Convenient estimators for the panel probit model: Further results

Bertschek and Lechner (1998) propose several variants of a GMM estimator based on the period specific regression functions for the panel probit model. The analysis is motivated by the complexity of maximum likelihood estimation and the possibly excessive amount of time involved in maximum simulated likelihood estimation. But, for applications of the size considered in their study, full likelihood estimation is actually straightforward, and resort to GMM estimation for convenience is unnecessary. In this note, we reconsider maximum likelihood based estimation of their panel probit model then examine some extensions which can exploit the heterogeneity contained in their panel data set. Empirical results are obtained using the data set employed in the earlier study. Helpful comments and suggestions by Irene Bertschek and Michael Lechner are gratefully acknowledged. This paper has also benefited from comments by two anonymous referees and from seminar participants at the Center for Health Economics at the University of York. Any remaining errors are the responsibility of the author.     

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 Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model

This paper is concerned with the estimation of the autoregressive parameter in a widely considered spatial autocorrelation model. The typical estimator for this parameter considered in the literature is the (quasi) maximum likelihood estimator corresponding to a normal density. However, as discussed in this paper, the (quasi) maximum likelihood estimator may not be computationally feasible in many cases involving moderate- or large-sized samples. In this paper we suggest a generalized moments estimator that is computationally simple irrespective of the sample size. We provide results concerning the large and small sample properties of this estimator.     

The Tobit model with serial correlation

This paper discusses the maximum likelihood solution of the probit regression model for limited dependent variables, when there is first order serial correlation among the residuals.     

When outcome heterogeneously matters for selection: a generalized selection correction estimator

The classical Heckman (1976, 1979) selection correction estimator (heckit) is misspecified and inconsistent, if an interaction of the outcome variable with an explanatory variable matters for selection. To address this specification problem, a full information maximum likelihood (FIML) estimator and a simple two-step estimator are developed. Monte Carlo (MC) simulations illustrate that the bias of the ordinary heckit estimator is removed by these generalized estimation procedures. Along with OLS and ordinary heckit, we apply these estimators to data from a randomized trial that evaluates the effectiveness of financial incentives for reducing obesity. Estimation results indicate that the choice of the estimation procedure clearly matters.     

A simple derivation of the limited information maximum likelihood estimator

The conventional derivation of the limited information maximum likelihood estimator is quite complicated. A simpler proof is provided in this note.     

Intra-industry trade and business cycles in ASEAN

The innovation adoption literature has focused primarily on a producer's decision of whether and how much to adopt. An equally pertinent question is when to adopt, because in the case of new technologies it often ‘pays to wait’ for more information. We propose a double-limit hurdle model to analyse adoption intensity and inertia in the context of a divisible technology. The proposed framework incorporates probit or Tobit models as testable special cases. A maximum likelihood estimation framework is set out and generalized to account for heteroscedastic errors. The empirical analysis, which uses household-level data from India's semi-arid tropics, provides new insights into the factors influencing adoption inertia and intensity.     

Robust estimation of the simplified multivariate GARCH model

In this paper, robust M-estimation of multivariate GARCH models are considered. The simplified GARCH model is chosen that involves the estimation of only univariate GARCH models, and hence easy to estimate, and does not put additional constraints on the model. The results of Monte Carlo simulations showed that accurate estimates of conditional correlations can be obtained using these robust estimators when the errors are heavy-tailed. We also investigate the forecasting performance of the class of robust estimators in predicting value-at-risk using various evaluation measures and collect empirical evidences of the better predictive potential of estimators such as LAD and B-estimator over the widely-used quasi-maximum likelihood estimator for the estimation and prediction of multivariate GARCH models. Applications to real data sets are also presented.     

Estimation of an endogenous switching regression model with discrete dependent variables: Monte-Carlo analysis and empirical application of three estimators

The performances of alternative two-stage estimators for the endogenous switching regression model with discrete dependent variables are compared, with regard to their usefulness as starting values for maximum likelihood estimation. This is especially important in the presence of large correlation coefficients, in which case maximum likelihood procedures have difficulties to converge. Monte-Carlo simulations indicate that an estimator that corrects for conditional heteroskedasticity of the residuals is superior in almost all instances, and especially when maximum likelihood is problematic. This result is also obtained in an empirical example in which off-farm work participation equations of farm women are conditional on farm work participation status. First version received: July 1995/final version received: March 1998     

Estimation of Discrete Choice Dynamic Programming Models

This study reviews estimation methods for the infinite horizon discrete choice dynamic programming models and conducts Monte Carlo experiments. We consider: the maximum likelihood estimator (MLE), the two‐step conditional choice probabilities estimator, sequential estimators based on policy iterations mapping under finite dependence, and sequential estimators based on value iteration mappings. Our simulation result shows that the estimation performance of the sequential estimators based on policy iterations and value iteration mappings is largely comparable to the MLE, while they achieve substantial computation gains over the MLE by a factor of 100 for a model with a moderately large state space.     

