Ejercicio Prueba de normalidad

library(wooldridge)

## Warning: package 'wooldridge' was built under R version 4.4.3

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. Estimacion del modelo

library(stargazer)
modelo_estimado <- lm(formula = price~lotsize+sqrft+bdrms, data = hprice1)
stargazer(modelo_estimado, title = "Estimacion del modelo", type = "html")

**Estimacion 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

2. Verificando supuestos de normalidad de los residuos

Prueba JB usando tseries

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 = 0.00000009794

library(fastGraph)

## Warning: package 'fastGraph' was built under R version 4.4.3

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)))

Usando normtest

library(normtest)
jb.norm.test(modelo_estimado$residuals)

## 
##  Jarque-Bera test for normality
## 
## data:  modelo_estimado$residuals
## JB = 32.278, p-value = 0.001

2. Prueba de kolmogorov Smirnov

Calculo manual

library(dplyr)

## 
## Adjuntando el paquete: 'dplyr'

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

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

library(gt)

## Warning: package 'gt' was built under R version 4.4.3

library(gtExtras)

## Warning: package 'gtExtras' was built under R version 4.4.3

residuos<-modelo_estimado$residuals  
residuos %>%  
  as_tibble() %>%  
  mutate(posicion=row_number()) %>%  
  arrange(value) %>%  
  mutate(dist1=row_number()/n()) %>%  
  mutate(dist2=(row_number()-1)/n()) %>%  
  mutate(zi=as.vector(scale(value,center=TRUE))) %>%  
  mutate(pi=pnorm(zi,lower.tail = TRUE)) %>%  
  mutate(dif1=abs(dist1-pi)) %>%  
  mutate(dif2=abs(dist2-pi)) %>%  
  rename(residuales=value) -> tabla_KS  


 tabla_KS %>%  
  gt() %>%  
  tab_header("Tabla para calcular el Estadistico KS") %>%  
  tab_source_note(source_note = "Fuente: Elaboración propia") %>%  
  tab_style(  
    style = list(
      cell_fill(color = "#A569BD"),  
      cell_text(style = "italic")  
      ),
    locations = cells_body(  
      columns = dif1,  
      rows = dif1==max(dif1)  
    )) %>%
   tab_style(  
    style = list(
      cell_fill(color = "#3498DB"),  
      cell_text(style = "italic")  
      ),
    locations = cells_body(  
      columns = dif2,  
      rows = dif2==max(dif2)  
    ))

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

matriz_X <- model.matrix(modelo_estimado)
n <- nrow(matriz_X)
print(n)

## [1] 88

Ya que las observaciones que se tienen son mayores a cincuenta se debe de usar la siguiente formula para obtener el valor critico:

$0,875897/\sqrt\ n$

al aplicarla se obtendra el valor critico de:

n_KS <- 88
resultado_ks <- 0.855897/sqrt(n_KS)
print(resultado_ks)

## [1] 0.09123893

Conclusion:

Dado que 0.0754392< 0.0912389 no se rechaza la hipotesis nula: $\epsilon~N(0,\sigma^2)$ , por lo tanto, los residuos siguen una distribucion 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

p.value <- prueba_KS$p.value

En este caso dado que 0.2496425> 0.05 No se rechaza la hipotesis nula: $\epsilon~N(0,\sigma^2)$ , los residuos siguen una distribucion normal.

