课件-高级计量下

Paneldat面板数据(Panel1 大学财政金2016年春季学内容简主要参考书：VerbeekM.(2012)AGuidetoModernWiley,Chichester.[Chapters8-Baltagi(2008)Econometric ysisofpaneldata,Wiley&Sons.CameronA.C.andTrivediP.K.(2005)Microeconometrics,CambridgeUniversityPress.LeeM.(2002)Paneldataeconometrics:methods-of-momentslimiteddependentvariables,AcademicPress,SanDiego.VerbeekM.(2004)AGuidetoModernEconometrics,Wiley&Wooldridge,J.(2010)Econometric ysisofcrosssectionandpaneldata,MITPress.

Paneldat混合截面(Pooledcross混合时间序列(Pooledtime面板数据(Panel

Paneldat

yit=x>β+i∈Nt,t=1,...,T，其中Nt为在时点t观测的(互不相交) kxitR对所有的i和t独立(但不一定是同分布E(it|xit)=应(cross-eects)

Paneldat例子政策分自然实验(naturalexperiments)考虑两个时期：t1为某个政策实施前，t2混合截面包括两个识别组(identiedgroups):对照组(controlgroup,C)和处理组(treatmentgroup,T)yit=α0+α1DT+β0I(t=1)+β1DTI(t=2)+双差分估计(dierence-in-dierencesestimation,ˆ1ˆ1

Paneldat面板数据(Panel : 能刻画不可观测的异质性(unobservedheterogeneity)或 应(individualeects)

Paneldat线性面板数据(Linearpaneldata一般线性面板模型(i1n,t1Tyit=αit+x>βit+yit=α+x>β+ yit αjDc

+x>β+

其中DcI(iRegionj)D

(tTimeintervals)。(

Paneldat效应(individualeects)面板模 yit=αi+x>β+ 称unobserved 和时间特定(individual-andtime-specic)效应模yit=αi+γt+x>β+

Paneldat效应(individualeects)的假如果E(yit|xi1xiTE(yit|xitx>βOLSE(yit|xi,...,xiT)=E(αi|xi,...,xiT)+x>β+E(it|xi,...,xiT 我们需要假定，对所有的i1n和st1TE(it|xi1xiT)=0或者至少E(itxis)=0E(αi|xi1xiT)=0或者至少E(αixit)=

Paneldat假设举)=估计工作培训(jobtraining,JT)对工资(wages)log(wageit)=αi+γt+z>β+δJTit+JTit可能是内生的，E(αiJTit)6=假设E(αixit0不成立，所以在OLS估计之前必须先消除掉αi⇒应(xed-eect)

Paneldat假设举对于假设E(αixit)=0，考 (taxes)对香烟消费(tobaccoconsumption,TC)的影log(TCit)=αi+γt+z>β+δTaxit+在t>1 关，E(αiTaxit)=0假设E(αixit0成立，所以在OLS估计之前不需要先消除掉αi⇒随机效应(random-eect)估计

Paneldat假设举对于假设E(itxis0，考虑R&Dpatentsit=αi+γt+δ0RDit+···+δ5RDit−5+E(itRDis)=假设E(itxis0成立，所以在消除掉αi之后，可以用OLS

Paneldat假设举对于假设E(itxis0，考虑静态工资模型(staticwage E(ittenureit)6=假设E(itxis0

Paneldat假设举对于假设E(itxis0，考虑动态工资模型(dynamicwage估计滞后依赖变量模型(laggeddependentvariablelog(wageit)=αi+βlog(wageit−1)+ it可以假设为i.i.d.，但是不管用何种方法消除 误差项˜it是序列相关的E(˜itwageit−1)6=

Paneldat面板模型主要内

Paneldat固定效应(xed-eects,FE)模 特定效应面板模型(i=1,...,n和s,t=1,...,Tyit=αi+x>β+ 效应，β为k×1向量。假E(it|xi1xiTαi)=0(严格外生，strictE(αi|xit)6= 须估计E(αi|xi1,...,xiT)6=0或者消除掉它。注 应(xedeect,FE)>E(yit|xi1,...,xiT,αi)=αi+x

