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
Threeaspectsofthelinearregressionmodelforaconditionallynormallydistributedresponseyare:
（1）Thelinearpredictor
throughwhich
.
（2）
is
（3）
GLMs:
extends
（2）and（3）tomoregeneralfamiliesofdistributionsfory.Specifically,
mayfollowadensity:
:
canonicalparameter,dependsonthelinearpredictor.
:
dispersionparameter,isoftenknown.
Alsoandarerelatedbyamonotonictransformation,
CalledthelinkfunctionoftheGLM.
SelectedGLMfamiliesandtheircanonicallink
Family
Canonicallink
Name
binomial
logit
gaussian
identity
poisson
log
1.2BinaryDependentVariables
Model:
Intheprobitcase:
equalsthestandardnormalCDF
Inthelogitcase:
equalsthelogisticCDF
Example:
（1）Data
Consideringfemalelaborparticipationforasampleof872womenfromSwitzerland.
Thedependentvariable:
participation
Theexplainvariables:
income,age,education,youngkids,oldkids,foreignyesandage^2.
R:
library（"AER"）
data（"SwissLabor"）
summary（SwissLabor）
participationincomeageeducation
no:
471Min.:
7.187Min.:
2.000Min.:
1.000
yes:
4011stQu.:
10.4721stQu.:
3.2001stQu.:
8.000
Median:
10.643Median:
3.900Median:
9.000
Mean:
10.686Mean:
3.996Mean:
9.307
3rdQu.:
10.8873rdQu.:
4.8003rdQu.:
12.000
Max.:
12.376Max.:
6.200Max.:
21.000
youngkidsoldkidsforeign
Min.:
0.0000Min.:
0.0000no:
656
1stQu.:
0.00001stQu.:
0.0000yes:
216
Median:
0.0000Median:
1.0000
Mean:
0.3119Mean:
0.9828
3rdQu.:
0.00003rdQu.:
2.0000
Max.:
3.0000Max.:
6.0000
（2）Estimation
R:
swiss_prob=glm（participation~.+I（age^2）,data=SwissLabor,family=binomial（link="probit"））
summary（swiss_prob）
Call:
glm（formula=participation~.+I（age^2）,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-0.666940.13196-5.0544.33e-07***
age2.075300.405445.1193.08e-07***
education0.019200.017931.0710.28428
youngkids-0.714490.10039-7.1171.10e-12***
oldkids-0.146980.05089-2.8880.00387**
foreignyes0.714370.121335.8883.92e-09***
I（age^2）-0.294340.04995-5.8933.79e-09***
---
Signif.codes:
0‘***’0.001‘**’0.01‘*’0.05‘.’0.1‘’1
（Dispersionparameterforbinomialfamilytakentobe1）
Nulldeviance:
1203.2on871degreesoffreedom
Residualdeviance:
1017.2on864degreesoffreedom
AIC:
1033.2
NumberofFisherScoringiterations:
4
（3）Visualization
Plottingparticipationversusage
R:
plot（participation~age,data=SwissLabor,ylevels=2:
1）
（4）Effects
Averagemarginaleffects:
Theaverageofthesamplemarginaleffects:
R:
fav=mean（dnorm（predict（swiss_prob,type="link"）））
fav*coef（swiss_prob）
（Intercept）incomeageeducationyoungkids
1.241929965-0.2209318580.6874661850.006358743-0.236682273
oldkidsforeignyesI（age^2）
-0.0486901700.236644422-0.097504844
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
1.495137092-0.2659758800.8276281450.007655177-0.284937521
oldkidsI（age^2）
-0.058617218-0.117384323
Foreign:
（Intercept）incomeageeducationyoungkids
1.136517140-0.2021795510.6291152680.005819024-0.216593099
oldkidsI（age^2）
-0.044557434-0.089228804
（5）Goodnessoffitandprediction
Pseudo-R2:
asthelog-likelihoodforthefittedmodel,
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$participation,pred=round（fitted（swiss_prob）））
pred
true01
no337134
yes146255
67.89%
ROCcurve:
TPR（c）:
thenumberofwomenparticipatinginthelaborforcethatareclassifiedasparticipatingcomparedwiththetotalnumberofwomenparticipating.
FPR（c）:
thenumberofwomennotparticipatinginthelaborforcethatareclassifiedasparticipatingcomparedwiththetotalnumberofwomennotparticipating.
