Estimating DemandChapter 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

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

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.

quantity

Price

2004

20092008 2006

2010

2007

2005

Is this curve demand or supply?

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

Statistical Estimation of Demand Functions

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., TastesTastes Time Time TrendTrend

Functional Forms: Linear

Linear Model 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.

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

time subscripts & error term Find “best fitting” line

t = Yt - a - b Xt

t2 = [Yt - a - b Xt] 2 .

mint 2= [Yt - a - b Xt] 2 .

Solution:

slope b = Cov(Y,X)/Var(X) and

intercept a = mean(Y) - b•mean(X)

_X

Y

_Y

a

XY

Simple Linear Regression: Assumptions & Solution Methods 1. The dependent

variable is random.2. A straight line

relationship exists.3. The error term has

a mean of zero and a finite variance: 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 For-Profit Mystat

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 XOne 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 t2

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 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.”

USING THE REGRESSION EQUATIONTO 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

As more variables are 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

Y

_Y

^Yt

Yt predicted ^

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 = .790Adjusted R2 = .684 Standard Error of Estimate = 17.417

Variable Coefficient Std error T P(2 tail)Constant310.245 95.075 3.263 .017PromExp .008 0.204 0.038 .971SellPrice -12.202 4.582 -2.663 .037DispInc 2.677 3.160 0.847 .429

Analysis of VarianceSource Sum of Squares DF Mean Squares F

pRegression 6829.8 3 2276.6

7.5 .019Residual 1820.1 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

  Coefficients Standard Error t StatIntercept 159.17 94.16 1.69Price -102.56 33.25 -3.08Income 1.00 1.77 0.57

Temperature 3.94 0.82 4.83

Regression StatisticsMultiple R 0.736R Square 0.541Adjusted R Square 0.510Standard Error 47.312Observations 48

Linear estimation yields:

Find The Linear Elasticities

Cans = 159.17 -102.56 Price +1.00 Income + 3.94 Temp

The price elasticity in Alabama is = ((Q/Q/P)(P/Q) = -102.56(2.19/200)= P)(P/Q) = -102.56(2.19/200)= --1.1231.123

The price elasticity in Nevada is The price elasticity in Nevada is = (= (Q/Q/P)(P/Q) = -102.56(2.19/166) = P)(P/Q) = -102.56(2.19/166) = --1.3531.353

The price elasticity in Wisconsin is = ((Q/Q/P)(P/Q) = -102.56(2.38/97)= P)(P/Q) = -102.56(2.38/97)= --2.5162.516

The estimated elasticities are The estimated elasticities are elasticelastic for individual states. for individual states.

We can estimate the elasticity from the whole samples as: 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.

Linear Specification write as an equation: