Economics 173Business Statistics

Lecture 26

© Fall 2001, Professor J. Petry

http://www.cba.uiuc.edu/jpetry/Econ_173_fa01/

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Components of a Time Series

• A time series can consists of four components.Long - term trend (T).Cyclical effect (C).– Seasonal effect (S).Random variation (R).

The seasonal component of the time-seriesexhibits a short term (less than one year) calendar repetitive behavior.

6-88 12-88 6-89 12-89 6-90

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20.6 Measuring the Seasonal effects

• Seasonal variation may occur within a year or even within a shorter time interval.

• To measure the seasonal effects we construct seasonal indexes.

• Seasonal indexes express the degree to which the seasons differ from one another.

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• Example 20.6: Computing seasonal indexes

• Calculate the quarterly seasonal indexes for hotel occupancy rate in order to measure seasonal variation.

• DataYear Quarter Rate Year Quarter Rate Year Quarter Rate

1991 1 0.561 1993 1 0.594 1995 1 0.6652 0.702 2 0.738 2 0.8353 0.8 3 0.729 3 0.8734 0.568 4 0.6 4 0.67

1992 1 0.575 1994 1 0.6222 0.738 2 0.7083 0.868 3 0.8064 0.605 4 0.632

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0 5 10 15 20 25

t

Rat

e

• Perform regression analysis for the modely = 0 + 1t + where t represents the chronological time, and y represents the occupancy rate. Time (t) Rate1 0.5612 0.7023 0.84 0.5685 0.5756 0.7387 0.8688 0.605 . . . .

t005246.639368.y

The regression line represents trend.

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ttt

ttt

t

t RST

RSTyy

• Now Consider the multiplicative model

tttt RSTy

The regression line represents trend.

(Assuming no cyclical effects).

Rate/Predicted rate

0

0.5

1

1.5

1 3 5 7 9 11 13 15 17 19

No trend is observed, butseasonality and randomnessstill exist.

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Rate/Predicted rate

0

0.5

1

1.5

1 3 5 7 9 11 13 15 17 19

Rate/Predicted rate

0

0.5

1

1.5

1 3 5 7 9 11 13 15 17 19

• To remove most of the random variation but leave the seasonal effects,average the terms StRt for each season.

Rate/Predicted rate0.8701.0801.2210.8600.8641.1001.2840.8880.8651.0671.0460.8540.8790.9931.1220.8740.9131.1381.1810.900

Rate/Predicted rate0.8701.0801.2210.8600.8641.1001.2840.8880.8651.0671.0460.8540.8790.9931.1220.8740.9131.1381.1810.900

(.870 + .864 + .865 + .879 + .913)/5 = .878Average ratio for quarter 1:

Average ratio for quarter 2: (1.080+1.100+1.067+.993+1.138)/5 = 1.076

Average ratio for quarter 3: (1.222+1.284+1.046+1.123+1.182)/5 = 1.171

Average ratio for quarter 4: (.861 +.888 + ..854 + .874 + .900)/ 5 = .875

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• Normalizing the ratios:– The sum of all the ratios must be 4, such that the average

ratio per season is equal to 1.– If the sum of all the ratios is not 4, we need to normalize

(adjust) them proportionately. Suppose the sum of ratios equaled 4.1. Then each ratio will be multiplied by 4/4.1

(Seasonal averaged ratio) (number of seasons) Sum of averaged ratiosSeasonal index =

In our problem the sum of all the averaged ratios is equal to 4:.878 + 1.076 + 1.171 + .875 = 4.0.No normalization is needed. These ratios become the seasonal indexes.

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• Example

• Assume that in the previous example that the averaged ratios were:Quarter 1: .878Quarter 2: 1.076Quarter 3: 1.171Quarter 4: .775

Determine the seasonal index.

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Quarter 2 Quarter 3Quarter 3Quarter 2

• Interpreting the results– The seasonal indexes tell us what is the ratio

between the time series value at a certain season, and the overall seasonal average.

– In our problem:

Annual averageoccupancy (100%)

Quarter 1 Quarter 4 Quarter 1 Quarter 4

87.8%107.6%

117.1%

87.5%12.2% below theannual average

7.6% above theannual average

17.1% above theannual average

12.5% below theannual average

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• The smoothed time series.– The trend component and the seasonality

component are recomposed using the multiplicative model.

0.5

0.6

0.7

0.8

0.9

1 3 5 7 9 11 13 15 17 19

tttt S)t0052.639(.STy

In period #1 ( quarter 1): 566.)878))(.1(0052.639(.STy 111 In period #2 ( quarter 2): 699.)076.1))(2(0052.639(.STy 222

Actual series Smoothed series

The linear trend (regression) line

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• Example

• Recompose the smoothed time series in the above example for periods 3 and 4 assuming our initial seasonal index figures of:Quarter 1: .878Quarter 2: 1.076Quarter 3: 1.171Quarter 4: .875 and equation of: tttt S)t0052.639(.STy

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• Deseasonalizing time series

– By removing the seasonality, we can identify changes in the other components of the time - series.

Seasonally adjusted time series = Actual time seriesSeasonal index