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This paper considers the problem of obtaining efficient estimates for the parameters of a system of M regression equations. The disturbance terms of this system are assumed to be related by both serial and contemporaneous correlation. Under the further assumption that the serial correlation is a first order autoregressive process, the paper develops an estimator that is consistent and has the same asymptotic normal distribution as the Aitken estimator which assumes the covariance matrix to be known. The paper concludes with a discussion of some alternative covariance specifications and points out certain difficulties with the standard single equation procedures for handling autoregressive schemes. [ABSTRACT FROM AUTHOR]

Most efforts to analyze and forecast interregional population growth and distribution are hampered by the general paucity of reliable data on interregional migration. This has led demographers to rely on crudely constructed measures of net migration. Recent efforts to express the population growth process in matrix form, however, suggest a means whereby the schedule of growth that operates in an interregional system may be estimated solely on the basis of historical data on interregional population distributions. Such a method is described and illustrated in this paper. The method estimates a growth regime which, when applied to the initial interregional population distribution in the time series, most closely reproduces the remaining population distributions in the historical sequence.

In regression analysis with heteroscedastic and/or correlated errors, the usual assumption is that the covariance matrix Sigma of the errors is completely specified, except perhaps for a scalar multiplier. This condition is relaxed in this paper by assuming only that Sigma has a certain pattern; for example, that Sigma is diagonal or partitionable into a diagonal matrix of sub-matrices. The method used for estimating Sigma is the standard procedure of equating certain quadratic forms of the observations (in this case, squares and products of residuals from regression) to their expectations, and solving for the unknown variances and covariances. A numerical example illustrates the method. [ABSTRACT FROM AUTHOR]

A linear programming food price index permits the possibility of routine substitutions in response to price changes, while a fixed weight index does not. This paper cites a variety of evidence indicating that consumers do trade off among foods in response to both long-term and seasonal price changes. Implications of this for food price indexes are discussed. A linear programming food price index is then developed, based on an empirical menu-planning model, and compared with corresponding fixed weight price indexes over a nine-month period. The linear programming index shows a significantly smaller increase in food prices over the nine-month period than the fixed weight index with initial period weights. [ABSTRACT FROM AUTHOR]

This paper deals with a multivariate extension of Friedman's chi[sup 2, sub tau]-test. A rank permutation distribution and the large sample properties of the criterion are studied. The asymptotic relative efficiency (A.R.E.) for a sequence of translation alternatives is studied and bounds are given for certain special cases. It is shown that, under specified conditions, the A.R.E. of this test with respect to the likelihood ratio test is largest, when the block dispersion matrices differ and can be greater than unity when the differences are large. [ABSTRACT FROM AUTHOR]

The purpose of this paper is the presentation of formulae for obtaining matrix derivatives of the second order to use in making tests for maxima and minima. The theory of such second order derivatives is presented. These formulae require the rearrangment of the parameter elements in vector form and the transformed results feature Kronecker products, which have certain desirable properties. Application is made to several types of problems. [ABSTRACT FROM AUTHOR]

Matrix differentiation, the procedure of finding partial derivatives of the elements of a matrix function with respect to the elements of the argument matrix, is the subject matter of this paper. The method followed here amounts to linking the differentials of the matrix function and the argument matrix, and then identifying matrices of partial derivatives. The basic assumption made is mathematical independence of the elements of the argument matrix. Matrix functions are divided into two categories: Kronecker matrix products and non-Kronecker (= ordinary) matrix products. Three definitions of partial derivatives matrices are used, to be denoted by D[sub 1], D[sub 2] and D[sub 3] in this abstract. D[sub 2] and D[sub 3] are applied to Kronecker products, D[sub 1] is applied to ordinary products. This is a matter of efficiency only. D[sub 1] is developed first. Transforming matrices into column vectors turns out to be very convenient. D[sub 2] and D[sub 3] are developed then. D[sub 3] is a generalisation of D[sub 1]. [ABSTRACT FROM AUTHOR]

A method is proposed to eliminate trends from a sample from a nonstationary system of linear stochastic difference equations. The autocovariance matrix and the spectral density matrix of the detrended component of the sample are derived. The latter matrix turns out to have the same form as the spectral density matrix for a stationary system when expressed in terms of the roots of the system. Since the parameters of the system are assumed known throughout the paper, the problem of statistical inference does not arise. [ABSTRACT FROM AUTHOR]

This paper contains a generalization of the Gauss Markov Theorem based on the properties of the generalized inverse of a matrix as defined by Penrose. A minimum variance vector estimate x of a parameter vector x is given for the linear model of less than full rank. Since linear unbiased estimates may not always exist for this case the unbiased constraint is replaced by the more general constraint that the norm ||E(x) --x|| is minimized. [ABSTRACT FROM AUTHOR]

Formulas for confidence, prediction, and tolerance regions for the multivariate normal distribution for the various cases of known and unknown mean vector and covariance matrix are assembled for easy reference in this expository paper. Tables are provided for the bivariate case. [ABSTRACT FROM AUTHOR]