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IntroductorymEconometrics
- Chapter m 1 ThemNature mofmEconometrics mand mEconomic mData JeffreymM.mWooldridge
- Part m 1 Regression mAnalysis mwith mCross-Sectional mData
- Chapter m 2 The mSimple mRegression mModel
- Chapter m 3 MultiplemRegression mAnalysis: mEstimation
- Chapter m 4 Multiple mRegression mAnalysis: mInference
- Chapter m 5 MultiplemRegression mAnalysis: mOLS mAsymptotics
- Chapter m 6 MultiplemRegression mAnalysis: mFurther mIssues
- Chapter m 7 MultiplemRegressionmAnalysismwithmQualitativemInformation:mBinarymvariables m
- Chapter m 8 Heteroskedasticity
- Chapter m 9 More mon mSpecification mand mData mproblems
- Part m 2 Regression mAnalysis mwith mTime mSeries mData
- Chapter m 10 Basic mRegression manalysis mwith mTime mSeries mData
- Chapter m 11 FurthermIssues min mUsing mOLS mwith mTime mSeries mData
- Chapter m 12 Serial mCorrelation mand mHeteroskedasticitymin mTimemSeries mRegression
- Part m 3 Advanced mTopics
- Chapter m 13 Pooling mCross mSections macross mTime. mSimple mPanel mData mMethods
- Chapter m 14 Advanced mPanel mData mMethods
- Chapter m 15 Instrumental mVariables mEstimation mand mTwo mStage mLeast mSquares
- Chapter m 16 SimultaneousmEquations mModels
- Chapter m 17 LimitedmDependentmVariablemModelsmand mSamplemSelectionmCorrections m
- mChapter m 18 Advanced mTime mSeries mTopics
- Chapter m 19 Carrying mOut man mEmpirical mProject
- Appendix: mSome mfundamentals mof mprobability
Chapterm 1 The mNature mof mEconometricsmandmEconomic mData
I. The mgoal mof manymeconometric manalysis mis mto mestimate mthe mparameters min mthe mmodel mand mto mtest mhypotheses mabout mthese mparameters; mthe mvalues mand msigns mof mthe mparameters mdeterminemthemvaliditymofman meconomicmtheorymand mthemeffects mof mcertain mpolicies. II. Panel mdata m- madvantages:
- Having mmultiple mobservations mon mthe msame munits mallows mus mto mcontrol mcertain munobservedmcharacteristicsmofmindividuals, mfirms, mand msomon.mThemusemofmmoremthan mone mobservation mcan mfacilitate mcausal minference min msituations mwhere minferring mcausality mwould mbe mverymhard mif monlyma msingle mcross msection mwere mavailable.
- Theymoften mallow mus mtomstudymthemimportancemofmlags min mbehaviormormthemresult mof mdecision mmaking.
Part m 1 RegressionmAnalysismwithmCross-Sectional mData
mChapter m 2 mThe mSimple mRegression mModel
I. Model: mY m=mb0 m+mb1x m+mu
- Populationmregressionmfunctionm(PRF) :mE(y|x) m=mb0 m+mb1x
- systematic mpart mof my: m b0 m+ mb1x
- unsystematic mpart: mu
II. Sample mregression mfunction m(SRF ): myhat m= mb0hat m+ mb1hat*x
- PRF mis msomething mfixed, mbut munknown, min mthe mpopulation. mSince mthe mSRF mis mobtainedmformamgiven msamplemofmdata,mamnewmsamplemwill mgeneratemamdifferent mslope mand mintercept.
III. Correlation :mit mismpossible mformu mtombemuncorrelated mwithmxmwhilembeing mcorrelated mwith mfunctions mof mx, msuch mas mx^2. E(u|x)m=mE(u) m mCov(u, mx)m=m0. mnot mvice mversa.
IV. Algebraic mproperties mof mOLS mstatistics
- The msum mof mthe mOLS mresiduals mis mzero.
- The msample mcovariance mbetween mthe m(each) mregressors mand mthe mresiduals mis mzero. mConsequently, mthemsample mcovariancembetween mthemfitted mvaluesmand mthe mresiduals mis mzero.
- The mpoint m(m x , m ym ) mis mon mthe mOLS mregression mline.
- the mgoodness-of-fit mof mthe mmodel mis m invariant m to mchanges minmthe munits mofmymor mx.
- Themhomoskedasticitymassumptionmplays mnomroleminmshowingmOLSmestimatorsmare munbiased.
V. Variance
- Var(b1) m= mvar(u)/SSTx a. moremvariation minmthemunobservables m(u)maffectingmymmakes mit mmoremdifficult mto mpreciselymestimate mb1.
SST (^) jm (1 mm Rm^2 m) j
SST mj (1mm Rm^2 m) j
j
j
1
j
VII. VariancemofmOLSmestimators: mA5: mhomoskedasticity
1. Gaussm–mMarkovmassumptions:mA1m–mA
m^22
- Var ( bj m ) m m SST m (1 mm Rm^2 m ) m
, mwhere mR m is mfrom mregressing mxj mon mall mother mindependent
variablesm(and mincludingmanmintercept). a. Themerror mvariance, mσ2, m is mamfeature mof mthempopulation, mit mhas mnothingmto mdo mwith mthe msample msize. b. SSTj: mthe mtotal msamplemvariation minmxj: ma msmall msample msizem msmall mvalue mofmSSTj mlarge mvar(bj)
c. R^2 m: mhigh mcorrelation mbetween mtwo mormmore mindependent mvariables mis mcalled
multicollinearity.
- Amhighmdegree mofmcorrelationmbetweenmcertainmindependentmvariablesmcanmbe mirrelevant mas mto mhow mwell mwe mcan mestimate mother mparameters min mthe mmodel : mY m= mb 0 m+ mb 1 x 1 m+ mb 2 x 2 m+ mb 3 x 3 m+ mu, mwhere mx2 mand mx3 mare mhighlymcorrelated. Themvar(b2)mand mvar(b3)mmaymbemlarge. mBut mthemamount mofmcorrelation mbetween mx2 mand mx mhas mno mdirect meffect mon mvar(b1). mIn mfact, mif mx1 mis muncorrelated mwith mx2 mand mx3, mthen
Rm^2 m=0 mand mvar(b1)m=
mand mx3.
m^2 SST 1
, mregardless mof mhow mmuch mcorrelation mthere mis mbetween mx
If mb1 mis mthe mparameter mof minterest, mwe mdo mnot mreallymcaremabout mthe mamount mof correlation mbetween mx2 mand mx3.
