Forecasting Models With Trend and Seasonal Effects

1. Forecasting Models With Trend and Seasonal Effects

2. Types of Seasonal Models Trend Effects Seasonal Effects Random Effects • Two possible models are: Additive Model yt = Tt + St + εt Multipicative Model yt = TtStεt

3. Additive ModelRegression Forecasting Procedure Tt εt St • Suppose a time series is modeled as having k seasons (Here we illustrate k = 4 quarters) • Problem is modeled with k-1 (4-1 = 3) dummy variables, S1, S2, and S3 corresponding to seasons 1, 2, and 3 respectively. The combination of 0’s and 1’s for each of the dummy variables at each period indicate the season corresponding to the time series value. • Season 1: S1 = 1, S2 = 0, S3 = 0 • Season 2: S1 = 0, S2 = 1, S3 = 0 • Season 3: S1 = 0, S2 = 0, S3 = 1 • Season 4: S1 = 0, S2 = 0, S3 = 0 • Multiple regression is then done on with t, S1, S2, and S3 as the independent variables and the time series values yt as the dependent variable. yt = β0 + β1t + β2S1 + β3S2 + β4S3 + εt

4. ExampleTroy’s Mobil Station YEAR SEASON 1 2 3 4 5 FALL 3497 3726 3989 4248 4443 WINTER 3484 3589 3870 4105 4307 SPRING 3553 3742 3996 4263 4466 SUMMER 3837 4050 4327 4544 4795 • Troy owns a gas station in a vacation resort city that has many spring and summer visitors. • Due to a steady increase in population Troy feels that average sales experience long term trend. • Troy also knows that sales vary by season due to the vacationers. • Based on the last 5 years data below with sales in 1000’s of gallons per season, Troy needs to predict total sales for next year (periods 21, 22, 23, and 24).

5. Scatterplot of Time Series Summer Fall Spring Winter General Pattern:Winterless than Fall, Springmore thanWinter, Summermore thanSpring, Fallless thanSummer

6. The Model Spring Winter Fall • There is also apparent long term trend. • The form of the model then is: yt = β0 + β1t + β2F + β3W + β4S + εt

7. The Excel Input

8. Add Dummy Variables In Fall, not Winter, not Spring Not Fall, In Winter, not Spring Not Fall, not Winter, In Spring Not Fall, not Winter, not Spring Pattern Repeats

9. Regression Intput

10. Regression Output Low p-value for F-test Low p-values for all t-tests Conclusion Good model – all factors significant

11. The Forecasts =$G$17+$G$18*B22+MMULT(C22:E22,$G$19:$G$21) =SUM(F22:F25) Drag F22 down to F25

12. What if Some of the p-values are high? • Would not just eliminate Spring or Winter • A test exists to decide if adding the dummy variables add value to the model H0: 2 = 3 = 4 = 0 HA: At least one of these ’s  0 • Run 2 models: • Full: Time + (3) Seasonal Variables • Reduced: Time Only • Test --- Reject H0 (Accept H1) if F > F,3,DFE(Full) F = ((SSEREDUCED-SSEFULL)/3)/MSEFULL • So if F >F,3,DFE(Full)---Include seasonal variables

13. Multiplicative ModelClassical Decomposition Approach • The time series is first decomposed into its components (trend, seasonal variation). • After these components have been determined, the series is re-composed by multiplying the components.

14. Smooth the time series to remove random effects and seasonality and isolate trend. Calculate moving averages to get values for Tt for each period t. Classical Decomposition • Determine “period factors” to isolate the (seasonal)·(error) factors. • Calculate the ratio yt/Tt. • Determine the “unadjusted seasonal factors” to eliminate the random component from the period factors • Average all the yt/Tt that correspond to the same season.

15. Determine the “adjusted seasonal factors”. Calculate: [Unadjusted seasonal factor] [Average seasonal factor] Classical Decomposition (Cont’d) • Determine “Deseasonalized data values”. Calculate: yt [Adjusted seasonal factors]t • Determine a deseasonalized trend forecast. Use linear regression on the deseasonalized time series. • Determine an “adjusted seasonal forecast”. Calculate:(Desesonalized values) · [Adjusted seasonal factors]).

