Lecture Managerial economics (Ninth edition): Chapter 4 – Thomas, Maurice

Managerial Economics
ninth edition

Thomas
Maurice

Chapter 4
Basic Estimation
Techniques
McGraw­Hill/Irwin
McGraw­Hill/Irwin
Managerial Economics, 9e
Managerial Economics, 9e

Copyright © 2008 by the McGraw­Hill Companies, Inc. All rights reserved.


Managerial Economics

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

Y

a bX

• I nt e r c e pt  par am e t e r  ( a)  g ive s  value  o f   Y

wh e r e  r e g r e s s io n line  c r o s s e s   Y ­ ax is  (value

o f   Y  wh e n  X  is  z e r o )

•  Slope parameter (b) gives the change in Y 
associated with a one­unit change in X,  

b

4­2

Y/ X


Managerial Economics

Method of Least Squares
• Parameter estimates are obtained by
choosing values of a & b that minimize
the sum of squared residuals
• T h e  r e s id ual is  t h e  d if f e r e nc e  b e t we e n t h e

ac t ual & f it t e d  value s  o f   Y ,  Yi

Yˆi

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

Yˆ
4­3

ˆ
aˆ bX


Managerial Economics

Sample Regression Line
(Figure 4.2)
S
70,000
60,000
ei

50,000

) sr all od( s el a S

20,000
10,000

0

•
•

40,000
30,000

60,000

Si

•

Sˆ i

•

•

•

46,376

•
2,000

4,000

6,000

8,000

Advertising expenditures (dollars)

4­4

Sample regression line
Sˆ i 11573
,

4. 9719 A

10,000

A


Managerial Economics

Unbiased Estimators
• The estimates of aˆ & bˆ do not generally
equal the true values of a & b
• aˆ  &  bˆ  ar e  r and o m  var iab le s  c o m put e d  us ing
d at a f r o m  a r and o m  s am ple

• 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
4­5


Managerial Economics

Relative Frequency Distribution*
(Figure 4.3)
Relative Frequency Distribution*
for  bˆ  when  b = 5

Relative frequency of bˆ
1

0

1

2

3

4

5

6

7

8

9

ˆ
Least-squares estimate of b (b)

4­6

*Also called a probability density function (pdf)

10


Managerial Economics

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

b = 0 it is possible that the
sample will produce an estimate bˆ

• Even if

that is different from zero

• Test for statistical significance
using t-tests or p-values
4­7


Managerial Economics

Performing a t-Test
• 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
4­8


Managerial Economics

Performing a t-Test
• t -ratio is computed as t

bˆ
Sbˆ

where Sbˆ is the standard error of the estimate bˆ

• 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
4­9


Managerial Economics

Performing a t-Test
• If absolute value of t-ratio is greater

than the critical t, the parameter
estimate is statistically significant

4­


Managerial Economics

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
4­

• Also the probability of finding significance when 
none exists


Managerial Economics

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


Managerial Economics

F-Test
• Used to test for significance of
overall regression equation
• Compare F-statistic to critical Fvalue 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


Managerial Economics

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


Managerial Economics

Quadratic Regression Models
• Use when curve fitting scatter plot
is U-shaped or -shaped
U

•

4­

Y

a bX

cX

2

•

Fo r  line ar  t r ans f o r m at io n c o m put e
ne w var iab le   Z X 2

•

Es t im at e   Y

a bX

cZ


Managerial Economics

Log-Linear Regression Models
• Use when relation takes the form: Y

b 

Pe r c e nt ag e  c h ang e  in  Y
Pe r c e nt ag e  c h ang e  in  X

•

c 

Pe r c e nt ag e  c h ang e  in  Y
Pe r c e nt ag e  c h ang e  in  Z

•

T r ans f o r m  b y  t ak ing  nat ur al lo g ar it h m s :

•

b  and   c  ar e  e las t ic it ie s

•

4­

aX b Z c

lnY

lna b ln X

c ln Z