Session 1: Dealing with EndogeneityBen Shepherd Session 1: Dealing with Endogeneity. Introduction...

IntroductionDealing with Endogeneity

Examples of IV Gravity ModelsSummary

Session 1: Dealing with Endogeneity

Ben Shepherd

Niehaus Center, Princeton UniversityGEM, Sciences Po

ARTNeT Capacity Building Workshop for Trade Research:“Behind the Border” Gravity Modeling

Thursday, December 18, 2008

Ben Shepherd Session 1: Dealing with Endogeneity



Outline

1 IntroductionOverviewEndogeneity and its Consequences

2 Dealing with EndogeneityAd Hoc SolutionsIV Estimation

3 Examples of IV Gravity Models“Trading on Time” (Djankov et al., 2008)“Contract Enforcement” (Ranjan & Lee, 2007)

4 Summary




OverviewEndogeneity and its Consequences

Outline




4 Summary





Overview

One of the most important OLS assumptions is that theerrors are uncorrelated with the dependent variables.

A number of possible reasons:

Measurement errorOmitted variablesEndogeneity

Endogeneity is particularly important in policy settings.Pervasive in trade facilitation settings.





Overview

Tariffs are an obvious example of an endogeneity problemin many gravity models.

Intuition suggests that tariffs should impact negatively onbilateral trade.

But political economy suggests that greater importpenetration is likely to lead to increased demand forprotection by industry lobbies (e.g., Grossman andHelpman, “protection for sale”)

What are the consequences for estimation and inference?How can we deal with the problem?





Outline




4 Summary





Conceptualizing Endogeneity

A simple example of endogeneity is the following:

A = XB+E

X = AC+W

Since the current value of X depends on the current valueof A, X must be influenced by current shocks to A:

A = (AC+W)B+E

Thus, X and E are correlated. This violates the OLSassumptions.





Consequences of Endogeneity

Endogeneity bias is not a “simple” violation to deal with.There is no equivalent to the robust estimators used toconduct inference in the presence of general patterns ofheteroskedasticity.

Moreover, it has serious consequences for our estimates.

In the presence of endogeneity, OLS can produce biasedand inconsistent parameter estimates. Hypotheses testscan be seriously misleading.

All it takes is one endogenous variable to seriously distortALL OLS estimates of a model.




Ad Hoc SolutionsIV Estimation

Outline




4 Summary





Introduction

We need some way of separating out genuinely exogenousvariation in independent variables that might beendogenous.

Ideally, we would also like some way of testing the extentto which endogeneity is a problem in our data, andensuring that the solution we have chosen is a good one.

Two groups of solutions:

Ad hoc approachesInstrumental variables estimation





Ad Hoc Solutions

If a dependent variable is potentially endogenous, it isintuitively appealing to look for a proxy that does not sufferfrom the same problem.

The most common approach is to lag the suspect variablesby one or more periods.

The argument is that although current values of (eg) tariffsmight be endogenous to import penetration, it is unlikelythat past values of tariffs are subject to the same problem.





Ad Hoc Solutions

Advantages of Lags/Proxies

Very simple to implement.

Additional data requirements limited.

Intuitively appealing.

Limitations of Lags/Proxies

Interpretation becomes a little more difficult since thevariable in the regression is only a proxy for the variable weare interested in.

Loss of precision in some cases.

No way of gauging empirically how serious theendogeneity problem is, and whether the solution isadequate to deal with it.





Outline




4 Summary





Introduction

The best way to deal with endogeneity concerns is throughinstrumental variables (IV) techniques.

The most common IV estimator is Two Stage LeastSquares (TSLS).

Intuitively, IV estimation works as follows:

Find a genuinely exogenous variable (instrument) that isstrongly correlated with the potentially endogenousregressor.Ensure that the instrument only influences the dependantvariable through the potentially endogenous independentvariable.





Introduction

IV estimation is intuitively appealing, and relatively simpleto implement on a technical level.

The most difficult part lies in the selection of appropriateinstruments.

The main advantages of IV estimation are:

Rigor and transparencyAmenability to empirical testing (appropriateness ofinstruments, extent of endogeneity, etc.)





Setting up the Problem

Recall the gravity model in matrix form:

X = AB+E

Assume that A contains some potentially endogenousvariables.

Let Z be the matrix of all exogenous variables in the model.Z includes any exogenous variables in A, as well as anumber of instruments equal to the number of potentiallyendogenous variables in A.





Two Stage Least Squares

We can therefore write:

A = ZBZ +W

We can run a first stage regression. The OLS estimate ofB∗

Z, B∗

Z, is:

B∗

Z =(Z′Z

)−1 Z′A

This estimate allows us to derive fitted values for A:

A = Z(Z′Z

)−1 Z′A

Since Z includes exogenous variables from the mainregression too, they are perfectly explained by their“instruments” (themselves). A for those variables only isidentical to the original A.






By construction, these fitted values must be exogenous tothe original dependent variable X.

Think of them as the part of X that is explained exclusivelyby exogenous variation (the instruments and otherexogenous variables).

We can therefore rewrite the original gravity model, andderive a new esimator based on a second OLS regression:

X = ABIV +E

BIV =(

A′A)−1

A′X






Most econometrics texts express BIV =(

A′A)−1

A′X in a

slightly different way.

Define PZ = Z(Z′Z)−1 Z′, which is a symmetric, idempotentmatrix. Therefore, PZ = P′

Z, and PZPZ ≡ P′

ZPZ = PZ.

Thus A = PZA andBIV =

[(PZA)′ PZA

]−1

(PZA)′ X = (A′PZA)−1 A′PZX.






If all of our assumptions hold, BIV should beuncontaminated by endogeneity bias.

If the normal OLS assumptions hold, and so do the IVassumptions, the TSLS estimator can also be shown tohave similar statistical properties to OLS (consistent,unbiased, efficient).

Interpretation of parameters and hypothesis testing can alltake place as usual, following the same procedures asOLS.






Caution is required, however:

1 IV estimates are only as good as the instruments used.We need to run some statistical tests and make sure themodel is working properly.

2 Although it’s called “two stage” least squares, avoidrunning the two stages separately. You will get incorrectstandard errors (too small), and you might mistakenlyexclude exogenous variables from the main model–acommon error.

3 Use STATA’s “canned” routes: ivreg, and ivreg2 (includesthe tests we are about to discuss). But make sure youknow what they’re telling you!





Testing TSLS ModelsOverview

TSLS provides a number of useful tests that can help indeciding whether IV estimation is necessary, and whether theinstruments chosen are valid.

Endogeneity test: Is there evidence that correlationbetween the potentially endogenous variables and theerror term is strong enough to result in substantivelybiased estimates?Instrument relevance test: Are the instruments sufficientlystrongly correlated with the potentially endogenousvariables?Exogeneity/excludability of instruments: Are theinstruments genuinely uncorrelated with the main equationresiduals?





Testing TSLS ModelsTesting for Instrument Relevance

TSLS is only as good as the instruments.

In the presence of weak instruments, the TSLS estimatorcan actually produce worse results than simple OLS.

So the first step in testing must be to ensure that theinstruments are strongly enough correlated with thepotentially endogenous variables.





Testing TSLS ModelsTesting for Instrument Relevance

To test for instrument relevance, make sure to run the firststage regressions of the potentially endogenous variableson all of the exogenous variables.

Are the instruments individually statistically significant?Are their signs and magnitudes sensible?

Are the instruments jointly statistically significant? Look fora high F-statistic.

Extension: Explore test statistics designed specifically todeal with weak instruments, e.g. Stock/Yogo.





Testing TSLS ModelsTesting for Endogeneity

In gravity models, endogeneity is nearly always a problemfor policies, at least in principle.

Estimating a TSLS model implies costs:

Loss of precision, due to two stage estimation;Possibility of incorrect inference when instruments areweak.

It would therefore be nice to have a test that allows us tosee how serious the endogeneity problem really is. Do thebenefits of TSLS outweigh the costs in a particularinstance?






We can test for the endogeneity of suspect independentvariables using a Hausman test.

H0: the regressor is exogenous; H1: it is endogenous.

Thus, a higher value of the test statistic indicates a moreserious endogeneity problem.

Start to worry if it rejects the null at the 10% level.






The mechanics of the test are cumbersome.

Intuitively, it is equivalent (in large samples) to the followingprocedure:

Run the first stage regressions and save the residuals;Include the first stage residuals as additional regressors inthe main equation;Test the joint significance of the first stage residuals.

In other words, the test looks at whether or not there is acorrelation between the dependent variable, and that partof the suspect variables’ variation that is not explained bygenuinely exogenous factors.





Testing TSLS ModelsTesting for Exogeneity and Excludability of Instruments

In addition to being relevant, instruments must also be:

Exogenous with respect to the dependent variable in themain equation; andExcludable from the main regression.

We can use another version of the Hausman test (in fact,Hansen’s J) to test whether or not these conditions jointlyhold.

The test can ONLY be run if the model is overidentified, i.e.if we have more excluded instruments than potentiallyendogenous variables.





Testing TSLS ModelsTesting for Exogeneity and Excludability of Instruments

H0 for Hansen’s J: the instruments are uncorrelated withthe main equation errors; H1: the instruments arecorrelated with the main equation errors.

Thus, a high value of the test statistic tends to indicate thatthere is a problem with the instruments:

Endogeneity: They are correlated with the main equationerrors because there is feedback running from thedependent variable to the instruments; and/orNon-excludability: The instruments should appear in themain regression, and the test is effectively picking up anomitted variables problem.

Be scared if Hansen’s J rejects the null at the 10% level.





Testing TSLS Models

Always test for instrument relevance first. If instrumentsare weak, tests like Hausman and Hansen’s J can givemisleading results.

Use the Hausman test to assess the extent to whichendogeneity is really a problem.

If at all possible, ensure the model is overidentified, andtest exogeneity and excludability via Hansen’s J.

If the model passes all of these tests, then it shouldprovide a reasonable guide for causal inference.




“Trading on Time” (Djankov et al., 2008)“Contract Enforcement” (Ranjan & Lee, 2007)

Outline




4 Summary





Examples of IV Gravity Models“Trading on Time” (Djankov et al., 2008)

The main independent variable of interest is the DoingBusiness measure of trading time.

But time is potentially endogenous to trade: an unexpectedincrease in trade volumes can lead to congestion, andlonger wait times.

Solution: Use the number of administrative formalities(documents) as an instrument for time.

Time and documents are very strongly correlated.Test results suggest that exogeneity/excludability is alsosatisfied. Intuitively, though, excludability is morequestionable: wouldn’t very complex and burdensomedocumentation requirements constitute an independentsource of trade costs?






In the latest version of their paper, they take a differentapproach.

They instrument for trading times using times inneighboring countries.

Their results remain robust to IV regression.





Outline




4 Summary





Examples of IV Gravity Models“Contract Enforcement” (Ranjan & Lee, 2007)

Contract enforcement can affect international tradetransaction costs.

But, institutional development might be endogenous totrade.

So they instrument for contract enforcement using mortalityfigures for European settlers during the colonial era.






Why use settler mortality as an instrument for currentcontract enforcement?

In a famous paper, Acemoglu Johnson and Robinson(AER, 2001) show that there is in fact a strong correlationbetween them, and that it is not just a statistical artefact:

European countries tended to operate extractive colonies inharsh (malarial) environments, and settler colonies in moretemperate ones;As a result, high mortality environments tended to seelesser importation of European governance institutions;And so we now observe a strong correlation betweensettler mortality in the colonial era, and current measures ofgovernace.




Summary

Endogeneity can be a very serious issue in gravity models.

It is not just “academic”: it can result in wrong coefficientsigns, and significant under- or over-estimation of policyeffects.

Basically two ways to deal with it:

Ad hoc solutions (proxies, lags);IV estimation




Summary

IV estimation is only useful if the instruments are good.They must be:

Strongly correlated with the potentially endogenousvariables;Genuinely exogenous to the model.

Whenever possible, overidentify the model (i.e., includemore instruments than potentially endogenous variables):it allows you to test for instrument exogeneity andexcludability.

Footnote: GMM is a cooler way of doing IV estimation.Identical to TSLS for just identified models, but can givemore precise estimates with over-identified models.


Session 1: Dealing with EndogeneityBen Shepherd Session 1: Dealing with Endogeneity. Introduction...

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