MANAGERIAL ECONOMICS 11th Edition

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MANAGERIAL ECONOMICS 11th Edition

• By
• Mark Hirschey

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Demand Estimation

• Chapter 6

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Chapter 6OVERVIEW

• Demand Curve Estimation
• Regression Analysis
• Measuring Regression Model Significance
• Measures of Individual Variable Significance

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Demand Curve Estimation

• Simple Linear Demand Curves
• The best estimation method balances marginal
  costs and marginal benefits.
• Simple linear relations are useful for demand
  estimation.
• Using Simple Linear Demand Curves
• Straight-line relations give useful
  approximations.

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• Example
• A business sells 2,000 units per month at a
  price of 10 each. It can sell 250 more items per
  month for each 0.25 reduction in price. What
  price per unit will maximize the monthly revenue?
  
• A price of p10 corresponds to x 2,000 and a
  price of p9.75 corresponds to x 2250. Using
  this information you can use the point-slope form
  to create the price equation.

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• m (10 - 9.75) / (2000 -2250) Find the slope
• m -.001
• p-10 -0.001(x-2000) Point-slope
  form
• p-0.001x 12
• Substituting this value into the revenue equation
  produces
• R x(-0.001x12)
• -0.001x2 12x
• Total Revenue at q200020,000
• Total Revenue at q225021,937.5
• To maximize the revenue function, find the
  critical numbers
• R 12 -.002x0

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Continued

• x 6,000 Critical number
•  The price that corresponds to this production
  level is
• p12-0.001x Demand function
• 12 - 0.001(6000) Substitute 6,000 for x
• 6 Price per unit

(6000, 36000)
Revenue
R- therefore x6000 maximum
6000
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Regression Analysis

• Regression analysis is a statistical technique
  that attempts to explain movements in one
  variable, the dependent variable, as a function
  of movements in a set of other variables, called
  independent (or explanatory) variables through
  the quantification of a single equation.
• However, a regression result no matter how
  statistically significant, cannot prove
  causality. All regression analysis can do is test
  whether a significant quantitative relationship
  exists.

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Specifying the Regression Model

• Linear Model Model Assumption X and Y are
  linearly related. Each coefficient will tell us
  how much quantity demanded will change by a one
  unit change in the coefficient.
• Multiplicative Model Nonlinear relation that
  involves X variable interactions. The coefficient
  estimates are interpreted as estimates on the
  constant elasticity of Y with respect of X, or
  the percentage change in Y due to a 1 percent
  change in X

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• The equation that describes how y is related
  to x and
• an error term is called the regression
  model.

• The simple linear regression model is

y b0 b1x e

• where
• b0 and b1 are called parameters of the model,
• e is a random variable called the error term.

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Assumptions About the Error Term ?
1. The error ? is a random variable with mean
of zero.
2. The variance of ? , denoted by ? 2, is the
same for all values of the independent
variable.
3. The values of ? are independent.
4. The error ? is a normally distributed
random variable.
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• The simple linear regression equation is

E(y) ?0 ?1x

• Graph of the regression equation is a straight
  line.

• b0 is the y intercept of the regression line.

• b1 is the slope of the regression line.

• E(y) is the expected value of y for a given x
  value.

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• Positive Linear Relationship

Regression line
Intercept b0
Slope b1 is positive
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• Negative Linear Relationship

Regression line
Intercept b0
Slope b1 is negative
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• No Relationship

Regression line
Intercept b0
Slope b1 is 0
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• 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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• Least Squares Criterion
• where
• yi observed value of the dependent variable
• for the ith observation
• yi estimated value of the dependent variable
• for the ith observation
• This regression technique that calculates the ?
  so as to minimize the sum of the squared
  residuals.

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• The Multiple Regression Model
• y ?0 ?1x1 ?2x2 . . . ?pxp ?
• The Multiple Regression Equation
• E(y) ?0 ?1x1 ?2x2 . . . ?pxp
• The Estimated Multiple Regression Equation
• y b0 b1x1 b2x2 . . . bpxp

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• Least Squares Criterion
• Computation of Coefficients Values
• The formulas for the regression coefficients
  b0, b1, b2, . . . bp involve the use of matrix
  algebra. We will rely on computer software
  packages to perform the calculations.
• A Note on Interpretation of Coefficients
• bi represents an estimate of the change in y
  corresponding to a one-unit change in xi when all
  other independent variables are held constant.

