Multistep forecasts in ARFIMA models with Monte-Carlo simulations

I have 3 questions about ARFIMA-* models forecasting.

Let's look at standard stationary non-seasonal ARFIMA model representation via coefficients.

$$\left(1 - \sum_{i=1}^p\phi_{i} L\right) \left(1 - L\right)^{d_{frac}}\left(x_{t} - \mu \right) = \left(1 + \sum_{j=1}^{q} \theta_{j}L\right)a_{t}$$, where $$a_{t}$$ - white noise.

Let's define $$\left(1 - \sum_{i=1}^p\phi_{i} L\right) = \Phi\left(L\right)$$, $$\left(1 + \sum_{j=1}^{q} \theta_{j}L\right) = \Theta\left(L\right)$$. In this case we could write $$x_{t} = \mu + \frac{\Theta\left(L\right)}{\Phi\left(L\right)\left(1-L\right)^{d_{frac}}}a_{t} = \mu + \sum_{k \geq 0}\psi_{k}a_{t-k}$$ - it's an infinite MA representation of our stationary process.

We could rewrite it as $$x_{t} = I_{k}\left(t - k\right) + C_{k}\left( t- k\right)$$ (according to (Box, Jenkins) notation), where $$C_{k}\left(t-k\right)$$ is a complimentary function and $$I_{k}\left(t - k\right)$$ is a particular integral an could be represented as $$I_{k}\left(t-k\right) = \sum_{j=0}^{t-k-1}\psi_{j}a_{t-j}$$.

So, the first question.

What's the representation of nonstationary non-seasonal ARFIMA model? We could write integrated process as: $$\left(1-L\right)^{d_{int}}\left(x_{t} - poly_{t}\right) = a_{t}$$, where $$poly_{t} = \sum_{i=0}^{d_{int}} \frac{\mu_{i}}{i!} t^i$$, but how to calculate $$\mu_{i}$$ if we have only $$\mu$$ when we calculate coefficients of our model for differenced series.

Second question:

Let's have $$\hat {x}_{t}\left(l\right)$$ - forecast value of model for step $$l$$. Is this correct that $$x_{t+l}$$ could be writted as $$x_{t+l} - poly_{t+l}= \hat {x}_{t}\left(l\right) + e_{t}\left(l\right)$$, where $$e_{t}\left(l\right)$$ is a forecast error or it could be written as $$x_{t+l} - poly_{t+l}= \hat {x}_{t}\left(l\right) - poly_{t} + e_{t}\left(l\right)$$

Third question. Each ARMA model with non-zero mean could be written as $$\left(1 - \sum_{i=1}^p\phi_{i} L\right) \left(x_{t} - \mu \right) = \left(1 + \sum_{j=1}^{q} \theta_{j}L\right)a_{t} =>$$ $$\left(1 - \sum_{i=1}^p\phi_{i} L\right) x_{t} = \theta_{0} + \left(1 + \sum_{j=1}^{q} \theta_{j}L\right)a_{t}$$ with $$\theta_{0} = \mu \left(1 - \sum_{i=1}^p\phi_{i} \right)$$. We could write model $$\left(1 - \sum_{i=1}^p\phi_{i} L\right) x_{t} = \theta_{0} + \left(1 + \sum_{j=1}^{q} \theta_{j}L\right)a_{t}$$ as a model with a non-zero white noise: $$\left(1 - \sum_{i=1}^p\phi_{i} L\right) x_{t} = \left(1 + \sum_{j=1}^{q} \theta_{j}L\right)\xi_{t}$$, where $$\xi_{t}$$ are i.i.d. with $$E\xi_{t} = \frac{\theta_{0}}{\left(1 + \sum_{j=1}^{q} \theta_{j} \right)}$$

So, how could it be calculated for stationary/non-stationary ARFIMA model with non-zero fractional differencing parameter?

I can't prove that the value of

$$\frac{\Theta\left(1\right)}{\Phi\left(1\right)\left(\sum_{k\geq0}{d \choose k}\left(-1\right)^{k}\right)}$$ will converge for $$-0.5 < d < 0.5$$. And other question - if it is, how many coefficients of $$\frac{1}{\sum_{k\geq0}{d \choose k}\left(-L\right)^{k}}$$ should I calculate for well definition of $$\theta_{0}$$?

Thank you.

• @RichardHardy I fear me that someone won't understand question if I'll post this by 3 independent question.((( – Dmitriy Jun 2 at 14:07
• I get it. Still I am not sure the current solution is optimal, but it might be better than the other one. – Richard Hardy Jun 2 at 14:09