Book Econometric Analysis of Cross Section and Panel Data By Wooldridge - Chapter 5 doc

Further, without more informa-tion, we cannot consistently estimate any of the parameters in equation 5.1.The method of instrumental variables IV provides a general solution to theproble

In this chapter we treat instrumental variables estimation, which is probably secondonly to ordinary least squares in terms of methods used in empirical economic re-search The underlying population model is the same as in Chapter 4, but we explic-itly allow the unobservable error to be correlated with the explanatory variables.

5.1 Instrumental Variables and Two-Stage Least Squares

5.1.1 Motivation for Instrumental Variables Estimation

To motivate the need for the method of instrumental variables, consider a linearpopulation model

but where xK might be correlated with u In other words, the explanatory variables

x1, x2; ; xK1 are exogenous, but xK is potentially endogenous in equation (5.1).The endogeneity can come from any of the sources we discussed in Chapter 4 To fixideas it might help to think of u as containing an omitted variable that is uncorrelatedwith all explanatory variables except xK So, we may be interested in a conditionalexpectation as in equation (4.18), but we do not observe q, and q is correlated with

xK

As we saw in Chapter 4, OLS estimation of equation (5.1) generally results in consistent estimators of all the bj if CovðxK; uÞ 0 0 Further, without more informa-tion, we cannot consistently estimate any of the parameters in equation (5.1).The method of instrumental variables (IV) provides a general solution to theproblem of an endogenous explanatory variable To use the IV approach with xKendogenous, we need an observable variable, z1, not in equation (5.1) that satisfiestwo conditions First, z1 must be uncorrelated with u:

In other words, like x1; ; xK1, z1is exogenous in equation (5.1)

The second requirement involves the relationship between z1 and the endogenousvariable, xK A precise statement requires the linear projection of xK onto all theexogenous variables:

xK¼ d0þ d1x1þ d2x2þ


þ dK1xK1þ y1z1þ rK ð5:4Þwhere, by definition of a linear projection error, EðrKÞ ¼ 0 and rK is uncorrelatedwith x , x ; ; x , and z The key assumption on this linear projection is that the

coe‰cient on z1is nonzero:

This condition is often loosely described as ‘‘z1 is correlated with xK,’’ but thatstatement is not quite correct The condition y100 means that z1 is partially corre-lated with xK once the other exogenous variables x1; ; xK1 have been netted out

If xK is the only explanatory variable in equation (5.1), then the linear projection is

xK¼ d0þ y1z1þ rK, where y1 ¼ Covðz1; xKÞ=Varðz1Þ, and so condition (5.5) andCovðz1; xKÞ 0 0 are the same.

At this point we should mention that we have put no restrictions on the tion of xK or z1 In many cases xK and z1 will be both essentially continuous, butsometimes xK, z1, or both are discrete In fact, one or both of xKand z1can be binaryvariables, or have continuous and discrete characteristics at the same time Equation(5.4) is simply a linear projection, and this is always defined when second moments ofall variables are finite

distribu-When z1 satisfies conditions (5.3) and (5.5), then it is said to be an instrumentalvariable (IV) candidate for xK (Sometimes z1 is simply called an instrument for xK.)Because x1; ; xK1 are already uncorrelated with u, they serve as their own instru-mental variables in equation (5.1) In other words, the full list of instrumental vari-ables is the same as the list of exogenous variables, but we often just refer to theinstrument for the endogenous explanatory variable

The linear projection in equation (5.4) is called a reduced form equation for theendogenous explanatory variable xK In the context of single-equation linear models,

a reduced form always involves writing an endogenous variable as a linear projectiononto all exogenous variables The ‘‘reduced form’’ terminology comes from simulta-neous equations analysis, and it makes more sense in that context We use it in all IVcontexts because it is a concise way of stating that an endogenous variable has beenlinearly projected onto the exogenous variables The terminology also conveys thatthere is nothing necessarily structural about equation (5.4)

