1. Ajustar el mejor modelo ARIMA(p,d,q) según n BIC. Recuerde que además de ajustar el mejor modelo debe probar que los residuales sean ruido blanco gaussiano preferiblemente. Escribir la ecuación que describe el proceso generador de datos basado en esta familia de modelos.

  2. Proyectar h = 4 pasos adelante cada serie sobre un fan chart y realizar una descripción del fenómeno.

Observamos la serie en el siguiente grafico

Pruebas de estacionalidad dickey fulley

H0: z1 no es estacionaria

Ha: z1 es estacionaria

adf.test(SeriesA)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  SeriesA
## Dickey-Fuller = -2.6562, Lag order = 5, p-value = 0.3014
## alternative hypothesis: stationary

ya que p-value = 0.3014 se confirma que la serie no es estacionaria.

a continuación se procedera a realizar la diferencia para convertir la serie en estacionaria.

adf.test(diff(SeriesA))
## Warning in adf.test(diff(SeriesA)): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  diff(SeriesA)
## Dickey-Fuller = -9.9271, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary

dado que el p-values=0.01 se confirma que la serie es estacionaria con la primera diferencia.

Identifiquemos p y q

ts_cor(diff(SeriesA))

De acuerdo al grafico identificamos los siguientes coeficiente p y q

Estimaciom MODELO 1

Se realizará la primera estimación con p, d, q de (1,1,0) modelo AR

modeloAR1=stats::arima(SeriesA, order = c(1,1,0),
                     fixed =c(NA))
modeloAR1
## 
## Call:
## stats::arima(x = SeriesA, order = c(1, 1, 0), fixed = c(NA))
## 
## Coefficients:
##           ar1
##       -0.4139
## s.e.   0.0650
## 
## sigma^2 estimated as 0.113:  log likelihood = -64.53,  aic = 133.06
BIC(modeloAR1)
## [1] 139.6137
tt1=modeloAR1$coef[which(modeloAR1$coef!=0)]/sqrt(diag(modeloAR1$var.coef))
tt1
##       ar1 
## -6.362761

Diagnostico (residual)

residuals(modeloAR1)
## Time Series:
## Start = 1 
## End = 197 
## Frequency = 1 
##   [1]  0.016999990 -0.364130283 -0.465540134 -0.324155100  0.917229933
##   [6]  0.213850334 -0.182770067  0.558614967 -0.051689800 -0.224155100
##  [11] -0.341385033  0.575844900  0.089695234  0.117229933  0.082770067
##  [16] -0.400000000  0.134459866  0.024155100  0.158614967 -0.517229933
##  [21]  0.051689800  0.424155100  0.124155100  0.100000000 -0.058614967
##  [26]  0.158614967 -0.117229933 -0.182770067 -0.341385033  0.675844900
##  [31]  0.031080267  0.475844900 -0.351689800 -0.348310200 -0.041385033
##  [36] -0.300000000  0.375844900  0.306925167 -0.258614967  0.275844900
##  [41] -0.034459866 -0.182770067 -1.041385033  0.886149666  0.038005434
##  [46] -0.206925167 -0.200000000  0.217229933 -0.375844900  0.193074833
##  [51]  0.465540134 -0.575844900 -0.489695234  0.017229933  0.041385033
##  [56]  0.400000000 -0.234459866  0.634459866 -0.068919733 -0.765540134
##  [61]  0.251689800  0.006925167 -0.382770067  1.275844900 -0.220609533
##  [66] -0.231080267 -0.258614967 -0.224155100  0.358614967 -0.334459866
##  [71]  0.293074833  0.306925167  0.341385033 -0.775844900 -0.272465300
##  [76]  0.141385033 -0.058614967  0.058614967 -0.358614967 -0.065540134
##  [81]  0.141385033 -0.058614967 -0.341385033 -0.024155100 -0.058614967
##  [86]  0.158614967 -0.017229933  0.158614967 -0.217229933 -0.124155100
##  [91] -0.200000000  0.117229933 -0.017229933  0.058614967  0.641385033
##  [96]  0.148310200  0.158614967  0.082770067 -0.400000000  0.034459866
## [101] -0.317229933  0.534459866 -0.510304766  0.268919733  0.248310200
## [106] -0.300000000 -0.624155100  0.193074833  0.465540134 -0.275844900
## [111] -0.065540134  0.041385033  0.400000000  0.265540134  0.041385033
## [116] -0.400000000 -0.065540134 -0.458614967  0.093074833  0.324155100
## [121]  0.182770067  0.241385033 -0.217229933  0.075844900  0.282770067
## [126]  0.182770067 -0.058614967  0.058614967 -0.058614967 -0.041385033
## [131]  0.300000000 -0.475844900 -0.248310200  0.000000000  0.100000000
## [136] -0.458614967 -0.006925167  0.182770067 -0.058614967 -0.041385033
## [141] -0.100000000 -0.141385033  0.458614967 -0.093074833 -0.124155100
## [146]  0.200000000  0.582770067 -0.093074833 -0.224155100 -0.241385033
## [151]  0.317229933  0.165540134  0.200000000 -0.117229933 -0.382770067
## [156] -0.224155100  0.158614967  0.482770067 -0.034459866 -0.082770067
## [161] -0.100000000 -0.041385033  0.000000000  0.300000000 -0.075844900
## [166] -0.382770067 -0.124155100  0.100000000 -0.258614967  0.075844900
## [171]  0.482770067  0.665540134  0.206925167 -0.200000000 -0.182770067
## [176] -0.541385033 -0.306925167  0.158614967  0.182770067  0.241385033
## [181]  0.182770067  0.441385033 -0.734459866 -0.372465300  0.000000000
## [186]  0.200000000  0.182770067  0.141385033  0.041385033 -0.400000000
## [191]  0.834459866  0.613850334 -0.517229933 -0.048310200 -0.017229933
## [196] -0.541385033 -0.006925167
yfit=fitted(modeloAR1)
par(mfrow=c(3,2))
plot(yfit, col="red")
lines(SeriesA)
et1=residuals(modeloAR1)
plot(et1)
acf(et1)
pacf(et1)
qqPlot(et1)
## [1] 64 43
acf(abs(et1))

test de autocorrelación Ljum-box

H0 No hay autocorrelación serial Ha hay autocorrelación serial

#H0: r1 =r2 =r3………rlag = 0 no son significativos #Ha: al menos uno es diferente de 0

