An examination of two-step estimators for models with lagged dependent variables and autocorrelated errors

The possible roles of the Durbin Equation and the first observation correction in improving the efficiency of parameter estimates in the lagged dependent variables-serial correlation model are examined. Unconstrained estimation of the Durbin Equation results in an estimate of ρ which is inefficient and its use in feasible generalized least squares does not provide asymptotically efficient estimates. Evidently, the first observation correction is a very important determinant of small sample properties in the present model. Asymptotically inefficient estimators which use a first observation correction frequently outperform Hatanaka's asymptotically efficient estimator in finite samples, essentially because it does not use the first observation.     

The behaviour of inconsistent instrumental variables estimators in dynamic systems with autocorrelated errors

To appropriately interpret time-series evidence when empirical relationships are incorrectly formulated, a general mis-specification framework is required. A linear, stationary, dynamic, simultaneous system with autoregressive errors is postulated to investigate instrumental variables ables estimators when the instruments are unknowingly correlated with the equation errors. The approach uses control variates (Hendry and Harrison, Journal of Econometrics, July 1974) to develop asymptotic distributions and exact moments for approximations to the econometric estimators. The accuracy of the asymptotic results for finite sample moments is corroborated by simulation. The analysis highlights the need for care in interpreting estimated equations and tests for predictive failure.     

Asymptotic properties of the efficient estimators for cointegrating regression models with serially dependent errors

In this paper, we analytically investigate three efficient estimators for cointegrating regression models: Phillips and Hansen’s [Phillips, P.C.B., Hansen, B.E., 1990. Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125] fully modified OLS estimator, Park’s [Park, J.Y., 1992. Canonical cointegrating regressions. Econometrica 60, 119–143] canonical cointegrating regression estimator, and Saikkonen’s [Saikkonen, P., 1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21] dynamic OLS estimator. We consider the case where the regression errors are moderately serially correlated and the AR coefficient in the regression errors approaches 1 at a rate slower than 1/T$1/T$, where T$T$ represents the sample size. We derive the limiting distributions of the efficient estimators under this system and find that they depend on the approaching rate of the AR coefficient. If the rate is slow enough, efficiency is established for the three estimators; however, if the approaching rate is relatively faster, the estimators will have the same limiting distribution as the OLS estimator. For the intermediate case, the second-order bias of the OLS estimator is partially eliminated by the efficient methods. This result explains why, in finite samples, the effect of the efficient methods diminishes as the serial correlation in the regression errors becomes stronger. We also propose to modify the existing efficient estimators in order to eliminate the second-order bias, which possibly remains in the efficient estimators. Using Monte Carlo simulations, we demonstrate that our modification is effective when the regression errors are moderately serially correlated and the simultaneous correlation is relatively strong.     

z-Transform and identification of linear econometric models with autocorrelated errors

The paper consists of two main parts. In the first part we derive the solution of systems of linear stochastic difference equations by means of thez-transform. In the second part thisz-transform is used to treat the problem of identification of linear econometric systems (the term econometric is used to stress the special aspects of the identification problem dealt with in econometrics). It is shown, that under suitable restrictions observationally equivalent structures are related by unimodular matrices. Using this result, we state (rank-) conditions which ensure, that the unimodular matrices are constant, such that the classical econometric identification theorems can be applied. These conditions are given for stationary errors in the general case as well as in the MA, AR and ARMA case.     

Finite sample properties of estimators for autoregressive moving average models

We analyze by simulation the properties of three estimators frequently used in the analysis of autoregressive moving average time series models for both nonseasonal and seasonal data. The estimators considered are exact maximum likelihood, exact least squares and conditional least squares. For samples of the size commonly found in economic applications, the estimators are compared in terms of bias, mean squared error, and predictive ability. The reliability of the usually calculated confidence intervals is assessed for the maximum likelihood estimator.     