From the structural equation (5.1) and the reduced form for xK, we obtain areduced form for y by plugging equation (5.4) into equation (5.1) and rearranging:

y¼ a0þ a1x1þ


þ aK1xK1þ l1z1þ v ð5:6Þwhere v¼ u þ bKrK is the reduced form error, aj¼ bjþ bKdj, and l1¼ bKy1 By ourassumptions, v is uncorrelated with all explanatory variables in equation (5.6), and soOLS consistently estimates the reduced form parameters, the ajand l1

Estimates of the reduced form parameters are sometimes of interest in their ownright, but estimating the structural parameters is generally more useful For example,

at the firm level, suppose that x is job training hours per worker and y is a measure

of average worker productivity Suppose that job training grants were randomlyassigned to firms Then it is natural to use for z1 either a binary variable indicatingwhether a firm received a job training grant or the actual amount of the grant perworker (if the amount varies by firm) The parameter bKin equation (5.1) is the e¤ect

of job training on worker productivity If z1 is a binary variable for receiving a jobtraining grant, then l1 is the e¤ect of receiving this particular job training grant onworker productivity, which is of some interest But estimating the e¤ect of an hour ofgeneral job training is more valuable

We can now show that the assumptions we have made on the IV z1 solve theidentification problem for thebj in equation (5.1) By identification we mean that wecan write the bjin terms of population moments in observable variables To see how,write equation (5.1) as

where Eðz0xÞ is K  K and Eðz0yÞ is K  1 Equation (5.9) represents a system of Klinear equations in the K unknowns b1, b2; ;bK This system has a unique solution

if and only if the K K matrix Eðz0xÞ has full rank; that is,

in which case the solution is

The expectations Eðz0xÞ and Eðz0yÞ can be consistently estimated using a randomsample onðx; y; z1Þ, and so equation (5.11) identifies the vector b.

It is clear that condition (5.3) was used to obtain equation (5.11) But where have

we used condition (5.5)? Let us maintain that there are no linear dependencies amongthe exogenous variables, so that Eðz0zÞ has full rank K; this simply rules out perfect

collinearity in z in the population Then, it can be shown that equation (5.10) holds ifand only if y100 (A more general case, which we cover in Section 5.1.2, is covered

in Problem 5.12.) Therefore, along with the exogeneity condition (5.3), assumption(5.5) is the key identification condition Assumption (5.10) is the rank condition foridentification, and we return to it more generally in Section 5.2.1

Given a random samplefðxi; yi; zi1Þ: i ¼ 1; 2; ; Ng from the population, the

in-strumental variables estimator of b is

When searching for instruments for an endogenous explanatory variable,

con-ditions (5.3) and (5.5) are equally important in identifying b There is, however, one

practically important di¤erence between them: condition (5.5) can be tested, whereascondition (5.3) must be maintained The reason for this disparity is simple: thecovariance in condition (5.3) involves the unobservable u, and therefore we cannottest anything about Covðz1; uÞ

Testing condition (5.5) in the reduced form (5.4) is a simple matter of computing a

t test after OLS estimation Nothing guarantees that rK satisfies the requisite skedasticity assumption (Assumption OLS.3), so a heteroskedasticity-robust t statis-tic for ^y1is often warranted This statement is especially true if xKis a binary variable

homo-or some other variable with discrete characteristics

A word of caution is in order here Econometricians have been known to say that

‘‘it is not possible to test for identification.’’ In the model with one endogenous able and one instrument, we have just seen the sense in which this statement is true:assumption (5.3) cannot be tested Nevertheless, the fact remains that condition (5.5)can and should be tested In fact, recent work has shown that the strength of the re-jection in condition (5.5) (in a p-value sense) is important for determining the finitesample properties, particularly the bias, of the IV estimator We return to this issue inSection 5.2.6

vari-In the context of omitted variables, an instrumental variable, like a proxy variable,must be redundant in the structural model [that is, the model that explicitly containsthe unobservables; see condition (4.25)] However, unlike a proxy variable, an IV for

xK should be uncorrelated with the omitted variable Remember, we want a proxyvariable to be highly correlated with the omitted variable