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Relationship Among SST, SSR, SSE
.

• R

observed
.
SSE

SST
estimated
SSR
mean

• where
• SST total sum of squares
• SSR sum of squares due to regression
• SSE sum of squares due to error

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• Relationship Among SST, SSR, SSE
• SST SSR SSE
• where
• SST total sum of squares
• SSR sum of squares due to
  regression
• SSE sum of squares due to error

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Measuring Regression Model Significance

• Standard Error of the Estimate increases with
  scatter about the regression line.

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Coefficient of Determination

• Multiple Coefficient of Determination
• R 2 SSR/SST
• Adjusted Multiple Coefficient of Determination

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Goodness of Fit, r and R2

• r 1 means perfect correlation r 0 means no
  correlation.
• R2 1 means perfect fit R2 0 means no
  relation.
• Corrected Coefficient of Determination, R2
• Adjusts R2 downward for small samples.

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F statistic

• Tells if R2 is statistically significant.

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Measures of Individual Variable Significance

• t statistics
• t statistics compare a sample characteristic to
  the standard deviation of that characteristic.
• A calculated t statistic more than two suggests a
  strong effect of X on Y (95 confidence).
• A calculated t statistic more than three suggests
  a very strong effect of X on Y (99 confidence).
  
• Two-tail t Tests
• Tests of effect.
• One-Tail t Tests
• Tests of magnitude or direction.

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Example

• An economist collected data for a sample of 20
  computer stores. A suggestion was made that
  regression analysis could be used to determine if
  sales was related to the years of experience of
  the manager and the score on the firms manager
  aptitude test. The years of experience, score on
  the aptitude test, and corresponding daily sales
  (1000s) for a sample of 20 stores is shown on
  the next slide.

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• Exper. Score Sales Exper.
  Score Sales
• 4 78 24 9 88 38
• 7 100 43 2 73 26.6
• 1 86 23.7 10 75 36.2
• 5 82 34.3 5 81 31.6
• 8 86 35.8 6 74 29
• 10 84 38 8 87 34
• 0 75 22.2 4 79 30.1
• 1 80 23.1 6 94 33.9
• 6 83 30 3 70 28.2
• 6 91 33 3 89 30

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• Excel Computer Output
• The regression is
• Sales 3.17 1.40 Exper 0.251 Score
• Predictor Coef Stdev
  t-ratio p
• Constant 3.174 6.156 .52 .613
• Exper 1.4039 .1986 7.07 .000
• Score .25089 .07735 3.24 .005
• s 2.419 R-sq 83.4
  R-sq(adj) 81.5

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Interpreting the Coefficients
b1 1. 404
Sales are expected to increase by 1,404
for each additional year of experience (when
the variable score on manager attitude test is
held constant).
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Interpreting the Coefficients
b2 0.251
Sales are expected to increase by 251 for
each additional point scored on the manager
aptitude test (when the variable years of
experience is held constant).
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• Excel Computer Output (continued)
• Analysis of Variance
• SOURCE DF SS MS
  F P
• Regression 2 500.33 250.16 42.76 0.000
• Error 17 99.46 5.85
• Total 19 599.79

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• F Test
• Hypotheses H0 ?1 ?2 0
• Ha One or both of the parameters
• is not equal to zero.
• Rejection Rule
• For ? .05 and d.f. 2, 17 F.05
  3.59
• Reject H0 if F gt 3.59.
• Test Statistic
• F MSR/MSE 250.16/5.85 42.76
• Conclusion
• We can reject H0.

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• t Test for Significance of Individual Parameters
• Hypotheses H0 ?i 0
• Ha ?i 0
• Rejection Rule DFn-p-1
• For ? .05 and d.f. 17, t.025 2.11
• Reject H0 if t gt 2.11
• Test Statistics
• Conclusions
• Reject H0 ?1 0 Reject H0
  ?2 0

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Self Test Problem 1
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