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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