The exact moments of the least squares estimator for the autoregressive model

Exact mean and variance of the least squares estimate of the stationary first-order autoregressive coefficient, i.e., β in y_t=α+βx_t+u_t are evaluated algebraically as well as numerically. It turns out that the least squares estimate is seriously biased for the sample of two-digits sizes typically dealt with in econometrics if the mean of the process is unknown, i.e., if the equation has a non-zero intercept (α≠0). Kendall's approximation to the mean and Barlett's approximation to the variance are shown to be fairly good. Also, our numerical results confirm Orcutt and Winokur's (Econometrica, Vol. 37) based on Monte Carlo experiments.     

CUE with many weak instruments and nearly singular design

This paper analyzes many weak moment asymptotics under the possibility of similar moments. The possibility of highly related moments arises when there are many of them. Knight and Fu (2000) designate the issue of similar regressors as the “nearly singular” design in the least squares case. In the nearly singular design, the sample variance converges to a singular limit term. However, Knight and Fu (2000) assume that on the nullspace of the limit term, the difference between the sample variance and the singular matrix converges in probability to a positive definite matrix when multiplied by an appropriate rate. We consider specifically Continuous Updating Estimator (CUE) with many weak moments under nearly singular design. We show that the nearly singular design affects the form of the limit of the many weak moment asymptotics that is introduced by Newey and Windmeijer (2009a). However, the estimator is still consistent and the Wald test has the standard χ²${\chi }^2$ limit.     

A comparison of the Box-Cox maximum likelihood estimator and the non-linear two-stage least squares estimator

This paper investigates the limiting behaviour of the ‘maximum likelihood estimator’(MLE) based on normality, as well as the nonlinear two-stage least squares estimator (NL2S), for the i.i.d. and regression models in which the Box-Cox transformation is applied to the dependent variable. Since the transformed variable cannot in general be normally distributed, the untransformed variable is assumed to have a two-parameter gamma distribution. Tables of probability limits and asymptotic variance demonstrate that, in this case, the inconsistency of the ‘normal MLE’ is often quite pronounced, while the NL2S is consistent and typically well behaved.     

The nonlinear limited-information maximum- likelihood estimator and the modified nonlinear two-stage least-squares estimator

Several limited-information type estimators of the nonlinear simultaneous equation model are considered and their asymptotic covariance matrices are compared. Amemiya (1974) proposed the general class of nonlinear two-stage least-squares estimators. In this paper, its two specific members are considered and, in addition, the nonlinear limited-information maximum- likelihood estimator and the modified nonlinear two-stage least-squares estimator are proposed. Both are shown to be asymptotically more efficient than the nonlinear two-stage least-squares estimator, and the second has the advantage of being computationally simple.     

Stationarity and the existence of moments of a family of GARCH processes

This paper investigates some structural properties of a family of GARCH processes. A simple sufficient condition for the existence of the αδ-order stationary solution of the processes is derived, where α∈(0,1] and δ>0. The solution is strictly stationary and ergodic, and the causal expansion of the family of GARCH processes is also established. Furthermore, the necessary and sufficient condition for the existence of the moments is obtained. The technique used in this paper for the moment conditions is different from that used in He and Terasvirta (J. Econom. 92 (1999a) 173), and avoids the assumption that the process started at some finite value infinitely many periods ago. Moreover, the conditions for the strict stationarity of the model and the existence of its moments are simple to check and should prove useful in practice.     

Another look at the naive estimator in a regression model

We consider the linear regression model where only a particular linear function of the dependent variables is observed, Stahlecker and Schmidt (1987) proposed a naive least squares (LS) estimator for regression coefficients in such a case. In this note we represent their estimator as a general ridge estimator. This observation leads to a view different from the previous work and provides an easy way of obtaining many important properties of the naive LS estimator. Our approach also gives some insight into the relationship between the naive LS estimator and the generalized least squares estimator.     

On the strong uniform consistency of a new kernel density estimator

Summary A new multivariate kernel probability density estimator is introduced and its strong uniform consistency is proved under certain regularity conditions. This result is then applied particularly to a kernel estimator whose mean vector and covariance matrix areμ _n andV _n, respectively, whereμ _n is an unspecified estimator of the mean vector andV _n, up to a multiplicative constant, the sample covariance matrix of the probability density to be estimated, respectively. Work supported by the Natural Sciences and Engineering Research Council of Canada and by the Fonds F.C.A.R. of the Province of Quebec.     

On the asymptotic optimality of the LIML estimator with possibly many instruments

We consider the estimation of the coefficients of a linear structural equation in a simultaneous equation system when there are many instrumental variables. We derive some asymptotic properties of the limited information maximum likelihood (LIML) estimator when the number of instruments is large; some of these results are new as well as old, and we relate them to results in some recent studies. We have found that the variance of the limiting distribution of the LIML estimator and its modifications often attain the asymptotic lower bound when the number of instruments is large and the disturbance terms are not necessarily normally distributed, that is, for the micro-econometric models of some cases recently called many instruments and many weak instruments.     

