IV/2SLS

Question 1

When is xi endogenous and the OLS estimator biased and inconsistent?

Answer 1

When E(εixi)≠0. Then we need a new estimator that has at least the desirable large sample, like IV, 2SLS, GMM.

Question 2

Omitted variable bias definition:

Answer 2

Omitted variable bias occurs when a relevant variable is left out of a regression model and is correlated with an included regressor, leading to biased and inconsistent OLS estimates. The bias arises because the omitted variable’s effect is incorrectly absorbed into the error term, violating the exogeneity assumption.

Question 3

Measurement error bias definition:

Answer 3

Measurement error bias arises when variables in a regression model—typically the regressors—are measured with error, causing the observed values to differ from the true ones. This leads to a correlation between the regressor and the error term, resulting in attenuation bias and inconsistent OLS estimates.

Question 4

Instrumental variable IV model assumptions:

Answer 4

Consider yi=xi’β+εi and E(εixi)≠0.
1: Linearity, zi is an Lx1 vector of instrumental variables
2: Relevance: E(zixi’) has full column rank K. For relevance we need: Cov(zi,xi)≠0 and Cov(zi,εi)=0
3: Exogeneity: E(ziεi)=0
4: Errors are homoskedastic and non-autocorrelated: Var(εi|zi)=σ^2 and Cov(εi,εj|zi)=0
5: Random sampling

Question 5

When do we use the IV estimator?
And when do we use the 2SLS estimator?

Answer 5

IV: When L=K, then E(zixi’) has full column rank and zixi’ is invertible (Exactly identified; as many instruments as there are endogenous variables)
2SLS: When L>K, then the inverse of Z’X does not exist (overidentified; more instruments than endogenous variables)

Question 6

IV form and properties:

Answer 6

βIV^=(Z’X)^-1Z’y
1: βIV^ is biased
2: βIV^ is consistent
3: βIV^ is asymptotically normal with Var(βIV^)=s.e.(βIV^)=s^2(Z’X)^-1Z’Z(X’Z)^-1

Question 7

2SLS form procedure:

Answer 7

1: Select K+1 optimal instruments, which is a linear combination of L+1 instruments
2: Pz=Z(Z’Z)^-1Z’, X^=PzX, so Pz projects X onto the span of Z and z is orthogonal to ε
3: Stage 1: Construct X^
Stage 2: Apply OLS on y and X^, so we get β2SLS^=(X^’X^)^-1X^’y

Question 8

If L=K what is then the relation between IV and 2SLS estimator?

Answer 8

β2SLS^=βIV^

Question 9

Hausman test:

Answer 9

H0: βOLS^ and β2SLS^ are consistent estimators
H1: β2SLS^ is consistent, βOLS^ is not consistent
Test-statistic:
H=(β2SLS^-βOLS^)/√(s.e.(β2SLS^)^2-s.e.(βOLS^)^2).
Reject H0 if |H|>φ(1-α)

If we reject H0, there is evidence of endogeneity, and therefore OLS is inconsistent –> use IV/2SLS