En este informe se presentan los resultados de la Actividad 2 el cual corresponde a conceptos de modelos SARIMA(p, d, q) × (P, D, Q) con la implementación de la identificación, ajuste, Diagnostico, residuales, las pruebas para la evaluación de los modelos y el pronóstico en R-Studio para las bases de datos Wineind y Precipitaciones.
Identificar los Modelos (Base Wineind)
Se utilizará la siguiente Base de datos del paquete forecast: Wineind: Australian total wine sales, la cual cuenta con información desde Enero de 1980 hasta diciembre de 1993.
## Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
## 1980 15136 16733 20016 17708 18019 19227 22893 23739 21133 22591 26786 29740
## 1981 15028 17977 20008 21354 19498 22125 25817 28779 20960 22254 27392 29945
## 1982 16933 17892 20533 23569 22417 22084 26580 27454 24081 23451 28991 31386
## 1983 16896 20045 23471 21747 25621 23859 25500 30998 24475 23145 29701 34365
## 1984 17556 22077 25702 22214 26886 23191 27831 35406 23195 25110 30009 36242
## 1985 18450 21845 26488 22394 28057 25451 24872 33424 24052 28449 33533 37351
## 1986 19969 21701 26249 24493 24603 26485 30723 34569 26689 26157 32064 38870
## 1987 21337 19419 23166 28286 24570 24001 33151 24878 26804 28967 33311 40226
## 1988 20504 23060 23562 27562 23940 24584 34303 25517 23494 29095 32903 34379
## 1989 16991 21109 23740 25552 21752 20294 29009 25500 24166 26960 31222 38641
## 1990 14672 17543 25453 32683 22449 22316 27595 25451 25421 25288 32568 35110
## 1991 16052 22146 21198 19543 22084 23816 29961 26773 26635 26972 30207 38687
## 1992 16974 21697 24179 23757 25013 24019 30345 24488 25156 25650 30923 37240
## 1993 17466 19463 24352 26805 25236 24735 29356 31234 22724 28496 32857 37198
## 1994 13652 22784 23565 26323 23779 27549 29660 23356Identificar los Modelos
Se realiza inicialmente la identificación de los Modelos por medio de los gráficos con una diferencia ordinaria, una diferencia estacional y una diferencia ordianria-estacional y luego aplicamos las pruebas de estacionariedad.
|Base / p-value |Dickey- Fuller| Fhillips Perron| Urca |
|Base Original |0,01 |0,01 |-0,05 |
|d Ordinaria |0,01 | 0,01 |-7,73 |
|D Estacional |0,01 |0,01 |-3,61 |
|(D,d) Ord-Est |0,01 |0,01 |-7,22 |
Tabla 1: Pruebas de estacionariedad Base Wineind
##
## Augmented Dickey-Fuller Test
##
## data: z1
## Dickey-Fuller = -6.6415, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary##
## Phillips-Perron Unit Root Test
##
## data: z1
## Dickey-Fuller Z(alpha) = -125.51, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary##
## Augmented Dickey-Fuller Test
##
## data: diff(z1)
## Dickey-Fuller = -8.4847, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary##
## Phillips-Perron Unit Root Test
##
## data: diff(z1)
## Dickey-Fuller Z(alpha) = -161.42, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary##
## Augmented Dickey-Fuller Test
##
## data: diff(z1, 12)
## Dickey-Fuller = -4.1834, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary##
## Phillips-Perron Unit Root Test
##
## data: diff(z1, 12)
## Dickey-Fuller Z(alpha) = -139.76, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary##
## Augmented Dickey-Fuller Test
##
## data: diff(diff(z1, 12))
## Dickey-Fuller = -8.7747, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary##
## Phillips-Perron Unit Root Test
##
## data: diff(diff(z1, 12))
## Dickey-Fuller Z(alpha) = -171.58, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12840.6 -3204.2 497.5 3726.3 11807.0
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 0.004801 0.015717 0.305 0.76045
## z.diff.lag1 -0.639427 0.081965 -7.801 9.13e-13 ***
## z.diff.lag2 -0.671234 0.090843 -7.389 9.17e-12 ***
## z.diff.lag3 -0.519117 0.105068 -4.941 2.03e-06 ***
## z.diff.lag4 -0.331069 0.109486 -3.024 0.00293 **
## z.diff.lag5 -0.188153 0.104598 -1.799 0.07403 .
## z.diff.lag6 -0.414008 0.090259 -4.587 9.34e-06 ***
## z.diff.lag7 -0.169670 0.080878 -2.098 0.03757 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5075 on 152 degrees of freedom
## Multiple R-squared: 0.4622, Adjusted R-squared: 0.4339
## F-statistic: 16.33 on 8 and 152 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: 0.3054
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12494.2 -3177.4 464.3 4068.3 12231.8
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 -4.22553 0.57286 -7.376 1.01e-11 ***
## z.diff.lag1 2.57567 0.53118 4.849 3.06e-06 ***
## z.diff.lag2 1.87418 0.47181 3.972 0.00011 ***
## z.diff.lag3 1.34134 0.40629 3.301 0.00120 **
## z.diff.lag4 0.98549 0.32684 3.015 0.00301 **
## z.diff.lag5 0.75966 0.24033 3.161 0.00190 **
## z.diff.lag6 0.29481 0.15641 1.885 0.06136 .
## z.diff.lag7 0.07648 0.08195 0.933 0.35213
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5075 on 151 degrees of freedom
## Multiple R-squared: 0.792, Adjusted R-squared: 0.7809
## F-statistic: 71.86 on 8 and 151 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -7.3762
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12607.3 -1209.5 419.3 1750.7 6899.1
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 -0.65236 0.18397 -3.546 0.000532 ***
## z.diff.lag1 -0.25003 0.18013 -1.388 0.167327
## z.diff.lag2 -0.36078 0.17296 -2.086 0.038805 *
## z.diff.lag3 -0.25273 0.16656 -1.517 0.131428
## z.diff.lag4 -0.22709 0.15423 -1.472 0.143151
## z.diff.lag5 -0.07470 0.13823 -0.540 0.589757
## z.diff.lag6 -0.07355 0.11143 -0.660 0.510317
## z.diff.lag7 0.02940 0.08554 0.344 0.731588
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2633 on 140 degrees of freedom
## Multiple R-squared: 0.4827, Adjusted R-squared: 0.4531
## F-statistic: 16.33 on 8 and 140 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -3.546
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12347 -1378 -75 1554 8045
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 -5.15399 0.69510 -7.415 1.09e-11 ***
## z.diff.lag1 3.33336 0.65431 5.094 1.12e-06 ***
## z.diff.lag2 2.45174 0.58537 4.188 4.96e-05 ***
## z.diff.lag3 1.75569 0.49481 3.548 0.000529 ***
## z.diff.lag4 1.12502 0.38895 2.892 0.004438 **
## z.diff.lag5 0.72254 0.27962 2.584 0.010798 *
## z.diff.lag6 0.35743 0.17279 2.069 0.040441 *
## z.diff.lag7 0.17936 0.08521 2.105 0.037103 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2705 on 139 degrees of freedom
## Multiple R-squared: 0.8054, Adjusted R-squared: 0.7942
## F-statistic: 71.93 on 8 and 139 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -7.4147
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62Con estos resultados concluimos que la base de datos tiene una tendencia nula es decir se evidencia un comportamiento constante lo que nos indica que la base es estacionaria lo cual también se confirma con las pruebas, gráficamente si se puede evidenciar la presencia de ciclos por lo que se decide continuar con una diferencia estacional ya que no se hace necesaria la diferencia ordinaria.
Procedemos a crear el grafico para identificar el modelo por medio del ACF y el PACF, con el cual se obtienes los siguientes modelos (p, d, q) x (P, D, Q):
- (0, 0 ,22) x (1,1,0)
- (22,0,22) x (1,1,3)
- (22,0, 0) x (0,1,3)
- (0 ,0,22) x (1,1,3)
Construcción de los Modelos
modelo1<-stats::arima(z1,
order=c(0,0,22),
seasonal=list(order=c(1,1,0),
period=12), fixed = c(NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA))
modelo2<-stats::arima(z1,
order=c(22,0,22),
seasonal=list(order=c(1,1,3),
period=12), fixed = c(NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA))
modelo3<-stats::arima(z1,
order=c(22,0,0),
seasonal=list(order=c(0,1,3),
period=12), fixed = c(NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA))
modelo4<-stats::arima(z1,
order=c(0,0,22),
seasonal=list(order=c(1,1,3),
period=12), fixed = c(NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA))Ajsutes de los Modelos
A continuación, se procede a realizar el ajuste de los modelos planteados a partir de los coeficientes más significativos.