- The mtradeoff mbetween mbias mand mvariance. If mthe mtrue mmodel mis mY m= mb 0 m+ mb 1 x 1 m+ mb 2 x 2 m+ mu, minstead, mwe mestimate mYm= mb 0 m+ mb’ 1 x 1 m+ mu a. when mb2 mis mnonzero, mb’1 mis mbiased, mb1 mis munbiased, mvar(b’1)<var(b1); b. when mb2 mismzero,mb’1mismunbiased,mb1 mis munbiased, mvar(b’1)<var(b1) mm a mhigher mvariance mfor mthe mestimator mof mb1 mis mthe mcost mof mincluding man mirrelevant mvariable min ma mmodel;
VIII. Estimating:mstandard merrorsmof mestimators.
- Under mA1-A5 : mE(σ’^2 ) m= mσ^2 , mwhere mσ’^2 m = m
u m
2 m (σ’ mis mσhat) ( n m m k m m1)
- Standard mdeviation m of mbj’, msd(bj’) m= m
- Standard merror m of mbj’: mse(bj’) m= m hat
Standard merror mof mbj’ mis mnot ma mvalid mestimator mof msd(bj’) mif mthe merrors mexhibit mheteroskedasticity.mThus,mwhilemthempresencemofmheteroskedasticitymdoesmnotmcause mbias min mthe mbj’, mit mdoes mlead mto mbias min mthe musual mformula mfor mVar(bj’), mwhich mthen minvalidates mthe mstandard merrors.
Chapterm 4 MultiplemRegressionmAnalysis:mInference
I. Classical mLinear mmodel m(CLM) massumptions:
- EvenmundermGauss-Markovmassumptionsm(A1-5), mthemdistributionmofmestimators mcan mhave mvirtuallymanymshape.
- A6:mNormality mu m~ mN(0, mσ^2 ).
- Under mCLM, mthe mOLS mestimators mare mthe mminimum mvariance munbiased mestimators ; mwemno mlongermhavemtomrestrict mourmcomparisonmtomestimatorsmthat mare mlinear min mthe m y.
- THM m4.1 m under mA1-A6: mb’j m~ mN(bj, mVar(b’j)); m m(b’j m– mbj)/sd(b’j) m~ mN(0,1).
- THM m4.2m under mA1-A6: m m(b’j m– mbj)/se(b’j)m~ mt(n-k-1).
II. Test m– mT-test:
- It mis ma mkeymto mremember mthat mwe mare mtesting mhypothesis mabout mthe mpopulation mparameters.mWemaremnotmtestingmhypothesis mabout mthe mestimates mfrommamparticular msample. mThus, mit mnever mmakes msense mto mstate ma mnull mhypothesis mas m“H0: mb’j m= m0”
- There mis mno m“correct” msignificance mlevel.
- P-value: a. Givenmthemobserved mvaluemofmthemt mstatistic, mwhat mis mthemsmallest msignificancemlevel mat mwhich mthe mnull mhypothesis mwould mbe mrejected? b. P-value mis mthe mprobabilitymof mobserving ma mt mstatistic mas mextreme mas mwe mdid mif mthe mnull mhypothesismis mtrue.mThis mmeansmthat msmall mp-valuesmaremevidencemagainst mthe mnull; mlarge mp-values mprovide mlittle mevidence magainst mH0. c. Tomobtain mthemone-sided mp-value: mjustmdivide mthe mtwo-sided mp-valuembym 2.
- Wemshould msaym“ wemfail mtomrejectmH0matmthemx% mlevel ,”mrathermthanm“H0mis maccepted mat mthe mx% mlevel”.
III. Economic mvs. mstatistic msignificance.
- The m statistical msignificance m of ma mvariable mxj mis mdetermined mentirely mby mthe msize mof mtbj, mwhereas mthem economic msignificancem or mpractical msignificance mof ma mvariable mis mrelated mto mthe msize mand msign mof mbj.
- Withmlargemsample msizes,mparametersmcan mbemestimatedmverymprecisely: mstandardmerrors mare moften mquite msmall mrelative mto mthe mcoefficient mestimates, mwhich musuallymresults min mstatistical msignificance.
- Some mresearchers minsist mon musing msmaller msignificance mlevels mas mthe msample msize mincreases, mpartlymas mamwaymtomoffset mthemfact mthat mstandard merrorsmaremgettingmsmaller. mMost mresearchers mare malso mwilling mto mentertain mlarger msignificance mlevels min mapplications mwith msmall msample msizes, mreflecting mthe mfact mthat mit mis mharder mto mfind msignificance mwith msmaller msample msizes.
IV. Confidence mintervals m(CI)
- meaning mof ma mCI: mif mrandom msamples mwere mobtained mover mand mover magain, mwith mthe mCI mcomputed meach mtime, mthen mthe munknown mpopulation mvalue mbj mwould mlie min mthe mCImfor m95% mof mthe msamples. mUnfortunately, mfor mthe msingle msample mthat mwe muse mto mconstruct mthe mCI,mwemdomnot mknow mwhethermbj mis mactuallymcontained min mtheminterval. mWemhopemwe mhave mobtained ma msample mthat mis mone mof mthe m95% mof mall msamples mwhere mthe minterval mestimate mcontains mbj, mbut mwe mhave mno mguarantee.
q
**2. withmlarge msample, mwe mdon’tmneed mA6 mto marrive matmnormality
- THM m5.2:** Under mthe mGauss-Markov massumptions mA1-A a. n m ( b '^ m m b m )m a m Nm (0,m^2 m/m a^2 m)m, mwhere m m^2 m/m a^2 m>0 mis mthemasymptotic mvariance mof j j j j n
n m ( b '^ m m b m )m; mfor mthe mslope mcoefficients, m a^2 m= m p m lim( n ^1 m r^2 m)m, mwhere mrmare mthe mresiduals
j j j ij jm 1 from mregressing mxj mon mthe mother mindependent mvariables.
b. σ’^2 mis ma mconsistent mestimator mof m σ^2 = mvar(u); c. formeach mj,m(b’j m– mbj)/se(b’j) m mN(0,1)mwheremse(b’j)mis mthemusualmOLS mstandard merror. mNote: mWe m don’t mneed mnormality massumption m here, mbut mwe m do mneed mto massume m the merror mhas mfinite mvariance, m zero mconditional mmean mand mhomoskedasticity.
