Chapter 2. Two-Variable Regression Analysis: Some Basic Ideas

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Presentation 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 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

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

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 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

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.

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

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

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 = 1 + 2 Xi + ui SRF: Estimator of 2 Estimator of 1

Example on SRF

PRF and SRF Compared SRF Error PRF Error