SERIES DE TIEMPO
Curso de Macroeconometria 2024-2 Unicesar Aguachica
Juan Andrés Guerrero Duran
2024-09-24
1 SERIES DE TIEMPO
1.1 Creamos el proyecto en R studio.
1.1.1 Importamos los datos.
1.1.2 Consultamos que tipo de datos son.
class(pasajeros)## [1] "tbl_df" "tbl" "data.frame"1.1.3 Convertimos los datos en serie de tiempo.
library(tseries)
attach(pasajeros)
Pasajerosts=ts(PASAJEROS, start = c(1949,1), frequency = 12)
class(Pasajerosts)## [1] "ts"1.1.4 Resumen estadistico de la serie de tiempo
summary(Pasajerosts)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 104.0 180.0 265.5 280.3 360.5 622.01.1.5 Graficas
plot(Pasajerosts) #por defecto nos envia una grafica de lineas#
plot(Pasajerosts,type="l",pch =20) #lineas
plot(Pasajerosts,type="p",pch =20) #puntos
plot(Pasajerosts,type="o",pch =20) #puntos y lineas juntos
plot(Pasajerosts,type="b",pch =20) #puntos y lineas separados
plot(Pasajerosts,type="c",pch =20) #lineas sin puntos
plot(Pasajerosts,type="h",pch =20) #lineas verticales o columnas
plot(Pasajerosts,type="s",pch =20) #lineas que se unen formando cuadros
plot(Pasajerosts,type="S",pch =20) #toma del ultimo dato
plot(Pasajerosts,type="n",pch =20) #nada
Opte usted por las que mas le convenga para trabajar la
grafica.
Como ya sabemos podemos agregar titulos, subtitulos, nombres a los ejes, color y leyenda.
plot(Pasajerosts,type="o",pch =20, main="PASAJEROS",
sub="Datos obtenidos de Rdata",xlab="TIEMPO", ylab="N° DE PASAJEROS",
col= "blue")
legend("bottomright", c("Pasajeros"),lwd=c(1),col=c("blue"))
grid()
Podriamos graficar tambien con "forecast"
library(forecast)
library(ggplot2)
library(ggfortify)autoplot(Pasajerosts, ts.colour = "blue", ts.linetype = "dashed")
autoplot(Pasajerosts, ts.colour = "red", ts.linetype = "dashed")
autoplot(Pasajerosts, ts.colour = "green", ts.linetype = "dashed")
autoplot(Pasajerosts, ts.colour = "blue", ts.linetype = "dashed",
xlab="TIEMPO", ylab="N° DE PASAJEROS" ) +
ggtitle("Pasajeros de Areolineas Internacionales",
subtitle = "Datos tomados de Rdata")
1.1.6 Descomposición temporal
#grafica de la descomposición temporal.
plot(decompose(Pasajerosts), col="blue") 
#por defecto nos genera un esquema aditivo a menos que le indiquemos lo contrario.
plot(decompose(Pasajerosts,type = c("multiplicative")),col="blue")
plot(decompose(Pasajerosts,type = c("multiplicative")),col="blue")
# También podemos representar de manera individual estos componentes
# Representación gráfica de la tendencia
plot(Pasajerosts)
lines (decompose(Pasajerosts,type = c("additive"))$trend,col=2)
lines (decompose(Pasajerosts,type = c("multiplicative"))$trend,col=2)
legend("topleft", c("SERIE TEMPORAL","TENDENCIA"),lwd=c(1,2),col=c("black",2))
grid()
# Representación gráfica de la estacionalidad
plot (decompose(Pasajerosts,type = c("additive"))$seasonal,col=2)
plot (decompose(Pasajerosts,type = c("multiplicative"))$seasonal,col=2)
legend("topleft", c("Estacionalidad"),lwd=c(2), col=c(2))
grid()
# Representación gráfica del componente irregular
plot(decompose(Pasajerosts,type = c("additive"))$random,col="red")
plot(decompose(Pasajerosts,type = c("multiplicative"))$random,col="red")
legend("topleft", c("RANDOM"),lwd=c(2), col=c(2))
grid()
