1. Economics 173 Business Statistics Lecture 25 © 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 consist of four components. – Long - term trend (T). –Cyclical effect (C). –Seasonal effect (S). Random variation (R). A trend is a long term relatively smooth pattern or direction, that persists usually for more than one year.

3. 3 20.4 Trend Analysis The trend component of a time series can be linear or non-linear. It is easy to isolate the trend component using linear regression. –For linear trend use the model y =  0 +  1 t +  –For non-linear trend with one (major) change in slope use the quadratic model y =  0 +  1 t +  2 t 2 + 

4. 4 Example 20.3 (Identify the trend component) Annual sales for a pharmaceutical company are believed to change linearly over time. Based on the last 10 year sales records, measure the trend component. Start by renaming your years 1, 2, 3, etc.

5. 5 Solution –Using Excel we have There is a relatively good fit. Be aware of possible cyclical effects and seasonality, that may reduce the fit of the model. 11 Forecast for period 11

6. 6 Example 20.4 Determine the long-term trend of the number of cigarettes smoked by Americans 18 years and older. Data Solution –Cigarette consumption increased between 1955 and 1963, but Decreased thereafter. –A quadratic model seems to fit this pattern.

7. 7 Using the computer The quadratic model fits the data very well.

8. 8 Example Forecast the trend components for the previous two examples for periods 12, and 41 respectively.

9. 9 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). A cycle is a wavelike pattern describing a long term behavior (for more than one year). Cycles are seldom regular, and often appear in combination with other components. 6-88 12-88 6-89 12-89 6-90

10. 10 20.5 Measuring the Cyclical Effects Although often unpredictable, cycles need to be isolated. To identify cyclical variation we use the percentage of trend. –Determine the trend line (by regression). –Compute the trend value for each period t. –Calculate the percentage of trend by

11. 11 Example 20.5 Does the demand for energy in the US exhibit cyclic behavior over time? Assume a linear trend and calculate the percentage of trend. Data –The collected data included annual energy consumption from 1970 through 1993.

12. 12 When groups of the percentage of trend alternate around 100%, the cyclic effects are present. (66.4/69.828)100

13. 13 Example Find the values for trend and percentage of trend for observations 6 (year 1975) and 7 (1976) in the above example. The actual observations were 70.6 and 74.4.