Seemingly unrelated nonlinear regressions

The article considers the estimation of the parameters of a set of nonlinear regression equations when the responses are contemporaneously but not serially correlated. Conditions are set forth such that the estimator obtained is strongly consistent, asymptotically normally distributed, and asymptotically more efficient than the single-equation least squares estimator. The methods presented allow estimation of the parameters subject to nonlinear restrictions across equations. The article includes a discussion of methods to perform the computations and a Monte Carlo simulation.     

On the relative efficiency of estimators which include the initial observations in the estimation of seemingly unrelated regressions with first-order autoregressive disturbances

The generalized least squares estimator for a seemingly unrelated regressions model with first-order vector autoregressive disturbances is outlined, and its efficiency is compared with that of an approximate generalized least squares estimator which ignores the first observation. A scalar index for the loss of efficiency is developed and applied to a special case where the matrix of autoregressive parameters is diagonal and the regressors are smooth. Also, for a more general model, a Monte Carlo study is used to investigate the relative efficiencies of various estimators. The results suggest that Maeshiro (1980) has overstated the case for the exact generalized least squares estimator, because, in many circumstances, it is only marginally better than the approximate generalized least squares estimator.     

A distribution-free least squares estimator for censored linear regression models

This paper describes a method for estimating simultaneously the parameter vector of the systematic component and the distribution function of the random component of a censored linear regression model. The estimator is obtained by minimizing the sum of the squares of the differences between the observed values of the dependent variable and the corresponding expected values of this variable according to the estimated parameter vector and distribution function. The resulting least squares parameter estimator incorporates information on the distribution of the random component of the regression model that is available from the estimation sample. Hence, it may often be more efficient than are parameter estimators that do not use such information. The results of numerical experiments with the least squares estimator tend to support this hypothesis.     

Single-equation estimators and aggregation restrictions when equations have the same sets of regressors

It is well known that linear equations subject to cross-equation aggregation restrictions can be ‘stacked’ and estimated simultaneously. However, if every equation contains the same set of regressors, a number of single-equation estimation procedures can be employed. The applicability of ordinary least squares is widely recognized but the article demonstrates that the class of applicable estimators is much broaders than OLS. Under specified conditions, the class includes instrumental variables, generalized least squares, ridge regression, two-stage least squares, k-class estimators, and indirect least squares. Transformations of the original equations and other related matters are discussed also.     

Mean estimation bias in least squares estimation of autoregressive processes

Estimation of the parameters of an autoregressive process with a mean that is a function of time is considered. Approximate expressions for the bias of the least squares estimator of the autoregressive parameters that is due to estimating the unknown mean function are derived. For the case of a mean function that is a polynomial in time, a reparameterization that isolates the bias is given. Using the approximate expressions, a method of modifying the least squares estimator is proposed. A Monte Carlo study of the second-order autoregressive process is presented. The Monte Carlo results agree well with the approximate theory and, generally speaking, the modified least squares estimators performed better than the least squares estimator. For the second-order process we also considered the empirical properties of the estimated generalized least squares estimator of the mean function and the error made in predicting the process one, two and three periods in the future.     

The asymptotic behaviour of the estimated generalized least squares method in the linear regression model

Abstract  In the linear regression model the generalized least squares (GLS) method is only applicable if the covariance matrix of the errors is known but for a scalar factor. Otherwise an estimator for this matrix has to be used. Then we speak of the estimated generalized least squares (EGLS) method. In this paper the asymptotic behaviour of both methods is compared. Results are applied to some standard models commonly used in econometrics     

A consistent estimator in general functional errors-in-variables models

The problem of estimation in nonlinear functional errors-in-variables model is considered. A modified least squares estimator is studied, its consistency and asymptotic normality is established. Simulation results are also presented showing the performance of the estimator in comparison with the naive ordinary least squares estimator. Received: June 1999     

Ridge regression estimation of the Rotterdam model

The best guesses of unknown coefficients specified in Theil's model of introspection are like predictions and not like de Finetti's prevision and therefore not the values taken by random variables. Constrained least squares procedures can be formulated which are free of these difficulties. The ridge estimator is a simple version of a constrained least squares estimator which can be made operational even when little prior information is available. Our operational ridge estimators are nearly minimax and are not less stable than least squares in the presence of high multicollinearity. Finally, we have presented the ridge estimates for the Rotterdam demand model.     

An application of approximate finite sample results to parameter estimation in a linear errors-in-variables model

In the simple errors-in-variables model the least squares estimator of the slope coefficient is known to be biased towards zero for finite sample size as well as asymptotically. In this paper we suggest a new corrected least squares estimator, where the bias correction is based on approximating the finite sample bias by a lower bound. This estimator is computationally very simple. It is compared with previously proposed corrected least squares estimators, where the correction aims at removing the asymptotic bias or the exact finite sample bias. For each type of corrected least squares estimators we consider the theoretical form, which depends on an unknown parameter, as well as various feasible forms. An analytical comparison of the theoretical estimators is complemented by a Monte Carlo study evaluating the performance of the feasible estimators. The new estimator proposed in this paper proves to be superior with respect to the mean squared error.     

