Lecture 5: Hypothesis testing with the classical linear model
Lecture 5: Hypothesis testing
with the classical linear model
Assumption MLR6: Normality
u ~ N (0, ? 2 )
E (u | x1 , x2 , ?, xk ) ? E (u ) ? 0
Var (u | x1 , x2 , ?, xk ) ? Var (u ) ? ? 2
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MLR6 is not one of the Gauss-Markov
assumptions. It¡¯s not necessary to assume the
error is normally distributed in order to obtain
the best linear unbiased estimator from OLS.
MLR6 makes OLS the best unbiased estimator
(linear or not), and allows us to conduct
hypothesis tests.
Assumption MLR6: Normality
Assumption MLR6: Normality
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But if Y takes on only n distinct values, for any set of
values of X, the residual can take on only n distinct values.
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Non-normal errors: we can no longer trust hypothesis tests
Heteroscedasticity
With a dichotomous Y, run a logit or probit model
With an ordered categorial Y, ordered probit/logit
With an unordered categorial Y, multinomial probit/logit
With non-negative integer counts, poisson or negative
binomial models
But . . . chapter 5 we¡¯ll see large sample size can
overcome this problem.
Assumption MLR6: Normality
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If we assume that the error is normally
distributed conditional on the x¡¯s, it follows:
?? j ~ N ( ? j ,Var ( ?? j )
( ?? j ? ? j ) / sd ( ?? j ) ~ N (0,1)
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In addition, any linear combination of beta
estimates is normally distributed, and multiple
estimates are jointly normally distributed.
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