On efficiency of the ratio estimator

Summary The exact mean square error for the ratio estimator of a finite population total based on simple random sampling without replacement is shown to have an expected value less than that of the variance of the ratio estimator based on Midzuno’s scheme, under a usual super-population model.     

On classes of estimators of the variance function of a linear estimator

Summary The variance function of a linear estimator can be expressed into a quadratic form. The present paper presents classes of estimators of this quadratic form along the lines implicitly suggested byHorvitz andThompson [1952] while formulating the classes of linear estimators. Accordingly it is noted that there exist nine principal classes of estimators out of which one principal class is examined in detail. Furthermore to illustrate the theory an example is considered where the expression for a unique estimator variance of the best estimator in theT ₁ class is derived.     

On the variance of the ratio estimator

Summary In an earlier paper [Rao 1966] an exact expression for the variance of the ratio estimator under theMidzuno-Sen sampling scheme is obtained and here we study some of the interesting properties of the coefficients involved in this expression which depend on the auxiliary information. Use of these coefficients is made of in finding out an exact expression for the Bias and Mean Square Error of the ratio estimator under Simple Random Sampling With-Out Replacement (SRSWOR) scheme.     

On the efficiency of the Cochrane-Orcutt estimator

Using analytic approximations, we reconcile some radically contradictory evidence and resolve an interesting paradox that occurs in a simple linear model with autocorrelated disturbances. In general, the behavior of conventional coefficient estimators is quite sensitive to the specification of the exogenous variables, or, equivalently, to whether the marginal efficiency or the conditional efficiency of the coefficient estimators is being compared.     

K-matrix-class estimators and the full-information maximum-likelihood estimator as a special case

This article presents a unified treatment of simultaneous system estimation. A general class of full-information estimators is proposed, called K-matrix-class (KMC). It is shown that the K-matrix-class includes both full-information maximum-likelihood and three-stage least- squares estimators as special cases and that the k-class can be regarded as a subclass of the K-matrix-class. Conditions under which KMC estimators are consistent (similar to those of the k-class estimators) are given. Furthermore, as a full information-generalization of the double k-class estimators, the double K-matrix-class estimators (DKMC) are proposed.     

Some properties of the LIML estimator in a dynamic panel structural equation

We investigate the finite sample and asymptotic properties of the within-groups (WG), the random-effects quasi-maximum likelihood (RQML), the generalized method of moment (GMM) and the limited information maximum likelihood (LIML) estimators for a panel autoregressive structural equation model with random effects when both T (time-dimension) and N (cross-section dimension) are large. When we use the forward-filtering due to Alvarez and Arellano (2003), the WG, the RQML and GMM estimators are significantly biased when both T and N are large while T/N is different from zero. The LIML estimator gives desirable asymptotic properties when T/N converges to a constant.     

A property of Stein's estimator for the mean of a multivariate normal distribution

A uniform bound on the risk (under squared error loss) of Stein's estimator Ψ₁ for the mean of the multivariate normal distribution is given. Using the bound, the asymptotic behaviour of the risk of Ψ₁ under a Bayesian assumption is obtained.     

Robust trend inference with series variance estimator and testing-optimal smoothing parameter

The paper develops a novel testing procedure for hypotheses on deterministic trends in a multivariate trend stationary model. The trends are estimated by the OLS estimator and the long run variance (LRV) matrix is estimated by a series type estimator with carefully selected basis functions. Regardless of whether the number of basis functions K is fixed or grows with the sample size, the Wald statistic converges to a standard distribution. It is shown that critical values from the fixed-K asymptotics are second-order correct under the large-K asymptotics. A new practical approach is proposed to select K that addresses the central concern of hypothesis testing: the selected smoothing parameter is testing-optimal in that it minimizes the type II error while controlling for the type I error. Simulations indicate that the new test is as accurate in size as the nonstandard test of Vogelsang and Franses (2005) and as powerful as the corresponding Wald test based on the large-K asymptotics. The new test therefore combines the advantages of the nonstandard test and the standard Wald test while avoiding their main disadvantages (power loss and size distortion, respectively).     

On the existence of bivariate moments of lower order given the existence of moments of higher order

Dit artikel geeft een voorbeeld van een twee-dimensionale waarschijnlijkheidsdcht-heid met de eigenschap, dat het bestaan van momenten van de vorm E(g^ky^l ) niet het bestaan garandeert van alle momenten van lagere orde, E(g^myⁿ ) met 0myⁿ ) met 0 < m < k en 0 < n < l is, dat de drie momenten E(x^k), E(y^l ) en E(x^ky^l ) bestaan, alsmede een iets algemener resultaat.