GMM estimation of panel data models with time-varying slope coefficients

A difference/system generalized method of moments (GMM) model that imposes time-constant coefficients is common in empirical studies using panel data. However, a rejection by the Sargan–Hansen test is sometimes a serious concern for researchers. We highlight the fact that the Sargan–Hansen test for GMM estimators applied to panel data is a joint test of valid orthogonality conditions and coefficient stability over time. A possible reason for a rejection is therefore that the slope coefficients vary over time. One solution is to estimate an empirical model in which the coefficients are time specific. We apply this solution to the system GMM estimator of simple nondynamic Cobb–Douglas production functions for a selection of Swedish industries and find that relaxing the assumption of constant slope coefficients results in more satisfactory outcomes of the Sargan–Hansen test.     

The asymptotic efficiency of semiparametric estimators for censored linear regression models

This paper presents numerical comparisons of the asymptotic mean square estimation errors of semiparametric generalized least squares (SGLS), quantite, symmetrically censored least squares (SCLS), and tobit maximum likelihood estimators of the slope parameters of censored linear regression models with one explanatory variable. The results indicate that the SCLS estimator is less efficient than the other two semiparametric estimators. The SGLS estimator is more efficient than quantile estimators when the tails of the distribution of the random component of the model are not too thick and the probability of censoring is not too large. The most efficient semiparametric estimators usually have smaller mean square estimation errors than does the tobit estimator when the random component of the model is not normally distributed and the sample size is 500–1,000 or more.     

Generalized least squares and maximum likelihood estimation of the logistic function for technology diffusion

Given a simple stochastic model of technology adoption, we derive a function for technological diffusion that is logistic in the deterministic part and has an error term based on the binomial distribution. We derive two estimators—a generalized least squares (GLS) estimator and a maximum likelihood (ML) estimator—which should be more efficient than the ordinary least squares (OLS) estimators typically used to estimate technological diffusion functions. We compare the two new estimators with OLS using Monte-Carlo techniques and find that under perfect specification, GLS and ML are equally efficient and both are more efficient than OLS. There was no evidence of bias in any of the estimators. We used the estimators on some example data and found evidence suggesting that under conditions of misspecification, the estimated variance-covariance of the ML estimator is badly biased. We verified the existence of the bias with a second Monte-Carlo experiment performed with a known misspecification. In the second experiment, GLS was the most efficient estimator, followed by ML, and OLS was least efficient. We conclude that the GLS estimator of choice.     

Fractional integration with drift: estimation in small samples

We examine the finite-sample behavior of estimators of the order of integration in a fractionally integrated time-series model. In particular, we compare exact time-domain likelihood estimation to frequency-domain approximate likelihood estimation. We show that over-differencing is of critical importance for time-domain maximum-likelihood estimation in finite samples. Overdifferencing moves the differencing parameter (in the over-differenced model) away from the boundary of the parameter space, while at the same time obviating the need to estimate the drift parameter. The two estimators that we compare are asymptotically equivalent. In small samples, however, the time-domain estimator has smaller mean squared error than the frequency-domain estimator. Although the frequency-domain estimator has larger bias than the time-domain estimator for some regions of the parameter bias, it can also have smaller bias. We use a simulation procedure which exploits the approximate linearity of the bias function to reduce the bias in the time-domain estimator.     

Moment Estimation of the Probit Model with an Endogenous Continuous Regressor

We propose a generalized method of moments (GMM) estimator with optimal instruments for a probit model that includes a continuous endogenous regressor. This GMM estimator incorporates the probit error and the heteroscedasticity of the error term in the first‐stage equation in order to construct the optimal instruments. The estimator estimates the structural equation and the first‐stage equation jointly and, based on this joint moment condition, is efficient within the class of GMM estimators. To estimate the heteroscedasticity of the error term of the first‐stage equation, we use the k‐nearest neighbour (k‐nn) non‐parametric estimation procedure. Our Monte Carlo simulation shows that in the presence of heteroscedasticity and endogeneity, our GMM estimator outperforms the two‐stage conditional maximum likelihood estimator. Our results suggest that in the presence of heteroscedasticity in the first‐stage equation, the proposed GMM estimator with optimal instruments is a useful option for researchers.     

A Note on Covariance Matrix Estimation in Quantile Regressions

This note discusses some issues related to bandwidth selection based on moment expansions of the mean squared error （MSE） of the regression quantile estimator. We use higher order expansions to provide a way to distinguish among asymptotically equivalent nonparametric estimators. We derive approximations to the （standardized） MSE of the covariance matrix estimation. This facilitates a comparison of different estimators at the second order level, where differences do occur and depend on the bandwidth choice. A method of bandwidth selection is defined by minimizing the second order effect in the mean squared error.     

