Spec Tests

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Econometrics I1

Specication tests

Marianna Belloc

Sapienza University of Romehttp://w3.uniroma1.it/belloc/

November-December, 2009PhD Economics, University of Siena

1These slides strongly rely on (and some parts are taken directly from):W1: J. Wooldridge: Introductory Econometrics: A Modern Approach, South-Western

W2: J. Wooldridge: Econometric Analysis of Cross Section and PanelData, MITPressBelloc (Sapienza) Lecture VII Nov-De c, 2009 1 / 21
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Testing for endogeneity

Call y1 the dependent variable of ther structural model, y2 theendogenous independent variable, and z1 the exogenous regressors in

the structural model,

y1 =z11+1y2+u1

where z1 (l1 1). Assume we have a subset of instruments z, which

is a l 1 vector, such thatE(zu1) =0

Identication requires that at least one instrument in z is excluded inz1 (order condition), and at least one instrument in z which is

excluded in z1 is partially correlated with y2 (rank condition)The Hausman test for endogeneity consists in comparing ols and 2slsestimators for 1 (1, 1): ify2 is uncorrelated with u1, then thetwo estimators should dier only by the sampling error. We will

consider the regression based form of the testBelloc (Sapienza) Lecture VII Nov-De c, 2009 2 / 21
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Testing for endogeneity: Hausman test I

Consider the linear projection ofy2 on z

y2 =z2+v2 where E(z0v2) =0

Since zis uncorrelated with u1, it follows that y2 is endogenous (thatis E(u1y2)6=0) if and only ifE(u1v2)6=0

We then want to test ifE(u1v2)6=0, and run

u1 =1v2+e1

where 1 =E(u1v2)/E(v22), E(e1v2) =0 and E(z

0e1) =0(since

E(z0

u1) =0and E(z0

v2) =0)Substitute u1 =1v2+e1 in y1 =z11+1y2+u1 and obtain

y1 =z11+1y2+1v2+e1

Belloc (Sapienza) Lecture VII Nov-De c, 2009 3 / 21
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Testing for endogeneity: Hausman test IIThe Hausman test consists in testing

H0 :1 =0

since v2 is not observable it can be obtained by the ols residuals ofthe rst stage, v2, so that we can consistently estimate regression

y1 =z11+1y2+1v2+e1

by ols and then compute tstatistic (the classical one or theheteroskedasticity robust if heresoskedasticity is suspected) for 1 inorder to the H0 :1 =0 (Nota bene: ols estimates from aboveequation are identical to estimates from the 2sls procedure)

When there is only one suspected endogenous regressor, we can use

directly a statistic that only compares ols and 2sls estimates of theparameter of interest, that is in our case

1,2sls 1,ols[se(1,2sls)2 se(1,ols)2]1/2

which is the Hausman t statisticBelloc (Sapienza) Lecture VII Nov-De c, 2009 4 / 21
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Testing for endogeneity: application I

To implement the Hausman procedure

I

reg educ exper exper2 motheduc fatheduc huseducI predict res, residualI reg ly educ exper exper2 res

The null that educ is exogenous can be rejected at the 10% level, sowe are induced to use 2sls rather than ols

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Testing for endogeneity: application II

Type

I ivreg ly exper exper2 (educ=motheduc fatheduc huseduc)I est store 2slsI reg ly educ exper exper2I hausman 2sls

where 0.027/0.0165= 1.65 which induces to reject the null at the10% level

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Testing for endogeneity: Hausman test III

Extensions to the multiple (lets say g1) endogenous regressors case isstraightforward

y1 =z11+y21+u1 with E(z0u1) =0

where 1 is g1 1, where y2 = (y12 , y22 ...yg12)

Reduced form for y2 isy2 =z2+v2

where v2 is g1 1, and v2 can be obtained from the ols residuals ofeach of the g1 rst stages (one for each endogenous regressor yh2 onz)

Estimatey1 =z11+y21+ v21+e

Run F test for H0 :1 =0 (test for multiple g1 restrictions), wherethe restricted model is that obtained setting 1 =0

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Testing for endogeneity: Hausman test IV

An alternative, as usual, is the LM test:

I Run ols regression y1 on z1 and y2,and obtain u1I Run ols regression ofy2 on z, and obtain v2I Run ols regression u1 on z1, y2 and v2, and obtain R

2u

I Compute nR2ua2g1

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Testing for endogeneity: application III

We want to estimate

log(wage) = 1+2exper+3exper2 +4educ+5black educ+ u

where we suspect educto be endogenous and consequently also theinteraction term educ blackto be endogenous

Suppose we exploit Card (1995)s idea to use nearc4 as an instrumentforeduc(which is a dummy variable if the individual grew up in theproximity of a four year college), and so nearc4 blackas aninstrument for educ black

We have

I use D:nLezioninDottoratonEconometrics_SienanstatanCARD.DTAI gen exper2=exper*experI gen educbl=educ*blackI gen nearc4bl=nearc4*blackI ivreg lwage exper exper2 (educ educbl=nearc4 nearc4bl)

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Testing for endogeneity: application IV

