ECON 301: Econometrics - Instrumental Variables Estimation Outline

This document provides a comprehensive outline of instrumental variables (IV) estimation in econometrics, covering essential concepts and techniques. It begins with an introduction to the identification problem, emphasizing the need for instruments and the characteristics of a good instrument. The outline then delves into the IV and two-stage least squares (2SLS) estimators, detailing their construction, consistency proofs, and a comparison of their performance relative to ordinary least squares (OLS). Key topics include the conditions for valid instruments (relevance and exogeneity), the weak instruments problem, and the application of IV estimation in multiple variable scenarios, including multiple exogenous and endogenous variables, and multiple instruments. The outline also covers testing for endogeneity using Hausman and overidentification tests, and addresses sources of biased coefficients and standard errors. This outline is a valuable resource for understanding and applying IV estimation in econometric analysis.

Outline Chapter 15
1. The identification problem—when do we need instruments
2. What makes a good instrument—conditions
3. The IV/2SLS estimator—single variable case
a. How IV estimator is constructed
b. Proof that is consistent
c. How 2SLS estimator is constructed
d. Proof is same as IV estimator with single var
4. Comparison of OLS and IV estimators
a. Comparing bias when have weak instruments
b. Comparing standard errors
5. Multiple variable case
a. Multiple exogenous vars
b. Multiple instruments
c. Multiple endogenous vars and multiple instruments
6. Testing for endogeneity
a. Endogeneity of X: Hausman test
b. Endogeneity of Z: overid tests

Primary Concerns in Estimation:
1. Biased coefficients—incorrect magnitude/sign
2. Biased standard errors—efficiency, incorrect inferences
Sources of biased coefficients
1. Mis-measured X / errors-in-variables—attenuation bias (bias to zero)
2. Omitted Variables (Z X, Z Y and therefore if omit Z is in error)
3. Reverse causation (X Y, Y X)
Chapter 13/14: panel models one way to deal with time invariant forms of
omitted variables.
Chapter 15: another method for dealing with omitted variables –
instrumental variables (IV). IV can be used to solve error-in-variables and
simultaneous causality problems as well as omitted variables.
The basic idea:
If x is correlated with u, we can think about decomposing x into two
components,
(1) the part that is uncorrelated with u and
(2) the part that is correlated with u.
If we can find information that allows us to isolate the first part we can use
that part of the variation in x to consistently estimate β1
In Chapter 13/14, relied on assumption that it is often the fixed (time
invariant) part of X that is correlated with u. Estimating with dummy
variables removed that variation.

The Basic Model
yi =β0 +β1xi +ui and (yi , xi, zi ) i= 1,..., n
where i denotes entities, y is the dependent variable, and x is an explanatory
variable for each entity and z is an instrument.
If Cov(xi ,ui ) ≠ 0 the OLS estimator is inconsistent.
IV uses an additional variable z to isolate the part of x that is
uncorrelated with u.
Conditions for Valid Instruments
(1) Instrument Relevance Cov(zi, xi) ≠ 0
(2) Instrument Exogeneity Cov(zi, ui) = 0
Together these imply that Z only affects Y through X
Note: We can test whether Cov(zi, xi) ≠ 0 (How?)
We usually cannot test whether Cov(zi, ui) = 0 (Why not?)

Identification—Construction of IV estimator in single variable case
Identification of a parameter in this context means that we can write β1 in
terms of population moments (parameters) that can be estimated using
sample data.
From
yi =β0 +β1xi +ui and (yi , xi, zi ) i= 1,..., n
Recall that β1 = Cov(yi, xi)/Var(xi) =  xy/ x2
We can write this in terms of how vary with z:
Cov(yi, zi) = β1Cov(zi, xi) + Cov(zi, ui)
Cov(zi, ui)=0 so
β1 = Cov(yi, zi) / Cov(zi, xi) =  zy/ zx
Sample analog: 




 ))((
1
1
))((
1
1
ˆ 1
xxzz
n
yyzz
n
ii
iiIV

Again note: if Z=X, then get OLS estimator
In matrix notation:

Large Sample Properties—Is this a consistent estimator of beta?
1
?
1 ˆ
ˆ
ˆ 


 p
zx
zyIV 
Work with numerator:




































))((
1
))((
1
))((
))((
ˆ
1
))((
ˆ
1
))((
1
ˆ)1(
1
))((
1
))((
))()()((
1
1
))((
1
1
1
1
1
1
1
1
xxzz
n
uzz
n
xxzz
uzz
So
n
uzz
n
uzz
n
n
n
uuzz
n
xxzz
uuxxzz
n
yyzz
n
ii
ii
ii
iiIV
ii
zx
iizx
iiii
iii
iizy







Now apply LLN:
zx
izip
IV
uz
n


  

))((
1
ˆ 1
As N gets large, second term gets small assuming that Cov(z,u) = 0 
consistent estimator
In small samples, in practice usually biased. This is because if x is
correlated with u, in practice is very rarely case that z and u have exactly
zero correlation. Underlines importance of large n with IV for consistency.