##
## Call:
## stats::arima(x = z1, order = c(0, 0, 22), seasonal = list(order = c(1, 1, 0),
## period = 12), fixed = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
## NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA))
##
## Coefficients:
## ma1 ma2 ma3 ma4 ma5 ma6 ma7 ma8 ma9
## 0.1184 -0.1627 0.3292 -0.0409 0.1760 0.0635 0.1453 -0.1130 0.1850
## s.e. 0.1337 0.1161 0.1448 0.1428 0.1088 0.1358 0.1352 0.1426 0.1468
## ma10 ma11 ma12 ma13 ma14 ma15 ma16 ma17
## 0.2267 -0.1339 -0.2061 -0.1062 0.0275 0.4004 -0.0948 0.3038
## s.e. 0.1121 0.1016 0.1764 0.1476 0.1279 0.1305 0.1187 0.1499
## ma18 ma19 ma20 ma21 ma22 sar1
## 0.0907 0.0886 0.1674 -0.1253 0.4311 -0.2827
## s.e. 0.1381 0.1013 0.1177 0.1193 0.1075 0.1304
##
## sigma^2 estimated as 3467940: log likelihood = -1411.33, aic = 2870.65## ma1 ma2 ma3 ma4 ma5 ma6
## 0.1886921877 0.0815979667 0.0123132528 0.3875955511 0.0540913149 0.3204694893
## ma7 ma8 ma9 ma10 ma11 ma12
## 0.1421454427 0.2148219727 0.1049421167 0.0225618470 0.0948914256 0.1224698614
## ma13 ma14 ma15 ma16 ma17 ma18
## 0.2365158942 0.4150978659 0.0013108302 0.2130041656 0.0223679712 0.2561280257
## ma19 ma20 ma21 ma22 sar1
## 0.1916115361 0.0786679363 0.1477218160 0.0000501747 0.0159561188##
## Call:
## stats::arima(x = z1, order = c(22, 0, 22), seasonal = list(order = c(1, 1, 3),
## period = 12), fixed = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
## NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
## NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA))
##
## Coefficients:
## ar1 ar2 ar3 ar4 ar5 ar6 ar7 ar8 ar9
## -0.0133 -0.4374 0.3486 -0.3603 0.0913 -0.361 0.5529 0.3119 0.3441
## s.e. 0.2552 0.2725 0.1530 0.1570 0.1563 0.113 0.1610 0.1785 0.2356
## ar10 ar11 ar12 ar13 ar14 ar15 ar16 ar17
## 0.0564 0.0656 -0.7404 0.2019 -0.2933 0.1916 -0.3047 0.1134
## s.e. 0.2151 0.1211 0.1310 0.2389 0.2251 0.1869 0.1521 0.1338
## ar18 ar19 ar20 ar21 ar22 ma1 ma2 ma3 ma4
## -0.0390 0.4395 0.3695 0.0521 0.4103 0.1704 0.3197 -0.1315 0.3421
## s.e. 0.1065 0.0880 0.1637 0.2019 0.1375 0.2632 0.2900 0.1631 0.1501
## ma5 ma6 ma7 ma8 ma9 ma10 ma11 ma12
## 0.3246 0.5199 -0.2313 -0.5689 -0.0559 0.1670 0.0027 -0.1879
## s.e. 0.1239 0.1890 0.2865 0.1791 0.2970 0.2516 0.1831 0.3147
## ma13 ma14 ma15 ma16 ma17 ma18 ma19 ma20
## -0.4072 -0.1567 0.4203 -0.0397 0.0532 -0.4130 0.1633 0.0167
## s.e. 0.1947 0.1668 0.2176 0.1686 0.2446 0.1831 0.2087 0.2441
## ma21 ma22 sar1 sma1 sma2 sma3
## 0.4736 -0.2144 0.5036 -0.1413 -0.454 -0.3968
## s.e. 0.1893 0.1921 0.1895 0.1859 NaN 0.1713
##
## sigma^2 estimated as 2292561: log likelihood = -1389.31, aic = 2876.61## ar1 ar2 ar3 ar4 ar5 ar6
## 4.792375e-01 5.565308e-02 1.233498e-02 1.186289e-02 2.800930e-01 9.170445e-04
## ar7 ar8 ar9 ar10 ar11 ar12
## 4.221483e-04 4.167881e-02 7.354420e-02 3.968078e-01 2.946563e-01 6.528463e-08
## ar13 ar14 ar15 ar16 ar17 ar18
## 1.999437e-01 9.772189e-02 1.538164e-01 2.383792e-02 1.992233e-01 3.575131e-01
## ar19 ar20 ar21 ar22 ma1 ma2
## 1.142391e-06 1.302779e-02 3.984384e-01 1.758826e-03 2.593991e-01 1.363126e-01
## ma3 ma4 ma5 ma6 ma7 ma8
## 2.110486e-01 1.230392e-02 5.041655e-03 3.482283e-03 2.105826e-01 9.739360e-04
## ma9 ma10 ma11 ma12 ma13 ma14
## 4.255750e-01 2.541670e-01 4.940917e-01 2.758386e-01 1.939341e-02 1.747870e-01
## ma15 ma16 ma17 ma18 ma19 ma20
## 2.801955e-02 4.072110e-01 4.141780e-01 1.308168e-02 2.178492e-01 4.728117e-01
## ma21 ma22 sar1 sma1 sma2 sma3
## 6.921971e-03 1.334465e-01 4.525619e-03 2.244530e-01 NaN 1.119795e-02##
## Call:
## stats::arima(x = z1, order = c(22, 0, 0), seasonal = list(order = c(0, 1, 3),
## period = 12), fixed = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
## NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA))
##
## Coefficients:
## ar1 ar2 ar3 ar4 ar5 ar6 ar7 ar8 ar9
## 0.0484 -0.1148 0.1919 -0.0221 0.2145 0.0295 0.1999 -0.1012 0.2323
## s.e. 0.0821 0.0867 0.0887 0.0852 0.0830 0.0850 0.0844 0.0960 0.0818
## ar10 ar11 ar12 ar13 ar14 ar15 ar16 ar17
## 0.0145 -0.0128 -0.6593 0.0129 -0.0388 0.1998 -0.0409 0.2233
## s.e. 0.0919 0.0649 0.1481 0.0873 0.0829 0.0952 0.0837 0.0937
## ar18 ar19 ar20 ar21 ar22 sma1 sma2 sma3
## 0.0335 0.1067 0.0492 0.1535 0.2117 0.0939 -0.5573 -0.1073
## s.e. 0.0928 0.0867 0.0901 0.0896 0.1098 0.1809 0.1957 0.1284
##
## sigma^2 estimated as 3980687: log likelihood = -1413.07, aic = 2878.13## ar1 ar2 ar3 ar4 ar5 ar6
## 0.2783510703 0.0939661842 0.0161596666 0.3979487431 0.0054327295 0.3644154264
## ar7 ar8 ar9 ar10 ar11 ar12
## 0.0096942895 0.1468678006 0.0026279102 0.4376492737 0.4218947879 0.0000089942
## ar13 ar14 ar15 ar16 ar17 ar18
## 0.4414176900 0.3201016605 0.0189080195 0.3129473909 0.0093159632 0.3594143650
## ar19 ar20 ar21 ar22 sma1 sma2
## 0.1102602090 0.2928292911 0.0446042967 0.0280669841 0.3023402152 0.0025570771
## sma3
## 0.2025410988##
## Call:
## stats::arima(x = z1, order = c(0, 0, 22), seasonal = list(order = c(1, 1, 3),
## period = 12), fixed = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
## NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA))
##
## Coefficients:
## ma1 ma2 ma3 ma4 ma5 ma6 ma7 ma8 ma9
## 0.1548 -0.1874 0.323 0.0199 0.1586 0.0394 0.1665 -0.0988 0.1838
## s.e. 0.1424 0.1168 0.154 0.1463 0.1103 0.1295 0.1337 0.1334 0.1436
## ma10 ma11 ma12 ma13 ma14 ma15 ma16 ma17
## 0.2355 -0.0882 -0.0906 -0.0597 0.0006 0.4325 -0.0527 0.2713
## s.e. 0.1127 0.1061 0.2066 0.1596 0.1310 0.1224 0.1435 0.1510
## ma18 ma19 ma20 ma21 ma22 sar1 sma1 sma2 sma3
## 0.1057 0.0809 0.1073 -0.1055 0.4493 0.0876 -0.5164 0.0897 -0.1852
## s.e. 0.1307 0.1173 0.1300 0.1234 0.1173 0.6893 0.6358 0.2563 0.1245
##
## sigma^2 estimated as 3388433: log likelihood = -1410, aic = 2873.99## ma1 ma2 ma3 ma4 ma5 ma6
## 1.394488e-01 5.550482e-02 1.895869e-02 4.461325e-01 7.633564e-02 3.807298e-01
## ma7 ma8 ma9 ma10 ma11 ma12
## 1.076271e-01 2.300606e-01 1.014360e-01 1.933939e-02 2.035942e-01 3.308429e-01
## ma13 ma14 ma15 ma16 ma17 ma18
## 3.545109e-01 4.981992e-01 2.851231e-04 3.569602e-01 3.737733e-02 2.101432e-01
## ma19 ma20 ma21 ma22 sar1 sma1
## 2.457103e-01 2.054459e-01 1.970857e-01 9.882473e-05 4.495428e-01 4.040152e-01
## sma2 sma3
## 2.329477e-01 5.275527e-03Se ajustan los modelos y se obtienen los siguientes a los cuales también se le excluyeron los coeficientes no significativos:
- (0, 0 ,22) x (1,1,0)
- (22,0,21) x (1,1,3)
- (22,0, 0) x (0,1,2)
- (0 ,0,22) x (0,1,3)
modelo1<-stats::arima(z1,
order=c(0,0,22),
seasonal=list(order=c(1,1,0),
period=12), fixed = c(0,0,NA,0,0,0,0,0,0,NA,0,0,0,0,NA,0,0,0,0,0,0,NA,NA))
modelo2<-stats::arima(z1,
order=c(22,0,21),
seasonal=list(order=c(1,1,3),
period=12), fixed = c(0,0,NA,NA,0,NA,NA,NA,0,0,0,NA,0,0,0,NA,0,0,NA,NA,0,NA,0,0,0,NA,NA,NA,0,NA,0,0,0,0,NA,0,NA,0,0,NA,0,0,NA,NA,0,0,NA))
modelo3<-stats::arima(z1,
order=c(22,0,0),
seasonal=list(order=c(0,1,2),
period=12), fixed = c(0,0,NA,0,NA,0,NA,0,NA,0,0,NA,0,0,NA,0,NA,0,0,0,NA,NA,0,NA))
modelo4<-stats::arima(z1,
order=c(0,0,22),
seasonal=list(order=c(0,1,3),
period=12), fixed = c(0,0,NA,0,0,0,0,0,0,NA,0,0,0,0,NA,0,NA,0,0,0,0,NA,0,0,NA))Se ajustan los modelos y se obtienen los siguientes a los cuales también se le excluyeron los coeficientes no significativos:
- (0, 0 ,22) x (1,1,0)
- (22,0,21) x (1,1,3)
- (22,0, 0) x (0,1,2)
- (0 ,0,22) x (0,1,3)
Diagnostico de los Modelos
## [1] 2889.339## [1] 2930.359## [1] 2892.762## [1] 2908.611Pruebas para los Residuales
Se procede a realizar el diagnostico de los modelos para elegir el mejor en términos de BIC y pruebas de los residuales:
Modelo 1
En este modelo podemos evidenciar en los residuales que se alinea a los datos reales pero en algunos intervalos se desvía del comportamiento real, se evidencia también una varianza constante lo que permite inferir en una media cercana a cero, en los gráficos del ACF y el PACF observamos que una barra se sale del límite y otra esta sobre este, en la gráfica de normalidad evidenciamos algunos datos alejados de la línea de normalidad lo que quizás nos dice que los residuales no son normales pero que se verificara con la prueba de normalidad.
## Jan Feb Mar Apr May
## 1980 15.135990 16.732989 20.015988 17.707988 18.018988
## 1981 -93.321570 1074.925229 -11.630846 3226.010387 1184.990612
## 1982 1381.456973 -322.515780 93.827659 2636.557947 2702.288789
## 1983 592.334931 517.279647 1917.149676 -2276.915747 3360.471403
## 1984 7.118951 2445.168112 1035.973144 -450.442626 2005.051781
## 1985 116.654822 1189.909043 -871.728786 536.327885 883.415560
## 1986 1118.185995 -519.581615 -1394.142930 2087.055739 -3141.971156
## 1987 2231.158188 -3665.572526 -2973.797995 3506.010496 -1118.022276
## 1988 1390.251143 1105.557660 -185.710608 -224.663564 -2010.168008
## 1989 -2812.242985 -2079.828137 1352.546714 -2531.049921 -3240.533736
## 1990 -3588.480769 -3623.281720 3537.916399 7262.550878 1752.035403
## 1991 -289.199534 2093.876143 -3924.697752 -9081.125127 2030.469992
## 1992 1586.739921 1292.219922 72.901200 -1121.074306 2352.418826
## 1993 2188.163892 850.343066 -209.394578 3154.705063 -347.287101
## Jun Jul Aug Sep Oct
## 1980 19.226986 22.892984 23.738982 21.132983 22.590982
## 1981 2685.348286 2306.043889 4450.732084 -412.277253 -266.494213
## 1982 172.181397 641.055451 -456.240016 2292.363749 438.095263
## 1983 366.115961 -1372.138128 1846.269487 1226.408693 -69.873105
## 1984 -937.211866 1518.561160 4405.053215 -1430.482306 991.972567
## 1985 291.161933 -2122.817181 -1231.329306 527.521424 2750.216949
## 1986 1343.909361 4513.768474 -219.845696 2149.086214 -2183.605560
## 1987 -878.753656 2715.352113 -8523.928200 247.602991 -168.463486
## 1988 2647.034557 531.719140 -1643.963958 -3750.209040 -492.985025
## 1989 -1383.173325 -3442.818593 1449.742618 -1675.383380 -1246.753226
## 1990 -420.454520 -2503.608471 913.070603 1868.022188 -1013.013760
## 1991 1431.041332 1713.967421 1159.234889 1426.129713 430.966439
## 1992 1100.140636 2681.086388 -2528.527428 -1808.035329 -2008.785436
## 1993 1138.659792 -1086.525385 5796.830468 -3027.378359 1615.843828
## Nov Dec
## 1980 26.785977 29.739975
## 1981 8.646766 -81.943532
## 1982 618.993113 945.404660
## 1983 587.246186 2636.289468
## 1984 -664.168242 2103.512377
## 1985 2759.975918 1013.398262
## 1986 -170.647693 1185.281455
## 1987 1301.629835 2421.907225
## 1988 1322.641151 -4013.908071
## 1989 -1350.552919 3513.083042
## 1990 1940.862282 -311.146493
## 1991 -1164.478471 2045.847713
## 1992 -308.575285 -1091.073538
## 1993 732.598473 -209.863319## Jan Feb Mar Apr May Jun Jul Aug
## 1980 15120.86 16716.27 19995.98 17690.29 18000.98 19207.77 22870.11 23715.26
## 1981 15121.32 16902.07 20019.63 18127.99 18313.01 19439.65 23510.96 24328.27
## 1982 15551.54 18214.52 20439.17 20932.44 19714.71 21911.82 25938.94 27910.24
## 1983 16303.67 19527.72 21553.85 24023.92 22260.53 23492.88 26872.14 29151.73
## 1984 17548.88 19631.83 24666.03 22664.44 24880.95 24128.21 26312.44 31000.95
## 1985 18333.35 20655.09 27359.73 21857.67 27173.58 25159.84 26994.82 34655.33
## 1986 18850.81 22220.58 27643.14 22405.94 27744.97 25141.09 26209.23 34788.85
## 1987 19105.84 23084.57 26139.80 24779.99 25688.02 24879.75 30435.65 33401.93
## 1988 19113.75 21954.44 23747.71 27786.66 25950.17 21936.97 33771.28 27160.96
## 1989 19803.24 23188.83 22387.45 28083.05 24992.53 21677.17 32451.82 24050.26
## 1990 18260.48 21166.28 21915.08 25420.45 20696.96 22736.45 30098.61 24537.93
## 1991 16341.20 20052.12 25122.70 28624.13 20053.53 22384.96 28247.03 25613.77
## 1992 15387.26 20404.78 24106.10 24878.07 22660.58 22918.86 27663.91 27016.53
## 1993 15277.84 18612.66 24561.39 23650.29 25583.29 23596.34 30442.53 25437.17
## Sep Oct Nov Dec
## 1980 21111.87 22568.41 26759.21 29710.26
## 1981 21372.28 22520.49 27383.35 30026.94
## 1982 21788.64 23012.90 28372.01 30440.60
## 1983 23248.59 23214.87 29113.75 31728.71
## 1984 24625.48 24118.03 30673.17 34138.49
## 1985 23524.48 25698.78 30773.02 36337.60
## 1986 24539.91 28340.61 32234.65 37684.72
## 1987 26556.40 29135.46 32009.37 37804.09
## 1988 27244.21 29587.99 31580.36 38392.91
## 1989 25841.38 28206.75 32572.55 35127.92
## 1990 23552.98 26301.01 30627.14 35421.15
## 1991 25208.87 26541.03 31371.48 36641.15
## 1992 26964.04 27658.79 31231.58 38331.07
## 1993 25751.38 26880.16 32124.40 37407.86## [1] 136 92
##
## Box-Ljung test
##
## data: et
## X-squared = 12.134, df = 15, p-value = 0.6688
##
## Runs Test
##
## data: as.factor(sign(et))
## Standard Normal = 0.13908, p-value = 0.8894
## alternative hypothesis: two.sided##
## Jarque Bera Test
##
## data: et
## X-squared = 78.072, df = 2, p-value < 2.2e-16Modelo 2
En este modelo podemos evidenciar en los residuales que se alinea a los datos reales un poco más ajustados, se evidencia también una varianza constante con algunos datos que se ven atípicos pero aun así se puede inferir que hay una media cercana a cero, en los gráficos del ACF y el PACF observamos todas las barras dentro del límite, en la gráfica de normalidad evidenciamos un poco menos de datos que el Modelo 1 alejados de la línea de normalidad lo que quizás nos dice que los residuales no son normales pero que se verificara con la prueba de normalidad.
## Jan Feb Mar Apr May
## 1980 15.135980 16.732977 20.015974 17.707975 18.018973
## 1981 -66.896189 771.865369 -21.795727 2287.022342 842.529420
## 1982 6.075327 -462.530104 -458.841478 2697.228849 1821.159723
## 1983 -238.065982 -537.887648 2155.185270 -1661.894330 3446.891866
## 1984 409.815609 2011.809040 1319.637275 846.350701 1974.348205
## 1985 -248.964098 133.482145 -894.572208 933.247455 -113.965139
## 1986 756.395518 -117.472311 -855.449482 1516.298612 -3410.469977
## 1987 303.764936 -2236.303272 -2660.792553 1732.433476 -1825.110698
## 1988 686.464595 523.544508 546.441203 -157.434854 -1042.114589
## 1989 -1638.748516 -1363.822452 2088.673300 -1541.854667 -2458.332181
## 1990 -2513.859351 -1820.744705 2205.263517 5864.532245 419.450768
## 1991 -1004.840063 3848.858193 -2040.199476 -5660.026987 -584.684221
## 1992 1714.940467 -1319.908297 -514.345827 -1931.970970 798.964521
## 1993 519.691367 -112.526199 1866.201046 3034.648687 2876.350267
## Jun Jul Aug Sep Oct
## 1980 19.226968 22.892960 23.738952 21.132956 22.590947
## 1981 1821.036350 1681.450262 3075.617632 -459.160531 -275.933899
## 1982 1303.202454 993.588943 41.449770 1109.918239 -617.714044
## 1983 -601.345543 -633.174747 899.100252 656.564079 -1109.934235
## 1984 -1180.803175 1422.483540 3497.453897 -1682.153710 402.577640
## 1985 855.484896 -2445.464184 -746.113034 -540.923242 1987.011486
## 1986 457.394615 1827.268737 1043.170579 1711.194436 -545.688452
## 1987 -911.863010 2487.320330 -5802.477297 808.555694 29.416971
## 1988 256.234221 -727.901771 -2193.126899 -2693.074415 800.685775
## 1989 -1457.056213 -1907.541554 -421.808024 -524.954518 1087.077136
## 1990 333.564680 -1109.399382 -11.587117 545.589196 -977.278530
## 1991 530.217792 900.453517 995.296811 1834.842831 1094.507570
## 1992 -664.435883 1294.109405 -1303.689684 -800.415182 -498.416153
## 1993 2517.573122 -808.514451 2744.605936 -1439.658343 768.411600
## Nov Dec
## 1980 26.785939 29.739935
## 1981 -417.449033 -345.387430
## 1982 -114.987617 -311.402163
## 1983 665.244559 1604.952173
## 1984 -800.620525 1357.897862
## 1985 1634.255551 2329.818786
## 1986 952.299587 2223.451345
## 1987 1684.601078 1666.445493
## 1988 2210.979535 -3143.342825
## 1989 180.530227 2341.906977
## 1990 1554.287446 136.723960
## 1991 -576.618417 437.339283
## 1992 166.863689 -883.584153
## 1993 -886.620833 -519.609834## Jan Feb Mar Apr May Jun Jul Aug
## 1980 15120.86 16716.27 19995.98 17690.29 18000.98 19207.77 22870.11 23715.26
## 1981 15094.90 17205.13 20029.80 19066.98 18655.47 20303.96 24135.55 25703.38
## 1982 16926.92 18354.53 20991.84 20871.77 20595.84 20780.80 25586.41 27412.55
## 1983 17134.07 20582.89 21315.81 23408.89 22174.11 24460.35 26133.17 30098.90
## 1984 17146.18 20065.19 24382.36 21367.65 24911.65 24371.80 26408.52 31908.55
## 1985 18698.96 21711.52 27382.57 21460.75 28170.97 24595.52 27317.46 34170.11
## 1986 19212.60 21818.47 27104.45 22976.70 28013.47 26027.61 28895.73 33525.83
## 1987 21033.24 21655.30 25826.79 26553.57 26395.11 24912.86 30663.68 30680.48
## 1988 19817.54 22536.46 23015.56 27719.43 24982.11 24327.77 35030.90 27710.13
## 1989 18629.75 22472.82 21651.33 27093.85 24210.33 21751.06 30916.54 25921.81
## 1990 17185.86 19363.74 23247.74 26818.47 22029.55 21982.44 28704.40 25462.59
## 1991 17056.84 18297.14 23238.20 25203.03 22668.68 23285.78 29060.55 25777.70
## 1992 15259.06 23016.91 24693.35 25688.97 24214.04 24683.44 29050.89 25791.69
## 1993 16946.31 19575.53 22485.80 23770.35 22359.65 22217.43 30164.51 28489.39
## Sep Oct Nov Dec
## 1980 21111.87 22568.41 26759.21 29710.26
## 1981 21419.16 22529.93 27809.45 30290.39
## 1982 22971.08 24068.71 29105.99 31697.40
## 1983 23818.44 24254.93 29035.76 32760.05
## 1984 24877.15 24707.42 30809.62 34884.10
## 1985 24592.92 26461.99 31898.74 35021.18
## 1986 24977.81 26702.69 31111.70 36646.55
## 1987 25995.44 28937.58 31626.40 38559.55
## 1988 26187.07 28294.31 30692.02 37522.34
## 1989 24690.95 25872.92 31041.47 36299.09
## 1990 24875.41 26265.28 31013.71 34973.28
## 1991 24800.16 25877.49 30783.62 38249.66
## 1992 25956.42 26148.42 30756.14 38123.58
## 1993 24163.66 27727.59 33743.62 37717.61## [1] 92 136
##
## Box-Ljung test
##
## data: et
## X-squared = 3.3524, df = 15, p-value = 0.9992
##
## Runs Test
##
## data: as.factor(sign(et))
## Standard Normal = -0.8352, p-value = 0.4036
## alternative hypothesis: two.sided##
## Jarque Bera Test
##
## data: et
## X-squared = 24.53, df = 2, p-value = 4.713e-06Modelo 3
En este modelo el comportamiento es muy similar al Modelo 2 podemos evidenciar en los residuales que se alinea a los datos reales un poco más ajustados, se evidencia también una varianza constante con algunos datos que se ven atípicos, pero aun así se puede inferir que hay una media cercana a cero.
En los gráficos del ACF y el PACF observamos todas las barras dentro del límite, en la gráfica de normalidad evidenciamos un poco menos de datos que el Modelo 1 alejados de la línea de normalidad lo que quizás nos dice que los residuales no son normales pero que se verificara con la prueba de normalidad.
## Jan Feb Mar Apr May Jun
## 1980 15.13599 16.73299 20.01598 17.70798 18.01898 19.22697
## 1981 -84.77985 982.43231 -56.92364 2884.14260 913.61558 2291.99687
## 1982 -83.76819 -470.17432 -524.97050 2376.17798 2270.31696 934.03967
## 1983 -714.10833 -179.19625 2048.07804 -1949.49359 3003.71465 -295.71805
## 1984 -369.37065 592.93453 1279.75563 -150.61572 1583.49897 -828.25175
## 1985 -1300.92529 239.66585 335.90010 -1442.30398 542.68860 201.54106
## 1986 -522.30335 -1007.13177 -1278.73190 1066.18780 -3671.85647 209.54883
## 1987 945.61154 -3545.91729 -4728.55222 2010.97339 -852.27435 -1691.07663
## 1988 570.29839 1254.10644 -777.96517 319.14517 -3027.75880 -1068.93907
## 1989 -2335.06599 -1730.03150 -1062.29031 -494.01723 -1526.47173 -1394.60778
## 1990 -2969.81844 -2004.71879 3018.06744 8400.78670 -533.41273 198.13462
## 1991 -1381.39492 2197.19020 -2390.17965 -7119.82681 221.64963 2343.26776
## 1992 923.37243 -638.08368 977.09352 -707.57895 1443.33336 806.33041
## 1993 2641.17651 1046.72895 329.44928 1719.04281 2066.47868 659.70404
## Jul Aug Sep Oct Nov Dec
## 1980 22.89297 23.73896 21.13296 22.59096 26.78595 29.73995
## 1981 1774.82109 3831.28213 -969.16974 -691.78703 -581.55362 -491.82861
## 1982 729.34469 -644.16675 1000.76304 -1012.98707 1029.17710 -173.60676
## 1983 -795.24080 1873.18313 661.69548 -1507.37978 -41.54534 1996.66582
## 1984 439.51526 3380.95536 -1865.11722 -332.68549 -1540.42368 2040.43570
## 1985 -3004.45980 44.31698 -968.26000 2547.50876 2419.56187 2234.68668
## 1986 3117.19116 655.22197 613.45926 -1450.43119 -896.52854 1998.47590
## 1987 3241.55274 -8662.66013 -551.00782 1067.77063 3416.33081 1626.04787
## 1988 2192.60455 -3482.30806 -3687.02506 54.21520 1749.46684 -3230.55690
## 1989 -2809.85106 -3361.82121 652.45864 372.23679 359.79730 2365.27224
## 1990 -2943.01871 -1100.53690 30.47923 857.30873 2578.63341 -1620.23572
## 1991 2267.95650 532.40063 2532.62618 257.54293 -932.47956 3111.64562
## 1992 925.08471 -1865.30125 280.61377 -1024.82040 -938.81347 -1664.23532
## 1993 -945.08110 4193.14725 -3386.95303 736.55801 748.61296 1070.68659## Jan Feb Mar Apr May Jun Jul Aug
## 1980 15120.86 16716.27 19995.98 17690.29 18000.98 19207.77 22870.11 23715.26
## 1981 15112.78 16994.57 20064.92 18469.86 18584.38 19833.00 24042.18 24947.72
## 1982 17016.77 18362.17 21057.97 21192.82 20146.68 21149.96 25850.66 28098.17
## 1983 17610.11 20224.20 21422.92 23696.49 22617.29 24154.72 26295.24 29124.82
## 1984 17925.37 21484.07 24422.24 22364.62 25302.50 24019.25 27391.48 32025.04
## 1985 19750.93 21605.33 26152.10 23836.30 27514.31 25249.46 27876.46 33379.68
## 1986 20491.30 22708.13 27527.73 23426.81 28274.86 26275.45 27605.81 33913.78
## 1987 20391.39 22964.92 27894.55 26275.03 25422.27 25692.08 29909.45 33540.66
## 1988 19933.70 21805.89 24339.97 27242.85 26967.76 25652.94 32110.40 28999.31
## 1989 19326.07 22839.03 24802.29 26046.02 23278.47 21688.61 31818.85 28861.82
## 1990 17641.82 19547.72 22434.93 24282.21 22982.41 22117.87 30538.02 26551.54
## 1991 17433.39 19948.81 23588.18 26662.83 21862.35 21472.73 27693.04 26240.60
## 1992 16050.63 22335.08 23201.91 24464.58 23569.67 23212.67 29419.92 26353.30
## 1993 14824.82 18416.27 24022.55 25085.96 23169.52 24075.30 30301.08 27040.85
## Sep Oct Nov Dec
## 1980 21111.87 22568.41 26759.21 29710.26
## 1981 21929.17 22945.79 27973.55 30436.83
## 1982 23080.24 24463.99 27961.82 31559.61
## 1983 23813.30 24652.38 29742.55 32368.33
## 1984 25060.12 25442.69 31549.42 34201.56
## 1985 25020.26 25901.49 31113.44 35116.31
## 1986 26075.54 27607.43 32960.53 36871.52
## 1987 27355.01 27899.23 29894.67 38599.95
## 1988 27181.03 29040.78 31153.53 37609.56
## 1989 23513.54 26587.76 30862.20 36275.73
## 1990 25390.52 24430.69 29989.37 36730.24
## 1991 24102.37 26714.46 31139.48 35575.35
## 1992 24875.39 26674.82 31861.81 38904.24
## 1993 26110.95 27759.44 32108.39 36127.31## [1] 92 124
##
## Box-Ljung test
##
## data: et
## X-squared = 8.0248, df = 15, p-value = 0.9228
##
## Runs Test
##
## data: as.factor(sign(et))
## Standard Normal = -0.71805, p-value = 0.4727
## alternative hypothesis: two.sided##
## Jarque Bera Test
##
## data: et
## X-squared = 80.813, df = 2, p-value < 2.2e-16Modelo 4
En este modelo podemos evidenciar en los residuales que se alinea a los datos reales un poco más ajustados, se evidencia también una varianza constante con algunos datos que se ven atípicos pero aun así se puede inferir que hay una media cercana a cero, en los gráficos del ACF y el PACF observamos que una barra se sale del límite y otra esta sobre este, en la gráfica de normalidad evidenciamos un poco menos de datos que el Modelo 1 alejados de la línea de normalidad lo que quizás nos dice que los residuales no son normales pero que se verificara con la prueba de normalidad.
