Download Document

Estimation of Demand
Chapter 4
• A chief uncertainty for managers is the future. They
fear what will happen to their product?
» Managers use forecasting, prediction &
estimation to reduce their uncertainty.
» The methods that they use vary from consumer
surveys or experiments at test stores to statistical
procedures on past data such as regression
analysis.
• Objective of the Chapter: Learn how to interpret the results
of regression analysis from demand data.
2005 South-Western Publishing
Slide 1
Demand Estimation
Using Marketing Research Techniques
• Consumer Surveys
» ask a sample of consumers their attitudes
• Consumer Focus Groups
» experimental groups try to emulate a market
(Hawthorne effect = behave differently in when
observed)
• Market Experiments in Test Stores
» get demand information by trying different prices
• Historical Data
» what happened in the past is guide to the future
Slide 2
Statistical Estimation of Demand Functions:
Plot Historical Data
Price
• Look at the relationship
of price and quantity
over time
• Plot it
» Is it a demand curve or
a supply curve?
» The problem is this does
not hold other things
equal or constant.
Is this curve demand
or supply?
2000
2004
2001
2003
1999
2002 2001
quantity
Slide 3
Statistical Estimation of Demand Functions
• Steps to take:
» Specification of the model -- formulate the
demand model, select a Functional Form
• linear
Q = a + b•P + c•Y
• double log
log Q = a + b•log P + c•log Y
• quadratic
Q = a + b•P + c•Y+ d•P2
» Estimate the parameters -• determine which are statistically significant
• try other variables & other functional forms
» Develop forecasts from the model
Slide 4
Specifying the Variables
• Dependent Variable -- quantity in units,
quantity in dollar value (as in sales
revenues)
• Independent Variables -- variables thought
to influence the quantity demanded
» Instrumental Variables -- proxy variables for the
item wanted which tends to have a relatively
high correlation with the desired variable: e.g.,
Tastes
Time Trend
Slide 5
Functional Forms: Linear
• Linear Q = a + b•P + c•Y
» The effect of each variable is constant, as in
Q/P = b and Q/Y = c, where P is price and Y is
income.
» The effect of each variable is independent of
other variables
» Price elasticity is: ED = (Q/P)(P/Q) = b•P/Q
» Income elasticity is: EY = (Q/Y)(Y/Q)= c•Y/Q
» The linear form is often a good approximation of
the relationship in empirical work.
Slide 6
Functional Forms: Multiplicative
or Double Log
• Multiplicative
Q = A • Pb • Yc
» The effect of each variable depends on all the other
variables and is not constant, as in Q/P = bAPb-1Yc
and Q/Y = cAPbYc-1
» It is double log (log is the natural log, also written as ln)
Log Q = a + b•Log P + c•Log Y
» the price elasticity, ED = b
» the income elasticity, EY = c
» This property of constant elasticity makes this
approach easy to use and popular among economists.
Slide 7
Simple Linear Regression
• Yt = a + b Xt + t
Y
• time subscripts & error term
• Find “best fitting” line
t = Yt - a - b Xt
t 2= [Yt - a - b Xt] 2 .
a
_
Y
• mint 2= [Yt - a - b Xt] 2 .
Solution:
slope b = Cov(Y,X)/Var(X) and
intercept a = mean(Y) - b•mean(X)
DY
DX
_
X
Slide 8
Simple Linear Regression:
Assumptions & Solution Methods
1. The dependent variable
is random
2. A straight line
relationship exists
3. error term has a mean
of zero and a finite
variance
4. the independent
variables are indeed
independent
• Spreadsheets - such as
» Excel, Lotus 1-2-3, Quatro
Pro, or Joe Spreadsheet
• Statistical calculators
• Statistical programs such as
»
»
»
»
»
Minitab
SAS
SPSS
ForeProfit
Mystat
Slide 9
Sherwin-Williams Case (p. 141)
• Ten regions with data on promotional expenditures (X)
and sales (Y)
• Result: Y = 120.755 + .434 X
• One use of a regression is to make predictions.
• If a region had promotional expenditures of 185, the
prediction is Y = 201.045, by substituting 185 for X
• The regression output will tell us also the standard
error of the estimate, se . In this case, se = 22.799
• Approximately 95% prediction interval is Y  2 se.
