Problema C17.4

En el ejemplo 9.1 se agregaron los términos cuadráticos pcnv^2, ptim86^2, inc86^2 para narr86

  1. Use los datos en CRIME1.RAW para agregar estos mismos términos a la regresión de Poisson en el ejemplo 17.3
data("crime1",package = "wooldridge")
pcnv_2<-(crime1$pcnv)^2
ptime86_2<-(crime1$ptime86)^2
inc86_2<-(crime1$inc86)^2

narr86_MCO<-lm(narr86~
                 pcnv+
                 avgsen+
                 tottime+
                 ptime86+
                 qemp86+
                 inc86+
                 black+
                 hispan+
                 born60+
                 pcnv_2+
                 ptime86_2+
                 inc86_2,
               data=crime1)
narr86_poisson<-glm(narr86~
                      pcnv+
                      avgsen+
                      tottime+
                 ptime86+
                 qemp86+
                 inc86+
                 black+
                 hispan+
                 born60+
                 pcnv_2+
                 ptime86_2+
                 inc86_2,
               crime1,
               family=poisson(link = "log"))
library("stargazer")
## 
## Please cite as:
##  Hlavac, Marek (2022). stargazer: Well-Formatted Regression and Summary Statistics Tables.
##  R package version 5.2.3. https://CRAN.R-project.org/package=stargazer
install.packages("stargazer")
## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.3'
## (as 'lib' is unspecified)
stargazer(narr86_poisson, type="text")
## 
## =============================================
##                       Dependent variable:    
##                   ---------------------------
##                             narr86           
## ---------------------------------------------
## pcnv                       1.153***          
##                             (0.282)          
##                                              
## avgsen                      -0.026           
##                             (0.021)          
##                                              
## tottime                      0.012           
##                             (0.016)          
##                                              
## ptime86                    0.684***          
##                             (0.091)          
##                                              
## qemp86                       0.023           
##                             (0.033)          
##                                              
## inc86                      -0.012***         
##                             (0.002)          
##                                              
## black                      0.591***          
##                             (0.074)          
##                                              
## hispan                     0.422***          
##                             (0.075)          
##                                              
## born60                      -0.093           
##                             (0.064)          
##                                              
## pcnv_2                     -1.795***         
##                             (0.307)          
##                                              
## ptime86_2                  -0.103***         
##                             (0.016)          
##                                              
## inc86_2                   0.00002***         
##                            (0.00001)         
##                                              
## Constant                   -0.710***         
##                             (0.070)          
##                                              
## ---------------------------------------------
## Observations                 2,725           
## Log Likelihood            -2,168.866         
## Akaike Inf. Crit.          4,363.733         
## =============================================
## Note:             *p<0.1; **p<0.05; ***p<0.01
  1. Calcule la estimación de \(\hat{\sigma}^2\) dada por \(\sigma^2=(n-k-1)^{-1}\Sigma_{i=1}^n\frac{\hat{u}_i^2}{\hat{y}_i}\) ¿Hay alguna evidencia de sobredispersión? ¿Cómo se debe ajustar los errores estándar de la EMV de Poisson?
residuales<- narr86_poisson$y-narr86_poisson$fitted.values
sigma2<-(sum(residuales^2/narr86_poisson$fitted.values))/narr86_MCO[["df.residual"]]
sigma2
## [1] 1.391154
raiz.sigma<-sqrt(sigma2)
raiz.sigma
## [1] 1.179472

Para que exista sobredispersión de ser \(\sigma^2>1\), en este caso si exite sobredispersión.

