The multivariable regression model

1. The multivariable regression model Airline sales is obviously a function of fares—but other factors come into play as well (e.g., income levels and fares of rivals). Multivariable regression is a technique that allows for more than one explanatory variable.

2. Model specification Recall from Chapter 3 we said that airline ticket sales were a function of three variables, that is: Q = f(P, PO, Y) [3.1] Again, Q is the airline’s coach seats sold per flight; P is the fare; P0 is the rival’s fare; and Y is a regional income index. Our regression specification can be written as follows:

3. The Data

4. Estimating multivariable regression models using OLS Let: Yi = 0 + 1X1i + 2X2i + i Computer algorithms find the ’s that minimize the sum of the squared residuals:

5. SPSS output We estimated the multivariable model using SPSS once again.

6. Results of the regression Our equation is estimated as follows:

7. Results of In-Sample Forecast

8. In-sample forecast for the multivariable model

9. Comparison of models • Notice that Adjusted R2 for the multivariable model is .720, compared to .557 for the single variable model. Hence we have a considerable increase in explanatory power. • The standard error of the regression has decreased from 18.6 to 14.8

10. Other results

11. The F test The F test provides another “goodness of fit” criterion for our regression equation. The F test is a test of joint significance of the estimated regression coefficients. The F statistic is computed as follows: Where K - 1 is degrees of freedom in the numerator and n – K is degrees of freedom in the denominator

12. We set up the following null hypothesis an alternative hypothesis: H0 : 1 = 2 = 3 = 0 HA: H0 is not true We adhere to the following decision rule: Reject H0 if F > FC, where FC is the critical value of F at the level of significance selected by the forecaster. Suppose we select the 5 percent significance level. The critical value of F (3 degrees of freedom in the numerator and 12 degrees of freedom in the denominator) is 3.49. Thus we can reject the null hypothesis since 13.9 > 3.49.

13. Example: The Demand for Coal COAL = 12,262 + 92.43FIS + 118.57FEU -48.90PCOAL + 118.91PGAS • COAL is monthly demand for bituminous coal (in tons) • FIS is the Federal Reserve Board Index of Iron and Steel production. • FEU the FED Index of Utility Production. • PCOAL is a wholesale price index for coal. • PGAS is a wholesale price index for naturalgas. Source: Pyndyck and Rubinfeld (1998), p. 218.