Language Reference: FARMACOV Call

FARMACOV Call

CALL FARMACOV( cov, d <*>, phi <*>, theta <*>, sigma <*>, p <*>, q <*>, lag ) ;

The FARMACOV subroutine computes the autocovariance function for an autoregressive fractionally integrated moving average (ARFIMA) model of the form ARFIMA( $\text{[math]}$ ).

The input arguments to the FARMACOV subroutine are as follows:

d

specifies a fractional differencing order. The value of $\text{[math]}$ must be in the open interval $\text{[math]}$ excluding zero. This input is required.

phi

specifies an $\text{[math]}$ -dimensional vector that contains the autoregressive coefficients, where $\text{[math]}$ is the number of the elements in the subset of the AR order. The default is zero. All the roots of $\text{[math]}$ should be greater than one in absolute value, where $\text{[math]}$ is the finite-order matrix polynomial in the backshift operator $\text{[math]}$ , such that $\text{[math]}$ .

theta

specifies an $\text{[math]}$ -dimensional vector that contains the moving average coefficients, where $\text{[math]}$ is the number of the elements in the subset of the MA order. The default is zero.

p

specifies the subset of the AR order. The quantity $\text{[math]}$ is defined as the number of elements of phi.

If you do not specify p, the default subset is p $\text{[math]}$ .

For example, consider phi=0.5.

If you specify p=1 (the default), the FARMACOV subroutine computes the theoretical autocovariance function of an ARFIMA( $\text{[math]}$ ) process as $\text{[math]}$

If you specify p=2, the FARMACOV subroutine computes the autocovariance function of an ARFIMA( $\text{[math]}$ ) process as $\text{[math]}$

q

specifies the subset of the MA order. The quantity $\text{[math]}$ is defined as the number of elements of theta.

If you do not specify q, The default subset is q $\text{[math]}$ .

The usage of q is the same as that of p.

lag

specifies the length of lags, which must be a positive number. The default is lag $\text{[math]}$ .

The FARMACOV subroutine returns the following value:

cov: is a lag $\text{[math]}$ vector that contains the autocovariance function of an ARFIMA( $\text{[math]}$ ) process.

As an example, consider the following ARFIMA( $\text{[math]}$ ) process:

$\text{[math]}$

In this process, $\text{[math]}$ . The following statements compute the autocovariance of this process:

d = 0.3;
phi = 0.5;
theta = -0.1;
sigma = 1.2;
call farmacov(cov, d, phi, theta, sigma) lag=5;
print cov;

Figure 23.109 Autocovariance of an ARFIMA Process

cov
4.2493033
3.5806774
2.9152846
2.4381017
2.1068697
1.8743199

For $\text{[math]}$ , the series $\text{[math]}$ represented as $\text{[math]}$ is a stationary and invertible ARFIMA( $\text{[math]}$ ) process with the autocovariance function

$\text{[math]}$

and the autocorrelation function

$\text{[math]}$

Notice that $\text{[math]}$ decays hyperbolically as the lag increases, rather than showing the exponential decay of the autocorrelation function of a stationary ARMA( $\text{[math]}$ ) process.

For $\text{[math]}$ , the series $\text{[math]}$ is a stationary and invertible ARFIMA( $\text{[math]}$ ) process represented as

$\text{[math]}$

where $\text{[math]}$ and $\text{[math]}$ and $\text{[math]}$ is a white noise process; all the roots of the characteristic AR and MA polynomial lie outside the unit circle.

Let $\text{[math]}$ , so that $\text{[math]}$ follows an ARFIMA( $\text{[math]}$ ) process; let $\text{[math]}$ , so that $\text{[math]}$ follows an ARMA( $\text{[math]}$ ) process; let $\text{[math]}$ be the autocovariance function of $\text{[math]}$ and $\text{[math]}$ be the autocovariance function of $\text{[math]}$ .

Then the autocovariance function of $\text{[math]}$ is as follows:

$\text{[math]}$

The explicit form of the autocovariance function of $\text{[math]}$ is given by Sowell (1992).