FARMALIK Call

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Fractionally Integrated Time Series Analysis

FARMALIK Call

computes the log-likelihood function of an ARFIMA(p,d,q) 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 d should be in the open interval (-1,1) excluding zero.

phi

specifies an m_p-dimensional vector containing 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 containing 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.

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 3-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 -0.5× (lnl[2]+lnl[3]). If the opt=0 is specified, only the weighted sum of squares of residuals returns in lnl[1].

To compute the log-likelihood function of an ARFIMA(1,0.3,1) model

$(1-0.5B)(1-B)^{0.3}y_t=(1+0.1B){\epsilon}_t$

where $\epsilon_t \sim NID(0, 1.2)$ ,you can specify

 
   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(p,d,q) model. The exact log-likelihood function is worked by Sowell (1992); the conditional sum of squares function is worked 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 = [y₁,y₂, ... ,y_T ]' 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 $\Sigma=[ \gamma_{i-j} ]$ for i,j = 1,2, ... ,T.

The conditional sum of squares function does not require the normality assumption. The initial observations y₀, y_-1, ... and ${\epsilon}_0$ , ${\epsilon}_{-1}, ...$ 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

$\ell=-{T \over 2}\log ( {1 \over T}\sum_{t=1}^T{\epsilon}_t^2 )$

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