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lect19_06apr10 - Imbens Lecture Notes 19 ARE213 Spring'06...

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Imbens, Lecture Notes 19, ARE213 Spring ’06 1 ARE213 Econometrics Spring 2006 UC Berkeley Department of Agricultural and Resource Economics Endogeneity III: Selection Models (W 17.4.1) 1. The Model In this lecture we study selection models . Typically they consist of two equations, one outcome equation describing the relation between an outcome of interest Y i and a vector of covariates X i , and the second, the selection equation, describing the relation between a binary participation decision D i and another vector of covariates Z i . There are various forms of these models. Here we consider a specific case, originally studied by Heckman (1979). Y i = X i β + ε i , (1) D i = 1 { Z i γ + η i > 0 } . (2) The parametric form of the model assumes that ε i η i X i , Z i ∼ N 0 0 , σ 2 ε ρ · σ ε ρ · σ ε 1 . (3) The variance for η i is normalized to one since we only observe the sign of X i γ + η i . For a random sample from the population we observe D i , Z i , and X i . Only for observations with D i = 1 do we observe Y i . This model is known as the Heckman selection model, or the type II Tobit model (Amemiya), or the probit selection model (Wooldridge). Variations include the case where Z i γ + η i is observed if D i = 1, so that the selection equation is not probit but tobit. That case is referred to as type II tobit by Amemiya and the tobit selection model by Wooldridge. The classic example is a wage equation, where we only observe the age if the individual decided to work ( D i = 1). Unlike in the Tobit case non participation does not imply that
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Imbens, Lecture Notes 19, ARE213 Spring ’06 2 Y i is negative. In fact, the Tobit model is a special case of this model with ε i = η i , γ = β , and thus D i = 1 { Y i 0 } . In the wage example we think that those who participate in the labor market get relatively high wages compared to those who decided not to participate. If the selection equation had hours worked, with the actual number of observed if hours is positive, we would have the tobit selection model. Another example is that of people buying life insurance (see Wooldridge). We are inter- ested in the relation between the price people pay for life insurance and their characteristics. However, we only observe the price of life insurance for those who purchase it. We do not know what price people who choose not to purchase life insurance would have paid, had they done so. The selection equation models the decision to purchase life insurance. Here we may be concerned that those who did purchase the life insurance (and thus who had relatively high values of η ) paid different prices from those who did not. Specifically, those
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