Estimating Demand Functions

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Estimating Demand Functions
Chapter 4
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1. Objectives of Demand Estimation

• determine the relative influence of demand
  factors

• forecast future demand

• make production plans and effective inventory
  controls

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2. Major approaches to Demand Estimation
a.Marketing Research

• Consumer survey) (telephone, questionnaire,
  interviews, online survey)

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Advantage provides useful data for the
introduction of new products

• Disadvantages
• It could be biased due to unrepresentative
  sampling size
• Consumers may provide socially acceptable
  response rather than true preferences

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• Consumer Clinic
• a sample of consumers is chosen either randomly,
  or based on socio-economic features of the market
• They are given some money to spend on goods
• Their purchases are being observed by a
  researcher

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• Advantages
• more realistic than consumer surveys
• avoids the shortcomings of market
  experiments(costs).

• Disadvantages
• participants know that they are in anartificial
  situation
• small sample because of high cost

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• Market Experiments
• Similar to consumer clinic, but are conducted in
  an actual market place
• Select several markets with similar
  socio-economic characteristics and change a
  different factor in each
• market
• Use census data for various markets and study
  the impacts of differences in demographic
  characteristics on buying habits

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Market Experiments Advantages

• Can be done on a large scale
• Consumers are not aware that they are part of an
  experiment

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Problems of Marketing Research

• the sample may not be representative
• Consumers may not be able to answer questions
  accurately-biased

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• 2b.Statistical Method
• Involves the use of regression analysis to
  determine the relative quantitative effect of
  each of the demand determinants.

• Regression Analysis is usually
• more objective than marketing research
• provides more complete information than market
  research
• less expensive

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3. Steps in regression analysis

• Specify the model (theory)
• Obtain data (types and sources)
• Specifying the form of the demand equation
  (linear, log linear)
• Specifying the form of the demand equation
  (linear, log linear)
• Estimate the regression coefficients (Finding the
  line of best fit by minimizing the error sum of
  squares)
• ?(Yt )2 ?(Yt - a - bXt)2 0

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Regression Parameters
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Steps in Regression Analysis contd

• Test the significance of the regression results
  (Overall tests and individual tests).
• Use the results of the regression analysis as a
  supporting evidence in making business policy
  decisions (change price, ad strategy, customer
  service)

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4 a.Given Sales (Yt in 000 units) and
Advertising Expenditures (Xt) (in mill. ) data
as follow
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Regression Example
Yt Xt
37 5 -7 -1 7 1
48 7 4 1 4 1
45 6 1 0 0 1
36 3 -8 -3 24 9
25 4 -19 -2 38 4
55 9 11 3 33 9
63 8 19 2 38 4
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a.
b.
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4c. Interpretation of Regression Coefficients
-- is the intercept term which represents the
value of the dependent variable when Xt0.
-- has no economic meaning when its value lies
outside the range of observed data for Yt. Note
Data Rangegt 25-63
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- the slope of the regression line -
represents the change in the dependent variable
(Yt) related to a unit change in the independent
variable
5.14 means that a 1 million dollar
increase in ad expenses will result in an
increase in sales by 5140 units.
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4d. Overall Measures of Model Performance

• R2coefficient of determination
• the ratio of the variation in sales
  explained by the variation in ad expenses.
• Explained Variation/Total Variation

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Notice that R2 is adjusted for the degrees of
freedom- the number of observations beyond the
minimum needed to calculate a given regression
statistic. For example, to calculate the
intercept term, at least one observation is
needed to calculate an intercept term plus one
slope coefficient, at least two observations are
required, and so on.
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Calculating R2

39 25 49
49 44 0
44 0 1
28 256 64
34 100 361
59 81 121
54 100 361
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gtExplained variation
gtTotal variation
.761 means that 76.1 of the variation of
in sales is explained by the variation in
advertising expenditures.
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Note One would like R2 to be as high as
possible. R2, however, depends on the type of
data used in the estimation. It is relatively
higher for time series and smaller for
cross-sectional data. For a cross-section data,
R2 of .5 is acceptable.
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(ii) F-Statistic

