The multivariable regression model

1
The multivariable regression model
Airline sales is obviously a function of
faresbut 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 airlines coach seats sold per
flight P is the fare P0 is the rivals 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 gt 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
gt 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.