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Instrumental variable

In statistics, econometrics, epidemiology and related disciplines, the method ofinstrumental variables (IV) is used to estimate

causal relationships when controlled experiments are not feasible.

Statistically, IV methods allowconsistent estimation when the explanatory variables (covariates) are correlated with the error

terms. Such correlation may occur when the dependent variable causes at least one of thecovariates ("reverse" causation),when there are relevant explanatory variables which are omitted from the model, or when the covariates are subject to

measurement error. In this situation, ordinarylinear regression generally produces biased and inconsistent estimates. However,if an instrumentis available, consistent estimates may still be obtained. An instrument is a variable that does not itself belong inthe explanatory equation and is correlated with theendogenous explanatory variables, conditional on the other covariates.Formal definitions of instrumental variables, using counterfactuals and graphical criteria, are given in (Pearl, 2000).

[1]Notions of

causality in econometrics, and their relationship with instrumental variables and other methods, are discussed in Heckman(2008).

[2]

In linear models, there are two main requirements for using an IV:

y The instrument must be correlated with the endogenous explanatory variables, conditional on the other covariates.y The instrument cannot be correlated with the error term in the explanatoryequation, that is, the instrument cannot

suffer from the same problem as the original predicting variable.

Example

Informally, in attempting to estimate the causal effect of some variablexon anothery, an instrument is a third variablezwhichaffectsyonly through its effect onx. For example, suppose a researcher wishes to estimate the causal effect of smoking ongeneral health (as in Leigh and Schembri 2004

[3]). Correlation between health and smoking does not imply that smoking causes

poor health because other variables may affect both health and smoking, or because health may affect smoking in addition to

smoking causing health problems. It is at best difficultand expensive to conduct controlled experiments on smoking status inthe general population. The researcher may proceed to attempt to estimate the causal effect of smoking on health fromobservational data by using t he tax rate on tobacco products as an instrument for smoking in a health regression. I f tobacco

taxes onlyaffect health because they affect smoking (holding other variables in the model fixed), correlation between tobaccotaxes and health is evidence that smoking causes changes in health. An estimate of t he effect of smoking on health can bemade by also making use of the correlation between taxes and smoking patterns.

Applications

IV methods are commonly used to estimate causal effects in contexts in which controlled experiments are not available.

Credibility of the estimates hinges on the selection of suitable instruments. Good instruments are often created by policychanges, for example, the cancellation of federal student aid scholarship program may reveal the effects of aid on some

students' outcomes. Othernatural and quasi-natural experiments of various types are commonly exploited, for example, Miguel,Satyanath, and Sergenti (2004)

[4]use weather shocks to identify the effect of changes in economic growth (i.e., declines) on

civil conflict. Angrist and Krueger (2001)[5]

presents a survey of the history and uses of instrumental variable techniques.

Estimation

Suppose the data are generated by a process of the form

where iindexes observations, yi is the dependent variable,xi is a covariate, is an unobserved error term representing allcauses ofyiother thanxi, and is an unobserved scalar parameter. The parameter is the causal effect onyiof a one unitchange inxi, holding all other causes ofyiconstant. The econometric goal is to estimate . For simplicity's sake assume thedraws of are uncorrelated and that they are drawn from distributions with the samevariance, t hat is, that the errors are

serially uncorrelated and homoskedastic.

Suppose also that a regression model of nominally the same form is proposed. Given a random sample ofTobservations from

this process, the ordinary least squares estimator is

wherex, yand denote column vectors of length T. Whenxand are uncorrelated, under certain regularity conditions thesecond term has an expected value conditional onxof zero and converges to zero in the limit, so the estimator isunbiased andconsistent. Whenxand the other unmeasured, causal variables collapsed into the term are correlated, however, the OLS

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estimator is generally biased and inconsistent for . In this case, it is valid to use the estimates to predict values ofygivenvalues ofx, but the estimate does not recover the causal effect ofxon y.

An instrumental variablezis one that is correlated with the independent variable but not with theerrorterm. Using the methodof moments, take expectations conditional onzto find

The second term on the right-hand side is zero by assumption. Solve for and write the resulting expression in terms of samplemoments,

When zand are uncorrelated, the final term, under certain regularity conditions, approaches zero in the limit, providing aconsistent estimator. Put another way, the causal effect ofxon ycan be consistently estimated from these data even thoughxis not randomly assigned through experimental methods.

The approach generalizes to a model with multiple explanatory variables. SupposeX is the T Kmatrix of explanatoryvariables resulting from Tobservations on Kvariables. Let Zbe a T Kmatrix of instruments. Then it can be shown that theestimator

is consistent under a mult ivariate generalization of the conditions discussed above. I f there are more instruments than t hereare

covariates in the equation of interest so thatZis a T Mmatrix with M > K, the generalized method of moments can be usedand the resulting IV estimator is

where PZ = Z(Z'Z) 1

Z'. The second expression collapses to t he first when the number of instruments is equal to t he number of

covariates in the equation of interest.

Interpretation as two-stage least squares

One computational method which can be used to calculate IV estimates is two-stage least-squares (2SLS). In the first stage,each endogenous covariate in the equation of interest is regressed on all of the exogenous variables in the model, including

both exogenous covariates in the equation of interest and the excluded instruments. The predicted values from theseregressions are obtained.

Stage 1: Regress each column ofX on Z, (X= Z + errors)

and save the predicted values:

In the second stage, the regression of interest is estimated as usual, except that in this stage each endogenous covariate isreplaced with the predicted values from its first stagemodel from the first stage.

