Estimating Demand

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Estimating Demand

• Outline
• Where do demand functions come from?
• Sources of information for demand estimation
• Cross-sectional versus time series data
• Estimating a demand specification using the
  ordinary least squares (OLS) method.
• Goodness of fit statistics.

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The goal of forecasting
To transform available data into equations
that provide the best possible forecasts of
economic variablese.g., sales revenues and costs
of productionthat are crucial for management.
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Demand for air travel Houston to Orlando
Recall that our demand function was estimated as
follows
Now we will explain how we estimated this demand
equation
4.1
Q 25 3Y PO 2P
Where Q is the number of seats sold Y is a
regional income index P0 is the fare charged by
a rival airline, and P is the airlines own fare.
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Questions managers should ask about a
forecasting equations

• What is the best equation that can be obtained
  (estimated) from the available data?
• What does the equation not explain?
• What can be said about the likelihood and
  magnitude of forecast errors?
• What are the profit consequences of forecast
  errors?

5
How do get the data to estimate demand
forecasting equations?

• Customer surveys and interviews.
• Controlled market studies.
• Uncontrolled market data.

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Campbells soup estimates demand functions from
data obtained from a survey of more than 100,000
consumers
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Survey pitfalls

• Sample bias
• Response bias
• Response accuracy
• Cost

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Types of data
Time -series data historical data--i.e., the
data sample consists of a series of daily,
monthly, quarterly, or annual data for variables
such as prices, income , employment , output ,
car sales, stock market indices, exchange rates,
and so on. Cross-sectional data All observations
in the sample are taken from the same point in
time and represent different individual entities
(such as households, houses, etc.)
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Time series data Daily observations, Korean Won
per dollar
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Example of cross sectional data
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Estimating demand equations using regression
analysis
Regression analysis is a statistical technique
that allows us to quantify the relationship
between a dependent variable and one or more
independent or explanatory variables.
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Regression theory
Y
X and Y are notperfectly correlated.However,
there is on average a positive relationshipbetwe
en Y and X
0
X
X1
X2
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We assume that expected conditional values of Y
associated with alternative values of X fall on
a line.
Y
E(Y Xi) ?0 ?1Xi
Y1
?1
?1 Y1 - E(YX1)
E(YX1)
X
0
X1
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Specifying a single variable model
Our model is specified as follows Q f (P)
where Q is ticket sales and P is the fare
Q is the dependent variablethat is, we think
that variations in Q can be explained by
variations in P, the explanatory variable.
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Estimating the single variable model
Since the datapoints are unlikely to
fallexactly on a line, (1)must be modifiedto
include a disturbanceterm (ei)
1
2

• ?0 and ?1 are called parameters or population
  parameters.
• We estimate these parameters using the data we
  have available

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Estimated Simple Linear Regression Equation

• The estimated simple linear regression equation

• The graph is called the estimated regression
  line.

• b0 is the y intercept of the line.

• b1 is the slope of the line.

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Estimation Process
Regression Model y b0 b1x e Regression
Equation E(y) b0 b1x Unknown Parameters b0, b1
b0 and b1 provide estimates of b0 and b1
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Least Squares Method

• Least Squares Criterion

where yi observed value of the dependent
variable for the ith observation
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Least Squares Method

• Slope for the Estimated Regression Equation

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Least Squares Method

• y-Intercept for the Estimated Regression Equation

where xi value of independent variable for
ith observation
yi value of dependent variable for ith
observation
n total number of observations
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Line of best fit
The line of best fit is the one that minimizes
the squared sum of the vertical distances of the
sample points from the line
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The 4 steps of demand estimation using regression

• Specification
• Estimation
• Evaluation
• Forecasting

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Table 4-2 Ticket Prices and Ticket Sales along an
Air Route
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Simple linear regression begins by plotting Q-P
values on a scatter diagram to determine if there
exists an approximate linear relationship
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Scatter plot diagram with possible line of best
fit
7

2
0
6
2
0
5
2
0
4
2
0
3
2
0
2
2
0
0
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Note that we use X to denote the
explanatoryvariable and Y is the dependent
variable.So in our example Sales (Q) is the Y
variable and Fares (P) is the X variable.
Q Y P X
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Computing the OLS estimators
We estimated the equation using the
statistical software package SPSS. It generated
the following output
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Reading the SPSS Output
From this table we see that our estimate of ?0 is
478.7 and our estimate of ?1 is 1.63.
Thus our forecasting equation is given by
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Step 3 Evaluation