- THMm5.3: m Asymptotic mefficiencym ofmOLS: mundermA1-A5,mthemOLS mestimators mhave mthe msmallest masymptotic mvariances.
III. Other mlarge msample mtests: mThe mLagrange mMultiplier mstatistic LM mrequires mestimation mof mthe mrestricted mmodel monly.
- Regressmymonmthe mrestricted msetm ofmindependentmvariablesmand msavemthe mresiduals, muhat.
- Regress muhat mon m all m of mthe mindependent mvariables mand mobtain mthe mR
- Compute m LM=n R^2 m (n mismthemoriginal msample msize).
- LM m~ 2 As mwith mthe mF mstatistic, mwe mmust mbe msure mto muse mthe msame mobservations min msteps m 1 mand m2. mIfmdatamaremmissingmformsomemofmthemindependent mvariables mthat maremexcluded munder mthe mnull mhypothesis, mthe mresiduals mfrom mstep m 1 mshould mbe mobtained mfrom ma mregression mon mthe mreduced mdata mset.
Chapterm 6 MultiplemRegressionmAnalysis:mFurthermIssues I. Beta mcoefficients :
- Computemthe mz-scoremfor meverymvariable min mthe msample.
ym m 0 m m 1 m x 1 m m... mm k m xk m m um , mwhere m b m m( m /m m ) z m m b m m bmx m m... mm b m x m m v j^ j^ y^ j 0 1 m 1 k m m k
- Ifmx1mincreasesmbymonemstandardmdeviation,mthenmymchanges mbymb1 mstandard mdeviations. mThis mmakes mthe mscale mof mthe mregressors mirrelevant.
- Whethermwemusem standardizedmormunstandardizedmvariablesmdoesmnotmaffect mstatistical msignificance: mthe mt mstatistics mare mthe msame m in mboth mcases.
II. Models mwith minteraction mterms. Ym= mb 0 m+ mb 1 x 1 m+mb 2 x 2 m+ mb3x 1 *x 2 m+ mu, mor Ym=ma 0 m+ ma 1 x 1 m+ma 2 x 2 m+ma3(x 1 m– mμ 1 )(x 2 m –μ 2 ) m+mu
III. Adjusted mR
m m x (^) k k m m x k k
- Them populationmR2 m ismdefined masm 1 m- m m^2 m/mm^2 m,mR2mestimates m m^2 m/mm^2 m by u y u y (SSR/n)/(SST/n). mThesemarembiased.
- adjusted mR2 m= m 1 m- m[SSR/(n-k-1)]/[SST/(n-1)].
- It mis mtemptingmtomthink mthatmadjusted mR2mcorrects mthembias minmR2 mformestimating mthe mpopulation mR2, mbut mit mdoes mnot: m the mratio mof mtwo munbiased mestimators mis mnot man munbiased mestimator.
- If mwe madd ma mnew mindependent mvariable mto ma mregression mequation, m adjusted mR mincreasesmif,mandmonlymif,mthemtmstatisticmon mthemnewmvariablemis mgreatermthan mone min mabsolute mvalue. m (an mextension mof mthis mis mthat madjusted mR2 mincreases mwhen ma mgroup mof mvariables mis madded mto ma mregression, miff mthe mF mstatistic mfor mjoint msignificance mof mthe mnew mvariables mis mgreater mthan munity.) mThus, mwe msee mthat musing madjusted mR2 mto mdecide mwhether ma mcertain mindependent mvariable m(or mset mof mvariables) mbelongs min mammodel mgivesmus mamdifferent manswermthan mstandard mt mormF mtesting.
- Comparing madjusted mR2 mto mchoose mamong mdifferent mnonnested msets mof mindependent mvariablesmcanmbemvaluablemwhenmthesemvariablesmrepresentmdifferent mfunctional mforms. Y m= mb 0 m+ mb 1 log(x) m+ mu, mor mY m=mb 0 m+mb 1 xm+mb 2 x^2 m+mu,mor
- There mis man mimportant mlimitation min musing madjusted mR2 mto mchoose mbetween mnonnestedmmodels: mwemcannotmusemitmtomchoosembetween mdifferent mfunctional mforms mfor mthe mdependent mvariables.
IV. Controllingmformtoo mmanymfactors min mregression
- Addingma mnew mindependent mvariable mto ma mregression mcan mexacerbate mthe mmulticollinearitymproblem.mOn mthemothermhand, msincemwemaremtakingmsomething mout mof mthe merror mterm, madding ma mvariable mgenerallymreduces mthe merror mvariance. mGenerally, mwe mcannot mknow mwhich meffect mwill mdominate.
- We m should malways minclude mindependent mvariables mthat maffect my mand mare muncorrelatedmwithmallmofmthemindependentmvariablesmofminterest. m msmaller msampling mvariance, mthough mwe mobtain man munbiased mand mconsistent mestimator mwith/without mthe mcontrolling mvariables.
V. Prediction mand mresidual manalysis
- Confidence mintervals mformpredictions : mCImformthemaveragemvaluemofmymformthe msubpopulation mwith ma mgiven mset mof mcovariates. Y m= mθ+ mb1(x1 m– mc1) m+ mb2(x2 m– mc2) m…m+ mbk(xk m– mck) m+ mu.
Whereθ mismexactlymthempredicted mvaluemofmymwhen mxj m=mcj. mWemcan mget mthe mCImofθ mfrom mthe mOLS.
- mBecause mthe m variance mof mthe mintercept mestimator mis msmallest mwhen meach mexplanatory mvariables mhas mzero msample mmean , mit mfollows mfrom mthe mregression mabovemthatmthemvariancemofmthemprediction mismsmallest mat mthemmean mvaluesmofmthemxj.
