econometrics

ECON 140

Section 13, November 28, 2013
ECON 140 - Section 13

1

The IV Estimator with a Single Regressor and a Single Instrument

1.1

The IV Model and Assumptions

ˆ Consider the univariate linear regression framework: Yi = β0 + β1 Xi + ui
ˆ Until now, it was assumed that E (ui |Xi ) = 0, i.e. conditional mean independence.
ˆ Let's relax this assumption and allow the covariance between Xi and ui to be dierent from zero.
ˆ Our problem here is that ui is not observed.
ˆ Doing OLS yields inconsistent estimates (remember the OVB formula).
ˆ In this case we refer to Xi as an endogenous variable.
ˆ The way to get consistent estimates is to use an instrument, which is a variable that satises the

following two properties:

1. Relevance: Cov (Zi , Xi ) = 0.
2. Exogeneity: Cov (Zi , ui ) = 0.
ˆ In words: since the variation of Xi is contaminated (it is correlated with the variation of ui ), it

follows that we need a variable that allows us to get variation in Xi that is clean, i.e. it holds ui
xed.

1.2

The Two Stage Least Squares Estimator

ˆ Since the OLS estimator doesn't yield consistent estimates, we need an estimator that uses the

instrument and yields consistent estimates.

ˆ This estimator is called Two Stage Least Squares (TSLS).
ˆ This is how it works:

1. In the rst stage, regress Xi on a constant term and Zi : Xi = π0 + π1 Zi + vi .
2. In the second stage, regress Yi on a constant term and the predicted values from the rst
ˆ
regression, Xi ≡ π0 + π1 Zi .
ˆ
ˆ
ˆ The idea is the following: in the rst stage, you want to get rid of the variation of Xi that is

correlated to ui .

ˆT
ˆT
ˆ In the second stage, you use this variation, which is clean, to get the estimates β0 SLS and β1 SLS ,

which are consistent.

1

ECON 140

Section 13, November 28, 2013

ˆ Consider what happens if any of the two conditions on the instrument fails:

1. No relevance: then π1 = 0, and it essentially means that