Estimation of Censored Regression Model: A Simulation Study

We investigate the finite sample performance of several estimators proposed for the panel data Tobit regression model with individual effects, including Honoré estimator, Hansen’s best two-step GMM estimator, the continuously updating GMM estimator, and the empirical likelihood estimator (ELE). The latter three estimators are based on more conditional moment restrictions than the Honoré estimator, and consequently are more efficient in large samples. Although the latter three estimators are asymptotically equivalent, the last two have better finite sample performance. However, our simulation reveals that the continuously updating GMM estimator performs no better, and in most cases is worse than Honoré estimator in small samples. The reason for this finding is that the latter three estimators are based on more moment restrictions that require discarding observations. In our designs, about seventy percent of observations are discarded. The insufficiently few number of observations leads to an imprecise weighted matrix estimate, which in turn leads to unreliable estimates. This study calls for an alternative estimation method that does not rely on trimming for finite sample panel data censored regression model.     

Relationships among estimators of triangular econometric models

Estimators for triangular models where one endogenous variable may be dichotomous are compared. The selection bias correction method utilized in the simultaneous probit model is applied to standard simultaneous equations models. The resulting estimator is shown to be equivalent to 2SLS in some cases and provides a Hausman (1978) test of endogeneity. Also, other comparisons of estimators are made.     

Instrumental variable estimation of a spatial autoregressive panel model with random effects

This paper extends the instrumental variable estimators of Kelejian and Prucha (1998) and Lee (2003) proposed for the cross-sectional spatial autoregressive model to the random effects spatial autoregressive panel data model. It also suggests an extension of the Baltagi (1981) error component 2SLS estimator to this spatial panel model.     

Small Sample Properties Of Existing And New Estimators Of The Autocorrelation Coefficient And Regression Coefficients In The Context Of Ar(1) Errors

This study systematically and comprehensively investigates the small sample properties of the existing and some new estimators of the autocorrelation coefficient and of the regression coefficients in a linear regression model when errors follow an autoregressive process of order one. The new estimators of autocorrelation coefficient proposed here are based on the jackknife procedure. The jackknife procedure is applied in two alternative ways: first to the regression itself, and second to the residuals of the regression model. Next, the performance of the existing and new estimators of autocorrelation coefficient (thirty-three in total) is investigated in terms of bias and the root mean squared errors. Finally, we have systematically compared all of the estimators of the regression coefficients (again thirty-three) in terms of efficiency and their performance in hypothesis testing. We observe that the performance of the autocorrelation coefficient estimators is dependent upon the degree of autocorrelation and whether the autocorrelation is positive or negative. We do not observe a direct link between the bias and efficiency of an estimator. The performance of the estimators of the regression coefficients also depends upon the degree of autocorrelation. If the efficiency of regression estimator is of concern, then the iterative Prais-Winsten estimator should be used since it is most efficient for the widest range of independent variables and values of the autocorrelation coefficient. If testing of the hypothesis is of concern, then the estimators based on jackknife technique are certainly superior and are highly recommended. However, for negative values of the autocorrelation coefficient, the estimators based on Quenouille procedure and iterative Prais-Winsten estimator are comparable. But, for computational ease iterative Prais-Winsten estimator is recommended.

     

Optimality of the Stein-type Estimator for Variance

This paper considers the optimality of the Stein-type estimator of the disturbance variance among the class of pre-test estimators after the pre-test for a linear hypothesis on coefficients. The Stein-type estimator is a member in this class with a critical value, say c_s . It is known that the Stein-type estimator is the best in the subclass which consists of the pre- test estimators with smaller critical values than c_s . It is analytically shown that an extension of the optimality to a larger class is not possible.     

An Improved Two‐step Regularization Scheme for Spot Volatility Estimation

We are concerned with the problem of spot volatility estimation in the presence of microstructure noise. We introduce an estimator based on the technique of multi‐step regularization. A preliminary form for such an estimator was proposed in Ogawa (2008) and was shown to work in a real‐time manner. However, the main drawback of this scheme is that it needs a lot of observation data. The aim of the present paper is to introduce an improvement to this scheme, such that the modified estimator can work more efficiently and with a data set of smaller size. The technical aspects of implementation of the proposed scheme and its performance on simulated data are analysed. The scheme is tested against other spot volatility estimators, namely a realized volatility type estimator, the Fourier estimator and three kernel estimators.     

Nonparametric regression estimation with general parametric error covariance: a more efficient two-step estimator

Recently Martins-Filho and Yao (J Multivar Anal 100:309–333, 2009) have proposed a two-step estimator of nonparametric regression function with parametric error covariance and demonstrate that it is more efficient than the usual LLE. In the present paper we demonstrate that MY’s estimator can be further improved. First, we extend MY’s estimator to the multivariate case, and also establish the asymptotic theorem for the slope estimators; second, we propose a more efficient two-step estimator for nonparametric regression function with general parametric error covariance, and develop the corresponding asymptotic theorems. Monte Carlo study shows the relative efficiency loss of MY’s estimator in comparison with our estimator in nonparametric regression with either AR(2) errors or heteroskedastic errors. Finally, in an empirical study we apply the proposed estimator to estimate the public capital productivity to illustrate its performance in a real data setting.     

Improved estimators in some linear errors-in-variables models in finite samples

The paper uses a Monte Carlo study to demonstrate the dominance under mean squared errors or quadratic loss of a new improved estimator for some linear errors-in-variables models in finite samples. The new estimator is non-linear and biased in a conventional sense and has a smaller risk than the least squares and the Stein estimators. Standard errors for this estimator can be conveniently obtained by bootstrapping methods.     

ORDINARY LEAST SQUARES ESTIMATION OF A DYNAMIC GAME MODEL

Estimation of dynamic games is known to be a numerically challenging task. A common form of the payoff functions employed in practice takes the linear‐in‐parameter specification. We show a least squares estimator taking a familiar OLS/GLS expression is available in such a case. Our proposed estimator has a closed form. It can be computed without any numerical optimization and always minimizes the least squares objective function. We specify the optimally weighted GLS estimator that is efficient in the class of estimators under consideration. Our estimator appears to perform well in a simple Monte Carlo experiment.     

A comparison of bias approximations for the two-stage least squares (2SLS) estimator

We consider the bias of the two-stage least squares (2SLS) estimator in linear instrumental variable regression with only one endogenous regressor. By using asymptotic expansion techniques, we approximate the 2SLS coefficient estimation bias under various scenarios regarding the number and strength of instruments.     

On Calculation of the Extended Gini Coefficient

The conventional formula for estimating the extended Gini coefficient is a covariance formula provided by Lerman and Yitzhaki (1989). We suggest an alternative estimator, obtained by approximating the Lorenz curve by a series of linear segments. In a Monte Carlo experiment designed to assess the relative bias and efficiency of the two estimators, we find that, when using grouped data with 20 or fewer groups, our new estimator has less bias and lower mean squared error than the covariance estimator. When individual observations are used, or the number of groups is 30 or more, there is little or no difference in the performance of the two estimators.     

Efficiency of two-step estimators for censored systems of equations: Shonkwiler and Yen reconsidered

This study analyses a parametric estimator for a system of equations with limited dependent variables that was recently proposed. Its performance is compared with those of alternative estimation procedures using Monte Carlo methods. The comparison shows that this new estimator is less efficient for a wide range of parameter regions than multivariate generalizations of the classical Heckman model. This result can be explained by its variance depending on the squared conditional mean of the dependent variables. Additionally, it turns out that within the class of generalized Heckman estimators, rather simple ones display the best performance.     

On Liu estimators for the logit regression model

This paper introduces a shrinkage estimator for the logit model which is a generalization of the estimator proposed by Liu (1993) for the linear regression. This new estimation method is suggested since the mean squared error (MSE) of the commonly used maximum likelihood (ML) method becomes inflated when the explanatory variables of the regression model are highly correlated. Using MSE, the optimal value of the shrinkage parameter is derived and some methods of estimating it are proposed. It is shown by means of Monte Carlo simulations that the estimated MSE and mean absolute error (MAE) are lower for the proposed Liu estimator than those of the ML in the presence of multicollinearity. Finally the benefit of the Lie estimator is shown in an empirical application where different economic factors are used to explain the probability that municipalities have net increase of inhabitants.     

Some simulation studies of nonparametric estimators

This paper constructs a number of Monte Carlo studies to assess the quality of various nonparametric estimators that have been proposed recently for the estimation of nonlinear econometric models. We consider both kernel and Fourier series based methods of estimation, and also examine techniques that have been suggested to improve the bias properties of the kernel estimator. The two models examined are a production function and a model emphasising the effects of risk. The Fourier estimator does very well in estimating the first of these, but not the second, while the kernel estimator shows substantial bias for the first, which is only partially alleviated by the procedures advocated for bias correction, and good results for the second.