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
Consider the univariate linear regression framework: Yi = β0 + β1 Xi + ui
Until now, it was assumed that E (ui |Xi ) = 0, i.e. conditional mean independence.
Let's relax this assumption and allow the covariance between Xi and ui to be dierent from zero.
Our problem here is that ui is not observed.
Doing OLS yields inconsistent estimates (remember the OVB formula).
In this case we refer to Xi as an endogenous variable.
The way to get consistent estimates is to use an instrument, which is a variable that satis es the
following two properties:
1. Relevance: Cov (Zi , Xi ) = 0.
2. Exogeneity: Cov (Zi , ui ) = 0.
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
Since the OLS estimator doesn't yield consistent estimates, we need an estimator that uses the
instrument and yields consistent estimates.
This estimator is called Two Stage Least Squares (TSLS).
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
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
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
Consider what happens if any of the two conditions on the instrument fails:
1. No relevance: then π1 = 0, and it essentially means that
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
Consider the univariate linear regression framework: Yi = β0 + β1 Xi + ui
Until now, it was assumed that E (ui |Xi ) = 0, i.e. conditional mean independence.
Let's relax this assumption and allow the covariance between Xi and ui to be dierent from zero.
Our problem here is that ui is not observed.
Doing OLS yields inconsistent estimates (remember the OVB formula).
In this case we refer to Xi as an endogenous variable.
The way to get consistent estimates is to use an instrument, which is a variable that satis es the
following two properties:
1. Relevance: Cov (Zi , Xi ) = 0.
2. Exogeneity: Cov (Zi , ui ) = 0.
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
Since the OLS estimator doesn't yield consistent estimates, we need an estimator that uses the
instrument and yields consistent estimates.
This estimator is called Two Stage Least Squares (TSLS).
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
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
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
Consider what happens if any of the two conditions on the instrument fails:
1. No relevance: then π1 = 0, and it essentially means that