Question 6 [23] The time series plot in the following graph shows Discount Soda's (Egdworth) for the recorded monthly Cola sales for the three years 2003 to 2005, i.e. from 1 January 2003 to 31December 2005 Monthly sales of Cola (in hundreds of cases) Monthly Sales M 10 30 20 Time Figure 1: Monthly cola sales (a) Which of the two models, Y; = TN, X CL, X SN, X IR, and Y, = TN, +CL: + SN, +IN, where TN, CL, SN, IR are trend, cyclical, seasonal and irregular components respec- tively, would you suggest adequately fits the untransformed and log transformed series data? Give a reason(s) for your answer. (3) (b) Which operator(s) of transformation would you apply to this series to make it stationary? (2) (c) A log transformation of the Cola sales data suggests that the seasonality is constant. Hence, the seasonal model using trigonometric functions, Z = Bo + Bit + B2 sin(t) + B3 cos (cor) + 6, 2x where Zx = log (Y;), o = is the time index, L is the seasonality period and BN (0,0%). can be used. The multiple regression equation in vector-matrix is Z=X8+. Write down the design matrix X for the above model. (3) (d) Table 6.1 gives the R-output for the model in (c) fitted for the series. Table 6.1: Estimation of deterministic linear trend and seasonality via sinusoidal functions fit