Lecture4

# Let us rewrite the first of the two decompositions

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Let us rewrite the first of the two decompositions, but this time including in our notation the parameters which enter each function: ( 29 ( 29 ( 29 f Y X f Y X f X , ; | ; ; φ φ φ = 1 1 2 2 (9) So φ are the parameters of the joint distribution and φ 1 and φ 2 are the parameters of the conditional and marginal distributions respectively. Suppose we were interested in estimating all the parameters of the joint distribution (or equivalently, estimating all the parameters of the conditional and marginal distributions). This could be done directly by using a system estimator to estimate the parameters of the joint distribution or by jointly estimating the conditional and marginal distributions. These approaches really amount to doing the same thing. But suppose our interest is only in the parameters of the conditional distribution, φ 1 (or possibly a subset of those parameters, or some function of them). Let us use the symbol λ to denote these as the “parameters of interest”. Under what circumstances can we obtain efficient estimates of the parameters of interest by estimating the conditional distribution alone? The answer is as follows: if the regressors of the conditional model are weakly exogenous for the parameters of interest , then we can efficiently estimate those parameters from the conditional distribution alone. In terms of equation (9), X is said to be weakly exogenous for the parameters of interest if the parameters of interest, λ , are functions only of φ 1 and not of φ 2 if the parameter sets φ 1 and φ 2 are variation free. The term “variation free” means that the values taken by the parameter set φ 1 impose no constraints on the values that can be taken by the parameter set φ 2 (and vice versa). Intuitively, if X is weakly exogenous for the parameters of interest λ , then the marginal distribution contains no information which is relevant for the estimation of λ and so λ can be estimated efficiently from the conditional distribution alone. Put another way, if X is weakly exogenous for the parameters of interest λ , we can regard the marginal distribution as being “determined outside the system” and so can ignore it for estimation purposes. AN EXAMPLE : It will be useful to give these ideas some support by means of a simple example. Consider the two regressions: Y X u X X v t t t t t t = + + = + + - β β α α 1 2 0 1 1 10 11 ( ) ( ) We will allow for the possibility that the disturbance terms in (10) and (11) are correlated: u v e t t t = + θ (12) with Cov(v t , e t ) = 0 We may think of (10) as a structural equation, suggested by some economic theory, and (11) as an equation explaining how X is generated (in this case, by a first-order autoregressive process). Equation (11) can be thought of as the marginal distribution of X. Is (10) the conditional distribution of Y given X? As things stand, it is not. To derive the conditional distribution of Y given X, we need to obtain an expression for Y (in terms of X) such that the disturbance term of this distribution is not correlated with the disturbance term in the marginal distribution. Equation

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13 (12) shows that, if θ 0, u and v are correlated, and so (10) does not qualify as a conditional distribution.
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