Bajo las condiciones del problema anterior ### 3.1 Utilice la libreria Sandwich para estimar la matriz de varianzas y covarianzas de \(\hat{\beta}\) suponiendo heterocedastidad utilizando la especificación \(HC2\)
library(sandwich)## Warning: package 'sandwich' was built under R version 3.6.3vcovHC(m2_5, type="HC2")## X1 X2
## X1 0.0008602218 -0.001046587
## X2 -0.0010465868 0.001333104Halle la matriz \(H = X(X^TX)^{−1}X^T\) y los residuos de la estimación de \(\beta\) por OLS con base en los datos originales simulados con T = 100 y verifique los resultados del punto anterior
dat_e3<-simulacion_e2(s_e2,d0,d1,b_xx,rho,100)
#la matriz H
H<-dat_e3$X%*%solve(t(dat_e3$X)%*%dat_e3$X)%*%t(dat_e3$X)
#La estimación con la simulación para T=100
dat_e4<-data.frame(dat_e3$Y,dat_e3$X)
m4_1<-lm(dat_e3.Y~X1+X2+0,dat_e4)
#La matriz de varianzas y covarianzas de los OLS
I<-diag(1,100,100)
Omg_ols<-(sigma(m4_1)^2)*I
const<-solve(t(dat_e3$X)%*%dat_e3$X)%*%t(dat_e3$X)%*%Omg_ols%*%dat_e3$X%*%solve(t(dat_e3$X)%*%dat_e3$X)
const## [,1] [,2]
## [1,] 0.009961178 -0.01249936
## [2,] -0.012499359 0.01632534#verificando los calculos del punto anterior para HC2
HC2<-function(H,modelo_lm){
n<-nrow(H)
SR<-residuals(modelo_lm)^2
X<-model.matrix(modelo_lm)
Omg<-matrix(0,n,n)
SOmg<-NULL
for (i in 1:n){
SOmg[i]<-H[i,i]
Omg[i,i] = SR[i]/(1-SOmg[i])
}
HC2_matrix<-solve(t(X)%*%X)%*%t(X)%*%Omg%*%X%*%solve(t(X)%*%X)
return(HC2_matrix)
}
HC2(H,m4_1)## X1 X2
## X1 0.01277930 -0.01598153
## X2 -0.01598153 0.02060172El resultado de HC2 es diferente, pero porque el tamaño de muestra del punto 3.1 era mil, y en el punto 3.2 son solo 100.
Repita el punto anterior para la opción HC4
# con paquete sandwich
vcovHC(m2_5, type="HC4")## X1 X2
## X1 0.0008655969 -0.001053365
## X2 -0.0010533649 0.001341717#programando a mano
HC4<-function(H,modelo_lm){
n<-nrow(H)
SR<-residuals(modelo_lm)^2
X<-model.matrix(modelo_lm)
Omg<-matrix(0,n,n)
SOmg<-NULL
hm<-mean(diag(H))
delta<-NULL
for (i in 1:n){
SOmg[i]<-H[i,i]
if(SOmg[i]/hm>4){
delta[i]<-4} else{delta[i]<-SOmg[i]/hm}
Omg[i,i] = SR[i]/((1-SOmg[i])^delta[i])
}
HC4_matrix<-solve(t(X)%*%X)%*%t(X)%*%Omg%*%X%*%solve(t(X)%*%X)
return(HC4_matrix)
}
HC4(H,m4_1) # para T=100## X1 X2
## X1 0.01412561 -0.01763404
## X2 -0.01763404 0.02263413nuevamente, existen amplias diferencias por el tamaño de muestra. pero como se verá en el siguiente ejercicio, los resultados son consistentes con la librería sandwich.
Bajo las condiciones del problema anterior replique CON LOS DATOS SIMULADOS PARA T = 100 los ejemplos ilustrativos de la viñeta Sandwich
library(lmtest)## Warning: package 'lmtest' was built under R version 3.6.3## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericcoeftest(m4_1, df=Inf, vcov=vcovHC(m4_1, type="HC0"))##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## X1 0.39926 0.11027 3.6207 0.0002938 ***
## X2 0.91261 0.14009 6.5143 7.304e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1coeftest(m4_1, df=Inf, vcov=vcovHC(m4_1, type="HC4"))##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## X1 0.39926 0.11885 3.3593 0.0007813 ***
## X2 0.91261 0.15045 6.0660 1.311e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1coeftest(m4_1, df=Inf, vcov=NeweyWest(m4_1, lag = 4, prewhite = FALSE))##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## X1 0.39926 0.11198 3.5654 0.0003633 ***
## X2 0.91261 0.14847 6.1467 7.91e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1coeftest(m4_1, df = Inf, vcov = NeweyWest)##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## X1 0.39926 0.12259 3.2568 0.001127 **
## X2 0.91261 0.16007 5.7013 1.189e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1parzenHAC <- function(x, ...) kernHAC(x, kernel = "Parzen", prewhite = 2, adjust = FALSE, bw = bwNeweyWest, ...)
coeftest(m4_1, df = Inf, vcov = parzenHAC)##
## z test of coefficients:
##
## Estimate Std. Error z value Pr(>|z|)
## X1 0.39926 0.12523 3.1882 0.001431 **
## X2 0.91261 0.16530 5.5211 3.369e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Use the data in WAGE2 for this exercise.
library(gmm)## Warning: package 'gmm' was built under R version 3.6.3library(wooldridge)## Warning: package 'wooldridge' was built under R version 3.6.3data('wage2')
#como GMM no funciona con nulos
wage<-na.omit(wage2)In Example 15.2, if \(sibs\) is used as an instrument for \(educ\), the IV estimate of the return to education is .122. To convince yourself that using \(sibs\) as an IV for \(educ\) is not the same as just plugging \(sibs\) in for \(educ\) and running an \(OLS\) regression, run the regression of \(log(wage)\) on \(sibs\) and explain your findings.