PaneldatLeastsquaresestimatorwithdummyLeastsquaresdummyvariables(LSDV)yit=αi+x>β+it αjDij+x>β+Dij=I(i=j)为 劣势：矩阵nT×(n+k)非常大附带参数问题(incidentalparameterproblem),参见Neymanand(1948),Lancaster但是OLS对β的估计是一

Paneldat组内最小二乘(within-groupLS)估计FE

yit=αi+x>β+ 求时间均值(v¯T1PTvi

t=1

yit−y¯i=(xit−x¯i)>β+it−=

Paneldat组内最小二乘(within-groupLS)估计FE模型的组内转换(within-group可以用LS来估E(it|xi1xiTαi)=0⇒E(˜it|xi1xiTαi)=0(严格外生性i·

ιT )yi·x˜i·=(IT T

ιTTTT其中IT为单位矩阵，ιT1

Paneldat组内最小二乘(within-groupLS)估计

Xn

!−

!Xn i=1 i=1

矩阵符号，定义I{(i,t):1≤i≤n,1≤t≤TnT×1向量：E˜˜itI

Paneldat组内最小二乘(within-groupLS)估计 对系数β的估计等价于LSDV当n→∞和T固定(或者T)，是一致的¯iinference:自由度与n(T1)成正比，而不是要求较强的识别假设E(it|xi1xiTαi

PaneldatConsiderthejobtrainingdataforstudyingtheeectofjobtraininggrantsonrmscraprates54rmsreportingscrapratesdatacollectedinyear1987-1989grantawardedonlyinyears1988and1989,butanyrmgotthematmostoncepresenceofunionsineachrmisreported(alongwithmanyothervariablesneglectedforthemoment)

PaneldatConsiderthejobtrainingdataforstudyingtheeectofjobtraininggrantsonrmscrapratesdummyunioncannotbeusedasitistime-invariantproposedxed-eectmodelyit=β1I(t=1988)+β2I(t=1989)+β3Grantit+β4Grantit−1+αi+xtreglscrapd88d89grantgrant_1,

PaneldatInference:自由度与n(T−1)成正比，而不是1

(T−1)Var(˜it)=Var(it

T

is)≈Var(it T(standardLSerrorsaretobemultiplied(nT−k)/{n(T−1)−去均值的误差(demeanederrors)˜it是负的序列相关(it是Cov(˜it,˜is)=Cov(it

1

is

1T

ir)=−Var

it)1<T r如果T较小，可用异方差和自相关一致的估计量(HAC)来估计方差矩

Paneldat可用于pooledLS,RE和FE(Arellano,i i i i假设E(x˜it˜it0，E(˜it˜jt0，对任意的ij1ni6= H−1PnPTP

ˆˆ˜

H−n i

s=1itisit ˜˜(˜˜)(˜˜H=PnP

i it =it 如果n固定T，一致性要求在中间的矩阵中引入权重

Paneldat H−1PnP

P 1P

ˆˆx˜

H−n

i s=1

i=1it

it H−1P

i=1i·ni =(˜>˜)−=nH=PnP

Σˆ=

nˆˆ>= i=1i·iit =it

Paneldat Paneldatxtreglscrapd88d89grantgrant_1,feFirst-dierence另一种估计FE模型的方法:对方程yitαix>β+it 4yit=yit−yit−1=(xit−xit−1)β+it−it−1=4xitβ+4LS是一致估计(FD,rst-dierenceE(it|xi1,...,xit+1,αi)=0⇒E(4it|xi1,...,xit,αi)=这个假设很像sequentialexogeneityassumption:E(it|xi1xitαi)=0LS不是有效的，因为误差项是负的序列相关(iti.i.d.)Cov(4it,4it−1)=Cov(it−it−1,it−1−it−2)=−σ并且Cov(4it,4it−s)=0对任意的s>1。(序列相关并不会随着T的增

Paneldat固定效应估计量的比当T=2时，这两种方法是等价的当T时，组内估计量是有效的Cov(4it,4it−1)=−σ2+2Cov(it,it−1)−Cov(it,it−2