- prediction minterval: mCImfor ma mparticular munit mfrom mthe mpopulation.
y^0 m =m (^0 10) m m 1
m...mmm^0 m x m u^0
e '0^ m m y^0 m m y '0^ m m( (^0 10) m m 1
m... mmm^0 m x m )mm u^0 m m y '
- Chow mtest : mjust man mF mtest, m only mvalid mundermhomoskedasticity, mbutmnormality mis mnot mneeded m for masymptotic manalysis. H0:mtwo mgroups:mg1 mand mg2: mthemintercept mand mall mslopes maremthemsamemacross mthemtwo mgroups. a. Run mOLS mfor mg1 mand mg2 mseparately, mget mSSR1 mand mSSR2 m mSSRur m= mSSR1 m+ mSSR2. b. ThemrestrictedmSSR mis mjust mthemSSR mfrommpoolingmthemgroupsmand mestimatingmamsingle mequation, msaymSSRp.
c. F m= m
[ SSRp m m( SSR 1 m m SSR 2 m )]m m
[ N m m2( k m 1)] SSR 1 m m SSR 2 km m 1
- Note: m one mimportant mlimitation mof mthe mChow mtest mis mthat mthe mnull mhypothesis mallows mfor mno mdifferences mat mall mbetween mthe mgroups. m mtwo mways mto mallow mthe mintercepts mto mdiffer munder mH0. a. onemis mtomincludemthe mgroupmdummymand mall minteraction mterms mand mthenmtest mjoint msignificance mof mthe minteraction mterms monly. b. Second: mtomformmFmstatisticmjust mlikemChowmtest, mbutmincludemamdummymgroup mvariable min mthe mrestricted mpooling mregression.
III. Ambinarymdependentmvariablemm Linearmprobabilitymmodelm(LPM). mModel: mP(y=1|x) m= mE(y|x) m= mb 0 m+ mb 1 x 1 m+ mb 2 x 2 m+ m… m+ mbkxk Limitations:
- predictions mcan mbe meither m>1 mor m<0.
- Heteroskedasticity: mVar(y|x) m= mp(x)[1-p(x)] m mexcept min mthe mcase mwhere mthe mprobabilitymdoes mnot mdepend mon manymof mthe mindependent mvariables, m there mmust mbe mheteroskedasticity minmamLPM. mThis mdoes mnot mcause mbiasmin mthemOLS mestimators. mBut mhomoskedasticitymis mcrucial mfor mjustifying mthe musual mt mand mF mstatistics, meven min mlarge msamples. Evenmwith mthesemproblems, mthemLPMmismuseful mand moftenmappliedmin meconomics. mIt musually mworks mwell mfor mvalues mof mthe mindependent mvariables mthat mare mnear mthe maverages min mthe msample.
IV. Self-selection Themtermmis musedmwhen mambinarymindicator mofmparticipation mmight mbemsystematicallymrelated munobserved mfactors. mE.g. Ym= mb0 m+ mb1*mparticipate m+ mu Where mparticipate mis ma mbinarymvariable mequal mto m 1 mif mthe mindividual mparticipates min ma mprogram. mThen mwe mare mworried mthat mthe maverage mvalue mof mu mdepends mon mparticipation: mE(u|participate=1)m≠E(u|participate=0)mmOLSmestimatorsmarembiased. mThemself-seleciton mproblem mis manother mwaymthat man mexplanatorymvariable mcan mbe mendogenous.
Chapterm 8 Heteroskedasticity
I. Consequences mof mheteroskedasticity:
- Heteroskedasticitymdoes mnotmcausembiasmorminconsistencymin mthemOLS mestimators, mwhereas msomething mlikemomitting man mimportant mvariable mwouldmhave mthis meffect.
( x^ m^ m x )^ m^
x n
x
m^ r^ m^ um '
j
- R^2 mand madjusted mR^2 maremalso munaffected mbymthempresencemofmheteroskedasticitymbecause
mpopulation mR^2 mis msimplym 1 m- m m^2 u^ m/ mm^2 m. y mBoth mvariances min mthe mpopulation mR^2 mare unconditionalmvariances, mitmismunaffected mbymthempresencemofmheteroskedasticitymin mVar(u|x). mFurther, mSSR/n mconsistentlymestimates mσ^2 u, mand mSST/n mconsistentlymestimates mσ^2 y, mwhether mor mnot mVar(u|x) mis mconstant.
- However,mthemusual mOLS mstandardmerrors maremno mlongermvalid mformconstructingmCIs mand mt mstatistics. mThe musual mOLS mt mstatistic mdo mnot mhave mt mdistributions min mthe mpresence mof mheteroskedasticity, mandmthemproblemmis mnot msolved mby musingmlargemsamplemsizes. m mOLS mis mno mlonger mBLUE.
II. Heteroskedasticity-robust mprocedures
- One mregressor: n 2 2 i i Var(b’j) m= m m^ i ^1 m, mThe mvalid mestimator mof mb’j, mfor mheteroskedasticitymof manymform SSTm^2
( x^ m^ m x )
(^2) m u m ' 2 i i (including mhomoskedasticity), mis m m^ i ^1 SSTm^2 n 2 2 ij i
- multiplemregressors: mVar(b’j) m= m m^ i ^1 m, ----------------- (1).mwheremrij mdenotes mthe mith SSR^2 residual mform mregressingmxj mon mall mothermindependent mvariables,mand mSSRj mismthe msummof msquared mresiduals mfrom mthis mregression. mThe msquare mroot mof mthe mquantity mis mcalled mthe m heteroskedasticity-robust mstandard merror mfor mb’j. a. Sometimes, mas mamdegreemofmfreedommcorrection,m(1)mismmultipliedmbymn/(n-k-1)mbefore mtaking mthe msquare mroot. mTypically, mwe muse mwhatever mform mis mcomputed mbymthe mregression mpackage mat mhand. b. Heteroskedasticity-robustmtmstatistic : mtm=m(OLSmestimatorm–mhypothesizedmvalue)/ mheteroskedasticity-robust mstandard merror.
- Heteroskedasticity-robust mstandard merrors mcan mbe meither mlarger mor msmaller mthan mthe musual mstandard merrors. mAs man mempirical mmatter, mthe mrobust mstandard merrors mare moften mfound mto mbe mlarger mthan mthe musual mstandard merrors.
- Why mdo mwe mstill muse musual mstandard merrors? m – mIf mthe mhomoskedasticity massumption mholds mand mthe merrors mare mnormallymdistributed, mthen mthe musual mt mstatistics mhave m exact m t mdistributions, mregardlessmofmthemsamplemsize.m Themrobustmstandardmerrors mandmrobust mt mstatistics mare mjustified monly mas mthe msample msize mbecomes mlarge.
