Chapter 04Chapter 04Chapter 04Chapter 04

8/19/2019 Chapter 04Chapter 04Chapter 04Chapter 04

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

McGraw-Hill"IrwinManagerial #conomics,

Managerial Economics ThomaMaurininth edition

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

McGraw-Hill"IrwinManagerial #conomics,

Managerial Economics ThomaMaurininth edition

Chapter 4

Basic EstimationTechniques


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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

Y

Y X

Intercept parameter ( ) gives value ofwhere regression line crosses -axis (value

of when is zero)• Slope parameter (b) gives the change in Y associated with a one-unit change in X ,

b Y / X = ∆ ∆


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4-3

Method of Least Squares

• The sample regression line is an

estimate of the true regression line

•

a b

Parameter estimates are obtained by

choosing values of & that minimize

the sum of squared residuals

•

i i ˆ Y Y Y −

The residual is the difference between theactual & fitted values of

ˆ ˆ ˆ Y a bX = +


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4-4

= +iŜ , . A

Sample regression line

11573 4 9719

Sample Regression Line

(Figure 4.!

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40,000

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60,000

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4-5

• The distribution of values the estimatesmight take is centered around the true

value of the parameter • n estimator is un"iased if its average

value (or expected value) is equal to thetrue value of the parameter

#n"iased Estimators

• ˆ ˆ a ba b

The estimates of & do not generally

equal the true values of &

• ˆ ˆ a b& are random variables computed using

data from a random sample


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4-6

Relati$e Frequenc% &istri"ution' (Figure 4.!

0 82 104 6

1

1 3 5 7 9

!elative "re#uen$% o" b̂

&east's#uares estimate o" ˆb (b)

!"lso called a probabilit# densit# function (pdf)

!elative (re#uen$% )istri*ution+,"or -.en 5b b =


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4-7

• !ust determine if there is sufficient

statistical evidence to indicate that

Y is truly related to X (i"e"# b $)

Statistical Significance

• b

ˆ b

%ven if $ it is possible that the

sample 'ill produce an estimate

that is different from zero

• Test for statistical significance

using t tests or pvalues

ll


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4-8

• irst determine the le$el of

significance

• $robabilit# of finding a parameterestimate to be statisticall# differentfrom zero when in fact it is zero

• $robabilit# of a T#pe I %rror

• * + le$el of significance le$el of

confidence

)erforming a t *Test

M i l E iM i l E i


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4-9


• ,se t table to choose critical t value

'ith n – k degrees of freedom for thechosen level of significance

• n number of observations• k number of parameters estimated

•

ˆ b

ˆ bt t

S =ratio is computed as

ˆ b

ˆ

S b'here is the standard error of the estimate

M i l E iM i l E i


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4-


• -f absolute value of t ratio is greater

than the critical t # the parameter

estimate is statistically significant

M i l E iM i l E i


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4-

#sing p*+alues

• Treat as statistically significant

only those parameter estimates

'ith pvalues smaller than the

maximum acceptable significance

level

• pvalue gives exact level of

significance

• "lso the probabilit# of findingsignificance when none exists

M i l E iM i l E i


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4-

Coefficient of &etermination

• R2 measures the percentage of total

variation in the dependent variable

that is explained by the regression

equation

• 'anges from to

• *igh R2 indicates Y and X are highl#correlated

M i l E iM i l E i


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4-

F *Test

• ,sed to test for significance ofoverall regression equation

• .ompare F statistic to critical F value from F table• Two degrees of freedom n – k & k – 1• +evel of significance

• -f F statistic exceeds the critical F #the regression equation overall isstatistically significant

M i l E iM i l E i


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4-

Multiple Regression

• ,ses more than one explanatory

variable

•

.oefficient for each explanatoryvariable measures the change in

the dependent variable associated

'ith a oneunit change in that

explanatory variable

M i l E iM i l E i


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4-

• ,se 'hen curve fitting scatter plot

,uadratic Regression Models

•

2

Y a bX cX = + +

•

2 Z X =

,or linear transformation compute

new variable

• Y a bX cZ = + +%stimate

is Ushaped or U

shaped



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4-

Log*Linear Regression Models

•b cY aX Z =,se 'hen relation takes the form/

•Y

b

X

=

$ercentage change in


•

Y c

Z =



•

lnY ln a b ln X c ln Z = + +

Transform b# ta-ing natural logarithms.

• b cand are elasticities

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