Advanced Econometrics I
EMET4314/8014
Semester 1, 2025
Assignment 7
(due: Tuesday week 8, 11:00am)
Exercises
Provide transparent derivations. Justify steps that are not obvious. Use self sufficient proofs. Make reasonable assumptions where necessary.
The linear model under endogeneity is
Y = Xβ + e
X = Zπ + v
where E(eiXi) ≠ 0 and E(eiZi) = 0. Notice dim X = N × K, dim β = K × 1, dim Z = N × L, dim π = L × K, and dim v = N × K.
The source of the endogeneity is correlation between the two error terms, write
e = vρ + w
where E(viwi) = 0. Notice dim ρ = K × 1, and dim w = N × 1.
Combining, we obtain
Y = Xβ + vρ + w (1)
(i) You have available a random sample (Xi
, Yi
, vi). You are running a regression of Y on X and v. Using linear algebra, define the OLS estimator of β in equation (1). Call it .
(Hint: Use the partitioned regression result on the next page.)
(ii) Prove that = β + op(1).
(iii) You do NOT have available a random sample (Xi
, Yi
, vi). Instead, you have available a random sample (Xi
, Yi
, Zi). You cannot run a regression of Y on X and v, but you can instead run a regression of Y on X and ˆv where ˆv is the first stage residual.
Using ˆv in place of v in equation (1), define the OLS estimator of β using linear alge-bra. Call it .
Prove or disprove: = (X′PZX)
−1X′PZY .
(iv) Which estimator do you prefer: or ? No need to prove anything here, just give a quick intuitive statement.
Partitioned Regression and Frisch-Waugh-Lovell Theorem
Partition the linear regression model like so:
Y = Xβ + e
= X1β1 + X2β2 + e
where X1 is of dimension N × K1 and X2 is of dimension N × K2 with K1 + K2 = K and X = [X1 X2]. Then how could you estimate β1? Write down the normal equations
Solving first for
Similarly
This has an interesting interpretation:
The OLS estimator results from regressing Y on X2 adjusted for X1. This ad-justment is crucial, obviously it wouldn’t be quite right to claim that results from regressing X2 on Y only. That would only be true of = 0 which means that the sample covariance between the two sets of regressors is zero. Now, doing the math by plugging into and letting and M2 = I − P2:
Multiplying both sides by and moving terms
The end result (and also symmetrically for ):
Remember that M1 and M2 are residual maker matrices:
At the same time M1 and M2 are symmetric and idempotent
(that is )
There’s a lot of intuition included here. This harks back all the way to Gram Schmidt orthogonalization. To obtain , you regress a version of Y on a version of X1. These versions are and . These are the versions of Y and X1 in which the influence of X2 has been removed, or partialled out or netted out. If X1 and X2 have zero sample covariance then = Y and = X1 and we only need to regress Y on X1 to obtain .