- In mlarge msample msizes , mwe mcan mmake ma mcase mfor m always mreporting monly mthe mheteroskedasticity-robustmstandardmerrorsminmcross-sectionalmapplications.
III. AmHeteroskedasticity-robust mLM mstatistic:
- Obtain mthe mresidual mu’ mfrommthe mrestricted mmodel.
- Regress meach mof mthe mindependent mvariables mexcluded munder mthe mnull mon mall mof mthe mincludedmindependent mvariables;mifmtheremaremqmexcluded mvariables,mthis mleads mtomq msets mof mresiduals m(r1, mr2,…, mrq).
fact, mifmVar(ui,e) m= m m i 2 m for mall mImand me, mthen m Var ( u im )mmm (^2) m/m m m , mwhere mmi mis mthe mnumber mof
mworkers min mfirm mi.
VI. Themheteroskedasticitymfunctionmmustmbemestimated: mFGLSm mestimatemthe mfunction mh(xi). Var(u|x)m= m m^2 m exp(m b m m bmx m m... mm b m x m ). 0 1 m 1 k m m k
- Run mthe mregression mof m ymon mx1, mx2, m…, mxk mand mobtain mthe mresiduals, mu’.
- Create mlog(u’^2 ).
- Run mthe mregression mof mlog(u’^2 ) mon mx1, m…, mxk mand mobtain mthe mfitted mvalues, mg’.
- let mh’ m= mexp(g’).
- Estimatemthemequation
mweights m1/h’. Note:
ym m 0 mm 1 x 1 m m...mm kmxk m u m , mbymWLS, musing
- Having mto mestimate mhi musing mthe msame mdata mmeans mthat mthe m FGLS mestimator mis mno mlonger munbiased. mNevertheless, mthe m FGLS mestimator mis mconsistent mand masymptotically mmore mefficientmthanmOLS .mAt manymrate, mformlargemsamplemsize, mFGLS mis man mattractive malternative mto mOLS mwhen mthere mis mevidence mof mheteroskedasticity mthat minflates mthe mstandard merrors mof mthe mOLS mestimates.
- WhilemcomputingmFmstatistic,mitmis mimportant mthat mthemsamemweights mbemusedmto mestimate mthe munrestricted mand mrestricted mmodels.
- OLS mand mWLS mestimates mcanmbemsubstantiallymdifferent.mWhenmthismhappens, mtypically, mthis mindicates mthat mone mof mthe mother mGauss-Markov massumptions mis mfalse, mparticularlymthe mzero mconditional mmean massumption mon mthe merror. mThe m Hauseman mtest mcan mbe mused mto mformally mcompare mthe mOLS mand mWLS mestimates mto msee mif mthey mdiffer mby mmore mthan mthe msampling merror msuggests.
Chapterm 9 Moremon mSpecificationmand mData mproblems
I. Functional mform mmisspecification
- Definition:mthemomitting mvariable mis mamfunction mof man mexplanatorymvariable min mthe mmodel.
- Ramsey’sm Regression mspecification merrormtest m(RESET) :
Originalmmodel: m^ ym m 0 m^ m 1 x 1 m^ m...mm k m^ xk m^ m u m , mifmit msatisfies mA3,mthen mno
nonlinearmfunctionsmofmthemindependent mvariablesmshouldmbemsignificant mwhen maddedmto mequation.
Testingmmodel: m y^ m^ ^ m^ ^ m^ m^ m^ x^ m^ m...mm^ m^ x^ m^ m^ m^ ym '
2 mm m ym ' 3 mm error
0 1 m 1 k m m k 1 2
RESET mis mthe mFmstatistic mfor mtesting mH0: m δ 1= δ 2=. Note:
- RESET mprovidesmno mreal mdirection mon mhow mto mproceed mif mthemmodel mis mrejected.
- It mcan mbe mshown mthat mRESET mhas mno mpower mfor mdetecting momitted mvariables mwhenever mtheymhave mexpectations mthat mare mlinear min mthe mincluded mindependent mvariables min mthe mmodel.mFurther,mifmthe mfunctional mform minmproperlymspecified,mRESET mhasmno mpowermfor
r
detectingmheteroskedasticity.mThembottommlinemismthat mRESETmismamfunctionalmformmtest, mand mnothing mmore.
- Tests magainst mnonnested malternatives:
ym m 0 mm 1 x 1 m m 2 m x 2 m m u (1)^ vs. ym m 0 mm 1 mlog( x 1 m )mm 2 mlog( x 2 m )mm u -------------------------------- (2)
Two mways: a. ym m b 0 m m b 1 m x 1 m m b 2 m x 2 m m b 3 m log( x 1 m )mm b 4 m log( x 2 m )mm u m ,mcanmfirstmtestmH0:mb1=b2=0, mcanmalsomtest
H0: mb3=b4=0. b. Davidson m– mMacKinnon m test: mifm(1)mis mtrue, mthen mthe mfitted mvalues mfrom m(2), mshould mbe insignificantminm(1). mT-testminmthemequation m ym m b 0 m m b 1 m x 1 m m b 2 m x 2 m mm ym ''m error m ,wmheremy’’mis mthe
mfitted mvalues mfrom m(2).
II. Lagged mdependent mvariables mas mproxymvariables.
- Usingmamlagged mdependent mvariablemcanmaccountmformhistoricalmfactors mthat mcause mcurrent mdifferences min mthe mdependent mvariable mthat mare mdifficult mto maccount mfor min mother mways.
- Using mlagged mdependent mvariables mas ma mgeneral mwaymof mcontrolling mfor munobserved mvariablesmismhardlymperfect.mBut mit mcan maid minmgettingmambettermestimatemofmthemeffects mof mpolicymvariables mon mvarious moutcomes.
III. Measurement merror:
- Measurementmerror min mthe mdependent mvariables : my: mtrue. m y: mobserved. m e m= mym– m y.
ym m 0 mm 1 x 1 m m... mm kmxk m m u m m em , mAssumeme mis mindependent mofmeach mexplanatorymvariable.
m OLSmestimatorsmaremunbiasedmandmconsistent. mVar(u+e) m= m m^2 m mm^2 m >m^2 m m larger merror mvariancemthan mwhen mno merror moccurs. u e u Bottom mline: mif mthe mmeasurement merror min mthe mdependent mvariable mis msystematicallymrelated mto mone mor mmore mof mthe mregressors, mthe mOLS mestimators mare mbiased. mIf mit mis mjust ma mrandom mreporting merrormthatmis mindependentmofmthemexplanatorymvariables, masmismoften massumed, mthen mOLS mis mperfectlymappropriate.
