Jan R. Magnus

On Differentiating Eigenvalues and Eigenvectors

Let X 0 be a square matrix (complex or otherwise) and u 0 a (normalized) eigenvector associated with an eigenvalue λ o of X 0 , so that the triple ( X 0 , u 0 , λ 0 ) satisfies the equations Xu = λ u , null. We investigate the conditions under which unique differentiable functions λ( X ) and u ( X ) exist in a neighborhood of X 0 satisfying λ( X 0 ) = λ O , u ( X 0 ) = u 0 , X = λ u , and null. We obtain the first and second derivatives of λ( X ) and the first derivative of u ( X ). Two alternative expressions for the first derivative of λ( X ) are also presented.

Multivariate error components analysis of linear and nonlinear regression models by maximum likelihood

Abstract This paper analyses the estimation by maximum likelihood of multivariate error component models, linear and nonlinear, under various assumptions on the errors. Special attention is given to testing and imposing positivity constraints, and equality and inequality restrictions. Also, the appropriate asymptotic covariance matrices are derived, enabling us, inter alia, to perform Wald tests of various relevant hypotheses.

Matrix differential calculus with applications to simple, Hadamard, and Kronecker products

Abstract Several definitions are in use for the derivative of an m × p matrix function F(X) with respect to its n × q matrix argument X. We argue that only one of these definitions is a viable one, and that to study smooth maps from the space of n × q matrices to the space of m × p matrices it is often more convenient to study the map from nq-space to mp-space. Also, several procedures exist for a calculus of functions of matrices. It is argued that the procedure based on differentials is superior to other methods of differentiation, and leads inter alia to a satisfactory chain rule for matrix functions.

L-structured matrices and linear matrix equations ∗

Conditions for the existence of solutions, and the general solution of linear matrix equations are given, when it is known a priori that the solution matrix has a given structure (e.g. symmetric, triangular, diagonal). This theory is subsequently extended to matrix equations that are linear in several unknown ‘structured’ matrices, and to partitioned matrix equations.

Consistent maximum-likelihood estimation with dependent observations : The general (non-normal) case and the normal case

Abstract In this paper we aim to establish intuitively appealing and verifiable conditions for the existence and weak consistency of ML estimators in a multi-parameter framework, assuming neither the independence nor the identical distribution of the observations. The paper has two parts. In the first part (Theorems 1 and 2) we assume that the joint density of the observations is known (except for the values of a finite number of parameters to be estimated), but we do not specify this distribution. In the second part (Theorems 3–6), we do specify the distribution and assume joint normality (but not independence) of the observations. Some examples are also provided.

On tests and significance in econometrics

Different aims of testing are investigated: theory testing, validity testing, simplification testing, and decision making. Different testing methodologies may serve these aims. In particular, the approaches of Fisher and Neyman-Pearson are considered. We discuss the meaning of statistical significance. Significance tests in the Journal of Econometrics are evaluated. The paper concludes with a challenge to ascertain the impact of statistical testing on economic thought.

The exact multi-period mean-square forecast error for the first-order autoregressive model

The finite-sample behaviour of the multi-period least-squares forecast is considered in the simple normal autoregressive model yt = βyt–1 + ut where ‖β‖ < 1. Necessary and sufficient conditions are established for the existence of the forecast bias and the mean-square forecast error (MSFE) and an exact expression for the MSFE is given. Exact numerical results are obtained for both the stationary and the fixed start-up case. Our main conclusions are that for small values of β the MSFE is a decreasing function of the number of forecast periods, and that the behaviour of the MSFE in the stationary and the fixed start-up case is very similar, except for values of ‖β‖ close to 1.

On levies to reduce the nitrogen surplus: The case of Dutch pig farms

Pig farms in the Netherlands pay a zero or low price for using the environment. As a consequence, the environment is overused. The Dutch government wants to reduce the emissions of nitrogen and phosphorus. Possible instruments are regulation and levies. In this study a levy on feed and a levy on the nitrogen surplus are investigated, by incorporating a bad output in the production model. The model is estimated using panel data of Dutch pig farms over the period 1975–1989. Levies on nitrogen turn out to be more cost-effective than levies on feed.

Asymptotic Normmality of Maximum Likelihood Estimators Obtained from Normally Distributed but Dependent Observations

In this article we aim to establish intuitively appealing and verifiable conditions for the first-order efficiency and asymptotic normality of ML estimators in a multi-parameter framework, assuming joint normality but neither the independence nor the identical distribution of the observations. We present five theorems (and a large number of lemmas and propositions), each being a special case of its predecessor.

The exact multi-period mean-square forecast error for the first-order autoregressive model with an intercept

Abstract We study the exact finite-sample behaviour of the mean-square forecast error ( MSFE ) of multi-period least-squares forecasts in the normal autoregressive model y t = α + β y t - 1 + u t . We obtain necessary and sufficient conditions for the existence of the MSFE and give an exact expression which we use to obtain numerical results for both the stationary and the fixed start-up model. We conclude, inter alia, that the behaviour of the MSFE in the model with intercept can be very different from that in the model without intercept, especially when β is close to unity.