Example 5.1 (Instrumental Variables for Education in a Wage Equation): Consider

a wage equation for the U.S working population

logðwageÞ ¼ b0þ b1experþ b2exper2þ b3educþ u ð5:12Þwhere u is thought to be correlated with educ because of omitted ability, as well asother factors, such as quality of education and family background Suppose that wecan collect data on mother’s education, motheduc For this to be a valid instrumentfor educ we must assume that motheduc is uncorrelated with u and that y100 in thereduced form equation

educ¼ d0þ d1experþ d2exper2þ y1motheducþ r

There is little doubt that educ and motheduc are partially correlated, and this lation is easily tested given a random sample from the population The potentialproblem with motheduc as an instrument for educ is that motheduc might be corre-lated with the omitted factors in u: mother’s education is likely to be correlated withchild’s ability and other family background characteristics that might be in u

corre-A variable such as the last digit of one’s social security number makes a poor IVcandidate for the opposite reason Because the last digit is randomly determined, it isindependent of other factors that a¤ect earnings But it is also independent of edu-cation Therefore, while condition (5.3) holds, condition (5.5) does not

By being clever it is often possible to come up with more convincing instruments.Angrist and Krueger (1991) propose using quarter of birth as an IV for education Inthe simplest case, let frstqrt be a dummy variable equal to unity for people born in thefirst quarter of the year and zero otherwise Quarter of birth is arguably independent

of unobserved factors such as ability that a¤ect wage (although there is disagreement

on this point; see Bound, Jaeger, and Baker, 1995) In addition, we must have y100

in the reduced form

educ¼ d0þ d1experþ d2exper2þ y1frstqrtþ r

How can quarter of birth be (partially) correlated with educational attainment?Angrist and Krueger (1991) argue that compulsory school attendence laws induce arelationship between educ and frstqrt: at least some people are forced, by law, to at-tend school longer than they otherwise would, and this fact is correlated with quarter

of birth We can determine the strength of this association in a particular sample byestimating the reduced form and obtaining the t statistic for H0: y1¼ 0

This example illustrates that it can be very di‰cult to find a good instrumentalvariable for an endogenous explanatory variable because the variable must satisfy

two di¤erent, often conflicting, criteria For motheduc, the issue in doubt is whethercondition (5.3) holds For frstqrt, the initial concern is with condition (5.5) Sincecondition (5.5) can be tested, frstqrt has more appeal as an instrument However, thepartial correlation between educ and frstqrt is small, and this can lead to finite sampleproblems (see Section 5.2.6) A more subtle issue concerns the sense in which we areestimating the return to education for the entire population of working people As wewill see in Chapter 18, if the return to education is not constant across people, the IVestimator that uses frstqrt as an IV estimates the return to education only for thosepeople induced to obtain more schooling because they were born in the first quarter

of the year These make up a relatively small fraction of the population

Convincing instruments sometimes arise in the context of program evaluation,where individuals are randomly selected to be eligible for the program Examplesinclude job training programs and school voucher programs Actual participation isalmost always voluntary, and it may be endogenous because it can depend on unob-served factors that a¤ect the response However, it is often reasonable to assume thateligibility is exogenous Because participation and eligibility are correlated, the lattercan be used as an IV for the former