1.1.7 Estacionariedad de la serie.
# Transformación logarítmica de los datos.
lnpasajerosts=log(Pasajerosts)
# Revisamos como queda
lnpasajerosts## Jan Feb Mar Apr May Jun Jul Aug
## 1949 4.718499 4.770685 4.882802 4.859812 4.795791 4.905275 4.997212 4.997212
## 1950 4.744932 4.836282 4.948760 4.905275 4.828314 5.003946 5.135798 5.135798
## 1951 4.976734 5.010635 5.181784 5.093750 5.147494 5.181784 5.293305 5.293305
## 1952 5.141664 5.192957 5.262690 5.198497 5.209486 5.384495 5.438079 5.488938
## 1953 5.278115 5.278115 5.463832 5.459586 5.433722 5.493061 5.575949 5.605802
## 1954 5.318120 5.236442 5.459586 5.424950 5.455321 5.575949 5.710427 5.680173
## 1955 5.488938 5.451038 5.587249 5.594711 5.598422 5.752573 5.897154 5.849325
## 1956 5.648974 5.624018 5.758902 5.746203 5.762051 5.924256 6.023448 6.003887
## 1957 5.752573 5.707110 5.874931 5.852202 5.872118 6.045005 6.142037 6.146329
## 1958 5.828946 5.762051 5.891644 5.852202 5.894403 6.075346 6.196444 6.224558
## 1959 5.886104 5.834811 6.006353 5.981414 6.040255 6.156979 6.306275 6.326149
## 1960 6.033086 5.968708 6.037871 6.133398 6.156979 6.282267 6.432940 6.406880
## Sep Oct Nov Dec
## 1949 4.912655 4.779123 4.644391 4.770685
## 1950 5.062595 4.890349 4.736198 4.941642
## 1951 5.214936 5.087596 4.983607 5.111988
## 1952 5.342334 5.252273 5.147494 5.267858
## 1953 5.468060 5.351858 5.192957 5.303305
## 1954 5.556828 5.433722 5.313206 5.433722
## 1955 5.743003 5.613128 5.468060 5.627621
## 1956 5.872118 5.723585 5.602119 5.723585
## 1957 6.001415 5.849325 5.720312 5.817111
## 1958 6.001415 5.883322 5.736572 5.820083
## 1959 6.137727 6.008813 5.891644 6.003887
## 1960 6.230481 6.133398 5.966147 6.068426# Graficamos la serie logarítmica.
plot(lnpasajerosts,type="o",pch =20, main="PASAJEROS",
sub="Datos obtenidos de Rdata",xlab="TIEMPO", ylab="N° DE PASAJEROS",
col= "blue")
legend("bottomright", c("Pasajeros"),lwd=c(1),col=c("blue"))
grid()
1.1.8 Diferenciación.
# cargamos el paquete
library(forecast)
# consultamos el número de diferenciaciones.
ndiffs(lnpasajerosts) ## [1] 1# aplicamos las diferenciaciones.
lndifpasajerosts=diff(lnpasajerosts)
# Consultamos los datos diferenciados.
lndifpasajerosts ## Jan Feb Mar Apr May
## 1949 0.052185753 0.112117298 -0.022989518 -0.064021859
## 1950 -0.025752496 0.091349779 0.112477983 -0.043485112 -0.076961041
## 1951 0.035091320 0.033901552 0.171148256 -0.088033349 0.053744276
## 1952 0.029675768 0.051293294 0.069733338 -0.064193158 0.010989122
## 1953 0.010256500 0.000000000 0.185717146 -0.004246291 -0.025863511
## 1954 0.014815086 -0.081678031 0.223143551 -0.034635497 0.030371098
## 1955 0.055215723 -0.037899273 0.136210205 0.007462721 0.003710579
## 1956 0.021353124 -0.024956732 0.134884268 -0.012698583 0.015848192
## 1957 0.028987537 -0.045462374 0.167820466 -0.022728251 0.019915310
## 1958 0.011834458 -0.066894235 0.129592829 -0.039441732 0.042200354
## 1959 0.066021101 -0.051293294 0.171542423 -0.024938948 0.058840500
## 1960 0.029199155 -0.064378662 0.069163360 0.095527123 0.023580943
## Jun Jul Aug Sep Oct
## 1949 0.109484233 0.091937495 0.000000000 -0.084557388 -0.133531393
## 1950 0.175632569 0.131852131 0.000000000 -0.073203404 -0.172245905
## 1951 0.034289073 0.111521274 0.000000000 -0.078369067 -0.127339422
## 1952 0.175008910 0.053584246 0.050858417 -0.146603474 -0.090060824
## 1953 0.059339440 0.082887660 0.029852963 -0.137741925 -0.116202008
## 1954 0.120627988 0.134477914 -0.030254408 -0.123344547 -0.123106058
## 1955 0.154150680 0.144581229 -0.047829088 -0.106321592 -0.129875081
## 1956 0.162204415 0.099191796 -0.019560526 -0.131769278 -0.148532688
## 1957 0.172887525 0.097032092 0.004291852 -0.144914380 -0.152090098
## 1958 0.180943197 0.121098097 0.028114301 -0.223143551 -0.118092489
## 1959 0.116724274 0.149296301 0.019874186 -0.188422419 -0.128913869
## 1960 0.125287761 0.150673346 -0.026060107 -0.176398538 -0.097083405
## Nov Dec
## 1949 -0.134732594 0.126293725
## 1950 -0.154150680 0.205443974
## 1951 -0.103989714 0.128381167
## 1952 -0.104778951 0.120363682
## 1953 -0.158901283 0.110348057
## 1954 -0.120516025 0.120516025
## 1955 -0.145067965 0.159560973
## 1956 -0.121466281 0.121466281
## 1957 -0.129013003 0.096799383
## 1958 -0.146750091 0.083510633
## 1959 -0.117168974 0.112242855
## 1960 -0.167251304 0.102278849# Revisamos los estadísticos generales de los datos diferenciados.
summary(lndifpasajerosts) ## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.22314 -0.08002 0.01482 0.00944 0.10588 0.22314# Graficamos la serie logarítmica.
plot(lndifpasajerosts,type="o",pch =20, main="PASAJEROS",
sub="Datos obtenidos de Rdata",xlab="TIEMPO", ylab="N° DE PASAJEROS",
col= "blue")
legend("bottomright", c("Pasajeros"),lwd=c(1),col=c("blue"))
grid()
## Revisamos la grafica descompuesta de la serie.
plot(decompose(lndifpasajerosts))
1.1.9 Raíces unitarias.
library(tseries) # Cargamos el paquete
# aplicamos los test para los datos originales (sin transformación).
adf.test(Pasajerosts)##
## Augmented Dickey-Fuller Test
##
## data: Pasajerosts
## Dickey-Fuller = -7.3186, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationarypp.test(Pasajerosts)##
## Phillips-Perron Unit Root Test
##
## data: Pasajerosts
## Dickey-Fuller Z(alpha) = -46.406, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary# aplicamos los test para los datos transformados.
adf.test(lndifpasajerosts)##
## Augmented Dickey-Fuller Test
##
## data: lndifpasajerosts
## Dickey-Fuller = -6.4313, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationarypp.test(lndifpasajerosts)##
## Phillips-Perron Unit Root Test
##
## data: lndifpasajerosts
## Dickey-Fuller Z(alpha) = -93.215, Truncation lag parameter = 4, p-value
## = 0.01
## alternative hypothesis: stationary1.1.10 Estacionalidad
# Consultamos el número de diferenciaciones estacionales.
nsdiffs(lndifpasajerosts)## [1] 1# Aplicamos las diferenciaciones.
slndifpasajerosts=diff(lndifpasajerosts,lag = 12)
#Revisamos los datos diferenciados.
slndifpasajerosts## Jan Feb Mar Apr May
## 1950 0.0391640254 0.0003606853 -0.0204955937 -0.0129391824
## 1951 0.0608438159 -0.0574482269 0.0586702728 -0.0445482375 0.1307053171
## 1952 -0.0054155517 0.0173917427 -0.1014149182 0.0238401918 -0.0427551544
## 1953 -0.0194192680 -0.0512932944 0.1159838078 0.0599468668 -0.0368526322
## 1954 0.0045585856 -0.0816780310 0.0374264055 -0.0303892058 0.0562346085
## 1955 0.0404006368 0.0437787584 -0.0869333465 0.0420982179 -0.0266605185
## 1956 -0.0338625981 0.0129425406 -0.0013259371 -0.0201613045 0.0121376128
## 1957 0.0076344124 -0.0205056421 0.0329361984 -0.0100296677 0.0040671175
## 1958 -0.0171530792 -0.0214318608 -0.0382276371 -0.0167134810 0.0222850448
## 1959 0.0541866435 0.0156009404 0.0419495935 0.0145027837 0.0166401455
## 1960 -0.0368219464 -0.0130853674 -0.1023790626 0.1204660714 -0.0352595574
## Jun Jul Aug Sep Oct
## 1950 0.0661483358 0.0399146358 0.0000000000 0.0113539840 -0.0387145122
## 1951 -0.1413434952 -0.0203308567 0.0000000000 -0.0051656631 0.0449064824
## 1952 0.1407198365 -0.0579370283 0.0508584172 -0.0682344071 0.0372785985
## 1953 -0.1156694702 0.0293034137 -0.0210054541 0.0088615490 -0.0261411837
## 1954 0.0612885480 0.0515902544 -0.0601073715 0.0143973778 -0.0069040505
## 1955 0.0335226920 0.0101033146 -0.0175746793 0.0170229552 -0.0067690233
## 1956 0.0080537348 -0.0453894333 0.0282685618 -0.0254476855 -0.0186576061
## 1957 0.0106831099 -0.0021597040 0.0238523779 -0.0131451021 -0.0035574105
## 1958 0.0080556723 0.0240660052 0.0238224494 -0.0782291716 0.0339976085
## 1959 -0.0642189225 0.0281982047 -0.0082401153 0.0347211322 -0.0108213792
## 1960 0.0085634870 0.0013770445 -0.0459342929 0.0120238806 0.0318304641
## Nov Dec
## 1950 -0.0194180859 0.0791502489
## 1951 0.0501609663 -0.0770628076
## 1952 -0.0007892377 -0.0080174844
## 1953 -0.0541223314 -0.0100156251
## 1954 0.0383852581 0.0101679673
## 1955 -0.0245519407 0.0390449480
## 1956 0.0236016842 -0.0380946915
## 1957 -0.0075467223 -0.0246668977
## 1958 -0.0177370877 -0.0132887505
## 1959 0.0295811174 0.0287322224
## 1960 -0.0500823303 -0.0099640062# Graficamos los datos diferenciados.
plot(slndifpasajerosts) 
# Graficamos la descomposición de los datos diferenciados.
plot(decompose(slndifpasajerosts)) 
