FARMASIM Call

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Fractionally Integrated Time Series Analysis

FARMASIM Call

generates an ARFIMA(p,d,q) 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 d should be in the open interval (-1,1) excluding zero.
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.
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.
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 n should be greater than or equal to the AR order. The default is n=100 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 initial=a₀, then y_-p+1, ... ,y₀ take the same value a₀. If the initial option is not specified, the initial values are set to zero.
seed: specifies the random number seed. If it is not supplied, the system clock is used to generate the seed. If it is negative, then the absolute value is used as the starting seed; otherwise, subsequent calls ignore the value of seed and use the last seed generated internally.

The FARMASIM subroutine returns the following value:

series: is an n vector containing the generated ARFIMA(p,d,q) process.

To generate an ARFIMA(1,0.3,1) process

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

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

 
   d    = 0.3;  
   phi  = 0.5;
   theta= -0.1;
   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 n from an ARFIMA(p,d,q) model. If the process is stationary and invertible, the initial values y_-p+1, ... , y₀ are produced using covariance matrices obtained from FARMACOV. If the process is nonstationary, the time series is recursively generated using the user-defined initial value or the zero initial value.

To generate an ARFIMA(p,d,q) process with $d\in [0.5,1)$ ,x_t is first generated for $d'\in (-0.5, 0)$ , where d'=d-1 and then y_t is generated by y_t = y_t-1 + x_t.

To generate an ARFIMA(p,d,q) process with $d\in (-1,-0.5]$ ,a two-step approximation based on a truncation of the expansion (1-B)^d is used; the first step is to generate an ARFIMA(0,d,0) 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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