Parcial 2

Ejercicio 1

insercion de datos

library(dplyr)
X<-matrix(data = c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
                   1,1,1,216,283,237,203,259,374,342,301,365,
                   384,404,426,432,409,553,572,506,528,501,
                   628,677,602,630,652),
          nrow = 24, ncol = 2, byrow = FALSE)
colnames(X) <-c("cte","x")
Y<-matrix(data = c(6.1,9.1,7.2,7.5,6.9,11.5,10.3,9.5,9.2,
                   10.6,12.5,12.9,13.6,12.8,16.5,17.1,15,
                   16.2,15.8,19,19.4,19.1,18,20.2),
          nrow = 24)
XY<- cbind(Y,X)
print(XY)

##            cte   x
##  [1,]  6.1   1 216
##  [2,]  9.1   1 283
##  [3,]  7.2   1 237
##  [4,]  7.5   1 203
##  [5,]  6.9   1 259
##  [6,] 11.5   1 374
##  [7,] 10.3   1 342
##  [8,]  9.5   1 301
##  [9,]  9.2   1 365
## [10,] 10.6   1 384
## [11,] 12.5   1 404
## [12,] 12.9   1 426
## [13,] 13.6   1 432
## [14,] 12.8   1 409
## [15,] 16.5   1 553
## [16,] 17.1   1 572
## [17,] 15.0   1 506
## [18,] 16.2   1 528
## [19,] 15.8   1 501
## [20,] 19.0   1 628
## [21,] 19.4   1 677
## [22,] 19.1   1 602
## [23,] 18.0   1 630
## [24,] 20.2   1 652

Obtencion matriz A

(X’.X)^{-1}*X’

A<-solve(t(X)%*%X)%*%t(X)
print(A)

##             [,1]          [,2]          [,3]          [,4]          [,5]
## cte  0.236906257  0.1776713023  0.2183400774  0.2483996068  0.1988897936
## x   -0.000446943 -0.0003113422 -0.0004044412 -0.0004732536 -0.0003599156
##              [,6]          [,7]          [,8]         [,9]        [,10]
## cte  0.0972178559  0.1255091777  0.1617574338  0.105174790  0.088376818
## x   -0.0001271679 -0.0001919325 -0.0002749121 -0.000145383 -0.000106929
##             [,11]         [,12]         [,13]         [,14]         [,15]
## cte  7.069474e-02  0.0512444579  4.593984e-02  6.627422e-02 -0.0610367256
## x   -6.645114e-05 -0.0000219255 -9.782148e-06 -5.633168e-05  0.0002351089
##             [,16]         [,17]         [,18]         [,19]         [,20]
## cte -0.0778346979 -0.0194838466 -0.0389341304 -0.0150633276 -0.1273445111
## x    0.0002735628  0.0001399859  0.0001845116  0.0001298665  0.0003869008
##             [,21]         [,22]         [,23]         [,24]
## cte -0.1706655976 -0.1043578121 -0.1291127187 -0.1485630024
## x    0.0004860716  0.0003342796  0.0003909486  0.0004354743

Obtencion de matriz P

P<-X%*%A
N <- nrow(P)
Iden<-diag(x=1,N,N)
head(P,6)

