Week 3

Question 1

What condition doesn’t hold when there is endogeneity?

Answer 1

Orthogonality condition:

plim(1/n X’ε) ≠ 0

Question 2

What are the causes of endogeneity?

Answer 2

• Omitted variables
• Measurement errors
• Simultaneity

Question 3

What are the consequences of endogeneity?

Answer 3

• OLS estimator inconsistent (plim(b) ≠ β)
• Standard asymptotic tests not applicable
• In general, correlation between X and ε (OLS biased)

Question 4

What is the criteria for a good instrument?

Answer 4

• It doesn’t directly influence y, correlated with X and only influences y through X

Question 5

What are the requirements of 2SLS?

Answer 5

1. Z and ε uncorrelated -> plim(1/n Z’ε) = 0
2. Z correlated with X: plim(1/n Z’X) = Q(zx) - rank(Q(zx)) = k - full rank - rank condition
3. Number of instruments m>= k - order condition
4. Z “stable” and not “multicollinear” : plim(1/n Z’Z) = Q(zz)
   rank(Q(zz)) = m

Question 6

2SLS estimator

Answer 6

β^ = (X^’X^)^(-1)X^’y

= (X’P(z)X)^(-1)X’P(z)y , where P(z) = Z(Z’Z)^(-1)Z’

Question 7

Is 2SLS consistent?

Answer 7

Yes, plim(b) = β

Question 8

Asymptotic distribution of 2SLS

Answer 8

sqrt(n) (b - β) -> N(0, σ^2 (Q(xz)’Q(zz)^(-1)Q(xz))^(-1))

In sample: b ~ N(β, σ^2 (X^’X^)^(-1))

Question 9

2SLS vs IV estimator

Answer 9

b(2SLS) = (X’P(z)X)^(-1)X’P(z)y

b(IV) = (Z’X)^(-1)Z’y when m=k

Question 10

Instrumental variables as GMM estimators

Answer 10

ε(i) = y(i) - x(i)’β
Moment condition: E(z(i)ε(i)) = E(z(i)(y(i) - x(i)’β)

Gn(β) = Σgi(β)
= Σz(i)(y(i) - x(i)’β)
= Z’(y - Xβ)
Set Z’(y - Xβ) = 0 -> Solve for β

Question 11

What test tests for whether explanatory variables are exogenous?

Answer 11

Durbin, Wu & Hausman test

Question 12

What test tests whether instruments are valid?

Answer 12

Sargan test

Question 13

Why can’t we just simply compare b(OLS) to b(IV)?

Answer 13

Test suffers from possible finite sample problems:

OLS/IV test can have non-psd covariance matrix

Question 14

Durbin-Wu-Hausman Test

Answer 14

H0: exogenous

1) a) Regress y on x(exo) and x(endo) -> e(i) = y(i) - b1xi(exo) - b2xi(endo)
b) Regress x(endo) on Z -> v(i)^ = x(endo) = z(i)’γ^
2) Regress e on x(exo) and x(endo) and residuals v^
3) LM = nR^2 ~ χ2(1)

Question 15

Sargan Test

Answer 15

H0: Valid/exogenous instruments
LM-test

1) Apply IV on y = Xβ + ε with instruments Z -> e(IV) = y - Xb(IV)
2) Regress e(IV) on X, that is, perform OLS in the model ei(IV) = Zi’γ + ηi
3) LM = nR^2 ~ χ2(m-k)
- m = # instruments
- k = # endogenous variables

Only works when m>k

Question 16

Explain how to interpret outcome of Sargan test when null of Durbin-Wu-Hausman not rejected

Answer 16

H0 not rejected => don’t have sufficient evidence to reject endogeneity
Hence, validity of instruments not so relevant as OLS is the optimal method