Fractionally Integrated Time Series

Fractionally Integrated Time Series Analysis

Fractionally Integrated Time Series

The fractional differencing enables the degree of differencing d to take any real value rather than being restricted to integer values. The fractionally differenced processes are capable of modeling long-term persistence. The process

$(1-B)^dy_{t}={\epsilon}_t$

is known as a fractional Gaussian noise process or an ARFIMA(0,d,0) process, where $d\in (-1,1) \backslash \{0\}$ , $\epsilon_t$ is a white noise process with mean zero and variance $\sigma_{\epsilon}^2$ , and B is the backshift operator such that B^jy_t = y_t-j. The extension of an ARFIMA(0,d,0) model combines fractional differencing with an ARMA(p,q) model, known as an ARFIMA(p,d,q) model.

Consider an ARFIMA(0,0.2,0) represented as $(1-B)^{0.2}y_{t}={\epsilon}_t$ where $\epsilon_t \sim NID(0, 1)$ . With the following statements you can

compute the auto-covariance function
generate the simulated data
compute the log-likelihood function
fit a fractionally integrated time series model to the data
obtain the fractionally differenced data

   d = 0.2;
   call farmacov(cov, d); print cov;  
   call farmasim(yt, d); print yt;  
   call farmalik(lnl, yt, d); print lnl;
   call farmafit(d, ar, ma, sigma, yt); print d sigma;
   call fdif(zt, yt, d); print zt;

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