Estimation of Sample Selection Models with Spatial Dependence

SUMMARYWe consider the estimation of a sample selection model that exhibits spatial autoregressive errors (SAE). Our methodology is motivated by a two‐step strategy where in the first step we estimate a spatial probit model and in the second step (outcome equation) we include an estimated inverse Mills ratio (IMR) as a regressor to control for selection bias. Since the appropriate IMR under SAE depends on a parameter from the second step, both steps are jointly estimated employing the generalized method of moments. We explore the finite sample properties of the estimator using simulations and provide an empirical illustration. Copyright © 2010 John Wiley & Sons, Ltd.

Estimation of spatial sample selection models: A partial maximum likelihood approach

A semiparametric likelihood method is proposed for the estimation of sample selection models. The method is a two-step semiparametric scoring estimation procedure based on an index restriction and kernel estimation. Under some regularity conditions, the estimator is square-root n-consistent and asymptotically normal. The estimator is also asymptotically efficient in the sense that its asymptotic covariance matrix attains the semiparametric efficiency bound under the index restriction. For the binary choice sample selection model, it also attains the efficiency bound under the independence assumption. This method can be applied to the estimation of general sample selection models.

Semiparametric and nonparametric estimation of sample selection models under symmetry

Two-step estimation of heteroskedastic sample selection models

When employing the spatial diagnostics examined in Chapter 4, social scientists will often find evidence of spatial autocorrelation (see, e.g., Eff 2004). As discussed in Chapter 1, this predisposition of social science data toward spatial autocorrelation often results from interdependence between the units studied by social scientists. In other cases, social science data exhibit spatial dependence not as a result of behavioral interdependence but as a consequence of spatial clustering in the sources of behaviors of interest to social scientists. The spatial dependence, in short, may be consistent with either a spatial lag model or a spatial error model. Substantive theory will often lead scholars to believe that a spatial lag specification or a spatial error specification is more appropriate for their particular substantive application. Scholars may, for example, expect that a spatial diffusion process is at work and thus believe that a spatial lag model is warranted. Although such a specification may seem appropriate, such a theoretical expectation should not go untested. It would be inappropriate to estimate a diffusion model with a spatially lagged dependent variable if the spatial dependence diagnosed via, for example, the univariate Moran's I , is instead produced by spatial clustering in the sources of otherwise independent behaviors. This model misspecification will lead the researcher to inappropriate substantive inferences about the nature of the spatial dependence in her data. Inappropriate spatial model specification is all the more problematic because of the close mathematical relationship between a spatial lag model and a spatial error model with spatial autoregressive error dependence. As this chapter will discuss, a spatial autoregressive error model can be rewritten as a spatial Durbin model with both spatially lagged dependent and independent variables if a set of nonlinear common factor constraints are valid. Because of this close relationship between spatial autoregressive dependence in a spatial lag model and spatial autoregressive dependence in a spatial error model, a significant spatial parameter in a spatial lag model may reflect spatial clustering in omitted sources of the behavior of interest rather than true spatial lag dependence consistent with a diffusion process.

The linear regression model is a statistical tool used to model the causal relationship of a dependent variable based on one or several independent or explanatory variables. In scenarios where the dependent variable is a censored variable and there is potential to exist sample selection, the sample selection model can be an alternative in analyzing this relationship. In the Heckman sample selection model, independent variables have the possibility of having an endogeneity effect, where they should be treated as endogenous variables in both the outcome equation and the selection equation instead of as exogenous variables. In result, by including endogenous covariates in the Heckman sample selection model, the sample selection model equation will have more than one equation and makes it a simultaneous equation. To estimate simultaneous equations, simple estimation methods such as the maximum likelihood estimator method are no longer appropriate. In this study, we will discuss the estimation of sample selection models with endogenous covariates utilizing the full information maximum estimator (FIML) approach. The sample selection model with endogenous covariates was then applied to the women labor supply data of Tomas Mroz's research and compared with several models. Based on the MSE and SSE values obtained from the linear regression model, Tobit regression model, Heckman sample selection model, and sample selection model with endogenous covariates, it was concluded that the Heckman sample selection model is the best model that fit the dataset since it yields the best results with the smallest MSE and SSE values