1.1.11 Construcción del modelo
1.1.11.1 Parte autorregresiva Integrada de Media Móvil
# Tomamos los datos antes de aplicar la diferenciación estacional.
lndifpasajerosts## Jan Feb Mar Apr May
## 1949 0.052185753 0.112117298 -0.022989518 -0.064021859
## 1950 -0.025752496 0.091349779 0.112477983 -0.043485112 -0.076961041
## 1951 0.035091320 0.033901552 0.171148256 -0.088033349 0.053744276
## 1952 0.029675768 0.051293294 0.069733338 -0.064193158 0.010989122
## 1953 0.010256500 0.000000000 0.185717146 -0.004246291 -0.025863511
## 1954 0.014815086 -0.081678031 0.223143551 -0.034635497 0.030371098
## 1955 0.055215723 -0.037899273 0.136210205 0.007462721 0.003710579
## 1956 0.021353124 -0.024956732 0.134884268 -0.012698583 0.015848192
## 1957 0.028987537 -0.045462374 0.167820466 -0.022728251 0.019915310
## 1958 0.011834458 -0.066894235 0.129592829 -0.039441732 0.042200354
## 1959 0.066021101 -0.051293294 0.171542423 -0.024938948 0.058840500
## 1960 0.029199155 -0.064378662 0.069163360 0.095527123 0.023580943
## Jun Jul Aug Sep Oct
## 1949 0.109484233 0.091937495 0.000000000 -0.084557388 -0.133531393
## 1950 0.175632569 0.131852131 0.000000000 -0.073203404 -0.172245905
## 1951 0.034289073 0.111521274 0.000000000 -0.078369067 -0.127339422
## 1952 0.175008910 0.053584246 0.050858417 -0.146603474 -0.090060824
## 1953 0.059339440 0.082887660 0.029852963 -0.137741925 -0.116202008
## 1954 0.120627988 0.134477914 -0.030254408 -0.123344547 -0.123106058
## 1955 0.154150680 0.144581229 -0.047829088 -0.106321592 -0.129875081
## 1956 0.162204415 0.099191796 -0.019560526 -0.131769278 -0.148532688
## 1957 0.172887525 0.097032092 0.004291852 -0.144914380 -0.152090098
## 1958 0.180943197 0.121098097 0.028114301 -0.223143551 -0.118092489
## 1959 0.116724274 0.149296301 0.019874186 -0.188422419 -0.128913869
## 1960 0.125287761 0.150673346 -0.026060107 -0.176398538 -0.097083405
## Nov Dec
## 1949 -0.134732594 0.126293725
## 1950 -0.154150680 0.205443974
## 1951 -0.103989714 0.128381167
## 1952 -0.104778951 0.120363682
## 1953 -0.158901283 0.110348057
## 1954 -0.120516025 0.120516025
## 1955 -0.145067965 0.159560973
## 1956 -0.121466281 0.121466281
## 1957 -0.129013003 0.096799383
## 1958 -0.146750091 0.083510633
## 1959 -0.117168974 0.112242855
## 1960 -0.167251304 0.102278849# Autocorrelación simple
acf(lndifpasajerosts)
# Autocorrelación parcial
pacf(lndifpasajerosts)
# Otra forma de ver los autocorrelogramas.
ggtsdisplay(lndifpasajerosts)
1.1.11.2 Parte estacional
# Autocorrelación simple
acf(slndifpasajerosts)
# Autocorrelación parcial
pacf(slndifpasajerosts)
# Otra forma de ver los autocorrelogramas.
ggtsdisplay(slndifpasajerosts)
1.1.11.3 Construcción del modelo mediante la función arima
# construimos el modelo
modelo1=arima(lnpasajerosts,order=c(2,1,1),seasonal = list(order=c(1,1,1)))
# Revisamos el modelo
modelo1##
## Call:
## arima(x = lnpasajerosts, order = c(2, 1, 1), seasonal = list(order = c(1, 1,
## 1)))
##
## Coefficients:
## ar1 ar2 ma1 sar1 sma1
## 0.5552 0.2530 -0.9653 -0.0598 -0.5168
## s.e. 0.0956 0.0949 0.0466 0.1551 0.1367
##
## sigma^2 estimated as 0.001305: log likelihood = 246.21, aic = -480.42# construimos los pronósticos
pronosticos1=forecast(modelo1,12,level = 95)
# revisamos los pronósticos
plot(pronosticos1)
