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

Language Reference

FARMAFIT Call

estimate the parameters of an ARFIMA() model

CALL FARMAFIT( d, phi, theta, sigma, series <, p, q, opt>);

The inputs to the FARMAFIT subroutine are as follows:

series

specifies a time series (assuming mean zero).

p

specifies the set or subset of the AR order. If you do not specify p, the default is p=0.

If you specify p=3, the FARMAFIT subroutine estimates the coefficient of the lagged variable $y_{t-3}$ .

If you specify p= $\{1,2,3\}$ , the FARMAFIT subroutine estimates the coefficients of lagged variables $y_{t-1}$ , $y_{t-2}$ , and $y_{t-3}$ .

q

specifies the subset of the MA order. If you do not specify q, the default is q=0.

If you specify q=2, the FARMAFIT subroutine estimates the coefficient of the lagged variable ${\epsilon}_{t-2}$ .

If you specify q= $\{1,2\}$ , the FARMAFIT subroutine estimates the coefficients of lagged variables ${\epsilon}_{t-1}$ and ${\epsilon}_{t-2}$ .

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 FARMAFIT subroutine returns the following values:

d: is a scalar containing a fractional differencing order.
phi: is a vector containing the autoregressive coefficients.
theta: is a vector containing the moving-average coefficients.
sigma: is a scalar containing a variance of the innovation series.

Consider the following ARFIMA(

) model:

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

In this model, ${\epsilon}_t \sim nid(0,1)$ . To estimate the parameters of this model, you can use the following code:

  
    d    = 0.3; 
    phi  = 0.5; 
    theta= -0.1; 
    call farmasim(yt, d, phi, theta); 
    call farmafit(d, ar, ma, sigma, yt) p=1 q=1; 
    print d ar ma sigma;

The FARMAFIT subroutine estimates parameters , $\phi(b)$ , $\theta(b)$ , and $\sigma_{\epsilon}^2$ of an ARFIMA() model. The log-likelihood function needs to be solved by iterative numerical procedures such as the quasi-Newton optimization. The starting value is obtained by the approach of Geweke and Porter-Hudak (1983); the starting values of the AR and MA parameters are obtained from the least squares estimates.

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