1. Economics 173 Business Statistics Lecture 19 Fall, 2001© Professor J. Petry http://www.cba.uiuc.edu/jpetry/Econ_173_fa01/

2. 2 The Regression Analysis Process A generalized procedure for combining art and science –Develop a model that has a sound basis. Theoretical and practical inputs into model basis –Gather data for the variables in the model. Gather data for dependent and independent variables –Draw the scatter diagram to determine whether a linear model (or other forms) appears to be appropriate. –Obtain the model coefficients and statistics using a statistical computer software.

3. 3 The Regression Analysis Process A generalized procedure for regression analysis –Assess the model fit and usefulness using the model statistics. Does the model hold promise? (overall model evaluation) Do are variables make sense? (significance, signs) –Diagnose violations of required conditions. Try to remedy problems when identified. –Assess the model fit and usefulness using the model statistics. –If the model passes the assessment tests, use it to interpret the coefficients and generate predictions.

4. 4 –La Quinta Motor Inns is planning an expansion. –Management wishes to predict which sites are likely to be profitable. –Several predictors of profitability which can be identified include: Competition Market awareness Demand generators Demographics Physical quality Example 18.1 Where to locate a new motor inn?

5. 5 Profitability Competition Market awareness Demand Generators Physical RoomsNearestOffice space College enrollment IncomeDisttown Distance to downtown. Median household income. Distance to the nearest La Quinta inn. Number of hotels/motels rooms within 3 miles from the site. Demographics

6. 6 –Data was collected from randomly selected 100 inns that belong to La Quinta, and ran for the following suggested model: Margin =     Rooms   Nearest   Office    College +  5 Income +  6 Disttwn +

7. 7 Excel output This is the sample regression equation (sometimes called the prediction equation) MARGIN = 72.455 - 0.008 ROOMS - 1.646 NEAREST + 0.02 OFFICE +0.212 COLLEGE - 0.413 INCOME + 0.225 DISTTWN Assessing this equation

8. 8 Standard error of estimate –We need to estimate the standard error of estimate –Compare s  to the mean value of y From the printout, Standard Error = 5.5121 Calculating the mean value of y we have –It seems s  is not particularly small. –Can we conclude the model does not fit the data well?

9. 9 Coefficient of determination –The definition is –From the printout, R 2 = 0.5251 –52.51% of the variation in the measure of profitability is explained by the linear regression model formulated above. –When adjusted for degrees of freedom, Adjusted R 2 = 1-[SSE/(n-k-1)] / [SST/(n-1)] = = 49.44%

10. 10 Testing the validity of the model: The F Test –We pose the question: Is there at least one independent variable linearly related to the dependent variable? –To answer the question we test the hypothesis H 0 :  1 =  2 = … =  k = 0 H 1 : At least one  i is not equal to zero. –If at least one  i is not equal to zero, the model is valid.

11. 11 To test these hypotheses we perform an analysis of variance procedure. The F test –Construct the F statistic –Rejection region F>F ,k,n-k-1 MSE MSR F  MSR=SSR/k MSE=SSE/(n-k-1) SST = SSR + SSE. Large F results from a large SSR. Then, much of the variation in y is explained by the regression model. The null hypothesis should be rejected; thus, the model is valid. Required conditions must be satisfied.

12. 12 Excel provides the following ANOVA results Example 18.1 - continued SSE SSR MSE MSR MSR/MSE

13. 13 Excel provides the following ANOVA results Example 18.1 - continued F ,k,n-k-1 = F 0.05,6,100-6-1 =2.17 F = 17.14 > 2.17 Also, the p-value (Significance F) = 3.03382(10) -13 Clearly,  = 0.05>3.03382(10) -13, and the null hypothesis is rejected. Conclusion: There is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. At least one of the  i is not equal to zero. Thus, at least one independent variable is linearly related to y. This linear regression model is valid

14. 14 Remember Our Armani Pizza Example from Simple Linear Regression? –Compare the p-values from the F-test, and the t-test. –In simple linear regression, testing the validity of the overall model, and the slope of our single independent variable should (and do!) provide identical results.

15. 15 Do the Variables Make Sense? –Interpreting the coefficients – This is the intercept, the value of y when all the variables take the value zero. Since the data range of all the independent variables do not cover the value zero, do not interpret the intercept. – In this model, for each additional 1000 rooms within 3 mile of the La Quinta inn, the operating margin decreases on the average by 7.6% (assuming the other variables are held constant).

16. 16 – In this model, for each additional mile that the nearest competitor is to La Quinta inn, the average operating margin decreases by 1.65% – For each additional 1000 sq-ft of office space, the average increase in operating margin will be.02%. – For additional thousand students MARGIN increases by.21%. – For additional $1000 increase in median household income, MARGIN decreases by.41% – For each additional mile to the downtown center, MARGIN increases by.23% on the average

17. 17 Testing the coefficients –The hypothesis for each  i –Excel printout H 0 :  i = 0 H 1 :  i = 0 Test statistic d.f. = n - k -1

18. 18 Example – Vacation Homes (18.1) –A developer who specializes in summer cottage properties is looking at a lakeside tract of land for possible development. –She wants to estimate the selling price for the individual lots. –She knows from experience that sale price depends upon lot size, number of mature trees, and distance to the lake. –She gathers data on 60 nearby lots which have sold recently, and conducts a regression analysis, with the following results:

19. 19

20. 20 Example – Vacation Homes (18.1) 1.What is the standard error of the estimate? Interpret its value. 2.What is the coefficient of determination? What does this statistic tell you? 3.What is the coefficient of determination, adjusted for degrees of freedom? Why does this value differ from the coefficient of determination? What does this tell you about the model? ========================================================= 1.Test the overall validity of the model. What does the p-value of the test statistic tell you? 2.Interpret each of the coefficients. 3.Test to determine whether each of the independent variables is linearly related to the price of the lot.