Language Reference

FARMALIK Call

CALL FARMALIK (lnl, series, d <*>, phi <*>, theta <*>, sigma <*>, p <*>, q <*>, opt );

The FARMALIK subroutine evaluates the log-likelihood function of an ARFIMA($p,d,q$) model for a given time series.

The input arguments 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 d should be in the open interval $(-1,1)$ excluding zero.

phi

specifies an $m_ p$-dimensional vector that contains the autoregressive coefficients, where $m_ p$ is the number of the elements in the subset of the AR order. The default is zero.

theta

specifies an $m_ q$-dimensional vector that contains the moving average coefficients, where $m_ q$ 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. The following are valid values:

0

requests the conditional sum of squares function. This is the default.

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. If opt=0 is specified, the conditional sum of squares function is evaluated and the result returns in lnl[1]. Otherwise, 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 $-0.5\times $ (lnl[2]+lnl[3]).

As an example, consider the following ARFIMA($1,0.3,1$) model:

\[ (1-0.5B)(1-B)^{0.3}y_ t = (1+0.1B){\epsilon }_ t \]

In this model, ${\epsilon }_ t \sim NID(0,1.2)$. The following statements compute the log-likelihood function of this model: 

d = 0.3;
phi = 0.5;
theta = -0.1;
sigma = 1.2;
call farmasim(yt, d, phi, theta, sigma) seed=1234;
call farmalik(lnl, yt, d, phi, theta, sigma);
print (lnl[1])[label="Conditional Sum of Squares"];

Figure 25.132: Log-Likelihood for an ARFIMA Model

Conditional Sum
of Squares
-16.67587 




            

            


            
The FARMALIK subroutine computes a log-likelihood function of the ARFIMA($p,d,q$) 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($p,d,q$) process with $d\in (-0.5,0.5)\backslash \{ 0\} $ represented as 
            

            

               
\[ \phi (B)(1-B)^ dy_ t = \theta (B){\epsilon }_ t \]

            

            
 where ${\epsilon }_ t \sim NID(0,\sigma ^2)$. 
            

            
Let $Y_ T=\left[y_1,y_2,\ldots ,y_ T \right]’$ and the log-likelihood function is as follows without a constant term: 
            

            

               
\[ \ell = -{1 \over 2} (\log |\Sigma | + Y_ T’\Sigma ^{-1}Y_ T ) \]

            

            
 where $\Sigma = \left[ \gamma _{i-j} \right]$ for $i,j=1,2,\ldots ,T$. 
            

            
The conditional sum of squares function does not require the normality assumption. The initial observations $y_0$, $y_{-1}, \ldots $ and ${\epsilon }_0$, ${\epsilon }_{-1}, \ldots $ are set to zero. 
            

            
Let $y_ t$ be an ARFIMA($p,d,q$) process represented as 
            

            

               
\[ \phi (B)(1-B)^ dy_ t = \theta (B){\epsilon }_ t \]

            

            
 Then the conditional sum of squares function is 

            

               
\[ \ell = -{T \over 2}\log \left( {1 \over T}\sum _{t=1}^ T{\epsilon }_ t^2 \right) \]