Distribution of the Test Statistic Sample Clauses

Distribution of the Test Statistic. ‌ To conduct statistical inference based on the entropy test of asymmetry, we need to know the sampling distribution of the test statistic under the null hypothesis of symmetry. There are in general two ways to derive the sampling distribution. One is to rely on asymptotic theory and the other way is to use bootstrap resampling method. The asymptotic theory for the class of entropy measures with similar functional forms has been developed by Xxxxx and Tjøstheim (1993); Tjøstheim (1996) and by Xxxx and White (2005). Under certain regularity conditions, Sˆρ(c) follows asymptotically normal distribution and the distribution derived under the null hypothesis does not depend on the choice of the bandwidths. This is partially because the bandwidth is a quantity that vanishes in the limit. However, for a given finite sample size, the computed value of the test statistic depends critically on the bandwidth selection (Maasoumi and Xxxxxx, 2008, see). It really raises concerns on using the simple asymptotic distribution to conduct statistically inference in empirical applications, since the results of such asymptotic-based tests tend to be highly sensitive to the bandwidth and there are many competing approaches for bandwidth selection. Therefore, following the suggestions of Xxxx and Xxxxx (2005) and many others, rather than relying on asymptotic distribution for inference, I use a bootstrap resampling approach to determine the empirical null distribution of Sˆρ(c) (see Xxxxx, 1982; Xxxx, 1992; Xxxxxxxx, 2001, for more discussions on bootstrap resampling approach). To construct a sample under the null hypothesis in the bootstrap resampling, let Z = {(xi,1, y1), (xi,2, y2), ...., (xi,T , yT ); (−xi,1, −y1), (−xi,2, −y2), ..., (−xi,T , −yT )}, which is a vector obtained by stacking together the original data pairs (xi, yi) with the rotated data pairs (−xi, −yi). Through bootstrapping samples from Z, we construct the empirical distribution of Sˆρ(c). I repeat the bootstrapping draws B times from Z, and then can obtain B resamples of Sˆρ(c). There are many different kinds of bootstrap resampling procedures, e.g. the simple bootstrap, wild bootstrap, block bootstrap, and so on. The choice of which bootstrap resampling procedure to use depends on the nature of the data. As stock return are known to be stationary and weakly dependent, the block bootstrap that takes such dependence structure into account seems to be the natural choice (see Ku¨nsch, 1989). Xxxxxxx and Xxxxx...
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