Paneldat PaneldatNijmanandVerbeek(1992)：nonresponseinpaneldataanditsimpactonlife-cycleconsumptionfunctionMundlak(1978)'sandChamberlain(1984)'sapproachtoindividualeects consumptionfunctionlnCit=β0+x>β+α∗+ unobservedindividualeects(E(αi|xi·)=E(αi|x¯i)=iicorrelated-randomeectmodel(whatabouttime-invariantvariables?)lnCit=β+x>β+x¯θ+α+yzingnon-

ysisusingsampleselectionmodel(weaklysignicant)Hausmantestforbalancedvs.unbalancedpanel(signicant)

Paneldat随机效应模型(random-eects 特定效应面板模型(i=1,...,n和s,t=1,...,Tyit=αi+x>β+ 效应，β为k×1向量。假E(it|xi1xiTαi)=0(严格外生，strictE(αi|xit0(而不是E(αi|xit)6= yit=x>β+(αi+itit是i.i.d.的，且Var(itσααi是i.i.d.的，且Var(αi)=σα

Paneldat随机效应模型(random-eectsyit=x>β+(αi+it对任意的t，vitαi+it是i.i.d.的，且均值为0可以用pooledLS来一致估计，因为E(vit|xit)=E(αi+it|xit)=LS估计

Xn

!−

Xn

!xit

=(X>X)−1Xi=1 i=12LS估计量不是有效的，因为Var(vi·Var(vi1viT6=σ

Paneldat随机效应模型(random-eectsE(vitvis

αE[(αi+it)(αi+isαi) E(α2)+E(αiis)+i)

itαi)+

itis)=σE(v2

E(αi=

it)2=E(α2)+2Eiασ2+iα

itαi)+E(因此，σ2ITσ2(ιTι σ2+σ σ σσααα σ2+σ σααα Ω

σα σα.. ασ ασα σα

σ2+σ σσα σ2+σσα

PaneldatGLS估α 1 σα

1

ιTι2 IT−2

ιT

I−(1−ψ) 2 +T 2T—1/ 1 ιTι2T IT−(1 ψ) 其中ψσ2/(σ2Tσ2)=(1Tσ2/σ21 −1/ Transformedmodelisquasi-demeanedby :forλ=1 yit−λy¯i=(xit−λx¯i)>β+[(1−λ)αi+(it−λ¯i√ p λ=1−σ/σ2+Tσ2=1−1/1+Tσ2 −

PaneldatFeasibleGLS估s s nˆ=1n

=1−αα

αασ2σ2σ2可由最初的PooledLS vv

nT−

itit

XnXT2nT− 2i=1对任意的t<s，由σ2E(vitvis)可得σ2

i=1ααα

XnXTnT(T−1)/2−

i=1t=1

PaneldatFeasibleGLS估LS估计模型yit=x>βvit，得到itnnn n =(nn

PaneldatFeasibleGLS估nininn

=Y−Xiˆni

i

ˆn

ιTι

ιTιY˜=In⊗(I−λˆ T)Y,X˜=I⊗(I− ) nnn nn

n n

⊗ˆ−

⊗ˆ−

iI⊗ˆ

X)−1X

iI⊗ˆ

PaneldatPaneldatPaneldat拟合度(Goodnessoft)Variationinyit=Withinvariation+Between1 1X 1(y−y¯)2 (y−y¯)2

between overall

t=1

PaneldatLSvs.FEvs.

p α其中λ11/1Tσ2α当λ1时，RE和FE →∞ α当λ0时，RE和pooledLS估计量是相同的：→时，λnα α是最有效的(除非σ2=0)α自Chamberlain(1984)处理

PaneldatLSvs.FEvs.检验LSvs.REBreuchs-PaganLM检验：H0:Var(αi0vs.H1:Var(αi)60seechapter4.2.1) P

1ˆ>(ιι>1LM= 2(T−其中ˆi·为第i个 检验LSvs.FE模型

1−i=P1i·TTini=1i

~Hχ固定效应F-检验：H0:α1···αnvs.H1:α1αnFF SSEwithin/(nT−n−

~H

Fn−1,nT

PaneldatFEvs.REHausman总结：在严格外生性下E(it|xi1xiTαiRE:GLS估计量是有效的，withinLS估计量是一致的如何检验假设H0:E(αixit)=0Hausman nn

