Case study 3: SEASONAL ARIMA MODEL Average monthly temperatures in Dubuque, Iowa, January 1964 – December 1975 Reference: Cryer [1986], pg. 268.

Example 1: Average monthly temperatures in Dubuque, Iowa (January 1964 – December 1975) … [Cryer [1986], pg. 268] Year Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec 1964 24.7 25.7 30.6 47.5 62.9 68.5 73.7 67.9 61.1 48.5 39.6 20.0 1965 16.1 19.1 24.2 45.4 61.3 66.5 72.1 68.4 60.2 50.9 37.4 31.1 1966 10.4 21.6 44.7 53.2 68.0 68.2 60.7 50.2 37.2 24.6 1967 21.5 14.7 35.0 48.3 54.0 69.6 65.7 60.8 49.1 33.2 26.0 … 1973 22.5 42.3 45.2 55.5 68.9 72.3 62.5 55.6 38.0 20.4 1974 17.6 20.5 34.2 49.2 54.8 63.8 74.0 57.7 50.8 36.8 25.5 1975 19.6 41.3 61.8 72.0 57.3 52.5 40.6 26.2 Low temperature Medium temperature High temperature

Example 1: IDENTIFICATION step [stationary, ACF and PACF] Nonstationer (seasonal) time series Dies down slowly

Example 1: IDENTIFICATION step … Seasonal Difference [Zt = Yt – Yt-12] Zt  MA(1)12 or Yt  IMA(1,1)12 Stationary time series ACF PACF Cuts off after lag 12 Dies down at seasonal lag

Example 1: ESTIMATION and DIAGNOSTIC CHECK step Yt = Yt-12 + at – 0.8633 at-12 Estimation and Testing parameter Diagnostic Check (white noise residual)

Example 1: DIAGNOSTIC CHECK step … [Normality test of residuals]

Example 1: FORECASTING step [MINITAB output]

Example 1: Comparison Actual v.s. Forecast … Forecast data Upper Actual data Lower