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

modelo <- lm(price ~ lotsize + sqrft + bdrms, data = hprice1)  # Crear modelo

1. Estime el siguiente modelo:

price = ˆα + ˆα1(lotsize) + ˆα2(sqrft) + ˆα3(bdrms) + E

library(stargazer)

## 
## Please cite as:

##  Hlavac, Marek (2022). stargazer: Well-Formatted Regression and Summary Statistics Tables.

##  R package version 5.2.3. https://CRAN.R-project.org/package=stargazer

stargazer(modelo, type = "text",
          title = "Resultados de la regresión",
          dep.var.labels = "Precio",
          covariate.labels = c("Tamaño del lote", "Tamaño de la casa", "Número de habitaciones"),
          digits = 4)

## 
## Resultados de la regresión
## ==================================================
##                            Dependent variable:    
##                        ---------------------------
##                                  Precio           
## --------------------------------------------------
## Tamaño del lote                 0.0021***         
##                                 (0.0006)          
##                                                   
## Tamaño de la casa               0.1228***         
##                                 (0.0132)          
##                                                   
## Número de habitaciones           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. Verique el supuesto de normalidad, a través de:

a) La prueba JB

library(moments)   # Para calcular asimetría y curtosis

residuos <- modelo$residuals   # Extrae los residuos del modelo

n <- length(residuos)          # Número de observaciones
S <- skewness(residuos)        # Calcula la asimetría
K <- kurtosis(residuos)        # Calcula la curtosis

JB <- (n/6)*(S^2 + ((K-3)^2)/4)   # Fórmula de Jarque-Bera

JB   # Muestra el estadístico JB

## [1] 32.27791

library(tseries)   # Librería para prueba JB

## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo

jb <- jarque.bera.test(residuos)   # Aplica prueba JB
JB <- (n/6)*(S^2 + ((K-3)^2)/4)

library(fastGraph)   # Para graficar distribución
gl <- 2
JB <- as.numeric(JB)

alpha <- 0.05                 # Nivel de significancia
gl <- 2                       # Grados de libertad
VC <- qchisq(1-alpha, gl)     # Valor crítico chi-cuadrado

shadeDist("dchisq",           
          parm1 = gl,
          xshade = JB,
          lower.tail = FALSE,
          sub = paste("VC:", round(VC,2), " JB:", round(JB,2)))

La gráfica de la distribución chi-cuadrado con 2 grados de libertad muestra la región crítica en la cola derecha. El estadístico de Jarque-Bera (JB) se ubica en comparación con el valor crítico (VC). Si el valor de JB es mayor que el valor crítico, se rechaza la hipótesis nula de normalidad. En caso contrario, no se rechaza. Por lo tanto, al observar la posición del estadístico JB en la gráfica, se concluye que [rechazamos / no rechazamos] la hipótesis de que los residuos siguen una distribución normal.

b) La prueba KS

library(dplyr)   # Manipulación de datos

## 
## Attaching package: '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)      # Tablas

tabla_ks <- data.frame(residuos) %>%          # Convierte residuos a data frame
  mutate(posicion = row_number()) %>%         # Crea índice
  arrange(residuos) %>%                       # Ordena de menor a mayor
  mutate(F_emp = row_number()/n()) %>%        # Función empírica
  mutate(F_emp_ant = (row_number()-1)/n()) %>% # Función empírica anterior
  mutate(z = (residuos - mean(residuos))/sd(residuos)) %>% # Estandariza
  mutate(F_teo = pnorm(z)) %>%                # Distribución normal teórica
  mutate(D1 = abs(F_emp - F_teo)) %>%         # Diferencia 1
  mutate(D2 = abs(F_emp_ant - F_teo)) %>%     # Diferencia 2
  rename(Residuos = residuos)                 # Renombra

tabla_ks %>%
  gt() %>%
  tab_header(title = "Cálculo manual KS") %>%
  tab_source_note(source_note = "Fuente: Elaboración propia")

Residuos	posicion	F_emp	F_emp_ant	z	F_teo	D1	D2
Cálculo manual 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

D <- max(c(tabla_ks$D1, tabla_ks$D2))   # Máxima diferencia
D

## [1] 0.0754392

ks <- ks.test(residuos, "pnorm", mean(residuos), sd(residuos))  # Prueba KS
ks

## 
##  Exact one-sample Kolmogorov-Smirnov test
## 
## data:  residuos
## D = 0.075439, p-value = 0.67
## alternative hypothesis: two-sided

c) La prueba SW

tabla_sw <- data.frame(residuos) %>%   # Data frame
  arrange(residuos) %>%                # Ordena
  mutate(posicion = row_number()) %>%  # Índice
  rename(Residuos = residuos)          # Renombra

tabla_sw %>%
  gt() %>%
  tab_header(title = "Ordenamiento residuos (SW)") %>%
  tab_source_note(source_note = "Fuente: Elaboración propia")

Residuos	posicion
Ordenamiento residuos (SW)
-120.026447	1
-115.508697	2
-107.080889	3
-91.243980	4
-85.461169	5
-77.172687	6
-74.702719	7
-65.502849	8
-63.699108	9
-62.566594	10
-59.845223	11
-54.466158	12
-54.300415	13
-52.129801	14
-51.441108	15
-48.704980	16
-48.350295	17
-47.855859	18
-45.639765	19
-43.142550	20
-41.749618	21
-40.869022	22
-37.749811	23
-36.663785	24
-36.646568	25
-33.801248	26
-29.766931	27
-26.696234	28
-24.271531	29
-23.651448	30
-19.683427	31
-17.817835	32
-16.762094	33
-16.596960	34
-16.271207	35
-13.815798	36
-13.462160	37
-12.081520	38
-11.629207	39
-11.312669	40
-8.236558	41
-7.662789	42
-6.752801	43
-6.707262	44
-6.402439	45
-5.446904	46
-3.537785	47
-2.824941	48
-2.745208	49
-0.195089	50
1.399296	51
5.363331	52
6.700640	53
7.386314	54
9.099900	55
12.433611	56
16.718018	57
18.093192	58
18.801816	59
19.168108	60
19.219211	61
20.334434	62
24.909926	63
26.236229	64
30.924022	65
32.253952	66
32.529367	67
32.675968	68
33.275839	69
36.031430	70
37.147186	71
40.320875	72
44.334467	73
46.907165	74
54.418366	75
55.091131	76
55.470305	77
62.939597	78
66.478628	79
67.426518	80
67.603959	81
69.707122	82
69.843246	83
74.848732	84
112.729191	85
163.795081	86
198.660139	87
209.375830	88
Fuente: Elaboración propia

sw <- shapiro.test(residuos)   # Prueba SW
sw

## 
##  Shapiro-Wilk normality test
## 
## data:  residuos
## W = 0.94132, p-value = 0.0005937

library(fastGraph)

qqnorm(residuos)   # Gráfico normal
qqline(residuos)   # Línea de referencia

Ejercicio de Pruebas de Normalidad, Tarea Individual.

Kevin Alexander Neira García

2026-04-16

1. Estime el siguiente modelo:

price = ˆα + ˆα1(lotsize) + ˆα2(sqrft) + ˆα3(bdrms) + E

2. Verique el supuesto de normalidad, a través de:

a) La prueba JB

b) La prueba KS

c) La prueba SW