ARIMA Modeling: Reproduction of a Manufacturing Case Study Example

data = read.csv('http://ucanalytics.com/blogs/wp-content/uploads/2015/06/Tractor-Sales.csv')
data = ts(data[,2],start = c(2003,1),frequency = 12)
plot(data, xlab='Years', ylab = 'Tractor Sales')

plot(diff(data),ylab='Differenced Tractor Sales')

plot(log10(data),ylab='Log (Tractor Sales)')

plot(diff(log10(data)),ylab='Differenced Log (Tractor Sales)')

par(mfrow = c(1,2))
acf(ts(diff(log10(data))),main='ACF Tractor Sales')
pacf(ts(diff(log10(data))),main='PACF Tractor Sales')
require(forecast)

## Loading required package: forecast

ARIMAfit = auto.arima(log10(data), approximation=FALSE,trace=FALSE)
summary(ARIMAfit)

## Series: log10(data) 
## ARIMA(0,1,1)(0,1,1)[12] 
## 
## Coefficients:
##           ma1     sma1
##       -0.4047  -0.5529
## s.e.   0.0885   0.0734
## 
## sigma^2 estimated as 0.0002571:  log likelihood=354.4
## AIC=-702.79   AICc=-702.6   BIC=-694.17
## 
## Training set error measures:
##                        ME       RMSE        MAE         MPE      MAPE
## Training set 0.0002410698 0.01517695 0.01135312 0.008335713 0.4462212
##                   MASE       ACF1
## Training set 0.2158968 0.01062604

par(mfrow = c(1,1))
pred = predict(ARIMAfit, n.ahead = 36)
pred

## $pred
##           Jan      Feb      Mar      Apr      May      Jun      Jul
## 2015 2.754168 2.753182 2.826608 2.880192 2.932447 2.912372 2.972538
## 2016 2.796051 2.795065 2.868491 2.922075 2.974330 2.954255 3.014421
## 2017 2.837934 2.836948 2.910374 2.963958 3.016213 2.996138 3.056304
##           Aug      Sep      Oct      Nov      Dec
## 2015 2.970585 2.847264 2.797259 2.757395 2.825125
## 2016 3.012468 2.889147 2.839142 2.799278 2.867008
## 2017 3.054351 2.931030 2.881025 2.841161 2.908891
## 
## $se
##             Jan        Feb        Mar        Apr        May        Jun
## 2015 0.01603508 0.01866159 0.02096153 0.02303295 0.02493287 0.02669792
## 2016 0.03923008 0.04159145 0.04382576 0.04595157 0.04798329 0.04993241
## 2017 0.06386474 0.06637555 0.06879478 0.07113179 0.07339441 0.07558934
##             Jul        Aug        Sep        Oct        Nov        Dec
## 2015 0.02835330 0.02991723 0.03140337 0.03282229 0.03418236 0.03549035
## 2016 0.05180825 0.05361850 0.05536960 0.05706700 0.05871534 0.06031866
## 2017 0.07772231 0.07979828 0.08182160 0.08379608 0.08572510 0.08761165

plot(data,type='l',xlim=c(2004,2018),ylim=c(1,1600),xlab = 'Year',ylab = 'Tractor Sales')
lines(10^(pred$pred),col='blue')
lines(10^(pred$pred+2*pred$se),col='orange')
lines(10^(pred$pred-2*pred$se),col='orange')

par(mfrow=c(1,2))
acf(ts(ARIMAfit$residuals),main='ACF Residual')
pacf(ts(ARIMAfit$residuals),main='PACF Residual')