Title: Forecasting
1- Forecasting
- Models
- With
- Trend and Seasonal Effects
2Types of Seasonal Models
Additive Model yt Tt St et
Multipicative Model yt TtStet
3Additive 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
4ExampleTroys 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).
5Scatterplot of Time Series
General Pattern Winter less than Fall, Spring
more than Winter, Summer more than Spring, Fall
less than Summer
6The Model
- There is also apparent long term trend.
- The form of the model then is
yt ß0 ß1t ß2F ß3W ß4S et
7The Excel Input
8Add Dummy Variables
9Regression Intput
10Regression Output
Conclusion Good model all factors significant
11The Forecasts
12What 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
13Multiplicative 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.
14Classical 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.
15Classical 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.
16CANADIAN 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.
17CFA - Solution
- Membership records from 1997 through 2000 were
collected and graphed.
The graph exhibits long term trend
The graph exhibits seasonality pattern
18Step 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
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20Step 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
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22Step 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
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24Step 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 -
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26Step 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
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28Step 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
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30Step 7The Forecast
- Re-seasonalize the forecast by multiplying the
unadjusted forecast by the adjusted seasonal
factor for each period.
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32Review
- 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