Some Decompositions of OLSEs and BLUEs Under a Partitioned Linear Model

We consider in this paper a partitioned linear model { y , X ₁ β ₁ + X ₂ β ₂ , σ ² σ } and two corresponding small models { y , X ₁ β ₁ , σ ² σ } and { y , X ₂ β ₂ , σ ² σ } . We derive necessary and sufficient conditions for (i) the ordinary least squares estimator under the full model to be the sum of the ordinary least squares estimators under the two small models; (ii) the best linear unbiased estimator under the full model to be the sum of the best linear unbiased estimators under the two small models; (iii) the best linear unbiased estimator under the full model to be the sum of the ordinary least squares estimators under the two small models. The proofs of the main results in this paper also demonstrate how to use the matrix rank method for characterizing various equalities of estimators under general linear models.     

The effect of a Durbin–Watson pretest on confidence intervals in regression

Consider a linear regression model and suppose that our aim is to find a confidence interval for a specified linear combination of the regression parameters. In practice, it is common to perform a Durbin–Watson pretest of the null hypothesis of zero first‐order autocorrelation of the random errors against the alternative hypothesis of positive first‐order autocorrelation. If this null hypothesis is accepted then the confidence interval centered on the ordinary least squares estimator is used; otherwise the confidence interval centered on the feasible generalized least squares estimator is used. For any given design matrix and parameter of interest, we compare the confidence interval resulting from this two‐stage procedure and the confidence interval that is always centered on the feasible generalized least squares estimator, as follows. First, we compare the coverage probability functions of these confidence intervals. Second, we compute the scaled expected length of the confidence interval resulting from the two‐stage procedure, where the scaling is with respect to the expected length of the confidence interval centered on the feasible generalized least squares estimator, with the same minimum coverage probability. These comparisons are used to choose the better confidence interval, prior to any examination of the observed response vector.     

Variable selection via composite quantile regression with dependent errors

We propose composite quantile regression for dependent data, in which the errors are from short‐range dependent and strictly stationary linear processes. Under some regularity conditions, we show that composite quantile estimator enjoys root‐n consistency and asymptotic normality. We investigate the asymptotic relative efficiency of composite quantile estimator to both single‐level quantile regression and least‐squares regression. When the errors have finite variance, the relative efficiency of composite quantile estimator with respect to the least‐squares estimator has a universal lower bound. Under some regularity conditions, the adaptive least absolute shrinkage and selection operator penalty leads to consistent variable selection, and the asymptotic distribution of the non‐zero coefficient is the same as that of the counterparts obtained when the true model is known. We conduct a simulation study and a real data analysis to evaluate the performance of the proposed approach.     

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.     

Exponential deconvolution: two asymptotically equivalent estimators

Two isotonic estimators for the distribution function in a specific deconvolution model, the exponential deconvolution model, are considered. The first estimator is a least squares projection of a naive estimator for the distribution function on the set of distribution functions. The second estimator is the well known maximum likelihood estimator. The two estimators are shown to be first order asymptotically equivalent at a fixed point.     

A note on amemiya's partially generalized least squares

This note explores the relationship between the generalized least squares estimator and Amemiya's partially generalized least squares estimator and establishes the conditions under which the two estimators are equal.     

Concentration behavior of the penalized least squares estimator

Consider the standard nonparametric regression model and take as estimator the penalized least squares function. In this article, we study the trade‐off between closeness to the true function and complexity penalization of the estimator, where complexity is described by a seminorm on a class of functions. First, we present an exponential concentration inequality revealing the concentration behavior of the trade‐off of the penalized least squares estimator around a nonrandom quantity, where such quantity depends on the problem under consideration. Then, under some conditions and for the proper choice of the tuning parameter, we obtain bounds for this nonrandom quantity. We illustrate our results with some examples that include the smoothing splines estimator.     

Model averaging based on James–Stein estimators

Existing model averaging methods are generally based on ordinary least squares (OLS) estimators. However, it is well known that the James–Stein (JS) estimator dominates the OLS estimator under quadratic loss, provided that the dimension of coefficient is larger than two. Thus, we focus on model averaging based on JS estimators instead of OLS estimators. We develop a weight choice method and prove its asymptotic optimality. A simulation experiment shows promising results for the proposed model average estimator.     

Functional Coefficient Cointegration Models Subject to Time–Varying Volatility with an Application to the Purchasing Power Parity

This paper analyses functional coefficient cointegration models with both stationary and non‐stationary covariates, allowing time‐varying (unconditional) volatility of a general form. The conventional kernel weighted least squares (KLS) estimator is subject to potential efficiency loss, and can be improved by an adaptive kernel weighted least squares (AKLS) estimator that adapts to heteroscedasticity of unknown form. The AKLS estimator is shown to be as efficient as the oracle generalized kernel weighted least squares estimator asymptotically, and can achieve significant efficiency gain relative to the KLS estimator in finite samples. An illustrative example is provided by investigating the Purchasing Power Parity hypothesis.     

Indirect inference in structural econometric models

This paper considers parametric inference in a wide range of structural econometric models. It illustrates how the indirect inference principle can be used in the inference of these models. Specifically, we show that an ordinary least squares (OLS) estimation can be used as an auxiliary model, which leads to a method that is similar in spirit to a two-stage least squares (2SLS) estimator. Monte Carlo studies and an empirical analysis of timber sale auctions held in Oregon illustrate the usefulness and feasibility of our approach.     

Efficiency of the OLSE for regressions on two-dimensional grids with sinusoidal regressors and spatially correlated errors

For spatial regressions with sinusoidal surfaces, the ordinary least squares estimator (OLSE) is shown to be asymptotically as efficient as the generalized least squares estimator (GLSE) in that the covariance matrices of the two estimators have the same nontrivial limit under the same normalization.