Transcript Methods of Economic Investigation: Lent Term: First Half

Omitted Variable Bias
Methods of Economic
Investigation
Lecture 7
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Today’s Lecture
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Review Regression Framework
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Linking last term to this term
Understanding conditional expectations
Covariates Added and Removed
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Why is this “Controlling”
What if you leave something out?
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Review: This Course So Far
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Do we know what the effect of X is on y?
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Defining concepts of “treatment” and “control”
Experimental settings: Explaining the
usefulness of Random Assignment
Non-Experimental settings: variation is `as if’
random
In Theory, all you need to do if you have
random assignment is take a difference in
means
 Go backwards a bit…
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General Goal in Causal Inference
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E(Y| T=1) – E(Y | T=0) or
E(Y| T=1, X) – E(Y | T=0, X)
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So we need to know how to estimate a
conditional expectation function
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We’ll worry about if can be causally interpreted
later—right now let’s just if we can even show
how we could estimate it
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What is a conditional expectation
The conditional expectation function (CEF)
is the population average of Y with X held
fixed
 CEF =E[Yi | Xi]
 It is a function of X and because X is a
random variable, the CEF is itself random
 We have been doing the special case
where the CEF only takes 2 values
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E[Yi | Ti=1] and E[Yi | Ti=0]
In practice, could be continuous but we’ll come back to
that
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Defining the CEF
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For a continuous Y with a density fy(. | X=xi)
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Let’s ground this in reality a bit:
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The expectation is a concept because we rarely
observe the entire population
Generally we’re going to use data to make
inferences about the sample CEF
Last term you proved under what conditions and
assumptions this sample CEF converges to the
population CEF
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How is Y related to the CEF
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The observed outcome Y can be
decomposed into the bit that is explained
by X (the CEF) and the unexplained bit ε
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Then last term you learned that
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CEF is the “best” (in a MMSE sense) summary
of the relationship between Y and X
Can also decompose the variance usefully into
explained variance and unexplained variance
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How is OLS related to this?
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If the true CEF is linear, it’s the sample
estimate
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Best Linear Predictor (BLUE)
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Example joint normality
Saturated model (different parameter for every
possible combination of values for X)
Best we can relative to other linear estimates
Common justification—local linear approximation
Minimizes Mean Squared Error
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Implies it is the best linear approximation to CEF
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This should be familiar
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You are probably used to thinking OLS as
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Turns out that is the thing is the
parameter we need to estimate the CEF
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And the sample analog that we estimate
will converge to the true β (under
appropriate conditions
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How do we establish Causality?
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Conditional Independence Assumption
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In words: on average, outcomes would be the
same in the treatment and control groups
In Math: Let Y1 be potential outcome if T=1 and
Y0 be potential outcome of T=0
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Can write Y = Y0 + (Y1 – Y0) *T
Then CIA: {Y0i,Y1i} independent of Ti (or Ti | Xi)
Or E[Yi | T=1 ] – E[Yi| T=0] = E[Y1i – Y0i]
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Intuition: since we never observe both Y0
and Y1 for any one individual, we want to
think on average, the two groups are the
same
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Selection Bias
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The Reason we need CIA: Selection bias
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In words: different people choose to be in the
treatment or control group. The underlying
differences in these people would generate
differences in outcomes regardless of the
treatment
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E(Y| T=1) – E(Y1 | T=0)
=E(Y - Y0 |Ti=1) +{E(Y0 | T=1) – E(Y0 | T=0)}
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What is generating the bias?
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If it’s selection on observables
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we can condition on X—estimate the CEF
Convenient that the regression can partial this out: let’s
see how
Simple example: 2 variables, T and X, each of
which are either zero or one. So, the CEF can
have 4 values
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E[Y|
E[Y|
E[Y|
E[Y|
T=0,
T=1,
T=0,
T=1,
X=0]=α
X=0]= α+β
X=1]= α+γ
X=1]= α+β+γ+δ
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Extracting β from the estimated CEF
Imagine we estimate each CEF’
 Write a single equation for the CEF
E[Y | T, X] = α + βT +γX+δ(T*X)
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Main effect of T Main effect of X
Interaction
What is β = E[Y| T=1, X=0]- E[Y| T=0, X=0]
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Why is X fixed? So we can get β out regression
Generalize this to a continuous X, and we just
hold it at some fixed point (e.g. it’s mean)
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What if T didn’t vary with X?
 If X has some fixed association with Y
independent of T, then the CEF only has 2
values:
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E[Y|
E[Y|
E[Y|
E[Y|
T=0,
T=1,
T=0,
T=1,
X=0]=α
X=0]= α+β
X=1]= α
X=1]= α+β
[for ease we can set γ and δ to zero…]
 In that case—we’re ok and can exclude X.
But we can increase precision because we
get 2 estimates of the CEF if we include X
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What T and X are correlated?
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Then we’ve got a problem
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E[Y|
E[Y|
E[Y|
E[Y|
T=0,
T=1,
T=0,
T=1,
X=0]=α
X=0]= α+β
X=1]= α+γ
X=1]= α+β+γ+δ
If we don’t have X: E[Y | T=1] has two
potential values and we cannot say:
E[Y | T=1] –E[Y | T=0 ≠ E[Y1 - Y0 | T=1]
 Another way to say this is we’ve got
“selection on unobservables”
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Omitted Variable Bias Formula
Simply from before, suppose δ=0 so our true
CEF is:
E[Y | T, X] = α + βT +γX
(REMEMBER: This means Y = E[Y | T, X] +e)
 We estimate
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What is happening when we estimate?
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What this is doing is given observed Y’s for the
various values of T it is choosing the α and β that
minimize the mean square error
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intuitively, the bias happens because the values
of Y’s are changing not only with respect to T but
also the unobserved X
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In more formal terms we get:
Should look
familiar
True β (this is
what we want)
Relationship
between X and Y
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What is relationship between X and T
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Pretend we could observe X…then w could
think about estimating
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Well, we know, since this is OLS and
assuming no OVB in this equation that
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So our omitted variable bias formula can
be re-written as:
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Signing your bias
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Generally, the concern with OVB is that it
biases your estimate upwards.
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The sign of the bias will be as follows:
γ<0
γ>0
θ<0
positive
negative
θ>0
negative
positive
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What did we learn today
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Conditional Expectation Functions (CEFs)
are how we determine “causal effects”
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Regressions allow the “best” way to
estimate these CEFs
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Selection on observables is not something
that will bias our estimates
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Selection on unobservables is a problem
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Next Class
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Review of OVB
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Error Correction Models
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Fixed Effects
Random Effects (interaction terms)
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