Chapter 4

Copyright © 2008 by the McGraw-Hill Companies, Inc. All rights reserved.

McGraw-Hill/IrwinManagerial Economics, 9e

Managerial Economics ThomasMauriceninth edition

Copyright © 2008 by the McGraw-Hill Companies, Inc. All rights reserved.

McGraw-Hill/IrwinManagerial Economics, 9e

Managerial Economics ThomasMauriceninth edition

Chapter 4

Basic Estimation Techniques

Managerial EconomicsManagerial Economics

4-2

Simple Linear Regression• Simple linear regression model relates dependent variable Y to one independent (or explanatory) variable X

Y a bX • a Y

YY X

I ntercept parameter ( ) gives value of where regression line crosses -axis (valueof when is zero)

• Slope parameter (b) gives the change in Y associated with a one-unit change in X, b Y / X


4-3

Method of Least Squares

• The sample regression line is an estimate of the true regression line

•a b

Parameter estimates are obtained bychoosing values of & that minimizethe sum of squared residuals•

i iˆY Y Y

The residual is the diff erence between theactual & fi tted values of ,

ˆˆ ˆY a bX


4-4

iS , . ASample regression line

11573 4 9719

Sample Regression Line (Figure 4.2)

A0 8,0002,00

010,000

4,000

6,000

10,000

20,000

30,000

40,000

50,000

60,000

70,000

Advertising expenditures (dollars)

Sale

s (d

olla

rs)

S

•

•• •

•

••

iS 46,376

ei

iS 60,000


4-5

• The distribution of values the estimates might take is centered around the true value of the parameter

• An estimator is unbiased if its average value (or expected value) is equal to the true value of the parameter

Unbiased Estimators• ˆa b

a bThe estimates of & do not generally equal the true values of & • ˆa b & are random variables computed usingdata f rom a random sample


4-6

Relative Frequency Distribution* (Figure 4.3)

0 82 104 6

1

1 3 5 7 9

Relative frequency of b

Least-squares estimate of ˆb (b)

*Also called a probability density function (pdf)

Relative Frequency Distribution*ˆfor when 5b b


4-7

• Must determine if there is sufficient statistical evidence to indicate that Y is truly related to X (i.e., b 0)

Statistical Significance

• bb

Even if = 0 it is possible that the

sample will produce an estimate that is different from zero

• Test for statistical significance using t-tests or p-values


4-8

• First determine the level of significance• Probability of finding a parameter

estimate to be statistically different from zero when, in fact, it is zero

• Probability of a Type I Error

• 1 – level of significance = level of confidence

Performing a t-Test


4-9

Performing a t-Test

• Use t-table to choose critical t-value with n – k degrees of freedom for the chosen level of significance• n = number of observations• k = number of parameters estimated

•

b

bt t

S-ratio is computed as

bˆS bwhere is the standard error of the estimate


4-10

Performing a t-Test

• If absolute value of t-ratio is greater than the critical t, the parameter estimate is statistically significant


4-11

Using p-Values

• Treat as statistically significant only those parameter estimates with p-values smaller than the maximum acceptable significance level

• p-value gives exact level of significance• Also the probability of finding

significance when none exists


4-12

Coefficient of Determination

• R2 measures the percentage of total variation in the dependent variable that is explained by the regression equation• Ranges from 0 to 1• High R2 indicates Y and X are highly

correlated


4-13

F-Test

• Used to test for significance of overall regression equation

• Compare F-statistic to critical F-value from F-table• Two degrees of freedom, n – k & k –

1• Level of significance

• If F-statistic exceeds the critical F, the regression equation overall is statistically significant


4-14

Multiple Regression

• Uses more than one explanatory variable

• Coefficient for each explanatory variable measures the change in the dependent variable associated with a one-unit change in that explanatory variable


4-15

• Use when curve fitting scatter plot

Quadratic Regression Models

• 2Y a bX cX •

2Z XFor linear transf ormation computenew variable

• Y a bX cZ Estimate

is U-shaped or

U

-shaped


4-16

Log-Linear Regression Models

• b cY aX ZUse when relation takes the form:

•Y

bX

Percentage change in


•Y

cZ



•

lnY lna b ln X c ln Z Transf orm by taking natural logarithms:

• b c and are elasticities

Chapter 4

Documents

multiple regression

exact level of significance

true value

true regression line

regression equation

chosen level of significance

sample regression linefigure

expected value