微观计量经济学模型ModelofMicroeconometrics

1、微观计量经济学模型（Model of Microeconometrics）1.1 Generalized Linear ModelsThree aspects of the linear regression model for a conditionally normally distributed response y are:(1) The linear predictor through which .(2) is (3)GLMs: extends (2)and(3) to more general families of distributions for y. Specifical

2、ly, may follow a density:canonical parameter, depends on the linear predictor.:dispersion parameter, is often known.Also and are related by a monotonic transformation,Called the link function of the GLM.Selected GLM families and their canonical linkFamilyCanonical linkNamebinomiallogitgaussianidenti

3、typoissonlog1.2 Binary Dependent VariablesModel:In the probit case: equals the standard normal CDFIn the logit case: equals the logistic CDFExample:(1)DataConsidering female labor participation for a sample of 872 women from Switzerland.The dependent variable: participationThe explain variables:inco

4、me,age,education,youngkids,oldkids,foreignyesandage2.R:library("AER")data("SwissLabor")summary(SwissLabor)participation income age education no :471 Min. : 7.187 Min. :2.000 Min. : 1.000 yes:401 1st Qu.:10.472 1st Qu.:3.200 1st Qu.: 8.000 Median :10.643 Median :3.900 Median : 9.0

5、00 Mean :10.686 Mean :3.996 Mean : 9.307 3rd Qu.:10.887 3rd Qu.:4.800 3rd Qu.:12.000 Max. :12.376 Max. :6.200 Max. :21.000 youngkids oldkids foreign Min. :0.0000 Min. :0.0000 no :656 1st Qu.:0.0000 1st Qu.:0.0000 yes:216 Median :0.0000 Median :1.0000 Mean :0.3119 Mean :0.9828 3rd Qu.:0.0000 3rd Qu.:

6、2.0000 Max. :3.0000 Max. :6.0000 (2) EstimationR:swiss_prob=glm(participation.+I(age2),data=SwissLabor,family=binomial(link="probit")summary(swiss_prob)Call:glm(formula = participation . + I(age2), family = binomial(link = "probit"), data = SwissLabor)Deviance Residuals: Min 1Q M

7、edian 3Q Max -1.9191 -0.9695 -0.4792 1.0209 2.4803 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 3.74909 1.40695 2.665 0.00771 * income -0.66694 0.13196 -5.054 4.33e-07 *age 2.07530 0.40544 5.119 3.08e-07 *education 0.01920 0.01793 1.071 0.28428 youngkids -0.71449 0.10039 -7.117

8、1.10e-12 *oldkids -0.14698 0.05089 -2.888 0.00387 * foreignyes 0.71437 0.12133 5.888 3.92e-09 *I(age2) -0.29434 0.04995 -5.893 3.79e-09 *-Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 1203.2 on 871 degrees of freedomResidual de

9、viance: 1017.2 on 864 degrees of freedomAIC: 1033.2Number of Fisher Scoring iterations: 4(3) VisualizationPlotting participation versus ageR:plot(participationage,data=SwissLabor,ylevels=2:1)(4) EffectsAverage marginal effects:The average of the sample marginal effects: R:fav=mean(dnorm(predict(swis

10、s_prob,type="link")fav*coef(swiss_prob)(Intercept) income age education youngkids 1.241929965 -0.220931858 0.687466185 0.006358743 -0.236682273 oldkids foreignyes I(age2) -0.048690170 0.236644422 -0.097504844The average marginal effects at the average regressor:R:av=colMeans(SwissLabor,-c(

11、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)-7av"foreign"*coef(swiss_prob)-7swiss: (Intercept) income age education youngkids 1.495137092

12、-0.265975880 0.827628145 0.007655177 -0.284937521 oldkids I(age2) -0.058617218 -0.117384323Foreign:(Intercept) income age education youngkids 1.136517140 -0.202179551 0.629115268 0.005819024 -0.216593099 oldkids I(age2) -0.044557434 -0.089228804(5) Goodness of fit and predictionPseudo-R2:as the log-

13、likelihood for the fitted model, as the log-likelihood for the model containing only a constant term.R: swiss_prob0=update(swiss_prob,formula=.1)1- as.vector(logLik(swiss_prob)/logLik(swiss_prob0)1 0.1546416Percent correctly predicted:R:table(true=SwissLabor$participation,pred=round(fitted(swiss_pro

