Prueba de Normalidad: Resultados del Modelo

                    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

Ajuste de los residuos a la distribucion normal

library(fitdistrplus)

## Warning: package 'fitdistrplus' was built under R version 4.5.3

## Cargando paquete requerido: MASS

## 
## Adjuntando el paquete: 'MASS'

## The following object is masked from 'package:wooldridge':
## 
##     cement

## Cargando paquete requerido: survival

fit_normal<-fitdist(data = modelo_estimado$residuals,distr = "norm")
plot(fit_normal)

summary(fit_normal)

## Fitting of the distribution ' norm ' by maximum likelihood 
## Parameters : 
##           estimate Std. Error
## mean -2.321494e-15   6.231625
## sd    5.845781e+01   4.406424
## Loglikelihood:  -482.8775   AIC:  969.7549   BIC:  974.7096 
## Correlation matrix:
##      mean sd
## mean    1  0
## sd      0  1

library(dplyr)  # Carga la librería dplyr para manipulación de datos

## Warning: package 'dplyr' was built under R version 4.5.3

## 
## Adjuntando el paquete: 'dplyr'

## The following object is masked from 'package:MASS':
## 
##     select

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(gt)  # Carga la librería gt para crear tablas de datos

## Warning: package 'gt' was built under R version 4.5.3

library(gtExtras)  # Carga la librería gtExtras para agregar funcionalidades a las tablas creadas con gt

## Warning: package 'gtExtras' was built under R version 4.5.3

## 
## Adjuntando el paquete: 'gtExtras'

## The following object is masked from 'package:MASS':
## 
##     select

residuos<-modelo_estimado$residuals  # Crea un vector con los residuos del modelo estimado
residuos %>%  # Utiliza el operador %>% para encadenar las operaciones siguientes al vector residuos
  as_tibble() %>%  # Convierte el vector residuos en una tibble (tabla) de una columna
  mutate(posicion=row_number()) %>%  # Agrega una columna llamada "posicion" con el número de fila
  arrange(value) %>%  # Ordena la tabla por los valores de residuos en orden ascendente
  mutate(dist1=row_number()/n()) %>%  # Agrega una columna "dist1" con los percentiles según su posición en la tabla (usando la función row_number() y n() para obtener el número de filas)
  mutate(dist2=(row_number()-1)/n()) %>%  # Agrega una columna "dist2" con los percentiles según su posición en la tabla, pero ajustando en una unidad para evitar problemas con los extremos de la distribución
  mutate(zi=as.vector(scale(value,center=TRUE))) %>%  # Agrega una columna "zi" con los valores de residuos escalados para tener media cero y varianza uno
  mutate(pi=pnorm(zi,lower.tail = TRUE)) %>%  # Agrega una columna "pi" con los valores de la función de distribución acumulada (CDF) de una distribución normal estándar evaluada en los valores de zi
  mutate(dif1=abs(dist1-pi)) %>%  # Agrega una columna "dif1" con las diferencias absolutas entre los percentiles según la posición y los valores de pi
  mutate(dif2=abs(dist2-pi)) %>%  # Agrega una columna "dif2" con las diferencias absolutas entre los percentiles ajustados según la posición y los valores de pi
  rename(residuales=value) -> tabla_JB  # Renombra la columna "value" como "residuales" y asigna la tabla resultante a la variable tabla_JB


#Formato
 tabla_JB %>%  # Utiliza el operador %>% para encadenar las operaciones siguientes a la tabla tabla_KS
  gt() %>%  # Crea una tabla con la función gt()
  tab_header("Tabla para calcular el Estadistico JB") %>%  # Agrega un encabezado a la tabla
  tab_source_note(source_note = "Fuente: Elaboración propia") %>%  # Agrega una nota de fuente a la tabla
  tab_style(  # Cambia el estilo de algunas celdas de la tabla
    style = list(
      cell_fill(color = "#A569BD"),  # Cambia el color de fondo de las celdas a un tono de morado
      cell_text(style = "italic")  # Cambia el estilo de texto de las celdas a itálico
      ),
    locations = cells_body(  # Aplica el estilo a las celdas del cuerpo de la tabla que cumplan las siguientes condiciones:
      columns = dif1,  # Que pertenezcan a la columna "dif1"
      rows = dif1==max(dif1)  # Que pertenezcan a la fila donde el valor de "dif1" es máximo
    )) %>%
   tab_style(  # Cambia el estilo de algunas celdas de la tabla
    style = list(
      cell_fill(color = "#3498DB"),  # Cambia el color de fondo de las celdas a un tono de azul
      cell_text(style = "italic")  # Cambia el estilo de texto de las celdas a itálico
      ),
    locations = cells_body(  # Aplica el estilo a las celdas del cuerpo de la tabla que cumplan las siguientes condiciones:
      columns = dif2,  # Que pertenezcan a la columna "dif2"
      rows = dif2==max(dif2)  # Que pertenezcan a la fila donde el valor de "dif2" es máximo
    ))

