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Transcript 
Chapter 2. Two-Variable Regression Analysis:
Some Basic Ideas
A Hypothetical Example
Imagine a hypothetical country with a total
population of 60 families.
Question:
To set a relationship between weekly family
consumption expenditure (Y) and and weekly
family income (X).
Conditional Distribution of Y with respect to X
Table
gives the distribution
of consumption
expenditure
Y corresponding
What2.1
is Conditional
Mean? How
do you calculate
it?
Conditional
Mean
Y given
55(1/5)+60(1/5)+65(1/5)+70(1/5)+75(1/5)
to
a fixed level
offor
income
X;that
thatX=80:
is conditional
distribution of Y conditional = 65
upon the given values of X.
Conditional Probabilities
Conditional
probability
of Y given X.
For
P (Yexample:
= 150Probability:
/ Xp =(Y=55
260) /=p(Y/X):
X1/7
= 80)
= 0,14
= 1/5 = 0,20
Conditional Mean or Conditional
Expectation
E (Y / X = Xi)
It is read as “the expected value of Y given that
X takes the specific value Xi”
Data of Table 2.1 on a Plot
Data of Table 2.2 on a Plot
The Concept of Population Regression Function
(PRF)
„ If E (Y / X = Xi),
„ then
Therefore,
„ E (Y / Xi) = f (Xi)
PRF
That is, if conditional mean of Y depends on each level of X variable,
then conditional mean of Y is said to be a function of given X
values.
On PRF More
PRF is E (Y / Xi) = f (Xi)
Therefore, PRF is also linear function of Xi,
that is:
E (Y / Xi) = β1 + β2 Xi
Where β1
and β2 are unknown parameters known
as the regression coefficients.
On PRF more
Dependent Variable
E (Y / Xi) = β1 + β2 Xi
Intercept
Slope
Independent Variable
Linear Population Regression Function
Stochastic Specification of PRF
E (Y / Xi) = β1 + β2 Xi
ui = Yi – E (Y/Xi)
Yi
Expected
Actual value
of Y or estimated value
=Stochastic
E (Y/XError
) + Term
u of Y with respect to X
i
i
Then,
Yi = β1 + β2 Xi + ui
Stochastic Error Term
E ( Yi / X) = E [E( Y / Xi)] + E (ui /Xi)
E ( Yi / X) = E( Y / Xi) + E (ui /Xi)
E (ui /Xi) = 0
Therefore,
E ( Yi / X) = E( Y / Xi)
The Sample Regression Function
(SRF)
PRF:
Yi = β1 + β2 Xi + ui
SRF:
Yˆ = βˆ1 + βˆ 2 X i + uˆ i
of β2
EstimatorEstimator
of β1
Example on SRF
PRF and SRF Compared
PRF Error
SRF Error
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