The Two Stage Least Squares Estimator
Assumptions:
1. Linear in parameters yi =β0 +β1xi +ui
2. (yi , xi, zi ) are iid draws—random sampling
3. No perfect collinearity—rank condition
4. E(ui) = 0 and Cov(zi, ui) = 0—Exogenous IVs
 1-4 give us consistency
5. E(u2i|zi) =  2 Homoskedasticy  Efficiency.
If the assumption are satisfied, β1 can be estimated using a particular IV
estimator called two stage least squares (2SLS or TSLS).
2SLS
First stage:
iiiii zxvzx 1010 ˆˆˆ  
zi is exogenous iz10   represents the part of xi that can be
predicted by zi  this part is therefore also exogenous
The other part of xi is the vi  this is the part that must be related to ui
So 2SLS uses the exogenous part and disregards the vi
Second stage:
iii uxy  ˆ10  This gives us the 2SLS estimates of β0 and β1

Consistency of 2SLS
Is the same as the formula for the IV estimator we already wrote down and
proved was consistent? Does TSLSIV  ˆˆ  ?
Does this equal




 ))((1
))((1
ˆ
1
xxzz
n
yyzz
n
ii
iiIV

Does it converge in probability to  zy/ zx ?




 21
)ˆˆ(1
))(ˆˆ(1
ˆ
xx
n
yxx
n
i
iiTSLS

ii zx 10 ˆˆˆ  
Work with numerator:
    i i
zyiiiiiiyxii yzzyzzyxxnyx  ˆˆ)(ˆ)ˆˆˆˆ)ˆˆ(/1ˆ),ˆcov( 111010ˆ
Work with denominator:
    i i
ziiixi zzzzxxnx 22
1
22
1
2
1010
2
ˆ
2 ˆˆ)(ˆ)ˆˆˆˆ)ˆˆ(/1ˆ)ˆvar( 
So,
zx
zy
z
zyTSLS




 ˆ
ˆ
ˆˆ
ˆ
ˆ 2
1
1  since 21 ˆ
ˆ
ˆ
z
zx


 
With a single var these two estimators are identical

Comparison of Bias in OLS and 2SLS & Importance of Testing For
Instrument Relevance
Again, recall 2 conditions for a valid instrument
1. Instrument Relevance Cov(zi, xi) ≠ 0
2. Instrument Exogeneity Cov(zi, ui) = 0
The Weak Instruments Problem
Question: What if (1) x and z are weakly correlated and (2) z and u are
weakly correlated? Is using the instrument better than OLS or not?
Result: Weak correlation between x and z can lead to large asymptotic bias
even if z and u are only moderately correlated.
Go back to Consistency proof. At end showed
x
u
zx
izip
TSLS
xiziCorr
uiziCorr
xiziCov
uiziCov
uz
n







.
),(
),(
),(
),(
))((
1
ˆ
1
11 
because Corr = Cov/stddev
Asymptotic bias will be big when
(1)zi and ui are highly correlated,
(2) zi and ui are not very correlated, but zi and xi are not very
correlated either
Recall that showed for OLS:
x
uii
p
OLS uxCorr


 .
),(ˆ 1 

So which one will be less biased depends on the relative magnitude of these
correlations.
Rule of Thumb Test: If F statistic for z vars (test that coeffs on zs are all
equal to zero in first stage where regress x on the z’s) is less than 10, you
have weak instruments
More complex forms of this test when have multiple instruments, multiple
endogenous variables (Bound, Jaeger Baker; Shea; Anderson test stats)

2SLS Estimation in Multiple Variable Case—One Endogenous
Explanatory Variable, 1 instrument, multiple exogenous vars
Digression:
How do we come up with estimators? One method is least squares method—OLS came from minimizing
sum of squared errors Another method is known as “method of moments”. A different class
of estimators. These come from matching up sample statistics
(functions of data) to some function of population parameters. Turns
out (surprise) OLS is also a MOM estimator (covariance/variance)
Going to show a MOM estimator here Later, we’ll also describe maximum likelihood estimators. There pick
estimators by choosing parameters that maximize likelihood of
drawing our particular sample. Will do later with binary dependent
variable models.
Notation: Here use y for the dependent var, x for endogenous independent
var, w for exogenous independent vars, and z for the instruments.
Wooldridge uses y for the dependent var and all endogenous independent
vars, z for exogenous independent vars and all instruments.
Again: endogenous means correlated with error
Suppose model is yi =β0 +β1x1i + β2w1i+ui
This is the structural model: β1 represents the causal effect of x1 on y
If x1 is endogenous (correlated with u), then all coefficients will be
biased.
Recall need an instrument z that is both exogenous and relevant
How do we express these conditions in the multiple variable case?

1st condition: Instrument Exogeneity: Need instrument for z for x1 where
E(u) =0, Cov(w1, u) = 0 and cov(z, x1) = 0
Express as “Moment conditions”—get estimators for β0 ,β1, β2 by solving



i
iiii
i
iiii
i
iii
wxyz
wxyw
wxy
0)ˆˆˆ(
0)ˆˆˆ(
0)ˆˆˆ(
12110
121101
12110



2nd condition: Instrument Relevance: Need z to be correlated with x1, though
now we have to also take w1 into account as well.
Easiest to write this relevance condition down by writing the reduced form:
iiii vzwx  2110  --Endogenous variables as functions of
ONLY exogenous variables
Need π2  0
Note that this reduced form is also the 1st stage. All of this generalizes
easily if have multiple ws.

2SLS Estimation in Multiple Variable Case—Single Endogenous
Explanatory Variable, Multiple Instruments
First stage:
iiijjiikkii xvzzwwx ˆ...... 111101  
This is also called the “Reduced Form”
Run a regression of endogenous independent var on ALL exogenous
vars (instruments and other exogenous vars in model)
Need to have at least one π  0
If we have just one instrument (as before), we say that the model is
“just identified” or “exactly identified”
If we have more than one instrument, we say the model is
“overidentified”
Second stage:
ikikiii uwwxy  11210 ...ˆ 