Transcript Much estimation theory is presented in a rather ad hoc fashion

Maximum Likelihood
Much estimation theory is presented in a rather ad hoc fashion. Minimising
squared errors seems a good idea but why not minimise the absolute error or the
cube of the absolute error?
The answer is that there is an underlying approach which justifies a particular
minimisation strategy conditional on certain assumptions.
This is the maximum likelihood principle.
The idea is to assume a particular model with unknown parameters, we
can then define the probability of observing a given event conditional on
a particular set of parameters. We have observed a set of outcomes in the
real world. It is then possible to choose a set of parameters which are
most likely to have produced the observed results.
This is maximum likelihood. In most cases it is both consistent and
efficient. It provides a standard to compare other estimation techniques.
An example
Suppose we sample a set of goods for quality and find 5 defective items
in a sample of 10. What is our estimate of the proportion of bad items in
the whole population.
Intuitively of course it is 50%. Formally in a sample of size n the
probability of finding B bad items is
n!
n- B
B
P=
(1

)

B!(n - B)!
 is the proportion of bad items in the population
If the true proportion=0.1, P=0.0015, if it equals 0.2, P=0.0254 etc, we could
search for the most likely value. Or we can solve the problem analytically,

P
nˆ !
n- B 
B -1
ˆ ) 
ˆ (1 - 
= B

ˆ

 B! (n - B)!

n!
n- B -1
B
ˆ
ˆ (1 -  ) = 0
- (n - B)

B! (n - B)!
ˆ )  (n - B) 
ˆ )
ˆ (1 - 
ˆ (1 - 
B
B -1
n- B
n - B -1
B
=0
ˆ )
ˆ = (n - B)(1 - 
B
-1
-1
ˆ ) = (n - B)
ˆ
B(1 - 
ˆ = B/n = 5/10 = 0.5

So the maximum likelihood estimate of the population proportion of bad
items is 0.5.
This basic procedure can be applied in many cases, once we can define
the probability density function for a particular event we have a general
estimation strategy.
A general Statement
Consider a sample (X1...Xn) which is drawn from a probability distribution
P(X|A) where A are parameters. If the Xs are independent with probability
density function P(Xi|A) the joint probability of the whole set is
n
P( X 1 ... X n | A) =  P( X i | A)
i=1
this may be maximised with respect to A to give the maximum likelihood
estimates.
It is often convenient to work with the Log of the likelihood function.
n
log (L(A)) =
 log (P( X
i
| A))
i=1
the advantage of this approach is that it is extremely general but if the
model is misspecified it may be particularly sensitive to this
misspecification.
The Likelihood function for the general non linear model
if Y is a vector of n endogenous variables and
Y = f(X, ) + e
e ~ N(0, )
Then the likelihood function for one period is
1
-1

L(  , ) =
exp[-0.5(Y - f(X, )  (Y - f(X, )]
0.5
0.5
(2 ) | |
and dropping some constants and taking logs
log(L(  ,  )) =  |  | -(Y - f(X,  ) -1 (Y - f(X,  ))
if the covariance structure is constant and has zero off diagonal elements this
reduces to single equation OLS
Two important matrices
The efficient score matrix
 log(L(  ))
= S(A)

this is made up of the first derivatives at each point in time. It is a measure of
dispersion of the maximum estimate.
The information matrix (Hessian)
This is defined as
   2 log(L(  )) 
E - 
 = I(  )
    
This is a measure of how `pointy' the likelihood function is.
The variance of the parameters is given either by the inverse Hessian or the
outer product of the score matrix
-1
ˆ
Var( ˆ ML ) = [I(  ) ] = S(  )S(  )
The Cramer-Rao Lower Bound
This is an important theorem which establishes the superiority of the ML
estimate over all others. 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.
if  is another estimate of 
*
-1
ˆ
Var(  )  I(  )
*
this is the Cramer-Rao inequality.
Concentrating the Likelihood function
suppose we split the parameter vector into two sub vectors
L(  ) = L(  1 ,  2 )
now suppose we knew 1 then sometimes we can derive a formulae for the ML
estimate of 2, eg
 2 = g(  1 )
then we could write the LF as
L(  1 ,  2 ) = L(  1 , g(  1 )) = L (  1 )
*
this is the concentrated likelihood function.
This process is often very useful in practical estimation as it reduces the
number of parameters which need to be estimated.
An example of concentrating the LF
The likelihood function for a standard single variable normal non-linear
model is
L(  ) = -Tlog(  ) -  e / 
2
2
2
we can concentrate this with respect to the variance as follows, the FOC for a
maximum with respect to the variance is
L
2
2
2 2
= -T/  +  e /(  ) = 0
2