A valid instrumental variable can also come from what is called a natural ment A natural experiment occurs when some (often unintended) feature of the setup

experi-we are studying produces exogenous variation in an otherwise endogenous tory variable The Angrist and Krueger (1991) example seems, at least initially, to be

explana-a good nexplana-aturexplana-al experiment Another exexplana-ample is given by Angrist (1990), who studiesthe e¤ect of serving in the Vietnam war on the earnings of men Participation in themilitary is not necessarily exogenous to unobserved factors that a¤ect earnings, evenafter controlling for education, nonmilitary experience, and so on Angrist used thefollowing observation to obtain an instrumental variable for the binary Vietnam warparticipation indicator: men with a lower draft lottery number were more likely toserve in the war Angrist verifies that the probability of serving in Vietnam is indeedrelated to draft lottery number Because the lottery number is randomly determined,

it seems like an ideal IV for serving in Vietnam There are, however, some potentialproblems It might be that men who were assigned a low lottery number chose toobtain more education as a way of increasing the chance of obtaining a draft defer-ment If we do not control for education in the earnings equation, lottery numbercould be endogenous Further, employers may have been willing to invest in jobtraining for men who are unlikely to be drafted Again, unless we can include mea-sures of job training in the earnings equation, condition (5.3) may be violated (Thisreasoning assumes that we are interested in estimating the pure e¤ect of serving inVietnam, as opposed to including indirect e¤ects such as reduced job training.)

Hoxby (1994) uses topographical features, in particular the natural boundariescreated by rivers, as IVs for the concentration of public schools within a school dis-trict She uses these IVs to estimate the e¤ects of competition among public schools

on student performance Cutler and Glaeser (1997) use the Hoxby instruments, aswell as others, to estimate the e¤ects of segregation on schooling and employmentoutcomes for blacks Levitt (1997) provides another example of obtaining instrumen-tal variables from a natural experiment He uses the timing of mayoral and guber-natorial elections as instruments for size of the police force in estimating the e¤ects ofpolice on city crime rates (Levitt actually uses panel data, something we will discuss

in Chapter 11.)

Sensible IVs need not come from natural experiments For example, Evans andSchwab (1995) study the e¤ect of attending a Catholic high school on various out-comes They use a binary variable for whether a student is Catholic as an IV forattending a Catholic high school, and they spend much e¤ort arguing that religion isexogenous in their versions of equation (5.7) [In this application, condition (5.5) iseasy to verify.] Economists often use regional variation in prices or taxes as instru-ments for endogenous explanatory variables appearing in individual-level equations.For example, in estimating the e¤ects of alcohol consumption on performance incollege, the local price of alcohol can be used as an IV for alcohol consumption,provided other regional factors that a¤ect college performance have been appropri-ately controlled for The idea is that the price of alcohol, including any taxes, can beassumed to be exogenous to each individual

Example 5.2 (College Proximity as an IV for Education): Using wage data for

1976, Card (1995) uses a dummy variable that indicates whether a man grew up inthe vicinity of a four-year college as an instrumental variable for years of schooling

He also includes several other controls In the equation with experience and itssquare, a black indicator, southern and urban indicators, and regional and urbanindicators for 1966, the instrumental variables estimate of the return to schooling is.132, or 13.2 percent, while the OLS estimate is 7.5 percent Thus, for this sample ofdata, the IV estimate is almost twice as large as the OLS estimate This result would

be counterintuitive if we thought that an OLS analysis su¤ered from an upwardomitted variable bias One interpretation is that the OLS estimators su¤er from theattenuation bias as a result of measurement error, as we discussed in Section 4.4.2.But the classical errors-in-variables assumption for education is questionable Anotherinterpretation is that the instrumental variable is not exogenous in the wage equation:location is not entirely exogenous The full set of estimates, including standard errorsand t statistics, can be found in Card (1995) Or, you can replicate Card’s results inProblem 5.4

5.1.2 Multiple Instruments: Two-Stage Least Squares

Consider again the model (5.1) and (5.2), where xK can be correlated with u Now,however, assume that we have more than one instrumental variable for xK Let z1,

z2; ; zM be variables such that

so that each zh is exogenous in equation (5.1) If each of these has some partial relation with xK, we could have M di¤erent IV estimators Actually, there are manymore than this—more than we can count—since any linear combination of x1,

cor-x2; ; xK1, z1, z2; ; zM is uncorrelated with u So which IV estimator should weuse?

