Estimating Demand
Outline
•Where do demand functions come from?
•Sources of information for demand estimation
•Cross-sectional versus time series data
•Estimating a demand specification using the
ordinary least squares (OLS) method.
•Goodness of fit statistics.
The goal of forecasting
To transform available data into
equations that provide the best
possible forecasts of economic
variables—e.g., sales revenues
and costs of production—that are
crucial for management.
Demand for air travel Houston to
Orlando
Now we will
explain how
we estimated
this demand
equation
Recall that our demand function was
estimated as follows:
Q = 25 + 3Y + PO – 2P
[4.1]
Where Q is the number of seats
sold; Y is a regional income
index; P0 is the fare charged by
a rival airline, and P is the
airline’s own fare.
Questions managers should
ask about a forecasting equations
1. What is the “best” equation that can be
obtained (estimated) from the available
data?
2. What does the equation not explain?
3. What can be said about the likelihood
and magnitude of forecast errors?
4. What are the profit consequences of
forecast errors?
How do get the data to estimate
demand forecasting equations?
•Customer surveys and interviews.
•Controlled market studies.
•Uncontrolled market data.
Campbell’s soup
estimates demand
functions from data
obtained from a survey of
more than 100,000
consumers
Survey pitfalls




Sample bias
Response bias
Response accuracy
Cost
Types of data
Time -series data: historical data--i.e., the data sample
consists of a series of daily, monthly, quarterly, or annual
data for variables such as prices, income , employment ,
output , car sales, stock market indices, exchange rates, and
so on.
Cross-sectional data: All observations in the sample are
taken from the same point in time and represent different
individual entities (such as households, houses, etc.)
Time series data: Daily observations,
Korean Won per dollar
Year Month Day Won per Dollar
1997
3
10
877
1997
3
11
880.5
1997
3
12
879.5
1997
3
13
880.5
1997
3
14
881.5
1997
3
17
882
1997
3
18
885
1997
3
19
887
1997
3
20
886.5
1997
3
21
887
1997
3
24
890
1997
3
25
891
Example of cross sectional data
Student ID
Sex
Age
Height
Weight
777672431
M
21
6’1”
178 lbs.
231098765
M
28
5’11”
205 lbs.
111000111
F
19
5’8”
121 lbs.
898069845
F
22
5’4”
98 lbs.
000341234
M
20
6’2”
183 lbs
Estimating demand equations
using regression analysis
Regression analysis is a
statistical technique that allows
us to quantify the relationship
between a dependent variable
and one or more independent or
“explanatory” variables.
Y
Regression theory
X and Y are not
perfectly correlated.
However, there is
on average a positive
relationship
between Y and X
0
X1
X2
X
We assume that
expected conditional values
of Y associated with
alternative values of X
fall on a line.
Y
E(Y |Xi) = 0 + 1Xi
Y1
1
1 = Y1 - E(Y|X1)
E(Y|X1)
0
X1
X
Specifying a single
variable model
Our model is specified as follows:
Q = f (P)
where Q is ticket sales and P is the fare
Q is the dependent
variable—that is, we think
that variations in Q can be
explained by variations in
P, the “explanatory”
variable.
Estimating the single variable model
Q i   0   1Pi
Q i   0   1 Pi   i
[1]
Since the data
points are unlikely to fall
exactly on a line, (1)
must be modified
to include a disturbance
term (εi)
[2]
0 and 1 are called parameters or
population parameters.
We estimate these parameters using
the data we have available
Estimated Simple Linear Regression Equation

The estimated simple linear regression equation
ŷ  b0  b1 x
• The graph is called the estimated regression line.
• b0 is the y intercept of the line.
• b1 is the slope of the line.
• ŷ is the estimated value of y for a given x value.
Estimation Process
Regression Model
y = 0 + 1x +
Regression Equation
E(y) = 0 + 1x
Unknown Parameters
0, 1
b0 and b1
provide estimates of
0 and 1
Sample Data:
x
y
x1
y1
.
.
.
.
xn yn
Estimated
Regression Equation
ŷ  b0  b1 x
Sample Statistics
b0, b1
Least Squares Method

