![]() ![]() It’s value is unobservable! And w e can’t use the critical value(s) from the non-central t distribution if we don’t know the value of the non-centrality parameter. In this case the usual t-statistic follows a non-central Student-t distribution, with a non-centrality parameter that increases monotonically with β 1 2, and which depends on the x data and the variance of the error term. ![]() We’ve omitted a (constant) regressor from the estimated model. However, now, in Table 1, β 1 = 1, so the DGP includes an intercept and the fitted model is under-specified. Recall that the estimated model omits the intercept – the model is fitted through the origin. Let’s take a look back at Table 1, and now focus on the second line of results (highlighted in orange). So far, all that we seem to have shown is that the permutation test and the t-test exhibit no “size-distortion” when the model is correctly specified, and the errors satisfy the assumptions needed for the t-test to be valid. More formally, using the uniftest package in R we find that the Kolmogorov-Smirnov test statistic for uniformity is D = 0.998 (p = 0.24) and the Kuiper test statistic is V = 1.961 (p = 0.23). One point that I covered was that if the null hypothesis is true, then this sampling distribution has to be Uniform on, regardless of the testing problem! This result gives another way of checking if the code for our simulation experiment is performing accurately, and that we’ve used enough replications and random selections of the permutations.Īs we can see, this distribution is “reasonably uniform”, as required. In an old post on this blog I discussed the sampling distribution of a p-value. we could obtain the 5% rejection rate correctly, even if the distribution of the p-values from which this rate was calculated is “weird”. However, the result for the randomization test could be misleading. Given the particular errors that were used in the DGP in the simulations, this had to happen for the t-test. Notice that the reported empirical significance levels match the anticipated 5%! That is, the DGP in (1) has no intercept, so the model is correctly specified, and the null hypothesis of a zero slope is true.īecause the null hypothesis is true, the power of the test is just its significance level (5%). ![]() This corresponds to the case where both β 1 and β 2 are zero. We’ll come back to this table shortly, but for now just focus on the row that is highlighted in light green. ![]()
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