Nota bene: R2 is missing because it is negative (indeed the explained sumof squares is negative) and stata suppresses it. Seehttp://www.stata.com/support/faqs/stat/2sls.html

Belloc (Sapienza) Lecture VII N ov-D ec, 2009 10 / 21
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Testing for endogeneity: application V

For HausmanI reg educ exper exper2 nearc4 nearc4blI predict res1, residualI reg educbl exper exper2 nearc4 nearc4blI predict res2, residualI

reg lwage educ educbl exper exper2 res1 res2I test res1 res2

which induces us to reject the null of exogeneity ofeduc and educbl

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Testing for endogeneity: application VI

For LM

I reg lwage exper exper2 educ educblI predict res, residualI reg educ exper exper2 nearc4 nearc4blI predict res1, residualI reg educbl exper exper2 nearc4 nearc4blI predict res2, residualI reg res exper exper2 educ educbl res1 res2I gen LM=0.0131 3010I

pval=chi2tail(2,39.43)

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Testing for endogeneity: application VI

compute: n R2 =0.0131 3010= 39.431, p-value is 0.0000. Thisinduces us again to reject the null of exogeneity ofeduc and educbl

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Testing for overidentifying restrictions I

If we have more instruments than required for exact identication, wecan test if they are exogenous, i.e. ifE(z, u1) =0

Considery1 =z11+y21+u1

and take the partition ofz= (z1, z2) where z2 is a 1 l2 vectorcontaining the excluded instruments

Hausmans general idea is to compare 2sls estimates with olsestimate, if all the instrument are valid they should dier only as a

result of the sampling error

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Testing for overidentifying restrictions II

We can implement the following procedureI Run 2sls regression y1 on z1 and y2,using zas instruments, and obtain

u1I Run ols regression ofu1 on z, and obtain R

2u

I Compute nR2uI Under H0 :E(z, u1 ) =0, nR2u a2q1 where q1 l2 g1 (i.e. number

of overidentifying restrictions)I (as always we can obtain the heteroskedasticity robust version)

IfH0 :E(z, u1) =0 is rejected, then some instrument is not valid

If, by contrast, H0 :E(z, u1) =0 is not rejected, then we can havesome condence that in the overall set of instruments

http://find/http://goback/


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Testing for overidentifying restrictions - application IType

I use D:nLezioninDottoratonEconometrics_Sienanstatanmroz.dta, gen

ly=log(wage), gen exper2=exper*experI ivreg ly exper exper2 (educ=motheduc fatheduc huseduc)I predict res, residualI reg res exper exper2 motheduc fatheduc huseducI gen pval=chi2tail(2, 1.11)

the test statistic is 0.026 428= 1.11 and the p-value is 0.574 so the

null hypothesis is not rejectedBelloc (Sapienza) Lecture VII N ov-D ec, 2009 16 / 21
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Testing for functional forms

Sometimes we may suspect that we are neglecting some nonlinearities

in the model estimated either by ols or by 2slsIn the case of all exogenous regressors we can just include nonlinearfunctions (squares or cross-products) of our regressions in theequation and then implement F orLM statistics

In the presence of suspected endogeneity we need instruments for allthe nonlinear functions we are chooseing of our (potentially)endogenous regressions. We might use as instruments thecorresponding nonlinear functions of all the instrument we are usingfor our (potentially) endogenous regressors, but that would consume

many degrees of freedomRamseys RESET test is conserving on the number of degrees offreedom (which is kregardless the number of nonlinear functions wewant to control for)


f f f S
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Testing for functional forms: Ramseys RESET test I

Consider

y=x +uwith E(ujx) =0

Condition E(ujx) =0 implies that u is uncorrelated with anyregressor in x and with any function ofx

Dene yi= xi, the tted values from the ols regression, and ui the

ols residuals. By denition of ols, cov(ui, yi) =0

But we may want to test ifui is correlated with any low-orderpolynomial ofyi : y

2i , y

3i , y

4i

There are two ways to test that

I Estimate by ols: yi = 1+2xi2...kxik+2y2i +3y3i +4y4i +ui,and compute standard F test for H0 :2 =3 =4 =0

I Regress ui on xi, y2i , y

3i ,y

4i , compute LM=nR

2u

a23


T i f f i l f R RESET II
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Testing for functional forms: Ramseys RESET test II

Nota bene: It is sometimes maintained that the RESET test can be

used as a test for any specication problem, such as heteroskedasticityor omitted variables. However it is a poor test for such problems


T i f f i l f li i I
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Testing for functional forms: application I

use D:nLezioninDottoratonEconometrics_Sienanstatanmroz.dta, genly=log(wage), gen exper2=exper*exper

reg ly exper exper2 educ

predict tted

predict res, residualgen tt2=tted*tted, gen tt3=tted*tted*tted, gentt4=tt2*tt2

reg res exper exper2 educ tt2 tt3 tt4

gen LM=0.0078*428gen pval=chi2tail(3,3.3384)


T ti f f ti l f li ti II
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Testing for functional forms: application II

LM=0.0078 428=3.3384, the corresponding p-value with 3 degrees offreedom is 0.3423 therefore we are not led to reject the null hypothesis

http://goforward/http://find/http://goback/

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