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

Language Reference

FARMASIM Call

generates an ARFIMA() process

CALL FARMASIM( series, d <, phi, theta, mu, sigma, n, p, q, initial, seed>);

The inputs to the FARMASIM subroutine are as follows:

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.
mu: specifies a mean value. The default is zero.
sigma: specifies a variance of the innovation series. The default is one.
n: specifies the length of the series. The value of should be greater than or equal to the AR order. The default is is used.
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.
initial: specifies the initial values of random variables. The initial value is used for the nonstationary process. If , then $y_{-p+1}, ... ,y_{0}$ take the same value . If the initial option is not specified, the initial values are set to zero.
seed: is a scalar containing the random number seed. At the first execution of the subroutine, the seed variable is used as follows:

If seed > 0, the input seed is used for generating the series.

If seed = 0, the system clock is used to generate the seed.

If seed < 0, the value (-1)(seed) is used for generating the series.

If the seed is not supplied, the system clock is used to generate the seed.

On subsequent calls of the subroutine in the DO loop - like environment, the seed variable is used as follows: If seed > 0, the seed remains unchanged. In other cases, after each execution of the subroutine, the current seed is updated internally.

The FARMASIM subroutine returns the following value:

series: is an vector containing the generated ARFIMA() process.

Consider the following ARFIMA(

) process:

$(1-0.5b)(1-b)^{0.3}(y_t-10) = (1+0.1b){\epsilon}_t$

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

  
    d    = 0.3; 
    phi  = 0.5; 
    theta= -0.1; 
    mu   = 10; 
    sigma= 1.2; 
    call farmasim(yt, d, phi, theta, mu, sigma, 100); 
    print yt;

The FARMASIM subroutine generates a time series of length from an ARFIMA() model. If the process is stationary and invertible, the initial values $y_{-p+1}, ... ,y_{0}$ are produced by using covariance matrices obtained from FARMACOV. If the process is nonstationary, the time series is recursively generated by using the user-defined initial value or the zero initial value.

To generate an ARFIMA() process with $d\in [0.5,1)$ , is first generated for $d'\in (-0.5, 0)$ , where and then $y_{t}$ is generated by $y_t = y_{t-1} + x_t$ .

To generate an ARFIMA() process with $d\in (-1,-0.5]$ , a two-step approximation based on a truncation of the expansion is used; the first step is to generate an ARFIMA() process $x_t=(1-b)^{-d}{\epsilon}_t$ , with truncated moving-average weights; the second step is to generate $y_t =\phi(b)^{-1}\theta(b)x_t$ .

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