endogeneity and instrumental variables

7/24/2019 Endogeneity and Instrumental Variables

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Endogeneity andInstrumental

VariablesPresented by Justin Balthrop

September 28, 2015


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What Exactly Are InstrumentalVariables?

Take the simple case of ordinary least squares regression with a singleexplanatory variable:

A fundamental assumption of estimating _1 is that the correlation betweenX and u is zero.

If this is not the case (X is endogenous), then using instrumental variables Zcan essentially detect the part of X which is *not* correlated with the errorterm.


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An Instrument must satisfyRelevance and Exclusion

Loosely speaking, the relevance restriction imposes that the instrument Z isnon-trivially related to the endogenous regressor X.

Exclusion, or exogeneity requires that Z not be systematically related to theerror term, u.


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Why do we need IV?

Before we worry about external validity and the big picture implications ofour results, we need to satisfy internal validity.

Three main sources of internal validity issues are: Omitted variables bias

Simultaneity bias

Errors-in-variables bias

Appropriately instrumenting for endogenous regressors can eliminate thesebiases


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Univariate IV Estimation: Two StLeast Squares

Stage 1: Identify the portion of X that is uncorrelated with the error, u

This gives estimates of _0 and _1, which are used to get predicted X values:

Stage 2: Replace X values with estimated X


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Underlying Assumptions of 2SLS

Instrument validity

_0 and _1 are well-estimated in the first stage (large samples)

Why it works:


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Careful about Standard Errors Second-stage OLS standard errors are not correct

They need to be adjusted for the fact that the explanatory variables areestimated

See Woolridgefor the math, STATA for the code- ivreg, robust

Other considerations: Heteroskedasticity

Appropriate clustering

Instrument relevance- more relevancelower estimator variance and higher R-squared in the first stage


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IV Regression in a Multivariate MoAside from messy algebra, estimation generalizes rather easily

Key identification criterion:at least as manyZ as endogenous X Dont forget to instrument for interactions between endogenous X

Underidentified = too few to estimate _vec (correctly, anyway)

Exactly identified = equal number of Z and endogenous X Overidentified = too many instruments


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Testing for Instrument RelevanceAssume one endogenous X

First stage regress is therefore:

Relevance comes fromat least one_i different from zero

If not, the instrument isweak


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Why are weak instruments so bad Back to the simple model:

With estimator:

Weak instruments leads to a near-zero denominator, and the resultingsampling distribution cannot be accurately approximated by its asymptoticdistribution


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Measuring the Strength of Instrum The first-stage F-test

Tests the hypothesis that instruments Z_i do not enter the first-stage regression

Small F-stat (less than 10) are the result of weak instruments

If the set of instruments is weak, get better instruments If that is impossible, consider dropping the weakest to improve the first-stage F

This is somewhat ad-hoc


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Too many Instruments = Tests fOveridentifying RestrictionsAssume we have multiple valid instruments with a single endogenousregressor.

Intuition: If we perform 2SLS using both instruments separately and arriveat completely different results, it shouldnt be that both instruments arevalid.

Statistics: J-test


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J-Test of Overidentifying Restrict Step 1: estimate the conditional expectation function using TSLS and both instru

Step 2: Compute predicted Y values using theactualXs

Step 3: Compute residuals

Step 4: Regress residuals against all instruments Z and exogenous regressors X

Step 5: Test the hypothesis that all coefficients on Z_i are zero, with J-statistic J=

Here, F is the F-stat from testing coefficients on Z_i

If some instruments are exogenous and others endogenous, J-stat will be large, rnull that all instruments are exogenous


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Sargan Test for Overidentification1. Estimate the 2SLS IV regression - Extract residuals

2. Regress these residuals on all exogenous variables and extract R2

3. Calculate nR2 which is 2 distributed

4. Compare the value with the critical value in the chi-square table withdegrees of freedom equal to # instruments less #

If the statistic (nR2) exceeds the critical 2 value, conclude the instruments

are invalid. They are not uncorrelated with the error term and hence has some explanatorypower in the main equation.

Be very careful: The test assumes that one instrument is valid.

If all instruments do not fulfill the criteria Cov(zi,ui) = 0, then the test mightsuggest that the instruments are valid, even when they are not


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Durbin-Wu-Hausman Test Balances the consistency of IV against the efficiency of LS

H0: IV and LS both consistent, but LS is efficient

H1: Only IV is consistent

DWH test for a single endogenous regressor:DWH = (bIV bLS) / (s2bIVs2bLS)~ N(0,1)

If |DWH| > 1.96, then X is endogenous and IV is the preferred estimator despite itsinefficiency

A roughly equivalent procedure for DWH:1. Estimate the first-stage model2. Include the first-stage residual in the structural model along with the endogenous X

3. Test for significance of the coefficient on residual

Note: Coefficient on endogenous X in this model is bIV(standard error issmaller, though) First-stage residual is a generated regressor


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The following example istaken from the University of

Albany Center for Social and

Demographic Analysispresentation on IVEstimation


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Angrist and Krueger (1991),J.L.E Returns to education (Y = wages)

Problem of omitted ability bias

Years of schooling vary by quarter of birth Compulsory schooling laws, age-at-entry rules

Someone born in Q1 is a little older and will be able to drop out sooner than someone born

Q.O.B. can be treated as a useful source of exogeneity in schooling


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Angrist and Krueger (1991),J.L.E. People born in Q1 do obtain lessschooling But pay close attention to the scale ofthe y-axis

Mean difference between Q1 and Q4is only 0.124, or 1.5 months

So...need large N since R2X,Zwill

be very smallA&K had over 300k for the 1930-39cohort

Source: Angrist and Krueger (1991), Figure I


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Angrist and Krueger (1991),J.L.E.

Final 2SLS model interacted QOB with year of birth (30), state of birth (150) OLS: b = .0628 (s.e. = .0003)

2SLS: b = .0811 (s.e. = .0109)

Least squares estimate does not appear to be badly biased by omitted variables But...replication effort identified some pitfalls in this analysis that are instructive


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Bound, Jaeger, and Baker (1995),J.A.S

Potential problems with QOB as an IV Correlation between QOB and schooling is weak

Small Cov(X,Z) introduces finite-sample bias, which will be exacerbated with the inclusion of many IV

QOB may not be completely exogenous Even small Cov(Z,e) will cause inconsistency, and this will be exacerbated when Cov(X,Z) is small

QOB qualifies as a weak instrument that may be correlated with unobserved determ

wages (e.g., family income)


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Bound, Jaeger, and Baker (1995),J.A.S

Even if the instrument is good, matters can be made far worse with IV as opposed Weak correlation between IV and endogenous regressor can pose severe finite-sample bias

Andreally large samples wont help, especially if there is even weak endogeneity between IV and er

First-stage diagnostics provide a sense of how good an IV is in a given setting F-test and partial-R2on IVs


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Lewbel (2012) Method of Identificati Mostly applicable to models with an unobserved common factor

Identification is achieved by having regressors that are uncorrelated with the product ofheteroskedastic errors

ConsiderY1,Y2 as observed endogenous variables, X a vector of observed exogenousregressors, and =(1,2) as unobserved error processes.

Consider a structural model of the form:

Y1 = X1+Y21+1 (1)

Y2 = X2+Y12+2 (2)

Higher-moment considerations (restricting correlations of with X);

In the presence of heteroskedasticity related to at least some elements of X,identification can be achieved.

endogeneity and instrumental variables

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