Regression Forecasting Procedure

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
Tt St εt

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 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

5. Scatterplot of Time Series
Gasoline Sales Over Five Year Period
3000
3200
3400
3600
3800
4000
4200
4400
4600
4800
5000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Period
Gasoline
Sales
(1000's
gallons)
Fall
Winter
Spring
Summer
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
Spring
Winter
Fall

7. The Excel Input

8. Add Dummy Variables
Not Fall, In Winter, not Spring
In Fall, not 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 forecasting additive model is:
Ft = 3610.625 + 58.33t – 155F – 323W –
248.27S
• Forecasts for year 6 are produced as follows:
• F(Year 6, Fall) = 3610.625+58.33(21) – 155(1) – 323(0) –
248.27(0)
• F(Year 6, Winter) = 3610.625+58.33(22) – 155(0) – 323(1) –
248.27(0)
• F(Year 6, Spring) = 3610.625+58.33(23) – 155(0) – 323(0) –
248.27(1)
• F(Year 6, Summer) = 3610.625+58.33(24) – 155(0) – 323(0)
– 248.27(0)
Troy’s Mobil Station –
Performing the forecast

12. The Forecasts
=$G$17+$G$18*B22+$G$19*C22+$G$20*D22+$G$21*E22
Drag F22 down to F25
=SUM(F22: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. • 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
Classical Decomposition
• Average all the yt/Tt that
correspond to the same season.

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

17. • 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.
CANADIAN FACULTY
ASSOCIATION (CFA)

18. CFA - Solution
• Membership records from 1997 through 2000
were collected and graphed.
YEAR PERIOD QUARTER
AVERAGE
MEMBERSHIP
1997 1 1 7130
2 2 6940
3 3 7354
4 4 7556
1998 5 1 7673
6 2 7332
7 3 7662
8 4 7809
1999 9 1 7872
10 2 7551
11 3 7989
12 4 8143
2000 13 1 8167
14 2 7902
15 3 8268
16 4 8436
1997 1998 1999 2000
The graph exhibits long term trend
The graph exhibits seasonality pattern

19. First moving average period is
centered at quarter (1+4)/ 2 = 2.5
Centered moving average of the first
two moving averages is
[7245.01 + 7380.75]/2 = 7312.875
• Smooth the time series to
remove random effects and
seasonality.
Calculate moving averages.
Step 1:
Isolating the Trend Component
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 location is t = 3
Trend value at period 3, T3

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

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

22. =C5/D5
Drag down to E16

23. 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 = {1.0056+1.0024+1.0079}/3 = 1.0053
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.

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

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

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

27. Step 5
The 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

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 = 7069.6677 + 78.4046t
Period (t)
17
18
19
20
Unadjusted Forecast (t)
8402.55
8480.95
8559.36
8637.76

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)
8402.55
8480.95
8559.36
8637.76
Period
17
18
19
20
Adjusted
Forecast (t)
8522.92
8186.26
8600.09
8773.06
Adjusted
Seasonal Factor
1.014325
.965252
1.004759
1.015663

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

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