In Section 5.2.3 we show that, under certain assumptions, the two-stage leastsquares (2SLS ) estimator is the most e‰cient IV estimator For now, we rely onintuition

To illustrate the method of 2SLS, define the vector of exogenous variables again by

z 1ð1; x1; x2; ; xK1; z1; ; zMÞ, a 1  L vector ðL ¼ K þ MÞ Out of all possiblelinear combinations of z that can be used as an instrument for xK, the method of2SLS chooses that which is most highly correlated with xK If xK were exogenous,then this choice would imply that the best instrument for xK is simply itself Rulingthis case out, the linear combination of z most highly correlated with xK is given bythe linear projection of xK on z Write the reduced form for xK as

xK¼ d0þ d1x1þ


þ dK1xK1þ y1z1þ


þ yMzMþ rK ð5:14Þwhere, by definition, rKhas zero mean and is uncorrelated with each right-hand-sidevariable As any linear combination of z is uncorrelated with u,

K is not a usable instrument However, as long as we make thestandard assumption that there are no exact linear dependencies among the exoge-nous variables, we can consistently estimate the parameters in equation (5.14) byOLS The sample analogues of the xiK for each observation i are simply the OLSfitted values:

^

x ¼ ^dd þ ^dd x þ


þ ^dd x þ ^y z þ


þ ^y z ð5:16Þ

Now, for each observation i, define the vector ^xi1ð1; xi1; ; xi; K1; ^xiKÞ, i ¼1; 2; ; N Using ^xias the instruments for xigives the IV estimator

^

xi0yi

!

where unity is also the first element of xi

The IV estimator in equation (5.17) turns out to be an OLS estimator To see thisfact, note that the N ðK þ 1Þ matrix ^XX can be expressed as ^X¼ ZðZ0ZÞ1Z0X¼

PZX, where the projection matrix PZ¼ ZðZ0ZÞ1Z0 is idempotent and symmetric.Therefore, ^X0X¼ X0PZX¼ ðPZXÞ0PZX¼ ^X0XX Plugging this expression into equa-^tion (5.17) shows that the IV estimator that uses instruments ^xi can be written as

^¼ ð ^X0XÞ^ 1X^0Y The name ‘‘two-stage least squares’’ comes from this procedure

To summarize, ^b b can be obtained from the following steps:

1 Obtain the fitted values ^xK from the regression

where the i subscript is omitted for simplicity This is called the first-stage regression

2 Run the OLS regression

This is called the second-stage regression, and it produces the ^bj

In practice, it is best to use a software package with a 2SLS command rather thanexplicitly carry out the two-step procedure Carrying out the two-step procedureexplicitly makes one susceptible to harmful mistakes For example, the following,seemingly sensible, two-step procedure is generally inconsistent: (1) regress xK on1; z1; ; zM and obtain the fitted values, say ~xK; (2) run the regression in (5.19) with

~

xK in place of ^xK Problem 5.11 asks you to show that omitting x1; ; xK1 in thefirst-stage regression and then explicitly doing the second-stage regression producesinconsistent estimators of the bj

Another reason to avoid the two-step procedure is that the OLS standard errorsreported with regression (5.19) will be incorrect, something that will become clearlater Sometimes for hypothesis testing we need to carry out the second-stage regres-sion explicitly—see Section 5.2.4

The 2SLS estimator and the IV estimator from Section 5.1.1 are identical whenthere is only one instrument for xK Unless stated otherwise, we mean 2SLS whenever

we talk about IV estimation of a single equation

What is the analogue of the condition (5.5) when more than one instrument isavailable with one endogenous explanatory variable? Problem 5.12 asks you to showthat Eðz0xÞ has full column rank if and only if at least one of the yjin equation (5.14)

is nonzero The intuition behind this requirement is pretty clear: we need at least oneexogenous variable that does not appear in equation (5.1) to induce variation in xKthat cannot be explained by x1; ; xK1 Identification of b does not depend on the

values of the dh in equation (5.14)

