INTRODUCTION TO STATISTICS AND ECONOMETRICS
Maximum Likelihood Estimators
In this section we show that if we assume the normality of {ut} in the model
(10.1.1) , the least squares estimators a, (3, and a are also the maximum likelihood estimators.
The likelihood function of the parameters (that is, the joint density of Уъ Уъ ■ ■ ■ , Ут) is given by
1 , (10.2.78) / = 0 ■ ■ — exp t=l V2tt ct |
- (yt~ 0L - fix,)2 L 2ct2 j |
= (2ttct2) r/2exp |
- - Ц - Z(yt - a - |3x()2 |
2ct2 |
Taking the natural logarithm of both sides of (10.2.78), we have
(10.2.79) log L = log 2tt - ^ log ct2 - Z(yt - cl - fix,)2.
2 2 2a2
Since log L depends on a and (3 only via the last term of the right-hand side of (10.2.79), the maximum likelihood estimators of a and (3 are identical to the least squares estimators.
Inserting a and (3 into the right-hand side of (10.2.79), we obtain the so-called concentrated log-likelihood function, which depends only on ct2.
(10.2.80) log L* = log 2tt - ^ log ct2 - - Ц Ий2.
2 2 2<т
9
Differentiating (10.2.80) with respect to ct and equating the derivative to zero yields
Solving (10.2.81) for a yields the maximum likelihood estimator, which is identical to the least squares estimator a2. These results constitute a generalization of the results in Example 7.3.3.
In Section 12.2.5 we shall show that the least squares estimators a and (3 are best unbiased if ut) are normal.