Box.test(et1, lag = 7, type = "Ljung-Box")
## 
##  Box-Ljung test
## 
## data:  et1
## X-squared = 20.82, df = 7, p-value = 0.004047
tsdiag(modeloAR1, gof.lag = 20)

Normalidad basado en Sesgo y curtosis

prueba de jarque-bera

H0: los residuales tienen distribución normal Ha: los residuales no tienen distribución normal

jarque.bera.test(et1)
## 
##  Jarque Bera Test
## 
## data:  et1
## X-squared = 10.024, df = 2, p-value = 0.006659

prueba de aleatoriedad

H0: los residuales son aleatorios es decir que no tienen comportamiento predecible Ha: Los residuales exhiben comportamiento (no son aleatorios)

Pronostico

forecast(modeloAR1, h=5)
##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 198       17.31723 16.88643 17.74803 16.65838 17.97608
## 199       17.35148 16.85214 17.85083 16.58780 18.11517
## 200       17.33731 16.74080 17.93381 16.42503 18.24958
## 201       17.34317 16.67737 18.00898 16.32492 18.36143
## 202       17.34075 16.60699 18.07450 16.21856 18.46293
plot(forecast(modeloAR1, h=5))
lines(yfit, col="red")

Estimaciom MODELO 2

Se realizara la segunda estimación con p, d, q de (6,1,0) Modelo AR

modeloAR2=stats::arima(SeriesA, order = c(6,1,0),
                     fixed =c(NA,NA,NA,NA,NA,NA))
modeloAR2
## 
## Call:
## stats::arima(x = SeriesA, order = c(6, 1, 0), fixed = c(NA, NA, NA, NA, NA, 
##     NA))
## 
## Coefficients:
##           ar1      ar2      ar3      ar4      ar5      ar6
##       -0.6208  -0.4183  -0.3796  -0.3321  -0.3367  -0.2299
## s.e.   0.0698   0.0802   0.0825   0.0832   0.0818   0.0715
## 
## sigma^2 estimated as 0.09451:  log likelihood = -47.41,  aic = 108.82
BIC(modeloAR2)
## [1] 131.7659
tt2=modeloAR2$coef[which(modeloAR2$coef!=0)]/sqrt(diag(modeloAR2$var.coef))
tt2
##       ar1       ar2       ar3       ar4       ar5       ar6 
## -8.893551 -5.214314 -4.599885 -3.992128 -4.117175 -3.216052
residuals(modeloAR2)
## Time Series:
## Start = 1 
## End = 197 
## Frequency = 1 
##   [1]  0.016999988 -0.332692352 -0.423992919 -0.392303658  0.702310317
##   [6]  0.147648362 -0.070127193  0.574421040  0.150484074  0.151105586
##  [11] -0.130430207  0.477641566  0.150524936  0.258528930  0.103920100
##  [16] -0.283809504  0.227900464  0.078952376  0.132937222 -0.490639981
##  [21] -0.061841365  0.286998556  0.185716065  0.184467631  0.019532658
##  [26]  0.242421896  0.090288661 -0.076262811 -0.369362443  0.551752278
##  [31]  0.011123604  0.579972254 -0.228638122 -0.193715498  0.015449473
##  [36] -0.287385883  0.209603571  0.187635245 -0.314280088  0.322759930
##  [41]  0.025807630 -0.038124281 -0.944888253  0.616263213 -0.149938311
##  [46] -0.154746111 -0.327448105  0.058037098 -0.355671126  0.269750667
##  [51]  0.271647222 -0.503944700 -0.468249787 -0.169649035 -0.167946754
##  [56]  0.326457031 -0.346859528  0.523902120  0.068829175 -0.509675906
##  [61]  0.265714045 -0.069461720 -0.398183851  1.169908659 -0.260261634
##  [66]  0.039096384 -0.093155126 -0.230165430  0.387148664 -0.321624483
##  [71]  0.069054667  0.241845726  0.411621523 -0.536486382 -0.205611819
##  [76] -0.013931275 -0.089436151 -0.057151118 -0.542651432 -0.284521369
##  [81]  0.056173089 -0.125388728 -0.404445964 -0.168599913 -0.226470124
##  [86]  0.089337523 -0.090014521  0.092820290 -0.210276288 -0.084797203
##  [91] -0.238436026  0.040700695 -0.114776057 -0.009360302  0.560763375
##  [96]  0.275424905  0.415010143  0.355600248 -0.144370537  0.219416030
## [101] -0.272450321  0.427861097 -0.543690276  0.176072700  0.146068477
## [106] -0.208918514 -0.580431868  0.054940824  0.243366646 -0.197877871
## [111] -0.138053684 -0.095885883  0.409338901  0.446442878  0.196907531
## [116] -0.264646272  0.045464230 -0.437346568  0.005252669  0.105205993
## [121]  0.058383693  0.235267367 -0.103836562  0.187837400  0.444119922
## [126]  0.340027495  0.112372955  0.167054998  0.022982286  0.088335939
## [131]  0.342580959 -0.429186845 -0.269517949 -0.147790439 -0.051116649
## [136] -0.536150705 -0.201618294 -0.085007116 -0.110837993 -0.076701347
## [141] -0.182824153 -0.214459936  0.442543325 -0.080065691 -0.071245785
## [146]  0.197414727  0.619659793  0.139804986  0.012787854 -0.200354065
## [151]  0.353520624  0.241428401  0.272158694 -0.093082758 -0.298005955
## [156] -0.205287544  0.094907704  0.369385094 -0.026954030 -0.061135934
## [161] -0.068072967  0.039186292  0.072432413  0.286677684 -0.092957795
## [166] -0.332339494 -0.179024304 -0.001784234 -0.317191640 -0.042392745
## [171]  0.289617777  0.682332365  0.487687769  0.149379143  0.096819405
## [176] -0.299022899 -0.267804770 -0.060633508 -0.108011462  0.062082311
## [181]  0.117341633  0.501468244 -0.456376024 -0.207360686 -0.101100819
## [186]  0.070875830  0.082972784  0.034670935 -0.027128668 -0.253795516
## [191]  0.890190037  0.766346805 -0.152693184  0.180920734  0.046479660
## [196] -0.395004317  0.021730750
yfit=fitted(modeloAR2)
par(mfrow=c(3,2))
plot(yfit, col="red")
lines(SeriesA)
et2=residuals(modeloAR2)
plot(et2)
acf(et2)
pacf(et2)
qqPlot(et2)
## [1] 64 43
acf(abs(et2))

