To learn more about the asymmetric relationship between empirical volatility and market returns, I thought we should study more data and more asset classes. To augment this analysis, however, I thought it worthwhile to introduce a more suitable parameterization than that available with the standard GJR AGARCH model. Let's call this Piecewise-Quadratic GARCH, or PQGARCH.
r_{t-1}^2\\\textrm{where}\;I_t&=1\;\textrm{if}\;r_t<0\;\textrm{and}\;0\;\textrm{otherwise}\\\textrm{and}\;J_t&=1\;\textrm{if}\;r_t>0\;\textrm{and}\;0\;\textrm{otherwise}\end{align*})
I fitted this model to daily interest rate changes of U.S. Three Month Treasury Bills. The data I used is available from the Federal Reserve Bank of St. Louis. The model was fitted independently to each year's daily data, from 1954 to date. This yields an annual time series of parameter estimates Dy and Ey. Our null hypothesis is for regular (i.e. symmetric) GARCH, which implies that the D's and E's should agree within sampling errors. The chart below illustrates the empirical distribution functions for both data series and the results of applying the Kolmogorov-Smirnov test to that data; which implies that we cannot reject the null hypothesis with a confidence greated than 82% — which is insufficient. (Years in which the regression did not converge without “tuning” are omitted.)
This demonstrates that asymmetric response does not appear to be a characteristic of U.S. interest rate changes over more than half a century of data. i.e. It is an attribute of stock markets not of all markets.