Australian Wine Sales Example
Australian total wine sales by wine makers in bottles <= 1 litre. Jan 1980 – Aug 1994.
Differencing
Seasonal Arima
We can extend the existing ARIMA model to handle these higher order lags (without having to include all of the intervening lags).
Seasonal \(\text{ARIMA}\,(p,d,q) \times (P,D,Q)_s\) : \[ \Phi_P(L^s) \, \phi_p(L) \, \Delta_s^D \, \Delta^d \, y_t = \delta + \Theta_Q(L^s) \, \theta_q(L) \, w_t\] . . .
where
\[
\begin{aligned}
\phi_p(L) &= 1-\phi_1 L - \phi_2 L^2 - \ldots - \phi_p L^p\\
\theta_q(L) &= 1+\theta_1 L + \theta_2 L^2 + \ldots + \theta_p L^q \\
\Delta^d &= (1-L)^d\\
\\
\Phi_P(L^s) &= 1-\Phi_1 L^s - \Phi_2 L^{2s} - \ldots - \Phi_P L^{Ps} \\
\Theta_Q(L^s) &= 1+\Theta_1 L + \Theta_2 L^{2s} + \ldots + \theta_p L^{Qs} \\
\Delta_s^D &= (1-L^s)^D\\
\end{aligned}
\]
Seasonal ARIMA - AR
Lets consider an \(\text{ARIMA}(0,0,0) \times (1,0,0)_{12}\) : \[
\begin{aligned}
(1-\Phi_1 L^{12}) \, y_t = \delta + w_t \\
y_t = \Phi_1 y_{t-12} + \delta + w_t
\end{aligned}
\]
m1.1 = forecast:: Arima (wineind, seasonal= list (order= c (1 ,0 ,0 ), period= 12 )))
Series: wineind
ARIMA(0,0,0)(1,0,0)[12] with non-zero mean
Coefficients:
sar1 mean
0.8780 24489.243
s.e. 0.0314 1154.487
sigma^2 = 6906536: log likelihood = -1643.39
AIC=3292.78 AICc=3292.92 BIC=3302.29
Fitted - Model 1.1
Seasonal Arima - Diff
Lets consider an \(\text{ARIMA}(0,0,0) \times (0,1,0)_{12}\) : \[
\begin{aligned}
(1 - L^{12}) \, y_t = \delta + w_t \\
y_t = y_{t-12} + \delta + w_t
\end{aligned}
\]
m1.2 = forecast:: Arima (seasonal= list (order= c (0 ,1 ,0 ), period= 12 )
Series: wineind
ARIMA(0,0,0)(0,1,0)[12]
sigma^2 = 7259076: log likelihood = -1528.12
AIC=3058.24 AICc=3058.27 BIC=3061.34
Fitted - Model 1.2
Residuals - Model 1.1 (SAR)
Residuals - Model 1.2 (SDiff)
Model 2
\(\text{ARIMA}(0,0,0) \times (0,1,1)_{12}\) :
\[
\begin{aligned}
(1-L^{12}) y_t = \delta + (1+\Theta_1 L^{12}) w_t \\
y_t = \delta + y_{t-12} + w_t + \Theta_1 w_{t-12}
\end{aligned}
\]
m2 = forecast:: Arima (wineind, order= c (0 ,0 ,0 ), seasonal= list (order= c (0 ,1 ,1 ), period= 12 )))
Series: wineind
ARIMA(0,0,0)(0,1,1)[12]
Coefficients:
sma1
-0.3246
s.e. 0.0807
sigma^2 = 6588531: log likelihood = -1520.34
AIC=3044.68 AICc=3044.76 BIC=3050.88
Fitted - Model 2
Residuals
Model 3
\(\text{ARIMA}(3,0,0) \times (0,1,1)_{12}\)
\[
\begin{aligned}
(1-\phi_1 L - \phi_2 L^2 - \phi_3 L^3) \, (1-L^{12}) y_t = \delta + (1 + \Theta_1 L)w_t \\
y_t = \delta + \sum_{i=1}^3 \phi_i y_{t-i} + y_{t-12} - \sum_{i=1}^3 \phi_i y_{t-12-i} + w_t + w_{t-12}
\end{aligned}
\]
m3 = forecast:: Arima (wineind, order= c (3 ,0 ,0 ), seasonal= list (order= c (0 ,1 ,1 ), period= 12 )))
Series: wineind
ARIMA(3,0,0)(0,1,1)[12]
Coefficients:
ar1 ar2 ar3 sma1
0.1402 0.0806 0.3040 -0.5790
s.e. 0.0755 0.0813 0.0823 0.1023
sigma^2 = 5948935: log likelihood = -1512.38
AIC=3034.77 AICc=3035.15 BIC=3050.27
Fitted model
Model - Residuals
