ID Endogeneity Conditional Moment Restrictions

ID Endogeneity Conditional Moment Restrictions -...

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Unformatted text preview: Conditional Moment Restrictions and Triangular Simultaneous Equations * Jinyong Hahn and Geert Ridder June, 2009 Abstract It is shown that in a nonparametric nonseparable triangular system the conditional moment restriction (CMR) does not identify the average structural function (ASF). The CMR identifies the ASF only if the model is structurally separable in observable covariates and unobservable random errors. This excludes for instance random coefficient models in which the CMR in general does not identify the average response. An implication of our results is that empirical researchers should use other methods than CMR if they want to estimate the average response. 1 Introduction Often we are interested in the relation between a (vector of) dependent variable(s) Y and a (vector of) independent variable(s) X . Because the list of independent variables is incomplete, the relation involves one or more unobservable errors that are treated as random variables. In the absence of prior knowledge of the nature of the relation, it is reasonable to adopt a nonparametric framework, and try to identify and estimate such a relationship. The literature has used two approaches within the nonparametric framework. In the first approach, the nonparametric relationship has an additively separable error that satisfies some conditional moment restriction (see, e.g. Newey and Powell (2003), Hall and Horowitz (2005), and Carrasco, Florens, and Renault (2006)). In the second approach, it has an error that is not separable, but * We are grateful for comments by the editor and two anonymous referees. Financial support for this research was generously provided through NSF SES 0819612 and 0819638. Department of Economics, UCLA Department of Economics, University of Southern California 1 satisfies some independence restriction (see, e.g. Imbens and Newey (2009)). It is of interest to clarify the relationship between the two nonparametric approaches. Common sense suggests that there could be a relationship between the two approaches. In order to understand why, consider the simple case where X is exogenous. If we adopt the nonseparable error approach, we would write Y = f ( X, ) , (1) and assume that X and are independent of each other. As argued by Blundell and Powell (2004) and others, the main function/parameter of interest is the average structural function (ASF) ( x ) defined by ( x ) = E [ f ( x, )] , (2) where the expectation is over the marginal distribution of . The ASF is the average outcome if the value X = x is assigned independently of the unobservable . Therefore we can think of the ASF as the average causal relation between Y and X . In model (1) the ASF can be obtained using the following conditional moment restriction (CMR) E [ Y- ( x ) | X = x ] = 0 . (3) This follows because by independence of X and we have 1 E [ Y | X = x ] = E [ f ( X, ) | X = x ] = E [ f ( x, )] = ( x ) . (4) The main result of this paper is that such a relationship does...
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This note was uploaded on 12/26/2011 for the course ECON 245a taught by Professor Staff during the Fall '08 term at UCSB.

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ID Endogeneity Conditional Moment Restrictions -...

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