# Construimos el modelo con los daros originales
modelo2=arima(Pasajerosts,order = c(2,1,1),seasonal = list(order=c(1,1,1)))
modelo2##
## Call:
## arima(x = Pasajerosts, order = c(2, 1, 1), seasonal = list(order = c(1, 1, 1)))
##
## Coefficients:
## ar1 ar2 ma1 sar1 sma1
## 0.5800 0.2287 -0.9782 -0.9016 0.8102
## s.e. 0.0892 0.0880 0.0289 0.2509 0.3456
##
## sigma^2 estimated as 124.5: log likelihood = -503.12, aic = 1018.25# Revisamos los pronósticos
pronosticos2=forecast(modelo2,12,level=95)
plot(pronosticos2)
# Revisemos autoarima
library(forecast) # Cargamos el paquete
# Con los datos logaritmicos
modelo3=auto.arima(lnpasajerosts)
modelo3## Series: lnpasajerosts
## ARIMA(0,1,1)(0,1,1)[12]
##
## Coefficients:
## ma1 sma1
## -0.4018 -0.5569
## s.e. 0.0896 0.0731
##
## sigma^2 = 0.001371: log likelihood = 244.7
## AIC=-483.4 AICc=-483.21 BIC=-474.77# con los dos datos originales
modelo4=auto.arima(Pasajerosts)
modelo4## Series: Pasajerosts
## ARIMA(2,1,1)(0,1,0)[12]
##
## Coefficients:
## ar1 ar2 ma1
## 0.5960 0.2143 -0.9819
## s.e. 0.0888 0.0880 0.0292
##
## sigma^2 = 132.3: log likelihood = -504.92
## AIC=1017.85 AICc=1018.17 BIC=1029.35##### revisemos que tuvo en cuenta autoarima
auto.arima(lnpasajerosts,trace=TRUE)##
## ARIMA(2,1,2)(1,1,1)[12] : Inf
## ARIMA(0,1,0)(0,1,0)[12] : -434.799
## ARIMA(1,1,0)(1,1,0)[12] : -474.6299
## ARIMA(0,1,1)(0,1,1)[12] : -483.2101
## ARIMA(0,1,1)(0,1,0)[12] : -449.8857
## ARIMA(0,1,1)(1,1,1)[12] : -481.5957
## ARIMA(0,1,1)(0,1,2)[12] : -481.6451
## ARIMA(0,1,1)(1,1,0)[12] : -477.2164
## ARIMA(0,1,1)(1,1,2)[12] : Inf
## ARIMA(0,1,0)(0,1,1)[12] : -467.4644
## ARIMA(1,1,1)(0,1,1)[12] : -481.582
## ARIMA(0,1,2)(0,1,1)[12] : -481.2991
## ARIMA(1,1,0)(0,1,1)[12] : -481.3006
## ARIMA(1,1,2)(0,1,1)[12] : -481.5633
##
## Best model: ARIMA(0,1,1)(0,1,1)[12]## Series: lnpasajerosts
## ARIMA(0,1,1)(0,1,1)[12]
##
## Coefficients:
## ma1 sma1
## -0.4018 -0.5569
## s.e. 0.0896 0.0731
##
## sigma^2 = 0.001371: log likelihood = 244.7
## AIC=-483.4 AICc=-483.21 BIC=-474.77#### revisemos los modelos --- analisis grafico ####
library(astsa)##
## Attaching package: 'astsa'## The following object is masked from 'package:forecast':
##
## gasmod1=sarima(lnpasajerosts,2,1,1,P=0,D=1,Q=1,S=12)## initial value -3.081350
## iter 2 value -3.216551
## iter 3 value -3.270868
## iter 4 value -3.273931
## iter 5 value -3.279327
## iter 6 value -3.279536
## iter 7 value -3.279862
## iter 8 value -3.283571
## iter 9 value -3.284586
## iter 10 value -3.292862
## iter 11 value -3.295196
## iter 12 value -3.302274
## iter 13 value -3.312399
## iter 14 value -3.313920
## iter 15 value -3.318981
## iter 16 value -3.320790
## iter 17 value -3.321546
## iter 18 value -3.322942
## iter 19 value -3.323293
## iter 20 value -3.323314
## iter 21 value -3.324807
## iter 21 value -3.324807
## iter 22 value -3.325012
## iter 22 value -3.325012
## iter 22 value -3.325012
## final value -3.325012
## converged
## initial value -3.295099
## iter 2 value -3.295907
## iter 3 value -3.297094
## iter 4 value -3.297191
## iter 5 value -3.297332
## iter 6 value -3.297573
## iter 7 value -3.297763
## iter 8 value -3.297826
## iter 9 value -3.297840
## iter 10 value -3.297840
## iter 10 value -3.297840
## final value -3.297840
## converged
## <><><><><><><><><><><><><><>
##
## Coefficients:
## Estimate SE t.value p.value
## ar1 0.5580 0.0955 5.8451 0.0000
## ar2 0.2470 0.0936 2.6388 0.0094
## ma1 -0.9646 0.0469 -20.5692 0.0000
## sma1 -0.5574 0.0780 -7.1483 0.0000
##
## sigma^2 estimated as 0.001306428 on 127 degrees of freedom
##
## AIC = -3.681468 AICc = -3.679044 BIC = -3.571727
## 
mod2=sarima(Pasajerosts,1,1,0,P=0,D=1,Q=0,S=12)## initial value 2.513683
