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 652A = \((X'.X)^{-1}*X'\)
#Matriz A
A<-solve(t(X)%*%X)%*%t(X)
head(A,10)## [,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#Matriz P
P<-X%*%A
N <- nrow(P)
Iden<-diag(x=1,N,N)
head(P,10)## [,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,] 0.08405176 0.07119228 0.08002118 0.08654688 0.07579866 0.05372643
## [8,] 0.10237642 0.08395731 0.09660327 0.10595028 0.09055520 0.05894031
## [9,] 0.07377207 0.06403141 0.07071903 0.07566205 0.06752060 0.05080156
## [10,] 0.06528015 0.05811591 0.06303464 0.06667023 0.06068221 0.04838537
## [,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
## [7,] 0.05986826 0.06773750 0.05545382 0.05180710 0.04796845 0.04374594
## [8,] 0.06773750 0.07900889 0.06141452 0.05619119 0.05069295 0.04464488
## [9,] 0.05545382 0.06141452 0.05211001 0.04934773 0.04644007 0.04324165
## [10,] 0.05180710 0.05619119 0.04934773 0.04731608 0.04517750 0.04282506
## [,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
## [7,] 0.04259434 0.04700879 0.0193705086 0.0157237913 0.028391336
## [8,] 0.04299541 0.04931839 0.0097310449 0.0045077151 0.022651913
## [9,] 0.04236935 0.04571316 0.0247780127 0.0220157364 0.031611012
## [10,] 0.04218349 0.04464286 0.0292450813 0.0272134303 0.034270744
## [,18] [,19] [,20] [,21] [,22]
## [1,] 0.0009203652 0.01298783 -0.043773933 -0.065674138 -0.032153415
## [2,] 0.0132826393 0.02168888 -0.017851577 -0.033107343 -0.009756681
## [3,] 0.0047951078 0.01571502 -0.035649015 -0.055466635 -0.025133543
## [4,] -0.0014782850 0.01129956 -0.048803643 -0.071993069 -0.036499050
## [5,] 0.0088543620 0.01857208 -0.027137197 -0.044773061 -0.017779392
## [6,] 0.0300731907 0.03350672 0.017356398 0.011125170 0.020662764
## [7,] 0.0241688210 0.02935100 0.004975572 -0.004429121 0.009965816
## [8,] 0.0166038473 0.02402647 -0.010887362 -0.024358055 -0.003739648
## [9,] 0.0284125867 0.03233793 0.013874291 0.006750526 0.017654248
## [10,] 0.0319183062 0.03480539 0.021225406 0.015985885 0.024005560
## [,23] [,24]
## [1,] -0.044667818 -0.0545005640
## [2,] -0.018474261 -0.0253237891
## [3,] -0.036457898 -0.0453556047
## [4,] -0.049750150 -0.0601617293
## [5,] -0.027857028 -0.0357751712
## [6,] 0.017102062 0.0143043678
## [7,] 0.004591707 0.0003691918
## [8,] -0.011437186 -0.0174852526
## [9,] 0.013583525 0.0103850996
## [10,] 0.021011548 0.0186591104#Matriz M
M<- (Iden-P)
head(M,10)## [,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
## [7,] -0.08405176 -0.07119228 -0.08002118 -0.08654688 -0.07579866
## [8,] -0.10237642 -0.08395731 -0.09660327 -0.10595028 -0.09055520
## [9,] -0.07377207 -0.06403141 -0.07071903 -0.07566205 -0.06752060
## [10,] -0.06528015 -0.05811591 -0.06303464 -0.06667023 -0.06068221
## [,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
## [7,] -0.05372643 0.94013174 -0.06773750 -0.05545382 -0.05180710
## [8,] -0.05894031 -0.06773750 0.92099111 -0.06141452 -0.05619119
## [9,] -0.05080156 -0.05545382 -0.06141452 0.94788999 -0.04934773
## [10,] -0.04838537 -0.05180710 -0.05619119 -0.04934773 0.95268392
## [,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
## [7,] -0.04796845 -0.04374594 -0.04259434 -0.04700879 -0.0193705086
## [8,] -0.05069295 -0.04464488 -0.04299541 -0.04931839 -0.0097310449
## [9,] -0.04644007 -0.04324165 -0.04236935 -0.04571316 -0.0247780127
## [10,] -0.04517750 -0.04282506 -0.04218349 -0.04464286 -0.0292450813
## [,16] [,17] [,18] [,19] [,20]
## [1,] 0.0187451258 -0.010753111 -0.0009203652 -0.01298783 0.043773933
## [2,] 0.0004164159 -0.020132167 -0.0132826393 -0.02168888 0.017851577
## [3,] 0.0130003063 -0.013692815 -0.0047951078 -0.01571502 0.035649015
## [4,] 0.0223014426 -0.008933294 0.0014782850 -0.01129956 0.048803643
## [5,] 0.0069819239 -0.016772505 -0.0088543620 -0.01857208 0.027137197
## [6,] -0.0244778019 -0.032870885 -0.0300731907 -0.03350672 -0.017356398
## [7,] -0.0157237913 -0.028391336 -0.0241688210 -0.02935100 -0.004975572
## [8,] -0.0045077151 -0.022651913 -0.0166038473 -0.02402647 0.010887362
## [9,] -0.0220157364 -0.031611012 -0.0284125867 -0.03233793 -0.013874291
## [10,] -0.0272134303 -0.034270744 -0.0319183062 -0.03480539 -0.021225406
## [,21] [,22] [,23] [,24]
## [1,] 0.065674138 0.032153415 0.044667818 0.0545005640
## [2,] 0.033107343 0.009756681 0.018474261 0.0253237891
## [3,] 0.055466635 0.025133543 0.036457898 0.0453556047
## [4,] 0.071993069 0.036499050 0.049750150 0.0601617293
## [5,] 0.044773061 0.017779392 0.027857028 0.0357751712
## [6,] -0.011125170 -0.020662764 -0.017102062 -0.0143043678
## [7,] 0.004429121 -0.009965816 -0.004591707 -0.0003691918
## [8,] 0.024358055 0.003739648 0.011437186 0.0174852526
## [9,] -0.006750526 -0.017654248 -0.013583525 -0.0103850996
## [10,] -0.015985885 -0.024005560 -0.021011548 -0.0186591104#Residuos 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,] 0.64215354Prueba de jarque Bera
library(normtest)
jb.norm.test(u_i)##
## Jarque-Bera test for normality
##
## data: u_i
## JB = 1.5606, p-value = 0.2205En caso de la prueba de Jarque-Bera la condicion de no rechazar de la Ho se puede evaluar por medio del p-value en la cual no se rechasa si p > \(\alpha\) y cuando el estadístico JB<V.C. JB(1.5606) > V.C(5.9915) p (0.2235) < \(\alpha\) (0.05)
SE RECHAZA LA HIPOTESIS NULA POR LO QUE NO HAY EVIDENCIA QUE LOS RESIDUOS SIGUEN UNA DISTRIBUCION NORMAL
Prueba de Kolmogorov Smirnov
library(nortest)
lillie.test(u_i)##
## Lilliefors (Kolmogorov-Smirnov) normality test
##
## data: u_i
## D = 0.14418, p-value = 0.2209-En caso de la prueba de Kolmogorov-Smirnov para un nivel de significancia del 5% y una muesta n=24 el V.C. = 0.1788 la condicion de no rechazar de la Ho es que el estadistico D < V.C., además también se puede evaluar por medio del p-value en la cual la condición de no rechazo es p-value > \(\alpha\) D (0.14418) < V.C.(0.1788) p-value (0.2209) > \(\alpha\)(0.05)
Prueba de Shapiro - Wilk
shapiro.test(u_i)##
## Shapiro-Wilk normality test
##
## data: u_i
## W = 0.95746, p-value = 0.3895SW (0.95746) < V.C.(1.644854) p-value (0.3895) > \(\alpha\)(0.05)
#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("Prn","Pac","rezago","nini", "educ", "ips")
rownames(R)<-c("Prn","Pac","rezago","nini", "educ", "ips")
print(R)## Prn Pac rezago nini educ ips
## Prn 1.00 -0.80 0.68 0.74 -0.57 -0.75
## Pac -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 -0.57 0.69 -0.77 -0.80 1.00 0.64
## ips -0.75 0.80 -0.82 -0.84 0.64 1.00determinante_R<- det(R)det_R<-det(R)
m<-2
chi_FG<--(N-1-(2*m+5)/6)*log(det_R)
print(chi_FG)## [1] 137.3059Calculo del valor crítico
gl<-m*(m-1)/2
VC<-qchisq(p = 0.95,df = gl)
print(VC)## [1] 3.841459Como \(X^2_{FG}\) > V.C. se rechaza H0, por lo tanto hay evidencia de colinealidad en los regresores
#Inversa de R
R_inv<-solve(R)
print(R_inv)## Prn Pac rezago nini educ ips
## Prn 3.12989194 1.8066302 -0.5430948 0.01312674 -0.2872837 0.6516651
## Pac 1.80663016 5.5207504 -0.8728955 3.04065276 -0.4001754 -0.9671414
## rezago -0.54309484 -0.8728955 4.6100427 -0.68684569 1.7897206 2.3488587
## nini 0.01312674 3.0406528 -0.6868457 7.70048852 2.2224574 2.0601470
## educ -0.28728369 -0.4001754 1.7897206 2.22245739 3.5016734 1.1980416
## ips 0.65166508 -0.9671414 2.3488587 2.06014702 1.1980416 5.1523030Obteniendo los VIF’s en la diagonal de la inversa de R
VIFs<-diag(R_inv)
print(VIFs)## Prn Pac rezago nini educ ips
## 3.129892 5.520750 4.610043 7.700489 3.501673 5.152303Las variables que se consideran colineales son Cuando VIF>2, se consideran variables colineales. Cuando VIF>5 o VIF>10, se consideran variables altamente colineales. - Respuesta: En este caso todas las variables se consideran colineales porque ningun VIF<2