residuales	posicion	dist1	dist2	zi	pi	dif1	dif2
Tabla para calcular el Estadistico JB
-120.026447	81	0.01136364	0.00000000	-2.041515459	0.02059981	0.0092361731	0.0205998094
-115.508697	77	0.02272727	0.01136364	-1.964673586	0.02472601	0.0019987418	0.0133623781
-107.080889	24	0.03409091	0.02272727	-1.821326006	0.03427866	0.0001877487	0.0115513850
-91.243980	48	0.04545455	0.03409091	-1.551957925	0.06033615	0.0148816002	0.0262452366
-85.461169	12	0.05681818	0.04545455	-1.453598781	0.07302879	0.0162106057	0.0275742421
-77.172687	32	0.06818182	0.05681818	-1.312620980	0.09465535	0.0264735301	0.0378371665
-74.702719	54	0.07954545	0.06818182	-1.270609602	0.10193378	0.0223883300	0.0337519664
-65.502849	39	0.09090909	0.07954545	-1.114130117	0.13261169	0.0417025941	0.0530662305
-63.699108	69	0.10227273	0.09090909	-1.083450505	0.13930425	0.0370315271	0.0483951634
-62.566594	83	0.11363636	0.10227273	-1.064187703	0.14362184	0.0299854747	0.0413491110
-59.845223	36	0.12500000	0.11363636	-1.017900230	0.15436269	0.0293626861	0.0407263225
-54.466158	13	0.13636364	0.12500000	-0.926408352	0.17711690	0.0407532663	0.0521169027
-54.300415	14	0.14772727	0.13636364	-0.923589260	0.17785010	0.0301228311	0.0414864675
-52.129801	15	0.15909091	0.14772727	-0.886669532	0.18762842	0.0285375141	0.0399011505
-51.441108	17	0.17045455	0.15909091	-0.874955638	0.19079902	0.0203444766	0.0317081129
-48.704980	47	0.18181818	0.17045455	-0.828417174	0.20371714	0.0218989601	0.0332625965
-48.350295	29	0.19318182	0.18181818	-0.822384375	0.20542908	0.0122472664	0.0236109028
-47.855859	11	0.20454545	0.19318182	-0.813974573	0.20782976	0.0032843043	0.0146479407
-45.639765	1	0.21590909	0.20454545	-0.776281294	0.21879146	0.0028823668	0.0142460032
-43.142550	9	0.22727273	0.21590909	-0.733806463	0.23153335	0.0042606233	0.0156242596
-41.749618	57	0.23863636	0.22727273	-0.710114247	0.23881665	0.0001802823	0.0115439187
-40.869022	27	0.25000000	0.23863636	-0.695136302	0.24348494	0.0065150566	0.0048485798
-37.749811	34	0.26136364	0.25000000	-0.642082009	0.26040997	0.0009536682	0.0104099682
-36.663785	71	0.27272727	0.26136364	-0.623609925	0.26644190	0.0062853771	0.0050782592
-36.646568	79	0.28409091	0.27272727	-0.623317083	0.26653809	0.0175528221	0.0061891857
-33.801248	37	0.29545455	0.28409091	-0.574921384	0.28267223	0.0127823120	0.0014186757
-29.766931	16	0.30681818	0.29545455	-0.506302171	0.30632227	0.0004959124	0.0108677240
-26.696234	22	0.31818182	0.30681818	-0.454073044	0.32488813	0.0067063089	0.0180699452
-24.271531	23	0.32954545	0.31818182	-0.412831567	0.33986501	0.0103195566	0.0216831929
-23.651448	86	0.34090909	0.32954545	-0.402284648	0.34373728	0.0028281851	0.0141918214
-19.683427	88	0.35227273	0.34090909	-0.334793052	0.36889060	0.0166178738	0.0279815102
-17.817835	10	0.36363636	0.35227273	-0.303061413	0.38092153	0.0172851663	0.0286488027
-16.762094	60	0.37500000	0.36363636	-0.285104441	0.38778206	0.0127820638	0.0241457002
-16.596960	21	0.38636364	0.37500000	-0.282295711	0.38885839	0.0024947507	0.0138583870
-16.271207	58	0.39772727	0.38636364	-0.276755010	0.39098411	0.0067431583	0.0046204781
-13.815798	56	0.40909091	0.39772727	-0.234991254	0.40710776	0.0019831485	0.0093804879
-13.462160	75	0.42045455	0.40909091	-0.228976273	0.40944368	0.0110108666	0.0003527698
-12.081520	4	0.43181818	0.42045455	-0.205493119	0.41859344	0.0132247451	0.0018611087
-11.629207	51	0.44318182	0.43181818	-0.197799788	0.42160086	0.0215809622	0.0102173258
-11.312669	74	0.45454545	0.44318182	-0.192415834	0.42370825	0.0308372092	0.0194735728
-8.236558	3	0.46590909	0.45454545	-0.140094626	0.44429261	0.0216164775	0.0102528411
-7.662789	70	0.47727273	0.46590909	-0.130335452	0.44815052	0.0291222111	0.0177585748
-6.752801	67	0.48863636	0.47727273	-0.114857588	0.45427900	0.0343573625	0.0229937262
-6.707262	31	0.50000000	0.48863636	-0.114083016	0.45458599	0.0454140074	0.0340503710
-6.402439	85	0.51136364	0.50000000	-0.108898313	0.45664157	0.0547220642	0.0433584278
-5.446904	82	0.52272727	0.51136364	-0.092645733	0.46309251	0.0596347676	0.0482711313
-3.537785	43	0.53409091	0.52272727	-0.060173762	0.47600862	0.0580822876	0.0467186512
-2.824941	61	0.54545455	0.53409091	-0.048049090	0.48083856	0.0646159857	0.0532523493
-2.745208	68	0.55681818	0.54545455	-0.046692922	0.48137899	0.0754391961	0.0640755598
-0.195089	65	0.56818182	0.55681818	-0.003318245	0.49867621	0.0695056040	0.0581419676
1.399296	55	0.57954545	0.56818182	0.023800450	0.50949411	0.0700513452	0.0586877088
5.363331	26	0.59090909	0.57954545	0.091224254	0.53634280	0.0545662924	0.0432026561
6.700640	53	0.60227273	0.59090909	0.113970383	0.54536936	0.0569033628	0.0455397265
7.386314	80	0.61363636	0.60227273	0.125632935	0.54998875	0.0636476093	0.0522839730
9.099900	41	0.62500000	0.61363636	0.154779103	0.56150227	0.0634977329	0.0521340965
12.433611	46	0.63636364	0.62500000	0.211481796	0.58374433	0.0526193043	0.0412556680
16.718018	62	0.64772727	0.63636364	0.284354766	0.61193074	0.0357965328	0.0244328965
18.093192	5	0.65909091	0.64772727	0.307744934	0.62086179	0.0382291219	0.0268654856
18.801816	38	0.67045455	0.65909091	0.319797835	0.62543921	0.0450153400	0.0336517036
19.168108	33	0.68181818	0.67045455	0.326028052	0.62779843	0.0540197476	0.0426561112
19.219211	72	0.69318182	0.68181818	0.326897255	0.62812720	0.0650546167	0.0536909803
20.334434	59	0.70454545	0.69318182	0.345865960	0.63527827	0.0692671805	0.0579035442
24.909926	78	0.71590909	0.70454545	0.423689939	0.66410402	0.0518050676	0.0404414312
26.236229	40	0.72727273	0.71590909	0.446248874	0.67229126	0.0549814685	0.0436178321
30.924022	25	0.73863636	0.72727273	0.525982978	0.70054998	0.0380863808	0.0267227444
32.253952	45	0.75000000	0.73863636	0.548603608	0.70836125	0.0416387548	0.0302751184
32.529367	49	0.76136364	0.75000000	0.553288104	0.70996693	0.0513967091	0.0400330727
32.675968	18	0.77272727	0.76136364	0.555781630	0.71081993	0.0619073452	0.0505437088
33.275839	20	0.78409091	0.77272727	0.565984762	0.71429793	0.0697929786	0.0584293423
36.031430	52	0.79545455	0.78409091	0.612854281	0.73001365	0.0654408934	0.0540772571
37.147186	84	0.80681818	0.79545455	0.631832029	0.73625168	0.0705665028	0.0592028664
40.320875	7	0.81818182	0.80681818	0.685812928	0.75358446	0.0645973596	0.0532337232
44.334467	30	0.82954545	0.81818182	0.754079634	0.77459930	0.0549461574	0.0435825211
46.907165	28	0.84090909	0.82954545	0.797838357	0.78751785	0.0533912405	0.0420276041
54.418366	87	0.85227273	0.84090909	0.925595465	0.82267187	0.0296008528	0.0182372164
55.091131	35	0.86363636	0.85227273	0.937038450	0.82563061	0.0380057535	0.0266421172
55.470305	44	0.87500000	0.86363636	0.943487765	0.82728426	0.0477157353	0.0363520989
62.939597	6	0.88636364	0.87500000	1.070532059	0.85781006	0.0285535797	0.0171899433
66.478628	50	0.89772727	0.88636364	1.130727018	0.87091500	0.0268122757	0.0154486394
67.426518	63	0.90909091	0.89772727	1.146849569	0.87427810	0.0348128083	0.0234491719
67.603959	19	0.92045455	0.90909091	1.149867648	0.87490081	0.0455537393	0.0341901029
69.707122	64	0.93181818	0.92045455	1.185640095	0.88211777	0.0497004123	0.0383367759
69.843246	8	0.94318182	0.93181818	1.187955411	0.88257451	0.0606073068	0.0492436705
74.848732	2	0.95454545	0.94318182	1.273093116	0.89850750	0.0560379553	0.0446743189
112.729191	66	0.96590909	0.95454545	1.917397313	0.97240626	0.0064971714	0.0178608078
163.795081	73	0.97727273	0.96590909	2.785970904	0.99733162	0.0200588896	0.0314225260
198.660139	42	0.98863636	0.97727273	3.378986513	0.99963623	0.0109998685	0.0223635048
209.375830	76	1.00000000	0.98863636	3.561248407	0.99981545	0.0001845478	0.0111790885
Fuente: Elaboración propia