Least Squares Criterion
min  (y i  y i ) 2
where:
yi = observed value of the dependent variable
for the ith observation
y^i = estimated value of the dependent variable
for the ith observation
Least Squares Method

Slope for the Estimated Regression
Equation
b1
( x  x )( y  y )


 (x  x )
i
i
2
i
Least Squares Method

y-Intercept for the Estimated Regression Equation
b0  y  b1 x
where:
xi = value of independent variable for ith
observation
yi = value of dependent variable for ith
_ observation
x = mean value for independent variable
_
y = mean value for dependent variable
n = total number of observations
Line of best fit
The line of best fit is the one
that minimizes the squared
sum of the vertical distances
of the sample points from the
line
The 4 steps of demand
estimation using regression
1. Specification
2. Estimation
3. Evaluation
4. Forecasting
Year and
Quarter
97-1
97-2
97-3
97-4
98-1
98-2
98-3
98-4
99-1
99-2
99-3
99-4
00-1
00-2
00-3
00-4
Mean
Std. Dev.
Average Number Average
Coach Seats
Fare
64.8
250
33.6
265
37.8
265
83.3
240
111.7
230
137.5
225
109.6
225
96.8
220
59.5
230
83.2
235
90.5
245
105.5
240
75.7
250
91.6
240
112.7
240
102.2
235
87.3
239.7
27.9
13.1
Table 4-2
Ticket Prices and Ticket
Sales along an Air Route
Simple linear regression
begins by plotting Q-P
values on a scatter
diagram to determine if
there exists an
approximate linear
relationship:
Scatter plot diagram
290
280
270
Fare
260
250
240
230
220
210
20
40
60
80
100
Passengers
120
140
160
Scatter plot diagram with possible line of best fit
Average One-way Fare
Demand curve: Q = 330-P
$ 27 0
26 0
25 0
24 0
23 0
22 0
0
50
100
150
Number of Seats Sold per Flight
Note that we use X to denote the explanatory
variable and Y is the dependent variable.
So in our example Sales (Q) is the “Y” variable
and Fares (P) is the “X” variable.
Q=Y
P=X
Computing the OLS
estimators
We estimated the equation using the statistical
software package SPSS. It generated the following
output:
Coefficientsa
Model
1
(Constant)
FARE
Unstandardized
Coefficients
B
Std. Error
478.690
88.036
-1.633
.367
a. Dependent Variable: PASS
Standardi
zed
Coefficien
ts
Beta
-.766
t
5.437
-4.453
Sig.
.000
.001
Reading the SPSS Output
From this table we see that
our estimate of 0 is 478.7
and our estimate of 1 is
–1.63.
Thus our forecasting equation is given
by:
Qˆ i  478.7  1.63Pi
Step 3: Evaluation
Now we will evaluate the forecasting equation
using standard goodness of fit statistics,
including:
1. The standard errors of the estimates.
2. The t-statistics of the estimates of the
coefficients.
3. The standard error of the regression (s)
4. The coefficient of determination (R2)
Standard errors of
the estimates
•We assume that the regression coefficients are
normally distributed variables.
•The standard error (or standard deviation) of the
estimates is a measure of the dispersion of the
estimates around their mean value.
•As a general principle, the smaller the standard error,
the better the estimates (in terms of yielding accurate
forecasts of the dependent variable).
The following rule-of-thumb is useful: The
standard error of the regression
coefficient should be less than half of the
size of the corresponding regression
coefficient.
Computing the standard error of 1
s̂ 1
Let
denote the standard error of our estimate of 1
Thus we have:
sˆ 1 
sˆ 1 2
s

