ForecastingForecasting

ModelsModelsWithWith

Trend and Seasonal EffectsTrend and Seasonal Effects

Types of Seasonal ModelsTypes of Seasonal Models

• Two possible models are:

Additive Modelyt = Tt + St + εt

Multiplicative Modelyt = TtStεt

Trend Effects

Seasonal Effects

Random Effects

Additive ModelAdditive ModelRegression Forecasting ProcedureRegression Forecasting Procedure• Suppose a time series is modeled as having k seasons

(Here we illustrate k = 4 quarters)– The following 4 equations represent time series value of 4

seasons

Season 1: yt = β0 + β1t + β2 + εt

Tt εt

Season 2: yt = β0 + β1t + β3 + εt

St

Season 3: yt = β0 + β1t + β4 + εt

Season 4: yt = β0 + β1t + β5 + εt

Additive ModelAdditive ModelRegression Forecasting ProcedureRegression Forecasting Procedure

• Combining the 4 equations into one, we can use 4 dummy variables, S1, S2, S3 and S4 corresponding to seasons 1, 2, 3 and 4 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, S4 = 0– Season 2: S1 = 0, S2 = 1, S3 = 0 ,S4 = 0– Season 3: S1 = 0, S2 = 0, S3 = 1, S4 = 0 – Season 4: S1 = 0, S2 = 0, S3 = 0, S4 = 0

• We can simplified the above equation by removing β5S4

yt = β0 + β1t + β2S1 + β3S2 + β4S3 + β5S4 + εt

Tt St εt

Additive ModelAdditive ModelRegression Forecasting ProcedureRegression Forecasting Procedure

– 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

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.

• 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 Stεt

ExampleExampleTroy’s Mobil StationTroy’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 5FALL 3497 3726 3989 4248 4443WINTER 3484 3589 3870 4105 4307SPRING 3553 3742 3996 4263 4466SUMMER 3837 4050 4327 4544 4795

Scatterplot of Time SeriesScatterplot 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

Gas

oli

ne

Sal

es

(100

0's

gal

lon

s)

Fall

WinterSpring

Summer

General PatternGeneral Pattern:: Winter less than Fall, Spring more than Winter, Summer more than Spring, Fall less than Summer

The ModelThe Model

• There is also apparent long term trend.

• The form of the model then is:

yt = β0 + β1t + β2F + β3W + β4S + εt

SpringWinterFall

The Excel InputThe Excel Input

Add Dummy VariablesAdd 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

Regression IntputRegression Intput

Regression OutputRegression Output

Low p-value for F-test

Low p-values for all t-tests

ConclusionGood model – all factors significant

• 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 – Troy’s Mobil Station – Performing the forecastPerforming the forecast

The ForecastsThe Forecasts

=$G$17+$G$18*B22+$G$19*C22+$G$20*D22+$G$21*E22

Drag F22 down to F25

=SUM(F22:F25)

What if Some of the p-values are high?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

Multiplicative ModelMultiplicative ModelClassical Decomposition ApproachClassical 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.

• 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 DecompositionClassical Decomposition

• Average all the yt/Tt that correspond to the same season.

• 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)Classical Decomposition (Cont’d)

Use linear regression on the deseasonalized time series.

Calculate:(Desesonalized values)

[Adjusted seasonal factors]).

• Determine an “adjusted seasonal forecast”.

• 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 CANADIAN FACULTY ASSOCIATION (CFA)ASSOCIATION (CFA)

CFA - SolutionCFA - Solution

• Membership records from 1997 through 2000 were collected and graphed.

YEAR PERIOD QUARTERAVERAGE

MEMBERSHIP

1997 1 1 71302 2 69403 3 73544 4 7556

1998 5 1 76736 2 73327 3 76628 4 7809

1999 9 1 787210 2 755111 3 798912 4 8143

2000 13 1 816714 2 790215 3 826816 4 8436

YEAR PERIOD QUARTERAVERAGE

MEMBERSHIP

1997 1 1 71302 2 69403 3 73544 4 7556

1998 5 1 76736 2 73327 3 76628 4 7809

1999 9 1 787210 2 755111 3 798912 4 8143

2000 13 1 816714 2 790215 3 826816 4 8436

Average Dues Paying Members (payroll deduction)

6500

7000

7500

8000

8500

9000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Per iod

Average Dues Paying Members (payroll deduction)

6500

7000

7500

8000

8500

9000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Per iod1997 1998 1999 2000

The graph exhibits long term trend

The graph exhibits seasonality pattern

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:Step 1:Isolating the Trend ComponentIsolating the Trend Component

Average membership for the first 4 periods = [7130+6940+7354+7556]/4 = 7245.01

Second moving average periodis 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

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

Since yt =TtStεt, then the period factorperiod factor, SSttεεtt is given by

SSttt t = y= ytt/T/Ttt

Step 2Step 2Determining the Period FactorsDetermining 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

=C5/D5Drag down to E16

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 + S3,98 3,98 + S3,99 3,99}/3 = {1.0056+1.0024+1.0079}/3 = 1.0053

Step 3Step 3Unadjusted Seasonal FactorsUnadjusted 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.

=AVERAGE(E3,E7,E11,E15)Drag down to F6

Paste Special(Values)

Copy F3:F6

Average seasonal factor = (1.01490+.96580+1.00533+1.01624)/4=1.00057

Step 4Step 4Adjusted Seasonal FactorsAdjusted Seasonal Factors

• Determine the “adjusted seasonal factors” so that average adjusted factor is 1

Calculate:

Unadjusted seasonal factors Average seasonal factor

Quarter1234

UnadjustedSeasonal Factor

1.01490 .965801.005331.01624

AdjustedSeasonal Factor

1.014325 .9652521.0047591.015663

Unadjusted Seasonal Factors/1.00057

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

Step 5Step 5The Deseasonalized Time SeriesThe Deseasonalized Time Series

Deseasonalized series value for Period 6Deseasonalized series value for Period 6(2(2ndnd quarter, 1998) quarter, 1998)

y6/(Quarter 2 Adjusted Seasonal Factor) = 7332/0.965252 = 7595.947595.94

• Determine “Deseasonalized data values”.

Calculate: yt

[Adjusted seasonal factors]t

=C3/G3Drag to cell H18

Step 6Step 6The Time Series Trend ComponentThe 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)17181920

Unadjusted Forecast (t)8402.558480.958559.368637.76

Run regressionDeseason vs. Period

=$L$18+$L$19*B19Drag to cell I22

Step 7Step 7The ForecastThe Forecast

Re-seasonalize the forecast by multiplying the unadjusted forecast by the adjusted seasonal factor for each period.

UnadjustedForecast (t)

8402.558480.958559.368637.76

Period17181920

AdjustedAdjustedForecast (t)Forecast (t)

8522.928522.928186.268186.268600.098600.098773.068773.06

AdjustedSeasonal Factor

1.014325 .9652521.0047591.015663

=I19*G3Drag down to J22

SeasonallyAdjustedForecasts

ReviewReview• 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