2. Airline ticket pricing Consider United Airlines Flight 815 from Chicago to LA on October 31, 1997 1 There were 27 different one-way fares, ranging from $1,248 for a first class ticket purchased the day of the flight to $87 for an advance purchase coach ticket. Some travelers cashed in frequent flier miles. Some qualified for senior citizen discounts. Some passengers traveled on restricted tickets that required Saturday stayovers. 1 So, How much did you pay for your ticket, New York Times, April 12, 1998

3. Assumptions 1.You are a manager for a regional airline offering non-stop service between Houston, TX and Orlando, FL. 2.Your airline makes one departure from each city per day (2 flights total). 3.One rival airline offers non-stop service on this route. 4.We ignore first class service and focus on the demand for coach-class travel.

4. The demand function Q = f(P, P O, Y) [1] [3.1] can be read as follows: The number of your airlines coach seats sold per flight (Q) is a function of the your airlines coach fare (P), its rivals fare (P O ), and income in the region (Y) Your forecasting unit has estimated the following demand function: Q = 25 + 3Y + P O – 2P [2]

5. Effect of changes in the explanatory variables 1.For each one point increase in the income index (Y), 3 additional seats will be sold, ceteris paribus. 2.For each $10 increase in the airlines fare, 20 fewer seats will be sold, ceteris paribus. 3.For each $10 increase in the competitors fare, 10 additional seats will be sold, ceteris paribus. Q is the dependent variable; P, P O, and Y are the independent or explanatory variables.

6. The multivariable regression model How did the Forecasting Unit estimate that equation? Multivariable regression is a technique that allows for more than one explanatory variable.

7. Model specification Suppose that airline ticket sales are a function of three variables, that is: Q = f(P, P O, Y) [3.1] Q is the airlines coach seats sold per flight; P is the fare; P 0 is the rivals fare; and Y is a regional income index. Our regression specification can be written as follows:

8. The Data

9. Estimating multivariable regression models using OLS Let: Y i = 0 + 1 X 1i + 2 X 2i + i Computer algorithms find the s that minimize the sum of the squared residuals:

10. Excel Output CoefficientsStandard Errort StatP-value Intercept29.13845472174.74268970.166750.8703 Fare (P)-2.123647330.340540892-6.23614E-05 Fare (P0)1.034455120.4667334692.216370.0467 Income (Y)3.0871388940.9993360183.089190.0094

11. Excel Output

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

13. Results of In-Sample Forecast

14. In-sample forecast for the multivariable model

15. Other results

16. 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

17. 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.

18. 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.