##            [,1]       [,2]       [,3]       [,4]       [,5]       [,6]
## [1,] 0.14036657 0.11042139 0.13098077 0.14617683 0.12114803 0.06974958
## [2,] 0.11042139 0.08956147 0.10388321 0.11446884 0.09703368 0.06122933
## [3,] 0.13098077 0.10388321 0.12248751 0.13623851 0.11358980 0.06707906
## [4,] 0.14617683 0.11446884 0.13623851 0.15232913 0.12582693 0.07140277
## [5,] 0.12114803 0.09703368 0.11358980 0.12582693 0.10567166 0.06428136
## [6,] 0.06974958 0.06122933 0.06707906 0.07140277 0.06428136 0.04965705
##            [,7]       [,8]       [,9]      [,10]      [,11]      [,12]
## [1,] 0.08405176 0.10237642 0.07377207 0.06528015 0.05634129 0.04650855
## [2,] 0.07119228 0.08395731 0.06403141 0.05811591 0.05188907 0.04503954
## [3,] 0.08002118 0.09660327 0.07071903 0.06303464 0.05494582 0.04604811
## [4,] 0.08654688 0.10595028 0.07566205 0.06667023 0.05720516 0.04679358
## [5,] 0.07579866 0.09055520 0.06752060 0.06068221 0.05348390 0.04556575
## [6,] 0.05372643 0.05894031 0.05080156 0.04838537 0.04584201 0.04304432
##           [,13]      [,14]         [,15]         [,16]       [,17]
## [1,] 0.04382689 0.05410658 -0.0102532092 -0.0187451258 0.010753111
## [2,] 0.04317149 0.05033236  0.0054990852 -0.0004164159 0.020132167
## [3,] 0.04362147 0.05292361 -0.0053159229 -0.0130003063 0.013692815
## [4,] 0.04395406 0.05483889 -0.0133096246 -0.0223014426 0.008933294
## [5,] 0.04340626 0.05168432 -0.0001435277 -0.0069819239 0.016772505
## [6,] 0.04228131 0.04520617  0.0268939925  0.0244778019 0.032870885
##              [,18]      [,19]       [,20]       [,21]        [,22]
## [1,]  0.0009203652 0.01298783 -0.04377393 -0.06567414 -0.032153415
## [2,]  0.0132826393 0.02168888 -0.01785158 -0.03310734 -0.009756681
## [3,]  0.0047951078 0.01571502 -0.03564902 -0.05546664 -0.025133543
## [4,] -0.0014782850 0.01129956 -0.04880364 -0.07199307 -0.036499050
## [5,]  0.0088543620 0.01857208 -0.02713720 -0.04477306 -0.017779392
## [6,]  0.0300731907 0.03350672  0.01735640  0.01112517  0.020662764
##            [,23]       [,24]
## [1,] -0.04466782 -0.05450056
## [2,] -0.01847426 -0.02532379
## [3,] -0.03645790 -0.04535560
## [4,] -0.04975015 -0.06016173
## [5,] -0.02785703 -0.03577517
## [6,]  0.01710206  0.01430437

Obtencion la matriz M

M<- (Iden-P)
head(M,6)