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.     

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.     

A COMPUTATIONALLY PRACTICAL SIMULATION ESTIMATION ALGORITHM FOR DYNAMIC PANEL DATA MODELS WITH UNOBSERVED ENDOGENOUS STATE VARIABLES*

This article develops a simulation estimation algorithm that is particularly useful for estimating dynamic panel data models with unobserved endogenous state variables. Repeated sampling experiments on dynamic probit models with serially correlated errors indicate the estimator has good small sample properties. We apply the estimator to a model of female labor supply and show that the rarely used Polya model fits the data substantially better than the popular Markov model. The Polya model also produces far less state dependence and many fewer race effects and much stronger effects of education, young children, and husband's income on female labor supply decisions.     

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.     

Estimation of a Tobit model with unknown censoring threshold

Conventional wisdom suggests that only the estimated intercept is affected by imposition of a zero censoring threshold on a Tobit model. This is true for Heckman-Lee estimation. For maximum likelihood (ML) estimation, however, it is only true if the censoring threshold is known and is subtracted from the dependent variable. Failure to properly transform the dependent variable prior to ML estimation of a zero threshold Tobit model will generally bias the coefficient estimates. A long neglected topic is ML estimation of a Tobit model with common, but unknown, censoring threshold. This paper shows that the ML estimator of the censoring threshold is the minimum order statistic from the observed subsample, and that existing software for estimation of a zero-threshold Tobit model is easily adapted to include estimation of the censoring threshold.     

Errors in variables in a random-effects probit model for panel data

In this paper we consider a probit model estimated on panel data with random person-specific effects when an independent variable is measured with error. For normally distributed measurement error, a structural maximum-likelihood estimate from a likelihood that is conditional on the observed independent variable but marginal on the incidental parameter is studied. A minimum-distance estimator without the use of external instruments is proposed.     

On the simultaneity of innovations and market structure

This paper examines the determinants of innovations and market structure within a simultaneous framework. From a competitive fringe model, quasi-conditional factor demand functions are derived that explain product and process innovations in terms of factor prices and market structure variables such as relative firm size, market size, and the concentration ratio where the latter set of variables result from the same optimizing process.Empirical evidence is gained from a cross section of 2276 West German firms in the manufacturing sector. In contrast to many other empirical studies, product and process innovations are measured by two dichotomous variables. An exogeneity test for the probit model is worked out and the conditional maximum likelihood estimator that emerges from this test is applied. The results show that simultaneity does matter, even if innovations are explained by market structure variables at the firm level. Accounting for endogeneity and cross-equation restrictions changes the results substantially.     

The comparative performance of the iterative instrumental variables estimator using structural and predictive criteria

The purpose of this paper is to give experimental evidence on the small-sample properties of the iterative instrumental variables estimator originally proposed byLyttkens [1970], relative to the more conventional methods including ordinary least squares, limited information single equation maximum likelihood and three stage least squares.     

The participation decision versus the level of participation in an environmental treaty: a spatial probit analysis

This paper represents treaty participation as a two-stage game, for which nations first decide whether or not to participate and then they choose their level of participation. The resulting subgame perfect equilibrium is used to derive a reduced-form equation for estimating and separating the influences of the variables at the two decision stages. This spatial probit equation forms the basis for a full-information maximum likelihood estimator that accounts for the simultaneity bias associated with public good spillins at both stages. When the procedure is applied to the Helsinki Protocol, we find that the strategic influence of a variable may drastically differ depending upon which stage is scrutinized.     

SEMINONPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATION OF CONDITIONAL MOMENT RESTRICTION MODELS*

This article studies estimation of a conditional moment restriction model with the seminonparametric maximum likelihood approach proposed by Gallant and Nychka (Econometrica 55 (March 1987), 363–90). Under some sufficient conditions, we show that the estimator of the finite dimensional parameter θ is asymptotically normally distributed and attains the semiparametric efficiency bound and that the estimator of the density function is consistent under L₂ norm. Some results on the convergence rate of the estimated density function are derived. An easy to compute covariance matrix for the asymptotic covariance of the θ estimator is presented.