Transcript estimated t

Estimating Demand
Chapter 4
• A chief uncertainty for managers is the future. Managers
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 based on demand data.
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 (but
beware of the Hawthorne effect = people often
behave differently in when being 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 using statistics is an alternative
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?
2004
2007
2010
2009
2008 2006
2005
quantity
Statistical Estimation of Demand Functions
 Steps to take:
 Specification
of the model -- formulate
the demand model, select a Functional Form
linear
 double log
c•log Y
 quadratic

Q = a + b•P + c•Y
log Q = a + b•log P +
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
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
Functional Forms: Linear
 Linear Model
 The
Q = a + b•P + c•Y
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.
Functional Forms: Multiplicative or
Double Log
 Multiplicative Exponential Model 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.
A Simple Linear Regression Model
 Yt = a + b Xt + t
Y
 time subscripts & error term
a
 Find “best fitting” line
t = Yt - a - b Xt
t2 = [Yt - a - b Xt] 2 .
_
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
_
X
DX
Simple Linear Regression:
Assumptions & Solution Methods
1. The dependent
 Spreadsheets - such as
 Excel, Lotus 1-2-3, Quatro
Pro, or Joe Spreadsheet
variable is
random.
 Statistical calculators
2. A straight line
relationship exists.  Statistical programs such
3. The error term
as
has a mean of
 Minitab
zero and a finite
 SAS
variance: the
 SPSS
independent
 For-Profit
variables are
 Mystat
indeed
Assumption 2: Theoretical
Straight-Line Relationship
Assumption 3: Error Term Has A
Mean Of Zero And A Finite Variance
Assumption 3: Error Term Has A
Mean Of Zero And A Finite Variance
FIGURE 4.4 Deviation of the Observations
about the Sample Regression Line
Sherwin-Williams Case
 Ten regions with data on promotional expenditures
(X) and sales (Y), selling price (P), and disposable
income (M)
 If look only at Y and X: 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.
Sherwin-Williams Case
Figure 4.5 Estimated Regression Line
Sherwin-Williams Case
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-2 at end of book,
usually column 0.05, & degrees of
freedom.
 D.F. = # observations, minus number
of independent variables, minus
one.
 N < 30
Sherwin-Williams Case
 In the simple linear
• The estimated t is:
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. •
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.”
USING THE REGRESSION EQUATION
TO MAKE PREDICTIONS
 A regression equation can be used to make
predictions concerning the value of Y, given
any particular value of X.
 A measure of the accuracy of estimation with
the regression equation can be obtained by
calculating the standard deviation of the errors
of prediction (also known as the standard error
of the estimate).
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.
Analysis of Variance
 R-squared is the percentage
of the variation in dependent
variable that is explained
Y
^
Yt

 As more variables are
^
Yt predicted
_
Y
included, R-squared rises
 Adjusted R-squared, however,
can decline


Adj R2 = 1 – (1-R2)[(N-1)/(N-K)]
As K rises, Adj
R2
may decline.
_
X
X
FIGURE 4.7 Partitioning the Total
Deviation
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 local
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.
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 (PromExp), 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
Figure 4.8 Computer Output: Sherwin-Williams Company
Dep var: Sales (Y)
N=10
R-squared = .790
Adjusted R2 = .684 Standard Error of Estimate = 17.417
Variable
Constant
PromExp
SellPrice
DispInc
Coefficient Std error
310.245
95.075
.008
0.204
-12.202
4.582
2.677
3.160
Analysis of Variance
Source
Sum of Squares
F
p
Regression
6829.8
3
7.5
.019
Residual
1820.1
6
T
P(2 tail)
3.263
.017
0.038
.971
-2.663
.037
0.847
.429
DF
Mean Squares
2276.6
303.4
Interpreting Multiple Regression Output
 Write the result as an equation:
Sales = 310.245 + .008 ProExp -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 4.8 to be -2.663 on SellPrice.
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!
Soft Drink Demand Estimation
A Cross Section Of 48 States
Linear estimation yields:
Intercept
Price
Income
Temperature
Coefficients Standard Error
159.17
94.16
-102.56
33.25
1.00
1.77
3.94
0.82
Regression Statistics
Multiple R
0.736
R Square
0.541
Adjusted R Square
0.510
Standard Error
47.312
Observations
48
t Stat
1.69
-3.08
0.57
4.83
Find The Linear Elasticities
Linear Specification write as an equation:
Cans = 159.17 -102.56 Price +1.00 Income + 3.94
Temp
The price elasticity in Alabama is = (DQ/DP)(P/Q) = -102.56(2.19/200)= -1.123
The price elasticity in Nevada is = (DQ/DP)(P/Q) = -102.56(2.19/166) = -1.353
The price elasticity in Wisconsin is = (DQ/DP)(P/Q) = -102.56(2.38/97)= -2.516
The estimated elasticities are elastic for individual states.
We can estimate the elasticity from the whole samples as:
(Q/P)  (Mean P/Mean Q) = 102.56 x ($2.22/160) = -1.423,
which is also elastic.