Paneldat

PaneldatExample:Example:jobtrainingandF- Paneldat PaneldatOwusu-Gyapong(1986):panel-datastudyonstrikeactivityinCanada.Paneldatacontainthenumberofstrikesperyearineachindustry(60changesinnominalwages,unioncoverage,proportionoffemales,four-rmconcertrationratio,averageageofemployees,exportandinventory-to-saleratios,pricechanges,andunemploymentrates LSvs.LSDVLSrejectedLSvs.RELSrejectedREvs.FEREaccepted

PaneldatPPaneldatEndogeneityandgeneralizedmethodof同时性(simultaneity):E(it|xit6FE模型：E(it|xitαi)6=0 过去的冲击对未来自变量的反馈：E(it|xis6=0,t<yit=γ0yit−1+αi+vitαi+it和xityit1是相关的：都包含

PaneldatEndogeneity：E(αi+it|xi1xiT)6=0(pooled)LS估计是有偏的组内LS估计通常也是有偏的，除非它的外生性条件FD估计量通常也是有偏的，除非它的外生性条Endogeneity：E(it|xi1,...,xiT,αi)6=用IV估计量，或者更一IV可以满足strict,weak,orcontemporaneousexogeneityconditions(next

Paneldat到目前为止，strictexogeneityE(it|xi1,...,xiT,αi)=weakorsequential)exogeneityE(it|xi1xitαi0contemporaneousexogeneity假设：E(it|xit,αi)=weakerexogeneityassumptionscanfacilitatenaturalifxit=yit−1,contemporaneousfeedback(viaαi)possibleintheofsequentialexogeneityE(it|yi1,...,yit−1,αi)=

PaneldatRandomeect

yit=x>β+(αi+it假设k个工具变量zit存在kdim(xitE(it|zi1,...,ziT,αi)=E(αi|zi1,...,ziT)= E(vit|zi1ziT)=E(αi+it|zi1ziT)=0，其中vitαi+itE[zitE(vit|zi1,...,ziT)]=E(zitvit)=E(zitvit)=E[zit(αi因此，E(zityit)=E(zitx

it)]=E[zit(yit−x>β)]=β=E(zitx>)−1E(zityit

PaneldatRandomeect

yit=x>β+(αi+it!− Xn

Xnnˆ ni=1 i=1

矩阵符号，定义I{(i,t):1≤i≤n,1≤t≤TnT×1向量：Y=(yitInT×k矩阵：XxInT×k向量：Zz

nn

1Z>

PaneldatRandomeect2SL 工具变量个数l>E(zitvit)=0:l个约束，k

nn

1X>Z(Z>Z)−1Z>(

ˆX>YˆZ2SLS估计量是一致

Z

PaneldatRandomeectIV自相;—1/ ιTι IT−λ p α其中，λ可由λ=1 αˆ−Z

Z ˆ−i·i·i

PaneldatRandomeectIV自相— i·in =Y i·iniˆni

i

ˆn ˆιι> Y=In⊗(IT−λnTT)Y；对X和Z n⊗ˆ−n

˜Z

(˜>˜˜>Y˜ Z˜

i− ⊗ˆn⊗ˆn⊗ˆn)⊗ˆn⊗ˆn⊗ˆn)

PaneldatExample:airfare=theticketdist=thelengthofthegivenrouteModelofthedemandforair2 β1log(fareit)+β2log(distit)+β3log(distit2+β4I(t=1998)+β5I(t=1999)+β6I(t= αi+

PaneldatExample:Example:air PaneldatRE-GLSvs.RE-IVHausman检验严格外生性：E(it|xi1xiTαi外生的:GLS估计量是有效的，IV估计量是一致的:如何检验假设H0:E(it|xi1xiTαi)=0Hausman nn

Paneldat PaneldatFixedeectyit−y¯i=(xit−x¯i)>β+(it−¯i假设k个工具变量zit存在kdim(xitE(it|zi1,...,ziT,αi)= 注意：假设E(αi|zi1ziT0不再需要E(it−¯i|zi1,...,ziT)=