#OLS
m9_1<-lm(lwage~sibs, data=wage2)
summary(m9_1)##
## Call:
## lm(formula = lwage ~ sibs, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.97662 -0.25857 0.02503 0.28572 1.22677
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.861076 0.022078 310.771 < 2e-16 ***
## sibs -0.027904 0.005908 -4.723 2.68e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4164 on 933 degrees of freedom
## Multiple R-squared: 0.02335, Adjusted R-squared: 0.0223
## F-statistic: 22.31 on 1 and 933 DF, p-value: 2.68e-06#GMM
m9_1_2<-gmm(wage$lwage~wage$sibs,x=wage$sibs,type="iterative")
summary(m9_1_2)##
## Call:
## gmm(g = wage$lwage ~ wage$sibs, x = wage$sibs, type = "iterative")
##
##
## Method: iterative
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.8844699 0.0274480 250.8187638 0.0000000
## wage$sibs -0.0246553 0.0072532 -3.3992542 0.0006757
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 4.61477680377028e-29 *******Como la variable dependiente se encuentra en escala logaritmica, se puede decir que el número de hermanos incrementa el salario en un 2.8%. Cuando se usaba sibs como instrumento para educ, el beta era -0.22 con una constante de 14.14, y al usar el educ estimado para estimar log(wage) el beta estimado era de 0.122. es decir, explicaba un 12.2%; es decir, usar sibs como instrumento para educ podría explicar mejor a log(wage) que usar directamente sibs
The variable \(brthord\) is birth order (\(brthord\) is one for a first-born child, two for a second-born child, and so on). Explain why \(educ\) and \(brthord\) might be negatively correlated. Regress \(educ\) on \(brthord\) to determine whether there is a statistically significant negative correlation.
cor.test(wage2$brthord,wage2$educ, method="pearson" )##
## Pearson's product-moment correlation
##
## data: wage2$brthord and wage2$educ
## t = -6.1062, df = 850, p-value = 1.551e-09
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.2684608 -0.1397521
## sample estimates:
## cor
## -0.2049925#OLS
m9_2<-lm(educ~brthord,data=wage2)
summary(m9_2)##
## Call:
## lm(formula = educ ~ brthord, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.8668 -1.5842 -0.7362 2.1332 6.1117
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.14945 0.12868 109.962 < 2e-16 ***
## brthord -0.28264 0.04629 -6.106 1.55e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.155 on 850 degrees of freedom
## (83 observations deleted due to missingness)
## Multiple R-squared: 0.04202, Adjusted R-squared: 0.04089
## F-statistic: 37.29 on 1 and 850 DF, p-value: 1.551e-09#GMM
m9_2_2<-gmm(wage$educ~wage$brthord,x=wage$brthord, type="iterative")
summary(m9_2_2)##
## Call:
## gmm(g = wage$educ ~ wage$brthord, x = wage$brthord, type = "iterative")
##
##
## Method: iterative
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.4256e+01 1.5362e-01 9.2802e+01 0.0000e+00
## wage$brthord -2.6435e-01 5.3592e-02 -4.9326e+00 8.1134e-07
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 1.99567293028392e-27 *******La correlación es negativa según el test de Pearson, así como tambien se obtiene un estimador con signo negativo en la regresión. Hacer supuestos sobre la relación entre ambas variables puede ser muy subjetivo, comenzando porque el coeficiente de correlación es bajo, y el $R^2" muestra que usar el valor esperado condicional como estimación del nivel educativo es algo muy débil. Si esta relación fuera cierta, es como si al hijo mayor le dieran más educación que al menor; eso sería entendible en sociedades antiguas, pero hoy en día incluso podrían haber casos dónde el hijo menor sea el consentido y se le brinden mejores oportunidades.
Use \(brthord\) as an IV for \(educ\) in equation (15.1). Report and interpret the results.
# IV Estimators
m9_3<-AER::ivreg(lwage~educ|brthord, data=wage2) #En vez de hacer dos regresiones, sugiero instalar la librería AER que cuenta con una función para usar IV estimators
summary(m9_3)##
## Call:
## AER::ivreg(formula = lwage ~ educ | brthord, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8532 -0.2557 0.0435 0.2970 1.3033
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.03040 0.43295 11.619 < 2e-16 ***
## educ 0.13064 0.03204 4.078 4.97e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4215 on 850 degrees of freedom
## Multiple R-Squared: -0.02862, Adjusted R-squared: -0.02983
## Wald test: 16.63 on 1 and 850 DF, p-value: 4.975e-05# GMM Two steps
m9_3_2<-gmm(wage$lwage~wage$educ, x=wage$brthord, type="twoStep") # cambiamos el tipo por "two steps", y ahora en X van los instrumentos
summary(m9_3_2)##
## Call:
## gmm(g = wage$lwage ~ wage$educ, x = wage$brthord, type = "twoStep")
##
##
## Method: twoStep
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.6389e+00 6.3194e-01 7.3407e+00 2.1254e-13
## wage$educ 1.5902e-01 4.5997e-02 3.4571e+00 5.4593e-04
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 1.6640610302406e-27 *******Parece que el orden de los hermanos es un mejor instrumento para educ cuando se estima el salario
Now, suppose that we include number of siblings as an explanatory variable in the wage equation; this controls for family background, to some extent: \(log(wage)=\beta_0+\beta_1educ+\beta_2sibs+u.\) Suppose that we want to use \(brthord\) as an IV for \(educ\), assuming that \(sibs\) is exogenous. The reduced form for \(educ\) is \(educ=\pi_0+\pi_1sibs+\pi_2brthord+v.\) State and test the identification assumption.
