What is the identification condition? how many exogenous variables must be excluded for the above equation to be identified

a.    What is the identification condition - how many exogenous variables must be excluded for the above equation to be identified (assuming that the equation is exactly identified)?

b.    What is/are the first stage regressions? ( assume the excluded exogenous are x4t etc. and proceed). 

c.     What is/are the second stage regressions, (please explain and demonstrate why 2SLS estimates are consistent). 

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 Q2. The linear regression model in vector form is given as,y1 = Y1 1 + X1β + o1 = Z161 + o1 where y1 is a T × 1, Y1 is a T × g1 , 1 is a g1 × 1, X1 is a T × k1, β is a k × 1 and o1 is a T × 1). Let X2  be a T × k2, matrix that represents the excluded exogenous variables which appear in the other equations in the system.

Let X = [X1, X2], Z1 = [Y1, X1] and 61 = ( 1, β1). The 2SLS estimator is given as, 61,2sls =

(Zr PXZ1)—1Zr PXy1 where PX = X(Xr X)—1Xr .

a.    Suppose the model is just identified show that the above estimator reduces to ^61,2sls = (Xr Z1)—1Xr y1

b.     Given the information in part "a", let W be a set of instruments and suppose W = X what can we conclude about 2SLS and the simple IV estimator when the model is just identified? 

c.    Explain how to conduct a Hausman's specification test. Discuss the properties of the estimators, H0,

H1, the level of significance, the test statistics, the distribution of the test statistics and the conclusion.

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 Q3. Assume the following model,  yt = βxt + ot, for t = 1,............ , T  assume that xt is endogenous and you find an instrument rt. The first stage regression is,

 xt = çrt + ut, for t = 1,....... , T

Assume that  N (0, σ2) and u iid N (0, σ2 ). Assume that the instrumental variable is irrelevant (using an F-test we cannot reject the null that ç = 0).