Stage 2: RegressY on the predicted values from the first stage:

The resulting estimator of is numerically identical to the expression displayed above. A small correction must be made to the

sum-of-squared residuals in the second-stage fitted model in order that the covariance matrix of is calculated correctly.

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Identification

In the instrumental variable regression, if we have multiple endogenous regressors and multiple instruments

the coefficients on the endogenous regressors are said to be:

Exactly identified ifm = k.Overidentified ifm > k.

Underidentified ifm < k.

The parameters are not identified if t here are fewer instruments than there are covariates or, equivalently, if there are fewerexcluded instruments than there are endogenous covariates in the equation of interest.

Non-parametric analysis

When the form of the structural equations is unknown, an instrumental variableZcan still be defined through the equations:

x= g(z,u)

y= f(x,u)

where fand gare two arbitrary functions and Zis independent ofU. Unlike linear models, however, measurements ofZ,XandYdo not allow for the identification of the average causal effect ofXon Y, denoted ACE

ACE = Pr(y| do(x)) = Eu[f(x,u)].

Balke and Pearl [1997][6]

derived tight bounds on ACE and showed that these can provide valuable information on the sign and

size of ACE.

In linear analysis, there is no test to falsify the assumption theZis instrumental relative to the pair (X,Y). This is not the casewhenX is discrete. Pearl (2000)

[1]has shown that, for all fand g, the following constraint, called "Instrumental Inequality" must

hold wheneverZsatisfies the two equations above:

[edit] On the interpretation of IV estimates

The exposition above assumes that the causal effect of interest does not vary across observations, that is, that is a constant.

Generally, different subjects will respond differently to changes in the "treatment"x. When this possibility is recognized, theaverage effect in the population of a change inxon ymay differ from the effect in a given subpopulation. For example, theaverage effect of a job training program may substantially differ across the group of people who actually receive the trainingand the group which chooses not to receive training. For these reasons, IV methods invoke implicit assumptions on behavioralresponse, or more generally assumptions over the correlation between the response to treatment and propensity to receivetreatment.

[7]

The standard IV estimator can recoverlocal average treatment effects (LATE) rather than average treatment effects (ATE).[8]

Imbens and Angrist (1994) demonstrate that the linear IV estimate can be interpreted under weak conditions as a weightedaverage of local average treatment effects, where the weights depend on the elasticity of the endogenous regressor to changesin the instrumental variables. Roughly, that means that the effect of a variable is only revealed for the subpopulations affectedby the observed changes in the instruments, and that subpopulations which respond most to changes in the instruments willhave the largest effects on the magnitude of the IV estimate.

For example, if a researcher uses presence of a land-grant college as an inst rument for college education in an earningsregression, she identifies the effect of college on earnings in the subpopulation which would obtain a college degree if a collegeis present but which would not obtain a degree if a college is not present. This empirical approach does not, without furtherassumptions, tell the researcher anything about the effect of college among peoplewho would either always or never get acollege degree regardless of whether a local college exists.

[edit] Potential problems

Instrumental variables estimates are generally inconsistent if the instruments are correlated with the error term in the equationof interest. Another problem is caused by the selection of "weak" instruments, instruments that are poor predictors oftheendogenous question predictor in t he first-stage equation. In this case, the predict ion of the question predictor by the instrument

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Generalised IV estimator s GIVE)

A moregeneral formof the IV estimatorwherewehavemore instrumental variables than endogenous X variables

GIVE ispotentiallymoreefficient thansimple IV, if instrumentsarewell -chosen Test whether instrumentsare validusing Sargans test

t

henmight aregressorvariablebecorrelatedwith theerror term?

u

orrelatedshocksacross linkedequations Simultaneousequations Errors invariables

v

odel hasa laggeddependent variableandaseriallycorrelatederrorterm

Instrumental variablesandpanel datamethods ineconomicsand finance

w

hat are instrumental variablesx

IV)methods?v

ost widelyknownasasolution toendogenousregressors: explanatoryariablescorrelatedwith theregressionerror term, IV methodsprovideaway to nonethelessobtainconsistent parameterestimates.Although IV estimatorsaddress issuesofendogeneity, theviolationof the

y

eroconditional meanassumptioncausedbyendogenousregressorscanalsoarise for twoothercommoncauses: measurement errorinregressors

x

errors-in-variables)

andomitted-variablebias. The lattermayarise insituationswhereavariableknown toberelevant for thedatageneratingprocess isnot measurable, andnogoodproxiescanbe found.

Theweak instrumentsproblem

Instrumental variablesmethodsrelyon twoassumptions: theexcluded instrumentsaredistributed independentlyof theerrorprocess, and theyaresufficientlycorrelatedwith the includedendogenousregressors. Testsofoveridentifyingrestrictions

address the first assumption, althoughweshouldnote that arejectionof theirnull maybe indicative that theexclusionrestrictions forthese instrumentsmaybe inappropriate. That is, someof the instrumentshavebeen improperlyexcluded fromtheregressionmodelsspecification.Thespecificationofan instrumental variablesmodel asserts that theexcluded instrumentsaffect thedependent variableonly indirectly, through theircorrelationswith the includedendogenousvariables. Ifanexcludedinstrument exertsbothdirect and indirect influenceson thedependent variable, theexclusionrestrictionshouldberejected.Thiscanbereadily testedby including thevariableasaregressor. To test thesecondassumptionthat theexcludedstrumentsaresufficientlycorrelatedwith the includedendogenousregressorsweshouldconsiderthegoodness-of-fit of thefirst stageregressionsrelatingeachendogenousregressorto theentireset of instruments.