In-sample estimation results

In this section, we show the estimation results and also investigate the model perfor- mance for in-sample fits. Like the standard HAR model, the TV-HAR model can be easily estimated by the standard OLS method.

[Tables3.3to3.5around here]

Tables 3.3 and3.4 show the parameter estimates, with standard errors in parentheses, of the standard HAR, the HARQ and the TV-HAR models for the S&P 500 and SPY indices. The adjusted R-squared values and Akaike information criterion (AIC) are followed by the estimated parameters in the table to compare the in-sample fit. The in-sample estimation results for the ten stocks are shown in the Appendix. Table3.5

summaries and compares the in-sample fits of different models for the ten stocks. We discuss the in-sample estimation results for the HAR, HARQ and TV-HAR models as follows.

HAR model places more weight on the weekly lags according to the full sample period estimates. Compared with other sampling frequencies, the HAR model fits the data better with the 300 seconds RV. In terms of different market regimes, according to the estimations for the HAR model, because the pre-crisis period is less volatile, the daily lags seems more informative than them in the crisis period, the HAR model tends to put more weight on the daily lags for the pre-crisis period than the crisis period. It seems that the market microstructure noise is greater for the crisis period as the HAR model fits the data best for the 150 seconds RV during pre-crisis period but for the crisis period the best fits sampling frequency is 300 seconds RV. For SPY, the estimation results share similar pattern compared with the S&P 500. The difference is that the HAR model does not place largest weight on the daily lags for the pre-crisis period.

B. The HARQ

Next we discuss the in-sample estimation results of the HARQ model. Compared with other sampling frequencies, the HARQ model fits the data better with the 150 seconds RV for both indices. In line withBollerslev et al.(2016), the values ofβqare all negative and significant. This indicates that the uninformative days with large measurement errors have a smaller impact on the forecasts than days where RV is estimated precisely. The value ofγ is the theoretical maximum weight of daily lags, which are higher for the crisis period than the pre-crisis period, because the crisis period has some days which the daily lags tend to be very uninformative and requires larger adjustment for their weights. Compared with the HAR model, the HARQ model tends to allocate lower weight on the weekly and monthly lags, but a greater average weight on the daily lags, so the HARQ model generally allows for a more rapid response, except when the signal

is poor.

C. The TV-HAR

In this section we focus on the in-sample estimation results of the TV-HAR model. Similar to the HARQ model, the TV-HAR model fits the data better with the 150 seconds RV for both indices. As expected, the value ofα of the TV-HAR model is negative and strongly significant for all sampling frequencies and subsample periods. Therefore, the absolute difference of daily and monthly lags is negatively related to the weight of daily lags, which is consistent with the inner working of the TV-HAR model. Like the HARQ model, the theoretical maximum weight of daily lags measured byγ is larger for the crisis period than the pre-crisis period. Compared with the HAR model, the TV-HAR model allocates lower weight to the weekly and monthly lags, but a greater average weight to the daily lags. One possible explanation for this is that the TV-HAR model allows the weight of daily lags to change according to the different scenarios, so the daily lags can offer more accurate information for forecasting future RV. The theoretical maximum weight of daily lags is generally larger for the TV-HAR model compared with the HARQ model.

D. Comparison

Finally, we compare the in-sample fits of the HAR, HARQ and TV-HAR models. For the S&P 500, the TV-HAR model generally performs better than the HAR and HARQ models. It fits the data best for the 150, 450 and 900 seconds RV. The HARQ model

R-square and AIC across different sampling frequencies. Based on the average values, the TV-HAR model performs best for the S&P 500. In terms of SPY, the HARQ model generally performs better than the HAR and TV-HAR models according to the average level. The TV-HAR model performs best for the 900 seconds RV. Then we compare the average in-sample fits for the 10 stocks shown in Table3.5. There are five stocks support the TV-HAR model and five stocks support the HARQ model. The HARQ model generally fits the data better for the pre-crisis period, and the TV-HAR model fits the data better for the crisis period. Therefore, in terms of the in-sample fits, the TV-HAR and HARQ models obtain similar gains compared with the HAR model. As the financial market participants care more about the out-of-sample forecasting, in order to further compare the TV-HAR and HARQ models, in the next section we investigate the forecasting performances.