Language Reference: FARMACOV Call :: SAS/IML(R) 9.2 User's Guide

Language Reference

FARMACOV Call

computes the autocovariance function for an ARFIMA() process

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

The inputs to the FARMACOV subroutine are as follows:

d: specifies a fractional differencing order. The value of must be in the open interval excluding zero. This input is required.
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. All the roots of $\phi(b)=0$ should be greater than one in absolute value, where $\phi(b)$ is the finite-order matrix polynomial in the backshift operator , such that $b^j y_{t}=y_{t-j}$ .
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.
p: specifies the subset of the AR order. The quantity is defined as the number of elements of phi.

If you do not specify p, the default subset is p $=\{1,2, ... ,m_p\}$ .

For example, consider phi=0.5.

If you specify p=1 (the default), the FARMACOV subroutine computes the theoretical autocovariance function of an ARFIMA() process as $y_t = 0.5 y_{t-1} + \epsilon_t.$

If you specify p=2, the FARMACOV subroutine computes the autocovariance function of an ARFIMA() process as $y_t = 0.5 y_{t-2} + \epsilon_t.$
q: specifies the subset of the MA order. The quantity is defined as the number of elements of theta.

If you do not specify q, the default subset is q $=\{1,2, ... ,m_q\}$ .

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 .

The FARMACOV subroutine returns the following value:

cov: is a vector containing the autocovariance function of an ARFIMA() process.

Consider the following ARFIMA(

) process:

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

In this process, $\epsilon_t \sim nid(0, 1.2)$ . To compute the autocovariance of this process, you can use the following code:

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

For $d\in (-0.5,0.5)\backslash \{0\}$ , the series $y_{t}$ represented as $(1-b)^dy_{t} = {\epsilon}_t$ is a stationary and invertible ARFIMA(

) process with the autocovariance function

$\gamma_k = e(y_{t}y_{t-k}) = { {(-1)^k \gamma (-2d+1)} \over {\gamma (k-d+1)\gamma (-k-d+1) }}$

and the autocorrelation function

$\rho_k = {\gamma_k \over \gamma_0} = {{ \gamma (-d+1)\gamma (k+d)} \over {\gam... ... (k-d+1) }} \sim { \gamma (-d+1) \over {\gamma (d)}} k^{2d-1}, karrow \infty$

Notice that $\rho_k$ decays hyperbolically as the lag increases, rather than showing the exponential decay of the autocorrelation function of a stationary ARMA(

) process.

The FARMACOV subroutine computes the autocovariance function of an ARFIMA(

) process.

For $d\in (0.5,0.5)\backslash \{0\}$ , the series $y_{t}$ is a stationary and invertible ARFIMA(

) process represented as

$\phi(b)(1-b)^dy_t = \theta(b){\epsilon}_t$

where $\phi(b)=1-\phi_1b-\phi_2b^2 - ... - \phi_pb^p$ and $\theta(b)=1-\theta_1b-\theta_2b^2 - ... - \theta_qb^q$ and ${\epsilon}_t$ is a white noise process; all the roots of the characteristic AR and MA polynomial lie outside the unit circle.

Let $x_t = \theta(b)^{-1}\phi(b)y_t$ , so that

follows an ARFIMA(

) process; let

, so that

follows an ARMA(

) process; let $\gamma_k^x$ be the autocovariance function of $\{x_t\}$ and $\gamma_k^z$ be the autocovariance function of $\{z_t\}$ .

Then the autocovariance function of $\{y_t\}$ is as follows:

$\gamma_k = \sum_{j=-\infty}^{j=\infty} \gamma_j^z\gamma_{k-j}^x$

The explicit form of the autocovariance function of $\{y_t\}$ is given by Sowell (1992, p. 175).

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