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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.

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:

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:

SPSS output

We estimated the multivariable model using SPSS once again.

Results of the regression

Our equation is estimated as follows:

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

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

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.

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.

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