Verifique el supuesto de Normaidad a traves de” a) La prueba de JB

library(tseries)
salida_JB<-jarque.bera.test(modelo_estimado$residuals)
salida_JB

## 
##  Jarque Bera Test
## 
## data:  modelo_estimado$residuals
## X-squared = 32.278, df = 2, p-value = 9.794e-08

Dado que el p-valor es de 0.00000009794 inferior a 0.05, existe evidencia estadística suficiente para rechazar la hipótesis de que los residuos no se distribuyen normalmente. Por lo tanto, el modelo no cumple con el supuesto de normalidad bajo la prueba de Jarque-Bera.

library(fastGraph)
alpha_sig<-0.05
JB<-salida_JB$statistic
gl<-salida_JB$parameter
VC<-qchisq(1-alpha_sig,gl,lower.tail = TRUE)
shadeDist(JB,ddist = "dchisq",
          parm1 = gl,
          lower.tail = FALSE,xmin=0,
          sub=paste("VC:",round(VC,2)," ","JB:",round(JB,2)))

Rechazar la Hipotesis nula H0 si JB >=V.C En este caso JB=32.28 mayor que VC=5.99 se rechaza la hipotesis nula y se concluye que “Los residuales no tienen una distribucion normal”.

Prueba de Kolmogorov Smirnov-Lilliefors

library(dplyr)  # Carga la librería dplyr para manipulación de datos
library(gt)  # Carga la librería gt para crear tablas de datos
library(gtExtras)  # Carga la librería gtExtras para agregar funcionalidades a las tablas creadas con gt

residuos<-modelo_estimado$residuals  # Crea un vector con los residuos del modelo estimado
residuos %>%  # Utiliza el operador %>% para encadenar las operaciones siguientes al vector residuos
  as_tibble() %>%  # Convierte el vector residuos en una tibble (tabla) de una columna
  mutate(posicion=row_number()) %>%  # Agrega una columna llamada "posicion" con el número de fila
  arrange(value) %>%  # Ordena la tabla por los valores de residuos en orden ascendente
  mutate(dist1=row_number()/n()) %>%  # Agrega una columna "dist1" con los percentiles según su posición en la tabla (usando la función row_number() y n() para obtener el número de filas)
  mutate(dist2=(row_number()-1)/n()) %>%  # Agrega una columna "dist2" con los percentiles según su posición en la tabla, pero ajustando en una unidad para evitar problemas con los extremos de la distribución
  mutate(zi=as.vector(scale(value,center=TRUE))) %>%  # Agrega una columna "zi" con los valores de residuos escalados para tener media cero y varianza uno
  mutate(pi=pnorm(zi,lower.tail = TRUE)) %>%  # Agrega una columna "pi" con los valores de la función de distribución acumulada (CDF) de una distribución normal estándar evaluada en los valores de zi
  mutate(dif1=abs(dist1-pi)) %>%  # Agrega una columna "dif1" con las diferencias absolutas entre los percentiles según la posición y los valores de pi
  mutate(dif2=abs(dist2-pi)) %>%  # Agrega una columna "dif2" con las diferencias absolutas entre los percentiles ajustados según la posición y los valores de pi
  rename(residuales=value) -> tabla_KS  # Renombra la columna "value" como "residuales" y asigna la tabla resultante a la variable tabla_KS


#Formato
 tabla_KS %>%  # Utiliza el operador %>% para encadenar las operaciones siguientes a la tabla tabla_KS
  gt() %>%  # Crea una tabla con la función gt()
  tab_header("Tabla para calcular el Estadistico KS") %>%  # Agrega un encabezado a la tabla
  tab_source_note(source_note = "Fuente: Elaboración propia") %>%  # Agrega una nota de fuente a la tabla
  tab_style(  # Cambia el estilo de algunas celdas de la tabla
    style = list(
      cell_fill(color = "#A569BD"),  # Cambia el color de fondo de las celdas a un tono de morado
      cell_text(style = "italic")  # Cambia el estilo de texto de las celdas a itálico
      ),
    locations = cells_body(  # Aplica el estilo a las celdas del cuerpo de la tabla que cumplan las siguientes condiciones:
      columns = dif1,  # Que pertenezcan a la columna "dif1"
      rows = dif1==max(dif1)  # Que pertenezcan a la fila donde el valor de "dif1" es máximo
    )) %>%
   tab_style(  # Cambia el estilo de algunas celdas de la tabla
    style = list(
      cell_fill(color = "#3498DB"),  # Cambia el color de fondo de las celdas a un tono de azul
      cell_text(style = "italic")  # Cambia el estilo de texto de las celdas a itálico
      ),
    locations = cells_body(  # Aplica el estilo a las celdas del cuerpo de la tabla que cumplan las siguientes condiciones:
      columns = dif2,  # Que pertenezcan a la columna "dif2"
      rows = dif2==max(dif2)  # Que pertenezcan a la fila donde el valor de "dif2" es máximo
    ))

