TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 18

CHAPTER 35 Least Squares as the Normal Maximum Likelihood Estimate. Now assume ε is multivariate normal. We will show that in this case the OLS ˆ estimator β is at the same time the Maximum Likelihood Estimator. For this we need to write down the density function of y. | CHAPTER 35 Least Squares as the Normal Maximum Likelihood Estimate Now assume e is multivariate normal. We will show that in this case the OLS estimator is at the same time the Maximum Likelihood Estimator. For this we need to write down the density function of y. First look at one yt which is yt K1 N x ß 2 where X . xt is the tth row of X. It is written as a LxnJ column vector since we follow the column vector convention. The marginal 855 856 35. LEAST SQUARES AS THE NORMAL MAXIMUM LIKELIHOOD ESTIMATE density function for this one observation is fyt yt . V 2na2 Since the yi are stochastically independent their joint density function is the product which can be written as fy y 2tu2 - 2 exp -2-2 y - X0 T y - X . To compute the maximum likelihood estimator it is advantageous to start with the log likelihood function log fy y a2 - n log 2n - n log a2 - -12 y - X T y - X . 2 2 2a2 Assume for a moment that a2 is known. Then the MLE of is clearly equal to the OLS 3. Since 3 does not depend on a2 it is also the maximum likelihood estimate when a2 is unknown. is a linear function of y. Linear transformations of normal variables are normal. Normal distributions are characterized by their mean vector and covariance matrix. The distribution of the MLE of is therefore N a2 XTX -1 . 35. LEAST SQUARES AS THE NORMAL MAXIMUM LIKELIHOOD ESTIMATE 857 If we replace 3 in the log likelihood function by 3 we get what is called the log likelihood function with 3 concentrated out. log fy y 3 3 a2 - log2n - 2 log a2 - 2-2 y - X 3 T y - X 3 . One gets the maximum likelihood estimate of a2 by maximizing this concentrated log likelihoodfunction. Taking the derivative with respect to a2 consider a2 the name of a variable not the square of another variable one gets d n 1 1 da2 log fy y 3 - 2 a2 2a4 y- X 3 T y- X 3 Setting this zero gives a 2 y - X 3 y X3 . n n This is a scalar multiple of the unbiased estimate s2 eT n - k which we had earlier.

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