test de autocorrelacion Ljum-box

H0 No hay autocorrelacion serial Ha hay autocorrelacion serial

H0: r1 =r2 =r3………rlag = 0 no son significativos Ha: al menos uno es diferente de 0

Box.test(et2, lag = 7, type = "Ljung-Box")
## 
##  Box-Ljung test
## 
## data:  et2
## X-squared = 0.42246, df = 7, p-value = 0.9997
tsdiag(modeloAR2,gof.lag=20)

Normalidad basado en Sesgo y curtosis

prueba de jarque-bera

H0: los residuales tienen distribución normal Ha: los residuales no tienen distribución normal

jarque.bera.test(et2)
## 
##  Jarque Bera Test
## 
## data:  et2
## X-squared = 9.6203, df = 2, p-value = 0.008147

prueba de aleatoriedad

H0: los residuales son aleatorios… no tienen comportamiento predecible Ha: Los residuales exhiben comportamiento (no son aleatorios)

runs.test(as.factor(sign(et2)))
## 
##  Runs Test
## 
## data:  as.factor(sign(et2))
## Standard Normal = -1.7829, p-value = 0.0746
## alternative hypothesis: two.sided
sign(et2)==0
## Time Series:
## Start = 1 
## End = 197 
## Frequency = 1 
##   [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [37] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [49] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [61] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [73] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [85] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [97] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [109] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [121] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [133] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [145] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [157] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [169] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [181] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [193] FALSE FALSE FALSE FALSE FALSE

Pronostico

forecast(modeloAR2, h=5)
##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 198       17.61258 17.21860 18.00656 17.01003 18.21513
## 199       17.69057 17.26921 18.11193 17.04616 18.33498
## 200       17.63104 17.18814 18.07393 16.95369 18.30839
## 201       17.67962 17.22618 18.13307 16.98614 18.37311
## 202       17.62180 17.15970 18.08389 16.91508 18.32851
plot(forecast(modeloAR2, h=5))
lines(yfit, col="red")

