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We use heteroskedasticit-consistent standard errors of variances to test whether variances are elevated during the Fed Time interval. The model estimated is:
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With four time intervals, there are eight equations in (1) and eight parameters. These parameters could be estimated with maximum likelihood. However,the standard errors would not be robust to conditional heteroskedasticity. Therefore,we use Hansen's(1982) generalized method of moments(GMM) to obtain heteroskedasticity-consistent standard errors of the variances and to conduct hypothesis tests on the parameters. The model is exactly identified when we condiction on a vector of ones.One advantage of Hansen's approach is that only weak distributional assumptions are required.
With the GMM methodology, it is straightforward totest the hypothesis that variances are different during a particular time period.For example,variance equality can be tested with:
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We also test for the effects of the open market operations on both the returns and the volatilities.This analysis is specialized to the Fed Time half-hour interval,denoted with the FT subscript.The following model is estimated:
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The formulation in (3) allows for both the mean and the variance processes to vary by year and by operation. Furthermore, it is straightforwardto conduct hypothesis tests on the parameters of interest. How ever, in practical terms,(3) may require a lot of data.With seven years of daily data and up to 15 observations per day, the estimation involves up to 20,000 observations with parameters for means and variances by year. Often, we specialize the estimation to examine one particular year or one particular operation.
We use(3) in a number of different ways. We estimate variances on the days when there are draining operations are equal to returns on days with adding operations.