#OLS
m9_4<-lm(educ~sibs+brthord,data=wage2)
summary(m9_4)##
## Call:
## lm(formula = educ ~ sibs + brthord, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.1438 -1.6854 -0.6852 2.0090 5.9950
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.29650 0.13329 107.260 < 2e-16 ***
## sibs -0.15287 0.03987 -3.834 0.000135 ***
## brthord -0.15267 0.05708 -2.675 0.007619 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.137 on 849 degrees of freedom
## (83 observations deleted due to missingness)
## Multiple R-squared: 0.05833, Adjusted R-squared: 0.05611
## F-statistic: 26.29 on 2 and 849 DF, p-value: 8.33e-12#GMM
m9_4_2<-gmm(wage$educ~wage$sibs+wage$brthord,x=cbind(wage$sibs,wage$brthord), type="iterative")
summary(m9_4_2)##
## Call:
## gmm(g = wage$educ ~ wage$sibs + wage$brthord, x = cbind(wage$sibs,
## wage$brthord), type = "iterative")
##
##
## Method: iterative
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.3911908 0.1589633 90.5315110 0.0000000
## wage$sibs -0.1424838 0.0438234 -3.2513182 0.0011487
## wage$brthord -0.1402304 0.0655217 -2.1402125 0.0323376
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 6.81915163182099e-27 *******En este caso, acorde a lo que se revisó en el ejercicio 8, se requeriría que los estimadores de los instrumentos sean significativos, o al menos el de el orden de los hermanos, pues siblings se considera exógeno y además se incluye en la ecuación del logaritmo del salario. La variable del número de hermanos es significativa al 5%
Estimate the equation from part (iv) using \(brthord\) as an IV for \(educ\) (and \(sibs\) as its own IV). Comment on the standard errors for \(\hat{\beta}_{educ}\) and \(\hat{\beta}_{sibs}\).
#IV Estimators
m9_5<-AER::ivreg(lwage~ educ+sibs|.-educ+brthord, data=wage2)
summary(m9_5)##
## Call:
## AER::ivreg(formula = lwage ~ educ + sibs | . - educ + brthord,
## data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.84808 -0.26227 0.03841 0.29901 1.30836
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.938527 1.055690 4.678 3.37e-06 ***
## educ 0.136994 0.074681 1.834 0.0669 .
## sibs 0.002111 0.017372 0.122 0.9033
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.427 on 849 degrees of freedom
## Multiple R-Squared: -0.05428, Adjusted R-squared: -0.05676
## Wald test: 10.9 on 2 and 849 DF, p-value: 2.124e-05#GMM Two Steps
m9_5_2<-gmm(wage$lwage~ wage$educ+wage$sibs,x=cbind(wage$brthord,wage$sibs), type="twoStep")
summary(m9_5_2)##
## Call:
## gmm(g = wage$lwage ~ wage$educ + wage$sibs, x = cbind(wage$brthord,
## wage$sibs), type = "twoStep")
##
##
## Method: twoStep
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.748128 1.804650 2.076928 0.037808
## wage$educ 0.220264 0.126439 1.742064 0.081497
## wage$sibs 0.018586 0.027029 0.687636 0.491682
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 3.30940832847922e-29 *******La variable de hermanos no es significativa, tal vez porque ya se encluía en la estimación de educ. El estimador Educ ahora es significativo solo al 10%, por lo cual, el error estándar es más grande.
Using the fitted values from part (iv), \(\hat{educ}\), compute the correlation between \(\hat{educ}\) and \(sibs\). Use this result to explain your findings from part (v).
cor(m9_4$fitted.values,m9_4$model, method = "pearson")## educ sibs brthord
## [1,] 0.2415094 -0.9294818 -0.848797Use the data in PHILLIPS for this exercise.
data('phillips')
phill<-na.omit(phillips)In Example 11.5, we estimated an expectations augmented Phillips curve of the form \(\Delta inf_t=\beta_0+\beta_1unem_t+e_t,\) where \(\Delta inf_t=inf_t-inf_{t-1}\). In estimating this equation by \(OLS\), we assumed that the supply shock, \(e_t\), was uncorrelated with \(unem_t\). If this is false, what can be said about the \(OLS\) estimator of \(\beta_1\)?
#OLS
m9_6<-lm(cinf~unem,data=phillips)
summary(m9_6)##
## Call:
## lm(formula = cinf ~ unem, data = phillips)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.0741 -0.9241 0.0189 0.8606 5.4800
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.8282 1.2249 2.309 0.0249 *
## unem -0.5176 0.2090 -2.476 0.0165 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.307 on 53 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.1037, Adjusted R-squared: 0.08679
## F-statistic: 6.132 on 1 and 53 DF, p-value: 0.0165#GMM
m9_6_2<-gmm(phill$cinf~phill$unem,x=phill$unem, type="iterative")
summary(m9_6_2)##
## Call:
## gmm(g = phill$cinf ~ phill$unem, x = phill$unem, type = "iterative")
##
##
## Method: iterative
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.8282017 1.1832122 2.3902743 0.0168358
## phill$unem -0.5176487 0.1911874 -2.7075458 0.0067783
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 9.86503197737205e-29 *******Si existiese correlación entre el error y la variable expplicativa entonces el estimador de unem sería sesgado
Suppose that \(e_t\) is unpredictable given all past information: \(E(e_t|inf_{t-1}, unem_{t-1},...)=0\). Explain why this makes \(unem_{t-1}\) a good IV candidate for \(unem_t\).