which implies that
 =  e /T
2
2
so the concentrated log likelihood becomes
2
(

)
=
Tlog(

e /T) - T
L

*
Prediction error decomposition
We assumed that the observations were independent in the statements
above. This will not generally be true especially in the presence of
lagged dependent variables. However the prediction error
decomposition allows us to extend standard ML procedures to
dynamic models.
From the basic definition of conditional probability
Pr( ,  ) = Pr( |  )Pr(  )
this may be applied directly to the likelihood function,
log(L( Y 1 ,Y 2 ,...Y T -1 ,Y T ))
= log(L( Y T | Y 1 ,Y 2 ,...,Y T -1 )) + log(L( Y 1 ,Y 2 ,...,Y T -1 ))
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
T -2
=  log(L( Y T -i | Y 1 ,...,Y T -i-1 )) + log(L( Y 1 ))
i=0
that is, a series of one step ahead prediction errors conditional on actual
lagged Y.
Testing hypothesis.
If a restriction on a model is acceptable this means that the reduction in
the likelihood value caused by imposing the restriction is not `significant'.
This gives us a very general basis for constructing hypothesis tests but
to implement the tests we need some definite metric to judge the tests
against, i.e. what is significant.
L
Lu
LR

*
Consider how the likelihood function changes as we move around the
parameter space, we can evaluate this by taking a Taylor series expansion
around the ML point
  log(L(  )) 
ˆ
ˆ
log(L(  )) = log(L(  )) + (  -  )




2

log(L(  ))  ˆ

ˆ
+ 0.5(  -  )
(  -  ) + O(1)
   
and of course
 log(L(  ))
= S(  ) = 0

 log(L(  )) = I(  )

2
So
log(L(  )) = log(L( ˆ )) + 0.5( ˆ -  )I(  )( ˆ -  )
r
r
r
ˆ
ˆ
ˆ
ˆ
log(L(  )) - log(L(  )) = 0.5.(  -  )I(  )(  -  )
it is possible to demonstrate that
r
2
ˆ
ˆ
ˆ

(  - r) I(  )(  -  ) ~  (m)
where m is the number of restrictions, and so
And so
r
2
ˆ
2[ log (L(  )) - log (L(  ))] ~  (m)
this gives us a measure for judging the significance of likelihood based
tests.
Three test procedures.
To construct the basic test we need an estimate of the likelihood value at
the unrestricted point and the restricted point and we compare these two.
There are three ways of deriving this.
The likelihood ratio test
we simply estimate the model twice, once unrestricted and once
restricted and compare the two.
The Wald test
This estimates only the unrestricted point and uses an estimate of the
second derivative to `guess' at the restricted point. Standard `t' tests are a
form of wald test.
The LaGrange multiplier test
this estimates only the restricted model and again uses an estimate of the
second derivatives to guess at the restricted point.
L
Lu
LR

*
If the likelihood function were quadratic then LR=LM=W. In general
however W>LR>LM
A special form of the LM test
The LM test can be calculated in a particularly convenient way under
certain circumstances.
The general form of the LM test is
2
-1

LM = S(  ) [I(  ) ] S(  ) ~  (m)
Now suppose
Y t = f( X t ,  1 ,  2 ) + et
where we assume that the subset of parameters 1 is fixed according to a
set of restrictions g=0 (G is the derivative of this restriction).
Now
S(  1 ) =  Ge
-2
I(  1 ) = (  E(GG) )
-1
-2
and so the LM test becomes
 eG[  E(GG) ]  Ge
-2
-2
-1
-2
And
if E(GG) = GG
-2
LM =  ee
which may be interpreted as TR2 from a regression of e on G
This is used in many tests for serial correlation heteroskedasticity
functional form etc.
e is the actual errors from a restricted model and G is the restrictions in
the model.
An Example: Serial correlation
Suppose
Y = X + u
u =  u-1 + e
The restriction that = 0 may be tested as an LM test as follows
estimate the model without serial correlation. save the residuals u. then
estimate the model
m
uˆ = X +  uˆt-i
i=1
then TR2 from this regression is an LM(m) test for serial correlation
Quasi Maximum Likelihood
ML rests on the assumption that the errors follow a particular distribution
(OLS is only ML if the errors are normal etc.) What happens if we make
the wrong assumption.
White(1982) Econometrica, 50,1,pg1. demonstrates that, under very broad
assumptions about the misspecification of the error process, ML is still a
consistent estimator. The estimation is then referred to as Quasi
Maximum Likelihood.
But the covariance matrix is no longer the standard ML one instead it is
given by
-1
-1
ˆ
ˆ
ˆ
ˆ
ˆ
C(  ) = I(  ) [S(  )S(  )]I(  )
Generally we may construct valid Wald and LM tests by using this
corrected covariance matrix but the LR test is invalid as it works directly
from the value of the likelihood function.
Numerical optimisation
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.
there are many algorithms and many computer programmes implement
these for you.
It is useful to understand the broad steps of the procedure.
1. set an arbitrary initial set of parameters.
2. determine a direction of movement
3. determine a step length to move
4. examine some termination criteria and either stop or go back to 2.
L
Lu