residuales	posicion	dist1	dist2	zi	pi	dif1	dif2
Tabla para calcular el Estadistico KS
-120.026447	81	0.01136364	0.00000000	-2.041515459	0.02059981	0.0092361731	0.0205998094
-115.508697	77	0.02272727	0.01136364	-1.964673586	0.02472601	0.0019987418	0.0133623781
-107.080889	24	0.03409091	0.02272727	-1.821326006	0.03427866	0.0001877487	0.0115513850
-91.243980	48	0.04545455	0.03409091	-1.551957925	0.06033615	0.0148816002	0.0262452366
-85.461169	12	0.05681818	0.04545455	-1.453598781	0.07302879	0.0162106057	0.0275742421
-77.172687	32	0.06818182	0.05681818	-1.312620980	0.09465535	0.0264735301	0.0378371665
-74.702719	54	0.07954545	0.06818182	-1.270609602	0.10193378	0.0223883300	0.0337519664
-65.502849	39	0.09090909	0.07954545	-1.114130117	0.13261169	0.0417025941	0.0530662305
-63.699108	69	0.10227273	0.09090909	-1.083450505	0.13930425	0.0370315271	0.0483951634
-62.566594	83	0.11363636	0.10227273	-1.064187703	0.14362184	0.0299854747	0.0413491110
-59.845223	36	0.12500000	0.11363636	-1.017900230	0.15436269	0.0293626861	0.0407263225
-54.466158	13	0.13636364	0.12500000	-0.926408352	0.17711690	0.0407532663	0.0521169027
-54.300415	14	0.14772727	0.13636364	-0.923589260	0.17785010	0.0301228311	0.0414864675
-52.129801	15	0.15909091	0.14772727	-0.886669532	0.18762842	0.0285375141	0.0399011505
-51.441108	17	0.17045455	0.15909091	-0.874955638	0.19079902	0.0203444766	0.0317081129
-48.704980	47	0.18181818	0.17045455	-0.828417174	0.20371714	0.0218989601	0.0332625965
-48.350295	29	0.19318182	0.18181818	-0.822384375	0.20542908	0.0122472664	0.0236109028
-47.855859	11	0.20454545	0.19318182	-0.813974573	0.20782976	0.0032843043	0.0146479407
-45.639765	1	0.21590909	0.20454545	-0.776281294	0.21879146	0.0028823668	0.0142460032
-43.142550	9	0.22727273	0.21590909	-0.733806463	0.23153335	0.0042606233	0.0156242596
-41.749618	57	0.23863636	0.22727273	-0.710114247	0.23881665	0.0001802823	0.0115439187
-40.869022	27	0.25000000	0.23863636	-0.695136302	0.24348494	0.0065150566	0.0048485798
-37.749811	34	0.26136364	0.25000000	-0.642082009	0.26040997	0.0009536682	0.0104099682
-36.663785	71	0.27272727	0.26136364	-0.623609925	0.26644190	0.0062853771	0.0050782592
-36.646568	79	0.28409091	0.27272727	-0.623317083	0.26653809	0.0175528221	0.0061891857
-33.801248	37	0.29545455	0.28409091	-0.574921384	0.28267223	0.0127823120	0.0014186757
-29.766931	16	0.30681818	0.29545455	-0.506302171	0.30632227	0.0004959124	0.0108677240
-26.696234	22	0.31818182	0.30681818	-0.454073044	0.32488813	0.0067063089	0.0180699452
-24.271531	23	0.32954545	0.31818182	-0.412831567	0.33986501	0.0103195566	0.0216831929
-23.651448	86	0.34090909	0.32954545	-0.402284648	0.34373728	0.0028281851	0.0141918214
-19.683427	88	0.35227273	0.34090909	-0.334793052	0.36889060	0.0166178738	0.0279815102
-17.817835	10	0.36363636	0.35227273	-0.303061413	0.38092153	0.0172851663	0.0286488027
-16.762094	60	0.37500000	0.36363636	-0.285104441	0.38778206	0.0127820638	0.0241457002
-16.596960	21	0.38636364	0.37500000	-0.282295711	0.38885839	0.0024947507	0.0138583870
-16.271207	58	0.39772727	0.38636364	-0.276755010	0.39098411	0.0067431583	0.0046204781
-13.815798	56	0.40909091	0.39772727	-0.234991254	0.40710776	0.0019831485	0.0093804879
-13.462160	75	0.42045455	0.40909091	-0.228976273	0.40944368	0.0110108666	0.0003527698
-12.081520	4	0.43181818	0.42045455	-0.205493119	0.41859344	0.0132247451	0.0018611087
-11.629207	51	0.44318182	0.43181818	-0.197799788	0.42160086	0.0215809622	0.0102173258
-11.312669	74	0.45454545	0.44318182	-0.192415834	0.42370825	0.0308372092	0.0194735728
-8.236558	3	0.46590909	0.45454545	-0.140094626	0.44429261	0.0216164775	0.0102528411
-7.662789	70	0.47727273	0.46590909	-0.130335452	0.44815052	0.0291222111	0.0177585748
-6.752801	67	0.48863636	0.47727273	-0.114857588	0.45427900	0.0343573625	0.0229937262