#IV Estimators
m9_7<-AER::ivreg(cinf~unem|unem_1,data=phillips)
summary(m9_7)##
## Call:
## AER::ivreg(formula = cinf ~ unem | unem_1, data = phillips)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.1642 -1.0120 0.2054 1.0858 6.3967
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.6338 1.6563 0.383 0.703
## unem -0.1304 0.2867 -0.455 0.651
##
## Residual standard error: 2.38 on 53 degrees of freedom
## Multiple R-Squared: 0.04568, Adjusted R-squared: 0.02767
## Wald test: 0.207 on 1 and 53 DF, p-value: 0.651#GMM Two Steps
m9_7_2<-gmm(phill$cinf~phill$unem, x=phill$unem_1,type = "twoStep")
summary(m9_7_2)##
## Call:
## gmm(g = phill$cinf ~ phill$unem, x = phill$unem_1, type = "twoStep")
##
##
## Method: twoStep
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.63382 1.62750 0.38944 0.69695
## phill$unem -0.13045 0.29777 -0.43808 0.66133
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 2.79099639581684e-31 *******Porque no está correlacionado. sin embargo, al usarlo como instrumento en la practica esto no parece muy verídico
cor(m9_6$residuals,phillips[-1,])# quitamos la primera observación de la base de datos porque esta es un valor nulo para cinf, y por tanto para los errores támbien## year unem inf inf_1 unem_1 cinf cunem
## [1,] 0.1397998 3.465309e-16 0.5007163 -0.2537467 0.1914138 0.9467324 -0.2737429Es unem en el periodo t el que tiene poca correlación con el error en el primer modelo, mientras que unem en el periodo t-1 tiene una correlación más alta.
Regress \(unem_t\) on \(unem_{t-1}\). Are \(unem_t\) and \(unem_{t-1}\) significantly correlated?
#OLS
m9_8<-lm(unem~unem_1,data=phillips)
summary(m9_8)##
## Call:
## lm(formula = unem ~ unem_1, data = phillips)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.1243 -0.6433 -0.2470 0.6098 2.8530
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.48968 0.52020 2.864 0.00599 **
## unem_1 0.74238 0.08929 8.314 3.54e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9986 on 53 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.566, Adjusted R-squared: 0.5578
## F-statistic: 69.12 on 1 and 53 DF, p-value: 3.537e-11#GMM
m9_8_2<-gmm(phill$unem~phill$unem_1,x=phill$unem_1)
summary(m9_8_2)##
## Call:
## gmm(g = phill$unem ~ phill$unem_1, x = phill$unem_1)
##
##
## Method: twoStep
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.4897e+00 5.4535e-01 2.7316e+00 6.3023e-03
## phill$unem_1 7.4238e-01 9.7920e-02 7.5815e+00 3.4160e-14
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 7.10370141584368e-28 *******La correlación es bastante alta, del 74%
Estimate the expectations augmented Phillips curve by IV. Report the results in the usual form and compare them with the \(OLS\) estimates from Example 11.5.
la estimación IV es la m9_7 mientras que la de OLS es la m9_6. Los resultados muestran que, a pesar de que \(unem_{t-1}\) es un buen predictor de \(unem\), no sirve como instrumento cuando se pretende explicar \(\Delta inf_t\)Use a \(2SLS\) routine to estimate the equation \(log(wage)=\beta_0+\beta_1educ+\beta_2exper+\beta_3tenure+\beta_4black+u,\) where \(sibs\) is the IV for \(educ\). Report the results in the usual form.
# IV Estimators
m10_1<-AER::ivreg(lwage~educ+exper+tenure+black|.-educ+sibs, data=wage2) #despues de la linea vertical se debe marcar cual variable a retirar de las regresoras, y cual instrumento usar para calcular la estimación del primer paso en su lugar
summary(m10_1)##
## Call:
## AER::ivreg(formula = lwage ~ educ + exper + tenure + black |
## . - educ + sibs, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8176 -0.2403 0.0139 0.2567 1.3225
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.215976 0.543451 9.598 < 2e-16 ***
## educ 0.093632 0.033719 2.777 0.00560 **
## exper 0.020922 0.008388 2.494 0.01279 *
## tenure 0.011548 0.002740 4.215 2.74e-05 ***
## black -0.183329 0.050136 -3.657 0.00027 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3848 on 930 degrees of freedom
## Multiple R-Squared: 0.1685, Adjusted R-squared: 0.165
## Wald test: 24.92 on 4 and 930 DF, p-value: < 2.2e-16#GMM Two Steps
m10_1_2<-gmm(wage$lwage~wage$educ+wage$exper+wage$tenure+wage$black,x=cbind(wage$sibs,wage$exper,wage$tenure,wage$black), type="twoStep") # se tienen que incluir todas las variables de los betas en el x, solo cambiando la que va con instrumento, de lo contrario el número de instrumentos será inferior al número de variables
summary(m10_1_2)##
## Call:
## gmm(g = wage$lwage ~ wage$educ + wage$exper + wage$tenure + wage$black,
## x = cbind(wage$sibs, wage$exper, wage$tenure, wage$black),
## type = "twoStep")
##
##
## Method: twoStep
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.1531e+00 6.9275e-01 7.4385e+00 1.0183e-13
## wage$educ 9.7994e-02 4.2009e-02 2.3327e+00 1.9665e-02
## wage$exper 2.4033e-02 1.1628e-02 2.0668e+00 3.8751e-02
## wage$tenure 8.2282e-03 3.3129e-03 2.4837e+00 1.3003e-02
## wage$black -1.5505e-01 6.7448e-02 -2.2988e+00 2.1518e-02
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 2.26999056478394e-26 *******Now, manually carry out \(2SLS\). That is, first regress \(educ_i\) on \(sibs_i\), \(exper_i\), \(tenure_i\), and \(black_i\) and obtain the fitted values, \(\hat{educ}_i, i=1,..., n\). Then, run the second stage regression \(log(wage_i )\) on \(\hat{educ}_i\), \(exper_i\), \(tenure_i\), and \(black_i, i=1,..., n\). Verify that the \(\hat{\beta}_j\) are identical to those obtained from part (i), but that the standard errors are somewhat different. The standard errors obtained from the second stage regression when manually carrying out \(2SLS\) are generally inappropriate.
