Look-back period | h = 1 | h = 5 | h = 10 | h = 22 |
|---|
k = 1 | < 0.001 | < 0.001 | < 0.001 | < 0.001 |
k = 5 | < 0.001 | < 0.001 | < 0.001 | < 0.001 |
k = 10 | < 0.001 | < 0.001 | < 0.001 | < 0.001 |
k = 22 | < 0.001 | < 0.001 | < 0.001 | < 0.001 |
- This table provides the p values of the Pesaran and Timmermann (2009) chi-square statistic that is used to test the existence of the momentum of jumps (MoJ). The MoJ refers to that the forecasting model with jump information (i.e., the HAR-CJ) which outperforms the benchmark model without jump information (i.e., the HAR-RV) over a recent past period is able to show better forecasting performance in the near future. Statistically, the future forecasting performance of the HAR-CJ relative to the HAR-RV for time \(t + 1:t + h\) is defined as
- \(fp_{t + 1:t + h} = I\left( {(RV_{t + 1:t + h} - \widehat{RV}_{t + 1:t + h}^{CJ} )^{2} - (RV_{t + 1:t + h} - \widehat{RV}_{t + 1:t + h}^{RV} )^{2} < 0} \right),\)
- where \(I( \cdot )\) refers to an indicator function, \(\widehat{RV}_{i + 1:i + h}^{CJ}\) and \(\widehat{RV}_{i + 1:i + h}^{RV}\) are the HAR-CJ and HAR-RV forecasts, respectively, for \(RV_{i + 1:i + h}\). Similarly, the past forecasting performance of the HAR-CJ relative to the HAR-RV for time \(t + 1:t + h\) is defined as
- \(pp_{t + 1:t + h} (k) = I\left( {\sum\limits_{i = t - h - k + 1}^{t - h} {(RV_{i + 1:i + h} - \widehat{RV}_{i + 1:i + h}^{CJ} )^{2} } - \sum\limits_{i = t - h - k + 1}^{t - h} {(RV_{i + 1:i + h} - \widehat{RV}_{i + 1:i + h}^{RV} )^{2} } < 0} \right),\)
- where k refers to the length of the look-back period. In a statistical sense, the cross-sectional dependence between \(pp_{t + 1:t + h} (k)\) and \(fp_{t + 1:t + h}\) equates with the existence of MoJ. The chi-square statistic of Pesaran and Timmermann (2009) is used to test the null hypothesis that \(pp_{t + 1:t + h} (k)\) and \(fp_{t + 1:t + h}\) are not cross-sectional dependent in the presence of serial dependencies for each series itself against the alternative hypothesis that the two time series are cross-sectional dependent. The corresponding p values are reported.
