Econometrics I
QM 6203
Mohd‐Pisal Zainal, Ph.D.
INCEIF
Week 4
February 5, 09
Basic Econometrics
Chapter 4: THE NORMALITY ASSUMPTION:
Classical Normal Linear Regression Model (CNLRM)
The Organization of the Presentation
• The Normality Assumption
• The Concept of Linearity and Stochastic Specification in PRF
• The Importance of the Error Term
• The Sample Regression Function (SRF)
• Summary & Conclusions
The Normality Assumption
• CNLR assumes that each u i is distributed normally u i
∼ N(0, σ2) with:
Mean = E(u i) = 0
Ass 3
Variance = E(u2i) = σ2
Ass 4
Cov(u i , u j ) = E(u i , u j) = 0 (i#j) Ass 5
• Note: For two normally distributed variables, the zero covariance or correlation means independence of them, so u i and u j are not only uncorrelated but also independently distributed. Therefore u i ∼
NID(0, σ2) is Normal and Independently Distributed
The Normality Assumption
•
1.
2.
3.
4.
Why the normality assumption?
With a few exceptions, the distribution of sum of a large number of independent and identically distributed random variables tends to a normal distribution as the number of such variables increases indefinitely
If the number of variables is not very large or they are not strictly independent, their sum may still be normally distributed
Under the normality assumption for ui , the OLS estimators β^1 and β^2 are also normally distributed
The normal distribution is a comparatively simple distribution involving only two parameters (mean and variance)
Properties of OLS Estimators under the Normality Assumption
•
With the normality assumption the OLS estimators β^1 , β^2 and σ^2 have the following properties:
1. They are unbiased
2. They have minimum variance. Combined 1 and 2, they are efficient estimators
3. Consistency, that is, as the sample size increases indefinitely, the estimators converge to their true population values
4. β^1 is normally distributed ∼
N(β1, σ^β12) and Z = (β^1‐ β1)/ σ^β1 is ∼ N(0,1)
5. β^2 is normally distributed ∼N(β2 ,σ^β22) And Z = (β^2‐ β2)/ σ^β2 is ∼ N(0,1)
6. (n‐2) σ^2/ σ2 is distributed as the χ2(n‐2)
Properties of OLS Estimators under
the Normality Assumption
7. β^1 and β^2 are distributed independently of σ^2. They have minimum variance in the entire class of unbiased estimators, whether linear or not. They are best unbiased estimators (BUE)
8. Let ui is ∼ N(0, σ2 ) then Yi is ∼
N[E(Yi); Var(Yi)] = N[β1+ β2X i ; σ2]
The Method of Maximum Likelihood (ML)
™ ML is point estimation method with some stronger theoretical properties than OLS (Appendix 4.A on pages 110‐114)
™ The estimators of coefficients β’s by OLS and ML are identical. They are true estimators of the β’s
™ (ML estimator of σ2) = Σu^i2/n (is biased estimator)
™ (OLS estimator of σ2) = Σu^i2/n‐2 (is unbiased estimator)
™ When sample size (n) gets larger the two estimators tend to be equal
Probability distributions related
to the Normal Distribution: The t, χ2,
and F distributions
See section (4.5) on pages 107‐108
with 8 theorems and Appendix A, on
pages 755‐776
Summary and Conclusions
See 10 conclusions on pages 109‐110
Summary and Conclusions
• For empirical purposes, it is the stochastic PRF that matters. The stochastic disturbance term ui plays a critical role in estimating the PRF.
• The PRF is an idealized concept, since in practice one rarely has access to the entire population of interest. Generally, one has a sample of observations from population and use the stochastic sample regression (SRF) to estimate the PRF.
Suggestions, Questions, and or Comments
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Chapter Five: TWO‐VARIABLE REGRESSION: Interval Estimation and Hypothesis Testing