### Estimaciom MODELO 3

Se realizara la tercera estimación con p, d, q de (5,1,0) Modelo AR

modeloAR3=stats::arima(SeriesA, order = c(5,1,0),
                     fixed =c(NA,NA,NA,NA,NA))
modeloAR3
## 
## Call:
## stats::arima(x = SeriesA, order = c(5, 1, 0), fixed = c(NA, NA, NA, NA, NA))
## 
## Coefficients:
##           ar1      ar2      ar3      ar4      ar5
##       -0.5754  -0.3612  -0.3101  -0.2510  -0.2013
## s.e.   0.0702   0.0804   0.0816   0.0815   0.0721
## 
## sigma^2 estimated as 0.09965:  log likelihood = -52.43,  aic = 116.86
BIC(modeloAR3)
## [1] 136.5288
tt3=modeloAR3$coef[which(modeloAR3$coef!=0)]/sqrt(diag(modeloAR3$var.coef))
tt3
##       ar1       ar2       ar3       ar4       ar5 
## -8.192799 -4.494747 -3.800834 -3.081088 -2.790361
residuals(modeloAR3)
## Time Series:
## Start = 1 
## End = 197 
## Frequency = 1 
##   [1]  0.016999988 -0.341797061 -0.436149592 -0.403120891  0.721590841
##   [6]  0.151021258 -0.071663584  0.669718471  0.157853213  0.064168498
##  [11] -0.345212812  0.528706212  0.108855780  0.159283082  0.164465368
##  [16] -0.274444186  0.222534103 -0.061924493  0.167048569 -0.528419046
##  [21] -0.009204647  0.353172052  0.125016936  0.191045795  0.005112586
##  [26]  0.314261064 -0.029657802 -0.148741349 -0.372745710  0.559313453
##  [31]  0.010993939  0.557992068 -0.210488591 -0.181099493 -0.002528347
##  [36] -0.431949080  0.266522609  0.133476605 -0.174991791  0.343243662
##  [41]  0.017923163 -0.037882563 -1.060929539  0.666480348 -0.113919098
##  [46] -0.193528614 -0.248638243  0.154924953 -0.263494768  0.058024589
##  [51]  0.392367349 -0.502876318 -0.435502373 -0.175148568 -0.075962336
##  [56]  0.258781867 -0.329910128  0.699182519  0.059987001 -0.564804006
##  [61]  0.238431329 -0.132774243 -0.359908500  1.079069202 -0.160075313
##  [66]  0.102811666 -0.212884326 -0.193537175  0.346053267 -0.534892083
##  [71]  0.270609533  0.245639492  0.463392601 -0.581212345 -0.253610296
##  [76]  0.051180374 -0.189984774 -0.055933943 -0.503605278 -0.079820142
##  [81]  0.039079187 -0.125394739 -0.370683772 -0.133138581 -0.136613268
##  [86]  0.080583605 -0.085461115  0.148422146 -0.164000951 -0.101307033
##  [91] -0.231203461  0.021969150 -0.092216345 -0.007686501  0.633232802
##  [96]  0.260306762  0.405361267  0.269981842 -0.188033631  0.127508766
## [101] -0.399339223  0.458303810 -0.580096822  0.238207856  0.213161035
## [106] -0.236125139 -0.546493771 -0.006476138  0.377271417 -0.313226770
## [111] -0.083635035  0.005843621  0.467893563  0.321145484  0.146633022
## [116] -0.219717201  0.001248101 -0.481351765 -0.055485871  0.122609706
## [121]  0.113007821  0.317437090 -0.112106100  0.241212170  0.334077797
## [126]  0.264629207  0.056754904  0.130423811  0.042876796  0.012926518
## [131]  0.289909987 -0.453412660 -0.241845602 -0.143842812 -0.010749356
## [136] -0.532684487 -0.172331727  0.065470483 -0.100153772 -0.064775687
## [141] -0.155542874 -0.123197198  0.401361273 -0.099555962 -0.048111112
## [146]  0.201444787  0.627431384  0.085276413 -0.090362583 -0.160669100
## [151]  0.321523130  0.152232836  0.197000997 -0.031212019 -0.282683374
## [156] -0.202349271  0.022272334  0.375982552 -0.044151623  0.005953767
## [161] -0.018139021  0.021095279 -0.005821808  0.228740399 -0.052479238
## [166] -0.326835165 -0.151843375  0.004912963 -0.325308462 -0.052049405
## [171]  0.377341062  0.734484003  0.439038108  0.094475755  0.080613497
## [176] -0.423783903 -0.385214099 -0.119367855 -0.041436369  0.153152389
## [181]  0.187492775  0.590871090 -0.506346645 -0.272033719 -0.135728460
## [186]  0.041451444  0.069680634  0.048657222  0.155680796 -0.282667355
## [191]  0.866198492  0.676134511 -0.227589996  0.136687887  0.030855475
## [196] -0.419886378 -0.172159173
yfit=fitted(modeloAR3)
par(mfrow=c(3,2))
plot(yfit, col="red")
lines(SeriesA)
et3=residuals(modeloAR3)
plot(et3)
acf(et3)
pacf(et3)
qqPlot(et3)
## [1] 43 64
acf(abs(et3))

test de autocorrelacion Ljum-box

H0 No hay autocorrelacion serial Ha hay autocorrelacion serial

H0: r1 =r2 =r3………rlag = 0 no son significativos Ha: al menos uno es diferente de 0

Box.test(et3, lag = 7, type = "Ljung-Box")
## 
##  Box-Ljung test
## 
## data:  et3
## X-squared = 8.8566, df = 7, p-value = 0.2631
tsdiag(modeloAR3, gof.lag=20)

Normalidad basado en Sesgo y curtosis

prueba de jarque-bera

H0: los residuales tienen distribución normal Ha: los residuales no tienen distribución normal

jarque.bera.test(et3)
## 
##  Jarque Bera Test
## 
## data:  et3
## X-squared = 6.2439, df = 2, p-value = 0.04407

prueba de aleatoriedad

H0: los residuales son aleatorios… no tienen comportamiento predecible Ha: Los residuales exhiben comportamiento (no son aleatorios)

runs.test(as.factor(sign(et3)))
## 
##  Runs Test
## 
## data:  as.factor(sign(et3))
## Standard Normal = -0.49966, p-value = 0.6173
## alternative hypothesis: two.sided
sign(et3)==0
## Time Series:
## Start = 1 
## End = 197 
## Frequency = 1 
##   [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [37] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [49] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [61] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [73] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [85] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [97] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [109] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [121] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [133] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [145] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [157] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [169] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [181] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [193] FALSE FALSE FALSE FALSE FALSE

Pronostico

forecast(modeloAR3, h=5)
##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 198       17.56710 17.16255 17.97164 16.94840 18.18580
## 199       17.53859 17.09909 17.97809 16.86643 18.21075
## 200       17.57824 17.11067 18.04582 16.86315 18.29333
## 201       17.56434 17.08029 18.04839 16.82404 18.30464
## 202       17.48466 16.98595 17.98338 16.72194 18.24738
plot(forecast(modeloAR3, h=5))
lines(yfit, col="red")