xi  Xi  X
and
ˆi
ei  Qi  Q
Where:
2
ˆ 1
Note that:
2
e
i

n  k  xi 2
and
k is the number of
estimated coefficients
Coefficientsa
Model
1
(Constant)
FARE
Unstandardized
Coefficients
B
Std. Error
478.690
88.036
-1.633
.367
Standardi
zed
Coefficien
ts
Beta
-.766
t
5.437
-4.453
Sig.
.000
.001
a. Dependent Variable: PASS
By reference to the SPSS output, we see
that the standard error of our estimate
of 1 is 0.367, whereas the (absolute value)our
estimate of 1 is 1.63 Hence our estimate is about 4 ½
times the size of its standard error.
The SPSS output tells us that the
t statistic for the the fare coefficient (P)
is –4.453 The t test is a way
of comparing the error
suggested by the null
hypothesis to the
standard error of the estimate.
The t test
 To test for the significance of our estimate of 1,
we set the following null hypothesis, H0, and the
alternative hypothesis, H1
H0: 1 0
H1: 1 < 0
The t distribution is used to
test for statistical significance of
the estimate:
ˆ 1   1  1.63  0
t

 4.45
sˆ
0.049
1
Coefficient of determination (R2)
The coefficient of determination, R2, is defined as the proportion of
the total variation in the dependent variable (Y) "explained" by the
regression of Y on the independent variable (X). The total variation in Y
or the total sum of squares (TSS) is defined as:
2
TSS   Yi  Y    yi 2
n
i 1
n
Note: yi  Yi  Y
i 1
The explained variation in the dependent variable(Y) is called
the regression sum of squares (RSS) and is given by:
RSS 
 Yˆ  Y 
2
n
i
i 1

n

i 1
ˆi 2
y
What remains is the unexplained variation in the dependent
variable or the error sum of squares (ESS)
n


2
n
ESS   Yi  Yˆ   ei 2
i 1
i 1
We can say the following:
•TSS = RSS + ESS, or
•Total variation = Explained variation + Unexplained variation
R2 is defined as:
n
R2
RSS
ESS

