Linear regression - Maximum Likelihood Estimation

This lecture shows how to perform maximum likelihood estimation of the parameters of a Normal Linear Regression Model, that is, of a linear regression model whose error terms are normally distributed conditional on the regressors.

In order to fully understand the material presented in this lecture, it might be useful to revise the lectures on Maximum likelihood estimation and on the Normal Linear Regression Model.

Assumptions

The objective is to estimate the parameters of the linear regression model

where is the dependent variable, is a vector of regressors, is the vector of regression coefficients to be estimated and is an unobservable error term.

We assume that our sample is made up of IID observations .

The regression equations can be written in matrix form as

where the vector of observations of the dependent variable is denoted by , the matrix of regressors is denoted by , and the vector of error terms is denoted by .

We also assume that the vector of errors has a multivariate normal distribution conditional on , with mean equal to and covariance matrix equal towhere is the identity matrix and

Note that also is a parameter to be estimated.

Furthermore, it is assumed that the matrix of regressors has full-rank.

The assumption that the covariance matrix of is diagonal implies that the entries of are mutually independent (i.e., is independent of for .). Moreover, they all have a normal distribution with mean and variance .

By the properties of linear transformations of normal random variables, we have that also the dependent variable is conditionally normal, with mean and variance . Therefore, the conditional probability density function of the dependent variable is

The likelihood function

The likelihood function is

Proof

Since the observations from the sample are independent, the likelihood of the sample is equal to the product of the likelihoods of the single observations:

The log-likelihood function

The log-likelihood function is

Proof

It is obtained by taking the natural logarithm of the likelihood function:

The maximum likelihood estimators

The maximum likelihood estimators of the regression coefficients and of the variance of the error terms are

Proof

The estimators solve the following maximization problem The first-order conditions for a maximum are where indicates the gradient calculated with respect to , that is, the vector of the partial derivatives of the log-likelihood with respect to the entries of . The gradient is which is equal to zero only ifTherefore, the first of the two equations is satisfied if where we have used the assumption that has full rank and, as a consequence, is invertible. The partial derivative of the log-likelihood with respect to the variance is which, if we assume , is equal to zero only ifThus, the system of first order conditions is solved byNote that does not depend on , so that this is an explicit solution.

Thus, the maximum likelihood estimators are:

1. for the regression coefficients, the usual OLS estimator;

2. for the variance of the error terms, the unadjusted sample variance of the residuals .

Asymptotic variance

The vector of parametersis asymptotically normal with asymptotic mean equal toand asymptotic covariance matrix equal to

Proof

The first entries of the score vector areThe -th entry of the score vector isThe Hessian, that is, the matrix of second derivatives, can be written as a block matrix Let us compute the blocks:andFinally, Therefore, the Hessian isBy the information equality, we have thatBut and, by the Law of Iterated Expectations,Thus,As a consequence, the asymptotic covariance matrix is

This means that the probability distribution of the vector of parameter estimates can be approximated by a multivariate normal distribution with mean and covariance matrix

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