Quantitative Demand Analysis Estimation of Demand

1
Session 3

• Quantitative Demand Analysis Estimation of
  Demand

2
Regression Analysis

• Used to estimate demand functions
• Important terminology
• Least Squares Regression Y a bX e
• Confidence Intervals
• t-statistic
• R-square or Coefficient of Determination
• F-statistic

3
Regression Analysis

• Statistical technique for estimating a
  relationship between the dependent variable and
  one or more independent variables
• Ex relationship between advertising expenditures
  and sales revenues.

4
Regression Analysis

• If we plot data for x and y on a graph--
• Regression analysis estimates the line that
  minimizes the sum of the squared vertical
  deviations of the observations from the line.

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Regression Analysis - Scatter Diagram

• Y

.
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Y a bX
.
.
.
.
.
X
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Objective Of Regression Analysis

• To obtain estimates of A B
• B (SLOPE COEFFICIENT) measures increase in
  dependent variable resulting from unit increase
  in independent variable
• a (INTERCEPT) gives value of Y when X 0.

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Simple Linear Regression
OLS -- ordinary least squares

• Qt a b Pt ??t
• time subscripts error term
• Find best fitting line
• ?t Qt - a - b Pt
• ?t 2 Qt - a - b Pt 2
• min????t 2 ??Qt - a - b Pt 2
• Solution b Cov(Q,P)/Var(P) and a mean(Q) -
  bmean(P)

Q
_ Q
_ P
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Ordinary Least Squares

• e1 is vertical deviation or error of actual Y
  from Y estimated from regression line in first
  year
• e1 Y1 - Y1
• errors occur because
• many variables have slight affect on Y
• error measurement in Y
• random human error

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Ordinary Least Squares

• Computer programs can be used to estimate
  regression line
• For example
• Yt 7.60 3.53Xt
• By substituting a value for Xt, the regression
  line can estimate Yt
• the b measures the marginal effect on Y from each
  unit change in X

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Ordinary Least Squares Assumptions
Solution Methods

• Spreadsheets -
• Statistical calculators
• Minitab, SAS, SPSS
• ForeProfit
• Excel, Lotus, Quatro Pro, Joe Spreadsheet
• tools/data analysis in Excel
• /Data/Regression in Lotus

• error term has a mean of zero and a finite
  variance
• dependent variable is random
• the independent variables are indeed independent

?1999 South-Western College Publishing
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Using a Spreadsheet to Perform a Regression

• Pages 94-95
• Homework Questions 22, 23 and 24 are regression
  analysis problems using Excel
• Spreadsheet program produces detailed information
  about regression

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Tests Of Significance

• to test whether independent variable positively
  affects the dependent variable
• To test for the statistical significance of b, we
  use standard error (deviation) of b given by Sb
• t-statistic b/Sb we get the t-statistic

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Tests Of Significance

• Rule of thumb - if the value of the t-statistic
  is greater than or equal to 2, then the parameter
  estimate is statistically different from zero.
• the higher the calculated t-ratio, the more
  confident we are that the true value of b is not
  equal to zero

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P-Values

• Regression packages report P-values
• much more precise measure of statistical
  significance
• If P-value for an estimated coefficient is .0009,
  this means there is only a 9 in 10,000 chance
  that the true coefficient of the variable is
  actually zero.
• P-values of .05 or below are considered
  significant.

15
Confidence Intervals

• Firm manager can construct upper and lower bounds
  on the true value of the estimated coefficient by
  constructing a 95 confidence interval.
• Rule of thumb for coefficients (a and b) is
• a or - 2 sa
• b or - 2 sb

16
Coefficient Of Determination (R2)

• Measures proportion of total variation in Y
  explained by variation in X.
• If R2 78, then variation in X explains 78 of
  variation in Y. (Ie 22 of variation is
  explained by something else.)

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Evaluates the overall fit of the regression line

• This tests for the overall explanatory power of
  the entire regression.
• If R2 1, all the variation in the dependent
  variable would be explained by the variation in
  independent variables.

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Test Of Goodness Of Fit And Correlation

• The square root of R2 is the coefficient of
  correlation (r).
• r is the measure of the degree of covariation
  that exists between x and y
• r ranges from -1 to 1.

19
Adjusted R2

• Problem with R2 is that as we add more
  coefficients, the R2 increases.
• We may have a high R2 because the number of
  observations is small relative to the number of
  estimated parameters.
• provides a misleading indicator of goodness of
  fit
• Better measure is adjusted R-square
• R2 1 - (1 - R2)(n-1)/n-k)

20
The F-Statistic

• An alternative measure of regression line good
  fit is F-Statistic
• F-Statistic provides measure of total variation
  explained by the regression relative to the total
  unexplained variation.
• the greater the F-statistic, the better the
  overall fit of the regression line.

21
Nonlinear Regression

• For example, if the relationship is a curve.
• For a log-linear demand function, you would take
  the logarithm of prices and quantities before
  executing the regression.
• You would regress the transformed Q on P to
  obtain parameter estimates.

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An Example

• Use a spreadsheet to estimate log-linear demand

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Summary Output
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Interpreting the Output

• Estimated demand function
• log Qx 7.58 - 0.84 logPx
• Own price elasticity -0.84 (inelastic)
• How good is our estimate?
• t-statistics of 5.29 and -2.80 indicate that the
  estimated coefficients are statistically
  different from zero
• R-square of .17 indicates we explained only 17
  percent of the variation
• F-statistic significant at the 1 percent level.

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Regression Analysis With Excel

• Page 3-7 in packet of handouts

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Regression Example

• Assume that bus ridership is a function of the
  price of the ticket, population density, per
  capita income levels in an area, and the number
  of parking places available.
• Given data, you could conduct a regression
  analysis to try to estimate demand for bus
  ridership.
• The results may look like those in the following
  slide.

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Demand Estimation Case
Riders 785 -2.14Price .110Pop .0015Income
.995Parking Predictor Coef St.dev t-ratio
p Constant 784.7 396.3 1.98 .083 Price -2.14 .
4890 -4.38 .002 Pop .1096 .2114 .520 .618 Incom
e .0015 .03534 .040 .966 Parking .9947 .5715 1.74
.120 R-sq 90.8 R-sq(adj) 86.2
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Regression example

• A public agency responsible for serving the
  commuter rail transportation needs of a large
  city is facing rising operating deficits on its
  system.
• The board of directors asked the system manger to
  explore alternatives to alleviate the financial
  plight of the system
• One suggestion is to institute a major cutback in
  service
• The board suggested that the system manager look
  into increasing fares and the conduct a study of
  the likely impact of a proposed fare hike.

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Regression example

• The system manager has collected data on
  important variables thought to have an impact on
  the demand for rides
• Price per ride (in cents) P -
• Population in the area serviced T
• Disposable per capita income I
• Parking rate per hour in the downtown area (in
  cents) H
• Perform a multiple regression on the data to
  determine the impact of the rate increase. Data
  follows on next slide.

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Regression Data
31
Demonstration Problem 3-5

• Page 101
• Multiple Regression

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SUMMARY

• Given market or survey data, regression analysis
  can be used to estimate
• Demand functions
• Elasticities
• A host of other things, including cost functions
• Managers can quantify the impact of changes in
  prices, income, advertising, etc.