• Hence, the predicted range is anywhere from 155.447
to 246.643.
Slide 10
T-tests
• Different
samples would
yield different
coefficients
• Test the
hypothesis
that
coefficient
equals zero
» Ho: b = 0
» Ha: b 0
• RULE: If absolute value of the
estimated t > Critical-t, then
REJECT Ho.
» We say that it’s significant!
• The estimated t = (b - 0) / b
• The critical t is:
» Large Samples, critical t2
• N > 30
» Small Samples, critical t is on Student’s t
Distribution, page B-3 at end of book, usually
column 0.05.
• D.F. = # observations, minus
number of independent variables,
minus one.
Slide 11
• N < 30
Sherwin-Williams Case
• In the simple linear
•
regression:
Y = 120.755 + .434 X •
• The standard error of the
slope coefficient is .14763.
(This is usually available
from any regression
program used.)
• Test the hypothesis that •
the slope is zero, b=0.
•
The estimated t is:
t = (.434 – 0 )/.14763 = 2.939
The critical t for a sample of 10, has
only 8 degrees of freedom
» D.F. = 10 – 1 independent variable – 1 for
the constant.
» Table B2 shows this to be 2.306 at the .05
significance level
Therefore, |2.939| > 2.306, so we reject
the null hypothesis.
We informally say, that promotional
expenses (X) is “significant.”
Slide 12
Correlation Coefficient
• We would expect more promotional expenditures to be
associated with more sales at Sherwin-Williams.
• A measure of that association is the correlation
coefficient, r.
• If r = 0, there is no correlation. If r = 1, the correlation
is perfect and positive. The other extreme is r = -1,
which is negative.
Y
Y
r = -1
Y
r=0
r = +1
X
X
X
Slide 13
2
Coefficient of Determination: R
• R-square is the percentage of
the variation in dependent
variable that is explained
^
Y
Yt
–
• R2 = [Yt -Yt] 2 / [Yt - Yt] 2
= SSR / SST
• As more variables are
included, R-square rises
• Adjusted R-square, however,
can decline
^
Yt predicted
_
Y
_
X
Slide 14
Association and Causation
• Regressions indicate association, but beware of jumping to
the conclusion of causation
• Suppose you collect data on the number of swimmers at a
beach and the temperature and find:
• Temperature = 61 + .04 Swimmers, and R2 = .88.
» Surely the temperature and the number of swimmers is positively
related, but we do not believe that more swimmers CAUSED the
temperature to rise.
» Furthermore, there may be other factors that determine the
relationship, for example the presence of rain or whether or not it
is a weekend or weekday.
• Education may lead to more income, and also more income
may lead to more education. The direction of causation is
often unclear. But the association is very strong.
Slide 15
Multiple Linear Regression
• Most economic relationships involve several
variables. We can include more independent
variables into the regression.
• To do this, we must have more observations (N) than
the number of independent variables, and no exact
linear relationships among the independent variables.
• At Sherwin-Williams, besides promotional expenses,
different regions charge different selling prices
(SellPrice) and have different levels of disposable
income (DispInc)
• The next slide gives the output of a multiple linear
regression, multiple, because there are three
independent variables
Slide 16
Figure 4.9 on page 151
Dep var: Sales (Y)
N=10
R-squared = .790
Adjusted R2 = .684 Standard Error of Estimate = 17.417
Variable Coefficient
Constant 310.245
Promotion
.008
SellPrice -12.202
DispInc
2.677
Std error
95.075
0.204
4.582
3.160
T
3.263
0.038
-2.663
0.847
P
.017
.971
.037
.429
Slide 17
Interpreting Multiple Regression Output
• Write the result as an equation:
Sales = 310.245 + .008 Promotion -12.202 SellPrice
+ 2.677 DispInc
• Does the result make economic sense?
» As promotion expense rises, so does sales. That makes sense.
» As the selling price rises, so does sales. Yes, that’s reasonable.
» As disposable income rises in a region, so does sales. Yup. That’s reasonable.
• Is the coefficient on the selling price statistically significant?
» The estimated t value is given in Figure 4.9 to be -2.663
» The critical t value, with 6 ( which is 10 – 3 – 1) degrees of freedom in table B2
is 2.447
» Therefore |-2.663| > 2.447, so reject the null hypothesis, and assert that the
selling price is significant!
Slide 18