## Jan Feb Mar Apr May
## 1980 15.135990 16.732989 20.015986 17.707986 18.018985
## 1981 -94.436576 1087.586037 -1.839909 3152.293397 1220.818323
## 1982 1165.848870 -810.520684 -142.760505 1715.325831 2236.134256
## 1983 -580.187300 1167.153848 2496.486547 -2508.634215 2544.826589
## 1984 622.104459 1449.689993 486.554498 968.707659 952.416249
## 1985 -343.297702 6.059110 -703.760747 126.240119 349.479165
## 1986 293.511373 -106.114660 -876.510431 2226.284061 -3469.442087
## 1987 2056.791621 -3515.690088 -3102.402475 2388.024914 411.070709
## 1988 308.781601 2077.398328 1646.603522 -907.959547 -2405.731357
## 1989 -1026.833980 -2514.430156 437.258129 -1934.998775 -3372.984395
## 1990 -2981.557742 -2208.303118 3747.681211 7541.164373 3230.080808
## 1991 510.796741 4227.550116 -4691.027625 -11190.035014 -505.255826
## 1992 247.363891 -1894.408936 2018.117391 2445.965874 1819.342562
## 1993 1812.115153 1237.768587 -129.596484 3139.199607 -684.170047
## Jun Jul Aug Sep Oct
## 1980 19.226982 22.892979 23.738976 21.132977 22.590975
## 1981 2428.394472 2161.698273 4350.861551 -880.273838 -738.323565
## 1982 -674.867027 -208.862734 -1740.334157 1887.932863 357.336403
## 1983 451.217450 -1131.604916 2920.462347 186.840667 -922.902905
## 1984 -867.835993 1995.690248 3467.483092 -969.369881 629.745806
## 1985 550.414090 -3426.180256 -2263.113531 864.149834 2008.506937
## 1986 812.050232 5639.009006 1329.787142 1942.345511 -2550.356226
## 1987 -1713.494337 1165.700305 -8228.638804 -1348.686856 1803.512123
## 1988 2068.200556 972.472542 1885.669162 -3459.879225 -828.349473
## 1989 -2565.448656 -3679.414770 166.353690 294.371520 -770.340900
## 1990 993.815036 210.327347 72.415187 2382.179235 791.180378
## 1991 1791.517063 1655.578708 -364.280421 -1310.037895 -565.890563
## 1992 795.945678 988.743077 -1206.435272 474.634974 -1919.827843
## 1993 633.139801 -1234.602223 6281.652535 -2717.043884 1826.208762
## Nov Dec
## 1980 26.785969 29.739967
## 1981 -259.420409 -179.429960
## 1982 161.408655 709.930554
## 1983 922.767905 2400.194001
## 1984 -640.089035 1473.603094
## 1985 3064.492188 408.902507
## 1986 -1149.623824 2303.791580
## 1987 998.840843 401.382956
## 1988 1548.520346 -4610.905540
## 1989 -2673.118827 4906.784202
## 1990 2981.737450 -1489.993451
## 1991 -1756.538551 2315.318818
## 1992 109.735839 -2170.248565
## 1993 1006.595216 -54.572363## Jan Feb Mar Apr May Jun Jul Aug
## 1980 15120.86 16716.27 19995.98 17690.29 18000.98 19207.77 22870.11 23715.26
## 1981 15122.44 16889.41 20009.84 18201.71 18277.18 19696.61 23655.30 24428.14
## 1982 15767.15 18702.52 20675.76 21853.67 20180.87 22758.87 26788.86 29194.33
## 1983 17476.19 18877.85 20974.51 24255.63 23076.17 23407.78 26631.60 28077.54
## 1984 16933.90 20627.31 25215.45 21245.29 25933.58 24058.84 25835.31 31938.52
## 1985 18793.30 21838.94 27191.76 22267.76 27707.52 24900.59 28298.18 35687.11
## 1986 19675.49 21807.11 27125.51 22266.72 28072.44 25672.95 25083.99 33239.21
## 1987 19280.21 22934.69 26268.40 25897.98 24158.93 25714.49 31985.30 33106.64
## 1988 20195.22 20982.60 21915.40 28469.96 26345.73 22515.80 33330.53 23631.33
## 1989 18017.83 23623.43 23302.74 27487.00 25124.98 22859.45 32688.41 25333.65
## 1990 17653.56 19751.30 21705.32 25141.84 19218.92 21322.18 27384.67 25378.58
## 1991 15541.20 17918.45 25889.03 30733.04 22589.26 22024.48 28305.42 27137.28
## 1992 16726.64 23591.41 22160.88 21311.03 23193.66 23223.05 29356.26 25694.44
## 1993 15653.88 18225.23 24481.60 23665.80 25920.17 24101.86 30590.60 24952.35
## Sep Oct Nov Dec
## 1980 21111.87 22568.41 26759.21 29710.26
## 1981 21840.27 22992.32 27651.42 30124.43
## 1982 22193.07 23093.66 28829.59 30676.07
## 1983 24288.16 24067.90 28778.23 31964.81
## 1984 24164.37 24480.25 30649.09 34768.40
## 1985 23187.85 26440.49 30468.51 36942.10
## 1986 24746.65 28707.36 33213.62 36566.21
## 1987 28152.69 27163.49 32312.16 39824.62
## 1988 26953.88 29923.35 31354.48 38989.91
## 1989 23871.63 27730.34 33895.12 33734.22
## 1990 23038.82 24496.82 29586.26 36599.99
## 1991 27945.04 27537.89 31963.54 36371.68
## 1992 24681.37 27569.83 30813.26 39410.25
## 1993 25441.04 26669.79 31850.40 37252.57## [1] 136 92
##
## Box-Ljung test
##
## data: et
## X-squared = 41.972, df = 15, p-value = 0.000227
##
## Runs Test
##
## data: as.factor(sign(et))
## Standard Normal = -0.36733, p-value = 0.7134
## alternative hypothesis: two.sided##
## Jarque Bera Test
##
## data: et
## X-squared = 156.89, df = 2, p-value < 2.2e-16Luego se realizan las pruebas a los residuales y por autocorrelación se descarta el Modelo 4 y se decide continuar con el Modelo 1 y 3 que son los que muestran los mejores resultados.
Base / p-value |BIC |Lyun Box |Runs test |Normalidad | Modelo 1 |2889,34 |0,6688 |0,8894 |2,2E-16 | Modelo 2 |2930,36 |0,9992 |0,4036 |0,000004713 | Modelo 3 |2892,76 |0,9228 |0,4727 |2,2E-16 | Modelo 4 |908,61 |0,000227 |0,7134 |2,2E-16 |
Pronóstico fuera de Muestra
Se realiza la proyección de los dos Modelos seleccionados con un h=6.
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 1994 17065.44 14169.13 19961.75 12635.92 21494.97
## Feb 1994 20864.96 17968.68 23761.25 16435.48 25294.45
## Mar 1994 24582.48 21686.20 27478.76 20153.00 29011.96
## Apr 1994 26807.59 23866.17 29749.01 22309.08 31306.11
## May 1994 25842.18 22900.79 28783.58 21343.71 30340.65
## Jun 1994 25214.98 22273.58 28156.37 20716.51 29713.45
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 1994 19872.74 17194.37 22551.10 15776.53 23968.95
## Feb 1994 22117.57 19439.21 24795.94 18021.37 26213.78
## Mar 1994 25196.82 22518.45 27875.19 21100.61 29293.03
## Apr 1994 26197.79 23454.23 28941.36 22001.87 30393.71
## May 1994 25490.97 22747.41 28234.54 21295.05 29686.89
## Jun 1994 23807.19 21013.08 26601.30 19533.97 28080.42Ecuaciones
Modelo 1: (0, 0 ,22) x (1,1,0)
Փ_1 (B^12 ) ∇_12=∝+ θ_22 (B)
(1+0,372B12)(1-B12)= ∝+(1+〖0,179B〗3+〖0,264B〗10+〖0,238B〗15+〖0,29B〗22)
Modelo 3: (22,0, 0) x (0,1,2)
∅_22 (B)∇_12=∝+ ʘ_2 (B^12)
(1-0,222B3-0,198B5-0,097B7-〖0,18B〗9+0,58B12-0,22B15-0,196B17-0,16B21-0,20B22)(1-B12)= ∝+(1-〖0,449B〗^24)
Identificar los Modelos (Base Precipitaciones)
Se utilizará la siguiente Base de datos “Precipitacioens.xlsx”, la cual cuenta con información desde enero de 2000 hasta diciembre del 2018, la cual corresponde a lasprecipitaciones del municipio de zipaquirá cuenca rio negro.