PaneldatFirst-dierence 4yit=yit−yit−1=(xit−xit−1)β+(it−it−1)=4xitβ+4假设k个工具变量zit存在k=dim(xit)E(it−it1|zitzitαi0ξit=(zitzitE(ξitξ和E(ξit(xit−xit−1E(it−it−1|ξi1,...,ξit)=E[ξit(it−it−1)]=E[ξit(yit−yit−1−(xit−xit−1)>β)]=E[ξit(yit−yit−1)]=E[ξit(xit−xit−1)>−β=E[ξit(xit−xit−1) E[ξit(yit−yit−1

PaneldatFixedeect−yit=x>β+αi+it⇒y˜it=x˜>β+固定效应工具变量(FEIV)

!− β=β=n

Xn

Xn

i=1 i=1矩阵符号，定义I{(i,t):1≤i≤n,1≤t≤TFEIV估计nn

PaneldatFixedeectIV2SLS 工具变量个数l>˜˜ ˜˜>˜−1˜>Z=Z(ZZ Z˜>

ˆ> nn

˜>

ˆ> Z

PaneldatExample:Example:air Paneldat转换后的面板模型(i=1n和t=1E(z˜is˜it0,1stTl个方程，k个参数(Z˜>E˜0)n Pn1

Pn1n

i

t=1it

nT

i=1Pt=1

i

Paneldat矩方程(lE[g(dit,β)]=0

1n

Xn1g(di·β)=01

g(dit,β)=iGMM其中dit代表所有数据"1

i=1{t# "1

g(di·,

g(di·, β∈B

i

il=k:β恰好被识别l<k:β不能被识别l>k:β

PaneldatGMM估计量的渐近n渐近正态，n(βˆGMM−β→N(0,VGMM)nVGMM=(Ω>WΩ)−1Ω>WΣWΩ(Ω>WΩ=E∂g(di·,β)/∂β

，Σ=Var{g(di·,VGMM=Ω−1Σ(Ω>)−最优权重WΣ1⇒VGMM1Σ(Ω>)−1n对某一给定的W(如WIl)，进行GMM估计计算ˆnΣˆ1nˆn

Paneldat估计协方差矩VGMM=(Ω>WΩ)−1Ω>WΣWΩ(Ω>W，Ω=E∂g(di·,i i

∂g(di·,i.i.d.:Σ=Eg(di·,β)g(di·,β)i og):Σˆ=1/ni

异方差(heteroscedasticity)和时间依赖(timedependence)(onlyifmomentconditionscontaintimeaverages)

1XnX

i=1{s}{txit

1XnXT

i=1t=1

PaneldatGMMinlinearGMM的矩条件E(zitit0，Qn(β)

1Xn

"1Xnz>(yit—x>

z>(yit—x> i=1

i=1

nQn(β)=1Z>(Y−Xβ)>nn

1Z>(Yn

n2∂Qn(β)

−2[Z>X]>Wn[Z>Y+2[Z>X]>Wn[Z>X]β=

PaneldatGMMinlinear1 βˆGMM=(X>ZWnZ>X)−XZWn1

渐近正态：

–β)→N(0, nVGMM=(X>ZWnZ>X)−1X>ZWnΣW>Z>X(X>ZWnZ>X)n估计Σ=Var(z>it

nnnVGMM≥(X>ZΣ−1Z>X)−

PaneldatGMMand1 βˆGMM=(X>ZWnZ>X)−XZWn1 none-stepGMM(2SLS):W=(Z>Zn1(it是i.i.d.的，Var(z>i·E(z>i·zi·σ2E(z>zi i i iβˆ2SLS=(X>Z(Z>Z)−1Z>X)−1X>Z(Z>Z)−1Zi对某一给定的W(如WnIn或者(Z>Z1)，进行GMM估计计算ˆi=yixβˆGMMinn

Paneldat过度识别约束检E[g(di·,β)]=0

1ni

过度识别条件l过度识别约束检验(Hansen,"1

1√

i innn注意：ˆn

PaneldatKiefer(1979):Eectsofpost-schoolvocationaltrainingonwageratesvariableofinterest:trainingdummyotherfactors:age,education,race,maritalLSvs.(GLS)within-groupIVvs.xed-eectsIVestimator(lacksofgood

