Chapter 4 Using Regression to Estimate Trends

Trend Models zLinear trend, zQuadratic trend zCubic trend zExponential trend

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

Mean Squared Error

Goodness of Fit Measures zCoefficient of Determination or R 2

Goodness of Fit Measures zAdjusted R 2

AIC and SBC

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!

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.

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

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!

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

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

LS // Dependent Variable is CARSALES Date: 02/17/98 Time: 13:44 Sample: 1976: :12 Included observations: 264 VariableCoefficientStd. Errort-StatisticProb. C TIME TIME2 2.52E E R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

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

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

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

LS // Dependent Variable is CARSALES Date: 02/17/98 Time: 14:00 Sample: 1976: :12 Included observations: 264 VariableCoefficientStd. Errort-StatisticProb. C TIME R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)