Ajuste de los errores estándar

library(sandwich)
ees.ajustados<-list(sqrt(diag(vcovHC(narr86_MCO,type="HC1"))),
                    raiz.sigma*sqrt(diag(vcovHC(narr86_poisson,type="HC1"))))
library("stargazer")
install.packages("stargazer")
## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.3'
## (as 'lib' is unspecified)
stargazer(narr86_MCO,narr86_poisson,type="text",se=ees.ajustados)
## 
## ========================================================
##                             Dependent variable:         
##                     ------------------------------------
##                                    narr86               
##                                OLS             Poisson  
##                                (1)               (2)    
## --------------------------------------------------------
## pcnv                        0.562***           1.153*** 
##                              (0.171)           (0.440)  
##                                                         
## avgsen                       -0.017             -0.026  
##                              (0.014)           (0.033)  
##                                                         
## tottime                       0.012             0.012   
##                              (0.013)           (0.027)  
##                                                         
## ptime86                     0.289***           0.684*** 
##                              (0.070)           (0.115)  
##                                                         
## qemp86                       -0.015             0.023   
##                              (0.017)           (0.043)  
##                                                         
## inc86                       -0.003***         -0.012*** 
##                              (0.001)           (0.002)  
##                                                         
## black                       0.293***           0.591*** 
##                              (0.058)           (0.117)  
##                                                         
## hispan                      0.163***           0.422*** 
##                              (0.040)           (0.109)  
##                                                         
## born60                       -0.038             -0.093  
##                              (0.032)           (0.095)  
##                                                         
## pcnv_2                      -0.738***         -1.795*** 
##                              (0.173)           (0.508)  
##                                                         
## ptime86_2                   -0.030***         -0.103*** 
##                              (0.006)           (0.019)  
##                                                         
## inc86_2                    0.00001***         0.00002***
##                             (0.00000)         (0.00001) 
##                                                         
## Constant                    0.517***          -0.710*** 
##                              (0.041)           (0.105)  
##                                                         
## --------------------------------------------------------
## Observations                  2,725             2,725   
## R2                            0.104                     
## Adjusted R2                   0.100                     
## Log Likelihood                                -2,168.866
## Akaike Inf. Crit.                             4,363.733 
## Residual Std. Error     0.815 (df = 2712)               
## F Statistic         26.201*** (df = 12; 2712)           
## ========================================================
## Note:                        *p<0.1; **p<0.05; ***p<0.01
  1. Use los resultados de las partes i) y ii) y la tabla 17.3 para calcular el estadístico de razón de cuasi verosimilitud para la significancia conjunta de los tres términos cuadráticos ¿Qué concluye usted?
narr86_poisson <- glm(narr86~
                   pcnv+
                   avgsen+
                   tottime+
                   ptime86+
                   qemp86+
                   inc86+
                   black+
                   hispan+
                   born60,
                 crime1,
                 family = poisson(link = "log"))

# Comparar el modelo Possion y MCO
library("stargazer")
install.packages("stargazer")
## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.3'
## (as 'lib' is unspecified)
stargazer(narr86_MCO, narr86_poisson, 
          type = "text",
          df=F,
          digits = 3,
          header = F,
          column.labels = c("MCO", "Poisson"),
          model.names = F)
## 
## ================================================
##                         Dependent variable:     
##                     ----------------------------
##                                narr86           
##                          MCO          Poisson   
##                          (1)            (2)     
## ------------------------------------------------
## pcnv                   0.562***      -0.402***  
##                        (0.154)        (0.085)   
##                                                 
## avgsen                  -0.017        -0.024    
##                        (0.012)        (0.020)   
##                                                 
## tottime                 0.012         0.024*    
##                        (0.009)        (0.015)   
##                                                 
## ptime86                0.289***      -0.099***  
##                        (0.044)        (0.021)   
##                                                 
## qemp86                  -0.015        -0.038    
##                        (0.017)        (0.029)   
##                                                 
## inc86                 -0.003***      -0.008***  
##                        (0.001)        (0.001)   
##                                                 
## black                  0.293***      0.661***   
##                        (0.045)        (0.074)   
##                                                 
## hispan                 0.163***      0.500***   
##                        (0.039)        (0.074)   
##                                                 
## born60                  -0.038        -0.051    
##                        (0.033)        (0.064)   
##                                                 
## pcnv_2                -0.738***                 
##                        (0.156)                  
##                                                 
## ptime86_2             -0.030***                 
##                        (0.004)                  
##                                                 
## inc86_2               0.00001***                
##                       (0.00000)                 
##                                                 
## Constant               0.517***      -0.600***  
##                        (0.038)        (0.067)   
##                                                 
## ------------------------------------------------
## Observations            2,725          2,725    
## R2                      0.104                   
## Adjusted R2             0.100                   
## Log Likelihood                      -2,248.761  
## Akaike Inf. Crit.                    4,517.522  
## Residual Std. Error     0.815                   
## F Statistic           26.201***                 
## ================================================
## Note:                *p<0.1; **p<0.05; ***p<0.01