Identificar los Modelos
Se realiza inicialmente la identificación de los Modelos por medio de los gráficos con una diferencia ordinaria, una diferencia estacional y una diferencia ordianria-estacional y luego aplicamos las pruebas de estacionariedad.
|Base / p-value |Dickey- Fuller| Fhillips Perron| Urca |
|Base Original |0,02 |0,01 |-1,63 |
|d Ordinaria |0,01 | 0,01 |-7,58 |
|D Estacional |0,01 |0,01 |-4,64 |
|(D,d) Ord-Est |0,01 |0,01 |-8,08 |
Tabla 1: Pruebas de estacionariedad Base precipitaciones
##
## Augmented Dickey-Fuller Test
##
## data: z1
## Dickey-Fuller = -3.8707, Lag order = 6, p-value = 0.01631
## alternative hypothesis: stationary##
## Phillips-Perron Unit Root Test
##
## data: z1
## Dickey-Fuller Z(alpha) = -148.87, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary##
## Augmented Dickey-Fuller Test
##
## data: diff(z1)
## Dickey-Fuller = -7.7694, Lag order = 6, p-value = 0.01
## alternative hypothesis: stationary##
## Phillips-Perron Unit Root Test
##
## data: diff(z1)
## Dickey-Fuller Z(alpha) = -238.01, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary##
## Augmented Dickey-Fuller Test
##
## data: diff(z1, 12)
## Dickey-Fuller = -4.9861, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary##
## Phillips-Perron Unit Root Test
##
## data: diff(z1, 12)
## Dickey-Fuller Z(alpha) = -182.9, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary##
## Augmented Dickey-Fuller Test
##
## data: diff(diff(z1, 12))
## Dickey-Fuller = -9.0435, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary##
## Phillips-Perron Unit Root Test
##
## data: diff(diff(z1, 12))
## Dickey-Fuller Z(alpha) = -277.34, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1278.09 -270.24 -23.89 279.88 1914.23
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 -0.07556 0.04645 -1.627 0.1053
## z.diff.lag1 -0.57317 0.07893 -7.261 7.17e-12 ***
## z.diff.lag2 -0.45770 0.08830 -5.184 5.06e-07 ***
## z.diff.lag3 -0.45887 0.09189 -4.993 1.24e-06 ***
## z.diff.lag4 -0.45008 0.09039 -4.979 1.32e-06 ***
## z.diff.lag5 -0.20132 0.08859 -2.273 0.0241 *
## z.diff.lag6 -0.01374 0.08194 -0.168 0.8670
## z.diff.lag7 0.08858 0.06887 1.286 0.1998
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 503.6 on 212 degrees of freedom
## Multiple R-squared: 0.3589, Adjusted R-squared: 0.3347
## F-statistic: 14.84 on 8 and 212 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -1.6267
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1305.19 -318.00 -67.93 239.01 1920.73
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 -3.67546 0.48511 -7.577 1.10e-12 ***
## z.diff.lag1 2.04708 0.45624 4.487 1.19e-05 ***
## z.diff.lag2 1.52793 0.40766 3.748 0.00023 ***
## z.diff.lag3 0.99738 0.34449 2.895 0.00419 **
## z.diff.lag4 0.46001 0.27137 1.695 0.09153 .
## z.diff.lag5 0.18087 0.19976 0.905 0.36628
## z.diff.lag6 0.09635 0.13301 0.724 0.46962
## z.diff.lag7 0.10517 0.06910 1.522 0.12953
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 505 on 211 degrees of freedom
## Multiple R-squared: 0.7591, Adjusted R-squared: 0.75
## F-statistic: 83.11 on 8 and 211 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -7.5766
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2406.86 -344.90 -16.47 366.14 2279.52
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 -0.63600 0.13701 -4.642 6.24e-06 ***
## z.diff.lag1 -0.19629 0.13234 -1.483 0.140
## z.diff.lag2 -0.02338 0.12623 -0.185 0.853
## z.diff.lag3 -0.04870 0.11960 -0.407 0.684
## z.diff.lag4 -0.05180 0.11047 -0.469 0.640
## z.diff.lag5 0.02988 0.10077 0.297 0.767
## z.diff.lag6 0.02046 0.09063 0.226 0.822
## z.diff.lag7 0.02808 0.07048 0.398 0.691
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 658.5 on 200 degrees of freedom
## Multiple R-squared: 0.4203, Adjusted R-squared: 0.3971
## F-statistic: 18.12 on 8 and 200 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -4.642
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62##
## ###############################################
## # Augmented Dickey-Fuller Test Unit Root Test #
## ###############################################
##
## Test regression none
##
##
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2503.02 -422.49 21.88 392.22 2432.86
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## z.lag.1 -3.90815 0.48369 -8.080 6.15e-14 ***
## z.diff.lag1 2.16776 0.45213 4.795 3.19e-06 ***
## z.diff.lag2 1.67502 0.40291 4.157 4.78e-05 ***
## z.diff.lag3 1.21655 0.34509 3.525 0.000525 ***
## z.diff.lag4 0.80625 0.27909 2.889 0.004295 **
## z.diff.lag5 0.53986 0.21036 2.566 0.011013 *
## z.diff.lag6 0.32028 0.14238 2.250 0.025575 *
## z.diff.lag7 0.14774 0.07132 2.071 0.039609 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 686.8 on 199 degrees of freedom
## Multiple R-squared: 0.7906, Adjusted R-squared: 0.7821
## F-statistic: 93.89 on 8 and 199 DF, p-value: < 2.2e-16
##
##
## Value of test-statistic is: -8.0799
##
## Critical values for test statistics:
## 1pct 5pct 10pct
## tau1 -2.58 -1.95 -1.62Con estos resultados concluimos que la base de datos tiene una tendencia nula es decir se evidencia un comportamiento constante lo que nos indica que la base es estacionaria lo cual también se confirma con las pruebas, gráficamente si se puede evidenciar la presencia de ciclos por lo que se decide continuar con una diferencia estacional ya que no se hace necesaria la diferencia ordinaria.
Procedemos a crear el grafico para identificar el modelo por medio del ACF y el PACF, con el cual se obtienes los siguientes modelos (p, d, q) x (P, D, Q):
- (2, 0 2,) x (3,1,1)
- (0, 0 2,) x (3,1,1)
- (2,0, 0) x (3,1,1)
- (17 ,0,0) x (3,1,1)
Construcción de los Modelos
modelo1<-stats::arima(z1,
order=c(2,0,2),
seasonal=list(order=c(3,1,1),
period=12), fixed = c(NA,NA,NA,NA,NA,NA,NA,NA))
modelo2<-stats::arima(z1,
order=c(0,0,2),
seasonal=list(order=c(3,1,1),
period=12), fixed = c(NA,NA,NA,NA,NA,NA))
modelo3<-stats::arima(z1,
order=c(2,0,0),
seasonal=list(order=c(3,1,1),
period=12), fixed = c(NA,NA,NA,NA,NA,NA))
modelo4<-stats::arima(z1,
order=c(17,0,0),
seasonal=list(order=c(3,1,1),
period=12), fixed = c(NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA,NA))Ajsutes de los Modelos
A continuación, se procede a realizar el ajuste de los modelos planteados a partir de los coeficientes más significativos.
##
## Call:
## stats::arima(x = z1, order = c(2, 0, 2), seasonal = list(order = c(3, 1, 1),
## period = 12), fixed = c(NA, NA, NA, NA, NA, NA, NA, NA))
##
## Coefficients:
## ar1 ar2 ma1 ma2 sar1 sar2 sar3 sma1
## -0.0407 0.8110 0.2743 -0.6912 -0.0620 0.0273 -0.0603 -1.0000
## s.e. 0.0970 0.0798 0.1252 0.1119 0.0737 0.0748 0.0738 0.0754
##
## sigma^2 estimated as 186215: log likelihood = -1636.8, aic = 3291.61## ar1 ar2 ma1 ma2 sar1 sar2
## 3.376731e-01 0.000000e+00 1.480528e-02 1.687172e-09 2.005780e-01 3.579354e-01
## sar3 sma1
## 2.074710e-01 0.000000e+00##
## Call:
## stats::arima(x = z1, order = c(0, 0, 2), seasonal = list(order = c(3, 1, 1),
## period = 12), fixed = c(NA, NA, NA, NA, NA, NA))
##
## Coefficients:
## ma1 ma2 sar1 sar2 sar3 sma1
## 0.2372 0.2073 -0.0353 0.0212 -0.0779 -1.000
## s.e. 0.0668 0.0671 0.0704 0.0730 0.0727 0.071
##
## sigma^2 estimated as 195222: log likelihood = -1640.91, aic = 3295.83## ma1 ma2 sar1 sar2 sar3 sma1
## 0.0002358367 0.0011388168 0.3083094610 0.3861477027 0.1423953852 0.0000000000##
## Call:
## stats::arima(x = z1, order = c(2, 0, 0), seasonal = list(order = c(3, 1, 1),
## period = 12), fixed = c(NA, NA, NA, NA, NA, NA))
##
## Coefficients:
## ar1 ar2 sar1 sar2 sar3 sma1
## 0.2406 0.1676 -0.0410 0.0245 -0.0675 -1.0000
## s.e. 0.0676 0.0678 0.0707 0.0734 0.0730 0.0713
##
## sigma^2 estimated as 193558: log likelihood = -1639.91, aic = 3293.81## ar1 ar2 sar1 sar2 sar3 sma1
## 0.0002294926 0.0070897195 0.2810484906 0.3694140112 0.1779887302 0.0000000000##
## Call:
## stats::arima(x = z1, order = c(17, 0, 0), seasonal = list(order = c(3, 1, 1),
## period = 12), fixed = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
## NA, NA, NA, NA, NA, NA, NA, NA, NA))
##
## Coefficients:
## ar1 ar2 ar3 ar4 ar5 ar6 ar7 ar8 ar9
## 0.2441 0.1156 0.0004 0.0185 0.1790 -0.0150 0.0656 -0.0438 0.0623
## s.e. 0.0679 0.0703 0.0705 0.0706 0.0701 0.0653 0.0670 0.0642 0.0638
## ar10 ar11 ar12 ar13 ar14 ar15 ar16 ar17 sar1
## -0.0752 0.0936 -0.4039 0.0550 0.0375 -0.0295 0.0286 0.1490 0.3022
## s.e. 0.0667 0.0683 0.2371 0.0831 0.0825 0.0738 0.0728 0.0807 0.2592
## sar2 sar3 sma1
## -0.1106 -0.0599 -1.0000
## s.e. 0.1729 0.1062 0.0806
##
## sigma^2 estimated as 177104: log likelihood = -1631.7, aic = 3307.41## ar1 ar2 ar3 ar4 ar5 ar6
## 2.046997e-04 5.078971e-02 4.979325e-01 3.968899e-01 5.733729e-03 4.090250e-01
## ar7 ar8 ar9 ar10 ar11 ar12
## 1.645787e-01 2.480457e-01 8.861967e-05 4.227305e-02 4.978662e-01 4.689736e-01
## ar13 ar14 ar15 ar16 ar17 sar1
## 1.623951e-02 4.277501e-01 1.875860e-01 2.741937e-01 1.411785e-03 3.280256e-01
## sar2 sar3 sma1
## 4.991572e-01 4.310215e-01 1.377423e-02Se ajustan los modelos y se obtienen los siguientes a los cuales también se le excluyeron los coeficientes no significativos:
- (0, 0 ,22) x (1,1,0)
- (22,0,21) x (1,1,3)
- (22,0, 0) x (0,1,2)
- (0 ,0,22) x (0,1,3)
modelo1<-stats::arima(z1,
order=c(2,0,2),
seasonal=list(order=c(0,1,1),
period=12), fixed = c(0,NA,NA,NA,NA))
modelo2<-stats::arima(z1,
order=c(0,0,2),
seasonal=list(order=c(0,1,1),
period=12), fixed = c(NA,NA,NA))
modelo3<-stats::arima(z1,
order=c(2,0,0),
seasonal=list(order=c(0,1,1),
period=12), fixed = c(NA,NA,NA))
modelo4<-stats::arima(z1,
order=c(17,0,0),
seasonal=list(order=c(0,1,1),
period=12), fixed = c(NA,0,0,0,NA,0,0,0,NA,NA,0,0,NA,0,0,0,NA,NA))Se ajustan los modelos y se obtienen los siguientes a los cuales también se le excluyeron los coeficientes no significativos:
- (2, 0,2) x (0,1,1)
- (0, 0,2) x (0,1,1)
- (2, 0, 0) x (0,1,1)
- (17,0,0) x (0,1,1)
Diagnostico de los Modelos
## [1] 3302.572## [1] 3304.903## [1] 3302.746## [1] 3319.069Pruebas para los Residuales
Se procede a realizar el diagnostico de los modelos para elegir el mejor en términos de BIC y pruebas de los residuales:
Modelo 1
En este modelo podemos evidenciar en los residuales que están alejados de los datos reales no siguen la alineación del histórico, se evidencia también una varianza no tan constante lo que permite inferir que no existe una media cercana a cero, en los gráficos del ACF y el PACF observamos todas las barras dentro del límite lo que quiere decir que los residuales son Nulos, en la gráfica de normalidad evidenciamos algunos datos alejados de la línea de normalidad lo que quizás nos dice que los residuales no son normales pero que se verificara con la prueba de normalidad.
