1. 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).

2. Conditional Distribution of Y with respect to X
Table 2.1 gives the distribution of consumption expenditure Y corresponding to a fixed level of income X; that is conditional distribution of Y conditional upon the given values of X.
What is Conditional Mean? How do you calculate it?
Conditional Mean for Y given that X=80: 55(1/5)+60(1/5)+65(1/5)+70(1/5)+75(1/5) = 65

3. Conditional Probabilities
Conditional Probability: p(Y/X): probability of Y given X.
For example: p (Y=55 / X = 80) = 1/5 = 0,20
P (Y = 150 / X = 260) = 1/7 = 0,14

4. 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”

5. Data of Table 2.1 on a Plot

6. Data of Table 2.2 on a Plot

7. The Concept of Population Regression Function (PRF)
If E (Y / X = Xi),
then
E (Y / Xi) = f (Xi)
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.
Therefore,
PRF

8. On PRF More PRF is E (Y / Xi) = f (Xi)
Therefore, PRF is also linear function of Xi,
that is:
E (Y / Xi) = 1 + 2 Xi
Where 1 and 2 are unknown parameters known as the regression coefficients.

9. On PRF more E (Y / Xi) = 1 + 2 Xi Slope Intercept
Dependent Variable
E (Y / Xi) = 1 + 2 Xi
Linear Population Regression Function
Slope
Intercept
Independent Variable

10. Stochastic Specification of PRF
E (Y / Xi) = 1 + 2 Xi
ui = Yi – E (Y/Xi)
Yi = E (Y/Xi) + ui
Then,
Yi = 1 + 2 Xi + ui
Actual value of Y
Expected or estimated value of Y with respect to X
Stochastic Error Term

11. 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)

12. The Sample Regression Function (SRF)
PRF:
Yi = 1 + 2 Xi + ui
SRF:
Estimator of 2
Estimator of 1

13. Example on SRF

14. PRF and SRF Compared
SRF Error
PRF Error