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Forecasting

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Troy also knows that sales vary by season due to the vacationers. ... Average all the yt/Tt that correspond to the same season. ... – PowerPoint PPT presentation

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Title: Forecasting


1
  • Forecasting
  • Models
  • With
  • Trend and Seasonal Effects

2
Types of Seasonal Models
  • Two possible models are

Additive Model yt Tt St et
Multipicative Model yt TtStet
3
Additive ModelRegression 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 0s and 1s 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 ß0 ß1t ß2S1 ß3S2 ß4S3 et
4
ExampleTroys 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 1000s of gallons per season, Troy needs to
    predict total sales for next year (periods 21,
    22, 23, and 24).

5
Scatterplot of Time Series

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 ß0 ß1t ß2F ß3W ß4S et
7
The Excel Input

8
Add Dummy Variables

9
Regression Intput

10
Regression Output

Conclusion Good model all factors significant
11
The Forecasts

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 gt
    F?,3,DFE(Full)
  • F ((SSEREDUCED-SSEFULL)/3)/MSEFULL
  • So if F gtF?,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
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.

15
Classical Decomposition (Contd)
  • Calculate Unadjusted seasonal factor
    Average seasonal factor
  • Determine the adjusted seasonal factors.
  • Determine Deseasonalized data values.

Calculate
yt Adjusted seasonal factorst
  • Determine a deseasonalized trend forecast.

Use linear regression on the deseasonalized time
series.
Calculate(Desesonalized values) Adjusted
seasonal factors).
  • Determine an adjusted seasonal forecast.

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

The graph exhibits long term trend
The graph exhibits seasonality pattern
18
Step 1Isolating the Trend Component
  • Smooth the time series to remove random effects
    and seasonality.

Calculate moving averages.
First moving average period is centered at
quarter (14)/ 2 2.5
Average membership for the first 4 periods
7130694073547556/4 7245.01
Second moving average period is centered at
quarter (25)/ 2 3.5
Average membership for periods 2, 5
6940735475567673/4 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
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20
Step 2Determining the Period Factors
  • Determine period factors to isolate the
    (Seasonal)(Random error) factor.

Calculate the ratio yt/Tt.
Since yt TtStet, then the period factor, Stet is
given by Stet yt/Tt
Example In period 7 (3rd quarter of 1998)S7e7
y7/T7 7662/7643.875 1.002371
21
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22
Step 3Unadjusted 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, Stet 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
    1.00561.00241.0079/3 1.0053

23
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24
Step 4Adjusted Seasonal Factors
Calculate Unadjusted seasonal
factors Average seasonal factor
  • Determine the adjusted seasonal factors so that
    average adjusted factor is 1
  • Average seasonal factor (1.01490.965801.00533
    1.01624)/41.00057

25
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26
Step 5The Deseasonalized Time Series
  • Determine Deseasonalized data values.

Calculate
yt Adjusted seasonal factorst
  • Deseasonalized series value for Period 6
  • (2nd quarter, 1998)
  • y6/(Quarter 2 Adjusted Seasonal Factor)
    7332/0.965252 7595.94

27
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28
Step 6The 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
29
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30
Step 7The Forecast
  • Re-seasonalize the forecast by multiplying the
    unadjusted forecast by the adjusted seasonal
    factor for each period.

31
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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 lt .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
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