point for GJR-GARCH but cannot remember where. I have also selected my error distribution as t-distribution since Jarque-Bera test indicated non-normality and financial time series generally exhibit fatter tails.

There are two issues I would like to address. Firstly, when running my GJR-GARCH regressions, nearly all of my ARCH effect terms have negative coefficients, this is obviously a problem since it breaches the non-negativity constraints requirements. Although in most cases, the ARCH term is not statistically significant, so I figured this actually wasn't an issue (see Screenshot 1). However, there are some cases where the ARCH term is negative and significant, which is a problem (see Screenshot 2). All GARCH effect terms and leverage terms are positive, and the majority are significant. What do you make of the negative ARCH effect coefficients?

Secondly, I want to ensure that my findings are robust, which I would ideally like to do by including more country-specific regressors. However, in nearly all academic literature I have seen, there is usually a maximum of one exogenous regressor included in the conditional mean and/or variance equation. I have seen a couple of papers where they conduct a GJR-GARCH without any exogenous regressors and then generate a GARCH variance series and use that in an OLS regression - but that instinctively feels wrong to me? What is the appropriate way to control for country specific effects in this context? I seem to find that if I include several exogenous regressors the GJR-GARCH model encounters similar issues as I have described above (e.g. significant negative ARCH effect coefficient - see Screenshot 3).

If anyone is able to provide additional clarification that would be great, or point me towards additional resources. I have looked at so many different papers and textbooks and am still none the wiser. Please let me know if I have been unclear anywhere or forgotten to mention something!

Many thanks!

I am doing a research project where I am examining the effect that Covid-19 has had on the returns and volatility of the financial markets. I have opted to use a GJR-GARCH-M (1,1) model to properly account for the leverage effect. I have purposely avoided using EGARCH and other asymmetric GARCH models because GJR is less explored in this context.

I am accounting for stock market volatility using 10 market indices for different countries, then I am taking my "volatility" to be ln(P_t/P_(t-1)), where P_t is the closing price of a particular index on day t, as specified in a number of different academic papers and text books (e.g. Brooks). Other independent variables I have currently collected are daily Covid-19 case data (cases and deaths) on a country-specific and global level. The Government Response Index from Oxford Covid Government Response Tracker which gives a measure of how stringent a particular government's covid measures have been (e.g. school closures, social distancing restrictions etc). Data has all been pre-formatted in Excel so that the dates all properly align and missing values (e.g. stock market holidays) were filled in using previous available value to ensure comparability cross-country.

As suggested in Brooks (2019) I ran an OLS regression of my "volatility" on all of my independent variables and then checked the correlogram of squared residuals which indicated presence of ARCH effects (All P-values zero up to 36 lags).

My current settings within EViews for GJR-GARCH-M are all "standard" apart from I selected 'EViews Legacy' - which was a recommendation I picked up at some point for GJR-GARCH but cannot remember where. I have also selected my error distribution as t-distribution since Jarque-Bera test indicated non-normality and financial time series generally exhibit fatter tails.

There are two issues I would like to address. Firstly, when running my GJR-GARCH regressions, nearly all of my ARCH effect terms have negative coefficients, this is obviously a problem since it breaches the non-negativity constraints requirements. Although in most cases, the ARCH term is not statistically significant, so I figured this actually wasn't an issue (see Screenshot 1). However, there are some cases where the ARCH term is negative and significant, which is a problem (see Screenshot 2). All GARCH effect terms and leverage terms are positive, and the majority are significant. What do you make of the negative ARCH effect coefficients?

Secondly, I want to ensure that my findings are robust, which I would ideally like to do by including more country-specific regressors. However, in nearly all academic literature I have seen, there is usually a maximum of one exogenous regressor included in the conditional mean and/or variance equation. I have seen a couple of papers where they conduct a GJR-GARCH without any exogenous regressors and then generate a GARCH variance series and use that in an OLS regression - but that instinctively feels wrong to me? What is the appropriate way to control for country specific effects in this context? I seem to find that if I include several exogenous regressors the GJR-GARCH model encounters similar issues as I have described above (e.g. significant negative ARCH effect coefficient - see Screenshot 3).

If anyone is able to provide additional clarification that would be great, or point me towards additional resources. I have looked at so many different papers and textbooks and am still none the wiser. Please let me know if I have been unclear anywhere or forgotten to mention something!

Many thanks!