Language Reference

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($p,d,q$).

The input arguments to the FARMACOV subroutine are as follows:

d

specifies a fractional differencing order. The value of d must be in the open interval $(-0.5,0.5)$ excluding zero. This input is required.

phi

specifies an $m_ p$-dimensional vector that contains the autoregressive coefficients, where $m_ p$ 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 B, such that $B^ j y_{t}=y_{t-j}$.

theta

specifies an $m_ q$-dimensional vector that contains the moving average coefficients, where $m_ q$ 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 $m_ p$ is defined as the number of elements of phi.

If you do not specify p, the default subset is p$=\{ 1,2,\ldots ,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($1,d,0$) 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($2,d,0$) process as $y_ t = 0.5~ y_{t-2} + \epsilon _ t.$

q

specifies the subset of the MA order. The quantity $m_ q$ is defined as the number of elements of theta.

If you do not specify q, The default subset is q$=\{ 1,2,\ldots ,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 lag=12.

The FARMACOV subroutine returns the following value:

cov

is a lag$+1$ vector that contains the autocovariance function of an ARFIMA($p,d,q$) process.

As an example, consider the following ARFIMA($1,0.3,1$) process:

\[ (1-0.5B)(1-B)^{0.3}y_ t = (1+0.1B){\epsilon }_ t \]

In this process, $\epsilon _ t \sim NID(0, 1.2)$. 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 25.130: Autocovariance of an ARFIMA Process

cov
4.2493033
3.5806774
2.9152846
2.4381017
2.1068697
1.8743199 




            

            


            
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($0,d,0$) process with the autocovariance function 
            

            

               
\[ \gamma _ k = {\mbox E}(y_{t}y_{t-k}) = { {(-1)^ k \Gamma (-2d+1)} \over {\Gamma (k-d+1)\Gamma (-k-d+1) }} \]

            

            
 and the autocorrelation function 

            

               
\[ \rho _ k = {\gamma _ k \over \gamma _0} = {{ \Gamma (-d+1)\Gamma (k+d)} \over {\Gamma (d)\Gamma (k-d+1) }} \sim { \Gamma (-d+1) \over {\Gamma (d)}} k^{2d-1}, ~ ~ k\rightarrow \infty \]

            

            
 Notice that $\rho _ k$ decays hyperbolically as the lag increases, rather than showing the exponential decay of the autocorrelation function of
               a stationary ARMA($p,q$) process. 
            

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

            

               
\[ \phi (B)(1-B)^ dy_ t = \theta (B){\epsilon }_ t \]

            

            
 where $\phi (B)=1-\phi _1B-\phi _2B^2 - \ldots - \phi _ pB^ p$ and $\theta (B)=1-\theta _1B-\theta _2B^2 - \ldots - \theta _ qB^ q$ 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 $x_ t$ follows an ARFIMA($0,d,0$) process; let $z_ t=(1-B)^ dy_ t$, so that $z_ t$ follows an ARMA($p,q$) 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).