Lecture4

# To obtain the conditional distribution proceed as

• Notes
• 17

This preview shows page 13 - 15 out of 17 pages.

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.

Subscribe to view the full document.

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.
You've reached the end of this preview.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern