The Instrumental Variables Estimator
The instrumental variables (IV) estimator is an alternative
to Ordinary Least Squares (OLS) which generates consistent
estimates when the RHS variables are correlated with the
error term.
Although the IV estimator will produce consistent estimates
they will typically be less efficient (have higher variance) than
the OLS estimates.
In the case of the simultaneous equations model, the equation
must be identified before we can make use of the IV estimator.
Instrumental Variables
We wish to estimate:
Yi   X i  ui
but cov  X i , ui   0
Suppose we find a variable Z such that:
cov  Zi , ui   0 but cov  X i , Zi   0
We define the instrumental variable estimator to be:
ˆIV 
Z Y
Z X
i i
i
i
Consistency of the IV estimator
We have
Zi   X i  ui 
1/ N  Ziui

ˆ
 IV 

1/ N  Zi X i
 Zi X i
Therefore:
cov  Z i , ui 
ˆ
plim  IV   

cov  X i , Z i 
Since Zi and ui are uncorrelated by assumption.
Variance of the IV estimator
 
var ˆIV 
 u2  Zi2
 Z X 
i
where ˆ ZX
i
2

 u2
2
2
ˆ ZX
X
 i
is the sample correlation between X and Z.
This shows that the IV estimator must always have a higher
variance than the OLS estimator.
The lower is the correlation between X and Z then the higher
will be the variance of the IV estimator.
The trade-off between bias and variance
It is not always obvious that we want an unbiased estimator at
all costs.
An alternative criterion is to look for the estimator with the
lowest mean square error.


 E  ˆ  E  ˆ  
MSE  E ˆ  
2
2

 
 E   E ˆ
 Variance + bias 2
2
The diagram shows the PDF for an unbiased estimator (the
red line) and the PDF for a biased estimator (the blue line).
The estimate obtained from the biased estimator is likely to
have a lower mean square error.
Example: Estimating the consumption function
The simplest Keynesian model has a pair of equations for
consumption and income:
ct  1   2 yt  ut1
yt  ct  it  ut 2
These are the structural equations of the model. The reduced
form equation for income can be derived as:

yt 
 1  it  ut1  ut 2 
1  2
The reduced form equation shows that the change in income
is correlated with the error term from the consumption function.
It follows that OLS estimation of the consumption function will
be inconsistent.
An alternative is to use the change in investment as an instrument
and use the IV estimator. The change in investment is correlated
with the change in income but there is no reason to think it should
be correlated with the error from the consumption function.
The Two Stage Least Squares Estimator
If we have more potential instruments then there are endogenous
variables then we can use the two stage least squares TSLS
estimator:
• Regress the endogenous variable on all the instruments and
compute the fitted values.
• Use these fitted values as instruments in a second stage
regression.
Stage 1:
Xˆ  ˆ1Z1  ˆ 2 Z2
Stage 2:
Xˆ Y


 Xˆ X
i i
i
i
The Hausman Specification Test
1. Regress the potentially endogenous variable X on the instrument
Z and calculate the residuals.
2. Regress Y on X and the residuals created at stage 1.
3. Test the significance of the residuals in the regression
estimated at stage 2.
Hausman, J. A. (1978) ‘Specification Tests in Econometrics’,
Econometrica, Vol 46, pp. 1251-1271.