EJERCICIO MULTICOLIEALIDAD

Datos del modelo

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

1. Estimación del modelo

options(scipen= 99999999)
library(stargazer)
modelo_es<- lm(price~lotsize+sqrft+bdrms,data = hprice1)
stargazer(modelo_es, title = "Modelo estimado", type = "text", digits = 4)
## 
## Modelo estimado
## ===============================================
##                         Dependent variable:    
##                     ---------------------------
##                                price           
## -----------------------------------------------
## lotsize                      0.0021***         
##                              (0.0006)          
##                                                
## sqrft                        0.1228***         
##                              (0.0132)          
##                                                
## bdrms                         13.8525          
##                              (9.0101)          
##                                                
## Constant                     -21.7703          
##                              (29.4750)         
##                                                
## -----------------------------------------------
## Observations                    88             
## R2                            0.6724           
## Adjusted R2                   0.6607           
## Residual Std. Error      59.8335 (df = 84)     
## F Statistic           57.4602*** (df = 3; 84)  
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01

2. Verificar si hay evidencia de la independencia de los regresores

a) Indice de condición y prueba FG

Indice de condición

Calculo manual

Matriz X

library(stargazer)
matriz_X<-model.matrix(modelo_es)
stargazer(head(matriz_X,n=6),type="text")
## 
## =================================
##   (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  
## ---------------------------------

Matriz XX

matriz_XX <-t(matriz_X)%*%matriz_X
stargazer(matriz_XX, type = "text")
## 
## ==============================================================
##             (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(matriz_XX))))
stargazer(Sn,type = "text")
## 
## ==========================
## 0.107    0      0      0  
## 0     0.00001   0      0  
## 0        0    0.0001   0  
## 0        0      0    0.029
## --------------------------

Matriz \({X^t X}\) normalizada

library(stargazer)
norma_XX<-(Sn%*%matriz_XX)%*%Sn
stargazer(norma_XX,type = "text",digits = 4)
## 
## ===========================
## 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   
## ---------------------------

Autovalores \({X^t X}\) normalizada

library(stargazer)
lambdas<-eigen(norma_XX,symmetric = TRUE)
stargazer(lambdas$values,type = "text")
## 
## =======================
## 3.482 0.455 0.039 0.025
## -----------------------

Cálculo de \({\kappa\left(x\right)=\sqrt{\frac{\lambda_{max}}{\lambda_{min}}}}\)

K<-sqrt(max(lambdas$values)/min(lambdas$values))
print(K)
## [1] 11.86778

K(X) ≤ 20 se considera que la multicolinealidad es leve

Cálculo del Indice de Condición usando librería “mctest”

library(mctest)
mat_x<-model.matrix(modelo_es)
mctest(mod = modelo_es)
## 
## 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 test

Cálculo del Indice de Condición usando librería “olsrr”

library(olsrr)
ols_eigen_cindex(model = modelo_es)
##   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 0.975025872

Prueba de Farrar-Glaubar

Cálculo manual

Calculo de |R|

library(stargazer)
Zn<-scale(matriz_X[,-1])
stargazer(head(Zn,n=6),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 
## -----------------------

Calcular la matriz R

library(stargazer)
n<-nrow(Zn)
R<-cor(matriz_X[,-1])
stargazer(R,type = "text",digits = 4)
## 
## =============================
##         lotsize sqrft  bdrms 
## -----------------------------
## lotsize    1    0.1838 0.1363
## sqrft   0.1838    1    0.5315
## bdrms   0.1363  0.5315   1   
## -----------------------------

Calcular |R|

det_R<-det(R)
print(det_R)
## [1] 0.6917931

Aplicando la prueba de Farrer Glaubar (Bartlett)

Estadistico \(\chi_{FG}^2\)

m<-ncol(matriz_X[,-1])
n<-nrow(matriz_X[,-1])
chi_FG<--(n-1-(2*m+5)/6)*log(det_R)
print(chi_FG)
## [1] 31.38122

Valor critico

grados_l<-m*(m-1)/2
VC<-qchisq(p = 0.95,df = grados_l)
print(VC)
## [1] 7.814728

Dado que \(\chi^2_{FG} \geq V.C.\) se rechaza Ho, por lo tanto hay evidencia de colinealidad en los regresores.

Cálculo de FG usando “mctest”

library(mctest)
mctest::omcdiag(mod = modelo_es)
## 
## Call:
## mctest::omcdiag(mod = modelo_es)
## 
## 
## 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 test

Cálculo de FG usando la “psych”

library(psych)
test_FG<-cortest.bartlett(matriz_X[,-1])
print(test_FG)
## $chisq
## [1] 31.38122
## 
## $p.value
## [1] 0.0000007065806
## 
## $df
## [1] 3

Grafica FG

library(fastGraph)
alpha_sig<-0.05
chi_FG<--(n-1-(2*m+5)/6)*log(det_R)
gl<-grados_l
vc<-qchisq(1-alpha_sig,gl,lower.tail=TRUE)
shadeDist(chi_FG,ddist = "dchisq",
          parm1 = gl,
          lower.tail = FALSE, xmin = 0,
          sub=paste("VC:",round(VC,2),"","chi_FG:",round(chi_FG,2)))

b) Factores inflacionarios de la varianza

Referencia entre \(R^2_j\)

library(dplyr)
R.cuadrado.regresores<-c(0,0.5,.8,.9)
as.data.frame(R.cuadrado.regresores) %>% mutate(VIF=1/(1-R.cuadrado.regresores))
##   R.cuadrado.regresores VIF
## 1                   0.0   1
## 2                   0.5   2
## 3                   0.8   5
## 4                   0.9  10

Calculo manual

Matriz de Correlación de los regresores del modelo

print(R)
##           lotsize     sqrft     bdrms
## lotsize 1.0000000 0.1838422 0.1363256
## sqrft   0.1838422 1.0000000 0.5314736
## bdrms   0.1363256 0.5314736 1.0000000

Inversa de la matriz de correlación \(R^{-1}\)

inver_R<-solve(R)
print(inver_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(inver_R)
print(VIFs)
##  lotsize    sqrft    bdrms 
## 1.037211 1.418654 1.396663

Cálculo de los VIF’s usando “performance”

library(performance)
VIFs<-multicollinearity(x = modelo_es,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(modelo_es)
print(VIFs_car)
##  lotsize    sqrft    bdrms 
## 1.037211 1.418654 1.396663

Cálculo de los VIF’s usando “mctest”

library(mctest)
mc.plot(mod = modelo_es,vif = 2)