## Jan Feb Mar Apr May
## 2000 0.6249993 1.3289982 1.0279980 0.7769980 0.7779978
## 2001 -230.3170061 -735.4359537 -408.9393696 -314.2592869 -275.2507135
## 2002 -144.9556663 -178.4374305 690.7351732 108.5631486 1013.8837612
## 2003 -110.2688753 -39.1881712 -249.5484542 1039.5808562 -554.3256713
## 2004 312.6364879 -245.8083006 -175.4417261 984.3737005 351.0822514
## 2005 -234.3464496 483.1111312 -599.0538461 242.4935562 177.5856602
## 2006 -0.5742214 -39.8377773 612.5876996 1224.7173005 866.4288679
## 2007 -539.7921145 -615.2005437 -10.4140561 -61.0216321 -541.6292567
## 2008 -233.9003597 -274.5709065 378.3586077 -469.1239922 1.9147645
## 2009 85.7326213 -163.9100277 -235.1434493 -421.9922590 -391.2394071
## 2010 -128.8730769 -340.0359797 -221.8447305 830.2460698 602.9940694
## 2011 -246.4198813 538.3083498 375.8313287 1175.8782883 387.1851272
## 2012 99.6065567 -299.7259647 144.5070094 414.8429386 -352.6590183
## 2013 -84.1414569 129.8106953 -264.5325784 -1114.2159644 70.2540877
## 2014 92.3510989 287.0865791 -312.8377130 -387.4797026 -172.7014343
## 2015 75.6116514 338.3450315 15.1044714 -637.1689896 -766.7757292
## 2016 67.6914666 49.5854290 375.9744798 344.7173523 -111.5017119
## 2017 -1.0263862 -321.9921770 -512.5007282 -791.6783413 267.8227729
## 2018 -48.4839503 -15.8299193 250.3593155 312.1916431 226.2702542
## Jun Jul Aug Sep Oct
## 2000 0.7439978 1.1109973 0.5819977 1.0639973 0.6459976
## 2001 -271.9483825 -509.5535552 197.0028072 -22.3946375 58.4822981
## 2002 -515.9783613 -152.6270374 -162.2882439 -495.6296689 714.7115448
## 2003 172.1465094 -60.6177771 44.7343276 -402.3796747 -624.1501876
## 2004 60.9446612 -79.2764737 -318.7035250 490.9565491 733.7145863
## 2005 275.8713600 116.8161463 -173.2805614 -118.7380281 -561.5380910
## 2006 790.6330778 -430.7446064 -341.2259689 -939.0336584 200.1357401
## 2007 575.2588269 -468.8012969 239.2263164 -345.2989823 1749.6211577
## 2008 575.7143489 66.6934673 -320.7881922 -25.1629581 -812.7660100
## 2009 -199.2255423 83.7565950 -53.2318136 -166.9796045 276.9559349
## 2010 100.3294246 1015.2987305 -91.8251445 229.1968354 -249.0481772
## 2011 -343.9427131 -582.7263249 -137.3692563 -163.7996175 644.9516775
## 2012 -320.0210428 281.3742882 -26.3490739 -471.5938538 686.8062356
## 2013 -313.5840943 -254.4897232 400.8280728 -450.9351417 -18.0078073
## 2014 161.0973967 -139.3895808 134.5242868 -327.7238237 276.2904784
## 2015 55.0683038 -35.7784812 55.0810231 -380.0677499 -263.2651326
## 2016 79.7383383 -166.9031180 -299.1513141 -29.0908352 143.9477464
## 2017 445.5212517 -85.9077318 481.0298464 -254.1767528 -170.7698989
## 2018 -267.8109303 -40.7561526 -206.6217809 18.0708978 2.3803696
## Nov Dec
## 2000 0.8019977 0.2119984
## 2001 -187.0231380 223.4535088
## 2002 -171.3379879 100.7509058
## 2003 -411.9025915 -147.7045179
## 2004 -553.8976518 -191.0305267
## 2005 655.5269071 156.4076723
## 2006 725.5663174 -181.9908489
## 2007 -1001.1033337 -74.9445520
## 2008 1438.1330684 148.0507647
## 2009 -158.8032561 -245.7564562
## 2010 579.0386923 159.9192937
## 2011 586.6485093 -112.4473256
## 2012 -200.0338847 -204.0666155
## 2013 -591.9238406 11.8177977
## 2014 104.8949705 133.0451044
## 2015 -505.6941452 49.6727451
## 2016 -617.7671542 311.6943338
## 2017 207.6443937 764.8307143
## 2018 -22.2194242 -412.6441016## Jan Feb Mar Apr May Jun
## 2000 624.375001 1327.671002 1026.972002 776.223002 777.222002 743.256002
## 2001 513.317006 923.435954 582.939370 348.259287 323.250713 305.948383
## 2002 292.955666 527.437431 615.264827 452.436851 681.116239 578.978361
## 2003 402.268875 620.188171 837.548454 574.419144 1026.325671 275.853491
## 2004 234.363512 438.808301 623.441726 690.626299 920.917749 406.055339
## 2005 461.346450 538.888869 831.053846 843.506444 1013.414340 371.128640
## 2006 433.574221 701.837777 739.412300 1290.282699 1405.571132 1013.366922
## 2007 560.792114 629.200544 711.414056 1155.021632 1035.629257 492.741173
## 2008 525.900360 295.570907 837.641392 1023.123992 1025.835236 557.285651
## 2009 382.267379 682.910028 788.143449 1174.992259 894.239407 644.225542
## 2010 193.873077 354.035980 545.844730 927.753930 1023.005931 787.670575
## 2011 481.419881 727.691650 992.168671 1545.121712 1586.814873 1205.942713
## 2012 597.393443 775.725965 953.492991 1508.157061 1355.659018 851.021043
## 2013 418.141457 499.189305 883.532578 1209.215964 885.745912 544.584094
## 2014 151.648901 281.913421 705.837713 969.479703 905.701434 399.902603
## 2015 345.388349 443.654968 863.895529 1131.168990 975.775729 389.931696
## 2016 153.308533 268.414571 652.025520 1030.282648 1033.501712 566.261662
## 2017 318.026386 363.992177 683.500728 852.678341 706.177227 422.478748
## 2018 485.483950 498.829919 877.640684 1163.808357 1215.729746 793.810930
## Jul Aug Sep Oct Nov Dec
## 2000 1109.889003 581.418002 1062.936003 645.354002 801.198002 211.788002
## 2001 557.553555 285.997193 828.394637 470.517702 642.023138 205.546491
## 2002 603.627037 656.288244 839.629669 766.288455 762.337988 445.249094
## 2003 736.617777 527.265672 863.379675 714.150188 518.902591 166.704518
## 2004 734.276474 557.703525 793.043451 931.285414 728.897652 376.030527
## 2005 811.183854 525.280561 929.738028 848.538091 470.473093 428.592328
## 2006 1248.744606 859.225969 1104.033658 890.864260 863.433683 602.990849
## 2007 696.801297 317.773684 740.298982 775.378842 1065.103334 158.944552
## 2008 761.306533 431.788192 712.162958 939.766010 544.866932 556.949235
## 2009 464.243405 392.231814 514.979604 885.044065 694.803256 288.756456
## 2010 786.701269 764.825144 824.803165 1281.048177 882.961308 725.080706
## 2011 1100.726325 686.369256 889.799617 1147.048322 1157.351491 711.447326
## 2012 823.625712 613.349074 760.593854 1081.193764 1079.033885 430.066615
## 2013 491.489723 273.171927 504.935142 862.007807 710.923841 57.182202
## 2014 581.389581 251.475713 513.723824 835.709522 804.105030 223.954896
## 2015 570.778481 256.918977 479.067750 810.265133 647.694145 8.327255
## 2016 703.903118 359.151314 480.090835 938.052254 752.767154 132.305666
## 2017 556.907732 291.970154 551.176753 903.769899 707.355606 296.169286
## 2018 784.756153 533.621781 575.929102 1081.619630 804.219424 417.644102## [1] 94 107
##
## Box-Ljung test
##
## data: et
## X-squared = 11.051, df = 15, p-value = 0.749
##
## Runs Test
##
## data: as.factor(sign(et))
## Standard Normal = -1.1258, p-value = 0.2603
## alternative hypothesis: two.sided##
## Jarque Bera Test
##
## data: et
## X-squared = 48.177, df = 2, p-value = 3.456e-11Modelo 2
En este modelo se puede evidenciar que el comportamiento de los residuales sigue estando alejado de los datos históricos, se evidencia también una varianza un poco más constante que permite inferir que existe una media cercana a cero, en los gráficos del ACF y el PACF observamos todas las barras dentro del límite y solo una fuera lo que quiere decir que casi todos los residuales son Nulos, en la gráfica de normalidad evidenciamos algunos datos alejados de la línea de normalidad lo que quizás nos dice que los residuales no son normales pero que se verificara con la prueba de normalidad.
