微观计量经济学模型ModelofMicroeconometrics.docx
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微观计量经济学模型ModelofMicroeconometrics.docx
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微观计量经济学模型ModelofMicroeconometrics
微观计量经济学模型(ModelofMicroeconometrics)
1.1GeneralizedLinearModels
Threeaspectsofthelinearregressionmodelforaconditionallynormally
distributedresponseyare:
(1)Thelinearpredictori^xT-throughwhich"二E(yi|xj.
2
(2)yi|XiisN(叫,二)
(3)i
GLMs:
extends
(2)and(3)tomoregeneralfamiliesofdistributionsfory.
Specifically,y|xmayfollowadensity:
f(y;日冲)=expty日+c(y;$)[
二:
canonicalparameter,dependsonthelinearpredictor.
:
dispersionparameter,isoftenknown.
Alsoiand叫arerelatedbyamonotonictransformation,
g(mi
CalledthelinkfunctionoftheGLM.
SelectedGLMfamiliesandtheircanonicallink
Family
Canonicallink
Name
binomial
log(曰(1一巴)
logit
gaussian
identity
poisson
log卩
log
1.2BinaryDependentVariables
Model:
E(%|人)=Pi=F(x「),i=1,2,……n
Intheprobitcase:
fequalsthestandardnormalCDF
Inthelogitcase:
FequalsthelogisticCDF
Example:
⑴Data
Consideringfemalelaborparticipationforasampleof872womenfrom
Switzerland.
Thedependentvariable:
participation
Theexplainvariables:
income,age,education,youngkids,oldkids,foreignyesandageH.
R:
library("AER")
data("SwissLabor")
summary(SwissLabor)
(2)Estimation
R:
swiss_prob=glm(participation~.+l(ageA2),data=SwissLabor,family=binomial(link="probit"))
summary(swiss_prob)
Call:
glm(formula=participation~.+I(ageA2),family=binomial(link="probit"),
data=SwissLabor)
DevianceResiduals:
Min1QMedian3QMax
-1.9191-0.9695-0.47921.02092.4803
Coefficients:
EstimateStd.ErrorzvaluePr(>|z|)
(Intercept)3.749091.406952.6650.00771**
income
-0.66694
0.13196
-5.0544.33e-07***
age
2.07530
0.40544
5.1193.08e-07***
education
0.01920
0.01793
1.0710.28428
youngkids
-0.71449
0.10039
-7.1171.10e-12***
oldkids
-0.14698
0.05089
-2.8880.00387**
foreignyes
0.71437
0.12133
5.8883.92e-09***
I(ageA2)
-0.29434
0.04995
-5.8933.79e-09***
Signif.codes:
0'***'0.001'**'0.01'*'0.05'.'0.1''1
(Dispersionparameterforbinomialfamilytakentobe1)
Nulldevianee:
1203.2on871degreesoffreedom
Residualdevianee:
1017.2on864degreesoffreedom
AIC:
1033.2
NumberofFisherSeoringiterations:
4
(3)Visualization
Plottingpartieipationversusage
R:
plot(partieipation~age,data=SwissLabor,ylevels=2:
1)
o
QU
scda
4
ega
(4)Effects
:
:
Xj
(x「)r
Averagemarginaleffects:
Theaverageofthesamplemarginaleffects:
R:
fav=mean(dnorm(predict(swiss_prob,type="link")))fav*coef(swiss_prob)
oldkidsforeignyesI(ageA2)
Theaveragemarginaleffectsattheaverageregressor:
R
av=colMeans(SwissLabor[,-c(1,7)])
av=data.frame(rbind(swiss=av,foreign=av),foreign=factor(c("no","yes"|))av=predict(swiss_prob,newdata=av,type="link")
av=dnorm(av)
av["swiss"]*coef(swiss_prob)[-7]
av["foreign"]*coef(swiss_prob)[-7]
swiss:
(Intercept)incomeageeducationyoungkids
oldkidsI(ageA2)
Foreign:
(Intercept)
incomeageeducationyoungkids
oldkids
I(ageA2)
(5)Goodnessoffitandprediction
Pseudo-R2:
R2=1』
C)
(?
