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# Random Coe¢ cient Model .

Wald Statistic Let  = r( ) : R k ! R q be any parameter vector of interest, b its estimator and Vb b its covariance matrix estimator. Consider the quadratic form W() =  b  0 Vb 1 b  b   = n  b  0 Vb 1   b   : (7.37) where Vb  = nVb b: When q = 1; then W() = T() 2 is the square of the t-ratio. When q > 1; W() is typically called a Wald statistic as it was proposed by Wald (1943). We are interested in its sampling distribution. The asymptotic distribution of W() is simple to derive given Theorem 7.9 and Theorem 7.10, which show that p n  b   d ! Z  N (0;V ) and Vb  p ! V : It follows that W() = p n  b  0 Vb 1  p n  b   d ! Z 0V 1  Z a quadratic in the normal random vector Z: As shown in Theorem 5.12, the distribution of this quadratic form is  2 q , a chi-square random variable with q degrees of freedom. Theorem 7.13 Under Assumptions 7.2, 7.3 and 7.4, as n ! 1; W() d !  2 q : Theorem 7.13 is used to justify multivariate conÖdence regions and multivariate hypo