815 from Chicago to LA on October 31 1997 1 There were 27 different oneway fares ranging from 1248 for a first class ticket purchased the day of the flight to 87 for an advance purchase coach ticket ID: 720146
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Slide1
Airline ticket pricing
Consider United Airlines Flight
815 from Chicago to LA on October 31, 19971There 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, 1998Slide2
Assumptions
You are a manager for a regional airline offering non-stop service between Houston, TX and Orlando, FL.
Your airline makes one departure from each city per day (2 flights total).One rival airline offers non-stop service on this route.We ignore first class service and focus on the demand for coach-class travel.Slide3
The demand function
Q = f(P, P
O, Y)[1]
[3.1] can be read as follows:
The number of your airline’s coach seats sold per flight
(Q)
is a function of the your airline’s coach fare
(P),
its rival’s fare
(PO), and income in the region (Y)
Your forecasting unit has estimated the following demand function:Q = 25 + 3Y + PO – 2P
[2]Slide4
Effect of changes in the explanatory variables
For each one point increase in the income index (Y), 3 additional seats will be sold, ceteris paribus.
For each $10 increase in the airline’s fare, 20 fewer seats will be sold, ceteris paribus.For each $10 increase in the competitor’s fare, 10 additional seats will be sold, ceteris paribus.Q is the dependent variable; P, PO, and Y are the independent or explanatory variables.Slide5
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. Slide6
Model specification
Suppose that airline ticket sales are a function of three variables, that is:
Q = f(P, PO, Y)[3.1]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:Slide7
The DataSlide8
Estimating multivariable regression models using OLS
Let:
Yi = 0 + 1X1i + 2X2i + iComputer algorithms find the ’s that minimize the sum of the squared residuals: Slide9
Excel Output
CoefficientsStandard Errort StatP-valueIntercept29.13845472174.74268970.166750.8703Fare (P)-2.123647330.340540892-6.23614E-05
Fare (P0)
1.03445512
0.466733469
2.21637
0.0467
Income (Y)
3.087138894
0.9993360183.089190.0094Slide10
Excel Output
Regression Statistics
Multiple R0.88112369R Square0.776378957
Adjusted R Square
0.720473696
Standard Error
14.77244284
Observations
16
ANOVA
df
SS
MS
F
Significance F
Regression
3
9091.739189
3030.58
13.887
0.0003
Residual
12
2618.700811
218.225
Total
15
11710.44
Slide11
Results of the regression
Our equation is estimated as follows:Slide12
Results of In-Sample ForecastSlide13
In-sample forecast for the multivariable modelSlide14
Other resultsSlide15
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 denominatorSlide16
We set up the following null hypothesis an alternative hypothesis:
H0 :
1 = 2 = 3 = 0HA: H0 is not trueWe 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. Slide17
Example: The Demand for Coal
COAL = 12,262 + 92.43FIS + 118.57FEU -
48.90PCOAL + 118.91PGASCOAL 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.