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, P O, Y) [3.1] Again, Q is the airline’s coach seats sold per flight; P is the fare; P 0 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: Y i =  0 +  1X 1i +  2 X 2i +  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 R 2 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: H 0 :  1 =  2 =  3 = 0 H A : H 0 is not true We adhere to the following decision rule: Reject H 0 if F > F C, where F C 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 natural gas. Source: Pyndyck and Rubinfeld (1998), p. 218.