##             [,1]        [,2]        [,3]        [,4]        [,5]
## [1,]  0.85963343 -0.11042139 -0.13098077 -0.14617683 -0.12114803
## [2,] -0.11042139  0.91043853 -0.10388321 -0.11446884 -0.09703368
## [3,] -0.13098077 -0.10388321  0.87751249 -0.13623851 -0.11358980
## [4,] -0.14617683 -0.11446884 -0.13623851  0.84767087 -0.12582693
## [5,] -0.12114803 -0.09703368 -0.11358980 -0.12582693  0.89432834
## [6,] -0.06974958 -0.06122933 -0.06707906 -0.07140277 -0.06428136
##             [,6]        [,7]        [,8]        [,9]       [,10]
## [1,] -0.06974958 -0.08405176 -0.10237642 -0.07377207 -0.06528015
## [2,] -0.06122933 -0.07119228 -0.08395731 -0.06403141 -0.05811591
## [3,] -0.06707906 -0.08002118 -0.09660327 -0.07071903 -0.06303464
## [4,] -0.07140277 -0.08654688 -0.10595028 -0.07566205 -0.06667023
## [5,] -0.06428136 -0.07579866 -0.09055520 -0.06752060 -0.06068221
## [6,]  0.95034295 -0.05372643 -0.05894031 -0.05080156 -0.04838537
##            [,11]       [,12]       [,13]       [,14]         [,15]
## [1,] -0.05634129 -0.04650855 -0.04382689 -0.05410658  0.0102532092
## [2,] -0.05188907 -0.04503954 -0.04317149 -0.05033236 -0.0054990852
## [3,] -0.05494582 -0.04604811 -0.04362147 -0.05292361  0.0053159229
## [4,] -0.05720516 -0.04679358 -0.04395406 -0.05483889  0.0133096246
## [5,] -0.05348390 -0.04556575 -0.04340626 -0.05168432  0.0001435277
## [6,] -0.04584201 -0.04304432 -0.04228131 -0.04520617 -0.0268939925
##              [,16]        [,17]         [,18]       [,19]       [,20]
## [1,]  0.0187451258 -0.010753111 -0.0009203652 -0.01298783  0.04377393
## [2,]  0.0004164159 -0.020132167 -0.0132826393 -0.02168888  0.01785158
## [3,]  0.0130003063 -0.013692815 -0.0047951078 -0.01571502  0.03564902
## [4,]  0.0223014426 -0.008933294  0.0014782850 -0.01129956  0.04880364
## [5,]  0.0069819239 -0.016772505 -0.0088543620 -0.01857208  0.02713720
## [6,] -0.0244778019 -0.032870885 -0.0300731907 -0.03350672 -0.01735640
##            [,21]        [,22]       [,23]       [,24]
## [1,]  0.06567414  0.032153415  0.04466782  0.05450056
## [2,]  0.03310734  0.009756681  0.01847426  0.02532379
## [3,]  0.05546664  0.025133543  0.03645790  0.04535560
## [4,]  0.07199307  0.036499050  0.04975015  0.06016173
## [5,]  0.04477306  0.017779392  0.02785703  0.03577517
## [6,] -0.01112517 -0.020662764 -0.01710206 -0.01430437

Obtencion de los residuos del modelo

E = (I-P)*Y

u_i<- M%*%Y
print(u_i)

##              [,1]
##  [1,] -0.50716765
##  [2,]  0.50270510
##  [3,] -0.03093888
##  [4,]  1.27897644
##  [5,] -0.98441350
##  [6,]  0.19969645
##  [7,] -0.04979501
##  [8,]  0.36804405
##  [9,] -1.83297302
## [10,] -0.99733747
## [11,]  0.30859470
## [12,]  0.05512008
## [13,]  0.57689973
## [14,]  0.46007774
## [15,] -0.11721068
## [16,] -0.08157512
## [17,] -0.22115126
## [18,]  0.32537412
## [19,]  0.72736569
## [20,]  0.15503494
## [21,] -0.90043126
## [22,]  1.02732313
## [23,] -0.90437184
## [24,]  0.64215354

Pruebas de normalidad

Aplicacion Jarque Bera

5% de nivel de significancia

library(normtest)
jb.norm.test(u_i)

## 
##  Jarque-Bera test for normality
## 
## data:  u_i
## JB = 1.5606, p-value = 0.2495

P-value(0.238)>Nivel de significancia(0.05), por lo que se concluye que no debemos rechazar la hipótesis nula(Ho), por lo que existe evidencia que los residuos siguen una distribución normal.

Prueba de Kolmogorov-Smirnov

5% de nivel de significancia

library(nortest)
lillie.test(u_i)

## 
##  Lilliefors (Kolmogorov-Smirnov) normality test
## 
## data:  u_i
## D = 0.14418, p-value = 0.2209

P-value(0.2209)>Nivel de significancia(0.05), por lo que se concluye que no debemos rechazar la hipótesis nula(Ho), por lo que existe evidencia que los residuos siguen una distribución normal.

Prueba de Shapiro - Wilk

5% de nivel de significancia

shapiro.test(u_i)

## 
##  Shapiro-Wilk normality test
## 
## data:  u_i
## W = 0.95746, p-value = 0.3895

P-value(0.3895)>Nivel de significancia(0.05), por lo que se concluye que no debemos rechazar la hipótesis nula(Ho), por lo que existe evidencia que los residuos siguen una distribución normal.