## Jan Feb Mar Apr May
## 2000 0.6249993 1.3289982 1.0279979 0.7769980 0.7779983
## 2001 -230.6391110 -734.8051901 -388.9817174 -281.0904051 -368.7271332
## 2002 -241.7942076 -302.8731198 695.8395179 26.1266420 894.8428224
## 2003 -21.6779379 -26.8449965 -203.8685865 1055.3490039 -524.8124602
## 2004 278.4562760 -402.6853668 -255.0762002 974.9068174 292.2961850
## 2005 20.8346255 479.7138479 -552.3753454 170.0293369 384.2671747
## 2006 -125.4534663 39.1982263 686.0308564 1271.8360005 818.2532396
## 2007 -464.9180605 -401.7422076 162.3762843 -34.7756430 -596.4453128
## 2008 254.8399498 -470.9384249 516.9595396 -603.3785644 33.7618304
## 2009 -201.7386978 43.8968972 -186.0512967 -295.6237192 -389.1993374
## 2010 -133.0384882 -353.1230040 -304.4919898 805.2417076 494.6070945
## 2011 -290.1319930 778.9326531 497.6042304 1243.5969381 502.3697025
## 2012 258.3992312 -71.3235961 273.7735877 585.6950890 -300.7569161
## 2013 95.8024925 158.9176599 -231.0206925 -1152.2961335 176.9285704
## 2014 66.9838451 61.5287432 -408.4020105 -538.5915691 -136.0104734
## 2015 81.6821814 246.2938792 40.1020093 -727.9983578 -670.2642154
## 2016 19.4629037 -196.4643522 294.9688832 199.9294589 -199.9734006
## 2017 77.3239351 -535.1532169 -484.6992803 -835.4792580 263.0617025
## 2018 -112.4145589 -132.1481831 423.7867222 300.7761504 262.4464185
## Jun Jul Aug Sep Oct
## 2000 0.7439984 1.1109980 0.5819984 1.0639980 0.6459984
## 2001 -356.2938698 -590.4602235 143.7480178 -92.5272059 -91.0684232
## 2002 -482.4166189 -178.8497453 111.8072681 -471.3978975 817.1382831
## 2003 47.6119544 219.4546402 -16.3456412 -282.0392439 -618.9271763
## 2004 -143.7195137 47.2695117 -243.7482104 597.8023892 785.4922867
## 2005 143.8829441 195.7622078 -187.6377110 20.6891117 -508.8865907
## 2006 840.0880734 -209.0556435 -67.2870943 -523.6022245 422.5833326
## 2007 584.6121540 -425.7309665 65.7978835 -216.2414661 1625.4833339
## 2008 565.0603909 61.2424677 -475.6957635 118.7682522 -788.3472061
## 2009 -99.6664445 19.5659182 -74.0537964 -290.7664862 296.6188442
## 2010 -92.0111341 1036.3218497 11.9456611 178.0396406 23.9259394
## 2011 -227.4540457 -260.1862522 206.3551513 55.2849882 733.7815241
## 2012 -230.5514278 489.5629859 59.1192153 -491.0019045 806.9423225
## 2013 -254.6062547 -468.9391918 364.4006170 -560.6724729 -183.2305084
## 2014 40.6576064 -240.8786201 -43.2906729 -345.3508129 126.9199361
## 2015 100.6093778 -36.1886790 -169.8087877 -416.4698691 -379.2300762
## 2016 3.0714913 -100.5705475 -369.1386110 13.6166796 111.3962848
## 2017 326.9082037 -329.9328708 332.3500666 -247.7931216 -315.3683151
## 2018 -255.5561972 83.8050112 -94.5984694 68.0357443 57.8240304
## Nov Dec
## 2000 0.8019984 0.2119995
## 2001 -205.8019861 155.2364492
## 2002 -132.7413882 -17.3214618
## 2003 -240.7933685 -185.3615728
## 2004 -590.8632800 -164.9323296
## 2005 754.3166103 185.0503102
## 2006 979.1672881 -255.5737713
## 2007 -926.3866936 -387.1761876
## 2008 1451.2734213 175.5345319
## 2009 -227.5029753 -322.1235860
## 2010 642.3582958 362.6136617
## 2011 710.8430371 -121.0900534
## 2012 -94.2720313 -321.1778538
## 2013 -578.4408329 -137.7667723
## 2014 117.6067634 -66.6512198
## 2015 -495.8685649 -104.7137216
## 2016 -667.6446254 225.3548680
## 2017 282.3473477 685.5975388
## 2018 -9.6114683 -388.9819042## Jan Feb Mar Apr May Jun
## 2000 624.37500 1327.67100 1026.97200 776.22300 777.22200 743.25600
## 2001 513.63911 922.80519 562.98172 315.09041 416.72713 390.29387
## 2002 389.79421 651.87312 610.16048 534.87336 800.15718 545.41662
## 2003 313.67794 607.84500 791.86859 558.65100 996.81246 400.38805
## 2004 268.54372 595.68537 703.07620 700.09318 979.70381 610.71951
## 2005 206.16537 542.28615 784.37535 915.97066 806.73283 503.11706
## 2006 558.45347 622.80177 665.96914 1243.16400 1453.74676 963.91193
## 2007 485.91806 415.74221 538.62372 1128.77564 1090.44531 483.38785
## 2008 37.16005 491.93842 699.04046 1157.37856 993.98817 567.93961
## 2009 669.73870 475.10310 739.05130 1048.62372 892.19934 544.66644
## 2010 198.03849 367.12300 628.49199 952.75829 1131.39291 980.01113
## 2011 525.13199 487.06735 870.39577 1477.40306 1471.63030 1089.45405
## 2012 438.60077 547.32360 824.22641 1337.30491 1303.75692 761.55143
## 2013 238.19751 470.08234 850.02069 1247.29613 779.07143 485.60625
## 2014 177.01615 507.47126 801.40201 1120.59157 869.01047 520.34239
## 2015 339.31782 535.70612 838.89799 1221.99836 879.26422 344.39062
## 2016 201.53710 514.46435 733.03112 1175.07054 1121.97340 642.92851
## 2017 239.67606 577.15322 655.69928 896.47926 710.93830 541.09180
## 2018 549.41456 615.14818 704.21328 1175.22385 1179.55358 781.55620
## Jul Aug Sep Oct Nov Dec
## 2000 1109.88900 581.41800 1062.93600 645.35400 801.19800 211.78800
## 2001 638.46022 339.25198 898.52721 620.06842 660.80199 273.76355
## 2002 629.84975 382.19273 815.39790 663.86172 723.74139 563.32146
## 2003 456.54536 588.34564 743.03924 708.92718 347.79337 204.36157
## 2004 607.73049 482.74821 686.19761 879.50771 765.86328 349.93233
## 2005 732.23779 539.63771 790.31089 795.88659 371.68339 399.94969
## 2006 1027.05564 585.28709 688.60222 668.41667 609.83271 676.57377
## 2007 653.73097 491.20212 611.24147 899.51667 990.38669 471.17619
## 2008 766.75753 586.69576 568.23175 915.34721 531.72658 529.46547
## 2009 528.43408 413.05380 638.76649 865.38116 763.50298 365.12359
## 2010 765.67815 661.05434 875.96036 1008.07406 819.64170 522.38634
## 2011 778.18625 342.64485 670.71501 1058.21848 1033.15696 720.09005
## 2012 615.43701 527.88078 780.00190 961.05768 973.27203 547.17785
## 2013 705.93919 309.59938 614.67247 1027.23051 697.44083 206.76677
## 2014 682.87862 429.29067 531.35081 985.08006 791.39324 423.65122
## 2015 571.18868 481.80879 515.46987 926.23008 637.86856 162.71372
## 2016 637.57055 429.13861 437.38332 970.60372 802.64463 218.64513
## 2017 800.93287 440.64993 544.79312 1048.36832 632.65265 375.40246
## 2018 660.19499 421.59847 525.96426 1026.17597 791.61147 393.98190## [1] 94 107
##
## Box-Ljung test
##
## data: et
## X-squared = 17.404, df = 15, p-value = 0.2953
##
## Runs Test
##
## data: as.factor(sign(et))
## Standard Normal = -0.13159, p-value = 0.8953
## alternative hypothesis: two.sided##
## Jarque Bera Test
##
## data: et
## X-squared = 35.922, df = 2, p-value = 1.584e-08Modelo 3
En este modelo se puede evidenciar que el comportamiento de los residuales sigue estando alejado de los datos históricos, se evidencia también una varianza un poco más constante que permite inferir que existe una media cercana a cero.
En los gráficos del ACF y el PACF observamos todas las barras dentro del límite y solo una fuera lo que quiere decir que casi todos los residuales son Nulos, en la gráfica de normalidad evidenciamos algunos datos alejados de la línea de normalidad lo que quizás nos dice que los residuales no son normales pero que se verificara con la prueba de normalidad.
## Jan Feb Mar Apr May
## 2000 0.6249993 1.3289981 1.0279976 0.7769976 0.7779977
## 2001 -228.1721203 -726.2972227 -369.6794206 -243.3620752 -288.1696365
## 2002 -214.7783137 -278.9349941 698.1553012 45.8985843 916.9750001
## 2003 -65.3133966 -40.3220504 -197.5787818 1058.8354059 -522.6689973
## 2004 353.9186432 -356.4918577 -234.2131798 964.0987951 319.0676095
## 2005 -53.1611575 508.9213477 -519.5838980 167.2061143 349.0264429
## 2006 -89.9976199 -11.3945977 644.9419735 1273.1316085 801.6658324
## 2007 -496.7915538 -496.8565654 160.6176764 22.8145005 -546.5488917
## 2008 137.9324695 -438.8338340 581.8939672 -606.3626732 59.9895487
## 2009 -158.6009599 -60.1073023 -250.1788034 -283.8340217 -381.7204265
## 2010 -141.3311938 -336.2294796 -263.6296464 836.1445121 533.4242433
## 2011 -308.8525663 715.5514758 444.0982183 1240.9033415 428.3853574
## 2012 180.2119082 -159.9912649 254.8431704 566.7547731 -310.9176795
## 2013 45.4647882 144.3790367 -201.0732683 -1151.5279336 167.5121414
## 2014 116.3278233 122.9447640 -376.7719178 -539.3434969 -134.7588365
## 2015 82.5060874 232.4447548 39.4352436 -737.6495419 -693.5395139
## 2016 79.5078570 -135.9025906 322.5518276 204.2186582 -189.0915121
## 2017 79.3385043 -482.7935681 -480.5071438 -839.1728759 320.4572426
## 2018 -81.2444691 -157.7973857 354.2236068 287.0543147 269.5582551
## Jun Jul Aug Sep Oct
## 2000 0.7439978 1.1109974 0.5819978 1.0639975 0.6459980
## 2001 -290.5649796 -546.7294690 189.1364363 -48.6581736 -50.0401597
## 2002 -540.0828463 -223.2232775 33.8661587 -466.6417056 832.4947630
## 2003 48.9725336 138.3654237 -12.3259303 -278.3611153 -652.5713759
## 2004 -125.6679693 -38.0576127 -305.7389546 595.2915718 778.5474393
## 2005 171.4185951 191.4559433 -234.2340903 -13.1383093 -536.6711365
## 2006 750.1024668 -367.6737029 -203.1412977 -628.4598223 407.8631207
## 2007 586.6558746 -421.0377998 110.2391801 -243.8282133 1636.2082856
## 2008 545.3012792 93.5500949 -469.2413046 64.0166973 -812.1971216
## 2009 -78.0851372 59.4951269 -28.0845631 -270.5860225 295.5815565
## 2010 -68.5607301 965.0733184 -65.5280473 147.4059889 -72.5077958
## 2011 -324.9993401 -399.3882405 120.0342048 59.3486149 757.1123585
## 2012 -262.4362646 432.3874595 66.3993821 -470.3904344 767.8018441
## 2013 -219.7862285 -359.1588771 395.0058242 -538.8832071 -139.6860394
## 2014 85.2606925 -175.7662267 -12.4760321 -341.4526852 145.9421324
## 2015 103.4760428 39.3481989 -85.6376962 -401.8615727 -378.0643317
## 2016 -19.6447944 -126.1515845 -358.7682726 22.1685534 123.9434136
## 2017 403.6377921 -243.3771911 333.0843107 -283.6738002 -305.1149064
## 2018 -295.7903903 37.2486398 -125.6774633 79.2127161 58.1422048
## Nov Dec
## 2000 0.8019981 0.2119992
## 2001 -231.8913133 140.6528862
## 2002 -151.9735026 3.8267932
## 2003 -258.4700561 -151.3525313
## 2004 -599.8760814 -224.2955551
## 2005 752.8150232 192.3604497
## 2006 989.7720279 -218.6139424
## 2007 -928.9745638 -396.3796090
## 2008 1483.9027516 189.7218710
## 2009 -225.4268998 -298.3414629
## 2010 593.0968629 342.3869570
## 2011 691.1232509 -149.5997569
## 2012 -117.6520510 -293.0657240
## 2013 -589.5344826 -95.0651931
## 2014 130.6258087 -37.0705720
## 2015 -475.0831292 -52.7552752
## 2016 -635.7937052 237.6199383
## 2017 265.3849557 700.1607204
## 2018 -9.6026620 -393.1019592## Jan Feb Mar Apr May Jun Jul
## 2000 624.3750 1327.6710 1026.9720 776.2230 777.2220 743.2560 1109.8890
## 2001 511.1721 914.2972 543.6794 277.3621 336.1696 324.5650 594.7295
## 2002 362.7783 627.9350 607.8447 515.1014 778.0250 603.0828 674.2233
## 2003 357.3134 621.3221 785.5788 555.1646 994.6690 399.0275 537.6346
## 2004 193.0814 549.4919 682.2132 710.9012 952.9324 592.6680 693.0576
## 2005 280.1612 513.0787 751.5839 918.7939 841.9736 475.5814 736.5441
## 2006 522.9976 673.3946 707.0580 1241.8684 1470.3342 1053.8975 1185.6737
## 2007 517.7916 510.8566 540.3823 1071.1855 1040.5489 481.3441 649.0378
## 2008 154.0675 459.8338 634.1060 1160.3627 967.7605 587.6987 734.4499
## 2009 626.6010 579.1073 803.1788 1036.8340 884.7204 523.0851 488.5049
## 2010 206.3312 350.2295 587.6296 921.8555 1092.5758 956.5607 836.9267
## 2011 543.8526 550.4485 923.9018 1480.0967 1545.6146 1186.9993 917.3882
## 2012 516.7881 635.9913 843.1568 1356.2452 1313.9177 793.4363 672.6125
## 2013 288.5352 484.6210 820.0733 1246.5279 788.4879 450.7862 596.1589
## 2014 127.6722 446.0552 769.7719 1121.3435 867.7588 475.7393 617.7662
## 2015 338.4939 549.5552 839.5648 1231.6495 902.5395 341.5240 495.6518
## 2016 141.4921 453.9026 705.4482 1170.7813 1111.0915 665.6448 663.1516
## 2017 237.6615 524.7936 651.5071 900.1729 653.5428 464.3622 714.3772
## 2018 518.2445 640.7974 773.7764 1188.9457 1172.4417 821.7904 706.7514
## Aug Sep Oct Nov Dec
## 2000 581.4180 1062.9360 645.3540 801.1980 211.7880
## 2001 293.8636 854.6582 579.0402 686.8913 288.3471
## 2002 460.1338 810.6417 648.5052 742.9735 542.1732
## 2003 584.3259 739.3611 742.5714 365.4701 170.3525
## 2004 544.7390 688.7084 886.4526 774.8761 409.2956
## 2005 586.2341 824.1383 823.6711 373.1850 392.6396
## 2006 721.1413 793.4598 683.1369 599.2280 639.6139
## 2007 446.7608 638.8282 888.7917 992.9746 480.3796
## 2008 580.2413 622.9833 939.1971 499.0972 515.2781
## 2009 367.0846 618.5860 866.4184 761.4269 341.3415
## 2010 738.5280 906.5940 1104.5078 868.9031 542.6130
## 2011 428.9658 666.6514 1034.8876 1052.8767 748.5998
## 2012 520.6006 759.3904 1000.1982 996.6521 519.0657
## 2013 278.9942 592.8832 983.6860 708.5345 164.0652
## 2014 398.4760 527.4527 966.0579 778.3742 394.0706
## 2015 397.6377 500.8616 925.0643 617.0831 110.7553
## 2016 418.7683 428.8314 958.0566 770.7937 206.3801
## 2017 439.9157 580.6738 1038.1149 649.6150 360.8393
## 2018 452.6775 514.7873 1025.8578 791.6027 398.1020## [1] 94 107
##
## Box-Ljung test
##
## data: et
## X-squared = 14.312, df = 15, p-value = 0.502
##
## Runs Test
##
## data: as.factor(sign(et))
## Standard Normal = -0.35731, p-value = 0.7209
## alternative hypothesis: two.sided##
## Jarque Bera Test
##
## data: et
## X-squared = 39.06, df = 2, p-value = 3.298e-09Modelo 4
En este modelo podemos evidenciar en los residuales que están alejados de los datos reales con picos bastante alejados, no siguen la alineación del histórico, se evidencia también una varianza no tan constante por varios datos alejados de la media y que sobresalen de los limites lo que permite inferir que no existe una media cercana a cero.
En los gráficos del ACF y el PACF observamos todas las barras dentro del límite lo que quiere decir que los residuales son Nulos, en la gráfica de normalidad evidenciamos algunos datos alejados de la línea de normalidad lo que quizás nos dice que los residuales no son normales pero que se verificara con la prueba de normalidad.
## Jan Feb Mar Apr May
## 2000 0.6249993 1.3289981 1.0279979 0.7769983 0.7779984
## 2001 -227.7837668 -725.2543049 -374.0005890 -357.2669684 -379.5593289
## 2002 -274.8922430 -230.1037518 683.9984332 21.9979858 929.0852108
## 2003 -47.1966671 2.9580415 -201.0051897 1060.2367137 -640.6725948
## 2004 162.2618370 -216.6825272 -152.4825132 978.8220086 297.2727011
## 2005 -168.8299408 446.8868308 -642.9667126 336.4900011 347.1285112
## 2006 -16.7544223 -24.0990071 705.0802935 1138.1211432 860.2534173
## 2007 -414.8759358 -335.2512156 110.4631334 -213.8907137 -571.1019032
## 2008 20.9444985 -335.4169914 160.3113786 -596.1843521 152.6384315
## 2009 199.5828513 -38.9087178 -221.9173992 -465.2263411 -416.7702144
## 2010 -118.7197917 -305.1896267 -295.1744306 691.7864742 469.1255371
## 2011 -355.5249734 758.8739685 413.3625756 1141.9758827 488.5291346
## 2012 206.8069519 -112.9817315 231.9075334 347.5487598 -382.7446889
## 2013 -14.4917586 260.6282629 -401.8436221 -1250.4575968 220.5680110
## 2014 35.3056441 119.8759814 -372.4795236 -376.6380373 -176.8775045
## 2015 113.5081375 326.8979338 33.1225493 -651.5131132 -646.9923932
## 2016 81.0630086 -44.9717529 350.9813636 276.7205914 -97.6378655
## 2017 56.2129715 -381.3291002 -445.5686018 -712.1142543 280.0501747
## 2018 -74.5187969 -85.1356921 405.1739556 320.0819051 120.3290153
## Jun Jul Aug Sep Oct
## 2000 0.7439982 1.1109976 0.5819981 1.0639981 0.6459982
## 2001 -329.9613571 -508.6200618 177.9030503 -165.0559438 -9.7387328
## 2002 -476.6184040 58.3438142 -94.2693711 -555.9869432 731.9531488
## 2003 310.1045417 -5.8470148 54.8472742 -513.3803784 -618.6595823
## 2004 -3.4026921 151.2117860 -254.4444076 338.7021603 612.3610186
## 2005 126.5247079 146.4645696 2.7033207 -83.9576948 -640.9828882
## 2006 944.0115946 -184.4588082 -204.3735954 -878.9515232 196.8416960
## 2007 717.6930923 -499.3128934 80.4381066 -338.1840816 1726.8065431
## 2008 508.7854736 48.6134037 -300.3216540 189.0267043 -837.1930363
## 2009 -176.9760306 93.4915946 -268.0187397 -95.1988229 422.3085748
## 2010 82.3266987 1101.7294947 59.8843125 219.8725580 -127.6448422
## 2011 -68.3308340 -342.0160435 1.2929276 -325.0454149 614.4603248
## 2012 -208.8638499 300.1836459 -101.9048829 -595.9042202 751.3604553
## 2013 -397.8309967 -490.5045805 406.6889981 -434.5537260 -57.9286862
## 2014 79.5129880 -286.9398256 103.3545776 -215.5029051 208.4582542
## 2015 33.1515120 -194.5787169 -121.1926407 -239.5264631 -199.6310850
## 2016 61.6677646 -118.0208464 -408.1902470 -26.6214024 128.6850905
## 2017 205.0665694 -159.6263131 531.7486356 -209.3789581 -212.6032207
## 2018 -213.6809172 157.9173119 -228.2434709 51.9457812 12.3580861
## Nov Dec
## 2000 0.8019981 0.2119990
## 2001 -132.9678233 248.6604711
## 2002 -152.7505642 158.3247833
## 2003 -241.5208321 -237.6165438
## 2004 -477.7742343 -83.3236783
## 2005 702.5070905 107.9424965
## 2006 648.4971665 -292.5417585
## 2007 -1155.6914476 -69.4602302
## 2008 1475.7494415 -88.5846358
## 2009 -234.5010294 -214.2318110
## 2010 685.5804738 118.3367630
## 2011 579.0854365 -56.7158813
## 2012 -110.4350311 -250.8206718
## 2013 -558.4950200 -7.6256033
## 2014 122.8175423 98.1775188
## 2015 -483.2530567 -49.1221897
## 2016 -612.9644473 262.9447688
## 2017 249.6859513 704.7080450
## 2018 34.5460911 -414.8419900## Jan Feb Mar Apr May Jun Jul
## 2000 624.3750 1327.6710 1026.9720 776.2230 777.2220 743.2560 1109.8890
## 2001 510.7838 913.2543 548.0006 391.2670 427.5593 363.9614 556.6201
## 2002 422.8922 579.1038 622.0016 539.0020 765.9148 539.6184 392.6562
## 2003 339.1967 578.0420 789.0052 553.7633 1112.6726 137.8955 681.8470
## 2004 384.7382 409.6825 600.4825 696.1780 974.7273 470.4027 503.7882
## 2005 395.8299 575.1132 874.9667 749.5100 843.8715 520.4753 781.5354
## 2006 449.7544 686.0990 646.9197 1376.8789 1411.7466 859.9884 1002.4588
## 2007 435.8759 349.2512 590.5369 1307.8907 1065.1019 350.3069 727.3129
## 2008 271.0555 356.4170 1055.6886 1150.1844 875.1116 624.2145 779.3866
## 2009 268.4171 557.9087 774.9174 1218.2263 919.7702 621.9760 454.5084
## 2010 183.7198 319.1896 619.1744 1066.2135 1156.8745 805.6733 700.2705
## 2011 590.5250 507.1260 954.6374 1579.0241 1485.4709 930.3308 860.0160
## 2012 490.1930 588.9817 866.0925 1575.4512 1385.7447 739.8638 804.8164
## 2013 348.4918 368.3717 1020.8436 1345.4576 735.4320 628.8310 727.5046
## 2014 208.6944 449.1240 765.4795 958.6380 909.8775 481.4870 728.9398
## 2015 307.4919 455.1021 845.8775 1145.5131 855.9924 411.8485 729.5787
## 2016 139.9370 362.9718 677.0186 1098.2794 1019.6379 584.3322 655.0208
## 2017 260.7870 423.3291 616.5686 773.1143 693.9498 662.9334 630.6263
## 2018 511.5188 568.1357 722.8260 1155.9181 1321.6710 739.6809 586.0827
## Aug Sep Oct Nov Dec
## 2000 581.4180 1062.9360 645.3540 801.1980 211.7880
## 2001 305.0969 971.0559 538.7387 587.9678 180.3395
## 2002 588.2694 899.9869 749.0469 743.7506 387.6752
## 2003 517.1527 974.3804 708.6596 348.5208 256.6165
## 2004 493.4444 945.2978 1052.6390 652.7742 268.3237
## 2005 349.2967 894.9577 927.9829 423.4929 477.0575
## 2006 722.3736 1043.9515 894.1583 940.5028 713.5418
## 2007 476.5619 733.1841 798.1935 1219.6914 153.4602
## 2008 411.3217 497.9733 964.1930 507.2506 793.5846
## 2009 607.0187 443.1988 739.6914 770.5010 257.2318
## 2010 613.1157 834.1274 1159.6448 776.4195 766.6632
## 2011 547.7071 1051.0454 1177.5397 1164.9146 655.7159
## 2012 688.9049 884.9042 1016.6395 989.4350 476.8207
## 2013 267.3110 488.5537 901.9287 677.4950 76.6256
## 2014 282.6454 401.5029 903.5417 786.1825 258.8225
## 2015 433.1926 338.5265 746.6311 625.2531 107.1222
## 2016 468.1902 477.6214 953.3149 747.9644 181.0552
## 2017 241.2514 506.3790 945.6032 665.3140 356.2920
## 2018 555.2435 542.0542 1071.6419 747.4539 419.8420## [1] 94 107
##
## Box-Ljung test
##
## data: et
## X-squared = 11.631, df = 15, p-value = 0.7067
##
## Runs Test
##
## data: as.factor(sign(et))
## Standard Normal = 0.67468, p-value = 0.4999
## alternative hypothesis: two.sided##
## Jarque Bera Test
##
## data: et
## X-squared = 45.204, df = 2, p-value = 1.528e-10Luego se realizan las pruebas a los residuales donde todos los modelos pasan las pruebas de autocorrelación y aleatoriedad, pero ninguno pasa las pruebas de normalidad aunque no se descartan por esta. Se decide elegir el Modelo 3 con el mejor BIC y el cual también tiene los p-value más altos.
Base / p-value |BIC |Lyun Box |Runs test |Normalidad | Modelo 1 |3321,99 |0,7796 |0,2603 |6,37E-14 | Modelo 2 |3319,46 |0,2953 |0,8953 |1,58E-08 | Modelo 3 |3317,44 |0,5000 |0,7209 |3,30E-09 | Modelo 4 |3381,67 |0,7067 |0,4999 |1,53E-10 |
Pronóstico fuera de Muestra
Se realiza la proyección del Modelo seleccionado con un h=6.
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2019 240.7144 -342.33807 823.7669 -650.9874 1132.416
## Feb 2019 409.0617 -190.41979 1008.5432 -507.7661 1325.890
## Mar 2019 731.9369 117.87910 1345.9948 -207.1835 1671.057
## Apr 2019 1089.1404 472.56246 1705.7183 146.1658 2032.115
## May 2019 1019.2519 401.61875 1636.8851 74.6635 1963.840
## Jun 2019 646.5068 28.60651 1264.4071 -298.4901 1591.504Ecuaciones
Modelo 3: (22,0, 0) x (0,1,2)
∅_2 (B)∇_12=∝+ ʘ_1 (B^12)
(1-0,2394B1-0,1711B3)(1-B^12)= ∝+(1-B^12)