3. Prueba de Shapiro Wilk

Calculo manual

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$mi^2)
n_SW<-nrow(hprice1)
theta<-1/sqrt(n)
tabla_SW$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") %>%  
  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	14406.34799223
-115.508697	0.018413598	-2.08767462	-0.226331231	26.143225495	13342.25903657
-107.080889	0.029745042	-1.88455395	-0.201511408	21.578020632	11466.31670225
-91.243980	0.041076487	-1.73832835	-0.185875811	16.960048752	8325.46388922
-85.461169	0.052407932	-1.62194155	-0.173430814	14.821600075	7303.61136157
-77.172687	0.063739377	-1.52411994	-0.162970954	12.576906330	5955.62354189
-74.702719	0.075070822	-1.43903134	-0.153872609	11.494702279	5580.49626206
-65.502849	0.086402266	-1.36324747	-0.145769197	9.548297773	4290.62326804
-63.699108	0.097733711	-1.29457343	-0.138426027	8.817614500	4057.57641853
-62.566594	0.109065156	-1.23151500	-0.131683320	8.238976839	3914.57869135
-59.845223	0.120396601	-1.17300649	-0.125427129	7.506214499	3581.45072682
-54.466158	0.131728045	-1.11825971	-0.119573169	6.512691096	2966.56233834
-54.300415	0.143059490	-1.06667420	-0.114057239	6.193355472	2948.53511008
-52.129801	0.154390935	-1.01778137	-0.108829231	5.673246083	2717.51610406
-51.441108	0.165722380	-0.97120790	-0.103849228	5.342119306	2646.18755812
-48.704980	0.177053824	-0.92665123	-0.099084876	4.825926905	2372.17509746
-48.350295	0.188385269	-0.88386232	-0.094509548	4.569564512	2337.75102457
-47.855859	0.199716714	-0.84263354	-0.090101040	4.311862673	2290.18324033
-45.639765	0.211048159	-0.80278966	-0.085840618	3.917745629	2082.98814155
-43.142550	0.222379603	-0.76418130	-0.081712307	3.525277277	1861.27961161
-41.749618	0.233711048	-0.72667986	-0.077702356	3.244043648	1743.03058469
-40.869022	0.245042493	-0.69017366	-0.073798824	3.016085791	1670.27697055
-37.749811	0.256373938	-0.65456498	-0.069991263	2.642156946	1425.04821452
-36.663785	0.267705382	-0.61976766	-0.066270458	2.429725818	1344.23312095
-36.646568	0.279036827	-0.58570518	-0.062628228	2.295109622	1342.97093753
-33.801248	0.290368272	-0.55230918	-0.059057264	1.996209250	1142.52439130
-29.766931	0.301699717	-0.51951819	-0.055550992	1.653582575	886.07020942
-26.696234	0.313031161	-0.48727661	-0.052103467	1.390966354	712.68890388
-24.271531	0.324362606	-0.45553386	-0.048709282	1.182248861	589.10722688
-23.651448	0.335694051	-0.42424369	-0.045363489	1.072912217	559.39099788
-19.683427	0.347025496	-0.39336354	-0.042061540	0.827915257	387.43729851
-17.817835	0.358356941	-0.36285409	-0.038799229	0.691318234	317.47522771
-16.762094	0.369688385	-0.33267878	-0.035572645	0.596272007	280.96778010
-16.596960	0.381019830	-0.30280344	-0.032378138	0.537378676	275.45909399
-16.271207	0.392351275	-0.27319601	-0.029212277	0.475319006	264.75217651
-13.815798	0.403682720	-0.24382619	-0.026071824	0.360203050	190.87627634
-13.462160	0.415014164	-0.21466524	-0.022953704	0.309006447	181.22976154
-12.081520	0.426345609	-0.18568573	-0.019854987	0.239878409	145.96311543
-11.629207	0.437677054	-0.15686137	-0.016772858	0.195055032	135.23845458
-11.312669	0.449008499	-0.12816677	-0.013704604	0.155035654	127.97648221
-8.236558	0.460339943	-0.09957734	-0.010647596	0.087699542	67.84088513
-7.662789	0.471671388	-0.07106908	-0.007599268	0.058231584	58.71832836
-6.752801	0.483002833	-0.04261848	-0.004557105	0.030773222	45.60032533
-6.707262	0.494334278	-0.01420234	-0.001518626	0.010185824	44.98736398
-6.402439	0.505665722	0.01420234	0.001518626	-0.009722911	40.99122172
