importacion de datos

library(tseries)
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
library(forecast)
d2s=AirPassengers
plot(d2s)

el comportamiento de los datos es de forma multiplictiva, por lo tanto procedemos a realizar la transformacion de los mismos

d2sl=log(d2s)
plot(d2sl)

adf.test(d2sl)
## Warning in adf.test(d2sl): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  d2sl
## Dickey-Fuller = -6.4215, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary

la serie en logaritmos es estacionario, por que el p-valor es menor que alfa. ## serie estacionaria

d2sld=diff(d2sl)
d2slds=diff(d2sld,12)
adf.test(d2slds)
## Warning in adf.test(d2slds): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  d2slds
## Dickey-Fuller = -5.1993, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary
pacf(d2slds,lag.max = 24,main="función de autocorrelación parcial")

## funcion de autocorrelacion parcial

pacf(d2slds,lag.max = 24,main="función de autocorrelación parcial")

Además en la función de autocorrelación parcial se observa igualmente un resago significativo en la parte regualr (AR(1)) y en la parte estacional luego del resago 12 niguno es claramente significativo (SAR(0)).

modelo sarima

mod=auto.arima(d2sl,trace = T)
## 
##  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]
mod
## Series: d2sl 
## 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

\((y_t-y_{t-1})-(y_{t-12}-y_{t-13})=-0.4018 E_{t-1}-0.5569/epsilon_{t-12}\)

mod1=arima(d2sl,order = c(1,1,1),seasonal = c(0,1,0))
mod1
## 
## Call:
## arima(x = d2sl, order = c(1, 1, 1), seasonal = c(0, 1, 0))
## 
## Coefficients:
##          ar1      ma1
##       0.1449  -0.5190
## s.e.  0.2455   0.2179
## 
## sigma^2 estimated as 0.001824:  log likelihood = 227.13,  aic = -448.25

evaluacion del modelo

Box.test(mod1$residuals)
## 
##  Box-Pierce test
## 
## data:  mod1$residuals
## X-squared = 0.026531, df = 1, p-value = 0.8706
d2slp=predict(mod1,4)
d2sp=exp(d2slp$pred)
d2sp
##           Jan      Feb      Mar      Apr
## 1961 449.6253 422.2509 452.5914 497.9750
ts.plot(d2s,d2sp,col=c("blue","red"))