I want to fit an ARFIMA (or FARMIA) model to a time-series. My problem is that I don't know how to estimate the confidence intervals of the estimated parameters. The sampling distribution seems to be difficult to derive. I could use a Bayesian approach but that will be computationally very expensive for my purpose. Therefore, I want to use the Frequentist method. Can you please suggest a fast method to do so? in fact, if there is an analytical expression to do so that would be great.
The mathematical representation of the model is given below.
Model:
$\phi_p(B)(1-B)^dX_t=\psi_q(B)\epsilon_t,$
where $B$ is backward shift operator, $\phi_p(B)$ and $\psi_q(B)$ are auto-regressive polynomial of order $p$ and moving-average polynomial of order $q$, and $\epsilon_t$ is a zero-mean uncorrelated noise with variance. The functional forms of $\phi_p(B)$ and $\psi_q(B)$ are as follows:
$\phi_p(B) = 1-\sum_{j=1}^{p}\phi_jB^j, $
$\psi_q(B) = 1-\sum_{j=1}^{q}\psi_jB^j.$
Assuming the $X_t$ to be a Gaussian process, and using Whittle's MLE approximation, the negative-log-likelihood to be minimized is:
$L(\theta)=\sum_{j=1}^{m}\log f_\theta(\omega_j) + \sum_{j=1}^{m}\frac{I(\omega_j)}{f_\theta(\omega_j)},$
where $f_\theta(\omega_j)$ is the power spectral density of the ARFIMA process, $I(\omega_j)$ is the periodogram of the observed data, and $\omega_j$ are the angular frequencies at which power spectral density and periodogram will be evaluated.