-6.707262	31	0.50000000	0.48863636	-0.114083016	0.45458599	0.0454140074	0.0340503710
-6.402439	85	0.51136364	0.50000000	-0.108898313	0.45664157	0.0547220642	0.0433584278
-5.446904	82	0.52272727	0.51136364	-0.092645733	0.46309251	0.0596347676	0.0482711313
-3.537785	43	0.53409091	0.52272727	-0.060173762	0.47600862	0.0580822876	0.0467186512
-2.824941	61	0.54545455	0.53409091	-0.048049090	0.48083856	0.0646159857	0.0532523493
-2.745208	68	0.55681818	0.54545455	-0.046692922	0.48137899	0.0754391961	0.0640755598
-0.195089	65	0.56818182	0.55681818	-0.003318245	0.49867621	0.0695056040	0.0581419676
1.399296	55	0.57954545	0.56818182	0.023800450	0.50949411	0.0700513452	0.0586877088
5.363331	26	0.59090909	0.57954545	0.091224254	0.53634280	0.0545662924	0.0432026561
6.700640	53	0.60227273	0.59090909	0.113970383	0.54536936	0.0569033628	0.0455397265
7.386314	80	0.61363636	0.60227273	0.125632935	0.54998875	0.0636476093	0.0522839730
9.099900	41	0.62500000	0.61363636	0.154779103	0.56150227	0.0634977329	0.0521340965
12.433611	46	0.63636364	0.62500000	0.211481796	0.58374433	0.0526193043	0.0412556680
16.718018	62	0.64772727	0.63636364	0.284354766	0.61193074	0.0357965328	0.0244328965
18.093192	5	0.65909091	0.64772727	0.307744934	0.62086179	0.0382291219	0.0268654856
18.801816	38	0.67045455	0.65909091	0.319797835	0.62543921	0.0450153400	0.0336517036
19.168108	33	0.68181818	0.67045455	0.326028052	0.62779843	0.0540197476	0.0426561112
19.219211	72	0.69318182	0.68181818	0.326897255	0.62812720	0.0650546167	0.0536909803
20.334434	59	0.70454545	0.69318182	0.345865960	0.63527827	0.0692671805	0.0579035442
24.909926	78	0.71590909	0.70454545	0.423689939	0.66410402	0.0518050676	0.0404414312
26.236229	40	0.72727273	0.71590909	0.446248874	0.67229126	0.0549814685	0.0436178321
30.924022	25	0.73863636	0.72727273	0.525982978	0.70054998	0.0380863808	0.0267227444
32.253952	45	0.75000000	0.73863636	0.548603608	0.70836125	0.0416387548	0.0302751184
32.529367	49	0.76136364	0.75000000	0.553288104	0.70996693	0.0513967091	0.0400330727
32.675968	18	0.77272727	0.76136364	0.555781630	0.71081993	0.0619073452	0.0505437088
33.275839	20	0.78409091	0.77272727	0.565984762	0.71429793	0.0697929786	0.0584293423
36.031430	52	0.79545455	0.78409091	0.612854281	0.73001365	0.0654408934	0.0540772571
37.147186	84	0.80681818	0.79545455	0.631832029	0.73625168	0.0705665028	0.0592028664
40.320875	7	0.81818182	0.80681818	0.685812928	0.75358446	0.0645973596	0.0532337232
44.334467	30	0.82954545	0.81818182	0.754079634	0.77459930	0.0549461574	0.0435825211
46.907165	28	0.84090909	0.82954545	0.797838357	0.78751785	0.0533912405	0.0420276041
54.418366	87	0.85227273	0.84090909	0.925595465	0.82267187	0.0296008528	0.0182372164
55.091131	35	0.86363636	0.85227273	0.937038450	0.82563061	0.0380057535	0.0266421172
55.470305	44	0.87500000	0.86363636	0.943487765	0.82728426	0.0477157353	0.0363520989
62.939597	6	0.88636364	0.87500000	1.070532059	0.85781006	0.0285535797	0.0171899433
66.478628	50	0.89772727	0.88636364	1.130727018	0.87091500	0.0268122757	0.0154486394
67.426518	63	0.90909091	0.89772727	1.146849569	0.87427810	0.0348128083	0.0234491719
67.603959	19	0.92045455	0.90909091	1.149867648	0.87490081	0.0455537393	0.0341901029
69.707122	64	0.93181818	0.92045455	1.185640095	0.88211777	0.0497004123	0.0383367759
69.843246	8	0.94318182	0.93181818	1.187955411	0.88257451	0.0606073068	0.0492436705
74.848732	2	0.95454545	0.94318182	1.273093116	0.89850750	0.0560379553	0.0446743189
112.729191	66	0.96590909	0.95454545	1.917397313	0.97240626	0.0064971714	0.0178608078
163.795081	73	0.97727273	0.96590909	2.785970904	0.99733162	0.0200588896	0.0314225260
198.660139	42	0.98863636	0.97727273	3.378986513	0.99963623	0.0109998685	0.0223635048
209.375830	76	1.00000000	0.98863636	3.561248407	0.99981545	0.0001845478	0.0111790885
Fuente: Elaboración propia