 1

TSS
RSS
n

ˆi
y

yi 2
2
 1
i 1
n
i 1
2
e
i

i 1
n

i 1
yi 2
ANOVAb
Model
1
Regres sion
Residual
Total
Sum of
Squares
6863.624
4846.816
11710. 440
Mean
Square
6863.624
346.201
df
1
14
15
F
19.826
a. Predic tors: (Constant), FARE
b. Dependent Variable: PASS
Mode l Summary
Model
1
R
R Square
.766a
.586
Adjust ed R
Square
.557
St d. Error
of the
Es timate
18.6065
a. Predic tors : (Const ant), FARE
We see from the SPSS model summary
table that R2 for this model is .586
Sig.
.001a
Notes on R2
Note that: 0 R2 1
If R2 = 0, all the sample points lie on a horizontal
line or in a circle
If R2 = 1, the sample points all lie on the
regression line
In our case, R2  0.586, meaning that 58.6 percent
of the variation in the dependent variable
(consumption) is explained by the regression.
This is not a particularly good fit based on
R2 since 41.4 percent of the
variation in the dependent
variable is unexplained.
Standard error of the
regression
The standard error of the regression (s) is
given by:
n
e
i
s
i 1
nk
2
Mode l Summary
Model
1
R
R Square
.766a
.586
Adjust ed R
Square
.557
St d. Error
of the
Es timate
18.6065
a. Predic tors : (Const ant), FARE
The model summary tells us that s = 18.6
Regression is based on the assumption that the error term
is normally distributed, so that 68.7% of the actual values of
the dependent variable (seats sold) should be within one
standard error ($18.6 in our example) of their fitted value.
Also, 95.45% of the observed values of seats sold should
be within 2 standard errors of their fitted values (37.2).
Step 4: Forecasting
Recall the equation obtained from the
regression results is :
Qˆ i  478.7  1.63Pi
Our first step is to
perform an “in-sample”
forecast.
At the most basic level, forecasting consists
of inserting forecasted values of the
explanatory variable P (fare) into the
forecasting equation to obtain forecasted
values of the dependent variable Q
(passenger seats sold).
In-Sample Forecast of Airline Sales
Year and
Quarter
97-1
97-2
97-3
97-4
98-1
98-2
98-3
98-4
99-1
99-2
99-3
99-4
00-1
00-2
00-3
00-4
Predicted
Sales (Q*)
64.8
33.6
37.8
83.3
111.7
137.5
109.6
96.8
59.5
83.2
90.5
105.5
75.7
91.6
112.7
102.2
Actual
Sales (Q)
70.44
45.94
45.94
86.77
103.1
111.26
111.26
119.43
103.1
94.94
78.61
86.77
70.44
86.77
86.77
94.94
Q* - Q (Q* - Q)sq
5.64
31.81
12.34
152.28
8.14
66.26
3.47
12.04
-8.6
73.96
-26.24
688.54
1.66
2.76
22.63
512.12
43.6
1900.96
11.74
137.83
-11.89
141.37
-18.73
350.81
-5.26
27.67
-4.83
23.33
-25.93
672.36
-7.26
52.71
Sum of Squared Errors
4846.80
In-Sample Forecast of Airline Sales
160
140
120
100
80
60
40
Actual
20
97.1
Fitted
97.3
98.1
98.3
99.1
Yea r/Quarter
99.3
00.1
00.3
Can we make a
good forecast?
Our ability to generate accurate forecasts of the dependent variable
depends on two factors:
•Do we have good forecasts of the explanatory variable?
•Does our model exhibit structural stability, i.e., will the causal
relationship between Q and P expressed in our forecasting equation
hold up over time? After all, the estimated coefficients are average
values for a specific time interval (1987-2001). While the past may be a
serviceable guide to the future in the case of purely physical phenomena,
the same principle does not necessarily hold in the realm of social
phenomena (to which economy belongs).
Single Variable Regression Using Excel
We will estimate an equation
and use it to predict home
prices in two cities. Our data
set is on the next slide
City
•Income (Y) is
average family
income in 2003
•Home Price
(HP) is the
average price of
a new or
existing home in
2003.
Income
Home Price
Akron, OH
74.1
114.9
Atlanta, GA
82.4
126.9
Birmingham, AL
71.2
130.9
Bismark, ND
62.8
92.8
Cleveland, OH
79.2
135.8
Columbia, SC
66.8
116.7
Denver, CO
82.6
161.9
Detroit, MI
85.3
145
Fort Lauderdale, FL
75.8
145.3
Hartford, CT
89.1
162.1
Lancaster, PA
75.2
125.9
Madison, WI
78.8
145.2
Naples, FL
100
173.6
Nashville, TN
77.3
125.9
87
151.5
Savannah, GA
67.8
108.1
Toledo, OH
71.2
101.1
Washington, DC
97.4
191.9
Philadelphia, PA
Model Specification

HP  b0  b1Y
Scatter Diagram: Income and Home Prices
200
Home Prices
180
160
140
120
100
80
50
60
70
80
Income
90
100
110
Excel Output
ANOVA
df
SS
1
9355.71550
2
Residual
16
2017.36949
8
Total
17
11373.085
Regression
Coefficients
Regression Statistics
Multiple R
0.906983447
R Square
0.822618973
Adjusted R
Square
0.811532659
Standard Error
11.22878416
Observations
18
Standard
Error
t Stat
Intercept
-48.11037724
21.58459326
-2.228922114
Income
2.332504769
0.270780116
8.614017895
Equation and prediction

HP  48.11  2.33Y
City
Income
Predicted HP
Meridian, MS
59,600
$ 138,819.89
Palo Alto, CA
121,000
$ 281,881.89