#OLS
m10_2<-lm(educ~sibs+exper+tenure+black,data=wage2)
wage2$educ2<-m10_2$fitted.values
m10_3<-lm(lwage~educ2+exper+tenure+black, data=wage2)
summary(m10_3)##
## Call:
## lm(formula = lwage ~ educ2 + exper + tenure + black, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.97409 -0.25720 0.00997 0.26147 1.25198
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.215976 0.568815 9.170 < 2e-16 ***
## educ2 0.093632 0.035293 2.653 0.008114 **
## exper 0.020922 0.008779 2.383 0.017368 *
## tenure 0.011548 0.002868 4.027 6.1e-05 ***
## black -0.183329 0.052476 -3.494 0.000499 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4028 on 930 degrees of freedom
## Multiple R-squared: 0.08912, Adjusted R-squared: 0.0852
## F-statistic: 22.75 on 4 and 930 DF, p-value: < 2.2e-16#GMM
m10_2_2<-gmm(wage$educ~wage$sibs+wage$exper+wage$tenure+wage$black,x=cbind(wage$sibs,wage$exper,wage$tenure,wage$black), type="iterative")
m10_3_2<-gmm(wage$lwage~m10_2_2$fitted.values+wage$exper+wage$tenure+wage$black, x=cbind(m10_2_2$fitted.values,wage$exper,wage$tenure,wage$black), type="iterative")
summary(m10_3_2)##
## Call:
## gmm(g = wage$lwage ~ m10_2_2$fitted.values + wage$exper + wage$tenure +
## wage$black, x = cbind(m10_2_2$fitted.values, wage$exper,
## wage$tenure, wage$black), type = "iterative")
##
##
## Method: iterative
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.1531e+00 6.9342e-01 7.4314e+00 1.0748e-13
## m10_2_2$fitted.values 9.7994e-02 4.2106e-02 2.3273e+00 1.9948e-02
## wage$exper 2.4033e-02 1.1567e-02 2.0778e+00 3.7728e-02
## wage$tenure 8.2282e-03 3.3990e-03 2.4208e+00 1.5488e-02
## wage$black -1.5505e-01 6.4033e-02 -2.4214e+00 1.5462e-02
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 2.17828765976177e-24 *******El error estándar de estimarlo manual es más alto
Now, use the following two-step procedure, which generally yields inconsistent parameter estimates of the \(\beta_j\), and not just inconsistent standard errors. In step one, regress \(educ_i\) on \(sibs_i\) only and obtain the fitted values, say \(\tilde{educ}_i\). (Note that this is an incorrect first stage regression.) Then, in the second step, run the regression of \(log(wage_i)\) on \(\tilde{educ}_i\), \(exper_i\), \(tenure_i\), and \(black_i, i=1,..., n\). How does the estimate from this incorrect, two-step procedure compare with the correct \(2SLS\) estimate of the return to education?
#OLS
m10_4<-lm(educ~sibs,data=wage2)
wage2$educ3<-m10_4$fitted.values
m10_5<-lm(lwage~educ3+exper+tenure+black, data=wage2)
summary(m10_5)##
## Call:
## lm(formula = lwage ~ educ3 + exper + tenure + black, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.97409 -0.25720 0.00997 0.26147 1.25198
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.771022 0.360376 16.014 < 2e-16 ***
## educ3 0.069975 0.026376 2.653 0.00811 **
## exper -0.000394 0.003121 -0.126 0.89957
## tenure 0.013975 0.002691 5.193 2.54e-07 ***
## black -0.241631 0.041528 -5.819 8.16e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4028 on 930 degrees of freedom
## Multiple R-squared: 0.08912, Adjusted R-squared: 0.0852
## F-statistic: 22.75 on 4 and 930 DF, p-value: < 2.2e-16#GMM
m10_4_2<-gmm(wage$educ~wage$sibs,x=wage$sibs, type="iterative")
m10_5_2<-gmm(wage$lwage~m10_4_2$fitted.values+wage$exper+wage$tenure+wage$black, x=cbind(m10_4_2$fitted.values,wage$exper,wage$tenure,wage$black), type="iterative")
summary(m10_5_2)##
## Call:
## gmm(g = wage$lwage ~ m10_4_2$fitted.values + wage$exper + wage$tenure +
## wage$black, x = cbind(m10_4_2$fitted.values, wage$exper,
## wage$tenure, wage$black), type = "iterative")
##
##
## Method: iterative
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.5779e+00 5.1129e-01 1.0909e+01 1.0396e-27
## m10_4_2$fitted.values 8.5334e-02 3.6658e-02 2.3279e+00 1.9920e-02
## wage$exper -3.7720e-04 3.5106e-03 -1.0745e-01 9.1444e-01
## wage$tenure 1.2396e-02 2.9860e-03 4.1515e+00 3.3025e-05
## wage$black -1.9812e-01 5.7235e-02 -3.4615e+00 5.3709e-04
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 1.13108248863633e-24 *******Use the data in CATHOLIC to answer this question. The model of interest is \(math_{12}=\beta_0+\beta_1cathhs+\beta_2lfaminc+\beta_3motheduc+\beta_4fatheduc+u,\) where \(cathhs\) is a binary indicator for whether a student attends a Catholic high school.
data('catholic')How many students are in the sample? What percentage of these students attend a Catholic high school?
# no todos en la muestra son estudiantes, solo los que no se habían graduado para 1994
students<-sum(catholic$hsgrad==0, na.rm=TRUE)
students## [1] 416#Estudiantes católicos
catolicos<-sum(catholic$cathhs==1 & catholic$hsgrad==0,na.rm=TRUE)
catolicos/students*100## [1] 1.442308Estimate the above equation by \(OLS\). What is the estimate of \(\beta_1\)? What is its 95% confidence interval?
