1. Economics 173 Business Statistics Lecture 28 © Fall 2001, Professor J. Petry http://www.cba.uiuc.edu/jpetry/Econ_173_fa01/

2. 2 20.9 Time-Series Forecasting with Linear regression Linear regression can be used to forecast time series with trend and seasonality. The model Linear trend value for period t, obtained from the linear regression Seasonal index for period t.

3. 3 Solution –The trend line was obtained from the regression analysis. –For 1996 we have: Example 20.9 –Use the seasonal indexes calculated in example 20.6 along with the trend line, to forecast each quarter occupancy rate in 1996. t Trend value Quarter SI Forecast 21.749 1.878.658 22.75521.076.812 23.76031.171.890 24.7654.875.670 F 21 =.749(.878) The process –Use simple linear regression to find the trend line. –Use the trend line to calculate the seasonal indexes. –To calculate F t multiply the trend value for period t by the seasonal index of period t.

4. 4 Forecasting Seasonal Time Series with Indicator Variables Use linear regression with indicator variables to forecast seasonal effects. If there are n seasons, we use n-1 indicator variables to define the season of period t: [Indicator variable n] = 1 if season n belongs to period t [Indicator variable n] = 0 if season n does not belong to period t

5. 5 Example 20.11 –Use regression analysis and indicator variables to forecast hotel occupancy rate in 1996 (see example 20.6) –Data Q i = 1 if quarter i occur at t 0 otherwise Quarter 1 belongs to t = 1 Quarter 2 does not belong to t = 1 Quarter 3 does not belong to t = 1 Quarter 1, 2, 3, do not belong to period t = 4.

6. 6 t = time in chronological order Q i = indicator variable (0, 1). The regression model y =  0 +  1 t +  2 Q 1 +  3 Q 2 +  4 Q 3 +  Good fit There is sufficient evidence to conclude that trend is present. There is insufficient evidence to conclude that seasonality causes occupancy rate in quarter 1 be different than this of quarter 4! Seasonality effects on occupancy rate in quarter 2 and 3 are different than this of quarter 4!

7. 7 The estimated regression model y =  0 +  1 t +  2 Q 1 +  3 Q 2 +  4 Q 3 +  1234 b2b2 b3b3 b4b4 Trend line b0b0

8. 8 Autoregressive models –Autocorrelation among the errors of the regression model provides opportunity to produce accurate forecasts. –Correlation between consecutive residuals leads to the following autoregressive model: y t =  0 +  1 y t-1 +  t

9. 9 Example 20.12 (Autoregressive model) –Forecast the increase in the Consumer Price Index (CPI) for the year 1994, based on the data collected for the years 1951 -1993. –Data

10. 10 Solution –Regressing the Percent increase (y t ) on the time variable (t), does not lead to good results. Y t = 2.14 +.098t r 2 = 15.0% Durbin-watson test statistic d =.47. This indicates the presence of first order autocorrelation between consecutive residuals. (-) (+) (-) The residuals form a pattern over time. Also,

11. 11 An autoregressive model appears to be a desirable technique. The increase in the CPI for periods 1, 2, 3,… are predictors of the increase in the CPI for periods 2, 3, 4,…, respectively.

12. 12 Compare