1 Ch 3DEMAND ESTIMATION In planning and in making policy decisions, managers must have some idea about the characteristics of the demand for their product(s)

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Ch 3 DEMAND ESTIMATION

In planning and in making policy decisions, managers must have some idea about the characteristics of the demand for their product(s) in order to attain the objectives of the firm or even to enable the firm to survive.

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Demand information about customer sensitivity to

modifications in price advertising packaging product innovations economic conditions etc.

are needed for product-development strategy

For competitive strategy details about customer reactions to changes in competitor prices and the quality of competing products play a significant role

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WHAT DO CUSTOMERS WANT?

How would you try to find out customer behavior?

How can actual demand curves be estimated?

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From Theory to Practice

D: Qx = f(px ,Y, pr , pe, , N)

What is the true quantitative relationship between demand and the factors that affect it?

How can demand functions be estimated?

How can managers interpret and use these estimations?

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Most common methods used are:

a) consumer interviews or surveys to estimate the demand for new

products to test customers reactions to

changes in the price or advertising to test commitment for established

products

b) market studies and experiments to test new or improved products in

controlled settings

c) regression analysis uses historical data to estimate

demand functions

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Consumer Interviews(Surveys)

Ask potential buyers how much of the commodity they would buy at different prices (or with alternative values for the non-price determinants of demand)

face to face approach telephone interviews

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Consumer Interviews continued

Problems:1. Selection of a representative

sample what is a good sample!

2. Response bias how truthful can they be?

3. Inability or unwillingness of the respondent to answer accurately

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Market Studies and Experiments

More expensive and difficult technique for estimating demand and demand elasticity is the controlled market study or experiment

Displaying the products in several different stores, generally in areas with different characteristics, over a period of time

for instance, changing the price, holding everything else constant

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Market Studies and Experiments continued

Experiments in laboratory or field• a compromise between market

studies and surveys

• volunteers are paid to stimulate buying conditions

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Market Studies and Experiments continued

Problems in conducting market studies and experiments:a) expensiveb) availability of subjectsc) do subjects relate to the problem, do

they take them seriously

BUT: today information on market behavior also collected by membership and award cards of stores

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Regression Analysis and Demand Estimation

A frequently used statistical technique in demand estimation

Estimates the quantitative relationship between the dependent variable and independent variable(s) quantity demanded being the dependent

variable

if only one independent variable (predictor) used: simple regression

if several independent variables used: multiple regression

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A Linear Regression Model

In practice the dependence of one variable on another might take any number of forms, but an assumption of linear dependency will often provide an adequate approximation to the true relationship

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Think of a demand function of general form:

Qi = + 1Y - 2 pi + 3ps - 4pc + 5Z + e

whereQi = quantity demanded of good iY = income

pi = price of good i

ps = price of the substitute(s)

pc = price of the complement(s)Z = other relevant determinant(s) of demande = error term

Values of and i ?

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and i has to be estimated from historical data

Data used in regression analysis cross-sectional data provide

information on variables for a given period of time

time series data give information about variables over a number of periods of time

New technologies are currently dramatically changing the possibilities of data collection!!!

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Simple Linear Regression Model

In the simplest case, the dependent variable Y is assumed to have the following relationship with the independent variable X:

Y = a + bX + uwhere

Y = dependent variableX = independent variablea = interceptb = slopeu = random factor

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Estimating the Regression Equation

Finding a line that ”best fits” the data

• The line that best fits a collection of X,Y data points, is the line minimizing the sum of the squared distances from the points to the line as measured in the vertical direction

• This line is known as a regression line, and the equation is called a regression equation

Estimated Regression Line:

Y= â + bXˆ

Y

ˆ

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Skatter Plot

0

100

200

300

400

500

600

0 100 200 300 400 500 600 700 800

L

Q

Observed Combinations of Output and Labor input

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X Variable 1 Line Fit Plot

0100200300400500600700800

0 200 400 600

X Variable 1

Y

Y

Predicted Y

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SUMMARY OUTPUT

Regression StatisticsMultiple R 0,959701R Square 0,921026Adjusted R Square0,917265Standard Error47,64577Observations 23

ANOVAdf SS MS F Significance F

Regression 1 555973,1 555973,1 244,9092 4,74E-13Residual 21 47672,52 2270,12Total 22 603645,7

CoefficientsStandard Error t Stat P-value Lower 95%Upper 95%Lower 95,0%Upper 95,0%Intercept -75,6948 31,64911 -2,39169 0,026208 -141,513 -9,87686 -141,513 -9,87686X Variable 11,377832 0,088043 15,64957 4,74E-13 1,194737 1,560927 1,194737 1,560927

Regression with ExcelEvaluate statistical significance of regression coefficients using t-test and statistical significance of R2 using F-test