Razón de verosimilitud \(RV=2(l_{nr}-L_r)\)

\(L_r = -2,248.761\) \(l_{nr} = -2,168.866\)

RV<- (2*(-2168.866-(-2248.761)))/sigma2
RV
## [1] 114.8614

##Problema C17.5

Consulte la tabla 13.1 en el capítulo 13. Ahí, se usaron los datos en FERTIL1.RAW para estimar un modelo lineal para kids, el número de niños que ha tenido una mujer

  1. Estime un modelo de regresión Poisson para \(\widehat{kids}\), empleando las mismas variables de la tabla 13.1. Interprete el coeficiente de y82
data("fertil1",package = "wooldridge")
kids.MCO <- lm(kids~
                   educ+
                   age+
                   {age^2}+
                   black+
                   east+
                   northcen+
                   west+
                   farm+
                     othrural+
                     town+
                     smcity+
                     y74+
                     y76+
                     y78+
                     y80+
                     y82+
                     y84,
                 data=fertil1)

kids.poisson <- glm(kids~
                   educ+
                   age+
                   {age^2}+
                   black+
                   east+
                   northcen+
                   west+
                   farm+
                     othrural+
                     town+
                     smcity+
                     y74+
                     y76+
                     y78+
                     y80+
                     y82+
                     y84,
                 data=fertil1,
                 family = poisson(link = "log"))
library("stargazer")
install.packages("stargazer")
## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.3'
## (as 'lib' is unspecified)
stargazer(kids.poisson, type="text")
## 
## =============================================
##                       Dependent variable:    
##                   ---------------------------
##                              kids            
## ---------------------------------------------
## educ                       -0.048***         
##                             (0.007)          
##                                              
## age                        0.204***          
##                             (0.055)          
##                                              
## }                          -0.002***         
##                             (0.001)          
##                                              
## black                      0.360***          
##                             (0.061)          
##                                              
## east                        0.088*           
##                             (0.053)          
##                                              
## northcen                   0.142***          
##                             (0.048)          
##                                              
## west                         0.080           
##                             (0.066)          
##                                              
## farm                        -0.015           
##                             (0.058)          
##                                              
## othrural                    -0.057           
##                             (0.069)          
##                                              
## town                         0.031           
##                             (0.049)          
##                                              
## smcity                       0.074           
##                             (0.062)          
##                                              
## y74                          0.093           
##                             (0.063)          
##                                              
## y76                         -0.029           
##                             (0.068)          
##                                              
## y78                         -0.016           
##                             (0.069)          
##                                              
## y80                         -0.020           
##                             (0.069)          
##                                              
## y82                        -0.193***         
##                             (0.067)          
##                                              
## y84                        -0.214***         
##                             (0.069)          
##                                              
## Constant                   -3.060**          
##                             (1.211)          
##                                              
## ---------------------------------------------
## Observations                 1,129           
## Log Likelihood            -2,070.226         
## Akaike Inf. Crit.          4,176.453         
## =============================================
## Note:             *p<0.1; **p<0.05; ***p<0.01

El coeficiente y82 representa la fertalidad, en este caso se pude ver un descenso de 19,3%

  1. ¿Cuál es la diferencia porcentual estimada en la fertilidad entre una mujer negra y una mujer no negra, manteniendo fijos los demás factores?

