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Transcript of Chapter 04Chapter 04Chapter 04Chapter 04
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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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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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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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= +iŜ , . A
Sample regression line
11573 4 9719
Sample Regression Line
(Figure 4.!
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• 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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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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• !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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• 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
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)erforming a t *Test
• ,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
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)erforming a t *Test
• -f absolute value of t ratio is greater
than the critical t # the parameter
estimate is statistically significant
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#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
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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
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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
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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
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• ,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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Log*Linear Regression Models
•b cY aX Z =,se 'hen relation takes the form/
•Y
b
X
=
$ercentage change in
$ercentage change in
•
Y c
Z =
$ercentage change in
$ercentage change in
•
lnY ln a b ln X c ln Z = + +
Transform b# ta-ing natural logarithms.
• b cand are elasticities