FARMACOV Call

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Fractionally Integrated Time Series Analysis

FARMACOV Call

computes the auto-covariance function for an ARFIMA(p,d,q) 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 d must be in the open interval (-0.5,0.5) excluding zero. This input is required.
phi: specifies an m_p-dimensional vector containing 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 containing 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, ... ,m_p}.

For example, consider phi=0.5.

If you specify p=1 (the default), the FARMACOV subroutine computes the theoretical auto-covariance 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 auto-covariance 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, ... ,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 containing the auto-covariance function of an ARFIMA(p,d,q) process.

To compute the auto-covariance of an ARFIMA(1,0.3,1) process

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

where $\epsilon_t \sim NID(0, 1.2)$ , you can specify

 
   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(0,d,0) process with the auto-covariance 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 auto-correlation function

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

Notice that $\rho_k$ decays hyperbolically as the lag increases, rather than showing the exponential decay of the auto-correlation function of a stationary ARMA(p,q) process.

The FARMACOV subroutine computes the auto-covariance function of an ARFIMA(p,d,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 $\phi(B)=1-\phi_1B-\phi_2B^2 - ... - \phi_pB^p$ 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 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 auto-covariance function of {x_t} and $\gamma_k^z$ be the auto-covariance function of {z_t}.

Then the auto-covariance 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 auto-covariance function of {y_t} is given by Sowell (1992, p. 175).

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