Fixed Effects and Random Effects Estimation of Higher-order Spatial Autoregressive Models with Spatial Autoregressive and Heteroscedastic Disturbances

Abstract

This paper develops a unified framework for fixed effects (FE) and random effects (RE) estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroscedasticity of unknown form in the idiosyncratic error component. We derive the moment conditions and optimal weighting matrix without distributional assumptions for a generalized moments (GM) estimation procedure of the spatial autoregressive parameters of the disturbance process and define both an RE and an FE spatial generalized two-stage least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators and derive their joint asymptotic distribution, which is robust to heteroscedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman test of the spatial random against the spatial FE model.     

Semiparametric estimation of a bivariate Tobit model

The existing semiparametric estimation literature has mainly focused on univariate Tobit models and no semiparametric estimation has been considered for bivariate Tobit models. In this paper, we consider semiparametric estimation of the bivariate Tobit model proposed by Amemiya (1974), under the independence condition without imposing any parametric restriction on the error distribution. Our estimator is shown to be consistent and asymptotically normal, and simulation results show that our estimator performs well in finite samples. It is also worth noting that while Amemiya’s (1974) instrumental variables estimator (IV) requires the normality assumption, our semiparametric estimator actually outperforms his IV estimator even when normality holds. Our approach can be extended to higher dimensional multivariate Tobit models.     

Interval censored regression with fixed effects

This paper considers identification and estimation of a fixed-effects model with an interval-censored dependent variable. In each time period, the researcher observes the interval (with known endpoints) in which the dependent variable lies but not the value of the dependent variable itself. Two versions of the model are considered: a parametric model with logistic errors and a semiparametric model with errors having an unspecified distribution. In both cases, the error disturbances can be heteroskedastic over cross-sectional units as long as they are stationary within a cross-sectional unit; the semiparametric model also allows for serial correlation of the error disturbances. A conditional-logit-type composite likelihood estimator is proposed for the logistic fixed-effects model, and a composite maximum-score-type estimator is proposed for the semiparametric model. In general, the scale of the coefficient parameters is identified by these estimators, meaning that the causal effects of interest are estimated directly in cases where the latent dependent variable is of primary interest (e.g., pure data-coding situations). Monte Carlo simulations and an empirical application to birthweight outcomes illustrate the performance of the parametric estimator.     

Semiparametric efficient adaptive estimation of asymmetric GARCH models

In this paper we derive a semiparametric efficient adaptive estimator of an asymmetric GARCH model. Applying some general results from Drost et al. [1997. The Annals of Statistics 25, 786–818], we first estimate the unknown density function of the disturbances by kernel methods, then apply a one-step Newton–Raphson method to obtain a more efficient estimator than the quasi-maximum likelihood estimator. The proposed semiparametric estimator is adaptive for parameters appearing in the conditional standard deviation model with respect to the unknown distribution of the disturbances.     

Indirect Estimation of Semiparametric Binary Choice Models*

One of the most cited studies within the field of binary choice models is that of Klein and Spady (1993) , in which the authors propose a semiparametric estimator for use when the distribution of the error term is unknown. However, although theoretically appealing, the estimator has been found to be difficult to implement, and therefore not very attractive from an applied point of view. The current study offers an indirect inference‐based solution to this problem. The new estimator is not only simple with good small‐sample properties, but also consistent and asymptotically normal.     

Censored regression quantiles with endogenous regressors

This paper develops a semiparametric method for estimation of the censored regression model when some of the regressors are endogenous (and continuously distributed) and instrumental variables are available for them. A “distributional exclusion” restriction is imposed on the unobservable errors, whose conditional distribution is assumed to depend on the regressors and instruments only through a lower-dimensional “control variable,” here assumed to be the difference between the endogenous regressors and their conditional expectations given the instruments. This assumption, which implies a similar exclusion restriction for the conditional quantiles of the censored dependent variable, is used to motivate a two-stage estimator of the censored regression coefficients. In the first stage, the conditional quantile of the dependent variable given the instruments and the regressors is nonparametrically estimated, as are the first-stage reduced-form residuals to be used as control variables. The second-stage estimator is a weighted least squares regression of pairwise differences in the estimated quantiles on the corresponding differences in regressors, using only pairs of observations for which both estimated quantiles are positive (i.e., in the uncensored region) and the corresponding difference in estimated control variables is small. The paper gives the form of the asymptotic distribution for the proposed estimator, and discusses how it compares to similar estimators for alternative models.     

