Chapter 5 Estimating Demand Functions

1
Chapter 5Estimating Demand Functions
2
Objectives

• The identification problem
• Consumer interviews
• Market experiments
• Regression analysis

3
Objectives

• Simple regression analysis
• Population and sample regression line
• Coefficient of determination (R-squared)
• Multiple regression analysis
• Excel regression package (handout)
• Interpreting the computer printout

4
Reading Material

• Material is on Chapter 5 of the textbook, pages
  153-186, and the handout on the mechanics of
  regression analysis using Excel.
• You are not responsible for the following
  material from Chapter 5
• Method of least squares, pages 165-167.
• Software packages, pages 175-176.

5
The Identification Problem

• This problem refers to situations where the data
  contain insufficient information to provide the
  required answer.
• The inability to distinguish between moves along
  a demand curve and shifts in supply and/or demand
  that reflect changes in behavior.
• Consider the straight forward approach to
  estimating a demand curve
• Plot the quantity of a product sold and its
  price for (say) three years (2004, 2005, 2006).

6
The Identification Problem
7
Underestimating the Price Elasticity of Demand
Can Create Problems
60
D 06
20
8
Dealing with the Identification Problem

• Because we are not controlling for changes in
  parameters that shift the demand curve, we can
  not be sure that the demand curve remained the
  same during the sample period.
• If the demand curve were fixed, changes in the
  supply curve would trace it correctly.

9
Alternative Methods of Estimating Demand Functions

• Consumer interviews.
• Market experiments.
• Regression analysis.

10
Consumer Interviews

• Firms frequently engage in consumer interviews
  and surveys concerning their buying habits,
  motives and intentions.
• One potential problem with this method is that
  frequently answers are not very accurate.
• Clever approaches can be used to remedy this
  situation.
• Despite the limitations of consumer interviews
  and surveys, many firms use these techniques
  often.

11
Market Experiments

• The idea of a market experiment is to vary the
  price of a product while keeping everything else
  the same.
• Controlled laboratory experiments can sometimes
  be carried out.
• Direct experimentation can be expensive or/and
  risky.
• Profits might be reduced permanently or
  temporarily.
• Customers might switch to competitors permanently
  if they face a price increase.

12
Regression Analysis

• Suppose that a products demand function is given
  by the following equation
• Y A B1X B2P B3I B4Pr, where
• Y quantity demanded
• X advertising expenses
• I consumers disposable income
• Pr prices of competing (rival) goods
• Regression analysis is a statistical technique
  that provides numerical values for the
  coefficients A, B1, B2, B3, and B4.

13
Regression Analysis

• The numerical values of the coefficients are
  extracted from historical data concerning the
  variables Y, X, P, I, and Pr.
• Suppose we have the following data (observations)
  on Miller Pharmaceutical Company
• Selling expense (millions) Sales (millions)
• 1 4
• 2 6
• 4 8
• 8 14
• 6 12
• 5 10
• 8 16
• 9 16
• 7 12

14
Simple Regression Analysis

• Assumes that the mean value of the dependent
  variable is a linear function of the independent
  variable
• Yi A BXi ei, where
• Yi the ith observed value of the dependent
  variable Y
• Xi the ith value of the independent variable X
• ei the error term that is added to (or
  subtracted from) the expression A B Xi to
  obtain the dependent variable Yi.

15
The Mean Value of X Falls on the Population
Regression Line
16
Sample Regression Line

• The general expression for the sample regression
  line is
• Yi a bXi
• where,
• Yi the value of the dependent variable
• predicted by the regression line.
• a the y-intercept of the regression line
• b the slope of the regression line.
• Xi the value of the independent variable
• evaluated at the ith observation.

17
Sample Regression Line

• The population regression line is based on the
  entire population sample, whereas the sample
  regression line is based only on the sample.
• The regression equation for the data concerning
  Miller Pharmaceutical Company data on slide 13 is
• Y-hat 2.533 1.504X
• Y-hat sales in millions of dollars
• X selling expenses in millions of dollars
• 2.533 value of a, the estimator of A
• 1.504 value of b, the estimator of B

18
Coefficient of Determination

• Commonly called the R-square.It measures
  goodness of fit of the estimated regression line
• It varies between 0 and 1. The closer R2 is to 1
  the better the fit.
• For example, if R2 0.80, then Xs
  variationexplains about 80 percent of Ys
  variation.