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t-test: test of statistical significance of each estimated regression coefficient

b: estimated coefficient SEb: standard error of the estimated

coefficient Rule of 2: if absolute value of t is

greater than 2, estimated coefficient is significant at the 5% level

If coefficient passes t-test, the variable has a true impact on demand

b̂SE

b̂ t

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Sum of Squares

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Sum of Squares continued

TSS = (Yi - Y)2

(total variability of the dependent variable about its mean)

RSS = (Ŷi - Y)2

(variability in Y explained by the sample regression)

ESS = (Yi - Ŷi)2

(variability in Y unexplained by the dependent variable x)

This regression line gives the minimum ESS among all possible straight lines.

where Y = mean of Y

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The Coefficient of Determination

Coefficient of determination R2 measures how well the line fits the scatter plot (Goodness of Fit)

R2 is always between 0 and 1 If it’s near 1 it means that the

regression line is a good fit to the data Another interpretation: the percentage

of variance ”accounted for”

TSSESS

1TSSRSS

R2

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Multiple Regression Procedure

1. Determine the appropriate predictors and the form of the regression model

2. Estimate the unknown a and b coefficients

3. Estimate the variance associated with the regression model

4. Check the utility of the model (R2, global F-test, individual t-test for each b coefficient)

5. Use the fitted model for predictions (and determine their accuracy)

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Specification of the Regression Model:

Proxy variables to present some other “real” variable, such

as taste or preference, which is difficult to measure

Dummy variables (X1= 0; X2= 1) for qualitative variable, such as gender or

location Linear vs. non-linear relationship

quadratic terms or logarithms can be used

Y = a + bX1 + cX12

QD=aIb logQD= loga + blogI

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Example: Specifying the Regression Equation for Pizza Demand

We want to estimate the demand for pizza by college students in USA

What variables would most likely affect their demand for pizza?

What kind of data to collect?

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Data: Suppose we have obtained cross-sectional data on college students of randomly selected 30 college campus (by a survey)

The following information is available: average number of slices consumed per

month by students average price of a slice of pizza sold

around the campus price of its complementary product (soft

drink) tuition fee (as proxy for income) location of the campus (dummy variable is

included to find out whether the demand for pizza is affected by the number of available substitutes); 1 urban, 0 for non-urban area

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Linear additive regression line:

Y = a + b1pp + b2 ps + b3T + b4L

where Y = quantity of pizza demandeda = the intercept

Pp = price of pizza

Ps = price of soft drinkT = tuition feeL = location

bi = coefficients of the X variables measuring the impact of the variables on the

demand for pizza

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Estimating and Interpreting the Regression Coefficients

Y = 26.27- 0.088pp - 0.076ps + 0.138T- 0.544 L(0.018) (0.018) (0.020) (0.087) (0.884)

R2 = 0.717 Standard error of Y = 1.64R2 = 0.67F = 15.8

Numbers in parentheses are standard errors of coefficients.

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Problems in the Use of Regression Analysis:

identification problem

multicollinearity (correlation of coefficients)

autocorrelation (Durbin-Watson test)

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Multicollinearity

A significant problem in multiple regression which occurs when there is a very high correlation between some of the predictor variables.

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Resulting problem:

Regression coefficients may be very misleading or meaningless because…

their values are sensitive to small changes in the data or to adding additional observations

they may even be opposite in sign from what ”makes sense”

their t-value (and the standard error) may change a lot depending upon which other predictors are in the model

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Multicollinearity continued

Solution:

Don’t use two predictors which are very highly correlated (however, x and x2 are O.K.)

Not a major problem if we are only trying to fit the data and make predictions and we are not interested in interpreting the numerical values of the individual regression coefficients.

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Multicollinearity continued

One way to detect the presence of multicollinearity is to examine the correlation matrix of the predictor variables. If a pair of these have a high correlation they both should not be in the regression equation – delete one.

Correlation Matrix

Y X1 X2 X3

Y 1.00 -.45 .81 .86

X1 -.45 1.00 -.82 -.59

X2 .81 -.82 1.00 .91

X3 .86 -.59 .91 1.00

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A test for Autocorrelated Errors:DURBIN-WATSON TEST

A statistical test for the presence of autocorrelation

Fit the time series with a regression model and then determine the residuals:

n

tt

n

ttt

ttt

d

yy

1

2

2

21

_

)

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The Durbin-Watson value d, will always be: 0 d 4.

The interpretation of d:

Possible values of d:

0 2 4

Strong +Correlation Uncorrelated Strong -Correlation

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Comments:

￭ As more variables are added to a multiple regression equation, R2 must increase (or stay the same); F may or may not increase

￭ The F test is a test of all the i= 0. We should expect a high F value (and low p value). If so, we can investigate further

￭ t-test for coefficients can determine which ones to delete from the model

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￭ OCCAM’S RAZOR. We want a model that does a good job of fitting the data using a minimum number of predictors. A high R2 is not the only goal; variables used should be ”meaningful”

￭ Don’t use more predictors in a regression model than 5% to 10% of n

￭ Would like a model with low MSE

￭ Results of t-tests for individual coefficients depend on which other predictors happen to be in the model