Estimaciom MODELO 4

Se realizara la cuarta estimación con p, d, q de (0,1,1) Modelo MA

modeloAR4=stats::arima(SeriesA, order = c(0,1,1),
                     fixed =c(NA))
modeloAR4
## 
## Call:
## stats::arima(x = SeriesA, order = c(0, 1, 1), fixed = c(NA))
## 
## Coefficients:
##           ma1
##       -0.6994
## s.e.   0.0645
## 
## sigma^2 estimated as 0.1007:  log likelihood = -53.51,  aic = 111.02
BIC(modeloAR4)
## [1] 117.5735
tt4=modeloAR4$coef[which(modeloAR4$coef!=0)]/sqrt(diag(modeloAR4$var.coef))
tt4
##       ma1 
## -10.84158
residuals(modeloAR4)
## Time Series:
## Start = 1 
## End = 197 
## Frequency = 1 
##   [1]  0.016999987 -0.327777824 -0.452833823 -0.478049299  0.666181275
##   [6]  0.256970669  0.078136671  0.653330233  0.156012989  0.009017867
##  [11] -0.293636784  0.494627643  0.145894056  0.302024083  0.211223611
##  [16] -0.252273745  0.123564286 -0.013581264  0.190501361 -0.466766225
##  [21] -0.026448986  0.281501968  0.196878110  0.237693503  0.066239164
##  [26]  0.246326648 -0.027722944 -0.119388999 -0.383498823  0.531786841
##  [31]  0.071923510  0.650302193 -0.145188681 -0.201542723 -0.140955870
##  [36] -0.398582361  0.221237647  0.254730197 -0.121845631  0.314783046
##  [41]  0.020154405 -0.085904320 -1.060080156  0.558596297 -0.109326370
##  [46] -0.076461176 -0.253475767  0.122722960 -0.414169455  0.110336274
##  [51]  0.377167487 -0.436214879 -0.505081955 -0.253246526 -0.177116712
##  [56]  0.276127304 -0.206880824  0.655310744  0.058314222 -0.559215933
##  [61]  0.108893006 -0.123841912 -0.386613122  1.129608748 -0.009969072
##  [66]  0.093027785 -0.234937803 -0.264311874  0.215144354 -0.349531359
##  [71]  0.255543161  0.278722944  0.494934526 -0.553850430 -0.287354444
##  [76] -0.100971264 -0.170617744 -0.019327417 -0.413517298 -0.189207617
##  [81] -0.032328888 -0.122610325 -0.385751769 -0.169788835 -0.218747691
##  [86]  0.047011240 -0.067121064  0.153056564 -0.192954601 -0.134949471
##  [91] -0.294381577 -0.005885933 -0.104116530  0.027182505  0.619011025
##  [96]  0.332926759  0.432843838  0.302724301 -0.188279295  0.068320366
## [101] -0.352217790  0.453664313 -0.482714180  0.262397151  0.183516518
## [106] -0.171651379 -0.620050326 -0.033653630  0.276463171 -0.206645925
## [111] -0.044524971 -0.031140078  0.378221110  0.364522008  0.254941068
## [116] -0.221698151 -0.055052266 -0.538502705 -0.076620483  0.146412817
## [121]  0.202398865  0.341554643 -0.061121953  0.157252249  0.309979796
## [126]  0.316795086  0.121561595  0.185018304  0.029398947  0.020561170
## [131]  0.314380165 -0.380127364 -0.265855213 -0.185935033 -0.030040093
## [136] -0.521009578 -0.164386059 -0.014969073 -0.110469139 -0.077260411
## [141] -0.154034739 -0.207729520  0.354717179 -0.051916278 -0.036309444
## [146]  0.174605735  0.622116557  0.135098720 -0.005514040 -0.203856434
## [151]  0.257425955  0.180039741  0.325917017  0.027941332 -0.280458263
## [156] -0.296148182 -0.007121468  0.395019355  0.076270441  0.053342370
## [161] -0.062693170 -0.043846636 -0.030665660  0.278552910 -0.005184393
## [166] -0.303625884 -0.212351258 -0.048515193 -0.333930778 -0.033546033
## [171]  0.376538422  0.763345162  0.533871827  0.173381718  0.021260498
## [176] -0.485130736 -0.439292951 -0.107234711  0.025001698  0.217485802
## [181]  0.252106214  0.576319196 -0.496931248 -0.347546047 -0.243068342
## [186]  0.030001752  0.120982763  0.184613477  0.129115817 -0.309698390
## [191]  0.783401725  0.747899078 -0.076930926  0.146195698  0.002247015
## [196] -0.498428472 -0.148593182
yfit=fitted(modeloAR4)
plot(residuals(modeloAR4))

summary(residuals(modeloAR4))
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -1.060080 -0.201543 -0.009969  0.011549  0.215144  1.129609

Realizando la grafica de los residuales del modelo escojido podemos concluir que contiotuyen un ruido blanco de media cercana a 0.

par(mfrow=c(3,2))
plot(yfit, col="red")
lines(SeriesA)
et4=residuals(modeloAR4)
plot(et4)
acf(et4)
pacf(et4)
qqPlot(et4)
## [1] 64 43
acf(abs(et4))

test de autocorrelacion Ljum-box

H0 No hay autocorrelacion serial Ha hay autocorrelacion serial

H0: r1 =r2 =r3………rlag = 0 no son significativos Ha: al menos uno es diferente de 0

Box.test(et4, lag = 7, type = "Ljung-Box")
## 
##  Box-Ljung test
## 
## data:  et4
## X-squared = 14.472, df = 7, p-value = 0.04339
tsdiag(modeloAR4,gof.lag=20)

Normalidad basado en Sesgo y curtosis

prueba de jarque-bera

H0: los residuales tienen distribución normal Ha: los residuales no tienen distribución normal

jarque.bera.test(et4)
## 
##  Jarque Bera Test
## 
## data:  et4
## X-squared = 5.0015, df = 2, p-value = 0.08202

prueba de aleatoriedad

H0: los residuales son aleatorios… no tienen comportamiento predecible Ha: Los residuales exhiben comportamiento (no son aleatorios)

runs.test(as.factor(sign(et4)))
## 
##  Runs Test
## 
## data:  as.factor(sign(et4))
## Standard Normal = -3.0485, p-value = 0.0023
## alternative hypothesis: two.sided
sign(et4)==0
## Time Series:
## Start = 1 
## End = 197 
## Frequency = 1 
##   [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [37] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [49] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [61] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [73] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [85] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [97] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [109] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [121] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [133] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [145] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [157] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [169] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [181] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [193] FALSE FALSE FALSE FALSE FALSE

