
The key innovation of the ARFIMA model is the generalization of the differencing parameter to allow for non-integer, fractional values. This generalization allows for a more flexible and nuanced representation of persistence in a time series.

Intuitively, the behavior of the process is fundamentally characterized by the fractional differencing parameter, $d$:
\begin{itemize}
    \item \textbf{If $d = 0$}, the model reduces to a standard stationary ARMA(p,q) process with short memory.
    \item \textbf{If $0 < d < 0.5$}, the process is stationary and exhibits long memory. Its autocorrelation function (ACF) decays hyperbolically, which is much slower than the exponential decay of a short-memory process. This slow decay implies that observations far in the past have a persistent, non-negligible correlation with current observations.
    \item \textbf{If $0.5 \le d < 1$}, the process is non-stationary but, unlike a unit-root process ($d=1$), it is mean-reverting. Shocks have a long-lasting effect, but they eventually dissipate, and the series will tend to return to its long-run mean.
\end{itemize}
This fractional parameter provides a continuum between stationary short-memory processes ($d=0$) and non-stationary unit-root processes ($d=1$), providing a much richer framework for modeling persistence.

Mathematically, the ARFIMA(p, d, q) model for a time series $y_t$ is defined as:
\begin{equation}
    \phi(L)(1-L)^d y_t = \theta(L)\epsilon_t
\end{equation}
where:
\begin{itemize}
    \item $L$ is the lag operator.
    \item $\phi(L) = 1 - \phi_1 L - \dots - \phi_p L^p$ is the autoregressive polynomial of order $p$.
    \item $\theta(L) = 1 + \theta_1 L + \dots + \theta_q L^q$ is the moving average polynomial of order $q$.
    \item $\epsilon_t$ is a white noise error term, $\epsilon_t \sim WN(0, \sigma^2)$.
    \item $(1-L)^d$ is the fractional differencing operator.
\end{itemize}
The fractional differencing operator is defined by its binomial series expansion, which involves the Gamma function ($\Gamma(\cdot)$):
\begin{equation}
    (1-L)^d = \sum_{k=0}^{\infty} \frac{\Gamma(k-d)}{\Gamma(k+1)\Gamma(-d)} L^k = 1 - dL - \frac{d(1-d)}{2!}L^2 - \frac{d(1-d)(2-d)}{3!}L^3 - \dots
\end{equation}
This expansion shows that the fractionally differenced series, $(1-L)^d y_t$, is a weighted sum of all past values of $y_t$. The weights decline hyperbolically as the lag increases, which is the mathematical source of the model's long memory property.

\subsection{Why Dynamic predictions Tend to a Flat Line?}

Dynamic (iterated) forecasts replace future realizations with their own predicted values. Since $\mathbb{E}(\varepsilon_{t+h}\mid\mathcal{F}_t)=0$ for all $h>0$, the conditional mean forecast increasingly depends only on the unconditional properties of the process. In ARIMA$(p,1,q)$ models without drift, this implies
\begin{equation}
\widehat{y}_{t+h\mid t} \approx y_t,
\quad h \text{ large},
\label{eq:arima_flat}
\end{equation}
which resembles a random-walk-type flat forecast.

In ARFIMA models with $0<d<0.5$, convergence to the unconditional mean occurs slower due to long memory. Although ARFIMA predictions may display greater persistence at short and medium horizons, dynamic multi-step predictions eventually become flat, similarly to ARIMA dynamic predictions. This behavior is a direct consequence of the long-memory structure described in \cite{granger1980introduction}, \cite{hosking1981fractional} and \cite{baillie1996long}, rather than a limitation of estimation or model specification. 

\subsection{Static Prediction and Empirical Usefulness}

Static (one-step-ahead) prediction conditions each prediction on realized observations as they become available. Formally, the one-step-ahead prediction is as follows:
\begin{equation}
\widehat{y}_{t+1\mid t} = \mathbb{E}(y_{t+1}\mid\mathcal{F}_t),
\label{eq:static}
\end{equation}
with $\mathcal{F}_t$ updated at each period using observed data. This approach avoids the accumulation of forecast errors inherent in dynamic forecasting and allows the long-memory component of ARFIMA models to remain informative in real time.

In applied settings such as exchange rates and financial returns, where forecasts are updated sequentially, static forecasts are often more useful for model comparison and decision-making. The advantage of ARFIMA models in such contexts lies primarily in their ability to capture persistent dependence at short horizons rather than in long-horizon unconditional forecasts.

\subsection{Forecast Intervals under Long Memory}

An important distinction between ARIMA and ARFIMA models arises in the behavior of forecast uncertainty. While point forecasts converge to a constant, forecast intervals behave differently in long memory. For stationary ARFIMA processes, the $h$-step-ahead forecast error variance increases with $h$ at a faster rate than in short-memory ARMA models:
\begin{equation}
\mathrm{Var}(y_{t+h} - \widehat{y}_{t+h\mid t})
\propto h^{2d},
\label{eq:variance}
\end{equation}
for large $h$ and $0<d<0.5$.

As a result, predicted intervals widen more rapidly in ARFIMA models, reflecting the persistent influence of past shocks. This feature is economically meaningful, as it implies greater long-horizon uncertainty even when point forecasts appear flat. \cite{baillie1996long} emphasizes that this divergence between mean forecasts and forecast uncertainty is a defining characteristic of long-memory processes and a key reason for their relevance in macroeconomic and financial time series analysis.

\subsection{Mathematical Formulations}
The ARIMA$(p,d,q)$ process is characterized by integer integration, while the ARFIMA$(p,d,q)$ process utilizes fractional differencing to model long memory:
\begin{center}
    ARIMA: $\phi(B)(1-B)^d y_t = \theta(B)\varepsilon_t, \quad d \in \{0, 1, 2\}$ \\
    ARFIMA: $\phi(B)(1-B)^d y_t = \theta(B)\varepsilon_t, \quad d \in (-0.5, 0.5)$
\end{center}
The fractional operator is expanded as:
\begin{equation}
(1-B)^d = \sum_{k=0}^{\infty} \frac{\Gamma(k-d)}{\Gamma(-d)\Gamma(k+1)} B^k
\end{equation}

The flat line behavior is a specific property of dynamic prediction in stationary processes. Because $\mathbb{E}(\varepsilon_{t+h}|\mathcal{F}_t)=0$, the model eventually reverts to the unconditional mean $\mu$. 

In contrast, static prediction avoids this convergence. Using actual observation $y_{t-1}$ to predict $y_t$, the model is updated with innovations in the real-world at each step. This prevents the memory from decaying into a constant mean, allowing the prediction to track the actual volatility of the series.

The following table summarizes the operational differences between the two forecasting approaches:

\begin{table}[H]
\centering
\begin{tabular}{lll}
\toprule
\textbf{Feature} & \textbf{Dynamic Forecast} & \textbf{Static Forecast} \\ \midrule
Input Data & Predicted values ($\hat{y}_{t+h-1}$) & Actual observations ($y_{t-1}$) \\
Horizon & Multi-step ($h > 1$) & One-step ahead ($h = 1$) \\
Visual Path & Converges to a "Flat Line" & Follows data fluctuations \\
Error Propagation & Errors accumulate over time & Errors are reset each step \\
Primary Use & Long-term strategic planning & Model validation/Nowcasting \\ \bottomrule
\end{tabular}
\caption{Dynamic and Static Forecasting Properties}
\end{table}
