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2013 | OriginalPaper | Chapter

10. Hypothesis Testing Methods and Confidence Regions

Author : Ron C. Mittelhammer

Published in: Mathematical Statistics for Economics and Business

Publisher: Springer New York

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Abstract

In this chapter we examine general methods for defining tests of statistical hypotheses and associated confidence intervals and regions for parameters or functions of parameters. In particular, the likelihood ratio, Wald, and Lagrange multiplier methods for constructing statistical tests are widely used in empirical work and provide well-defined procedures for defining test statistics, as well as for generating rejection regions via a duality principle that we will examine.

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previous chapter Hypothesis Testing Theory

Recall that the supremum of g(w) for w $ \in $ A is the smallest upper bound for the value of g(w) when w $ \in $ A. If the supremum is attainable for some value w $ \in $ A, the supremum is the maximum.

A more elegant solution procedure for this inequality constrained maximization problem could be formulated in terms of Kuhn-Tucker conditions.

R.G. Bartle, Real Analysis, p. 371.

Excellent references for additional details include L.G. Godfrey, (1988), Misspecification Tests in Econometrics, Cambridge Univ. Press, New York, pp. 5–20; and R.F. Engle, (1984), “Wald, Likelihood Ratio and Lagrange Multiplier Tests,” in Handbook of Econometrics vol. 2, Z. Giliches and M. Intriligata, Amsterdam: North Holland, pp. 775–826.

Λ _r is the random vector whose outcome is λ _r.

Note that under H ₀ W/q is the ratio of two independent central χ² random variables each divided by their respective degrees of freedom.

Recall that inf denotes infimum, which is the largest lower bound to the set of values under consideration. The infimum is the minimum if the minimum is contained in the set of values.

We are suppressing a technical condition for this inheritance in that convergence of the test statistic’s probability distribution to a limiting distribution should be uniform in Θ∈Ω. This will occur for the typical PDFs used in practice. For further details, see C.R. Rao, op. cit., pp. 350–351.

See M.A. Stephens, (1974), JASA, p. 730; and Chandra, et. al, (1981), JASA, p. 729.

As always, sup can be replaced by max when the maximum exists.

C. Jarque and A. Bera, “A Test for Normality of Observations and Regression Residuals”, International Statistical Review, pp.163–172, 1987.

Analysis of the JB-Test in MATLAB. MathWorks.

http://www.mathworks.com/access/helpdesk/help/toolbox/stats/jbtest.html. Retrieved May 2, 2012.

Wald, A. and Wolfowitz, J., (1940), On a test whether two samples are from the same population. Ann. Math. Statist. 11, pp. 147–162.

R.G. Bartle, op. cit., p. 371.

We must alert the reader to a technical point that we suppress notationally regarding the use of Taylor series representations of vector functions. Specifically, such a representation is actually a collection of Taylor series representations, one for each entry in the vector function, say f(z). As such, the point of evaluation of the final derivative terms in each Taylor series can differ for each coordinate function. For example, if f(z) is (j×1), then in the Taylor series representation

$$ {{\bf f}({\bf z}) = {\bf f}\left( {{{{\bf z}}_0}} \right) + \frac{{\partial {\bf f}({{{\bf z}}_{*}})\prime}}{{\partial {\bf z}}}\left( {{\bf z} - {{{\bf z}}_0}} \right)} $$

it can be that each row of $ \frac{{\partial {\bf f}({{{\bf z}}_{*}})\prime}}{{\partial {\bf z}}} $ must be evaluated at a different z _* = τz + (1 − τ)z ₀, τ∈[0,1]. Having alerted the reader to this situation, we will tacitly assume henceforth that this is understood. What is most important for our purposes is that z _* → z ₀ as z → z ₀, so in this case, in the limit, all rows of $ \frac{{\partial {\bf f}({{{\bf z}}_{*}})\prime}}{{\partial {\bf z}}} $ will be evaluated at the same point.

One can use the following result for symmetric matrices (Theil, H., (1971), Principles of Econometrics, John Wiley, NY, p. 18:

$$ {\left[\bf \begin{array}{ccc} {A} &\vline & {C} \cr \hline {{C}\prime} &\vline & {B} \end{array} \right]}^{{ - 1}} = \left[ \begin{array}{ccc} {{{A}}^{{ - 1}}} + {{{A}}^{{ - 1}}}{C}{{{\left( {{B} - {C}\prime{{{A}}^{{ - 1}}}{C}} \right)}}^{{ - 1}}}{C}^\prime{{{A}}^{{ - 1}}} &\vline & { - {{{A}}^{{ - 1}}}{C}{{{\left( {{B} - {C}\prime{{{A}}^{{ - 1}}}{C}} \right)}}^{{ - 1}}}} \cr \hline { - {{{\left( {{B} - {C}\prime{{{A}}^{{ - 1}}}{C}} \right)}}^{{ - 1}}}{C}\prime{{{A}}^{{ - 1}}}} &\vline & {{{{\left( {{B} - {C}\prime{{{A}}^{{ - 1}}}{C}} \right)}}^{{ - 1}}}}\end{array} \right] $$

Title: Hypothesis Testing Methods and Confidence Regions
Author: Ron C. Mittelhammer
Publisher: Springer New York
Book: Mathematical Statistics for Economics and Business
Print ISBN: 978-1-4614-5021-4

Electronic ISBN: 978-1-4614-5022-1

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-1-4614-5022-1_10

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