1. Forecasting Models With Trend and Seasonal Effects

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

3. Additive Model Regression Forecasting Procedure
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 εt Tt St

4. Example Troy’s Mobil Station
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).
YEAR
SEASON
FALL
WINTER
SPRING
SUMMER

5. Scatterplot of Time Series
Summer
Fall
Spring
Winter
General Pattern: Winter less than Fall, Spring more than Winter, Summer more than Spring, Fall less than Summer

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

7. The Excel Input

8. Add Dummy Variables Pattern Repeats 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. Good model – all factors significant
Regression Output
Low p-value for F-test
Low p-values for all t-tests
Conclusion
Good model – all factors significant

11. Troy’s Mobil Station – Performing the forecast
The forecasting additive model is: Ft = t – 155F – 323W – S
Forecasts for year 5 are produced as follows:
F(Year 5, Fall) = (21) – 155(1) – 323(0) – (0)
F(Year 5, Winter) = (22) – 155(0) – 323(1) – (0)
F(Year 5, Spring) = (23) – 155(0) – 323(0) – (1)
F(Year 5, Summer) = (24) – 155(0) – 323(0) – (0)

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

13. 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 HA) if F > F,3,DFE(Full)
F = ((SSEREDUCED-SSEFULL)/3)/MSEFULL
So if F >F,3,DFE(Full) ---Include seasonal variables

14. Multiplicative Model Classical 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.

15. Classical Decomposition
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.
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.

16. Classical Decomposition (Cont’d)
Calculate: [Unadjusted seasonal factor] [Average seasonal factor]
Determine the “adjusted seasonal factors”.
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]).

17. 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.

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

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

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

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

22. =C5/D5
Drag down to E16

23. Step 3 Unadjusted Seasonal Factors
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.
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 = {S3,97 e3,97 + S3,98 e3,98 + S3,99 e3,99}/3 = { }/3 =

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

25. Step 4 Adjusted Seasonal Factors
Determine the “adjusted seasonal factors” so that average adjusted factor is 1
Calculate:
Unadjusted seasonal factors Average seasonal factor
Average seasonal factor = ( )/4=
Quarter 1 2 3 4
Unadjusted
Seasonal Factor
.96580
Adjusted
Seasonal Factor
Unadjusted Seasonal Factors/

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

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

28. =C3/G3
Drag to cell H18

29. Step 6 The Time Series Trend Component
Regress on the Deseasonalized Time Series
Determine a deseasonalized forecast from the resulting regression equation
(Unadjusted Forecast)t = t
Unadjusted Forecast (t)
Period (t) 17 18 19 20

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

31. Step 7 The Forecast
Re-seasonalize the forecast by multiplying the unadjusted forecast by the adjusted seasonal factor for each period.
Unadjusted
Forecast (t)
Period 17 18 19 20
Adjusted
Seasonal Factor
Adjusted
Forecast (t)

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

33. 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