#OLS
m10_6<-lm(math12~cathhs+lfaminc+motheduc+fatheduc, data=catholic)
summary(m10_6)##
## Call:
## lm(formula = math12 ~ cathhs + lfaminc + motheduc + fatheduc,
## data = catholic)
##
## Residuals:
## Min 1Q Median 3Q Max
## -27.0824 -6.1371 0.4904 6.4524 26.7728
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.14862 1.31098 8.504 < 2e-16 ***
## cathhs 1.47723 0.41794 3.535 0.000411 ***
## lfaminc 1.84867 0.14263 12.961 < 2e-16 ***
## motheduc 0.71635 0.06208 11.539 < 2e-16 ***
## fatheduc 0.89125 0.05615 15.872 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.544 on 7425 degrees of freedom
## Multiple R-squared: 0.1846, Adjusted R-squared: 0.1841
## F-statistic: 420.2 on 4 and 7425 DF, p-value: < 2.2e-16#GMM
m10_6_2<-gmm(catholic$math12~catholic$cathhs+catholic$lfaminc+catholic$motheduc+catholic$fatheduc, x=cbind(catholic$cathhs,catholic$lfaminc,catholic$motheduc,catholic$fatheduc), type="iterative")
summary(m10_6_2)##
## Call:
## gmm(g = catholic$math12 ~ catholic$cathhs + catholic$lfaminc +
## catholic$motheduc + catholic$fatheduc, x = cbind(catholic$cathhs,
## catholic$lfaminc, catholic$motheduc, catholic$fatheduc),
## type = "iterative")
##
##
## Method: iterative
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.1149e+01 1.2634e+00 8.8243e+00 1.1017e-18
## catholic$cathhs 1.4772e+00 4.2225e-01 3.4985e+00 4.6790e-04
## catholic$lfaminc 1.8487e+00 1.4036e-01 1.3171e+01 1.2888e-39
## catholic$motheduc 7.1635e-01 6.2587e-02 1.1446e+01 2.4723e-30
## catholic$fatheduc 8.9125e-01 5.6567e-02 1.5756e+01 6.2654e-56
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 1.10517958274478e-18 *******El estimador \(\beta_1\) es 1.47 por OLS o 1.74 por GMM y sí es significativo al 5% en ambos
Using \(parcath\) as an instrument for \(cathhs\), estimate the reduced form for \(cathhs\). What is the t statistic for \(parcath\)? Is there evidence of a weak instrument problem?
#OLS
m10_7<-lm(cathhs~parcath+lfaminc+motheduc+fatheduc, data=catholic)
summary(m10_7)##
## Call:
## lm(formula = cathhs ~ parcath + lfaminc + motheduc + fatheduc,
## data = catholic)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.23375 -0.12118 -0.02169 0.00332 1.07310
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.321081 0.034755 -9.238 < 2e-16 ***
## parcath 0.143058 0.005566 25.701 < 2e-16 ***
## lfaminc 0.018351 0.003793 4.839 1.33e-06 ***
## motheduc 0.003923 0.001655 2.370 0.0178 *
## fatheduc 0.006585 0.001493 4.411 1.04e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2273 on 7425 degrees of freedom
## Multiple R-squared: 0.09594, Adjusted R-squared: 0.09545
## F-statistic: 197 on 4 and 7425 DF, p-value: < 2.2e-16#GMM
m10_7_2<-gmm(catholic$cathhs~catholic$parcath+catholic$lfaminc+catholic$motheduc+catholic$fatheduc, x=cbind(catholic$parcath,catholic$lfaminc,catholic$motheduc,catholic$fatheduc), type="iterative")
summary(m10_7_2)##
## Call:
## gmm(g = catholic$cathhs ~ catholic$parcath + catholic$lfaminc +
## catholic$motheduc + catholic$fatheduc, x = cbind(catholic$parcath,
## catholic$lfaminc, catholic$motheduc, catholic$fatheduc),
## type = "iterative")
##
##
## Method: iterative
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.2108e-01 5.2292e-02 -6.1402e+00 8.2410e-10
## catholic$parcath 1.4306e-01 1.4336e-02 9.9792e+00 1.8800e-23
## catholic$lfaminc 1.8351e-02 4.6713e-03 3.9286e+00 8.5458e-05
## catholic$motheduc 3.9228e-03 1.7177e-03 2.2838e+00 2.2383e-02
## catholic$fatheduc 6.5846e-03 1.7201e-03 3.8281e+00 1.2915e-04
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 3.4993953409197e-22 *******Por el momento, la variable parcath es significativa para estimar cathhs
Estimate the above equation by IV, using \(parcath\) as an IV for \(cathhs\). How does the estimate and 95% CI compare with the OLS quantities?