Testing the rank condition with a single endogenous explanatory variable andmultiple instruments is straightforward In equation (5.14) we simply test the nullhypothesis

against the alternative that at least one of the yj is di¤erent from zero This test gives

a compelling reason for explicitly running the first-stage regression If rKin equation(5.14) satisfies the OLS homoskedasticity assumption OLS.3, a standard F statistic orLagrange multiplier statistic can be used to test hypothesis (5.20) Often a hetero-skedasticity-robust statistic is more appropriate, especially if xK has discrete charac-teristics If we cannot reject hypothesis (5.20) against the alternative that at least one

yhis di¤erent from zero, at a reasonably small significance level, then we should haveserious reservations about the proposed 2SLS procedure: the instruments do not pass

a minimal requirement

The model with a single endogenous variable is said to be overidentified when M >

1 and there are M 1 overidentifying restrictions This terminology comes from thefact that, if each zh has some partial correlation with xK, then we have M 1 moreexogenous variables than needed to identify the parameters in equation (5.1) Forexample, if M ¼ 2, we could discard one of the instruments and still achieve identi-fication In Chapter 6 we will show how to test the validity of any overidentifyingrestrictions

5.2 General Treatment of 2SLS

5.2.1 Consistency

We now summarize asymptotic results for 2SLS in a single-equation model withperhaps several endogenous variables among the explanatory variables Write thepopulation model as in equation (5.7), where x is 1 K and generally includes unity.Several elements of x may be correlated with u As usual, we assume that a randomsample is available from the population

assumption2SLS.1: For some 1 L vector z, Eðz0uÞ ¼ 0.

Here we do not specify where the elements of z come from, but any exogenous ments of x, including a constant, are included in z Unless every element of x is ex-ogenous, z will have to contain variables obtained from outside the model The zeroconditional mean assumption, Eðu j zÞ ¼ 0, implies Assumption 2SLS.1

ele-The next assumption contains the general rank condition for single-equationanalysis

assumption2SLS.2: (a) rank Eðz0zÞ ¼ L; (b) rank Eðz0xÞ ¼ K

Technically, part a of this assumption is needed, but it is not especially important,since the exogenous variables, unless chosen unwisely, will be linearly independent inthe population (as well as in a typical sample) Part b is the crucial rank condition foridentification In a precise sense it means that z is su‰ciently linearly related to x sothat rank Eðz0xÞ has full column rank We discussed this concept in Section 5.1 forthe situation in which x contains a single endogenous variable When x is exogenous,

so that z¼ x, Assumption 2SLS.1 reduces to Assumption OLS.1 and Assumption2SLS.2 reduces to Assumption OLS.2

Necessary for the rank condition is the order condition, L b K In other words, wemust have at least as many instruments as we have explanatory variables If we do

not have as many instruments as right-hand-side variables, then b is not identified.

However, L b K is no guarantee that 2SLS.2b holds: the elements of z might not beappropriately correlated with the elements of x

We already know how to test Assumption 2SLS.2b with a single endogenous planatory variable In the general case, it is possible to test Assumption 2SLS.2b,given a random sample onðx; zÞ, essentially by performing tests on the sample ana-logue of Eðz0xÞ, Z0X=N The tests are somewhat complicated; see, for example Craggand Donald (1996) Often we estimate the reduced form for each endogenous ex-planatory variable to make sure that at least one element of z not in x is significant.This is not su‰cient for the rank condition in general, but it can help us determine ifthe rank condition fails

ex-Using linear projections, there is a simple way to see how Assumptions 2SLS.1 and

2SLS.2 identify b First, assuming that Eðz0zÞ is nonsingular, we can always writethe linear projection of x onto z as x¼ zP, where P is the L  K matrix P ¼