14、b) predtrue 0 1no 337 134yes 146 25567.89%ROC curve:TPR(c):the number of women participating in the labor force that are classified as participating compared with the total number of women participating.FPR(c):the number of women not participating in the labor force that are classified as participat

15、ing compared with the total number of women not participating.R:library("ROCR")pred=prediction(fitted(swiss_prob),SwissLabor$participation)plot(performance(pred,"acc")plot(performance(pred,"tpr","fpr")abline(0,1,lty=2)l Extensions: Multinomial responsesFor ill

16、ustrating the most basic version of the multinomial logit model, a model with only individual-specific covariates,.data("BankWages")It contains, for employees of a US bank, an ordered factor job with levels "custodial", "admin"(for administration), and "manage"

17、; (for management), to be modeled as afunction of education (in years) and a factor minority indicating minority status. There also exists a factor gender, but since there are no women in the category "custodial", only a subset of the data corresponding to males is used for parametric mode

18、ling below.summary(BankWages) job education gender minority custodial: 27 Min. : 8.00 male :258 no :370 admin :363 1st Qu.:12.00 female:216 yes:104 manage : 84 Median :12.00 Mean :13.49 3rd Qu.:15.00 Max. :21.00 summary(BankWages)edcat <- factor(BankWages$education)edcatlevels(edcat)3:10 <- re

19、p(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

20、)1.3 Regression Models for Count DataWe begin with the standard model for count data, a Poisson regression.Poisson Regression Model:Canonical link: the log linkExample:Trips to Lake Somerville,Texas,1980. based on a survey administered to 2,000 registered leisure boat owners in 23 counties in easter

21、n Texas.The dependent variable is trips, and we want to regress it on all further variables: a (subjective) quality ranking of the facility (quality), a factor indicating whether the individual engaged in water-skiing at the lake (ski),household income (income), a factor indicating whether the indiv

22、idual paid a users fee at the lake (userfee), and three cost variables (costC, costS,costH) representing opportunity costs.(1)Datadata("RecreationDemand")summary(RecreationDemand) trips quality ski income userfee Min. : 0.000 Min. :0.000 no :417 Min. :1.000 no :646 1st Qu.: 0.000 1st Qu.:0

23、.000 yes:242 1st Qu.:3.000 yes: 13 Median : 0.000 Median :0.000 Median :3.000 Mean : 2.244 Mean :1.419 Mean :3.853 3rd Qu.: 2.000 3rd Qu.:3.000 3rd Qu.:5.000 Max. :88.000 Max. :5.000 Max. :9.000 costC costS costH Min. : 4.34 Min. : 4.767 Min. : 5.70 1st Qu.: 28.24 1st Qu.: 33.312 1st Qu.: 28.96 Medi

24、an : 41.19 Median : 47.000 Median : 42.38 Mean : 55.42 Mean : 59.928 Mean : 55.99 3rd Qu.: 69.67 3rd Qu.: 72.573 3rd Qu.: 68.56 Max. :493.77 Max. :491.547 Max. :491.05 head(RecreationDemand) trips quality ski income userfee costC costS costH1 0 0 yes 4 no 67.59 68.620 76.8002 0 0 no 9 no 68.86 70.93

25、6 84.7803 0 0 yes 5 no 58.12 59.465 72.1104 0 0 no 2 no 15.79 13.750 23.6805 0 0 yes 3 no 24.02 34.033 34.5476 0 0 yes 5 no 129.46 137.377 137.850(2) Estimationrd_pois=glm(trips.,data=RecreationDemand,family=poisson)coeftest(rd_pois)z test of coefficients: Estimate Std. Error z value Pr(>|z|) (In

26、tercept) 0.2649934 0.0937222 2.8274 0.004692 * quality 0.4717259 0.0170905 27.6016 < 2.2e-16 *skiyes 0.4182137 0.0571902 7.3127 2.619e-13 *income -0.1113232 0.0195884 -5.6831 1.323e-08 *userfeeyes 0.8981653 0.0789851 11.3713 < 2.2e-16 *costC -0.0034297 0.0031178 -1.1001 0.271309 costS -0.04253

27、64 0.0016703 -25.4667 < 2.2e-16 *costH 0.0361336 0.0027096 13.3353 < 2.2e-16 *Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 R:logLik(rd_pois)the log-likelihood of the fitted model:'log Lik.' -1529.431 (df=8)rbind(obs = table(RecreationDemand$trips)1:10, exp = round(+ sapply(0:9, functio

28、n(x) sum(dpois(x, fitted(rd_pois) 0 1 2 3 4 5 6 7 8 9obs 417 68 38 34 17 13 11 2 8 1exp 277 146 68 41 30 23 17 13 10 7table(true=RecreationDemand$trips,pred=round(fitted(rd_nb)NOT WELL（3）Dealing with overdispersionPoisson distribution has the property that the variance equals the mean. In econometri

29、cs, Poisson regressions are often plagued by overdispersion.One way of testing for overdispersion is to consider the alternative hypothesis(Cameron and Trivedi 1990)Var(yi|xi) = i + a*h(i)where h is a positive function of i.Overdispersion corresponds to a > 0 and underdispersion to a < 0. Comm

30、on specifications of the transformation function h are h() = 2 or h() = . The former corresponds to a negative binomial (NB) model (see below) with quadratic variance function (called NB2 by Cameron and Trivedi 1998), the latter to an NB model with linear variance function (called NB1 by Cameron and

31、 Trivedi 1998). In the statistical literature, the reparameterization Var(yi|xi) = (1 + a) · i = dispersion · iof the NB1 model is often called a quasi-Poisson model with dispersion parameter.R: dispersiontest(rd_pois) Overdispersion testdata: rd_pois z = 2.4116, p-value = 0.007941alternat

32、ive hypothesis: true dispersion is greater than 1 sample estimates:dispersion 6.5658R:dispersiontest(rd_pois, trafo = 2) Overdispersion testdata: rd_pois z = 2.9381, p-value = 0.001651alternative hypothesis: true alpha is greater than 0 sample estimates: alpha 1.316051Both suggest that the Poisson m

33、odel for the trips data is not well specified.One possible remedy is to consider a more flexible distribution that does not impose equality of mean and variance.The most widely used distribution in this context is the negative binomial. It may be considered a mixture distribution arising from a Pois

34、son distribution with random scale, the latter following a gamma distribution. Its probability mass function isR: library("MASS")rd_nb <- glm.nb(trips ., data = RecreationDemand)coeftest(rd_nb)z test of coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -1.1219363 0.21430

35、29 -5.2353 1.647e-07 *quality 0.7219990 0.0401165 17.9976 < 2.2e-16 *skiyes 0.6121388 0.1503029 4.0727 4.647e-05 *income -0.0260588 0.0424527 -0.6138 0.53933 userfeeyes 0.6691676 0.3530211 1.8955 0.05802 . costC 0.0480087 0.0091848 5.2270 1.723e-07 *costS -0.0926910 0.0066534 -13.9314 < 2.2e-1

36、6 *costH 0.0388357 0.0077505 5.0107 5.423e-07 *-Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 R:logLik(rd_nb)'log Lik.' -825.5576 (df=9) 0 1 2 3 4 5 6 7 8 9obs 417 68 38 34 17 13 11 2 8 1exp 370 87 37 26 21 17 14 11 9 8(4) Zero-inflated Poisson and negative binomial modelsrbind(obs = table(Re

37、creationDemand$trips)1:10, exp = round(+ sapply(0:9, function(x) sum(dpois(x, fitted(rd_pois) 0 1 2 3 4 5 6 7 8 9obs 417 68 38 34 17 13 11 2 8 1exp 277 146 68 41 30 23 17 13 1 0 7One such model is the zero-inflated Poisson (ZIP) model (Lambert 1992),which suggests a mixture specification with a Pois

38、son count component and an additional point mass at zero. With IA(y) denoting the indicator function, the basic idea isfzeroinfl(y) = pi · I0(y) + (1 pi) · fcount(y; i),we consider a regression of trips on all further variables for the count part (using a negative binomial distribution) and model the inflation part as a function of quality and income:library(pscl)rd_zinb = zeroinfl(trips . | quality + income,data=RecreationDemand, dist="negbin")summary(