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 Y 0 and Y 1 We only observe 1 of these for any given individuals  Define a counterfactual S now: S 1 is the value of S if Z = 1 S 0 is the value of S if Z = 0 We only ever observe one of these for any given individual

The Counterfactual Individual U S 1U =1 S 0U =0 Y 1U Y 0U Y 1U Y 0U Z=1 Z=0 S 1U =0 S 0U =1 Y 1U Y 0U Y 1U Y 0U 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[Y 1U – Y 0U ]: which node doesn’t matter if homogeneous effects. With heterogeneous effects, it’s the average across the nodes LATE is E[Y | S 1U =1, Z=1] – E[Y | S 0U =1, Z=0]

Writing the first stage as counterfactuals  We can now define our variable of interest S as follows: S = S 0i + (S 1i – S 0i )z i =π 0 + π 1i Z i + ν i In this specification: π 0 = E[S 0i ] π 1i = S 1i – S 0i E[π 1i ] = E[S 1i – S 0i ]: 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) Y 1i = Y(1,1) = Y(1,0) = Y(S=1) Y 0i = Y(0,1) = Y(0,0) = Y(S=0)  Rewrite the potential outcome as: Y i = Y i (0,z i ) + [Y i (1,z i ) – Y i (0,z i )]S i = Y 0i + (Y 1i – Y 0i )S i = α 0 + ρ i S i + 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 π 1i >0 or π 1i <0 for all i where S i = π 0 + π 1i Z i + ν i In terms of the counterfactual, it must be the case that S 1i ≥ S 0i (or S 1i ≤ S 0i ) for all i

Back to LATE  Given these assumptions  To see why note the following: E[Y i | Z i =1] = E[Y 0i + (Y 1i – Y 0i )S i | Z i =1] = E[Y 0i + (Y 1i – Y 0i )S 1i ] E[Y i | Z i =0] = E[Y 0i + (Y 1i – Y 0i )S 0i ] E[S |Z i =1] – E[S |Z i =0] = E[S 1i – S 0i ] = Pr[S 1i >S 0i ]

LATE continued  Substituting these equalities in to our formula we get:  We are left with our LATE estimate

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

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 Σz i x i is small more likely when instruments are weak  Similar issue for variance estimation

Finite Sample Problem 3  z i non-stochastic  (ε i,u i ) have joint normal distribution with mean zero, variances σ 2 ε,σ 2 u, and covariance σ 2 εu  If σ 2 εu =0 then no endogeneity problem and OLS estimator consistent  If σ 2 εu ≠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

A Very Special Case: π= σ 2 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: 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.)