A semiparametric model for binary response and continuous outcomes under index heteroscedasticity

This paper formulates a likelihood‐based estimator for a double‐index, semiparametric binary response equation. A novel feature of this estimator is that it is based on density estimation under local smoothing. While the proofs differ from those based on alternative density estimators, the finite sample performance of the estimator is significantly improved. As binary responses often appear as endogenous regressors in continuous outcome equations, we also develop an optimal instrumental variables estimator in this context. For this purpose, we specialize the double‐index model for binary response to one with heteroscedasticity that depends on an index different from that underlying the ‘mean response’. We show that such (multiplicative) heteroscedasticity, whose form is not parametrically specified, effectively induces exclusion restrictions on the outcomes equation. The estimator developed exploits such identifying information. We provide simulation evidence on the favorable performance of the estimators and illustrate their use through an empirical application on the determinants, and affect, of attendance at a government‐financed school. Copyright © 2009 John Wiley & Sons, Ltd.     

Distribution free estimation of heteroskedastic binary response models using Probit/Logit criterion functions

In this paper estimators for distribution free heteroskedastic binary response models are proposed. The estimation procedures are based on relationships between distribution free models with a conditional median restriction and parametric models (such as Probit/Logit) exhibiting (multiplicative) heteroskedasticity. The first proposed estimator is based on the observational equivalence between the two models, and is a semiparametric sieve estimator (see, e.g. Gallant and Nychka (1987), Ai and Chen (2003) and Chen et al. (2005)) for the regression coefficients, based on maximizing standard Logit/Probit criterion functions, such as NLLS and MLE. This procedure has the advantage that choice probabilities and regression coefficients are estimated simultaneously. The second proposed procedure is based on the equivalence between existing semiparametric estimators for the conditional median model (,  and ) and the standard parametric (Probit/Logit) NLLS estimator. This estimator has the advantage of being implementable with standard software packages such as Stata. Distribution theory is developed for both estimators and a Monte Carlo study indicates they both perform well in finite samples.     

Variance estimation using auxiliary information: An almost unbiased multivariate ratio estimator

The goal of this paper is to investigate the repeated substitution method (seeSrivastava, 1967) estimating population variance in finite population sample surveys. We propose an almost unbiased multivariate ratio estimator that has a smaller mean squared error than the conventional biased multivariate ratio estimator (established byIsaki (1983)) and with the same precision as the multivariate regression estimator. Furthermore, it is a computationally much more interesting estimator since to compute it we only need to have knowledge of correlation among available variables, which it is common to have in several practical situations. A comparison of the multivariate ratio estimator proposed and the multivariate regression estimator is given.     

Partially varying coefficient instrumental variables models

In this article, we study a new class of semiparametric instrumental variables models, in which the structural function has a partially varying coefficient functional form. Under this specification, the model is linear in the endogenous/exogenous components with unknown constant or functional coefficients. As a result, the ill‐posed inverse problem in a general non‐parametric model with continuous endogenous variables can be avoided. We propose a three‐step estimation procedure for estimating both constant and functional coefficients and establish their asymptotic properties such as consistency and asymptotic normality. We develop consistent estimators for their error variances. We demonstrate that the constant coefficient estimators achieve the optimal ‐convergence rate, and the functional coefficient estimators are oracle. In addition, efficiency issue of the parameter estimation is discussed and a simple efficient estimator is proposed. The proposed procedure is illustrated via a Monte Carlo simulation and an application to returns to education.     

The semiparametric efficiency bound for models of sequential moment restrictions containing unknown functions

This paper computes the semiparametric efficiency bound for finite dimensional parameters identified by models of sequential moment restrictions containing unknown functions. Our results extend those of Chamberlain (1992b) and Ai and Chen (2003) for semiparametric conditional moment restrictions with identical information sets to the case of nested information sets, and those of Chamberlain (1992a) and Brown and Newey (1998) for models of sequential moment restrictions without unknown functions to cases with unknown functions of possibly endogenous variables. Our results are applicable to semiparametric panel data models and two stage plug-in problems. As an important example, we compute the efficiency bound for a weighted average derivative of a nonparametric instrumental variables regression (NPIV), and find that simple plug-in NPIV estimators are not efficient. We present an optimally weighted, orthogonalized, sieve minimum distance estimator that achieves the semiparametric efficiency bound.     

Semiparametric Estimator for Binary‐outcome Sample Selection: Prejudice Matters in Election

a semiparametric estimator for binary‐outcome sample‐selection models is proposed that imposes only single index assumptions on the selection and outcome equations without specifying the error term distribution. I adopt the idea in Lewbel (2000) using a ‘special regressor’ to transform the binary response Y so that the transformed Y becomes linear in the latent index, which then makes it possible to remove the selection correction term by differencing the transformed Y equation. There are various versions of the estimator, which perform differently trading off bias and variance. A simulation study is conducted, and then I apply the estimators to US presidential election data in 2008 and 2012 to assess the impact of racial prejudice on the elections, as a black candidate was involved for the first time ever in the US history.     