Pronostico

forecast(modeloAR4, h=5)
##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 198       17.50392 17.09718 17.91067 16.88187 18.12598
## 199       17.50392 17.07920 17.92865 16.85437 18.15348
## 200       17.50392 17.06195 17.94590 16.82798 18.17986
## 201       17.50392 17.04535 17.96250 16.80259 18.20525
## 202       17.50392 17.02933 17.97852 16.77809 18.22976
plot(forecast(modeloAR4, h=5))
lines(yfit, col="red")

### Estimaciom MODELO 5

Se realizara la quinta estimación con p, d, q de (1,1,1) Modelo ARMA

modeloAR5=stats::arima(SeriesA, order = c(1,1,1),
                     fixed =c(NA,NA))
modeloAR5
## 
## Call:
## stats::arima(x = SeriesA, order = c(1, 1, 1), fixed = c(NA, NA))
## 
## Coefficients:
##          ar1      ma1
##       0.2155  -0.8193
## s.e.  0.1011   0.0631
## 
## sigma^2 estimated as 0.09851:  log likelihood = -51.37,  aic = 108.74
BIC(modeloAR5)
## [1] 118.5766
tt5=modeloAR5$coef[which(modeloAR5$coef!=0)]/sqrt(diag(modeloAR5$var.coef))
tt5
##        ar1        ma1 
##   2.130938 -12.993605
tt5 <- modeloAR5$coef[which(modeloAR5$coef!=0)]/sqrt(diag(modeloAR5$var.coef))
tt2
##       ar1       ar2       ar3       ar4       ar5       ar6 
## -8.893551 -5.214314 -4.599885 -3.992128 -4.117175 -3.216052
residuals(modeloAR5)
## Time Series:
## Start = 1 
## End = 197 
## Frequency = 1 
##   [1]  0.0169999883 -0.3402074378 -0.4140621401 -0.4251136220  0.6900836271
##   [6]  0.1300453162  0.0469681135  0.6540056002  0.1014584755  0.0471522325
##  [11] -0.2393618190  0.5680871316  0.1136810922  0.3358981560  0.2317862161
##  [16] -0.2101131553  0.2140716366  0.0106942235  0.2302915482 -0.4544095635
##  [21]  0.0570313604  0.2820553803  0.1664263398  0.2363533452  0.0720959593
##  [26]  0.2806226027 -0.0131845468 -0.0676944774 -0.3339099207  0.5910765728
##  [31]  0.0118592237  0.6743781487 -0.1767803851 -0.1155191114 -0.0730951967
##  [36] -0.3598896070  0.2697906776  0.2132799611 -0.1468056095  0.3443784379
##  [41] -0.0040535016 -0.0602133003 -1.0277810831  0.6734395253 -0.2284267919
##  [46] -0.0793889927 -0.2650464034  0.1259453450 -0.4614700451  0.1296698565
##  [51]  0.3200276446 -0.5024510777 -0.4607995671 -0.2344426248 -0.2136416513
##  [56]  0.2249553159 -0.3019013750  0.6388565590 -0.0489922067 -0.5539253953
##  [61]  0.1754716347 -0.1639991694 -0.3912628352  1.1440853923 -0.1643627762
##  [66]  0.1377629065 -0.2086795841 -0.2063172455  0.2525104217 -0.3793244348
##  [71]  0.2969748613  0.2355530190  0.4714435531 -0.5783902979 -0.1799115560
##  [76] -0.0689622845 -0.1780573506 -0.0243351778 -0.4414926745 -0.1755158715
##  [81] -0.0653607340 -0.1751064641 -0.4219174049 -0.1810310559 -0.2698795331
##  [86]  0.0004314357 -0.1427543946  0.1045898692 -0.2574134511 -0.1462467457
##  [91] -0.3198254893 -0.0189372318 -0.1586238763 -0.0084125974  0.5715532978
##  [96]  0.2389715724  0.4173523894  0.2988447092 -0.1551452285  0.1590994196
## [101] -0.3127517152  0.5299664650 -0.5166560354  0.3491157168  0.1567200505
## [106] -0.1715933360 -0.5759310800  0.0358876020  0.2431882971 -0.2654084609
## [111] -0.0312434165 -0.0471528557  0.3613658822  0.3098649605  0.2323301370
## [116] -0.2096431329  0.0144472256 -0.5097167848 -0.0098605220  0.1272590722
## [121]  0.1611602842  0.3104907722 -0.0887110549  0.1919775073  0.3141865454
## [126]  0.3143170319  0.1359778875  0.2329657685  0.0693237209  0.0783534892
## [131]  0.3641979767 -0.3662606543 -0.1707675462 -0.1399163084 -0.0146387226
## [136] -0.5335480021 -0.1293863395 -0.0491190070 -0.1617990031 -0.1110140988
## [141] -0.1909580493 -0.2349052080  0.3290872229 -0.1381361088 -0.0485183085
## [146]  0.1602471151  0.5881886352  0.0741554765  0.0254202154 -0.1576183131
## [151]  0.3139652417  0.1710278232  0.3401295632  0.0355731274 -0.2277457053
## [156] -0.2219388396  0.0397110589  0.3894288861  0.0328580408  0.0700297216
## [161] -0.0426220008 -0.0133678740 -0.0109528047  0.2910259529 -0.0262132605
## [166] -0.2783696294 -0.1634169351 -0.0338936747 -0.3493243125 -0.0215527902
## [171]  0.3392330978  0.6917307333  0.4589914331  0.1760690385  0.0873679077
## [176] -0.4068622262 -0.3255878926 -0.0452125298  0.0198477784  0.1947080923
## [181]  0.2164238170  0.5557702745 -0.5308521692 -0.2409617694 -0.1974290899
## [186]  0.0382388806  0.0882226747  0.1507302306  0.1019450349 -0.3164725777
## [191]  0.8269178216  0.6619856114 -0.1007180409  0.2468015588  0.0591059641
## [196] -0.4300183054 -0.0445605439
yfit=fitted(modeloAR5)
par(mfrow=c(3,2))
plot(yfit, col="red")
lines(SeriesA)
et5=residuals(modeloAR5)
plot(et5)
acf(et5)
pacf(et5)
qqPlot(et5)
## [1] 64 43
acf(abs(et5))

test de autocorrelacion Ljum-box

H0 No hay autocorrelacion serial Ha hay autocorrelacion serial

H0: r1 =r2 =r3………rlag = 0 no son significativos Ha: al menos uno es diferente de 0

