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Maximum Likelihood(ML) Estimation

The basic idea

Assume a particular model with unknown

parameters. Determine how the likelihood of a

given event varies with model model

parameters Choose the parameter values that

maximize the likelihood of the observed event

A general mathematical formulation

Consider a sample (X1, ..., Xn) which is drawn

from a probability distribution P(X?) where ?

are parameters. If the Xs are independent with

probability density function P(Xi?) then the

joint probability of the whole set is Find

the parameters that maximize this function

- Consider a sample (X1, ..., Xn) which is drawn

from a probability distribution P(XA) where A

are parameters. - If the Xs are independent with probability

density function P(XiA) the joint probability of

the whole set is

The likelihood function for the general

non-linear model

Assume that Then the likelihood function

is Note that the ML-estimator of ? is

identical to the mean square estimator if ?

?2I, where I is the identity matrix.

Large sample properties of ML estimators

Consistency As the sample size increases, the

ML estimator converges to the true parameter

value Invaríance If f(?) is a function of the

unknown parameters of the distribution, then the

ML estimator of f(?) is f( ) Asymptotic

normality As the sampe size increases, the

sampling distribution of an ML estimator

converges to a normal distribution Variance For

large sample sizes, the variance of an ML

estimator (assuming a single unknown parameter)

is approximately the negative of the reciprocal

of the second derivative of the log-likelihood

function evaluated at the ML estimate. Note

that the ML-estimator of ? is identical to the

mean square estimator if ? ?2I, where I is the

identity matrix.

The information matrix (Hessian)

- The matrix
- is a measure of how pointy' the likelihood

function is. - The variance of the ML estimator is given by the

inverse Hessian

The Cramer-Rao lower bound

- The Cramer-Rao lower bound is the smallest

theoretical variance which can be achieved. - ML gives this, so any other estimation technique

can at best only equal it. - Do we need estimators other than ML estimators?

ML estimators for dynamic models

- A general decomposition technique for the log

likelihood function allows us to extend standard

ML procedures to dynamic models (time series

models). - From the basic definition of conditional

probability - This may be applied directly to the likelihood

function

Prediction error decomposition

- Consider the decomposition
- The first term is the conditional probability of

Y given all past values. We can then condition

the second term and so on to give - that is, a series of one step ahead prediction

errors conditional on actual lagged Y.

Numerical optimization

- In simple cases (e.g. OLS) we can calculate the

maximum likelihood estimates analytically. - But in many cases we cannot, then we resort to

numerical optimisation of the likelihood

function. - This amounts to hill climbing in parameter space.

- set an arbitrary initial set of parameters.
- determine a direction of movement
- determine a step length to move
- examine some termination criteria and either stop

or go back to 2

L

Lu

Gradient methods for determining the maximum of a

function

- These methods base the direction of movement on

the first derivatives of the likelihood function

with respect to the parameters. - Often the step length is also determined by (an

approximation to) the second derivatives. So - The class of gradient methods include Newton,

Quasi Newton, Steepest descent etc.

Qualitative response models

- Assume that we have a quantitative model
- but we only observe certain limited information,

e.g. - Then we can group the data into two groups and

form a likelihood function with the following

form - where F is the cumulative distribution function

of the error terms ut

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