Ej 2 multicolinealidad
JAIRO RODRIGO MENDES CARRILLO
2023-05-13
library(wooldridge)
data(hprice1)
head(force(hprice1),n=5)## price assess bdrms lotsize sqrft colonial lprice lassess llotsize lsqrft
## 1 300 349.1 4 6126 2438 1 5.703783 5.855359 8.720297 7.798934
## 2 370 351.5 3 9903 2076 1 5.913503 5.862210 9.200593 7.638198
## 3 191 217.7 3 5200 1374 0 5.252274 5.383118 8.556414 7.225482
## 4 195 231.8 3 4600 1448 1 5.273000 5.445875 8.433811 7.277938
## 5 373 319.1 4 6095 2514 1 5.921578 5.765504 8.715224 7.829630
- estime el siguiente modelo.
library(stargazer)
estimacion_modelo<-lm(price ~ lotsize + sqrft + bdrms,data = hprice1)
stargazer(estimacion_modelo,title = "Modelo Ejemplo", type = "text")##
## Modelo Ejemplo
## ===============================================
## Dependent variable:
## ---------------------------
## price
## -----------------------------------------------
## lotsize 0.002***
## (0.001)
##
## sqrft 0.123***
## (0.013)
##
## bdrms 13.853
## (9.010)
##
## Constant -21.770
## (29.475)
##
## -----------------------------------------------
## Observations 88
## R2 0.672
## Adjusted R2 0.661
## Residual Std. Error 59.833 (df = 84)
## F Statistic 57.460*** (df = 3; 84)
## ===============================================
## Note: *p<0.1; **p<0.05; ***p<0.01
- Verifique si hay evidencia de la independencia de los regresores (no colinealidad), a través de:
a) Indice de condición.
- Cálculo “manual”
library(stargazer)
x_matriz<-model.matrix(estimacion_modelo)
stargazer(head(x_matriz),type = "text",title = "matriz x")##
## matriz x
## =================================
## (Intercept) lotsize sqrft bdrms
## ---------------------------------
## 1 1 6,126 2,438 4
## 2 1 9,903 2,076 3
## 3 1 5,200 1,374 3
## 4 1 4,600 1,448 3
## 5 1 6,095 2,514 4
## 6 1 8,566 2,754 5
## ---------------------------------xx_matriz<- t(x_matriz)%*%x_matriz
stargazer(xx_matriz,type = "text",title = "matriz xtx")##
## matriz xtx
## ==============================================================
## (Intercept) lotsize sqrft bdrms
## --------------------------------------------------------------
## (Intercept) 88 793,748 177,205 314
## lotsize 793,748 16,165,159,010 1,692,290,257 2,933,767
## sqrft 177,205 1,692,290,257 385,820,561 654,755
## bdrms 314 2,933,767 654,755 1,182
## --------------------------------------------------------------- Cálculo de la matriz de normalización:
library(stargazer)
options(scipen = 999)
sn<-solve(diag(sqrt(diag(xx_matriz))))
stargazer(sn,type = "text",title = "diagonal xx")##
## diagonal xx
## ==========================
## 0.107 0 0 0
## 0 0.00001 0 0
## 0 0 0.0001 0
## 0 0 0 0.029
## --------------------------library(stargazer)
xx_normalizada<-(sn%*%xx_matriz)%*%sn
stargazer(xx_normalizada,type = "text",
digits = 4,
title = "XX normalizada",
align = TRUE,
notes = "Elaboracion propia")##
## XX normalizada
## ===========================
## 1 0.6655 0.9617 0.9736
## 0.6655 1 0.6776 0.6712
## 0.9617 0.6776 1 0.9696
## 0.9736 0.6712 0.9696 1
## ---------------------------
## Elaboracion propialibrary(stargazer)
lambdas<-eigen(xx_normalizada,symmetric = TRUE)
stargazer(lambdas$values,type = "text",title = "autovalores de xx normalizada")##
## autovalores de xx normalizada
## =======================
## 3.482 0.455 0.039 0.025
## ------------------------ Cálculo de k(x)
K<-sqrt(max(lambdas$values)/min(lambdas$values))
print(K)## [1] 11.86778Como κ(x)<20 la multicolinealidad es leve, no se considera un problema.