• Now we will evaluate the forecasting equation
  using standard goodness of fit statistics,
  including
• The standard errors of the estimates.
• The t-statistics of the estimates of the
  coefficients.
• The standard error of the regression (s)
• The coefficient of determination (R2)

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Standard errors of the estimates

• We assume that the regression coefficients are
  normally distributed variables.
• The standard error (or standard deviation) of the
  estimates is a measure of the dispersion of the
  estimates around their mean value.
• As a general principle, the smaller the standard
  error, the better the estimates (in terms of
  yielding accurate forecasts of the dependent
  variable).

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The following rule-of-thumb is useful The
standard error of the regression coefficient
should be less than half of the size of the
corresponding regression coefficient.
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Computing the standard error of ?1
Let
denote the standard error of our estimate of ?1
Note that
Thus we have
and
Where
and
k is the number of estimated coefficients
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By reference to the SPSS output, we see that the
standard error of our estimateof ?1 is 0.367,
whereas the (absolute value)our estimate of ?1
is 1.63 Hence our estimate is about 4 ½ times
the size of its standard error.
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The SPSS output tells us that the t statistic
for the the fare coefficient (P) is 4.453 The t
test is a wayof comparing the errorsuggested by
the nullhypothesis to the standard error of the
estimate.
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The t test

• To test for the significance of our estimate of
  ?1, we set the following null hypothesis, H0, and
  the alternative hypothesis, H1
• H0 ?1? 0
• H1 ?1 lt 0
• The t distribution is used to test for
  statistical significance of the estimate

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Coefficient of determination (R2)

• The coefficient of determination, R2, is defined
  as the proportion of the total variation in the
  dependent variable (Y) "explained" by the
  regression of Y on the independent variable (X).
  The total variation in Y or the total sum of
  squares (TSS) is defined as

Note
?The explained variation in the dependent
variable(Y) is called the regression sum of
squares (RSS) and is given by
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What remains is the unexplained variation in the
dependent variable or the error sum of squares
(ESS)

• We can say the following
• TSS RSS ESS, or
• Total variation Explained variation
  Unexplained variation

R2 is defined as
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We see from the SPSS model summary table that R2
for this model is .586
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Notes on R2

• Note that 0? R2 ?1
• If R2 0, all the sample points lie on a
  horizontal line or in a circle
• If R2 1, the sample points all lie on the
  regression line
• In our case, R2 ? 0.586, meaning that 58.6
  percent of the variation in the dependent
  variable (consumption) is explained by the
  regression.

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This is not a particularly good fit based on R2
since 41.4 percent of the variation in the
dependent variable is unexplained.
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Standard error of the regression

• The standard error of the regression (s) is given
  by

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• The model summary tells us that s 18.6
• Regression is based on the assumption that the
  error term is normally distributed, so that 68.7
  of the actual values of the dependent variable
  (seats sold) should be within one standard error
  (?18.6 in our example) of their fitted value.
• Also, 95.45 of the observed values of seats sold
  should be within 2 standard errors of their
  fitted values (?37.2).

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Step 4 Forecasting
Recall the equation obtained from the regression
results is
Our first step is to perform an in-sample
forecast.
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At the most basic level, forecasting consists of
inserting forecasted values of the explanatory
variable P (fare) into the forecasting equation
to obtain forecasted values of the dependent
variable Q (passenger seats sold).
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In-Sample Forecast of Airline Sales
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Can we make a good forecast?

• Our ability to generate accurate forecasts of
  the dependent variable depends on two factors
• Do we have good forecasts of the explanatory
  variable?
• Does our model exhibit structural stability,
  i.e., will the causal relationship between Q and
  P expressed in our forecasting equation hold up
  over time? After all, the estimated coefficients
  are average values for a specific time interval
  (1987-2001). While the past may be a serviceable
  guide to the future in the case of purely
  physical phenomena, the same principle does not
  necessarily hold in the realm of social phenomena
  (to which economy belongs).

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Single Variable Regression Using Excel
We will estimate an equation and use it to
predict home prices in two cities. Our data set
is on the next slide
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• Income (Y) is average family income in 2003
• Home Price (HP) is the average price of a new or
  existing home in 2003.

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Model Specification
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Excel Output
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Equation and prediction