Simulated root mean square error for three estimators of the effective spread (Y-axis) as a function of the logarithm of the intraday number of observations (X-axis) used in the estimation. The functions are plotted as a curve and the X-axis has logarithmic scale. The three estimators are: ES1, the Roll estimator, and the realized volatility based estimator. The chart has three panels that only differ in the value of the true effective spread being estimated. The left panel shows the root mean square error functions when the true effective spread equals 120 basis points. The root mean square error declines rapidly for all three estimators as the number of observations increases, but it approaches zero only for the Roll and realized volatility based estimators. When the number of intraday observations is less than 1,000, the ES1 and the Roll estimator exhibit similar root mean squared error, and both lie significantly below the root mean square error function of the Roll estimators. When the number of intraday observations exceeds 1,000, the root mean square error of the ES1 estimator no longer declines, while those of the Roll and realized volatility estimators continue to decline towards zero. The middle panels plots the root mean square error functions when the true effective spread equals 50 basis points. The pattern of the root mean square error functions is similar to that in the left panel, except that the root mean square error function for the ES1 estimator stops declining when the number of observations exceeds around 100, and at that point the function also lies significantly above those for the other two estimators. The right panel shows the same functions when the true effective spread equals 10 basis points. The functions for the Roll and Realized volatility based estimators are quite similar and rapidly decline towards zero. The root mean square error of the ES1 estimator only declines when the intraday number of observations is less than 100, and the curve lies significantly above the curves for the other two estimators when the number of observations exceeds around 10.
Expected squared range of the price process p and its three approximations based on equations (27), (28), and (29) (Y-axis) as a function of the number of intraday observations (X-axis). The expected squared range and its approximations are plotted as curves and the X-axis has logarithmic scale. The chart has two panels that only differ in the value of the effective spread parameter. The volatility of the efficient price equals 35 basis points in both panels. The left chart, where the true effective spread equals 10 basis points, shows that all four function smoothly increase as the number of intraday observations increases. Approximations based on equations (27) and (29) are practically identical and both lie above the expected squared range and the approximation based on equation (28). The differences between the four lines decline and the lines converge as the number of intraday observations increases. The approximation based on equation (28) is almost identical to the expected squared range when the number of observations is larger than 100. When it is smaller than one hundred, the approximation based on equation (28) lies slightly below the expected squared range, and the difference increases as the number of observations decreases. The right panel shows the same four functions under the assumption that the true effective spread equals 5 basis points. The picture is qualitatively very similar to the chart on the left, the only difference is that the there is a larger gap between the approximation based on equation (28) and the expected squared range when the number of observations is smaller than 100.
The figure shows the root mean square error of alternative estimators of the effective spread expressed as a fraction of the true effective spreads (Y-axis) separately for ten deciles based on the signal-to-noise ratio (X-axis). The root mean square error is plotted as a curve. The figure is produced as follows. Every month, the stocks in the sample are sorted into deciles by their signal-to-noise ratio, from smallest (decile 1) to largest (decile 10). The root mean square error for each decile is then calculated by averaging across all stock-month observations in the decile. The figure shows that the root mean square error of the ES1, ES2, and ES3 estimators gradually declines as the signal-to-noise ratio increases and approaches the root mean square error of the Roll and realized volatility based estimators. The root mean square error curves of the other estimators are essentially constant, but the curves of the Roll and realized volatility based estimators lie significantly above those of the RS1, RS2, and RS3 estimators.
Time-varying estimates of the effective spread averaged over the 147 stocks in the sample between September 2006 and April 2014. The Y-axis displays the time-varying estimates produced by the ES1, ES2, ES3, Roll and realized volatility based time-varying estimators together with the true average effective spread observed in the data. The X-axis displays time. The left panels shows estimates from the original data, while the right panels shows estimates from the artificial data created from the original data by inflating the effective spreads by a factor of 10. Both panels show that the Roll and realized volatility based estimators closely track the true effective spread, although the Roll estimator lies slightly below the true spread. In the left panel, the estimators ES1, ES2 and ES3 lie significantly above the true spread and do not track the dynamics of the true spread very closely. In the right panel, they still lie above the true spread, but the differences are smaller, and the estimators generally capture the peaks and troughs of the true spread.