#IV Estimators
m10_8<-AER::ivreg(math12~cathhs+lfaminc+motheduc+fatheduc|.-cathhs+parcath, data=catholic)
summary(m10_8)##
## Call:
## AER::ivreg(formula = math12 ~ cathhs + lfaminc + motheduc + fatheduc |
## . - cathhs + parcath, data = catholic)
##
## Residuals:
## Min 1Q Median 3Q Max
## -28.3521 -6.1166 0.4572 6.4546 26.8221
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.91472 1.37627 8.657 < 2e-16 ***
## cathhs 4.11742 1.46615 2.808 0.00499 **
## lfaminc 1.78453 0.14703 12.137 < 2e-16 ***
## motheduc 0.71331 0.06227 11.455 < 2e-16 ***
## fatheduc 0.87501 0.05696 15.361 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.567 on 7425 degrees of freedom
## Multiple R-Squared: 0.1802, Adjusted R-squared: 0.1797
## Wald test: 416.8 on 4 and 7425 DF, p-value: < 2.2e-16#GMM Two Step
m10_8_2<-gmm(catholic$math12~catholic$cathhs+catholic$lfaminc+catholic$motheduc+catholic$fatheduc, x=cbind(catholic$parcath,catholic$lfaminc,catholic$motheduc,catholic$fatheduc), type="twoStep")
summary(m10_8_2)##
## Call:
## gmm(g = catholic$math12 ~ catholic$cathhs + catholic$lfaminc +
## catholic$motheduc + catholic$fatheduc, x = cbind(catholic$parcath,
## catholic$lfaminc, catholic$motheduc, catholic$fatheduc),
## type = "twoStep")
##
##
## Method: twoStep
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.1915e+01 1.3121e+00 9.0809e+00 1.0768e-19
## catholic$cathhs 4.1174e+00 1.4891e+00 2.7651e+00 5.6908e-03
## catholic$lfaminc 1.7845e+00 1.4296e-01 1.2483e+01 9.2950e-36
## catholic$motheduc 7.1331e-01 6.2804e-02 1.1358e+01 6.7956e-30
## catholic$fatheduc 8.7501e-01 5.7618e-02 1.5186e+01 4.3473e-52
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 9.91406636788409e-19 *******Sigue siendo estadísticamente significativo, parece que parcath es un buen instrumento, aunque sin instrumentos cathhs era significativa al 0.01% y ahora solo lo es al 1%.
Test the null hypothesis that \(cathhs\) is exogenous. What is the p-value of the test?
Tanto gmm como iv estimatos nos dan el p value. En este caso es de 0.00569 con GMM o de 0.0049 con IV
Suppose you add the interaction between \(cathhs * motheduc\) to the above model. Why is it generally endogenous? Why is \(pareduc*motheduc\) a good IV candidate for \(cathhs*motheduc\)?
#OLS
m10_9<-lm(math12~cathhs+lfaminc+motheduc+fatheduc+cathhs*motheduc, data=catholic)
summary(m10_9)##
## Call:
## lm(formula = math12 ~ cathhs + lfaminc + motheduc + fatheduc +
## cathhs * motheduc, data = catholic)
##
## Residuals:
## Min 1Q Median 3Q Max
## -27.1084 -6.1397 0.4865 6.4471 26.8147
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.05641 1.31736 8.393 <2e-16 ***
## cathhs 3.76228 3.22718 1.166 0.244
## lfaminc 1.84739 0.14265 12.951 <2e-16 ***
## motheduc 0.72516 0.06330 11.456 <2e-16 ***
## fatheduc 0.89038 0.05617 15.852 <2e-16 ***
## cathhs:motheduc -0.16401 0.22968 -0.714 0.475
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.544 on 7424 degrees of freedom
## Multiple R-squared: 0.1846, Adjusted R-squared: 0.1841
## F-statistic: 336.2 on 5 and 7424 DF, p-value: < 2.2e-16#Sin instrumentos GMM
m10_9_2<-gmm(catholic$math12~catholic$cathhs+catholic$lfaminc+catholic$motheduc+catholic$fatheduc+catholic$cathhs*catholic$motheduc, x=cbind(catholic$cathhs,catholic$lfaminc,catholic$motheduc,catholic$fatheduc,catholic$cathhs*catholic$motheduc), type="iterative")
summary(m10_9_2)##
## Call:
## gmm(g = catholic$math12 ~ catholic$cathhs + catholic$lfaminc +
## catholic$motheduc + catholic$fatheduc + catholic$cathhs *
## catholic$motheduc, x = cbind(catholic$cathhs, catholic$lfaminc,
## catholic$motheduc, catholic$fatheduc, catholic$cathhs * catholic$motheduc),
## type = "iterative")
##
##
## Method: iterative
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value
## (Intercept) 1.1056e+01 1.2700e+00 8.7057e+00
## catholic$cathhs 3.7623e+00 2.8550e+00 1.3178e+00
## catholic$lfaminc 1.8474e+00 1.4036e-01 1.3162e+01
## catholic$motheduc 7.2516e-01 6.4195e-02 1.1296e+01
## catholic$fatheduc 8.9038e-01 5.6582e-02 1.5736e+01
## catholic$cathhs:catholic$motheduc -1.6401e-01 2.0278e-01 -8.0879e-01
## Pr(>|t|)
## (Intercept) 3.1569e-18
## catholic$cathhs 1.8757e-01
## catholic$lfaminc 1.4585e-39
## catholic$motheduc 1.3714e-29
## catholic$fatheduc 8.5631e-56
## catholic$cathhs:catholic$motheduc 4.1863e-01
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 1.09707624445294e-18 *******Es generalmente endogeno porque ya se incluyeron las dos variables previamente, de hecho incluirla afecta la significancia de de cathhs.
