To obtain the conditional distribution proceed as

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To obtain the conditional distribution, proceed as follows: (A) Substitute (12) into (10) giving Y X v e t t t t = + + + β β θ 1 2 13 ( ) ( ) (B) Rearrange (11) to give an expression for v t : v X X t t t = - - - α α 0 1 1 14 ( ) (C) Substitute (14) into (13): [ ] Y X X X e t t t t t = + + - - + - β β θ α α 1 2 0 1 1 15 ( ) which can be reexpressed as Y X X e t t t t = - + + - + - ( ) ( ) ( ) β α θ β θ θα 1 0 2 1 1 16 Note that: Equation (16) satisfies our requirement for a conditional distribution; the error term is given by e t which, by assumption, is not correlated with v t . Also, the regressors X t and X t-1 are both uncorrelated with e t , and so the “orthogonality of regressors and disturbance term” is satisfied. OLS will give consistent estimates of ( ) , ( ) β α θ β θ θα 1 0 2 1 - + and , but not of β 1 and β 2. If we run a regression of Y on an intercept and X, our regression model will be misspecified. This can be seen by looking at equation (16) which is the equation that should be estimated if the OLS estimator were being used. By estimating Y X t t t = + + θ θ η 1 2 we are assuming that η t is a white noise random variable whereas in fact η t = [ ] θ α α X X e t t t - - + - 0 1 1 A variable would be wrongly excluded (the term in square brackets above which includes the lagged value of X). Thus: plim ( ) θ β 1 1 plim ( ) θ β 2 2 IMPLICATIONS Q: Under what circumstance, if any, is equation (10) the conditional distribution for Y? A: If θ = 0 To see why, note that if θ = 0, the conditional model (16) collapses to (10). Q: Is X t weakly exogenous for the parameter of interest β in equation (10)? A: Only if θ = 0 You have already seen that if θ = 0, equation (10) is the conditional model. X t in (10) is weakly exogenous for β 1 and β 2 because those parameters enters the conditional distribution alone, and because the parameters of the marginal and conditional distributions are variation free (particular values of the α parameters do not place any restrictions on admissible values of the β parameters). On the other hand, if θ 0, then the conditional model is (16) not (10), and it is evident that (16) contains parameters from the marginal distribution.
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14 Q: If one estimated equation (10) by OLS, would one obtain consistent estimates of β β 1 2 and ? A: Only if θ = 0. It should be clear from the previous argument that when θ = 0 estimation of (10) involves estimation of the conditional model in which the necessary weak exogeneity assumption is satisfied. OLS will, therefore, yield consistent estimates of β β 1 2 and . ONE POSSIBLE INTERPRETATION OF GIVE AS A TWO STAGE OLS ESTIMATOR Suppose we have the model Y X u = + β in which X is a (T*k) matrix. Let the instrument set be W, a (T*p) matrix, where p k. The IV estimator may be thought of as the following two step estimator: Step 1: REGRESS X ON W BY OLS TO OBTAIN THE FITTED VALUES OF X, X ie Run the regression X W v = + θ which gives ( ) θ = - W W W X 1 and ( ) ( ) X W W W W W X P X where P W W W W W W = = = = - - θ 1 1 Step 2: REGRESS Y ON X Y X e = + β ( ) β IV X X X Y = - 1 which, after substitution, is identical to the GIVE estimator.
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