1. Economics 173 Business Statistics Lecture 26 © Fall 2001, Professor J. Petry http://www.cba.uiuc.edu/jpetry/Econ_173_fa01/

2. 2 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-series exhibits a short term (less than one year) calendar repetitive behavior. 6-88 12-88 6-89 12-89 6-90

3. 3 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.

4. 4 Example 20.6: Computing seasonal indexes Calculate the quarterly seasonal indexes for hotel occupancy rate in order to measure seasonal variation. Data

5. 5 Perform regression analysis for the model y =  0 +  1 t +  where t represents the chronological time, and y represents the occupancy rate. Time (t) Rate 1 0.561 2 0.702 3 0.8 4 0.568 5 0.575 6 0.738 7 0.868 8 0.605.. The regression line represents trend.

6. 6 Now Consider the multiplicative model The regression line represents trend. (Assuming no cyclical effects). No trend is observed, but seasonality and randomness still exist.

7. 7 To remove most of the random variation but leave the seasonal effects,average the terms S t R t for each season. (.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

8. 8 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 ratios Seasonal 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.

9. 9 Example Assume that in the previous example that the averaged ratios were: Quarter 1:.878 Quarter 2:1.076 Quarter 3:1.171 Quarter 4:.775 Determine the seasonal index.

10. 10 Quarter 2 Quarter 3 Quarter 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 average occupancy (100%) Quarter 1 Quarter 4 Quarter 1 Quarter 4 87.8% 107.6% 117.1% 87.5% 12.2% below the annual average 7.6% above the annual average 17.1% above the annual average 12.5% below the annual average

11. 11 The smoothed time series. –The trend component and the seasonality component are recomposed using the multiplicative model. In period #1 ( quarter 1): In period #2 ( quarter 2): Actual series Smoothed series The linear trend (regression) line

12. 12 Example Recompose the smoothed time series in the above example for periods 3 and 4 assuming our initial seasonal index figures of: Quarter 1:.878 Quarter 2:1.076 Quarter 3:1.171 Quarter 4:.875 and equation of:

13. 13 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 series Seasonal index