Dada \(\widehat{\beta}_k\), se obtiene \(exp(\widehat{\beta}_k)-1\) y se multiplica por el 100 para transformar el cambio proporcional en un cambio porcentual. donde \(\widehat{\beta}_k=0,360\)

cam.porcen<- exp(0.360)-1
print(cam.porcen)
## [1] 0.4333294

Calcualdo el cambio porcentual se puede decir que una mujer negra tiene un 43,33% más de hijos que una mujer no negra.

  1. Obtenga \(\hat{\sigma}\) Hay evidencia de sobre o subdispersión?
residuales2 <- kids.poisson$y- kids.poisson$fitted.values

sigma3<-(sum(residuales2^2/kids.poisson$fitted.values))/kids.MCO[["df.residual"]]
sigma3
## [1] 0.8914788
raiz.sigma2 <- sqrt(sigma3)
raiz.sigma2
## [1] 0.9441815

La estimacion es \(\hat{\sigma}=9.44\), como se puede observar \(\hat{\sigma}\) es menor a 1 lo que quiere decir que existe una subdisperción.

  1. Compare los valores ajustados de la regresión de Poisson y obtenga la R-cuadrada como la correlación cuadrada entre \(kids_i\) y \(kids_i\). Compare esto con la R-cuadrada del modelo de regresión lineal
library(sandwich)
ees.ajustados2<-list(sqrt(diag(vcovHC(kids.MCO,type="HC1"))),
                    raiz.sigma2*sqrt(diag(vcovHC(kids.poisson,type="HC1"))))
library("stargazer")
install.packages("stargazer")
## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.3'
## (as 'lib' is unspecified)
stargazer(kids.MCO,kids.poisson,type="text",se=ees.ajustados2)
## 
## =======================================================
##                             Dependent variable:        
##                     -----------------------------------
##                                    kids                
##                               OLS             Poisson  
##                               (1)               (2)    
## -------------------------------------------------------
## educ                       -0.128***         -0.048*** 
##                             (0.021)           (0.008)  
##                                                        
## age                         0.532***          0.204*** 
##                             (0.139)           (0.050)  
##                                                        
## }                          -0.006***         -0.002*** 
##                             (0.002)           (0.001)  
##                                                        
## black                       1.076***          0.360*** 
##                             (0.201)           (0.056)  
##                                                        
## east                         0.217*            0.088*  
##                             (0.127)           (0.046)  
##                                                        
## northcen                    0.363***          0.142*** 
##                             (0.117)           (0.041)  
##                                                        
## west                         0.198             0.080   
##                             (0.163)           (0.058)  
##                                                        
## farm                         -0.053            -0.015  
##                             (0.146)           (0.051)  
##                                                        
## othrural                     -0.163            -0.057  
##                             (0.181)           (0.065)  
##                                                        
## town                         0.084             0.031   
##                             (0.128)           (0.045)  
##                                                        
## smcity                       0.212             0.074   
##                             (0.154)           (0.052)  
##                                                        
## y74                          0.268             0.093   
##                             (0.188)           (0.058)  
##                                                        
## y76                          -0.097            -0.029  
##                             (0.200)           (0.065)  
##                                                        
## y78                          -0.069            -0.016  
##                             (0.198)           (0.065)  
##                                                        
## y80                          -0.071            -0.020  
##                             (0.194)           (0.063)  
##                                                        
## y82                        -0.522***         -0.193*** 
##                             (0.188)           (0.065)  
##                                                        
## y84                        -0.545***         -0.214*** 
##                             (0.186)           (0.066)  
##                                                        
## Constant                    -7.742**         -3.060*** 
##                             (3.071)           (1.107)  
##                                                        
## -------------------------------------------------------
## Observations                 1,129             1,129   
## R2                           0.130                     
## Adjusted R2                  0.116                     
## Log Likelihood                               -2,070.226
## Akaike Inf. Crit.                            4,176.453 
## Residual Std. Error    1.555 (df = 1111)               
## F Statistic         9.723*** (df = 17; 1111)           
## =======================================================
## Note:                       *p<0.1; **p<0.05; ***p<0.01
R2<- summary(kids.poisson)$r.squared
R2
## NULL