FARMALIK Call
computes the log-likelihood function of an ARFIMA(
) model
- CALL FARMALIK( lnl, series, d <, phi, theta, sigma,
p, q, opt>);
The inputs to the FARMALIK subroutine are as follows:
- series
- specifies a time series (assuming mean zero).
- d
- specifies a fractional differencing order.
This argument is required;
the value of
should be in the open interval
excluding zero.
- phi
- specifies an
-dimensional vector
containing the autoregressive coefficients,
where
is the number of the elements in the subset
of the AR order. The default is zero.
- theta
- specifies an
-dimensional vector
containing the moving-average coefficients,
where
is the number of the elements in the subset
of the MA order. The default is zero.
- sigma
- specifies a variance of the innovation series. The default is one.
- p
- specifies the subset of the AR order.
See the FARMACOV subroutine for additional details.
- q
- specifies the subset of the MA order.
See the FARMACOV subroutine for additional details.
- opt
- specifies the method of computing the log-likelihood function.
- opt=0
- requests the conditional sum of squares function.
This is the default.
- opt=1
- requests the exact log-likelihood function.
This option requires that the time series be stationary
and invertible.
The FARMALIK subroutine returns the following value:
- lnl
- is a three-dimensional vector. lnl[1] contains
the log-likelihood function of the model; lnl[2] contains
the sum of the log determinant of the innovation variance;
and lnl[3] contains the weighted
sum of squares of residuals. The log-likelihood function is computed
as
(lnl[2]+lnl[3]).
If the opt=0 is specified, only the weighted
sum of squares of residuals returns in lnl[1].
Consider the following ARFIMA(

) model:

In this model,

.
To compute the log-likelihood function of this model,
you can use the following code:
d = 0.3;
phi = 0.5;
theta= -0.1;
sigma= 1.2;
call farmasim(yt, d, phi, theta, sigma);
call farmalik(lnl, yt, d, phi, theta, sigma);
print lnl;
The FARMALIK subroutine computes a log-likelihood function of
the ARFIMA(

) model.
The exact log-likelihood function was proposed by Sowell (1992);
the conditional sum of squares function was proposed by Chung (1996).
The exact log-likelihood function only considers
a stationary and invertible
ARFIMA(

) process with

represented as

where

.
Let
![y_t=[y_1,y_2, ... ,y_t ]'](images/langref_langrefeq302.gif)
and
the log-likelihood function is as follows without a constant term:

where
![\sigma = [ \gamma_{i-j} ]](images/langref_langrefeq304.gif)
for

.
The conditional sum of squares function does
not require the normality assumption. The initial observations

,

and

,

are set to zero.
Let

be an ARFIMA(

) process represented as

then the conditional sum of squares function is

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