cálculo del indice de condición usando libreria “mctest”
library(mctest)
X_mat<-model.matrix(estimacion_modelo)
mctest(mod = estimacion_modelo)##
## Call:
## omcdiag(mod = mod, Inter = TRUE, detr = detr, red = red, conf = conf,
## theil = theil, cn = cn)
##
##
## Overall Multicollinearity Diagnostics
##
## MC Results detection
## Determinant |X'X|: 0.6918 0
## Farrar Chi-Square: 31.3812 1
## Red Indicator: 0.3341 0
## Sum of Lambda Inverse: 3.8525 0
## Theil's Method: -0.7297 0
## Condition Number: 11.8678 0
##
## 1 --> COLLINEARITY is detected by the test
## 0 --> COLLINEARITY is not detected by the testcálculo del indice de condición usando librería “olsrr”
library(olsrr)
ols_eigen_cindex(model =estimacion_modelo)## Eigenvalue Condition Index intercept lotsize sqrft bdrms
## 1 3.48158596 1.000000 0.003663034 0.0277802824 0.004156293 0.002939554
## 2 0.45518380 2.765637 0.006800735 0.9670803174 0.006067321 0.005096396
## 3 0.03851083 9.508174 0.472581427 0.0051085488 0.816079307 0.016938178
## 4 0.02471941 11.867781 0.516954804 0.0000308514 0.173697079 0.975025872PRUEBA DE FARRAR-GLAUBAR(BARTLETT)
- cálculo “manual”
library(stargazer)
Zn<-scale(X_mat[,-1])
stargazer(head(Zn),type = "text")##
## =======================
## lotsize sqrft bdrms
## -----------------------
## 1 -0.284 0.735 0.513
## 2 0.087 0.108 -0.675
## 3 -0.375 -1.108 -0.675
## 4 -0.434 -0.980 -0.675
## 5 -0.287 0.867 0.513
## 6 -0.045 1.283 1.702
## ------------------------ Calculo de Matriz R
library(stargazer)
n<-nrow(Zn)
R<-(t(Zn)%*%Zn)*(1/(n-1))
stargazer(R,type = "text")##
## ===========================
## lotsize sqrft bdrms
## ---------------------------
## lotsize 1 0.184 0.136
## sqrft 0.184 1 0.531
## bdrms 0.136 0.531 1
## ---------------------------- Calculo de |R|
determinante_R<-det(R)
print(determinante_R)## [1] 0.6917931Apliando la prueba de FARRAR-GLAUBAR(BARTLETT)
- Estadistico χ2FG
m<-ncol(X_mat[,-1])
n<-nrow(X_mat[,-1])
chi_fg<-(-(n-1-(2*m+5)/6)*log(determinante_R))
print(chi_fg)## [1] 31.38122- Valor critico
gl<-m*(m-1)/2
V.C<-qchisq(p = 0.95,df = gl)
print(V.C)## [1] 7.814728Como χ2FG(31.38122)≥V.C.(7.814728) se rechaza H0, por lo tanto hay evidencia de colinealidad en los regresores
Cálculo de FG usando “mctest”
library(mctest)
mctest::omcdiag(mod = estimacion_modelo)##
## Call:
## mctest::omcdiag(mod = estimacion_modelo)
##
##
## Overall Multicollinearity Diagnostics
##
## MC Results detection
## Determinant |X'X|: 0.6918 0
## Farrar Chi-Square: 31.3812 1
## Red Indicator: 0.3341 0
## Sum of Lambda Inverse: 3.8525 0
## Theil's Method: -0.7297 0
## Condition Number: 11.8678 0
##
## 1 --> COLLINEARITY is detected by the test
## 0 --> COLLINEARITY is not detected by the testCálculo de FG usando “psych”
library(psych)
fg_test<-cortest.bartlett(X_mat[,-1])
print(fg_test)## $chisq
## [1] 31.38122
##
## $p.value
## [1] 0.0000007065806
##
## $df
## [1] 3- resultados de forma gráfica usando la librería fastGraph
library(fastGraph)
shadeDist(chi_fg)
- Factores inflacionarios de la varianza, presente sus resultados de forma tabular y de forma gráfica.
- Cálculo “manual” inversa de la matriz de correlación R−1
inversa_R<-solve(R)
print(inversa_R)## lotsize sqrft bdrms
## lotsize 1.03721145 -0.1610145 -0.05582352
## sqrft -0.16101454 1.4186543 -0.73202696
## bdrms -0.05582352 -0.7320270 1.39666321## VIF’s para el modelo estimado:
VIFs<-diag(inversa_R)
print(VIFs)## lotsize sqrft bdrms
## 1.037211 1.418654 1.396663- Cálculo de los VIF’s usando “performance”
library(performance)
VIFs<-multicollinearity(x = estimacion_modelo,verbose = FALSE)
VIFs## # Check for Multicollinearity
##
## Low Correlation
##
## Term VIF VIF 95% CI Increased SE Tolerance Tolerance 95% CI
## lotsize 1.04 [1.00, 11.02] 1.02 0.96 [0.09, 1.00]
## sqrft 1.42 [1.18, 1.98] 1.19 0.70 [0.51, 0.85]
## bdrms 1.40 [1.17, 1.95] 1.18 0.72 [0.51, 0.86]plot(VIFs)
- Cálculo de los VIF’s usando “car”
library(car)
VIFs_car<-vif(estimacion_modelo)
VIFs_car## lotsize sqrft bdrms
## 1.037211 1.418654 1.396663- Cálculo de los VIF’s usando “mctest”
library(mctest)
mc.plot(mod = estimacion_modelo,vif = 2)