error component models methods of economic investigation lecture 8 1
TRANSCRIPT
Today’s Lecture Review of Omitted Variables Bias
Error component models Fixed Effects Random Effects
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Omitted Variable Bias Formula
True Relationship Y = α + βT +γX+ε We estimate
We get a bias from the omitted variable
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A specific form of this… Suppose we had to estimate
Where the error term can be decomposed into:
ijijij XY
ijjiij
Individual specific factor Going to set this to zero for ease of exposition
Group specific factor
Random error term uncorrelated with everything else
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What are the problems with this? Errors have cross-correlation across terms
OLS is not efficient Still get consistent estimates, but may be
harder to do inference
If X is correlated with γ or δ and we cannot observe this factor, then our OLS estimates will be biased
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Unobserved effects are not correlated with X’s
This can happen if the group effect is ‘random’ Maybe different geographic areas, not correlated
with X Different within-group correlations, violate OLS
assumptions that errors uncorrelated with each other
Don’t have to worry about OVB
Still have problems with standard error OLS not efficient Can’t do inference with our estimates
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OLS standard errors Usually estimate
Standard way to estimate variance of OLS estimates not valid—won’t necessarily be the same across j’s:
ijijijjijij XXY
1
,
2 ')ˆ(
jiijijXXV
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Robust Standard Errors Estimate the residuals:
Then estimate standard errors
Consistent, but may be able to do better…
ijijij XY
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'''')ˆ(
ijijij
ijijijijij
jjj XXXXXXV
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Exploiting the Random Effects Structure We have more information than we’re
using in the robust standard errors
The composite error means there is correlation within groups j but not between groups
Define where]'[ jjE ),...,( KJiJj
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FGLS Estimates—2 steps Estimate OLS and get
Get weighting matrix
Can now adjust the regression for the known correlation
OLSijijij XY ˆˆ
j
jjJ]ˆ'ˆ[
1ˆ
jjj
jjjOLS YXXX 1
1
1 ˆˆ̂
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FGLS Variance Estimate With a variance
If our model is correctly specified, this will simplify to:
1
111
1
1 '')ˆ(
jjj
jjjjj
jjj XXXXXXV
1
1)ˆ(
jjj XXV
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Simplify, let So we need to estimate:
When might this happen? Individuals are part of some “group” and that
group has a unique relationship to the outcomes (relative to other groups)
Looking in a time period—that time period is related to the outcome differently than other time periods
ijjij
ijjijij XY
Unobserved effects are correlated with X’s
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How might we fix this? If we could observe multiple individuals in
a group j, then we could difference out the group effect
So for example: look at individuals 1 & 2, both of whom are in group 1
1111111 XY
2112121 XY14
Differencing out Group Effects Difference between individuals 1 and 2,
gives us the following estimate:
How do we interpret this? β tells us how a change in the difference in X’s
between people 1 and 2, changes the difference in outcomes between people 1 and two
Often don’t really care about difference between 2 people…
)()( 112111211121 XXYY
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Average Group Effect Take the sample average outcome for
each group
Crucial assumption: average group effect is the group effect
This means: NO within group variation
jjjj XY
jNi
jjj
j N 1
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Estimating Deviation from the Average Group Effect Difference individual and group average
equations:
Define ~ terms as deviation from means so we estimate:
So we can estimate this with our usual OLS (more on the standard errors in a few minutes…)
)()()( jijjjjijjij XXYY
ijijij XY ~~~
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How do ‘fixed effects’ help? Define δj=1 for group j and zero for all
groups
Then we estimate:
And we will get an estimate for β1, our effect of X on Y, and δ our group effect
ijjijij XY ˆˆˆˆ10
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Estimation To see how, imagine we have two groups:
men and women
If we estimated two equations then its:
We won’t recover β0 separate from the δ’s
imimmim XY ˆˆˆˆ10 ififfif XY ˆˆˆˆ
10
constant, cm constant, cf
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Multiple Equation Estimation Let men be the base group, then for men
we estimate:
What does this give us? CEF: E[Y| X, j=m] = δm + β*E[X | j=m]
δm is the intercept for men or E[Y |X=0, j=m] If X is de-meaned, then this is the average effect
for men at the mean of the sample
imimmim XY ˆˆˆˆ 0
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Why estimate a single equation?
Single equation easy to estimate: run a regression for each group separately BUT
Hard to do inference on size of group effect Lose some power if, conditional on group—the
effect of various X’s is the same
For example: Men and women on average have different starting
wages Conditional on starting level, each year of
additional experience, increases wages by some fixed %
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Single equation estimation Now estimate for everyone:
For men this estimates
For women
imimmim XY ˆˆ)ˆˆ( 10
ijijmjfjmjij XY 10 )()(
ijijjm Xfemalec 1)(
ififmfmif XY ˆˆ)ˆˆ()ˆˆ( 10
Constant: E[Y |j=m, X=0]
Additional “female” effect
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How to interpret your “fixed effect”? The additional effect of women, relative to
men
Do we care about the fixed effects? Sometimes they have an interpretation e.g.
controlling for all observable factors, is there a different level of wages for men and women
Sometimes they are a just differencing out a bunch of things, and don’t have an interpretation, e.g. repeated observations on an individual, difference out an individual effect—not much interpretation for idiosyncratic individual effect 23
What did we learn today When have unobserved group effects can
be two issues: Uncorrelated with X’s: OLS not efficient, can fix
this with GLS Correlated with X’s: OVB, can include “fixed
effects”
Fixed effects, within-group differences, and deviation from means differences can all remove bias from unobserved group effect
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