1. Chapter 4 Using Regression to Estimate Trends

2. Trend Models • Linear trend, • Quadratic trend • Cubic trend • Exponential trend

3. Choosing a trend • Plot the data, choose possible models • Use goodness of fit measures to evaluate models • Try to Minimize the AIC and SBC • Choose a model

4. Mean Squared Error

5. Goodness of Fit Measures • Coefficient of Determination or R2

6. Goodness of Fit Measures • Adjusted R2

7. AIC and SBC

8. AIC and SBC(continued) • Choose the model that minimizes the AIC and SIC • Examples • choose AIC=3 over AIC=7 • choose SIC=-7 over SIC=-5 • The SIC has a larger penalty for extra parameters!

9. F-Test The F-test tests the hypothesis that the coefficients of all explanatory variables are zero. A p-value less than .05 rejects the null and concludes that our model has some value.

10. Testing the slopes • T-test tests a hypothesis about a coefficient. • A common hypothesis of interest is:

11. Steps in a T-test • 1. Specify the null hypothesis • 2. Find the rejection region • 3. Calculate the statistic • 4. If the test statistic is in the rejection region then reject!

12. Figure 5.1 Student-t Distribution f(t) () /2 /2 0 tc t -tc red area = rejection region for 2-sided test

13. An Example,n=264 f(t) .95 .025 .025 0 t -1.96 1.96 red area = rejection region for 2-sided test

14. LS // Dependent Variable is CARSALES Date: 02/17/98 Time: 13:44 Sample: 1976:01 1997:12 Included observations: 264 Variable Coefficient Std. Error t-Statistic Prob. C 13.10517 0.311923 42.01413 0.0000 TIME 0.000882 0.005479 0.160947 0.8723 TIME2 2.52E-05 2.02E-05 1.248790 0.2129 R-squared 0.107295 Mean dependent var 13.80292 Adjusted R-squared 0.100454 S.D. dependent var 1.794726 S.E. of regression 1.702197 Akaike info criterion 1.075139 Sum squared resid 756.2412 Schwarz criterion 1.115774 Log likelihood -513.5181 F-statistic 15.68487 Durbin-Watson stat 0.370403 Prob(F-statistic) 0.000000

15. Using our results Plugging in our estimates: Not in the rejection region, don’t reject!

16. P-Value=lined area=.8725 f(t) .95 .025 .025 0 t -1.96 1.96 .016 red area = rejection region for 2-sided test

17. Ideas for model building • F-stat is large, p-value=.000000 implies our model does explain something • “Fail to reject” does not imply accept in a t-test • Idea, drop one of the variables

18. LS // Dependent Variable is CARSALES Date: 02/17/98 Time: 14:00 Sample: 1976:01 1997:12 Included observations: 264 Variable Coefficient Std. Error t-Statistic Prob. C 12.81594 0.209155 61.27481 0.0000 TIME 0.007506 0.001376 5.454057 0.0000 R-squared 0.101961 Mean dependent var 13.80292 Adjusted R-squared 0.098533 S.D. dependent var 1.794726 S.E. of regression 1.704014 Akaike info criterion 1.073520 Sum squared resid 760.7597 Schwarz criterion 1.100611 Log likelihood -514.3044 F-statistic 29.74674 Durbin-Watson stat 0.368210 Prob(F-statistic) 0.000000