On the performance of tests by Wu and by Hausman for detecting the ordinary least squares bias problem

We first consider the performance of the Wu (1973) - Hausman (1978) (W-H) specification error test as a test for the existence of ordinary least squares (OLS) bias. We discuss power properties of the test under alternative null hypotheses, one of which has not previously been considered. We next consider how the W-H test performs as an indicator of the extent (rather than the existence) of an OLS bias problem, since this usage of the test seems common in applied studies. Finally Monte Carlo methods are used to evaluate Wu's two-step estimation procedure involving the W-H test as a pretest.     

Gaussian semiparametric estimation of multivariate fractionally integrated processes

This paper analyzes the semiparametric estimation of multivariate long-range dependent processes. The class of spectral densities considered is motivated by and includes those of multivariate fractionally integrated processes. The paper establishes the consistency of the multivariate Gaussian semiparametric estimator (GSE), which has not been shown in other work, and the asymptotic normality of the GSE estimator. The proposed GSE estimator is shown to have a smaller limiting variance than the two-step GSE estimator studied by Lobato [1999. A semiparametric two-step estimator in a multivariate long memory model. Journal of Econometrics 90, 129–153]. Gaussianity is not assumed in the asymptotic theory. Some simulations confirm the relevance of the asymptotic results in samples of the size used in practical work.     

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.     

Model specification test with correlated but not cointegrated variables

Many macroeconomic and financial variables show highly persistent and correlated patterns but are not necessarily cointegrated. Recently,  Sun et al. (2011) propose using a semiparametric varying coefficient approach to capture correlations between integrated but non cointegrated variables. Due to the complication arising from the integrated disturbance term and the semiparametric functional form, consistent estimation of such a semiparametric model requires stronger conditions than usually needed for consistent estimation for a linear (spurious) regression model, or a semiparametric varying coefficient model with a stationary disturbance. Therefore, it is important to develop a testing procedure to examine for a given data set, whether linear relationship holds or not, while allowing for the disturbance being an integrated process. In this paper we propose two test statistics for detecting linearity against semiparametric varying coefficient alternative specification. Monte Carlo simulations are used to examine the finite sample performances of the proposed tests.     

Random non-response in multi-character surveys

In the present investigation, we have made first shot to propose an alternative estimator of population total, in the presence of random non-response, for multi-characteristics by using probability proportional to size and without replacement (PPSWOR) sampling when the selection probabilities are poorly correlated with the study variables. The mean square error (MSE) expressions are derived for the proposed estimator. The behavior of the proposed estimator has been examined under super population model. An empirical study has also been carried out to look into the performance of the proposed estimator. The proposed estimator has been applied to real data set.     

Model assisted survey sampling strategy in two phases

Postulating a super-population regression model connecting a size variable, a cheaply measurable variable and an expensively observable variable of interest, an asymptotically optimal double sampling strategy to estimate the survey population total of the third variable is specified. To render it practicable, unknown model-parameters in the optimal estimator are replaced by appropriate statistics. The resulting generalized regression estimator is then shown to have a model-cum-asymptotic design based expected square error equal to that of the asymptotically optimum estimator itself. An estimator for design variance of the estimator is also proposed.     

Finite sample properties for the semiparametric estimation of the intercept of a censored regression model

Financial support for this paper was provided by a C.A. Anderson Fellowship of the Cowles Foundation. I wish to thank Donald Andrews, Moshe Buchinsky, Oliver Linton, and Peter Robinson for helpful discussions. I also wish to thank three anonymous referees for their comments and suggestions. I am, of course, responsible for any remaining errors. A popular two-step estimator of the intercept of a censored regression model is compared with consistent asymptotically normal semiparametric alternatives. Using a root mean squared error criterion, the semiparametric estimators perform better for a range of bandwidth parameter choices for a variety of distributions of the errors and regressors. For error distributions that are close to the normal, however, the two-step parametric estimator performs better.     

Mean square error estimation in multi-stage sampling

Summary: Suppose for a homogeneous linear unbiased function of the sampled first stage unit (fsu)-values taken as an estimator of a survey population total, the sampling variance is expressed as a homogeneous quadratic function of the fsu-values. When the fsu-values are not ascertainable but unbiased estimators for them are separately available through sampling in later stages and substituted into the estimator, Raj (1968) gave a simple variance estimator formula for this multi-stage estimator of the population total. He requires that the variances of the estimated fsu-values in sampling at later stages and their unbiased estimators are available in certain `simple forms'. For the same set-up Rao (1975) derived an alternative variance estimator when the later stage sampling variances have more ‘complex forms’. Here we pursue with Raj's (1968) simple forms to derive a few alternative variance and mean square error estimators when the condition of homogeneity or unbiasedness in the original estimator of the total is relaxed and the variance of the original estimator is not expressed as a quadratic form.  We illustrate a particular three-stage sampling strategy and present a simulation-based numerical exercise showing the relative efficacies of two alternative variance estimators. Received: 19 February 1999