We estimate the TVP-GARCH-MIDAS model defined in the equations (3.4)-(3.7) with the daily 22-days rolling window versions of realized volatility, ADS, and neg ADS using quasi-maximum likelihood methods. We include one MIDAS lag
year in the MIDAS filter, i.e. we set K = 252 in Eq. (3.6).16 The estimation re-
sults are presented in Table 3.5 along with the benchmark GJR-GARCH(1,1) model estimates.
In terms of likelihood criteria, only the model with RVt(22)yields a lower Bayesian
information criterion than the benchmark GJR-GARCH. Accordingly, estimates of
16We find K = 252 to be sufficiently large for our application. As demonstrated in Engle et
al. (2013) and Conrad and Loch (2014), the beta weighting function seems to be robust to the maximum number of lags K included in the MIDAS filter, as long as it is chosen large enough. See also Footnote 14 on the MIDAS lag length choice.
3.4.2 Estimation results 91
the model extension parameters are highly significant for the RV model, whereas
only the β2 parameter is found to be significant at the 5% level for the neg ADS
variable. However, due to the identification issue under the null discussed in Section 3.3.1, the (in)significance of the model extension parameters have to be taken with a pinch of salt. We therefore add the LM test statistic presented in Eq. (3.14) and Eq. (3.15) for testing the null hypothesis that the variable x has no explanatory power for time variation in the GARCH coefficient. The test statistic is significant at the 1% level for both ADS variables and lies slightly above the 10% significance
level for the RV variable.17 The estimated (restricted) MIDAS weighting schemes
are plotted in Figure 3.8.18 The schemes roughly imply vanishing weights for lags
beyond half a year.
Next, we have a closer look at the time variation in persistence that is im- plied by the model estimations. The estimated time-varying GARCH coefficients,
ˆ
βt = ˆβ1 + ˆβ2F (ˆγ, ˆΦ0xt−1), are shown in Figure 3.9 and some descriptive statistics
are summarized in Table 3.6. First, the signs of the transition parameter γ confirm our intuition from Section 3.2.1. A negative γ for RV implies that the time-varying GARCH coefficient is positively related to realized volatility, i.e. we see high (low) persistence during high (low) volatility regimes. Correspondingly, positive γ es- timates for the ADS variables imply increasing (decreasing) persistence for weak (strong) business conditions. The RV model implies a greater time variation in the GARCH coefficient than the ADS models. For RV , the coefficient lies in the range of [0.78, 0.90], whereas it lies in the range [0.91, 0.93] for the ADS models. Accordingly, we see a higher standard deviation of the GARCH coefficient in the
RV model.19 Both versions of the ADS variable yield a similar time variation in
persistence, though the neg ADS version yields a slightly smoother variation. How- ever, note that both models imply a lower persistence than the GJR-GARCH model on average.
For the ADS variables, particularly for the negative ADS, we see essentially two persistence regimes in Figure 3.9 that roughly correspond to recession and expansion periods with not much variation in between. For realized volatility on the other
17The financial crisis period seems to have distorting effects on the LM test for the realized
volatility model. The test statistic for the subsample ending 2007 is calculated as 4.06 and is significant at the 5% level.
18For all three variables, including an unrestricted scheme in Eq. (3.7) instead yielded no sig-
nificant improvements in terms of the likelihood (as measured by means of a likelihood ratio test).
19Note that the range of the time-varying GARCH coefficient implied by the model estimates
ˆ
β1, ˆβ2 differs for the strictly positive RV /neg ADS, and the ADS, since in the first case, the
hand, there is more variation in the GARCH coefficient during expansion periods. This suggests that there are other factors than the U.S. business conditions affecting volatility persistence, which are reflected in realized volatility but not in the ADS. For instance, monetary policy is an important driver of realized volatility. The term spread (which is not included in the ADS) has strong predictive power for realized volatility, as shown in Paye (2012), and is a leading indicator for financial volatility, as argued in Conrad and Loch (2014).
We illustrate the underlying transitions between high and low persistence regimes implied by the explantory variables in Figure 3.10. The figure plots estimates of
the transition function F (ˆγ, ˆΦ0xt−1) and the time-varying GARCH coefficient ˆβt,
which corresponds to a linear transformation of the transition function, against
the weighted average of the respective explanatory variable x, ˜xt = ˆΦ0xt−1 =
P252
k=1ϕˆkxt−k. To illustrate how the distribution of the explanatory variables re-
lates to the time variation in persistence, the figure includes histograms of ¯xt and of
the transition / GARCH coefficient. The skewness of realized volatility is evident and in line with the descriptive statistics in Table 3.4. Combined with some large outliers, this translates into a very steep transition function for realized volatility, which results in the GARCH coefficient being almost flat during very high volatility regimes, such as the financial crises 2008/09. On the contrary, the distribution of the ADS is symmetric around zero with few outliers and its translation function is smoother.
Finally, we find similar results if we let the explanatory variables govern time variation in the ARCH coefficient. The corresponding estimates of the time-varying ARCH coefficient as well as a summary of their descriptive statistics can be found in the Appendix. However, we note that these model estimates imply a higher level of average volatility persistence, compared to the specification with a time-varying GARCH coefficient. This finding is perfectly in line with the argument by Hillebrand (2005) that the effect of overestimating volatility persistence is stronger if changes in the GARCH parameter are not accounted for (see also the discussion on page 80). This reconfirms the specific choice of our model specification in Eq. (3.4).