R:
library（"ROCR"）
pred=prediction（fitted（swiss_prob）,SwissLabor$participation）
plot（performance（pred,"acc"））
plot（performance（pred,"tpr","fpr"））
abline（0,1,lty=2）
●Extensions:
Multinomialresponses
Forillustratingthemostbasicversionofthemultinomiallogitmodel,amodelwithonlyindividual-specificcovariates,.
data（"BankWages"）
Itcontains,foremployeesofaUSbank,anorderedfactorjobwithlevels"custodial","admin"（foradministration）,and"manage"（formanagement）,tobemodeledasafunctionofeducation（inyears）andafactorminorityindicatingminoritystatus.Therealsoexistsafactorgender,butsincetherearenowomeninthecategory"custodial",onlyasubsetofthedatacorrespondingtomalesisusedforparametricmodelingbelow.
summary（BankWages）
jobeducationgenderminority
custodial:
27Min.:
8.00male:
258no:
370
admin:
3631stQu.:
12.00female:
216yes:
104
manage:
84Median:
12.00
Mean:
13.49
3rdQu.:
15.00
Max.:
21.00
summary（BankWages）
edcat<-factor（BankWages$education）
edcat
levels（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:
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.
（1）Data
data（"RecreationDemand"）
summary（RecreationDemand）
tripsqualityskiincomeuserfee
Min.:
0.000Min.:
0.000no:
417Min.:
1.000no:
646
1stQu.:
0.0001stQu.:
0.000yes:
2421stQu.:
3.000yes:
13
Median:
0.000Median:
0.000Median:
3.000
Mean:
2.244Mean:
1.419Mean:
3.853
3rdQu.:
2.0003rdQu.:
3.0003rdQu.:
5.000
Max.:
88.000Max.:
5.000Max.:
9.000
costCcostScostH
Min.:
4.34Min.:
4.767Min.:
5.70
1stQu.:
28.241stQu.:
33.3121stQu.:
28.96
Median:
41.19Median:
47.000Median:
42.38
Mean:
55.42Mean:
59.928Mean:
55.99
3rdQu.:
69.673rdQu.:
72.5733rdQu.:
68.56
Max.:
493.77Max.:
491.547Max.:
491.05
head（RecreationDemand）
tripsqualityskiincomeuserfeecostCcostScostH
100yes4no67.5968.62076.800
200no9no68.8670.93684.780
300yes5no58.1259.46572.110
400no2no15.7913.75023.680
500yes3no24.0234.03334.547
600yes5no129.46137.377137.850
（2）Estimation
rd_pois=glm（trips~.,data=RecreationDemand,family=poisson）
coeftest（rd_pois）
ztestofcoefficients:
EstimateStd.ErrorzvaluePr（>|z|）
（Intercept）0.26499340.09372222.82740.004692**
quality0.47172590.017090527.6016<2.2e-16***
skiyes0.41821370.05719027.31272.619e-13***
income-0.11132320.0195884-5.68311.323e-08***
userfeeyes0.89816530.078985111.3713<2.2e-16***
costC-0.00342970.0031178-1.10010.271309
costS-0.04253640.0016703-25.4667<2.2e-16***
costH0.03613360.002709613.3353<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）=μi+a*h（μi）
wherehisapositivefunctionofμi.
Overdispersioncorrespondstoa>0andunderdispersiontoa<0.Commonspecificationsofthetransformationfunctionhareh（μ）=μ2orh（μ）=μ.Theformercorrespondstoanegativebinomial（NB）model（seebelow）withquadraticvariancefunction（calledNB2byCameronandTrivedi1998）,thelattertoanNBmodelwithlinearvariancefunction（calledNB1byCameronandTrivedi1998）.Inthestatisticalliterature,thereparameterization
Var（yi|xi）=（1+a）·μi=dispersion·μi
oftheNB1modelisoftencalledaquasi-Poissonmodelwithdispersionparameter.
R:
dispersiontest（rd_pois）
Overdispersiontest
data:
rd_pois
z=2.4116,p-value=0.007941
alternativehypothesis:
truedispersionisgreaterthan1
sampleestimates:
dispersion
6.5658
R:
dispersiontest（rd_pois,trafo=2）
Overdispersiontest
data:
rd_pois
z=2.9381,p-value=0.001651
alternativehypothesis:
truealphaisgreaterthan0
sampleestimates:
alpha
1.316051
Bo
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