Ejercicio 2

Insercion de datos Matriz R

R<-matrix(data = c(1,-0.8,0.68,0.74,-0.57,-0.75,-0.8, 1, -0.71, -0.87, 0.69,0.8,
                   0.68,-0.71,1,.81,-.77,-.82,.74,-.87,.81,1,.8,-.84,-.57,.69,-.77,
                   -.8,1,.64,-.75,.8,-.82,-.84,.64,1),
          nrow = 6, ncol = 6, byrow = TRUE)
colnames(R) <-c("Pprn","Ppac","rezago","nini", "educ_s", "ips")
rownames(R)<-c("Pprn","Ppac","rezago","nini", "educ_s", "ips")
print(R)

##         Pprn  Ppac rezago  nini educ_s   ips
## Pprn    1.00 -0.80   0.68  0.74  -0.57 -0.75
## Ppac   -0.80  1.00  -0.71 -0.87   0.69  0.80
## rezago  0.68 -0.71   1.00  0.81  -0.77 -0.82
## nini    0.74 -0.87   0.81  1.00   0.80 -0.84
## educ_s -0.57  0.69  -0.77 -0.80   1.00  0.64
## ips    -0.75  0.80  -0.82 -0.84   0.64  1.00

Calcular \(|R|\)

determinante_R<-det(R)
print(determinante_R)

## [1] 0.007674191

Aplicando la prueba de Farrer Glaubar (Bartlett)

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

m<-ncol(R[,-1])
n<-nrow(R[,-1])
chi_FG<--(n-1-(2*m+5)/6)*log(determinante_R)
print(chi_FG)

## [1] 12.17473

Valor Critico

gl<-m*(m-1)/2
VC<-qchisq(p = 0.95,df = gl)
print(VC)

## [1] 18.30704

Como \(\chi_{FG}^2 < V.C.\) No se rechaza Ho, por lo tanto no hay evidencia de colinealidad en los regresores

Cálculando los VIF para el modelo estimado

Matriz de Correlación de los regresores del modelo

print(R)

##         Pprn  Ppac rezago  nini educ_s   ips
## Pprn    1.00 -0.80   0.68  0.74  -0.57 -0.75
## Ppac   -0.80  1.00  -0.71 -0.87   0.69  0.80
## rezago  0.68 -0.71   1.00  0.81  -0.77 -0.82
## nini    0.74 -0.87   0.81  1.00   0.80 -0.84
## educ_s -0.57  0.69  -0.77 -0.80   1.00  0.64
## ips    -0.75  0.80  -0.82 -0.84   0.64  1.00

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

inversa_R<-solve(R)
print(inversa_R)

##               Pprn        Ppac     rezago         nini     educ_s
## Pprn    3.13121631  1.80847497 -0.5513454  0.002881242 -0.3034264
## Ppac    2.11340559  5.94807701 -2.7840457  0.667405238 -4.1394317
## rezago -0.61239160 -0.96942324  5.0417478 -0.150758552  2.6343721
## nini    0.79003908  4.12286240 -5.5268558  1.690211537 -7.2472527
## educ_s -0.06305707 -0.08783612  0.3928330  0.487816209  0.7685965
## ips     0.85951600 -0.67761291  1.0539886  0.452190045 -1.3354334
##               ips
## Pprn    0.6461421
## Ppac   -2.2464682
## rezago  2.6378428
## nini   -1.1797630
## educ_s  0.2629630
## ips     4.2855150

VIF’s para el modelo estimado

VIFs<-diag(inversa_R)
print(VIFs)

##      Pprn      Ppac    rezago      nini    educ_s       ips 
## 3.1312163 5.9480770 5.0417478 1.6902115 0.7685965 4.2855150