Email: support@essaywriterpros.com
Call Us: US - +1 845 478 5244 | UK - +44 20 7193 7850 | AUS - +61 2 8005 4826

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