19
Multiple Regression Analysis

• Suppose that in the case of Miller
  Pharmaceutical Company sales depend on its price
  and selling expenses
• Yi a b1Xi b2Pi ei
• where Yi sales
• xi selling expense
• Pi price
• The goal of multiple regression analysis is to
  estimate the coefficients a, b1, and b2 .

20
Sales, Selling Expense and Price, Miller
Pharmaceutical Company

• Selling expense Sales Price
• (millions) (millions) (Less 10)
• 1 4 1
• 2 6 0
• 4 8 5
• 8 14 8
• 6 12 4
• 5 10 3
• 8 16 2
• 9 16 7
• 7 12 6
• The computer output generates the estimated
  regression equation Yi 2.53 1.76Xi - 0.35Pi.

21
Standard error of estimate or mean square error

• A measure of the amount of scatter of individual
  observations about the regression line.
• Useful for constructing prediction intervals.
• Y-hat /- 2 standard errors.
• In the case of Miller Pharmaceutical, the MSE
  0.37 million units of sales, based on the
  multiple regression.
• For example, if the predicted value of Miller
  Pharmaceutical Companys sales is 11 million
  units, with probability 95 percent the firms
  sales will be between
• 10.26 11 - 2(0.37) and 11.74 11
  2(0.37).

22
F-Statistic

• Tells us whether or not the group of independent
  variables explains a statistically significant
  portion of the variation in the dependent
  variable.
• Large values of the F-statistic indicate that at
  least one of the independent variables is helping
  to explain variation in the dependent variable.
• Tables of the F-distribution are used to
  determine the probability that an observed value
  of the F-statistic could have arisen by chance,
  if non of the independent variables has any
  effect on the dependent variable.

23
t-Statistic

• Tells us whether or not each particular
  independent variable explains a statistically
  significant portion of the variation in the
  dependent variable
• All else equal, larger values for the t-statistic
  are better
• Rule of Thumb as long as the t-statistic is
  greater than 2, the independent variable belongs
  to the regression equation.
• In the case of Miller Pharmaceuticals the
  t-statistic for selling expenses is about 25.35,
  which means that this variable is highly
  significant.

24
Multicollinearity

• This is a situation in which two or more
  independent variables are very highly correlated.
• In this case the t statistics are meaningless.
• We can estimate the effects of the correlated
  variables as a group on the dependent variable,
  but not the effects of each of the independent
  variables separately.
• To cope with this situation one might have to
  alter the independent variables, or use variables
  that are not highly correlated to each other.
• Sometimes one might have to collect new data to
  deal with the problem of multicollinearity.

25
Serial Correlation of Error Terms
26
Serial Correlation

• In this case the error terms are not independent
  from each other.
• This problem usually arises in time series
  samples.
• The Durbin-Watson test is used to detect whether
  or not serial correlation is present in the error
  terms in a regression.
• The rule of thump is that in the absence of
  serial correlation the Durbin-Watson statistic d
  must be close to the value of 2.
• One way to deal with the problem of serial
  correlation is to take first differences of all
  the independent and dependent variables in the
  regression.

27
Further Analysis of the Residuals
28
Further Analysis of the Residuals

• In this case, the residuals (estimated errors) of
  a regression indicate that the relationship
  between the dependent and independent variables
  is not linear.
• One might want to try a quadratic functional form
  instead of a linear one

• The textbook describes another case where the
  variation of the residuals varies with the level
  of the independent variable.

29
Summary

• The identification problem refers to situations
  where there is no sufficient information to
  estimate the true parameters of an equation.
• Regression analysis is useful in estimating
  demand functions and other economic relations
  from available data.
• A simple regression includes only one independent
  variable, whereas a multiple regression includes
  more than one independent variables.
• In a simple regression, the coefficient of
  determination is used to measure the closeness of
  a fit in a regression line.

30
Summary

• In a multiple regression analysis, the multiple
  coefficient of determination, R2, measures the
  goodness of a fit.
• The F statistic can be used to test whether any
  of the independent variables has an effect on the
  dependent variable.
• The standard error of the estimate can be used to
  construct prediction intervals for the dependent
  variable.
• The t-statistic for a regression coefficient of
  each independent variable can be used to test
  whether this independent variable has any effect
  on the dependent variable.
• Some times we encounter problems or
  multicollinearity, serial correlation and
  residual patterns that needed to be treated
  appropriately.