16. CANADIAN FACULTY ASSOCIATION (CFA) • The CFA is the exclusive bargaining agent for public Canadian college faculty. • Membership in the organization has grown over the years, but in the summer months there was always a decline. • To prepare the budget for the 2001 fiscal year, a forecast of the average quarterly membership covering the year 2001 was required.

17. CFA - Solution 1997 1998 1999 2000 • Membership records from 1997 through 2000 were collected and graphed. The graph exhibits long term trend The graph exhibits seasonality pattern

18. Step 1:Isolating the Trend Component Trend value at period 3, T3 • Smooth the time series to remove random effects and seasonality. Calculate moving averages. First moving average period is centered at quarter (1+4)/2 = 2.5 Average membership for the first 4 periods = [7130+6940+7354+7556]/4 = 7245.01 Second moving average period is centered at quarter (2+5)/2 = 3.5 Average membership for periods [2, 5]= [6940+7354+7556+7673]/4 = 7380.75 Centered moving average of the first two moving averages is [7245.01 + 7380.75]/2 = 7312.875 Centered location is t = 3

19. =AVERAGE(C3:C6,C4:C7) Drag down to D16

20. Step 2Determining the Period Factors • Determine “period factors” to isolate the (Seasonal)·(Random error) factor. Calculate the ratio yt/Tt. Since yt =TtStεt, then theperiod factor, Stεtis given by Stet = yt/Tt Example: In period 7 (3rd quarter of 1998):S7ε7= y7/T7 = 7662/7643.875 = 1.002371

21. =C5/D5 Drag down to E16

22. Step 3Unadjusted Seasonal Factors This eliminates the random factor from the period factors, Stεt This leaves us with only the seasonality component for each season. Example: Unadjusted Seasonal Factor for the third quarter. S3 = {S3,97e3,97+ S3,98e3,98 + S3,99e3,99}/3= {1.0056+1.0024+1.0079}/3 = 1.0053 • Determine the “unadjusted seasonal factors” to eliminate the random component from the period factors Average all the yt/Tt that correspond to the same season.

23. =AVERAGE(E3,E7,E11,E15) Drag down to F6 Copy F3:F6 Paste Special(Values)

24. Step 4Adjusted Seasonal Factors Quarter 1 2 3 4 Adjusted Seasonal Factor 1.014325 .965252 1.004759 1.015663 Unadjusted Seasonal Factor 1.01490 .96580 1.00533 1.01624 Unadjusted Seasonal Factors/1.00057 Calculate: Unadjusted seasonal factors Average seasonal factor • Determine the “adjusted seasonal factors” so that average adjusted factor is 1 Average seasonal factor = (1.01490+.96580+1.00533+1.01624)/4=1.00057

25. F3/AVERAGE($F$3:$F$6) Drag down to G18

26. Step 5The Deseasonalized Time Series Deseasonalized series value for Period 6 (2nd quarter, 1998) y6/(Quarter 2 Adjusted Seasonal Factor) = 7332/0.965252= 7595.94 • Determine “Deseasonalized data values”. Calculate: yt [Adjusted seasonal factors]t

27. =C3/G3 Drag to cell H18

28. Step 6The Time Series Trend Component Period (t) 17 18 19 20 Unadjusted Forecast (t) 8402.55 8480.95 8559.36 8637.76 • Regress on the Deseasonalized Time Series • Determine a deseasonalized forecast from the resulting regression equation (Unadjusted Forecast)t = 7069.6677 + 78.4046t

29. Run regression Deseason vs. Period =$L$18+$L$19*B19 Drag to cell I22

30. Step 7The Forecast Period 17 18 19 20 Unadjusted Forecast (t) 8402.55 8480.95 8559.36 8637.76 Adjusted Forecast (t) 8522.92 8186.26 8600.09 8773.06 Adjusted Seasonal Factor 1.014325 .965252 1.004759 1.015663 Re-seasonalize the forecast by multiplying the unadjusted forecast by the adjusted seasonal factor for each period.

31. Seasonally Adjusted Forecasts =I19*G3 Drag down to J22

32. Review • Additive Model for Time Series with Trend and Seasonal Effects • Use of Dummy Variables • 1 less than the number of seasons • Use of Regression • Modified F test if all p-values not < .05 • Multiplicative Model for Time Series with Trend and Seasonal Effects • Determine a set of adjusted period factors to deseasonalize data • Do regression to obtain unadjusted forecasts • Reseasonalize results to give seasonally adjusted forecasts. • Excel