½Eðz0zÞ1Eðz0xÞ Since each column of P can be consistently estimated by regressing

the appropriate element of x onto z, for the purposes of identification of b, we can

treat P as known Write x¼ xþ r, where Eðz0rÞ ¼ 0 and so Eðx 0rÞ ¼ 0 Now, the2SLS estimator is e¤ectively the IV estimator using instruments x Multiplying

equation (5.7) by x 0, taking expectations, and rearranging gives

since Eðx 0uÞ ¼ 0 Thus, b is identified by b ¼ ½Eðx 0xÞ1Eðx 0yÞ provided Eðx 0xÞ isnonsingular But

Eðx 0xÞ ¼ P0Eðz0xÞ ¼ Eðx0zÞ½Eðz0zÞ1Eðz0xÞ

and this matrix is nonsingular if and only if Eðz0xÞ has rank K; that is, if and only ifAssumption 2SLS.2b holds If 2SLS.2b fails, then Eðx 0xÞ is singular and b is notidentified [Note that, because x¼ xþ r with Eðx 0rÞ ¼ 0, Eðx 0xÞ ¼ Eðx 0xÞ So b

is identified if and only if rank Eðx 0xÞ ¼ K.]

The 2SLS estimator can be written as in equation (5.17) or as

zi0zi

!1

XN i¼1

zi0xi

!2

4

35

1

XN i¼1

xi0zi

!

XN i¼1

zi0zi

!1

XN i¼1

zi0yi

!

ð5:22Þ

We have the following consistency result

theorem 5.1 (Consistency of 2SLS): Under Assumptions 2SLS.1 and 2SLS.2, the

2SLS estimator obtained from a random sample is consistent for b.

zi0zi

!1

N1XN i¼1

zi0xi

!2

4

35

zi0zi

!1

N1XN i¼1

assumption2SLS.3: Eðu2z0zÞ ¼ s2Eðz0zÞ, where s2¼ Eðu2Þ.

This assumption is the same as Assumption OLS.3 except that the vector of ments appears in place of x By the usual LIE argument, su‰cient for Assumption2SLS.3 is the assumption

ð ^b bÞ is asymptotically normally distributed with mean zero and variance matrix

The proof of Theorem 5.2 is similar to Theorem 4.2 for OLS and is therefore omitted.The matrix in expression (5.24) is easily estimated using sample averages To esti-mate s2 we will need appropriate estimates of the ui Define the 2SLS residuals as

Note carefully that these residuals are not the residuals from the second-stage OLSregression that can be used to obtain the 2SLS estimates The residuals from thesecond-stage regression are yi ^xib Any 2SLS software routine will compute equa-^tion (5.25) as the 2SLS residuals, and these are what we need to estimate s2

Given the 2SLS residuals, a consistent (though not unbiased) estimator of s2underAssumptions 2SLS.1–2SLS.3 is

^

s21ðN  KÞ1XN

i¼1

Many regression packages use the degrees of freedom adjustment N K in place of

N, but this usage does not a¤ect the consistency of the estimator

is a valid estimator of the asymptotic variance of ^b b under Assumptions 2SLS.1–

2SLS.3 The (asymptotic) standard error of ^bjis just the square root of the jth onal element of matrix (5.27) Asymptotic confidence intervals and t statistics areobtained in the usual fashion

diag-Example 5.3 (Parents’ and Husband’s Education as IVs): We use the data on the

428 working, married women in MROZ.RAW to estimate the wage equation (5.12)

We assume that experience is exogenous, but we allow educ to be correlated with u.The instruments we use for educ are motheduc, fatheduc, and huseduc The reducedform for educ is

educ¼ d0þ d1experþ d2exper2þ y1motheducþ y2fatheducþ y3huseducþ r

Assuming that motheduc, fatheduc, and huseduc are exogenous in the logðwageÞequation (a tenuous assumption), equation (5.12) is identified if at least one of y1, y2,and y3 is nonzero We can test this assumption using an F test (under homoskedas-ticity) The F statistic (with 3 and 422 degrees of freedom) turns out to be 104.29,which implies a p-value of zero to four decimal places Thus, as expected, educ isfairly strongly related to motheduc, fatheduc, and huseduc (Each of the three t sta-tistics is also very significant.)

