Comparisons of estimators are made on the basis of their mean squared errors and their concentrations of probability computed by means of asymptotic expansions of their distributions when the disturbance variance tends to zero and alternatively when the sample size increases indefinitely. The estimators include k-class estimators (limited information maximum likelihood, two-stage least squares, and ordinary least squares) and linear combinations of them as well as modifications of the limited information maximum likelihood estimator and several Bayes' estimators. Many inequalities between the asymptotic mean squared errors and concentrations of probability are given. Among medianunbiasedestimators, the limited information maximum likelihood estimator dominates the median-unbiased fixed k-class estimator.

In this paper we bring together those properties of the Kronecker product, the vec operator, and 0-1 matrices which in our view are of interest to researchers and students in econometrics and statistics. The treatment of Kronecker products and the vec operator is fairly exhaustive; the treatment of 0–1 matrices is selective. In particular we study the “commutation” matrix K (defined implicitly by K vec A = vec A′ for any matrix A of the appropriate order), the idempotent matrix N = ½ (I + K), which plays a central role in normal distribution theory, and the “duplication” matrix D, which arises in the context of symmetry. We present an easy and elegant way (via differentials) to evaluate Jacobian matrices (first derivatives), Hessian matrices (second derivatives), and Jacobian determinants, even if symmetric matrix arguments are involved. Finally we deal with the computation of information matrices in situations where positive definite matrices are arguments of the likelihood function.

The primary concern is to establish a fairly general framework in which estimators resulting from estimating equations gnθ = 0 are not consistent. This leads on to consistency by an intuitive route. Asymptotic distributions of consistent estimators are also touched upon, and the results are applied to various examples.

Under normality, we obtain higher-order approximations to the distributions of the periodogram and related statistics. Our approach is based on the theorem which decomposes the periodogram into the sum of two independent random variables. It is seen that this decomposition enables us to study fairly closely the higher-order properties of not only the periodogram, but also periodogram-based statistics such as the estimators of the spectrum and prediction error variance. Some of the approximation results are graphically presented together with the exact results based on simulations.

The strong consistency of regression quantile statistics (Koenker and Bassett [4]) in linear models with iid errors is established. Mild regularity conditions on the regression design sequence and the error distribution are required. Strong consistency of the associated empirical quantile process (introduced in Bassett and Koenker [1]) is also established under analogous conditions. However, for the proposed estimate of the conditional distribution function of Y, no regularity conditions on the error distribution are required for uniform strong convergence, thus establishing a Glivenko-Cantelli-type theorem for this estimator.

This article proposes the use of spectral methods to pool cross-sectional replications (N) of time series data (T) for time series analysis. Spectral representations readily suggest a weighting scheme to pool the data. The asymptotically desirable properties of the resulting estimators seem to translate satisfactorily into samples as small as T = 25 with N = 5. Simulation results, Monte Carlo results, and an empirical example help confirm this finding. The article concludes that there are many empirical situations where spectral methods canbe used where they were previously eschewed.

Exact expressions are derived for the density function, variance, and kurtosis of a linear combination of the elements of a two-stage estimator for the coefficients in a single equation of a SUR system. The estimator is the first iterate in the iterative generalized least squares procedure described by Telser [14]. Our results generalize all previously known results for this estimator and, in certain special cases, also generalize some earlier exact results for Zellner's unrestricted covariance matrix estimator, to which it reduces in these special cases.

This article extends recent work on the Gaussian or quasi-maximum likelihood estimation of the parameters of a closed higher-order continuous time dynamic model by introducing exogenous variables into the model The method presented yields exact maximum likelihood estimates when the innovations are Gaussian and the exogenous variables are polynomials in time of degree not exceeding two, and it can be expected to yield very good estimates under more general conditions. It is applicable, in principle, to a system of any order with mixed stock and iow data. The precise formulas for its implementation are derived, in this article, for a second-order system in which both the endog-enous and exogenous variables are a mixture of stock and flow variables.

This paper derives the exact distribution and moments of the Stein-ruleestimator in a model where the disturbances follow a non-normal distribution of the Edgeworth or Gram-Charlier type. The results are achieved by combining the approach of Davis [4] for examining non-normality with the fractional calculus techniques of Phillips [11].

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.

This article proposes a small sample bounds test for equality between sets of coefficients in two linear regressions with unequal disturbance variances. The probability that our test is inconclusive is given under the null hypothesis. It is also shown that our test is more powerful than the Jayatissa test when the regression coefficients differ substantially.

This paper examines the sensitivity of the distributions of OLS and 2SLS estimators to the assumption of normality of disturbances in a structural equation with two included endogenous variables. The approach taken is that ofimposing Edgeworth distributed errors on the reduced form equations and deriving the pdf of the estimators via the technique of Davis [11]. The sensitivity of the pdf s to changes in the non-normality parameters, i.e., skewness and kurtosis i s examined via extensive numerical computations.

Invariant polynomials with matrix arguments have been defined by the theory of group representations, generalizing the zonal polynomials. They have developed as a useful tool to evaluate certain integrals arising in multivariate distribution theory, which were expanded as power series in terms of the invariant polynomials. Some interest in the polynomials has been shown by people working in the field of econometric theory. In this paper, we shall survey the properties of the invariant polynomials and their applications in multivariate distribution theory including related developments in econometrics.

We consider testing for autoregressive disturbances in the linear regression model with a lagged dependent variable. An approximation to the null distribution of the Durbin—Watson statistic is developed using small-disturbance asymptotics, and is used to obtain test critical values. We also obtain nonsimilar critical values for the Durbin—Watson and Durbin's h and t tests. Monte Carlo results are reported comparing the performances of the tests under the null and alternative hypotheses. The Durbin–Watson test is found to be more powerful and to perform more consistently than either of Durbin's tests under Ho.

In the context of a linear dynamic model with moving average errors, we consider a heteroscedastic model which represents an extension of the ARCH model introduced by Engle [4]. We discuss the properties of maximum likelihood and least squares estimates of the parameters of both the regression and ARCH equations, and also the properties of various tests of the model that are available. We do not assume that the errors are normally distributed.

Two different methods for pooling time series of cross section data are used by economists. The first method, described by Kmenta, is based on the idea that pooled time series of cross sections are plagued with both heteroskedasticity and serial correlation.The second method, made popular by Balestra and Nerlove, is based on the error components procedure where the disturbance term is decomposed into a cross-section effect, a time-period effect, and a remainder.Although these two techniques can be easily implemented, they differ in the assumptions imposed on the disturbances and lead to different estimators of the regression coefficients. Not knowing what the true data generating process is, this article compares the performance of these two pooling techniques under two simple setting. The first is when the true disturbances have an error components structure and the second is where they are heteroskedastic and time-wise autocorrelated.

First, the strengths and weaknesses of the two techniques are discussed. Next, the loss from applying the wrong estimator is evaluated by means of Monte Carlo experiments. Finally, a Bartletfs test for homoskedasticity and the generalized Durbin-Watson test for serial correlation are recommended for distinguishing between the two error structures underlying the two pooling techniques.

In the case of regression models, one robust estimation procedure which has recently emerged is that of functional least squares. The procedure is based on the use of characteristic functions for which the tail behavior is relected by the behavior of these functions at the origin. Its attraction is that it is applicable to situations where the distribution of the disturbances may be long-tailed and/or asymmetric.

This paper extends this theory to include a large class of regression models of importance in econometrics. Indeed the regression models considered here include lagged dependent variables and deterministic exogenous variables.