Case study 4: Multiplicative Seasonal ARIMA Model Monthly total international airline passengers (thousands of passengers), 1949-1960 … Ref.: Bowerman & O’Connell [1993], pg. 539

Example 2: Monthly total international airline passengers (thousands of passengers), 1949-1960 … [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

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

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

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

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

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

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

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

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)12  (001)(100)12  (100)(001)12  (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

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 – 0.3271 at-1 – 0.6268 at-12 + (0.3271)(0.6268) at-13 Estimation and Testing parameter Diagnostic Check (white noise residual)

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 – 0.3899 at-1 Estimation and Testing parameter Diagnostic Check (white noise residual)

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 – 0.6378 at-12 Estimation and Testing parameter Diagnostic Check (white noise residual)

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)

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

Example 2: FORECASTING step [MINITAB output]

Example 2: FORECASTING step at the original scale

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 97.7342 146.8580 215.4570 88.6444 88.8618 7.30200 9.40560 11.47000 7.38689 7.33226 3.18330 4.05600 5.05900 2.95393 2.92610 12096.80 3447.82 1354.88 1693.68 1527.03 101.5010 52.1094 29.9744 37.4012 35.3060 21.7838 11.4550 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

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

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

IDENTIFICATION Step: Box-Cox … [continued]

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

IDENTIFICATION Step: Time Series Plot …

IDENTIFICATION Step: ACF data …

IDENTIFICATION Step: PACF data …

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

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

MINITAB command: Normality test for residual