## iter 2 value 2.463246
## iter 3 value 2.463246
## iter 3 value 2.463246
## iter 3 value 2.463246
## final value 2.463246
## converged
## initial value 2.460429
## iter 2 value 2.460427
## iter 2 value 2.460427
## iter 2 value 2.460427
## final value 2.460427
## converged
## <><><><><><><><><><><><><><>
##
## Coefficients:
## Estimate SE t.value p.value
## ar1 -0.3076 0.0828 -3.7164 3e-04
##
## sigma^2 estimated as 137.0157 on 130 degrees of freedom
##
## AIC = 7.789266 AICc = 7.789502 BIC = 7.833162
## 
mod3=sarima(lnpasajerosts,0,1,1,P=0,D=1,Q=1,S=12)## initial value -3.086228
## iter 2 value -3.267980
## iter 3 value -3.279950
## iter 4 value -3.285996
## iter 5 value -3.289332
## iter 6 value -3.289665
## iter 7 value -3.289672
## iter 8 value -3.289676
## iter 8 value -3.289676
## iter 8 value -3.289676
## final value -3.289676
## converged
## initial value -3.286464
## iter 2 value -3.286855
## iter 3 value -3.286872
## iter 4 value -3.286874
## iter 4 value -3.286874
## iter 4 value -3.286874
## final value -3.286874
## converged
## <><><><><><><><><><><><><><>
##
## Coefficients:
## Estimate SE t.value p.value
## ma1 -0.4018 0.0896 -4.4825 0
## sma1 -0.5569 0.0731 -7.6190 0
##
## sigma^2 estimated as 0.001348035 on 129 degrees of freedom
##
## AIC = -3.690069 AICc = -3.689354 BIC = -3.624225
## 
##### revisemos mediante la funcion checkresiduals ####
checkresiduals(modelo1)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(2,1,1)(1,1,1)[12]
## Q* = 27.162, df = 19, p-value = 0.1009
##
## Model df: 5. Total lags used: 24checkresiduals(modelo2)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(2,1,1)(1,1,1)[12]
## Q* = 34.926, df = 19, p-value = 0.01425
##
## Model df: 5. Total lags used: 24checkresiduals(modelo3)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(0,1,1)(0,1,1)[12]
## Q* = 26.446, df = 22, p-value = 0.233
##
## Model df: 2. Total lags used: 24checkresiduals(modelo4)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(2,1,1)(0,1,0)[12]
## Q* = 37.784, df = 21, p-value = 0.01366
##
## Model df: 3. Total lags used: 24#### CONSTRUYAMOS LOS PRONOSTICOS ####
pronosticosf=forecast(modelo3,12,level=95)
pronosticosf## Point Forecast Lo 95 Hi 95
## Jan 1961 6.110186 6.037607 6.182764
## Feb 1961 6.053775 5.969203 6.138347
## Mar 1961 6.171715 6.076650 6.266779
## Apr 1961 6.199300 6.094792 6.303809
## May 1961 6.232556 6.119388 6.345724
## Jun 1961 6.368779 6.247569 6.489988
## Jul 1961 6.507294 6.378544 6.636044
## Aug 1961 6.502906 6.367034 6.638779
## Sep 1961 6.324698 6.182058 6.467338
## Oct 1961 6.209008 6.059908 6.358109
## Nov 1961 6.063487 5.908195 6.218780
## Dec 1961 6.168025 6.006778 6.329272plot(pronosticosf)
plot(forecast(modelo3,12,level = 95)) 
#### ver los pronosticos en sus valores reales ####
exp(pronosticosf$mean)## Jan Feb Mar Apr May Jun Jul Aug
## 1961 450.4224 425.7172 479.0068 492.4045 509.0550 583.3449 670.0108 667.0776
## Sep Oct Nov Dec
## 1961 558.1894 497.2078 429.8720 477.2426plot(exp(pronosticosf$mean))