-5.446904	0.516997167	0.04261848	0.004557105	-0.024822110	29.66876028
-3.537785	0.528328612	0.07106908	0.007599268	-0.026884576	12.51592288
-2.824941	0.539660057	0.09957734	0.010647596	-0.030078835	7.98029397
-2.745208	0.550991501	0.12816677	0.013704604	-0.037621996	7.53616965
-0.195089	0.562322946	0.15686137	0.016772858	-0.003272200	0.03805971
1.399296	0.573654391	0.18568573	0.019854987	0.027782994	1.95802794
5.363331	0.584985836	0.21466524	0.022953704	0.123108313	28.76531940
6.700640	0.596317280	0.24382619	0.026071824	0.174697904	44.89857663
7.386314	0.607648725	0.27319601	0.029212277	0.215771059	54.55763860
9.099900	0.618980170	0.30280344	0.032378138	0.294637808	82.80817401
12.433611	0.630311615	0.33267878	0.035572645	0.442296424	154.59467612
16.718018	0.641643059	0.36285409	0.038799229	0.648646203	279.49212715
18.093192	0.652974504	0.39336354	0.042061540	0.761027520	327.36359375
18.801816	0.664305949	0.42424369	0.045363489	0.852915978	353.50828232
19.168108	0.675637394	0.45553386	0.048709282	0.933664777	367.41636183
19.219211	0.686968839	0.48727661	0.052103467	1.001387528	369.37806665
20.334434	0.698300283	0.51951819	0.055550992	1.129598008	413.48922446
24.909926	0.709631728	0.55230918	0.059057264	1.471112049	620.50439009
26.236229	0.720963173	0.58570518	0.062628228	1.643128534	688.33970624
30.924022	0.732294618	0.61976766	0.066270458	2.049349072	956.29510728
32.253952	0.743626062	0.65456498	0.069991263	2.257494854	1040.31742689
32.529367	0.754957507	0.69017366	0.073798824	2.400629035	1058.15970869
32.675968	0.766288952	0.72667986	0.077702356	2.538999708	1067.71890359
33.275839	0.777620397	0.76418130	0.081712307	2.719045583	1107.28147309
36.031430	0.788951841	0.80278966	0.085840618	3.092960242	1298.26396526
37.147186	0.800283286	0.84263354	0.090101040	3.347000059	1379.91339592
40.320875	0.811614731	0.88386232	0.094509548	3.810707636	1625.77293960
44.334467	0.822946176	0.92665123	0.099084876	4.392875123	1965.54494196
46.907165	0.834277620	0.97120790	0.103849228	4.871272904	2200.28216686
54.418366	0.845609065	1.01778137	0.108829231	5.922308882	2961.35853839
55.091131	0.856940510	1.06667420	0.114057239	6.283542333	3035.03273452
55.470305	0.868271955	1.11825971	0.119573169	6.632760113	3076.95468678
62.939597	0.879603399	1.17300649	0.125427129	7.894332885	3961.39282116
66.478628	0.890934844	1.23151500	0.131683320	8.754126443	4419.40796540
67.426518	0.902266289	1.29457343	0.138426027	9.333585010	4546.33534619
67.603959	0.913597734	1.36324747	0.145769197	9.854574914	4570.29533539
69.707122	0.924929178	1.43903134	0.153872609	10.726016772	4859.08292257
69.843246	0.936260623	1.52411994	0.162970954	11.382420482	4878.07906512
74.848732	0.947592068	1.62194155	0.173430814	12.981076532	5602.33268291
112.729191	0.958923513	1.73832835	0.185875811	20.953629849	12707.87061041
163.795081	0.970254958	1.88455395	0.201511408	33.006577315	26828.82842547
198.660139	0.981586402	2.08767462	0.226331231	44.962993843	39465.85101402
209.375830	0.992917847	2.45306927	0.286093929	59.901153719	43838.23810785
Fuente: Elaboración propia

Calculo del estadistico W

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

## [1] 0.9413208

Calculo del $W_n$ 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 hipotesis nula: $\epsilon~N(0,\sigma^2)$ , por lo que, los residuos no siguen una distribucion normal

Usando la libreria stats

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

Calculando nuevamente el valor de $W_n$

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

## [1] 3.241867