#IV Estimators
m10_10<-AER::ivreg(math12~cathhs+lfaminc+motheduc+fatheduc+cathhs*motheduc|.-cathhs+parcath-cathhs*motheduc+parcath*motheduc, data=catholic)
summary(m10_10)##
## Call:
## AER::ivreg(formula = math12 ~ cathhs + lfaminc + motheduc + fatheduc +
## cathhs * motheduc | . - cathhs + parcath - cathhs * motheduc +
## parcath * motheduc, data = catholic)
##
## Residuals:
## Min 1Q Median 3Q Max
## -30.4258 -6.2590 0.5254 6.5728 28.0794
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.32144 1.47286 6.329 2.61e-10 ***
## cathhs 72.64904 15.78263 4.603 4.23e-06 ***
## lfaminc 1.73374 0.15262 11.360 < 2e-16 ***
## motheduc 0.97506 0.08612 11.323 < 2e-16 ***
## fatheduc 0.84569 0.05932 14.257 < 2e-16 ***
## cathhs:motheduc -4.88146 1.08382 -4.504 6.77e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.816 on 7424 degrees of freedom
## Multiple R-Squared: 0.132, Adjusted R-squared: 0.1314
## Wald test: 317.8 on 5 and 7424 DF, p-value: < 2.2e-16#GMM Two Step
m10_10_2<-gmm(catholic$math12~catholic$cathhs+catholic$lfaminc+catholic$motheduc+catholic$fatheduc+catholic$cathhs*catholic$motheduc, x=cbind(catholic$parcath,catholic$lfaminc,catholic$motheduc,catholic$fatheduc,catholic$parcath*catholic$motheduc), type="twoStep")
summary(m10_10_2)##
## Call:
## gmm(g = catholic$math12 ~ catholic$cathhs + catholic$lfaminc +
## catholic$motheduc + catholic$fatheduc + catholic$cathhs *
## catholic$motheduc, x = cbind(catholic$parcath, catholic$lfaminc,
## catholic$motheduc, catholic$fatheduc, catholic$parcath *
## catholic$motheduc), type = "twoStep")
##
##
## Method: twoStep
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value
## (Intercept) 9.3214e+00 1.4134e+00 6.5948e+00
## catholic$cathhs 7.2649e+01 1.7570e+01 4.1348e+00
## catholic$lfaminc 1.7337e+00 1.5162e-01 1.1435e+01
## catholic$motheduc 9.7506e-01 8.6316e-02 1.1296e+01
## catholic$fatheduc 8.4569e-01 6.0129e-02 1.4065e+01
## catholic$cathhs:catholic$motheduc -4.8815e+00 1.2035e+00 -4.0559e+00
## Pr(>|t|)
## (Intercept) 4.2575e-11
## catholic$cathhs 3.5525e-05
## catholic$lfaminc 2.7976e-30
## catholic$motheduc 1.3659e-29
## catholic$fatheduc 6.2559e-45
## catholic$cathhs:catholic$motheduc 4.9934e-05
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 1.12121069393814e-18 *******Al incluir el instrumento se elimina la endogeneidad
Before you create the interactions in part (vi), first find the sample average of \(motheduc\) and create \(cathhs *(motheduc - \bar{motheduc})\) and \(parcath*(motheduc-\bar{motheduc})\). Add the first interaction to the model and use the second as an IV. Of course, \(cathhs\) is also instrumented. Is the interaction term statistically significant?
#las interacciones
catholic$interaccion1<-catholic$cathhs*(catholic$motheduc-mean(catholic$motheduc))
catholic$interaccion2<-catholic$parcath*(catholic$motheduc-mean(catholic$motheduc))
#Modelo IV Estimators
m10_11<-AER::ivreg(math12~cathhs+lfaminc+motheduc+fatheduc+interaccion1|.-cathhs+parcath-interaccion1+interaccion2, data=catholic)
summary(m10_11)##
## Call:
## AER::ivreg(formula = math12 ~ cathhs + lfaminc + motheduc + fatheduc +
## interaccion1 | . - cathhs + parcath - interaccion1 + interaccion2,
## data = catholic)
##
## Residuals:
## Min 1Q Median 3Q Max
## -30.4258 -6.2590 0.5254 6.5728 28.0794
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.32144 1.47286 6.329 2.61e-10 ***
## cathhs 7.44802 1.88842 3.944 8.09e-05 ***
## lfaminc 1.73374 0.15262 11.360 < 2e-16 ***
## motheduc 0.97506 0.08612 11.323 < 2e-16 ***
## fatheduc 0.84569 0.05932 14.257 < 2e-16 ***
## interaccion1 -4.88146 1.08382 -4.504 6.77e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.816 on 7424 degrees of freedom
## Multiple R-Squared: 0.132, Adjusted R-squared: 0.1314
## Wald test: 317.8 on 5 and 7424 DF, p-value: < 2.2e-16#GMM Two Step
m10_11_2<-gmm(catholic$math12~catholic$cathhs+catholic$lfaminc+catholic$motheduc+catholic$fatheduc+catholic$interaccion1, x=cbind(catholic$parcath,catholic$lfaminc,catholic$motheduc,catholic$fatheduc,catholic$interaccion2), type="twoStep")
summary(m10_11_2)##
## Call:
## gmm(g = catholic$math12 ~ catholic$cathhs + catholic$lfaminc +
## catholic$motheduc + catholic$fatheduc + catholic$interaccion1,
## x = cbind(catholic$parcath, catholic$lfaminc, catholic$motheduc,
## catholic$fatheduc, catholic$interaccion2), type = "twoStep")
##
##
## Method: twoStep
##
## Kernel: Quadratic Spectral
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.3214e+00 1.4134e+00 6.5948e+00 4.2576e-11
## catholic$cathhs 7.4480e+00 2.0454e+00 3.6413e+00 2.7127e-04
## catholic$lfaminc 1.7337e+00 1.5162e-01 1.1435e+01 2.7943e-30
## catholic$motheduc 9.7506e-01 8.6316e-02 1.1296e+01 1.3674e-29
## catholic$fatheduc 8.4569e-01 6.0130e-02 1.4065e+01 6.2728e-45
## catholic$interaccion1 -4.8815e+00 1.2035e+00 -4.0560e+00 4.9911e-05
##
## J-Test: degrees of freedom is 0
## J-test P-value
## Test E(g)=0: 1.12742185615146e-18 *******El resultado es el mismo que antes para los betas pero con errores estándar más pequeños
Compare the coefficient on \(cathhs\) in (vii) to that in part (iv). Is including the interaction important for estimating the average partial effect?
No es importante porque al incluir la interacción se incrementan los errores estándar y se reduce el \(R^2\)