2. Chapter 4 Using Regression to Estimate Trends

3. Trend Models zLinear trend, zQuadratic trend zCubic trend zExponential trend

4. Choosing a trend zPlot the data, choose possible models zUse goodness of fit measures to evaluate models zTry to Minimize the AIC and SBC zChoose a model

5. Mean Squared Error

6. Goodness of Fit Measures zCoefficient of Determination or R 2

7. Goodness of Fit Measures zAdjusted R 2

8. AIC and SBC

9. AIC and SBC(continued) zChoose the model that minimizes the AIC and SIC zExamples ychoose AIC=3 over AIC=7 ychoose SIC=-7 over SIC=-5 zThe SIC has a larger penalty for extra parameters!

10. 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.

11. Testing the slopes zT-test tests a hypothesis about a coefficient. zA common hypothesis of interest is:

12. Steps in a T-test z1. Specify the null hypothesis z2. Find the rejection region z3. Calculate the statistic z4. If the test statistic is in the rejection region then reject!

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

14. An Example,n=264.9 5 t 0 f(t) -1.961.96.025  red area = rejection region for 2-sided test

15. LS // Dependent Variable is CARSALES Date: 02/17/98 Time: 13:44 Sample: 1976:01 1997:12 Included observations: 264 VariableCoefficientStd. Errort-StatisticProb. 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

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

17. P-Value=lined area=.8725.9 5 t 0 f(t) -1.961.96.025  red area = rejection region for 2-sided test.016

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

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