1
Economics: Empirical Study and Forecasting III (part 2) Homework 1
Please submit it in group of maximum 4 persons in 2nd December, 2016.
Problem 1
Let y1 and y2 be scalars, and suppose the structural model is
1 1 1 ( 2) 1 1,
y z g y u E u z
1 0,where g(y2) is a 1G1 vector of functions of y2 and z contains at least one element not in z1. For example, g y( 2)(y y2, 22) allows for a quadratic. Or g(y2) might be a vector of dummy variables indicating different intervals that y2 falls into.
Assume that y2 has a linear conditional expectation, written as
2 2 2,
y z v E v z
2 0,(Remember, this is much stronger than simply written down a linear projection.) Further, assume (u1, v2) is independent of z. (This pretty much rules out y2 with discrete characteristics.)
a. Show that E y z y( 1 , 2)E y z v( 1 , 2)z1 1 g y( 2)1E u v( 1 2).
b. Now add the assumption E u v( 1 2)1 2v . Propose a consistent control function estimator of 1, 1 and 1.
c. How would you test the null hypothesis that y2 is exogenous? Be very specific.
d. How would you modify the CF (Control Function) approach if
2 2
1 2 1 2 1 2 2
( ) ( ),
E u v v v where 22 E v( 22) ? How would you test the null hypothesis that y2 is exogenous?
