Modeling Time Series Data

1
Modeling Time Series Data
Module 5
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A Composite Model
We can fit a composite model of the form Sales
(Trend) (Seasonality) (Cyclicality) (Error)
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Trend
Trend is the long term level and the pattern of
change in the dependent variable. It is estimated
as a simple function of the period number (time).
Linear regression or method of least squares is
used to estimate the trend.
A linear model captures the general upward (or
downward) trend with steady growth.
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Seasonality
Seasonality is a cycle with a period of exactly
one year. We estimate it as a proportion of trend
for each season. Data must be available on
seasonal basis. Time series decomposition is a
method to estimate seasonal component.
Seasonality captures regular, predictable
deviations from the trend. Typical seasons are
quarters, weeks, or days.
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Cyclicality
Cyclicality captures the effects of long-term
macroeconomic boom-bust cycles. It is often
difficult to get enough data to measure
accurately.
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Composite Model
Any residual deviations are attributed to random
error.
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Time Series Decomposition

• Start with raw data (y)
• Estimate Seasonal Indices
• Compute base trend using centered moving averages
  (t)
• Estimate seasonal ratios (y/t)
• Average seasonal ratios to get raw seasonal
  indices
• Normalize seasonal indices (s)
• De-seasonalize the raw data (y/s)
• Estimate the trend equation using de-seasonalized
  data (t)
• Forecast y t s
• Calculate error y (ts)

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Example Modeling Trend and Seasonality
Toys R Us Revenue (millions )
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Example Computing Moving Averages
Calculate Moving Average with span of 4
(1026 1056 1182 2861) 4
1531.3
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Example Using centered moving averages to
estimate base demand
Center Moving Average if using even number of
data points
(1531.3 1567.8) 2
1549.5
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Example Computing Seasonal Ratios
Calculate the ratio of the revenue to the
centered moving average
1182 1549.5
.7628
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Example Calculating raw Seasonal Indices
Calculate the average ratio for each season
(quarter).
.7162 .6949 .7006 .7424
4
Raw Seasonal Index .7135
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Example Normalizing Seasonal Indices
Normalize to make sure Seasonal Indices average
to 1.0 (or add up to 4 in this case)
.7135
. .7135.7077.74061.844
.7124
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Example De-Seasonalizing raw data
Deseasonalize observations.
1026 .7124
1440.2
y y/s
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Example De-Seasonalizing
Fit a regression line to the deseasonalized
observations y (using time as the independent
variable).
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Example De-Seasonalizing
56.93 (1) 1373.4 1430.3
Use trend to make deseasonalized predictions - T
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Example De-Seasonalizing
2568.9 .71241 1830.2
Reseasonalize predictions (TS) to make forecasts
into the future.
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Example De-Seasonalizing
Plot the forecasts TS
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Example De-Seasonalizing
(1026 1018.98)
2
97.0
As an alternative goodness of fit measure,
calculate Root Mean Square Error.
RMSE 9865.2 99.3
Average square error
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Example De-Seasonalizing with Statpro
http//www.indiana.edu/mgtsci/StatPro.html
Statpro can be used to calculate seasonal
indices. Click on Statpro -gt Forecast.
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Example De-Seasonalizing with Statpro
Select the dependent variable.
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Example De-Seasonalizing with Statpro
Select quarterly data.
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Example De-Seasonalizing with Statpro
Select a span of 4 and a moving average method of
deseasonalizing.
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Example De-Seasonalizing with Statpro
Statpro generates the same values that we
calculated manually.
(Statpro output)