Omitted variables,variability of estimated parameters and the appearance of autocorrelated disturbances

This paper shows that in a standard regression model with omitted variables, the OLS formula for the estimated variance matrix of the regression coefficients is more likely to underestimate the appropriate criterion of estimator reliability which is the Mean Square Errors matrix. Using examples of two and three regressor models, we show that overestimation, though possible, occurs in rather special cases. Throughout, our analysis is contrasted with that of Chaudhuri (1977) and clarifies some ambiguities of that paper. Finally, we disagree with Chaudhuri who distinguishes between the corresponding coefficients in the correct and the misspecified models. This distinction is inappropriate and leads to a misplaced criticism of some GLS variants when errors are serially correlated. A first-order Markov process is an inexact representation of serial correlation which is due to omitted regressors.     

Asymptotic properties of minimum contrast estimators for parameters of boolean models

Summary This paper presents a method for the estimation of parameters of random closed sets (racs’s) in ℝ ^d based on a single realization within a (large) convex sampling window. The essential idea first applied by Diggle (1981) in a special case consists in defining the estimation by minimizing a suitably defined distance (called contrast function) between the true and the empirical contact distribution function of the racs under consideration, where the most relevant case of Boolean models is discussed in details. The resulting estimates are shown to be strongly consistent (if the racs is ergodic) and asymptotically normal (if the racs is Boolean) when the sampling window expands unboundedly.     

Several efficient two-step estimators for the dynamic simultaneous equations model with autoregressive disturbances

Several asymptotically efficient methods are suggested on both the full and the limited information approach to estimate the simultaneous equations model in which the lagged endogenous variables and the autoregressive disturbances coexist. They are two-step procedures and do not involve iterations. A method is suggested also for the case where any portion of the autoregressive parameter matrix is specified to be zero. Since the consistency and efficiency depend upon the asymptotic, local identifiability, the necessary and sufficient condition is derived for it. It does not depend on the exclusion of the lagged endogenous variables.     

Efficiency of iterative estimators in the regression model with AR(1) disturbances

The mean square error approximation method of Nagar is applied to the iterated Prais-Winsten and (iterated) maximum likelihood estimators of regression coefficients in the model with AR(1) disturbances. Their mean square errors are found to equal that of the two-stage Prais-Winsten estimator at the second-order level of approximation.     

Some small sample properties of estimators and test statistics in the multivariate logit model

In this paper we compare three estimators for the multivariate logit model: two asymptotically efficient methods and a consistent method. The most interesting result is that at sample sizes of more than one hundred, the simple consistent estimator performs almost as well as the asymptotically efficient estimators.     

Assessing misspecified asset pricing models with empirical likelihood estimators

Hansen and Jagannathan (1997) compare misspecified asset pricing models based on least-square projections on a family of admissible stochastic discount factors. We extend their fundamental contribution by considering Minimum Discrepancy projections where misspecification is measured by a family of convex functions that take into account higher moments of asset returns. The Minimum Discrepancy problems are solved on dual spaces producing a family of estimators that captures the least-square problem as a particular case. We derive the asymptotic distributions of the estimators for the Cressie–Read family of discrepancies, and illustrate their use with an assessment of the Consumption Asset Pricing Model.     

Instrumental variables estimators of nonparametric models with discrete endogenous regressors

This paper discusses estimation of nonparametric models whose regressor vectors consist of a vector of exogenous variables and a univariate discrete endogenous regressor with finite support. Both identification and estimators are derived from a transform of the model that evaluates the nonparametric structural function via indicator functions in the support of the discrete regressor. A two-step estimator is proposed where the first step constitutes nonparametric estimation of the instrument and the second step is a nonparametric version of two-stage least squares. Linear functionals of the model are shown to be asymptotically normal, and a consistent estimator of the asymptotic covariance matrix is described. For the binary endogenous regressor case, it is shown that one functional of the model is a conditional (on covariates) local average treatment effect, that permits both unobservable and observable heterogeneity in treatments. Finite sample properties of the estimators from a Monte Carlo simulation study illustrate the practicability of the proposed estimators.     