- Measurement merror min mthe mindependent mvariables: mx: mtrue. mx: mobserved. m e m= mx m– mx. a. cov(e,x)m=m 0 m mym=b0 m+mb1x* m+mum= mb0m+mb1x m+m(u-b1e).m mOLS mis mconsistent mwith mlarger mvariance. b. Classicalmerrors-in-variablesm(CEV):mcov(e,x)m= m 0 m cov(e, mx)m=mvar(e),mcov(x,mu- mb1e) m= m-b1Var(e) m mOLS mis mbiased mand minconsistent mestimator. m^2 c. Under mCEV: mplim(b’1) m=b1( 1 * m^2 m mm^2 ),^ mwhere^ mr1^ mis^ mthempopulation^ merror^ min^ mthe 1 ^ e equation mx1 m= ma0 m+ ma1x2 m+… m+ makxk m+ mr1 m mplim(b’1) mis malso mcloser mto mzero mthan mis mb1. mThis mis mcalled mthe m attenuation mbias m in mOLS mdue mto mclassical merrors-in-variables: mon maverage m(or min mlarge msamples), mthe mestimated mOLS meffect mwill mbe mattenuated. d. There mare mless mclear-cut mfor mestimating mthe mbj mon mthe mvariablesmnot mmeasured mwith merror.
IV. Nonrandommsamples :
- Exogenous msample mselection : msamplemselection mbased mon mthemindependent mvariables m no mproblem.
r
II. Finite mSample m Properties mof mOLS munder mClassical mAssumptions.
- TS.A1: mLinear min mparameters : The mstochastic mprocess m{(xt1, m…, mxtk, myt): mt=1,2,…,n} mfollows mthe mlinear mmodel:
yt m m 0 mm 1 x 1 t m m... mm kmxkt m m ut
- TS.A2: mZero mconditional mmean m – mE(ut| X )=0, mt=1,2,…,n m mExplanatorymvariables mare strictly mexogenous. a. Themerrormat mtimemt,mut, mismuncorrelatedmwithmeach mexplanatorymvariable min meverymtime mperiod. b. Ifmut mis mindependent mof mXmandmE(ut) m=0, mthen mTS.A2 mautomaticallymholds. c. If mEm(ut| xt )=0, mwemsaymthatmthemxtj mare mcontemporaneously mexogenous .m mSufficient mfor mconsistency, mbut mstrict mexogeneitymneeded mfor munbiasedness. d. TS.A2 mputs mno mrestriction mon mcorrelation min mthe mindependent mvariables mor min mthe mut macrossmtime.mIt monlymsays mthat mthemaveragemvaluemofmut mis munrelated mtomthemindependent mvariables min mall mtime mperiods. e. IfmTS.A2 mfails, momitted mvariablesmand mmeasurement merrorminmregressors maremtwo mleading mcandidates mfor mfailure. f. Other mreasons: mX mcan mhave mno mlagged meffect mon m y. mif mX mdoes mhave ma mlagged meffect mon my, mthen mwe mshould mestimate ma mdistributed mlagmmodel. mA2 malso mexcludes mthe mpossibilitymthat mchanges minmthemerrormterm mtodaymcan mcausemfuturemchanges minmX. me.g. mhighermut m mhigher m yt m mhigher mXt+1. mExplanatorymvariables mthat mare mstrictlymexogenous mcannot mreact mto mwhat mhas mhappened mto m ymin mthe mpast.
- TS.A3: mno mperfect mcollinearity.
- THMm10.1m– mUnbiasedness mofmOLS:m UndermA1-A3,mthemOLS mestimators maremunbiased mconditional mon m X , mand mtherefore munconditionallymas mwell: mE(b’j) m= mbj, mj m= m0,1,…,k.
- TS.A4:mHomoskedasticity:m Var(ut| mX )=Var(ut), mt=1,2,…,k
- TS.A5:mNo mserial mcorrelation : mCorr(ut, mus| mX )=0,mfor mall mt≠s.m mA5massumes mnothing mabout mtemporal mcorrelation min mthe mindependent mvariable.
- THM m10.2 m–m(OLS mSampling mVariance) : mUnder mthe mTS mGauss-Markov mA1-A5, mthe
m^2 variance mof mb’j, mconditional mon mX, mis mVar(b’j|X)= m SST m (1m R^2 m ) m
, mj m=1,…,k, mwhere mSSTj
is mthemtotal msummofmsquares mofmXtj mand mR^2 j mis mthemR^2 mfrommthemregression mofmXj mon mthe mother mindependent mvariables.
- THMm10.3 m–mUnbiasedmestimationmof m^ m
2 m : mundermA1-5, mthemestimatormσ’2=SSR/dfmis man munbiased mestimator mof mm^2 m , mwhere mdf m= mn m– mk m-
- THM m10.4 m– mGauss-Markov mTheorem:m Under mA1-5, mOLS mestimators mare mthe mBLUE.
- TS.A6 : mut mare mindependent mof mXmand mare miid m~mN(0, m m^2 m ).
- THMm10.5m– mNormal msampling mdistributions:m undermA1-6, mthemCLMmassumptionsmfor mtime mseries, mthe mOLS mestimators mare mnormallymdistributed, mconditional mon mX. mFurther, munder mthe mnull mhypothesis, meach mt mstatistic mhas ma mt mdistribution, mand meach mF mstatistic mhas man mF mdistribution. mThe musual mconstruction mof mconfidence mintervals mis malso mvalid.
III. Trends mand mseasonality
- Phenomenon:minmmanymcases,mtwomtimemseries mappearmtombemcorrelatedmonlymbecause mtheymare mboth mtrending mover mtime mfor mreasons mrelated mto mother munobserved mfactors m– m spurious mregression.