)asthelog-likelihoodforthefittedmodel,0)(?
)asthelog-likelihoodforthemodelcontainingonlyaconstantterm.
R:
swiss_prob0=update(swiss_prob,formula=.~1)
1-as.vector(logLik(swiss_prob)/logLik(swiss_prob0))[1]0.1546416
Percentcorrectlypredicted:
R:
table(true=SwissLabor$participation,pred=round(fitted(swiss_prob)))
pred
true
0
1
no
337
134
yes
146
255
67.89%
ROCcurve:
TPR(c):
thenumberofwomenparticipatinginthelaborforcethatare
classifiedasparticipatingcomparedwiththetotalnumberofwomenparticipating.
FPR(c):
thenumberofwomennotparticipatinginthelaborforcethatareclassifiedasparticipatingcomparedwiththetotalnumberofwomennotparticipating.
R:
lbrary("ROCR")
pred=prediction(fitted(swiss_prob),SwissLabor$participation)
plot(performance(pred,"acc"))
plot(performance(pred,"tpr","fpr"))
abline(0,1,lty=2)
Cutoff
010s.o
iggoA昌」nCJov
0^.0
O
QLIcoo
go寸ofo欝」cltAESUdCItnH
Extensions:
Multinomialresponses
Forillustratingthemostbasicversionofthemultinomiallogitmodel,amodelwithonlyindividual-specificcovariates,.
data("BankWages")
Itcontains,foremployeesofaUSbank,anorderedfactorjobwithlevels
"custodial","admin"(foradministration),and"manage"(for
management),tobemodeledasafunctionofeducation(inyears)andafactorminorityindicatingminoritystatus.Therealsoexistsafactorgender,butsincetherearenowomeninthecategory"custodial",onlyasubsetofthedatacorrespondingtomalesisusedforparametricmodelingbelow.
summary(BankWages)
jobeducationgenderminoritycustodial:
27Min.:
8.00male:
258no:
370admin:
3631stQu.:
12.00female:
216yes:
104manage:
84Median:
12.00
Mean:
13.49
3rdQu.:
15.00Max.:
21.00
summary(BankWages)edcat<-factor(BankWages$education)edcatlevels(edcat)[3:
10]<-rep(c("14-15","16-18","19-21"),+c(2,3,3))head(edcat)
tab<-xtabs(~edcat+job,data=BankWages)head(tab)prop.table(tab,1)head(BankWages)
library("nnet")bank_mn2<-multinom(job~education+minority+gender,data=BankWages,trace=FALSE)
summary(bank_mn2)
1.3RegressionModelsforCountData
Webeginwiththestandardmodelforcountdata,aPoissonregression.
PoissonRegressionModel:
E®m二exp(XjT)
Canonicallink:
theloglink
Example:
TripstoLakeSomerville,Texas,1980.basedonasurveyadministeredto2,000registeredleisureboatownersin23countiesineasternTexas.Thedependentvariableistrips,andwewanttoregressitonallfurthervariables:
a(subjective)qualityrankingofthefacility(quality),afactorindicatingwhethertheindividualengagedinwater-skiingatthelake(ski),householdincome(income),afactorindicatingwhethertheindividualpaidauser'sfeeatthelake(userfee),andthreecostvariables(costC,costS,costH)representingopportunitycosts.
⑴Data
data("RecreationDemand")
tripsqualityski
Min.:
0.000Min.:
0.000no:
417
1stQu.:
0.0001stQu.:
0.000yes:
242
Median:
0.000Mean:
2.2443rdQu.:
2.000
Max.:
88.000
costC
Median:
0.000
Mean:
1.419
3rdQu.:
3.000
Max.:
5.000
costS
incomeuserfee
Min.:
1.000no:
6461stQu.:
3.000yes:
13
Median:
3.000
Mean3853
3rdQu.:
5.000
Max.:
9.000
costH
Min.
4.34Min.:
4.767Min.
