Bootstrap Tests for Regression Models (Palgrave Texts in by L. Godfrey

By L. Godfrey

This quantity includes an available dialogue studying computationally-intensive recommendations and bootstrap tools, delivering how you can increase the finite-sample functionality of recognized asymptotic checks for regression types. The book uses the linear regression version as a framework for introducing simulation-based checks to assist practice econometric analyses.

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1. Introduction The merits of tests are usually discussed by considering their behaviour under both null and alternative hypotheses. 21) of Chapter 1. In the latter situation, the probability of detecting a departure from the null hypothesis, that is, the power of the test, is emphasized. Given the decision rule, these probabilities are implied by the sampling distributions of test statistics under null and alternative hypotheses, respectively. As discussed in the previous chapter, the exact form of a sampling distribution under the null hypothesis can sometimes be derived for certain tests, under very restrictive assumptions.

36) is rarely available in practical situations because, in general, is unknown and it is not feasible to ˇ calculate the GLS estimator β. When the elements of are continuous functions of the elements of an unknown parameter vector ψ, estimates of the parameters of the generalized regression model can be obtained by minimizing the Nonlinear Least Squares (NLS) criterion QNLS (β, ψ) = (y − Xβ) [ (ψ)]−1 (y − Xβ), with respect to both β and ψ. Alternatively, if some consistent estimator ˆ is available and necessary regularity conditions are of ψ, denoted by ψ, satisfied, β can be estimated by minimizing the Feasible Generalized Least Squares (FGLS) function ˆ −1 (y − Xβ).

38) implies that n1/2 R(βˆ − β) ∼a N(0q , plim nR(X X)−1 X X(X X)−1 R ), and so, if the null hypothesis is true, n1/2 (Rβˆ − r) ∼a N(0q , plim nR(X X)−1 X X(X X)−1 R ). Consequently, if the restrictions of Rβ = r are valid, standard asymptotic theory implies that n(Rβˆ − r) [plim nR(X X)−1 X X(X X)−1 R ]−1 (Rβˆ − r) ∼a χ 2 (q). However, this result does not yield a feasible test procedure because it concerns a random variable that depends upon the probability limit of a matrix that is, in part, determined by the unknown matrix .

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