Working Papers
Relaxing Conditional Independence in an Endogenous Binary Response Model (Job Market Paper, PDF)
For binary response models, the control function approach to address endogeneity has been popularized by the works of Smith and Blundell (1986), Rivers and Vuong (1988), Blundell and Powell (2004), and Rothe (2009). The control function method proposes a function that when conditioned on, the covariates are no longer endogenous with the latent error. This method is generally easy to implement as one simply includes the control function as an additional covariate. But these papers utilize a Control Function assumption that imposes Conditional Independence (CF-CI) to obtain identification. However, CF-CI places restrictions on the relationship between the latent error and the instruments that are unlikely to hold in an empirical context. In particular, the literature has noted that CF-CI imposes homoskedasticity with respect to the instruments. This paper identifies the consequences of CF-CI, provides examples to motivate relaxing CF-CI, and proposes a new consistent estimator under weaker assumptions than CF-CI. By applying the weaker Conditional Mean Restriction of Kim and Petrin (2017) to the binary response setting, parameters and structural objects of interest can be recovered. The proposed method is illustrated in an application, estimating the effect of non-wife income on married women's labor supply. The CF-CI has testable implications in which a LR statistic is used. In the empirical application, there is strong statistical evidence that CF-CI is violated which necessitates the proposed approach. The proposed estimator is presented in a parametric framework but in some empirical contexts, the distributional assumptions may be unrealistic. Therefore, this paper also provides a semi-parametric extension that proposes a new distribution free estimator. Using the observational equivalence results of Khan (2013), the proposed sieve semi-parametric estimator is shown to be consistent under weaker assumptions than those found in the literature.
For binary response models, the control function approach to address endogeneity has been popularized by the works of Smith and Blundell (1986), Rivers and Vuong (1988), Blundell and Powell (2004), and Rothe (2009). The control function method proposes a function that when conditioned on, the covariates are no longer endogenous with the latent error. This method is generally easy to implement as one simply includes the control function as an additional covariate. But these papers utilize a Control Function assumption that imposes Conditional Independence (CF-CI) to obtain identification. However, CF-CI places restrictions on the relationship between the latent error and the instruments that are unlikely to hold in an empirical context. In particular, the literature has noted that CF-CI imposes homoskedasticity with respect to the instruments. This paper identifies the consequences of CF-CI, provides examples to motivate relaxing CF-CI, and proposes a new consistent estimator under weaker assumptions than CF-CI. By applying the weaker Conditional Mean Restriction of Kim and Petrin (2017) to the binary response setting, parameters and structural objects of interest can be recovered. The proposed method is illustrated in an application, estimating the effect of non-wife income on married women's labor supply. The CF-CI has testable implications in which a LR statistic is used. In the empirical application, there is strong statistical evidence that CF-CI is violated which necessitates the proposed approach. The proposed estimator is presented in a parametric framework but in some empirical contexts, the distributional assumptions may be unrealistic. Therefore, this paper also provides a semi-parametric extension that proposes a new distribution free estimator. Using the observational equivalence results of Khan (2013), the proposed sieve semi-parametric estimator is shown to be consistent under weaker assumptions than those found in the literature.
Parametric Identification of Multiplicative Exponential Heteroskedasticity, Oxford Bulletin of Economics and Statistics, 2018 (PDF)
Harvey (1976) first proposed multiplicative exponential heteroskedasticity in the context of a linear regression. These days it is more commonly seen in non-linear models such as a binary response model where correctly modeling the heteroskedasticity is imperative for consistent parameter estimates. However, there doesn't appear to be a formal proof of point identification for the parameters in the model. This paper presents several examples that show the conditions presumed throughout the literature are not sufficient for identification. The major contribution of this paper is to provide additional conditions and show identification for such a pervasive model.
Harvey (1976) first proposed multiplicative exponential heteroskedasticity in the context of a linear regression. These days it is more commonly seen in non-linear models such as a binary response model where correctly modeling the heteroskedasticity is imperative for consistent parameter estimates. However, there doesn't appear to be a formal proof of point identification for the parameters in the model. This paper presents several examples that show the conditions presumed throughout the literature are not sufficient for identification. The major contribution of this paper is to provide additional conditions and show identification for such a pervasive model.
Behavior of Pooled and Joint Estimators in Probit Model with Random Coefficients and Serial Correlation (Primary Author, joint with Jeffrey Wooldridge and Ying Zhu)
We propose a pooled maximum likelihood estimator (PMLE) for dealing with potential individual-specific heterogeneity and serial/temporal correlation in a Probit/Logit Mixture model (with binary outcomes). We compare the performance of our procedure with a joint (full) maximum likelihood estimator (JMLE), which is the dominant estimation method for mixture models in practice. The JMLE is more statistically efficient but computationally demanding; the implementation becomes even more difficult if one tries to model the serial correlation over time and allows the individual-specific heterogeneity to be correlated with the covariates (e.g., participation in some treatment). On the other hand, the PMLE is computationally simple, while robust to arbitrary forms of serial correlation and allows the individual-specific heterogeneity to be correlated with the covariates. Our result suggests that, in terms of the computation time, the JMLE takes 60 times longer than the PMLE (if the JMLE actually converges and in many instances, it simply fails to converge). We also find that, in actual implementation (as opposed to "in theory" ), the parameter estimates from the JMLE tend to be highly biased while the PMLE is consistent in the presence of serial correlation and individual-specific heterogeneity. However, if one focuses on a different statistics (the estimates for the Average Treatment Effect or the Average Partial Effects) instead of the parameter estimates themselves, we find that the JMLE can produce quite satisfactory estimates that are robust to serial correlation and individual-specific heterogeneity even under misspecification of the likelihood function with regard to the time series and correlation with the covariates. This result has important implications as when it comes to evaluating policy interventions, the ultimate interest usually concerns the treatment effect.
We propose a pooled maximum likelihood estimator (PMLE) for dealing with potential individual-specific heterogeneity and serial/temporal correlation in a Probit/Logit Mixture model (with binary outcomes). We compare the performance of our procedure with a joint (full) maximum likelihood estimator (JMLE), which is the dominant estimation method for mixture models in practice. The JMLE is more statistically efficient but computationally demanding; the implementation becomes even more difficult if one tries to model the serial correlation over time and allows the individual-specific heterogeneity to be correlated with the covariates (e.g., participation in some treatment). On the other hand, the PMLE is computationally simple, while robust to arbitrary forms of serial correlation and allows the individual-specific heterogeneity to be correlated with the covariates. Our result suggests that, in terms of the computation time, the JMLE takes 60 times longer than the PMLE (if the JMLE actually converges and in many instances, it simply fails to converge). We also find that, in actual implementation (as opposed to "in theory" ), the parameter estimates from the JMLE tend to be highly biased while the PMLE is consistent in the presence of serial correlation and individual-specific heterogeneity. However, if one focuses on a different statistics (the estimates for the Average Treatment Effect or the Average Partial Effects) instead of the parameter estimates themselves, we find that the JMLE can produce quite satisfactory estimates that are robust to serial correlation and individual-specific heterogeneity even under misspecification of the likelihood function with regard to the time series and correlation with the covariates. This result has important implications as when it comes to evaluating policy interventions, the ultimate interest usually concerns the treatment effect.
Works in Progress
Estimating Unreported Taxi Tips in Chicago: Addressing Sample Selection Bias for Machine Learning Methods (Joint with Dylan Brewer)
Understanding Demand for Organic Food: A General Control Function Approach (Joint with Katherine Harris)
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