5.70
1stQu.:
28.24
1stQu.:
33.312
1stQu.:
28.96
Median:
41.19
Median:
47.000
Median:
42.38
Mean
:
55.42Mean:
59.928
Mean:
55.99
summary(RecreationDemand)
3rdQu.:
69.673rdQu.:
72.5733rdQu.:
68.56
Max.:
493.77Max.:
491.547Max.:
491.05
head(RecreationDemand)
trips
qualityskiincome
userfee
costC
costS
costH
1
0
0yes
4
no
67.59
68.620
76.800
2
0
0no
9
no
68.86
70.936
84.780
3
0
0yes
5
no
58.12
59.465
72.110
4
0
0no
2
no
15.79
13.750
23.680
5
0
0yes
3
no
24.02
34.033
34.547
6
0
0yes
5
no129.46
137.377
137.850
(2)Estimationrd_pois=glm(trips~.,data=RecreationDemand,family=poisson)coeftest(rd_pois)
ztestofcoefficients:
EstimateStd.ErrorzvaluePr(>|z|)
(Intercept)
0.26499340.09372222.82740.004692**
quality
0.47172590.017090527.6016<2.2e-16***
skiyes
0.41821370.05719027.31272.619e-13***
income
-0.11132320.0195884-5.68311.323e-08***
userfeeyes
0.89816530.078985111.3713<2.2e-16***
costC
-0.00342970.0031178-1.10010.271309
costS
-0.04253640.0016703-25.4667<2.2e-16***
costH
0.03613360.002709613.3353<2.2e-16***
Signif.codes:
0‘***'0.001‘**'0.01‘*'0.05‘.'0.1
R:
logLik(rd_pois)
thelog-likelihoodofthefittedmodel:
'logLik.'-1529.431(df=8)
rbind(obs=table(RecreationDemand$trips)[1:
10],exp=round(
+sapply(0:
9,function(x)sum(dpois(x,fitted(rd_pois))))))
0123456789
obs417683834171311281
exp277146684130231713107
table(true=RecreationDemand$trips,pred=round(fitted(rd_nb)))
NOTWELL
(3)Dealingwithoverdispersion
Poissondistributionhasthepropertythatthevarianceequalsthemean.Ineconometrics,Poissonregressionsareoftenplaguedbyoverdispersion.Onewayoftestingforoverdispersionistoconsiderthealternativehypothesis(CameronandTrivedi1990)
Var(yi|xi)=a*h(+^i)
wherehisapositivefunctionof.
Overdispersioncorrespondstoa>0andunderdispersiontoa<0.
Commonspecificationsofthetransformationfunctionhareh(2or卩)=卩
h(□)=□.Theformercorrespondstoanegativebinomial(NB)model(see
below)withquadraticvariancefunction(calledNB2byCameronand
Trivedi1998),thelattertoanNBmodelwithlinearvariancefunction
(calledNB1byCameronandTrivedi1998).Inthestatisticalliterature,
thereparameterization
Var(yi|xi)=(1+a)•卩i=dispersion•
oftheNB1modelisoftencalledaquasi-Poissonmodelwithdispersionparameter.
R:
dispersiontest(rd_pois)
Overdispersiontestdata:
rd_poisz=2.4116,p-value=0.007941alternativehypothesis:
truedispersionisgreaterthan1sampleestimates:
dispersion
6.5658
R:
dispersiontest(rd_pois,trafo=2)Overdispersiontest
data:
rd_poisz=2.9381,p-value=0.001651alternativehypothesis:
truealphaisgreaterthan0sampleestimates:
alpha1.316051
BothsuggestthatthePoissonmodelforthetripsdataisnotwell
specified.
Onepossibleremedyistoconsideramoreflexibledistributionthatdoesnotimposeequalityofmeanandvariance.
Themostwidelyuseddistributioninthiscontextisthenegativebinomial.ItmaybeconsideredamixturedistributionarisingfromaPoissondistributionwithrandomscale,thelatterfollowingagammadistribution.Itsprobabilitymassfunctionis
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