Joint Tests for Zero Restrictions on Non-negative Regression Coefficients

Three tests for zero restrictions on regression coefficients that are known to be nonnegative are considered: the classical F test, the likelihood ratio test, and a one-sided t test in a particular direction. Critical values for the likelihood ratio test are given for the cases of two and three restrictions, and the power function is calculated for the case of two restrictions. The analysis is conducted in terms of a characterization of the clas all similar tests for the problem, of which each of the above tests is a member. The likelihood ratio test emerges as the preferred test.Likelihood ratio test; One-sided alternative; Regression; Similar regions.

On the conditional likelihood ratio test for several parameters in IV regression

For the problem of testing the hypothesis that all m coefficients of the RHS endogenous variables in an IV regression are zero, the likelihood ratio (LR) test can, if the reduced form covariance matrix is known, be rendered similar by a conditioning argument. To exploit this fact requires knowledge of the relevant conditional cdf of the LR statistic, but the statistic is a function of the smallest characteristic root of an ( m + 1)−square matrix, and is therefore analytically difficult to deal with when m > 1. We show in this paper that an iterative conditioning argument used by Hillier (2006) and Andrews, Moreira, and Stock (2007) to evaluate the cdf in the case m = 1 can be generalized to the case of arbitrary m . This means that we can completely bypass the difficulty of dealing with the smallest characteristic root. Analytic results are obtained for the case m = 2, and a simple and efficient simulation approach to evaluating the cdf is suggested for larger values of m .

Spatial circular matrices, with applications

The cumulants of the quadratic forms associated to the so-called spatial design matrices are often needed for inference in the context of isotropic processes on uniform grids. Unfortunately, because the eigenvalues of the matrices involved are generally unknown, the computation of the cumulants may be very demanding if the grids are large. This paper constructs circular counterparts, with known eigenvalues, to the spatial design matrices. It then studies some of their properties, and analyzes their performance in a number of applications.

Spatial design matrices and associated quadratic forms: structure and properties

The paper provides significant simplifications and extensions of results obtained by Gorsich, Genton, and Strang (J. Multivariate Anal. 80 (2002) 138) on the structure of spatial design matrices. These are the matrices implicitly defined by quadratic forms that arise naturally in modelling intrinsically stationary and isotropic spatial processes.We give concise structural formulae for these matrices, and simple generating functions for them. The generating functions provide formulae for the cumulants of the quadratic forms of interest when the process is Gaussian, second-order stationary and isotropic. We use these to study the statistical properties of the associated quadratic forms, in particular those of the classical variogram estimator, under several assumptions about the actual variogram.Cumulant; Intrinsically stationary process; Kronecker product; Quadratic form; Spatial design matrix; Variogram

Ill-posed Problems and Instruments' Weakness

Potscher (Econometrica, 2002) has pointed out that several estimation problems in econometrics are ill-posed. This paper further studies the nature of ill-posed problems in parametric models. Our starting point is that both parameters and estimators may be seen as maps from the manifold of density functions to an m-dimensional Euclidean space, and we investigate the properties that these maps have to transmit perturbations. In the special case of structural equations models, we argue that this framework provides coherent measures of instruments' weaknessIll-posed Problems, Weak Instruments, Parametric Models

Spatial design matrices and associated quadratic forms: structure and properties

The paper provides significant simplifications and extensions of results obtained by Gorsich, Genton, and Strang (J. Multivariate Anal. 80 (2002) 138) on the structure of spatial design matrices. These are the matrices implicitly defined by quadratic forms that arise naturally in modelling intrinsically stationary and isotropic spatial processes. We give concise structural formulae for these matrices, and simple generating functions for them. The generating functions provide formulae for the cumulants of the quadratic forms of interest when the process is Gaussian, second-order stationary and isotropic. We use these to study the statistical properties of the associated quadratic forms, in particular those of the classical variogram estimator, under several assumptions about the actual variogram.Cumulant, Intrinsically Stationary Process, Kronecker

Computationally efficient recursions for top-order invariant polynomials with applications

The top-order zonal polynomials Ck(A),and top-order invariant polynomials Ck1,...,kr(A1,...,Ar)in which each of the partitions of ki,i = 1,..., r,has only one part, occur frequently in multivariate distribution theory, and econometrics - see, for example Phillips (1980, 1984, 1985, 1986), Hillier (1985, 2001), Hillier and Satchell (1986), and Smith (1989, 1993). However, even with the recursive algorithms of Ruben (1962) and Chikuse (1987), numerical evaluation of these invariant polynomials is extremely time consuming. As a result, the value of invariant polynomials has been largely confined to analytic work on distribution theory. In this paper we present new, very much more efficient, algorithms for computing both the top-order zonal and invariant polynomials. These results should make the theoretical results involving these functions much more valuable for direct practical study. We demonstrate the value of our results by providing fast and accurate algorithms for computing the moments of a ratio of quadratic forms in normal random variables.

Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors

Using generating functions, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other symmetric functions (power-sum and elementary symmetric functions, Ruben (1962), Hillier, Kan, and Wang (2009)). Typically, in a recursion of this type the k-th object of interest, dk say, is expressed in terms of all lower-order dj ’s. In Hillier, Kan, and Wang (2009) we pointed out that, in the case of top-order zonal polynomials (and generalizations of them), a shorter (i.e., fixed length) recursion can be deduced. The present paper shows that the argument in Hillier, Kan, and Wang (2009) generalizes to a large class of objects/generating functions. The results thus obtained are then applied to various problems involving quadratic forms in noncentral normal vectors Keywords; generating functions, invariant polynomials, non-central normal distribution, recursions, symmetric functions, zonal polynomials

Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors

Using generating functions, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other symmetric functions (power-sum and elementary symmetric functions, Ruben, Hillier, Kan, and Wang). Typically, in a recursion of this type the k -th object of interest, d k say, is expressed in terms of all lower-order d j's. In Hillier, Kan, and Wang we pointed out that, in the case of top-order zonal polynomials (and generalizations of them), a shorter (i.e., fixed length) recursion can be deduced. The present paper shows that the argument in generalizes to a large class of objects/generating functions. The results thus obtained are then applied to various problems involving quadratic forms in noncentral normal vectors.

Exact properties of the conditional likelihood ratio test in an IV regression model

This paper was revised in May 2007. For a simplified structural equation/IV regression model with one right-side endogenous variable, we obtain the exact conditional distribution function for Moreira's (2003) conditional likelihood ratio (CLR) test. This is then used to obtain the critical value function needed to implement the CLR test, and reasonably comprehensive graphical versions of the function are provided for practical use. The analogous functions are also obtained for the case of testing more than one right-side endogenous coefficient, but only for an approximation to the true likelihood ratio test. We then go on to provide an exact analysis of the power functions of the CLR test, the Anderson-Rubin test, and the LM test suggested by Kleibergen (2002). The CLR test is shown to clearly conditionally dominate the other two tests for virtually all parameter configurations, but none of these test is either inadmissible or uniformly superior to the other two.