Question 3. Consider the following population regression model relating the dependent variable Yi and regressor Xi : y1i
Posted: Thu May 05, 2022 6:29 am
Question 3. Consider the following population regression model
relating the dependent variable Yi and regressor Xi : y1i = α1y2i +
β1z1i + u1i for i = 1, ..., n. Also, assume that y2i = α2y1i +
β2z2i + u2i . You observe (y1i , y2i , z1i , z2i) for i = 1, ...,
n.
a) Suppose you are interested in learning the value of α1.
Explain why you should not run a regression of y1i on (y2i , z1i)
to do so.
b) Explain how you can get a consistent estimator of α1 using
the data described above.
c) Solve for the “reduced form” of this model by finding
coefficients π11, π12, π21, π22 and error terms v1i and v2i such
that you obtain y1i = π11z1i + π12z2i + v1i and y2i = π21z1i +
π22z2i + v2i .
d) Explain how you could use estimates of π11, π12, π21 and π22
to get an estimate of α1.
relating the dependent variable Yi and regressor Xi : y1i = α1y2i +
β1z1i + u1i for i = 1, ..., n. Also, assume that y2i = α2y1i +
β2z2i + u2i . You observe (y1i , y2i , z1i , z2i) for i = 1, ...,
n.
a) Suppose you are interested in learning the value of α1.
Explain why you should not run a regression of y1i on (y2i , z1i)
to do so.
b) Explain how you can get a consistent estimator of α1 using
the data described above.
c) Solve for the “reduced form” of this model by finding
coefficients π11, π12, π21, π22 and error terms v1i and v2i such
that you obtain y1i = π11z1i + π12z2i + v1i and y2i = π21z1i +
π22z2i + v2i .
d) Explain how you could use estimates of π11, π12, π21 and π22
to get an estimate of α1.