When equation (5.12) is estimated by 2SLS, we get the following:

logð ^wwageÞ ¼ :187

ð:285Þ

þ :043ð:013Þ

exper  :00086

ð:00040Þ

exper2þ :080

ð:022Þeduc

where standard errors are in parentheses The 2SLS estimate of the return to tion is about 8 percent, and it is statistically significant For comparison, whenequation (5.12) is estimated by OLS, the estimated coe‰cient on educ is about 107with a standard error of about 014 Thus, the 2SLS estimate is notably below theOLS estimate and has a larger standard error

educa-5.2.3 Asymptotic E‰ciency of 2SLS

The appeal of 2SLS comes from its e‰ciency in a class of IV estimators:

theorem 5.3 (Relative E‰ciency of 2SLS): Under Assumptions 2SLS.1–2SLS.3,the 2SLS estimator is e‰cient in the class of all instrumental variables estimatorsusing instruments linear in z

Proof: Let ^b b be the 2SLS estimator, and let ~ b b be any other IV estimator using

instruments linear in z Let the instruments for ~b b be ~xx 1 zG, where G is an L Knonstochastic matrix (Note that z is the 1 L random vector in the population.)

We assume that the rank condition holds for ~xx For 2SLS, the choice of IVs ise¤ectively x ¼ zP, where P ¼ ½Eðz0zÞ1Eðz0xÞ 1 D1C (In both cases, we can re-place G and P with ffiffiffiffiffi

N

p-consistent estimators without changing the asymptotic vari-ances.) Now, under Assumptions 2SLS.1–2SLS.3, we know the asymptotic variance

It follows that Eð~x0xÞ ¼ Eð~x0xÞ, and so

Eðx 0xÞ  Eðx0~xÞ½Eð~x0~xÞ1Eð~x0xÞ

¼ Eðx 0xÞ  Eðx 0xÞ½Eð~~ x0~xÞ1Eð~x0xÞ ¼ Eðs 0sÞ

where s¼ x Lðxj ~xÞ is the population residual from the linear projection of x

on ~xx Because Eðs 0sÞ is p.s.d, the proof is complete

Theorem 5.3 is vacuous when L¼ K because any (nonsingular) choice of G leads

to the same estimator: the IV estimator derived in Section 5.1.1

When x is exogenous, Theorem 5.3 implies that, under Assumptions 2SLS.1–2SLS.3, the OLS estimator is e‰cient in the class of all estimators using instrumentslinear in exogenous variables z This statement is true because x is a subset of z and

so Lðx j zÞ ¼ x

Another important implication of Theorem 5.3 is that, asymptotically, we always

do better by using as many instruments as are available, at least under skedasticity This conclusion follows because using a subset of z as instruments cor-responds to using a particular linear combination of z For certain subsets we mightachieve the same e‰ciency as 2SLS using all of z, but we can do no better This ob-servation makes it tempting to add many instruments so that L is much larger than

homo-K Unfortunately, 2SLS estimators based on many overidentifying restrictions cancause finite sample problems; see Section 5.2.6

Since Assumption 2SLS.3 is assumed for Theorem 5.3, it is not surprising thatmore e‰cient estimators are available if Assumption 2SLS.3 fails If L > K, a moree‰cient estimator than 2SLS exists, as shown by Hansen (1982) and White (1982b,1984) In fact, even if x is exogenous and Assumption OLS.3 holds, OLS is not gen-

erally asymptotically e‰cient if, for x H z, Assumptions 2SLS.1 and 2SLS.2 hold but

Assumption 2SLS.3 does not Obtaining the e‰cient estimator falls under the rubric

of generalized method of moments estimation, something we cover in Chapter 8.5.2.4 Hypothesis Testing with 2SLS

We have already seen that testing hypotheses about a single bj is straightforward ing an asymptotic t statistic, which has an asymptotic normal distribution under thenull; some prefer to use the t distribution when N is small Generally, one should be