1. Case study 4: Multiplicative Seasonal ARIMA Model
Monthly total international airline passengers (thousands of passengers), …
Ref.: Bowerman & O’Connell [1993], pg. 539

2. Example 2: Monthly total international airline passengers (thousands of passengers), … [Bowerman & O’Connell [1993], pg. 539]
Year
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sept
Oct
Nov
Dec
1949
112
118
132
129
121
135
148
136
119
104
1950
115
126
141
125
149
170
158
133
114
140
1951
145
150
178
163
172
199
184
162
146
166
1952
171
180
193
181
183
218
230
242
209
191
194 …
1958
340
318
362
348
363
435
491
505
404
359
310
337
1959
360
342
406
396
420
472
548
559
463
407
405
1960
417
391
419
461
535
622
606
508
390
432
Peak conditions

3. Example 2: The international airline passenger data
Training data (in sample) n = 120
Testing data (out sample) n = 24

4. Example 2: IDENTIFICATION step [check stationary data]
Nonstationer (variance) time series Yt
is an appropriate pre-differencing transformation

5. Example 2: IDENTIFICATION step [stationary data]
Box-Cox Transformation:  transformation to stabilize (make stationer) variance of time series data

6. Example 2: IDENTIFICATION step [transformation data]
Nonstationer (mean and ariance) time series
Stationer variance, nonstationer (mean) time series

7. Example 2: IDENTIFICATION step [difference data]
ACF of Yt*
Dies down slowly
Stationer variance, nonstationer (mean) time series
Wt = Yt* – Yt-1*

8. Example 2: IDENTIFICATION step [stationary, ACF and PACF]
Wt (difference data, d=1)
Dies down slowly at seasonal lag

9. Example 2: IDENTIFICATION step [difference data]Wt
Wt = Y*t – Y*t-1
Stationer variance, nonstationer (mean) time series Zt
Zt = Wt – Wt-12
Zt = Y*t – Y*t-1 – Y*t-12 + Y*t-13

10. Stationary time series
Example 2: Nonseasonal & Seasonal Difference [Zt = Y*t – Y*t-1 – Y*t-12 + Y*t-13] Zt
Stationary time series
 (001)(001)  (001)(100)  (100)(001)  (100)(100)12
ACF
PACF
Cuts off after lag 1
Cuts off after lag 1
Cuts off after lag 12
Cuts off after lag 12

11. Example 2: ESTIMATION and DIAGNOSTIC CHECK step
Example 2: ESTIMATION and DIAGNOSTIC CHECK step  ARIMA(0,1,1)(0,1,1)12
Y*t = Y*t-1 – Y*t-12 + Y*t-13 + at – at – at-12 + (0.3271)(0.6268) at-13
Estimation and Testing parameter
Diagnostic Check (white noise residual)

12. Example 2: ESTIMATION and DIAGNOSTIC CHECK step
Example 2: ESTIMATION and DIAGNOSTIC CHECK step  ARIMA(0,1,1)(1,1,0)12
Y*t = Y*t-1 – … + at – at-1
Estimation and Testing parameter
Diagnostic Check (white noise residual)

13. Example 2: ESTIMATION and DIAGNOSTIC CHECK step
Example 2: ESTIMATION and DIAGNOSTIC CHECK step  ARIMA(1,1,0)(0,1,1)12
Y*t = Y*t-1 – … + at – at-12
Estimation and Testing parameter
Diagnostic Check (white noise residual)

14. Example 2: ESTIMATION and DIAGNOSTIC CHECK step
Example 2: ESTIMATION and DIAGNOSTIC CHECK step  ARIMA(1,1,0)(1,1,0)12
Y*t = Y*t-1 – … + at
Estimation and Testing parameter
Diagnostic Check (white noise residual)

15. Example 2: DIAGNOSTIC CHECK step … [Normality test of residuals]

16. Example 2: FORECASTING step [MINITAB output]

17. Example 2: FORECASTING step at the original scale

18. In sample (Training Data) Out sample (Testing Data)
Example 2: Comparison result between forecasting models by using MSE, MAE and MAPE
Model
In sample (Training Data)
Out sample (Testing Data)
MSE
MAE
MAPE
 Winter’s (*)
a. Model 1
b. Model 2
 Decomposition (*)
 ARIMA
a. [011][011]12
b. [110][011]12
6.1753
8.0342
7.5795
(* ) : error model is not white noise
 Winter’s Models
Model 1 : =0.9; =0.1; =0.3
Model 2 : =0.1; =0.2; =0.4

19. MINITAB command: IDENTIFICATION Step
Plot Data  stationarity data
To make (mean) stationarity data
ACF & PACF data  to find tentative ARIMA model

20. MINITAB command: Box-Cox Transformation
Box-Cox Transformation  to stabilize variance of data (stationarity in variance)

21. IDENTIFICATION Step: Box-Cox … [continued]

22. IDENTIFICATION Step: Difference Process …
Seasonal differencing (D=1, L=12)  Wt  Wt-12
Nonseasonal differencing (d=1)  Yt*  Yt-1*

23. IDENTIFICATION Step: Time Series Plot …

24. IDENTIFICATION Step: ACF data …

25. IDENTIFICATION Step: PACF data …

26. MINITAB command: ESTIMATION, DIAGNOSTIC CHECK & FORECASTING Step
Estimation, Diagnostic Check and Forecasting

27. FORECASTING Step: Transformation to original scale data …
Calculation to original scale data command

28. MINITAB command: Normality test for residual