Calculo del estadistico D

D<-max(max(tabla_KS$dif1),max(tabla_KS$dif2))
print(D)

## [1] 0.0754392

Calculo del valor Critico Para n>30 y un nivel de signficancia de 5% , la formula estandar es VC=0.886/Raiz cuadrada de la muestra VC=0.886/88=0.0944 Conclusión:

En este caso dado que 0.0754382 <0.0.0944 se rechaza la Hipótesis Nula: ϵ∼N(0,σ2) , por lo que los residuos no siguen una distribución normal.

Usando Nortest

library(nortest)
prueba_KS<-lillie.test(modelo_estimado$residuals)
prueba_KS

## 
##  Lilliefors (Kolmogorov-Smirnov) normality test
## 
## data:  modelo_estimado$residuals
## D = 0.075439, p-value = 0.2496

En este caso dado que 0.2496<=0.05 No se rechaza la Hipótesis Nula: ϵ∼N(0,σ2) , por lo que los residuos siguen una distribución normal.

Prueba de Shapiro Wilk

library(dplyr)
library(gt)
residuos<-modelo_estimado$residuals
residuos %>%  
  as_tibble() %>%
  rename(residuales=value) %>%
  arrange(residuales) %>%
  mutate(pi=(row_number()-0.375)/(n()+0.25)) %>%
  mutate(mi=qnorm(pi,lower.tail = TRUE)) %>% 
  mutate(ai=0)->tabla_SW

m<-sum(tabla_SW$mi^2)
n<-nrow(hprice1)
theta<-1/sqrt(n)
tabla_SW$ai[n]<- -2.706056*theta^5+4.434685*theta^4-2.071190*theta^3-0.147981*theta^2+0.2211570*theta+tabla_SW$mi[n]/sqrt(m)
tabla_SW$ai[n-1]<- -3.582633*theta^5+5.682633*theta^4-1.752461*theta^3-0.293762*theta^2+0.042981*theta+tabla_SW$mi[n-1]/sqrt(m)
tabla_SW$ai[1]<- -tabla_SW$ai[n]
tabla_SW$ai[2]<- -tabla_SW$ai[n-1]
omega<-(m-2*tabla_SW$mi[n]^2-2*tabla_SW$mi[n-1]^2)/(1-2*tabla_SW$ai[n]^2-2*tabla_SW$ai[n-1]^2)
tabla_SW$ai[3:(n-2)]<-tabla_SW$mi[3:(n-2)]/sqrt(omega)