1

2

*
Important classes of maximisation techniques.
Gradient methods. These base the direction of movement on the first
derivatives of the LF with respect to the parameters. Often the step length
is also determined by (an approximation to) the second derivatives. So
-1
  L   L 
  
 i+1 =  i + 
2 
      
2
These include, Newton, Quasi Newton, Scoring, Steepest descent,
Davidson Fletcher Powel, BHHH etc.
Derivative free techniques. These do not use derivatives and so they are
less efficient but more robust to extreme non-linearity’s. e.g. Powell or
non-linear Simplex.
These techniques can all be sensitive to starting values and `tuning'
parameters.
Some special LFs
Qualitative response models.
These are where we have only partial information (insects and poison) in
one form or another.
We assume an underlying continuous model,
Y t =  X t + ut
but we only observe certain limited information, eg z=1 or 0 related to y
z = 1 if Y > 0
z = 0 if Y < 0
then we can group the data into two groups and form a likelihood function
with the following form
L =  F(- X t )  F(1 -  X t )
z=0
z=1
where F is a particular density function eg. the standard normal
Cumulative function or perhaps the logistic (logit model) function
ARCH and GARCH
These are an important class of models which have time varying
variances
Suppose
Y t =  X t + et
et ~ N(0, ht )
2
ht =  0 +  1 ht -1 +  2 et -1
then the likelihood function for this model is
T
log(L(  , )) =  - | ht | -( e / ht )
2
t
t=1
which is a specialisation of the general Normal LF with a time varying
variance.
An alternative approach
Method of moments
A widely used technique in estimation is the Generalised Method of
Moments (GMM), This is an extension of the standard method of
moments.
The idea here is that if we have random drawings from an unknown
probability distribution then the sample statistics we calculate will
converge in probability to some constant. This constant will be a function
of the unknown parameters of the distribution. If we want to estimate k of
these parameters,
1 ,..., k
we compute k statistics (or moments) whose probability limits are known
functions of the parameters
m1 ,..., mk
These k moments are set equal to the function which generates the
moments and the function is inverted.
  f (m)
1
A simple example
Suppose the first moment (the mean) is generated by the following
distribution, f ( x |  ) . The observed moment from a sample of n
1
observations is
n
m1  (1/ n) xi
i 1
Hence
m1  f ( x | 1 )
And
1  f (m1 )  m1
1
Method of Moments Estimation (MM)
This is a direct extension of the method of moments into a much more
useful setting.
The idea here is that we have a model which implies certain things about
the distribution or covariance’s of the variables and the errors. So we
know what some moments of the distribution should be. We then invert
the model to give us estimates of the unknown parameters of the model
which match the theoretical moments for a given sample.
So suppose we have a model
Y  f ( , X )
where  are k parameters. And we have k conditions (or moments)
which should be met by the model.
E ( g (Y , X |  ))  0
then we approximate E(g) with a sample measure and invert g.
  g (Y , X ,0)
1
Examples
OLS
In OLS estimation we make the assumption that the regressors (Xs) are
orthogonal to the errors. Thus
E ( Xe)  0
The sample analogue for each xi is
(1 / n)t 1 xit et  0
n
and so
(1/ n)t 1 xit et  0  (1/ n)t 1 xit ( yt  xt '  )
n
n
and so the method of moments estimator in this case is the value of 
which simultaneously solves these i equations. This will be identical to
the OLS estimate.
Maximum likelihood as an MM estimator
In maximum likelihood we have a general likelihood function.
Ln( L)   Ln( f ( y, x |  ))
and this will be maximised when the following k first order conditions are
met.
E ( ln( f ( y, x |  )) /  )  0
This give rise to the following k sample conditions
(1/ n)i1 ( ln( f ( y, x |  )) /  )  0
n

Simultaneously solving these equations for
maximum likelihood.
gives the MM equivalent of
Generalised Method of Moments (GMM)
In the previous conditions there are as many moments as unknown
parameters, so the parameters are uniquely and exactly determined. If
there were less moment conditions we would not be able to solve them
for a unique set of parameters (the model would be under identified). If
there are more moment conditions than parameters then all the
conditions can not be met at the same time, the model is over identified
and we have GMM estimation.
Basically, if we can not satisfy all the conditions at the same time we have
to trade them of against each other. So we need to make them all as close
to zero as possible at the same time. We need a criterion function to
minimise.
Suppose we have k parameters but L moment conditions L>k.
E (m j ( ))  0  (1 / n)t 1 m j ( )  0
n
j  1,...L
Then we need to make all L moments as small as possible
simultaneously. One way is a weighted least squares criterion.
Min(q)  m ( )' Am ( )
That is, the weighted squared sum of the moments.
This gives a consistent estimator for any positive definite matrix A (not a
function of
)

The optimal A
If any weighting matrix is consistent they clearly can not all be equally
efficient so what is the optimal estimate of A.
Hansen(1982) established the basic properties of the optimal A and how
to construct the covariance of the parameter estimates.
The optimal A is simply the covariance matrix of the moment conditions.
(just as in GLS)
Thus
optimal A  W  asy. var(m )
The parameters which solve this criterion function then have the
following properties.
 gmm ~ N (0,Vgmm )
Where
Vgmm  (1/ n)(G'  G)
1
1
where G is the matrix of derivatives of the moments with respect to the
parameters and
  var(n (m  ))
1/ 2
 is the true moment value.
Conclusion
• Both ML and GMM are very flexible
estimation strategies
• They are equivalent ways of
approaching the same problem in many
instances.