2 e = 0. so in particular E (ei ) = 0 , E (zi ei ) = 0 and E zi i
(a) Should x2 i be treated as endogenous or exogenous? (b) Suppose we have a scalar instrument zi which satis…es xi =
2. Consider the structural equation yi =
0
+
1 xi
+Βιβλιοθήκη 2 2 xi+ ei
(1)
with xi treated as endogenous so that E (xi ei ) 6= 0. Assume yi and xi are scalar. Suppose we also have a scalar instructment zi which satis…es E (ei jzi ) = 0
0
+
1 zi
+ ui
(2)
with ui independent of zi and mean zero. 2 ) as instruments. Is this a su¢ cient number of instruments? Consider using (1; zi ; zi (Would this be just-identi…ed, over-identi…ed, or under-identi…ed)? (c) Write out the reduced form equation for x2 i . Under what condition on the reduced form parameters (2) are the parameters in (1) identi…ed?
2
Econometrics 710 Final Exam, Spring 2015 1. Consider the model yi = E (ei ) = 0 E (xi ei ) = 0 with both yi and xi scalar. Assuming > 0 and < 0, suppose the parameter of interest is 2 =2 . the area under the regression curve (e.g. consumer surplus), which is A = p !d N (0; V ) Let ^ = (^ ; ^ )0 be the least-squares estimates of = ( ; )0 so that n ^ ^ be a standard consistent estimate for V . You do not need to write out these and let V estimators. (a) Given the above, describe an estimator of A (b) Construct an asymptotic (1 ) con…dence interval for A ) percentile interval for A (c) Describe how to construct a bootstrap (1 + xi + ei
4. Consider the structural equation yi = x 0 1i E (zi ei ) = 0 where x2i is k2 1 and treated as endogenous. The variables zi = (x1i ; z2i ) are treated as exogenous, where z2i is `2 1 and `2 k2 . You are interested in testing the hypothesis H0 : Consider the reduced form equation for yi yi = x0 1i
(a) Write out this estimator ^ explicitly as a function of the sample (b) Find its probability limit as n ! 1 (c) In general, is ^ consistent for ? Is there a reasonable condition under which ^ is consistent?
3. Consider the structural equation and reduced form yi = xi = x2 i + ei zi + ui
2 with x2 i treated as endogenous so that E xi ei 6= 0. For simplicity we assume no intercepts. Assume yi ; zi ; and xi are scalar, and assume 6= 0. Consider the following estimator. First, estimate by OLS of xi on zi and construct the …tted values x ^i = ^ zi . Second, estimate 2 by OLS of yi on x ^i . [Added after the exam: Assume that E (zi ei ) = 0 and E (zi ui ) = 0 and consider adding extra conditions if helpful to answer the questions.]
1 0 + z2 i 2 2 1
+ x0 2i
2
+ ei
= 0:
+ vi
(3)
Show how to test H0 using only the OLS estimates of (3). Hint: This will require an analysis of the reduced form equations and their relation to the structural equation.