tabla_SW %>% 
  mutate(ai_ui=ai*residuales,ui2=residuales^2) ->tabla_SW

tabla_SW %>%
  gt() %>% tab_header("Tabla para calcular el Estadistico W") %>%  # Agrega un encabezado a la tabla
  tab_source_note(source_note = "Fuente: Elaboración propia")

residuales	pi	mi	ai	ai_ui	ui2
Tabla para calcular el Estadistico W
-120.026447	0.007082153	-2.45306927	-0.286093929	34.338837782	1.440635e+04
-115.508697	0.018413598	-2.08767462	-0.226331231	26.143225495	1.334226e+04
-107.080889	0.029745042	-1.88455395	-0.201511408	21.578020632	1.146632e+04
-91.243980	0.041076487	-1.73832835	-0.185875811	16.960048752	8.325464e+03
-85.461169	0.052407932	-1.62194155	-0.173430814	14.821600075	7.303611e+03
-77.172687	0.063739377	-1.52411994	-0.162970954	12.576906330	5.955624e+03
-74.702719	0.075070822	-1.43903134	-0.153872609	11.494702279	5.580496e+03
-65.502849	0.086402266	-1.36324747	-0.145769197	9.548297773	4.290623e+03
-63.699108	0.097733711	-1.29457343	-0.138426027	8.817614500	4.057576e+03
-62.566594	0.109065156	-1.23151500	-0.131683320	8.238976839	3.914579e+03
-59.845223	0.120396601	-1.17300649	-0.125427129	7.506214499	3.581451e+03
-54.466158	0.131728045	-1.11825971	-0.119573169	6.512691096	2.966562e+03
-54.300415	0.143059490	-1.06667420	-0.114057239	6.193355472	2.948535e+03
-52.129801	0.154390935	-1.01778137	-0.108829231	5.673246083	2.717516e+03
-51.441108	0.165722380	-0.97120790	-0.103849228	5.342119306	2.646188e+03
-48.704980	0.177053824	-0.92665123	-0.099084876	4.825926905	2.372175e+03
-48.350295	0.188385269	-0.88386232	-0.094509548	4.569564512	2.337751e+03
-47.855859	0.199716714	-0.84263354	-0.090101040	4.311862673	2.290183e+03
-45.639765	0.211048159	-0.80278966	-0.085840618	3.917745629	2.082988e+03
-43.142550	0.222379603	-0.76418130	-0.081712307	3.525277277	1.861280e+03
-41.749618	0.233711048	-0.72667986	-0.077702356	3.244043648	1.743031e+03
-40.869022	0.245042493	-0.69017366	-0.073798824	3.016085791	1.670277e+03
-37.749811	0.256373938	-0.65456498	-0.069991263	2.642156946	1.425048e+03
-36.663785	0.267705382	-0.61976766	-0.066270458	2.429725818	1.344233e+03
-36.646568	0.279036827	-0.58570518	-0.062628228	2.295109622	1.342971e+03
-33.801248	0.290368272	-0.55230918	-0.059057264	1.996209250	1.142524e+03
-29.766931	0.301699717	-0.51951819	-0.055550992	1.653582575	8.860702e+02
-26.696234	0.313031161	-0.48727661	-0.052103467	1.390966354	7.126889e+02
-24.271531	0.324362606	-0.45553386	-0.048709282	1.182248861	5.891072e+02
-23.651448	0.335694051	-0.42424369	-0.045363489	1.072912217	5.593910e+02
-19.683427	0.347025496	-0.39336354	-0.042061540	0.827915257	3.874373e+02
-17.817835	0.358356941	-0.36285409	-0.038799229	0.691318234	3.174752e+02
-16.762094	0.369688385	-0.33267878	-0.035572645	0.596272007	2.809678e+02
-16.596960	0.381019830	-0.30280344	-0.032378138	0.537378676	2.754591e+02
-16.271207	0.392351275	-0.27319601	-0.029212277	0.475319006	2.647522e+02
-13.815798	0.403682720	-0.24382619	-0.026071824	0.360203050	1.908763e+02
-13.462160	0.415014164	-0.21466524	-0.022953704	0.309006447	1.812298e+02
-12.081520	0.426345609	-0.18568573	-0.019854987	0.239878409	1.459631e+02
-11.629207	0.437677054	-0.15686137	-0.016772858	0.195055032	1.352385e+02
-11.312669	0.449008499	-0.12816677	-0.013704604	0.155035654	1.279765e+02
-8.236558	0.460339943	-0.09957734	-0.010647596	0.087699542	6.784089e+01
-7.662789	0.471671388	-0.07106908	-0.007599268	0.058231584	5.871833e+01
-6.752801	0.483002833	-0.04261848	-0.004557105	0.030773222	4.560033e+01
-6.707262	0.494334278	-0.01420234	-0.001518626	0.010185824	4.498736e+01
-6.402439	0.505665722	0.01420234	0.001518626	-0.009722911	4.099122e+01
-5.446904	0.516997167	0.04261848	0.004557105	-0.024822110	2.966876e+01
-3.537785	0.528328612	0.07106908	0.007599268	-0.026884576	1.251592e+01
-2.824941	0.539660057	0.09957734	0.010647596	-0.030078835	7.980294e+00
-2.745208	0.550991501	0.12816677	0.013704604	-0.037621996	7.536170e+00
-0.195089	0.562322946	0.15686137	0.016772858	-0.003272200	3.805971e-02
1.399296	0.573654391	0.18568573	0.019854987	0.027782994	1.958028e+00
5.363331	0.584985836	0.21466524	0.022953704	0.123108313	2.876532e+01
6.700640	0.596317280	0.24382619	0.026071824	0.174697904	4.489858e+01
7.386314	0.607648725	0.27319601	0.029212277	0.215771059	5.455764e+01
9.099900	0.618980170	0.30280344	0.032378138	0.294637808	8.280817e+01
12.433611	0.630311615	0.33267878	0.035572645	0.442296424	1.545947e+02
16.718018	0.641643059	0.36285409	0.038799229	0.648646203	2.794921e+02
18.093192	0.652974504	0.39336354	0.042061540	0.761027520	3.273636e+02
18.801816	0.664305949	0.42424369	0.045363489	0.852915978	3.535083e+02
19.168108	0.675637394	0.45553386	0.048709282	0.933664777	3.674164e+02
19.219211	0.686968839	0.48727661	0.052103467	1.001387528	3.693781e+02
20.334434	0.698300283	0.51951819	0.055550992	1.129598008	4.134892e+02
24.909926	0.709631728	0.55230918	0.059057264	1.471112049	6.205044e+02
26.236229	0.720963173	0.58570518	0.062628228	1.643128534	6.883397e+02
30.924022	0.732294618	0.61976766	0.066270458	2.049349072	9.562951e+02
32.253952	0.743626062	0.65456498	0.069991263	2.257494854	1.040317e+03
32.529367	0.754957507	0.69017366	0.073798824	2.400629035	1.058160e+03
32.675968	0.766288952	0.72667986	0.077702356	2.538999708	1.067719e+03
33.275839	0.777620397	0.76418130	0.081712307	2.719045583	1.107281e+03
36.031430	0.788951841	0.80278966	0.085840618	3.092960242	1.298264e+03
37.147186	0.800283286	0.84263354	0.090101040	3.347000059	1.379913e+03
40.320875	0.811614731	0.88386232	0.094509548	3.810707636	1.625773e+03
44.334467	0.822946176	0.92665123	0.099084876	4.392875123	1.965545e+03
46.907165	0.834277620	0.97120790	0.103849228	4.871272904	2.200282e+03
54.418366	0.845609065	1.01778137	0.108829231	5.922308882	2.961359e+03
55.091131	0.856940510	1.06667420	0.114057239	6.283542333	3.035033e+03
55.470305	0.868271955	1.11825971	0.119573169	6.632760113	3.076955e+03
62.939597	0.879603399	1.17300649	0.125427129	7.894332885	3.961393e+03
66.478628	0.890934844	1.23151500	0.131683320	8.754126443	4.419408e+03
67.426518	0.902266289	1.29457343	0.138426027	9.333585010	4.546335e+03
67.603959	0.913597734	1.36324747	0.145769197	9.854574914	4.570295e+03
69.707122	0.924929178	1.43903134	0.153872609	10.726016772	4.859083e+03
69.843246	0.936260623	1.52411994	0.162970954	11.382420482	4.878079e+03
74.848732	0.947592068	1.62194155	0.173430814	12.981076532	5.602333e+03
112.729191	0.958923513	1.73832835	0.185875811	20.953629849	1.270787e+04
163.795081	0.970254958	1.88455395	0.201511408	33.006577315	2.682883e+04
198.660139	0.981586402	2.08767462	0.226331231	44.962993843	3.946585e+04
209.375830	0.992917847	2.45306927	0.286093929	59.901153719	4.383824e+04
Fuente: Elaboración propia