Optimal instruments when the disturbances are small

Single-equation instrumental variable estimators (e.g., the k-class) are frequently employed to estimate econometric equations. This paper employs Kadane's (1971) small-σ method and a squared-error matrix loss function to characterize a single-equation class of optimal instruments, $\text{A}$. $\text{A}$ is optimal (asymptotically for a small scalar multiple, σ, of the model's disturbance) in that all of its members are preferred to all non-members. From this characterization it is shown all k-class estimators and certain iterative estimators belong to $\text{A}$. However, non-iterative principal component estimators [e.g., Kloek and Mennes (1960)] are unlikely to belong to $\text{A}$. These latter instrumental variable estimators have been advocated [see Amemiya (1966) and Kloek and Mennes (1960)] for estimating ‘large’ econometric models.     

Dynamic spatial autoregressive models with autoregressive and heteroskedastic disturbances

We propose a new class of models specifically tailored for spatiotemporal data analysis. To this end, we generalize the spatial autoregressive model with autoregressive and heteroskedastic disturbances, that is, SARAR(1, 1), by exploiting the recent advancements in score‐driven (SD) models typically used in time series econometrics. In particular, we allow for time‐varying spatial autoregressive coefficients as well as time‐varying regressor coefficients and cross‐sectional standard deviations. We report an extensive Monte Carlo simulation study in order to investigate the finite‐sample properties of the maximum likelihood estimator for the new class of models as well as its flexibility in explaining a misspecified dynamic spatial dependence process. The new proposed class of models is found to be economically preferred by rational investors through an application to portfolio optimization.     

Properties of shrinkage estimators in linear regression when disturbances are not normal

This paper considers a class of recently developed biased estimators of regression coefficients and studies its sampling properties when the disturbances are not normally distributed. It has been found that the conditions of dominance of these estimators over the least squares estimator, under non-normality, are quite different than their well-known dominance conditions under normality. Some implications of the results are also discussed.     

Estimating the autocorrelated error model with trended data

A Monte Carlo study of the small sample properties of various estimators of the linear regression model with first-order autocorrelated errors. When independent variables are trended, estimators using Ttransformed observations (Prais-Winsten) are much more efficient than those using T–1 (Cochrane–Orcutt). The best of the feasible estimators isiterated Prais-Winsten using a sum-of-squared-error minimizing estimate of the autocorrelation coefficient ?. None of the feasible estimators performs well in hypothesis testing; all seriously underestimate standard errors, making estimated coefficients appear to be much more significant than they actually are.     

GMM estimation of spatial autoregressive models with moving average disturbances

In this paper, we introduce the one-step generalized method of moments (GMM) estimation methods considered in Lee (2007a) and Liu, Lee, and Bollinger (2010) to spatial models that impose a spatial moving average process for the disturbance term. First, we determine the set of best linear and quadratic moment functions for GMM estimation. Second, we show that the optimal GMM estimator (GMME) formulated from this set is the most efficient estimator within the class of GMMEs formulated from the set of linear and quadratic moment functions. Our analytical results show that the one-step GMME can be more efficient than the quasi maximum likelihood (QMLE), when the disturbance term is simply i.i.d. With an extensive Monte Carlo study, we compare its finite sample properties against the MLE, the QMLE and the estimators suggested in Fingleton (2008a).     

On the efficiency of least squares estimators in non-linear models

Summary  The identity of least squares estimators å and maximum likelihood estimators â is studied in non-linear models of the type z = g ( a ), where z are observable quantities with a probability density function pr ( z ). This identity was proved for independent random variables z and for distributions pr ( z ), of which the arithmetic sample mean is an optimal estimate.     

On the efficiency of least squares estimators in non-linear models

Summary The identity of least squares estimators å and maximum likelihood estimators â is studied in non-linear models of the type z=g(a), where z are observable quantities with a probability density function pr(z). This identity was proved for independent random variables z and for distributions pr(z), of which the arithmetic sample mean is an optimal estimate.