• F-Statistic- a statistical test of significance
  of the regression model.
• F- Test of Hypotheses

Decision Rule Accept Ho if F-calculated lt
F-table Reject HO if F-calculatedgt F-table
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F-table is defined for df1k-1, df2n-k)
at a .05 (conventional) or a.01, or any other
level of significance. k of parameters (2),
n of observations (7) F(1, 5) at a .05
6.61, F-cal R2/k-1/(1-R2)/(n-k)
.751/.249/515.1 Reject Ho since
F-calgtF-table, i.e. the regression model exhibits
a statistically significant relationship.
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4e. The t-statistic test is a test of the
individual independent variable.

• T- test of hypotheses

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Decision Rule Accept Ho if t-lowerltt-calltt-uppe
r critical Value. Reject Ho if t-cal lt t-lower
or t-calgt t-upper critical value.
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t-Statistic test

• t-table( d.f.n-k 5, a .05 or at .01)
• t-table (5, a.05)2.571, p. A-56-Table 4
• t-cal 5.14/1.45 3.54 gt2.51.
• Therefore we reject the Ho that there is no that
  advertising does not affect. It does increase
  sales.

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Accept H0
Reject H0
Reject H0
t
0
-2.751
2.751
Decision Reject Ho since t-calgt t-upper value
from the table or t-calltt-lower value. There is
a statistically significant relationship between
sales and advertising
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Multiple Regression has more than one independent
variable.

• Example Earningsf(Age, ED, JOB Exp.)
• How do we estimate the regression coefficients in
  this case?

Use a variety of statistical software (Minitab,
Excel, SAS, SPSS, ET, Limdep, Shazam, TSP)
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-72.06 -.21Age 2.25ED
1.02JEXP (-2.1) (-1.93) (8.86)
(4.07) (The numbers in parenthesis are
t-values). R2 .874 F-cal 37.05 Test the
significance of each of the variables. Interpret
the meaning of the coefficients.
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5.The regression coefficients which are obtained
from a linear demand equation represent slopes
(the effect of a one unit change in the
independent variable on the dependent variable

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Problems in
regression Analysis
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• Problems in Regression Analysis arise due to
• the violation(s) of one or
• more of the classical assumptions of the linear
  regression model.

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Assumptions Linear Model

• The model is linear in parameters and in the
  error term
• The error term has a zero population mean µ0 and
  s2 s1 gt Normal Distribution Assumption
• All regressors are uncorrelated gt violation of
  this assumption results in multicollinearity gt
  biased estimates

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• E(etet-1) 0 gt no autocorrelation
• (time series)
• The error term for one period is
• systematically uncorrelated with the
• error term in another period
• If this assumption is violated i.e.
• E(etet-1)0 gt autocorrelation problem
• The variance of the error term et is the same for
  each
• observation
• E(Set)2 s2 1 gtheteroscadasticity

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a. Multicollinearity

• A situation where two or more explanatory
  variables in the regression are highly correlated
  which leads to large standard errors hence the
  insignificance of the slope coefficient.

• To reduce multcollinearity
• increase sample size.
• express one variable in terms of the other, or
• transform the functional relationship.
• Drop one of the highly collinear variables.

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b. Hetetroscedasticity

• Arises when the variance of the error terms is
  non-constant

• usually occurs in cross-sectional data
• (large std errors)

• leads to biased standard

• problem may be overcome by using log of the
  explanatory variables that lead to
  heteroscedastic disturbances, or by running a
  weighted least squares regression

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c. Autocorrelation

• occurs whenever consecutive errors or residuals
  are correlated(positive vs
• negative correlation
• The standard errors are biased downward making
  t-cal value to be larger
• We tend to reject the Ho more

• occurs in time series data