Box.test(et5, lag = 7, type = "Ljung-Box")
## 
##  Box-Ljung test
## 
## data:  et5
## X-squared = 7.3015, df = 7, p-value = 0.3982
tsdiag(modeloAR5,gof.lag=20)

Normalidad basado en Sesgo y curtosis

prueba de jarque-bera

H0: los residuales tienen distribución normal Ha: los residuales no tienen distribución normal

jarque.bera.test(et5)
## 
##  Jarque Bera Test
## 
## data:  et5
## X-squared = 6.0729, df = 2, p-value = 0.048

prueba de aleatoriedad

H0: los residuales son aleatorios… no tienen comportamiento predecible Ha: Los residuales exhiben comportamiento (no son aleatorios)

runs.test(as.factor(sign(et5)))
## 
##  Runs Test
## 
## data:  as.factor(sign(et5))
## Standard Normal = -3.0689, p-value = 0.002148
## alternative hypothesis: two.sided
sign(et5)==0
## Time Series:
## Start = 1 
## End = 197 
## Frequency = 1 
##   [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [37] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [49] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [61] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [73] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [85] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [97] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [109] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [121] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [133] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [145] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [157] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [169] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [181] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [193] FALSE FALSE FALSE FALSE FALSE

Pronostico

forecast(modeloAR5, h=5)
##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 198       17.47962 17.07739 17.88185 16.86446 18.09477
## 199       17.49678 17.06413 17.92943 16.83510 18.15846
## 200       17.50048 17.05479 17.94616 16.81886 18.18210
## 201       17.50127 17.04542 17.95713 16.80411 18.19844
## 202       17.50145 17.03614 17.96675 16.78983 18.21306
plot(forecast(modeloAR5, h=5))
lines(yfit, col="red")

Estimaciom MODELO 6

Se realizara la sexto estimación con p, d, q de (1,1,6) Modelo ARMA

modeloAR6=stats::arima(SeriesA, order = c(1,1,6),
                     fixed =c(NA,NA,NA,NA,NA,NA,NA))
modeloAR6
## 
## Call:
## stats::arima(x = SeriesA, order = c(1, 1, 6), fixed = c(NA, NA, NA, NA, NA, 
##     NA, NA))
## 
## Coefficients:
##          ar1      ma1     ma2      ma3     ma4      ma5     ma6
##       0.1851  -0.8053  0.0933  -0.1130  0.0112  -0.0799  0.0973
## s.e.  0.4009   0.3956  0.2601   0.0976  0.1133   0.1066  0.0710
## 
## sigma^2 estimated as 0.09677:  log likelihood = -49.64,  aic = 115.28
modeloAR6=stats::arima(SeriesA, order = c(1,1,6),
                     fixed =c(NA,NA,NA,NA,0,NA,NA))
modeloAR6
## 
## Call:
## stats::arima(x = SeriesA, order = c(1, 1, 6), fixed = c(NA, NA, NA, NA, 0, NA, 
##     NA))
## 
## Coefficients:
##          ar1     ma1     ma2      ma3  ma4      ma5     ma6
##       0.1719  -0.792  0.0839  -0.1081    0  -0.0737  0.0953
## s.e.  0.3898   0.384  0.2482   0.0842    0   0.0863  0.0679
## 
## sigma^2 estimated as 0.09677:  log likelihood = -49.65,  aic = 113.29
BIC(modeloAR6)
## [1] 136.2404
tt6=modeloAR6$coef[which(modeloAR6$coef!=0)]/sqrt(diag(modeloAR6$var.coef))
tt6
##        ar1        ma1        ma2        ma3        ma5        ma6 
##  0.4410030 -2.0626866  0.3381733 -1.2837468 -0.8538197  1.4044787
residuals(modeloAR6)
## Time Series:
## Start = 1 
## End = 197 
## Frequency = 1 
##   [1]  0.0169999880 -0.3367530086 -0.4394406559 -0.4009382942  0.7165251926
##   [6]  0.1763198726 -0.0739880602  0.6046605240  0.1050392802  0.0678785443
##  [11] -0.2243876658  0.5581702149  0.1999542398  0.2721120081  0.2196562829
##  [16] -0.2502428040  0.2441566418  0.0481480986  0.2088021114 -0.4564019987
##  [21] -0.0100387846  0.3432035988  0.1520432158  0.2013271099  0.0130489573
##  [26]  0.2698258827  0.0262367521 -0.0875911652 -0.3248815724  0.5862171363
##  [31]  0.0632157259  0.5935335707 -0.1839653765 -0.2011448012  0.0116403995
##  [36] -0.3450049937  0.2933259212  0.2064444497 -0.2128753983  0.3173753862
##  [41] -0.0037543798 -0.0637314721 -1.0114203429  0.6404195829 -0.0945588328
##  [46] -0.1825323410 -0.2717574336  0.0557810888 -0.3606887476  0.0982089325
##  [51]  0.3408888176 -0.5314331958 -0.4885519576 -0.2030119886 -0.1527794632
##  [56]  0.2590015743 -0.3444061159  0.5723928776  0.0043308306 -0.6049826383
##  [61]  0.2191355131 -0.1112185337 -0.3624621413  1.1432672425 -0.1617760168
##  [66]  0.0480817137 -0.1710648412 -0.2215466533  0.3800897905 -0.3885274251
##  [71]  0.2413540113  0.2617020891  0.4278132239 -0.5594883918 -0.2609068519
##  [76]  0.0241907241 -0.1403166979 -0.0175858848 -0.4987427573 -0.2058287937
##  [81] -0.0135909995 -0.1772240765 -0.4321948995 -0.2124060123 -0.2359170527
##  [86]  0.0200829139 -0.1333917407  0.0694039528 -0.2404972968 -0.1562862350
##  [91] -0.2721230969 -0.0057621769 -0.1151604436 -0.0272833831  0.5816538697
##  [96]  0.2422234108  0.3827791607  0.3033737824 -0.1567112662  0.2060133504
## [101] -0.2628743966  0.5314527043 -0.4692168316  0.2524083903  0.2237118016
## [106] -0.2337266458 -0.5608150994  0.0003284434  0.3166479496 -0.2689910246
## [111] -0.1093772726 -0.0660697724  0.3812692556  0.3502394826  0.1710496566
## [116] -0.2351392182  0.0115767852 -0.4354019829  0.0041899189  0.1687648268
## [121]  0.1182428073  0.2860168230 -0.1327314110  0.1770318024  0.3599192588
## [126]  0.3141011602  0.1303098371  0.1958796306  0.0866556024  0.0931116286
## [131]  0.3764781223 -0.3721858383 -0.2111586916 -0.1076004097 -0.0091173379
## [136] -0.5193283268 -0.1995451445 -0.0298969290 -0.1680435007 -0.1253710529
## [141] -0.2258258110 -0.2345019186  0.3536875607 -0.1200822252 -0.0917861454
## [146]  0.1709152226  0.5999580541  0.1133787235 -0.0330884661 -0.1490130796
## [151]  0.3527354489  0.2474646327  0.3014450568  0.0084698653 -0.2652991632
## [156] -0.1864820668  0.0772891185  0.4124417600  0.0031457410 -0.0097542131
## [161] -0.0518698437  0.0007397008  0.0269139646  0.2765622309 -0.0357272823
## [166] -0.3171144716 -0.1617017491 -0.0034001971 -0.3227636747 -0.0502487152
## [171]  0.3325824006  0.6822986559  0.4362553410  0.1007264810  0.0783389166
## [176] -0.3527741110 -0.2705616854  0.0080878198  0.0224478426  0.1668433414
## [181]  0.1632848132  0.5142446115 -0.5307600915 -0.2902927240 -0.1196340883
## [186]  0.0683840162  0.1207856621  0.0716656739  0.0660257434 -0.3218132111
## [191]  0.8325275517  0.7239997066 -0.1718563734  0.1942631101  0.0821323675
## [196] -0.3606045591 -0.0115591263
yfit=fitted(modeloAR6)
par(mfrow=c(3,2))
plot(yfit, col="red")
lines(SeriesA)
et6=residuals(modeloAR6)
plot(et6)
acf(et6)
pacf(et6)
qqPlot(et6)
## [1] 64 43
acf(abs(et6))

