Error Component Models

Methods of Economic Investigation

Lecture 8

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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

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,

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:

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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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What does a group specific term do?

]2|[ jYE ij

]3|[ jYE ij

]1|[ jYE ij

X

Y

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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

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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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Next Class Application: The effect of Schooling on

wages Ability Bias Fixing this with “twins” and “siblings” models

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