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Maximum Likelihood Estimation (MLE) for Multiple Regression

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II.II.2 Maximum Likelihood Estimation (MLE) for Multiple Regression

MLE is needed when one introduces the following assumptions


(in this work we only focus on the use of MLE in cases where y and e are normally distributed).

The pdf of y is given by


and the log likelihood function


Maximizing the log likelihood function is (here) equivalent to minimizing the SSR


Therefore it follows


and of course


so that the ML estimator of ß is a BLUE and also a best unbiased estimator (for nonlinear models).

The ML estimator for the variance parameter however is biased


so that we have to use


The large sample properties of the ML estimator can be deduced on using a Taylor expansion of the likelihood around the true parameter value


This expression can be shown to be true under the so-called regularity condition which implies that the information matrix times 1/T converges to a probability matrix in the limit


Now it follows from (II.II.2-9) that


From (II.II.2-1)it can be seen that y is identically and independently distributed. Thus the pdf can be written as a derivative of the log likelihood


and from the derivation (c.q. proof) of the Cramér-Rao lower bound it follows that each of the T observations has a zero expected value and finite variance. Therefore


According to (II.II.2-10) and the central limit theorem it can be shown that


Using (II.II.2-11) and (II.II.2-14) it is easily derived that


Applying Cramér's theorem (I.VI-36) and (I.VI-37), proves that


(Q.E.D.) since


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Multiple Regression
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