Dowjones

nit$set(width=75)

## Visualisation of  *dowjones* time series



```r
require(fma)
tsdisplay(dowjones)

dowjones time series

• has a trend: the time plot shows an upward trend, the ACF shows very high correlations that are not decreasing quickly in amplitude as the lag \(k\) increases; the PACF indicates very high correlation between \(y_t\) and \(y_{t-1}\) but all other lags \(k>1\) have PACF equivalent to 0.

• but no seasonality,

• and has a noise component.

Fitting a constant model to the time series: \[ y_t=c+\epsilon_t \]

and visualising the residuals \(\lbrace \epsilon_t\rbrace_{1,\cdots,n}\) as follow:

require(fma)
Arima(dowjones,order=c(0,0,0))

## Series: dowjones 
## ARIMA(0,0,0) with non-zero mean 
## 
## Coefficients:
##           mean
##       115.6833
## s.e.    0.6194
## 
## sigma^2 = 30.32:  log likelihood = -243.23
## AIC=490.46   AICc=490.62   BIC=495.17

tsdisplay(Arima(dowjones,order=c(0,0,0))$residuals)

Removing the trend in dowjones by differencing

To remove the trend, first order differencing is applied:

require(fma)
Arima(dowjones,order=c(0,1,0))

## Series: dowjones 
## ARIMA(0,1,0) 
## 
## sigma^2 = 0.1979:  log likelihood = -46.86
## AIC=95.73   AICc=95.78   BIC=98.07

tsdisplay(Arima(dowjones,order=c(0,1,0))$residuals)

Note that the model fitted here is (\(c=0\)): \[ y_t=y_{t-1}+\epsilon_t \]

The PACF indicates PACF(1) to be non-zero whereas all other \(PACF(k)=0\) for \(k>1\). On the ACF a quick decrease to 0 value can be observed as \(k\) increases. A AR(1) model can be suggested to fit this residuals.

Suggesting ARIMA(1,1,0)

require(fma)
Arima(dowjones,order=c(1,1,0))

## Series: dowjones 
## ARIMA(1,1,0) 
## 
## Coefficients:
##          ar1
##       0.4992
## s.e.  0.1001
## 
## sigma^2 = 0.1515:  log likelihood = -36.19
## AIC=76.38   AICc=76.54   BIC=81.07

tsdisplay(Arima(dowjones,order=c(1,1,0))$residuals)

Nothing more can be read with the time plot, ACF and PACF of the residuals. So the final model identified using visualisation is \[ (y_t-y_{t-1})=\phi_1 (y_{t-1}-y_{t-2}) +\epsilon_t \]

Using AIC and BIC

Goodness of fit criteria associated with ARIMA models are given in the R output. We note that starting with ARIMA(0,0,0) model, to ARIMA(0,1,0) model, and concluding with ARIMA(1,1,0) model then we have:

• a decrease of BIC values (indicates an improvement in the goodness of fit of the model to the time series data)

• a decrease of AIC values (indicates an improvement in the goodness of fit of the model to the time series data)

Using an automatic function in R a better model (based on AIC) is found ARIMA(1,1,1) (marginal improvement of AIC) but our ARIMA(1,1,0) is the best when BIC is chosen.

require(fma)
auto.arima(dowjones,trace=TRUE)

## 
##  ARIMA(2,1,2) with drift         : 81.9124
##  ARIMA(0,1,0) with drift         : 90.60466
##  ARIMA(1,1,0) with drift         : 76.62097
##  ARIMA(0,1,1) with drift         : 81.31565
##  ARIMA(0,1,0)                    : 95.78286
##  ARIMA(2,1,0) with drift         : 77.78036
##  ARIMA(1,1,1) with drift         : 77.27229
##  ARIMA(2,1,1) with drift         : 79.78963
##  ARIMA(1,1,0)                    : 76.54332
##  ARIMA(2,1,0)                    : 76.97825
##  ARIMA(1,1,1)                    : 75.70703
##  ARIMA(0,1,1)                    : 83.58248
##  ARIMA(2,1,1)                    : 79.20627
##  ARIMA(1,1,2)                    : 76.577
##  ARIMA(0,1,2)                    : 79.96298
##  ARIMA(2,1,2)                    : 78.66149
## 
##  Best model: ARIMA(1,1,1)

## Series: dowjones 
## ARIMA(1,1,1) 
## 
## Coefficients:
##          ar1      ma1
##       0.8510  -0.5263
## s.e.  0.1383   0.2548
## 
## sigma^2 = 0.1474:  log likelihood = -34.69
## AIC=75.38   AICc=75.71   BIC=82.41

Arima(dowjones,order=c(1,1,1))

## Series: dowjones 
## ARIMA(1,1,1) 
## 
## Coefficients:
##          ar1      ma1
##       0.8510  -0.5263
## s.e.  0.1383   0.2548
## 
## sigma^2 = 0.1474:  log likelihood = -34.69
## AIC=75.38   AICc=75.71   BIC=82.41

Exercises: investigate R commands Arima, arima, arima.sim. Read https://otexts.com/fpp2/arima-r.html

Forecasts and prediction intervals

Once the best model is selected, it can be used for computing forecasts and prediction intervals:

require(fma)
plot(forecast(Arima(dowjones,order=c(1,1,0)),h=20,level = c(80, 95)))

forecast(Arima(dowjones,order=c(1,1,0)),h=20,level = c(80, 95))

##    Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 79       120.8456 120.3469 121.3444 120.0829 121.6084
## 80       120.6538 119.7550 121.5526 119.2792 122.0284
## 81       120.5580 119.3057 121.8103 118.6428 122.4732
## 82       120.5102 118.9480 122.0724 118.1210 122.8994
## 83       120.4863 118.6501 122.3226 117.6781 123.2946
## 84       120.4744 118.3929 122.5560 117.2909 123.6579
## 85       120.4685 118.1643 122.7727 116.9445 123.9925
## 86       120.4655 117.9568 122.9742 116.6288 124.3022
## 87       120.4640 117.7656 123.1624 116.3372 124.5909
## 88       120.4633 117.5873 123.3393 116.0649 124.8617
## 89       120.4629 117.4196 123.5063 115.8085 125.1173
## 90       120.4627 117.2607 123.6648 115.5656 125.3599
## 91       120.4627 117.1094 123.8160 115.3342 125.5911
## 92       120.4626 116.9646 123.9606 115.1128 125.8124
## 93       120.4626 116.8256 124.0996 114.9003 126.0249
## 94       120.4626 116.6917 124.2335 114.6955 126.2296
## 95       120.4626 116.5624 124.3627 114.4978 126.4273
## 96       120.4626 116.4373 124.4878 114.3064 126.6187
## 97       120.4626 116.3159 124.6092 114.1208 126.8043
## 98       120.4626 116.1980 124.7271 113.9405 126.9846