PaneldatAbrevaya(2006):Eectsofsmokingon smokingduringpregnancyendogenousforbirthweight(e.g.,duetopoorernutrition,drinking,...)traditionalapproach:IV(e.g.,usingcounsellingorcigarettetaxes)alternativeapproach:paneldataestimationhere:constructionofmatchedpaneldata

Paneldat 效应αi无yit=x>

+x>

+z>δ+z>δ+α1,it

2,it

1i

2i 而x2,it和z2i与αi相关，所有解释变量均与it不相关。

Paneldat −显然，x2,it与x2,it−x¯2,i相关E[(x2,it−x¯2,i)αi = E[(x2,it−x¯2,i)αi|αi E{αiE[(x2,it−x¯2,i)|αi E{αi[E(x2,it|αi)−E(x¯2,i|αi E{αi·0}= andTaylor,1981)AmemiyaandMaCurdy(1986)则提出将x1,it−x¯1,i也作为IV，以增加估计

PaneldatPPaneldatDynamiclinearpanel-data动态面板模 yit=γyit

+x>β+αi+Cor(yityit1)6=0因为yit依赖于αitruestatedependence(γ6=0)unobservedheterogeneity(γ=0)的LSyit1与αi+it相关性随着T→∞而减弱)FD估计量：yit−1和y˜it−1=yit1yit2it−it1

Paneldat动态面板模yit=γyit−1+αi+自回归系数|γ|<sequentialexogeneity：E(it|yit−1yi1αi)=0误差项不允许序列相(random-eect)LS不可用：yit−1和vit=αi+it都包含

Paneldat动态面板模型的组内估计PnP

−¯ −

=γ

it−

i

(yit−

underregularityassumptions,BnT/nT因为yit−1=αi(1−γt−1)/(1−γ) it−1+···+γt−2i A σ

1−nT→

=

=O(T−1 (1− (1− nn

Paneldat动态面板模型的FD估计C PnP

∆it

nT=γ

PnP

—1i underregularityassumptions,DnT/nT因为yit−1=αi(1−γt−1)/(1−γ) it−1+···+γt−2i

，∆yit−1γt−2αi

it−1+(γ−

i

→E

=E[∆it

n但是很好找工具变量(e.g.,yit2

PaneldatFD估计量Iyitγyit1αi+it的一阶差分为(t2T∆yit=γ∆yit−1+∆AndersonandHsiaoE[yit−2∆it]=0并且E[yit−2∆yit−1]6=用yit2作为∆yit−1AndersonandHsiao矩条件：对t≥E[yit−2∆it]=0=E{yit−2(∆yit−γ∆yit−1)}=

PaneldatArellanoandBond1991)增加矩条当T=5时，我们有6个矩条件：对t=3： E[(i3−i2)yi1]=对t= E[(i4−i3)yi1]=E[(i4−i3)yi2]=对t

E[(

—i4)yi1]=E[(i5−i4)yi2]=E[(i5−i4)yi3]=

Paneldati∆Y=(∆yi·)n，其中∆yi·=(yi3−yi2,...,yiT−yiTii∆Ei)n，其中∆i=(i3−i2iT−iTi工具变量矩阵Z=(Z>Z[yi1 [yi1,yi2] Zi.0

.

..0

.[yi1,...,yiT−2E[ZiE[Zyi·γ∆yi·−10 者E[ZiE[Z1−γL)∆yi]· h

i QnT(γ)

Z>(∆Y−γL∆Y

W

Z>(∆Y−γL∆Y

PaneldatiGMMΣ1Var(Z>∆ii—

1

> 1

=

∆i)=

i

GZi=nZdiag(∆ˆi∆ˆiiGE(∆i(误差和工具变量独立)GMM允许异方差(自相关除外)iit iG=E(∆i∆>)=i

PaneldatArellanoandBond估计当n固定，T增大时，是有偏的Kiviet,Z>

.i

i数(Zi和Z都可以)i

PaneldatExample:airfare=theticketpricedist=thelengthofthegivenrouteDynamicmodelofthepriceofair ++

β1log(fareit−1)+β2log(concenitβ3I(t=1999)+β4I(t=αi+

PaneldatExample:Example:air

Paneldat idt矩条ArellanoandBondα如果γ→1或者σ2/σ2→∞α∆yit=(γ−1)yit−1+αi+Ahnand idtE[∆it−1