Abstract
We consider an extension of the recursive bivariate probit model for estimating the effect of a binary variable on a binary outcome in the presence of unobserved confounders, nonlinear covariate effects and overdispersion. Specifically, the model consists of a system of two binary outcomes with a binary endogenous regressor which includes smooth functions of covariates, hence allowing for flexible functional dependence of the responses on the continuous regressors, and arbitrary random intercepts to deal with overdispersion arising from correlated observations on clusters or from the omission of non-confounding covariates. We fit the model by maximizing a penalized likelihood using an Expectation-Maximisation algorithm. The issues of automatic multiple smoothing parameter selection and inference are also addressed. The empirical properties of the proposed algorithm are examined in a simulation study. The method is then illustrated using data from a survey on health, aging and wealth.
Highlights
Quantifying the effect of a predictor of interest on a particular response variable is a challenging task in observational studies
We consider the case in which the researcher is interested in estimating the effect of a binary endogenous variable on a binary outcome in the presence of unobserved confounders, nonlinear covariate-response relationships and overdispersion
A likely explanation is that the parameter that links the two equations of the bivariate model captures correlations due to both unobserved confounders and cluster or ‘litter’ effect
Summary
Quantifying the effect of a predictor of interest ( referred to as treatment) on a particular response variable is a challenging task in observational studies. This is because it is often the case that confounders which are associated with both treatment and response are either unknown or not readily quantifiable (this problem is known in econometrics as endogeneity of the variable of interest). We consider the case in which the researcher is interested in estimating the effect of a binary endogenous variable on a binary outcome in the presence of unobserved confounders, nonlinear covariate-response relationships and overdispersion. Marra & Radice (2011a) considered the same model and introduced a penalized likelihood based procedure which permits reliable estimation of the model coefficients at reasonably small sample sizes As the authors point out, very large sample sizes are required to obtain reasonable estimates of the binary treatment effect, undermining the utility of the method for practical modeling. Marra & Radice (2011a) considered the same model and introduced a penalized likelihood based procedure which permits reliable estimation of the model coefficients at reasonably small sample sizes
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