- Linear mtime mtrend : myt m= ma0 m+ ma1*t m+ met, mt=1,2,…
- Manymeconomicmtimemseriesmarembettermapproximatedmbymanm exponentialmtrend ,mwhich mfollows mwhen ma mseries mhas mthe msame maverage mgrowth mrate mfrom mperiod mto mperiod. Log(yt) m= ma0 m+ ma1*t m+ met, mt=1,2,… 4. Accountingmfor mexplained mor mexplanatorymvariables mthat maremtrending mis mstraightforward - mjust mputma mtime mtrend minmthe mregression. yt m m 0 m m 1 m x 1 t m m...mm k m xkt m m t m m ut - --------------------- (1)
- In msome mcases, madding ma mtime mtrend mcan mmake ma mkey mexplanatorymvariable mmore msignificant. mThis mcan mhappenmifmthemdependent mand mindependent mvariables mhavemdifferent mkinds mof mtrends m(say, mone mupward mand mone mdownward), mbut mmovement min mthe mindependent mvariable mabout mits mtrend mline mcauses mmovement min mthe mdependent mvariable maway mfrom mits mtrend mline.
- Interpretations: mbetas mfrom m(1) mare mjust mthe msamembetas mfrom: a. linearlymdetrend my, mall mx m(residual mfrom m ym= ma m+bt m+ mu) b. run mlinearlymdetrended mymon mall mlinearlymdetrended mx. c. This minterpretation mof mbetas mshows mthat m it mis ma mgood midea mto minclude ma mtrend min mthe mregression mif many mindependent mvariable mis mtrending, meven mif my mis mnot. m If m ymhas mno mnoticeable mtrend, mbut, msay, mx mis mgrowing mover mtime, mthen mexcluding ma mtrend mfrom mthe mregression mmaymmakemit mlook mas mifmx mhas mno meffect mon my, meven mthough mmovements mofmx mabout mits mtrend mmaymaffect my.
- Themusualmand madjusted mR^2 smformtimemseries mregressionsmcanmbemartificiallymhighmwhen mthe mdependent mvariable mis mtrending. m We mcan mget mR^2 mfrommthe mregression: mtime mdetrended my mon mx mand mt.
- Formseasonality: mjustmincludemamsetmofmseasonal mdummymvariables mtomaccount mfor mseasonalitymin mthe mdependent mvariable, mthe mindependent mvariables, mor mboth.
Chapterm 11 Further mIssues minmUsing mOLS mwithmTime mSeries mData
I. Stationarymand mweak mdependence
- Stationarymstochasticmprocess : m{xt:mtm=m1,2,…} mismstationarymifmformeverymcollection mof mtime mindices m 1 ≦t 1 <t 2 <…<tm, mthe mjoint mdistribution mof m(xt1, mxt2, m…, mxtm) mis mthe msame mas mthe mjoint mdistribution mof m(xt1+h,mxt2+h,…,xtm+h) mfor mall mintegers mh≧ 1 a. The msequence m{xt: mt m= m1,2,…} mis midenticallymdistributed, mwhen mm=1 mand mt1=1. b. Amprocessmwithma mtimemtrendmismclearly mnonstationary : mat mamminimum,mitsmmean mchanges mover mtime.
- C ovariancemstationarymprocess : mE(xt),mVar(xt)maremconstantmand mformanymt,mh≧1, mCov(xt, mxt+h) mdepends monly mon mh mand mnot mon mt. mifmamstationarymprocess mhas mamfinitemsecond mmoment, mthenmit mmust mbemcovariance mstationary, mbut mthe mconverse mis mnot mtrue. m Stationarity msimplifies mstatements mof mthe mLLN mand mCLT. mIf mwe mallow mthe mrelationship mbetween mtwo mvariables m(say, m yt mand mxt) mto mchange marbitrarilymin meach mtime mperiod, mthenmwe mcannot mhopemtomlearnmmuch mabout mhowmamchangeminmonemvariablemaffects mthe mother mvariable mif mwe monlymhave maccess mto ma msingle mtime mseries mrealization.
- Weaklymdependent : mamstationarymtimemseries mprocess mismsaidmtombemweaklymdependentmif mxt mand mxt+h mare m“almost mindependent” mas mh mincreases mwithout mbound. mA msimilar
- Themprevious msection mshows mthat,mprovidedmthemtimemseries maremweaklymdependent,musual mOLS minference mprocedures mare mvalid munder massumptions mweaker mthan mthe mCLM massumptions.
- Randommwalk: myt m=myt-1m+ met, mwhere, met mis miidmwithmzerommean mand mvarianceσ^2. mAssume mthe minitial mvalue, my0, mis mindependent mof met, mfor manymt.
- yt m=met m+met-1 m+m… m+ me1 m+my0 m m E(yt) m= mE(y0)m m does mnot mdepend mon mt.
- Var(yt)m= σ^2 t
- Amrandom mwalk mdisplays mhighlympersistent mbehavior mbecausemthemvaluemofmymtodaymis msignificant mfor mdetermining mthe mvalue mof m ymin mthe mvery mdistant mfuture. Yt+h m= met+hm+ met+h-1 m+…+ met+1 m+ myt m mE( y t+h| y t) m= myt, mfor mall mh≧ 1
- Corr(yt, myt+h) m= msqrt(t/(t+h)). mThe mcorrelation mdepends mon mthe mstarting mpoint, mt. Therefore, m amrandommwalkmdoesmnotmsatisfymthemrequirement mof manmasymptotically muncorrelated msequence.
- A mrandom mwalk mis ma mspecial mcase mof ma munit mroot mprocess: myt m= m yt-1 m+ met, mwhere, met mis mallowed mto mbe mgeneral, mweaklymdependent mseries. m{et} mcan mfollow man mMA mor mAR mprocess. mWhen m{et} mis mnot man miid msequence, mthe mproperties mof mthe mrandom mwalk mno mlongermhold. mBut mthe mkeymfeature mofm{yt} mis mpreserved: mthemvalue mof mymtodaymis mhighly mcorrelated mwith m ymeven min mthe mdistant mfuture.
- Not mto mconfuse mtrending mand mhighly mpersistent mbehaviors : ma mseries mcan mbe mtrending mbut mnot mhighly mpersistent. mFactors msuch mas minterest mrates, minflation mrates, mand munemployment mrates mare mthought mto mbe mhighlympersistent, mbut mthey mhave mno mobvious mupward mor mdownward mtrend. mHowever, mit mis moften mthe mcase mthat ma mhighlympersistent mseries malso mcontains mamclear mtrend.mOnemmodel mthat mleadsmtomthis mbehaviormis mthe mrandommwalk mwith mdrift.
