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Online Econometrics Textbook - Regression Extensions - Assumption Violations of Linear Regression

[Home] [Up] [SUR] [Simultaneity] [Multicollinearity] [Restricted LS] [Distributed Lags] [Assumption Violations]

[Heteroskedasticity] [Autocorrelation] [Nonlinearities] [Nonnormality] [Misspecification]

In the previous chapter we have discussed linear regression analysis under some specific assumptions. In this section we will investigate what can be done in case the assumptions of OLS or MLE are violated.

Suppose that

Online Econometrics Textbook - Regression Extensions - Assumption Violations of Linear Regression

(III.I-1)

with

(III.I-2)

(III.I-3)

which implies that the parameters are unbiased but inefficient.

A solution to this problem might be found by using Generalized Least Squares (GLS).

We know that

(III.I-4)

and therefore

(III.I-5)

In a first step we use the matrix P to transform y, X, and e as follows

(III.I-6)

It then follows

(III.I-7)

The unbiased GLS estimator can be written as

(III.I-8)

with

(III.I-9)

and an unbiased estimator for the variance

(III.I-10)

(III.I-11)

Suppose that

(III.I-12)

The log likelihood function is now equal to

(III.I-13)

Hence the GMLE estimator is

(III.I-14)

which is BLUE and

(III.I-15)

which is obviously biased. This problem can be solved by replacing T by T - K (cfr. eq. (III.I-10)).
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Heteroskedasticity
Autocorrelation
Nonlinearities
Nonnormality
Misspecification
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