mators
In this section, we construct the covariance matrices of 19 stocks for each of the 2769 trading days using the duration based, RK, and TS methods and compare them with the 5min, 30min, and open-to-close (OtoC) daily realized covariance matrix estimates. Two duration based variance/covariance estimators are included: COV1, which is the average of 401 estimates using price durations based on threshold
values from 2 times the daily average spread to 6 times the spread with increment 0.01 times the spread; and COV2, which is the average of 151 estimates using price durations based on threshold values from 2.5 times the daily average spread to 4 times the spread with increment 0.01 times the spread.
Table 3.5 presents the benchmark 5min and 30min realised variance/covariance estimates averaged over all trading days, where in the upper diagonal are the aver- age 5min realised covariances and in the lower diagonal the average 30min realised covariances.3 The diagonal elements are average realised variance estimates based on 5min returns and are in italics. Table 3.10 presents the benchmark average OtoC realised covariance estimates and the diagonal average realised variance estimates are in italics.
In Tables 3.6 and 3.7, the average daily estimates from COV1, COV2, RK and TS are compared with the average daily 5min realised variance/covariance estimates. COV1 estimates are in the upper diagonal, COV2 in the lower and the main diag- onal. In Table 3.7, RK estimates are in the upper and main diagonal and the TS estimates in the lower diagonal. Elements that are significantly4 different from the
5min realised variance/covariance benchmark are in bold. Overall, the duration based estimates are more close to the 5min estimates than RK but less close than TS, which is shown explicitly in Table 3.13.
In Tables 3.8 and 3.9, we compare COV1, COV2, RK, and TS estimates with the 30min realised variance/covariance estimates. The duration based estimators
3For illustration purpose, Tables 3.5 to 3.12 present covariance matrices for the first 15 stocks
only, due to limitation of space. Tables 3.13, 3.14, and 3.15 present statistics based on all 19 stocks.
4All significance tests in this section are Newey-West type HAC tests with 1% significance level,
produce estimates that are closer to the 30min benchmark than both the RK and TS estimators, as shown explicitly in Table 3.13.
Finally, in Tables 3.11 and 3.12, the four candidate estimates are compared with the OtoC realised variance/covariance estimates as a benchmark. Both RK and TS estimates are less different from the OtoC estimates than the duration based estimates, as shown explicitly in Table 3.13.
Table 3.13 summarizes the comparison between the candidate group including COV1,COV2, RK, and TS estimators and the benchmark group including the 5min, 30min, and OtoC realised variance/covariance estimators. Results are presented as the proportion of the matrix elements that are significantly different at 1% signifi- cance level (using Newey-West type HAC significance test) for the pair of estimators in comparison. Comparisons are done respectively on the whole matrix, on the off- diagonal elements, and on the main diagonal elements. On the variance part, the duration based estimates are closer to the 30min and OtoC estimates than RK and TS, but less close to the 5min estimates. On the covariance part, the candidate es- timators generally show lower levels of difference to the benchmark estimators than on the variance part. Specifically, duration based covariances are more similar to the 5min estimates than RK but less so than TS; they are also more close to the 30min estimates but less close to the OtoC covariances than RK and TS estimates. Table 3.14 presents summary statistics including mean, standard deviation and the first order autocorrelation for the four candidate variance/covariance estimators and the three benchmarks. The statistics are again presented for the whole ma- trix, and also separately for the variance and covariance parts. The overall mean
estimates of the four candidate estimators are quite close, with RK slightly higher. The mean variance estimates from the two duration based estimators are lower than those from the RK and TS estimators, while the mean covariance estimates from the four candidate estimators are quite close. For the variation of the estimates, both the variance and covariance estimates from the duration based estimators give smaller STD than those from the RK and TS estimators. The mean and stan- dard deviation features of the duration based estimates as compared to the RK and TS estimates are consistent with the findings from the simulation study in Section 3.4: the duration based covariance estimates exhibit variation smaller than and bias comparable with the competing estimates. It is also interesting to note that the duration based variance/covariance estimates have higher first order autocorrelation than other estimators.
Table 3.15 presents the average correlations between the candidate variance/covariance estimators and the benchmark estimators. As expected, the four high-frequency variance/covariance estimators are the most correlated with the 5min realised vari- ance/covariance estimates, followed by the 30min estimates, and the least correlated with the daily OtoC realised variance/covariance estimates. The levels of correlations with the benchmark estimates are very close among the four candidate estimators, with the duration based estimators showing slightly higher correlations.