- myt m= ma m+ m yt-1 m+ met, mif my0=0 mthenmE(yt)=at, mE( y t+h| y t) m=ah+ m yt
- Transformations mon mhighlympersistent mtimemseries: a. Weaklymdependent mprocesses mare msaid mto mbe m integrated mof morder mzero, mor mI(0). mThis mmeans mthat mnothing mneeds mto mbe mdone mto msuch mseries mbefore musing mthem min mregression manalysis: maveragesmofmsuch msequences malreadymsatisfymthemstandardmlimitmtheorems.mUnit mroot mprocesses mare msaid mto mbe mintegrated mof morder mone, mor mI(1). m This mmeans mthat mthe mfirst mdifference mof mthe mprocess mis mweaklymdependent m(and moften mstationary) b. Manymtime mseries myt mthat mare mstrictlympositivemare msuch mthat mlog(yt) mis mI(1). c. Differencing mtime mseries mbefore musing mthemmin mregression manalysis mhas manother mbenefit:mitmremoves manymlinearmtimemtrend. m mrathermthanmincludingmamtimemtrend min ma mregression, mwe mcan minstead mdifference mthose mvariables mthat mshow mobvious mtrends.
IV. Dynamicallymcomplete mmodels:
- yt m m 0 mm 1 x 1 t m m... mm kmxkt m m ut m , mlet m xt m = m ( x 1 t m ,...,m xkt m )m, mmaymor mmaymnot mincluded mlags mof m ymor
mz.mifmwemassumemthatmE(ut| xt , myt-1, mxt-1 ,…)m=m 0 mor, mequivalentlymE(yt| xt , myt-1, mxt-1 ,…)m= mE(yt| mxt ).
- The mabove mmodel mis mdynamicallymcomplete mmodel m mwhatever mis min m xt , menough mlags mhavembeen mincludedmso mthat mfurthermlags mofmymand mthemexplanatorymvariablesmdo mnot mmatter mfor mexplaining myt.
- Dynamicallymcomplete mmodel mmust msatisfymTS.A5’.m merrorsmare mseriallymuncorrelated.
Chapter m 12 SerialmCorrelationmandmHeteroskedasticityminmTimemSeries mRegression I. Chapterm 11 mshows mthatmwhen min man mappropriatemsense, mthemdynamics mofmammodel mhave mbeen mcompletelymspecified, mthe merrors mwill mnot mbe mseriallymcorrelated. mThus, mtesting mfor mserial mcorrelation mcan mbe mused mto mdetect mdynamic mmisspecification. mFurthermore, mstatic mand mfinite mdistributed mlagmmodels moften mhave mserially mcorrelated merrors meven mif mthere mis mno munderlying mmisspecification mof mthe mmodel.
II. OLS mwith mseriallymcorrelated merrors.
- Unbiasedness: mas mlongmas mthemexplanatory mvariablesmaremstrictlymexogenous,mthe mb’j mare munbiased, mregardless mof mthe mdegree mof mserial mcorrelation min mthe merrors m(Thmm10.1 massumed mnothing mabout mserial mcorrelation).m This mis manalogous mto mthe mobservation mthat mheteroskedasticitymin mthe merrors mdoes mnot mcause mbias min mthe mb’j.
- Efficiency mand mInference: m OLS mis mno mlonger mBLUE. mThe musual mOLS mstandard merrors mand mtest mstatistics mare mnot mvalid, meven masymptotically. m mif massume merror mtermmis mAR(1), merrors marempositivelymcorrelated, mand mx mismpositivelymcorrelated. mOLS mestimators mare munderestimated.
- Goodness-of-fit:m ourmusual mR2 mandmadjustedmR2 maremstill mvalid ,mprovidedmthe mdata mare mstationarymand mweaklymdependent.
- Thempopulation mR2 minmamcross-sectional mcontext mis m 1 m- m m^2 m/mm^2 m. mThis mdefinition mis mstill u y appropriate min mtime mseries mwith mstationary, mweaklymdependent mdata: mthe mvariances mof mboth mthemerrors mand mymdo mnot mchangemovermtime.mBymLLN, mR2mconsistentlymestimatemthe mpopulation mR2.
- Themargument mis messentiallymthemsamemas minmthemcross-sectionalmcase, mwhethermormnot mthere mis mheteroskedasticity.
- Since mthere mis mnever man munbiased mestimator mof mthe mpopulation mR2, mit mmakes mno msense mto mtalk mabout mbias min mR2 mcaused mby mserial mcorrelation. mThis margument mdoes mnot mgo mthrough mif m{yt} mis man m I(1).
- Serial mcorrelation min mthe mpresence mof mlagged mdependent mvariable: a. Wrongmstatement: m“OLS mis minconsistent min mthe mpresence mof mlagged mdependent mvariables mand mseriallymcorrelated merrors”. m mAssume m yt m= ma m+ mbyt-1 m+ mUt mand mE(Ut|yt-1)m=m 0 m msatisfies mTS.A2’mmOLSmestimatorsmaremconsistent.mBut m{ut}mcan mbe mseriallymcorrelated mbecause meven mthough mut mis muncorrelated mwith m yt-1, mit mcan mbe mcorrelated mwith m yt-2 m mcov(ut, mut-1) mis mnonzero. mTherefore, mthe mserial mcorrelation min mthe merrors mwill mcause mthe mOLS mstatistics mto mbe minvalid mfor mtesting mpurposes, mbut mit mwill mnot maffect mconsistency. b. But mifmwemassumem{ut} mfollowsmAR(1)m mOLS minconsistent. mBut minmthis mcase, myt mcan mbe mexpressed mbymAR(2). c. The mbottom mline mis mthat myou m need ma mgood mreason mfor mhaving mboth ma mlagged mdependentmvariableminma mmodel mandmamparticular mmodel mof mserial mcorrelation min mthe merrors. mOften, mserial mcorrelation min mthe merrors mof ma mdynamic mmodel msimply mindicates mthat mthe mdynamic mregression mfunction mhas mnot mbeen mcompletelymspecified.
III. Testing mfor mserial mcorrelation
- Casem1: mt m–mtest mformAR(1)mserial mcorrelation mwith m strictly mexogenous mregressors.