Calculo del estadistico W

W<-(sum(tabla_SW$ai_ui)^2)/sum(tabla_SW$ui2)
print(W)

## [1] 0.9413208

Cálculo del Wn y su p-value

mu<-0.0038915*log(n)^3-0.083751*log(n)^2-0.31082*log(n)-1.5861
sigma<-exp(0.0030302*log(n)^2-0.082676*log(n)-0.4803)
Wn<-(log(1-W)-mu)/sigma
print(Wn)

## [1] 3.241867

p.value<-pnorm(Wn,lower.tail = FALSE)
print(p.value)

## [1] 0.0005937472

library(fastGraph)
shadeDist(Wn,ddist = "dnorm",lower.tail = FALSE)

En este caso dado que 0.0005937 <=0.05 se rechaza la Hipótesis Nula: ϵ∼N(0,σ2) , por lo que los residuos no siguen una distribución normal.

Usando la liberia stats (Precargada al inicar R)

salida_SW<-shapiro.test(modelo_estimado$residuals)
print(salida_SW)

## 
##  Shapiro-Wilk normality test
## 
## data:  modelo_estimado$residuals
## W = 0.94132, p-value = 0.0005937

Mismos resultados que en el calculo Manual Importante, a partir de esta salida se puede calcular el Wn si se llegará a necesitar:

Wn_salida<-qnorm(salida_SW$p.value,lower.tail = FALSE)
print(Wn_salida)

## [1] 3.241867

Ejercicio Pruebas de Normalidad

KENNEDY_ULISES_VASQUEZ_PEREZ

Prueba de Normalidad: Resultados del Modelo