Estimation of sample selection models with two selection mechanisms

If the ordinary least squares (OLS) diagnostics discussed in the previous chapter indicate the existence of spatial lag or spatial error dependence, the researcher will wish to model the type of dependence indicated by these diagnostics. If the OLS diagnostics indicate the presence of a diffusion process, the researcher will wish to estimate a spatial lag model via maximum likelihood (ML) estimation or an instrumental variables specification incorporating instruments for the spatially lagged dependent variable. Alternatively, if the OLS diagnostics indicate the existence of spatial error dependence, the researcher may choose to estimate a more fully specified OLS model to model the spatial dependence or may choose to employ a ML or generalized method of moments (GMM) approach incorporating the spatial dependence in the errors. The spatial dependence diagnosed via the diagnostics discussed in Chapter 5 may alternatively be produced by spatial heterogeneity in the effects of covariates. If this is the only source of spatial dependence, modeling this heterogeneity will be sufficient to capture the spatial dependence. As a consequence, any specification search should also consider the possibility of spatial heterogeneity, which is the focus of Chapter 7. This chapter will first, however, examine alternative approaches for modeling spatial dependence if spatial heterogeneity is not present. This chapter begins by examining ML estimation of spatial lag models that derives from Ord (1975). Next, I explore alternative instrumental variables and GMM estimators for spatial lag dependence. Next, I turn to approaches for estimating spatial error models. I conclude by considering areas of concern in the estimation of spatial models. These include estimators for large sample sizes and diagnostics for continued spatial dependence. MAXIMUM LIKELIHOOD SPATIAL LAG ESTIMATION The mixed regressive, spatial autoregressive model, or spatial lag model, extends the pure spatial autoregressive model considered in Section 3.2 to include also the set of covariates and associated parameters: y = ρ W y +Xβ+e where X is again an N by K matrix of observations on the covariates, β is a K by 1 vector of parameters, and the remaining notation is as discussed in Section 3.2.

This paper investigates the origins of the collinearity problems encountered in the two-step estimation method for sample selection models. The analysis reveals several critical misconceptions and deficiencies in the literature. Remedies to the collinearity problems are proposed and evaluated. Monte Carlo experiments as well as an empirical example based on Mroz's (1987) data are used to illustrate the practical relevance and importance of the analysis.

On the choice between sample selection and two-part models

Summary Sample selection models are important for correcting the effects of non-random sampling. This paper is about semiparametric estimation using a series approximation to the correction term. Regression spline and power series approximations are considered. Asymptotic normality and consistency of an asymptotic variance estimator are shown.

Declining survey response rates have increased the costs of travel survey recruitment. Recruiting respondents based on their expressed willingness to participate in future surveys, obtained from a preceding survey, is a potential solution but may exacerbate sample biases. In this study, we analyze the self-selection biases of survey respondents recruited from the 2017 U.S. National Household Travel Survey (NHTS), who had agreed to be contacted again for follow-up surveys. We apply a probit with sample selection (PSS) model to analyze (1) respondents’ willingness to participate in a follow-up survey (the selection model) and (2) their actual response behavior once contacted (the outcome model). Results verify the existence of self-selection biases, which are related to survey burden, sociodemographic characteristics, travel behavior, and item non-response to sensitive variables. We find that age, homeownership, and medical conditions have opposing effects on respondents’ willingness to participate and their actual survey participation. The PSS model is then validated using a hold-out sample and applied to the NHTS samples from various geographic regions to predict follow-up survey participation. Effect size indicators for differences between predicted and actual (population) distributions of select sociodemographic and travel-related variables suggest that the resulting samples may be most biased along age and education dimensions. Further, we summarized six model performance measures based on the PSS model structure. Overall, this study provides insight into self-selection biases in respondents recruited from preceding travel surveys. Model results can help researchers better understand and address such biases, while the nuanced application of various model measures lays a foundation for appropriate comparison across sample selection models.

Bias in maximum likelihood estimator of disequilibrium and sample selection model with error-ridden observations

Nonparametric identification and estimation of sample selection models under symmetry