Working Papers
Relaxing Conditional Independence in an Endogenous Binary Response Model (Job Market Paper)
In the absence of instrumental variables as a valid method to address endogeneity in latent variable models, the control function method has become a popular approach. 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. For binary response models in particular, the control function approach has been popularized by the works of Smith and Blundell (1986), Rivers and Vuong (1988), Blundell and Powell (2004), and Rothe (2009). These papers utilize a Conditional Independence (CI) assumption to obtain identification. However, CI places fairly strict assumptions on the relationship between the latent error and the instruments. In particular, the literature has noted that CI requires no heteroskedasticity with respect to the instruments. This paper aims to identify the consequences of CI, provide examples to motivate relaxing CI, and propose an alternative estimation framework. Following the work of Kim and Petrin (working paper), I propose a general control function approach which requires a weaker Conditional Mean Restriction (CMR) for identification. I apply the proposed method to estimate the effect of non-wife income on married women's labor supply and find evidence that CI is violated. The proposed estimator is presented in a parametric framework to illustrate that it is computationally simple and flexible. Then I extend the estimator to a distribution free setting utilizing the results of Khan (2013) with a semi-parametric approach. This extension presents an estimator that is strictly weaker in assumption compared to prior estimators of the literature.
In the absence of instrumental variables as a valid method to address endogeneity in latent variable models, the control function method has become a popular approach. 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. For binary response models in particular, the control function approach has been popularized by the works of Smith and Blundell (1986), Rivers and Vuong (1988), Blundell and Powell (2004), and Rothe (2009). These papers utilize a Conditional Independence (CI) assumption to obtain identification. However, CI places fairly strict assumptions on the relationship between the latent error and the instruments. In particular, the literature has noted that CI requires no heteroskedasticity with respect to the instruments. This paper aims to identify the consequences of CI, provide examples to motivate relaxing CI, and propose an alternative estimation framework. Following the work of Kim and Petrin (working paper), I propose a general control function approach which requires a weaker Conditional Mean Restriction (CMR) for identification. I apply the proposed method to estimate the effect of non-wife income on married women's labor supply and find evidence that CI is violated. The proposed estimator is presented in a parametric framework to illustrate that it is computationally simple and flexible. Then I extend the estimator to a distribution free setting utilizing the results of Khan (2013) with a semi-parametric approach. This extension presents an estimator that is strictly weaker in assumption compared to prior estimators of the literature.
Behavior of Pooled and Joint Estimators in Probit Model with Random Coefficients and Serial Correlation (Primary Author, joint with Jeffrey Woodridge 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.
Parametric Identification of Multiplicative Exponential Heteroskedasticity
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.
Works in Progress
Estimating Unreported Taxi Tips in Chicago: Addressing Sample Selection Bias for Machine Learning Methods (Joint with Dylan Brewer)