Unformatted text preview: s. Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 7 / 50 Discussion of the normality assumption
The error term is the sum of “many” di¤erent unobserved factors
Sums of independent factors are normally distributed (CLT)
Problems:
How many di¤erent factors? Number large enough?
Possibly very heterogeneous distributions of individual factors
How independent are the di¤erent factors? The normality of the error term is an empirical question
At least the error distribution should be “close” to normal
In many cases, normality is questionable or impossible by de…nition Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 8 / 50 Discussion of the normality assumption (cont.)
Examples where normality cannot hold:
Interest rate (nonnegative; also currently at zero in many countries)
Number of arrests (takes on a small number of integer values)
Unemployment (indicator variable, takes on only 1 or 0) In some cases, normality can be achieved through transformations of
the dependent variable (e.g. use log(interest rate) instead of interest
rate)
Under normality, OLS is the best (even nonlinear) unbiased estimator
Important: For the purposes of statistical inference, the assumption of
normality can be replaced by a large sample size Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 9 / 50 Terminology: GaussMarkov assumptions
SLR 1. Linear in the parameters = β0 + β1 xi + εi
yi = β0 + β1 xi2 + εi
ln (yi ) = β0 + β1 xi + εi
yi SLR
SLR
SLR
SLR 2.
3.
4.
5. Random sampling (yi , xi ) is a random draw from the population
The variable xi takes on at least two di¤erent values
E (εi jxi ) = E (εi ) = 0
Homoskedasticity V (εi jxi ) = σ2 Next we add the assumption
SLR 6. εi s N 0, σ2 L ecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 10 / 50 If SLR 6. εi s N 0, σ2 holds, then since = β0 + β1 xi + εi
then
s N β0 + β1 xi , σ2 yi
yi jxi Further, since
n b=
β1 ∑ (xi x ) (yi y) i =1 n = n ∑ (xi x) 2 ∑ ci yi i =1 i =1 b s N β , Var b
β1
β1
1
L ecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 11 / 50 Although we know
b
β
βi
ri
β...
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This note was uploaded on 06/15/2013 for the course EMET 2007 taught by Professor Strachan during the Two '13 term at Australian National University.
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