Instrumental Variables: Problems
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Transcript of Instrumental Variables: Problems
Instrumental Variables: Problems
Methods of Economic Investigation
Lecture 16
Last Time IV
Monotonic Exclusion Restriction
Can we test our exclusion restriction? Overidentification test Separate Regression Tests
Today’s Class Issues with Instrumental Variables
Heterogeneous Treatment Effects LATE framework interpretation
Weak Instruments Bias in 2SLS Asymptotic properties Problems when the first stage is not very big
Heterogeneous Treatment Effects Recall our counterfactual worlds
Individual has two potential outcomes Y0 and Y1
We only observe 1 of these for any given individuals
Define a counterfactual S now: S1 is the value of S if Z = 1
S0 is the value of S if Z = 0 We only ever observe one of these for any given
individual
The CounterfactualIndividual U
S1U =1 S0U=0
Y1U Y0U Y1U Y0U
Z=1 Z=0
S1U=0 S0U=1
Y1U Y0UY1U Y0U
ITT is E[Y | Z=1] – E[Y | Z=0] (red vs. orange)
Observed difference is E[Y | S=1] – E[Y | S=0] (light blue vs dark blue)
ATE is the E[Y1U – Y0U]: which node doesn’t matter if homogeneous effects. With heterogeneous effects, it’s the average across the nodes
LATE is E[Y | S1U=1, Z=1] – E[Y | S0U=1, Z=0]
Writing the first stage as counterfactuals We can now define our variable of interest
S as follows:
S = S0i + (S1i – S0i)zi=π0 + π1i Zi + νi
In this specification: π0 = E[S0i]
π1i = S1i – S0i
E[π1i] = E[S1i – S0i]: The average effect of Z on S—this is just our ATE for the first stage regression
Exclusion Restriction The instrument operates only through the
channel of the variable of interest With homogeneous effects we describe this as E[ηZ]=0
For any value of S (i.e. S = 0, 1) Y(S, 0) = Y(S, 1)
Another way to think of this is that the exclusion restriction says we only want to look at the part of S that is varying with Z
The set of potential outcomes Use the exclusion restriction to define
potential outcomes with Y(S,Z) Y1i = Y(1,1) = Y(1,0) = Y(S=1)
Y0i = Y(0,1) = Y(0,0) = Y(S=0)
Rewrite the potential outcome as:Yi= Yi(0,zi) + [Yi(1,zi) – Yi(0,zi)]Si
= Y0i + (Y1i – Y0i)Si
= α0 + ρiSi + η
S is the unique Channel through which the instrument operates
Monotonicity For the set of individuals affected by the
instrument, the instrument must have the same effect It can have no effect on some people (e.g.
always takers, never takers) For those it has an effect on (e.g. complier) it
must be that π1i >0 or π1i <0 for all i
where Si = π0 + π1i Zi + νi
In terms of the counterfactual, it must be the case that S1i ≥ S0i (or S1i ≤ S0i ) for all i
Back to LATE Given these assumptions
To see why note the following: E[Yi | Zi=1] = E[Y0i + (Y1i – Y0i)Si | Zi=1]
= E[Y0i + (Y1i – Y0i)S1i ]
E[Yi | Zi=0] = E[Y0i + (Y1i – Y0i)S0i ]
E[S |Zi =1] – E[S |Zi =0] = E[S1i – S0i] = Pr[S1i>S0i]
]0|[]|[]0|[]1|[
]0|[]1|[0101
iiiiii ESSYYEZSEZSE
ZYEZYE
LATE continued Substituting these equalities in to our
formula we get:
We are left with our LATE estimate
)Pr(
)]Pr()|[(
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)])([(
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01
010101
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0101
ii
iiiiii
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SS
SSSSYYE
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)|( 0101 iiii SSYYE
How to Interpret the LATE Remember we thought of the LATE as useful
because Y(S=1, Z=0) = Y(S=1, Z=1)= Y(S=1) In the case of heterogeneous effects this is not
true The LATE will not be the same as the ATE
Our estimate is “local” to the set of people our instrument effects (the compliers) Is this group we care about on it’s own? Is there a theory on how this group’s effect size
might relate to other group’s effect
Finite Sample Problems This is a very complicated topic
Exact results for special cases, approximations for more general cases
Hard to say anything that is definitely true but can give useful guidance
With sufficiently strong instruments in a sufficiently large finite sample—you’re fine
Weak Instruments generate 3 problems: Bias Incorrect measurement of variance Non-normal distribution
Some Intuition for why Strength of Instruments is Important Consider very strong instrument
Z can explain a lot of variation in s Z very close to s-hat
Think of limiting case where correlation perfect – then s-hat=s IV estimator identical to OLS estimator Will have same distribution If errors normal then this is same as
asymptotic distribution
What if we have weak instrument… Think of extreme case where true
correlation between s and Z is useless First-stage tries to find some correlation so
estimate of coefficients will not normally be zero and will have some variation in X-hat
No reason to believe X-hat contains more ‘good’ variation than X itself
So central tendency is OLS estimate But a lot more noise – so very big variance
A Simple Example One endogenous variable, no exogenous
variables, one instrument All variables known to be mean zero so
estimate equations without intercepts
i i iy x
i i ix z u
ˆ IV i i
i i
z y
z x
Finite Sample Problems 1 and 2 To address issue of bias want to take
expectation of final term – would like it to be zero.
Problem – mean does not exist ‘fat tails’ i.e. sizeable probability of getting vary
large outcome This happens when Σzixi is small more likely when instruments are weak
Similar issue for variance estimation
Finite Sample Problem 3
zi non-stochastic
(εi,ui) have joint normal distribution with mean zero, variances σ2
ε,σ 2u, and
covariance σ2εu
If σ2εu =0 then no endogeneity problem and
OLS estimator consistent
If σ2εu ≠0 then endogeneity problem and
OLS estimator is inconsistent
IV Estimator for this special case..
Both numerator and denominator of final term are linear combinations of normal random variables so are also normally distributed
So deviation of IV estimator from β is ratio of two (correlated) normal random variables
Sounds simple but isn’t
2
1
ˆ1 1
i ii i i iIV
i i ii i i
zz z u nz z u z z u
n n
A Very Special Case: π= σ2
εu =0 X exogenous and Z useless (basically, OLS would
be okay but maybe you don’t know this In this case numerator and denominator in:
2
1
ˆ1 1
i ii i i iIV
i i ii i i
zz z u nz z u z z u
n n
• Ε and u are independent with mean zero• The IV estimator has a Cauchy distribution –
this has no mean (or other moments)
Rules-of-Thumb Mean of IV estimator exists if more than
two over-identifying restrictions Where mean exists:
• Probably can use as measure of central tendency of IV estimator where mean does not exist
• This is where rule-of-thumb on F-stat comes from
What to do - 1 Report the first stage and think about whether
it makes sense. Are the magnitude and sign as you would
expect, are the estimates too big or large but wrong-
signed?
Report the F-statistic on the instruments. The bigger this is, the better. General suggestion: F-statistics above about 10
put you in the safe zone
What to do – 2 Pick your best single instrument and report
just-identified estimates using this one only. Just-identified IV is median-unbiased and therefore
unlikely to be subject to a weak-instruments critique.
Look at the coefficients, t-statistics, and F-
statistics for excluded instruments in the reduced-form regression of dependent variables on instruments. Remember that the reduced form is proportional to
the causal effect of interest. Most importantly, the reduced-form estimates, since
they are OLS, are unbiased.
Next time Maximum Likelihood Estimation
Two uses: LIML as an alternative to 2SLS Discrete Choice Models (logit, probit, etc.)