1. M.Sc. in Economics Econometrics Module I Topic 4: Maximum Likelihood Estimation Carol Newman

2. Maximum Likelihood Estimation  Assumes probability distribution of function of the dependent variable is known except for a finite number of unknown parameters that we would like to estimate  MLE will produce the most efficient estimator but if the probability distribution is mis-specified estimator is inconsistent  Define probability density function for y which will depend on unknown parameters :  Describes data generating process that underlies the sample data  Joint pdf of n independently and identically distributed observations will be the produce of the individual densities:  This is the likelihood function

3. Maximum Likelihood Estimation  The likelihood principle: Choose estimator for the parameter  that maximises the likelihood of observing the actual sample given the probability density function for y  Usually work with the log-likelihood function and usually require that we condition on exogenous variables in X and a vector of parameters   This is referred to as the conditional log-likelihood equation  Application to the General Linear Model (see homework)

4. Maximum Likelihood Estimation  Asymptotic properties of the ML estimator Consistency Asymptotic efficiency (smallest variance among all consistent asymptotically normal estimators) Invariance: If is the ML estimator of and if is a continuous function, the ML estimator of is  Note: estimate of the Asymptotic Variance of the ML Estimator is the inverse of the second order derivatives of the likelihood equation

5. Maximum Likelihood Estimation  Key Terms Likelihood principle Conditional likelihood equation Gradient vector/score vector Hessian matrix Information matrix Regularity conditions MLE will produce the most efficient estimator of all consistent asymptotically normal estimators Inconsistent if likelihood equation is mis-specified Property of invariance allow re-parameterisation for simplification of estimation

6. Three Specification Testing Procedures

7. Wald Test Large values of this test statistic are inconsistent with the null hypothesis Only requires that you estimate the unrestricted model

8. Likelihood Ratio Test Estimate restricted and unrestricted model using MLE Compute the test statistic: where: Requires that you estimate the restricted and unrestricted models

9. Lagrange Multiplier Test Approach one: If the null hypothesis is true and the restrictions are valid then the lagrange multipliers associated with them will be close to zero Maximize subject to Lagrangean: LM Test statistic: Only requires that you estimate the restricted model

10. Lagrange Multiplier Test Approach two: Test whether the derivatives of the likelihood equation are close to zero when evaluated at the restricted estimates