test de autocorrelacion Ljum-box

H0 No hay autocorrelacion serial Ha hay autocorrelacion serial

H0: r1 =r2 =r3………rlag = 0 no son significativos Ha: al menos uno es diferente de 0

Box.test(et6, lag = 7, type = "Ljung-Box")
## 
##  Box-Ljung test
## 
## data:  et6
## X-squared = 2.748, df = 7, p-value = 0.9073
tsdiag(modeloAR6,gof.lag=20)

Normalidad basado en Sesgo y curtosis

prueba de jarque-bera

H0: los residuales tienen distribución normal Ha: los residuales no tienen distribución normal

jarque.bera.test(et6)
## 
##  Jarque Bera Test
## 
## data:  et6
## X-squared = 6.3182, df = 2, p-value = 0.04246

prueba de aleatoriedad

H0: los residuales son aleatorios… no tienen comportamiento predecible Ha: Los residuales exhiben comportamiento (no son aleatorios)

runs.test(as.factor(sign(et6)))
## 
##  Runs Test
## 
## data:  as.factor(sign(et6))
## Standard Normal = -2.2067, p-value = 0.02734
## alternative hypothesis: two.sided
sign(et6)==0
## Time Series:
## Start = 1 
## End = 197 
## Frequency = 1 
##   [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [37] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [49] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [61] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [73] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [85] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##  [97] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [109] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [121] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [133] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [145] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [157] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [169] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [181] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [193] FALSE FALSE FALSE FALSE FALSE

Pronostico

forecast(modeloAR6, h=5)
##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 198       17.48607 17.08741 17.88474 16.87637 18.09578
## 199       17.50817 17.08171 17.93463 16.85595 18.16039
## 200       17.52568 17.07607 17.97530 16.83805 18.21331
## 201       17.56310 17.10297 18.02322 16.85940 18.26680
## 202       17.53601 17.06714 18.00487 16.81894 18.25307
plot(forecast(modeloAR6, h=5))
lines(yfit, col="red")

Modelo escogido

Dado que se escogio el modelo 4 se gera la ecuacion para el mismo.

\[xt=μ+△1wt+(−10.84)∗△1wt−1\]

plot(forecast(modeloAR4,h=4, fan=T))
lines(fitted(modeloAR4), col="red")

podemos observar en el grafico cómo la linea roja correspondiente al pronostico del modelo se ajusta bastante al comportamiento de data incialmente analizada.

hchart(forecast(modeloAR4,h=4, fan=T))

En el anterior grafico observamos como los cuatro pasos proyectados corresponden al comportamiento de la curva pronosticada.