Neural ARFIMA model for forecasting BRIC exchange rates with long memory under oil shocks and policy uncertainties

Neural ARFIMA model for forecasting BRIC exchange rates with long memory under oil shocks and policy uncertainties

\journaltitle \copyrightyear \appnotes{Paper}

\authormark{Chakraborty et al.}

\address[1]{SAFIR, Sorbonne University Abu Dhabi, UAE} \address[2]{Sorbonne Center for Artificial Intelligence, Sorbonne University, Paris, France}

\corresp[$\ast$]{Corresponding author Email: \href{email:[email removed]}{[email removed]}.}

\abstract{ Accurate forecasting of exchange rates remains a persistent challenge, particularly for emerging economies such as Brazil, Russia, India, and China (BRIC). These series exhibit long memory, nonlinearity, and non-stationarity properties that conventional time series models struggle to capture. Additionally, there exist several key drivers of exchange rate dynamics, including global economic policy uncertainty, US equity market volatility, US monetary policy uncertainty, oil price growth rates, and country-specific short-term interest rate differentials. These empirical complexities underscore the need for a flexible modeling framework that can jointly accommodate long memory, nonlinearity, and the influence of external drivers. To address these challenges, we propose a Neural AutoRegressive Fractionally Integrated Moving Average (NARFIMA) model that combines the long-memory representation of ARFIMA with the nonlinear learning capacity of neural networks, while flexibly incorporating exogenous causal variables. We establish theoretical properties of the model, including asymptotic stationarity of the NARFIMA process using Markov chains and nonlinear time series techniques. We quantify forecast uncertainty using conformal prediction intervals within the NARFIMA framework. Empirical results across six forecast horizons show that NARFIMA consistently outperforms various state-of-the-art statistical and machine learning models in forecasting BRIC exchange rates. These findings provide new insights for policymakers and market participants navigating volatile financial conditions. The narfima R package provides an implementation of our approach.}

Introduction

{Exchange rates significantly influence macroeconomic outcomes, shaping trade balances, capital flows, inflationary pressures, and financial stability, thereby becoming a central focus for policymakers, central banks, and market participants in an increasingly interconnected global economy eichengreen1998exchange, hausmann1999financial, stoica2016exchange. Their role in assessing countries' financial stability has long been established taylor2001role, with accurate forecasts proving indispensable for guiding monetary policy, designing capital controls, and implementing macro-prudential measures wieland2013forecasting, lubik2007central. For commodity-dependent economies, where exchange rate fluctuations directly impact inflation forecasts and broader economic stability, reliable projections are particularly vital rossi2013exchange. Within this context, spot exchange rates, reflecting current market prices for immediate currency delivery, are especially important as they directly influence trade prices, international capital flows, external debt obligations, and monetary policy decisions pilbeam2015forecasting. Unlike aggregate measures such as the real effective exchange rate, spot rates respond swiftly to short-term market conditions, policy shifts, and uncertainty shocks, with their continuous trading facilitating rapid assimilation of new information bartsch2019economic. This makes spot markets an ideal setting for analyzing the transmission of oil price shocks and policy uncertainty to exchange rate dynamics across multiple horizons. Given the rising global prominence of the BRIC economies\footnote{As of January 1, 2024, the BRICS has expanded to include eleven members (Brazil, Russia, India, China, South Africa, Argentina, Egypt, Ethiopia, Iran, Saudi Arabia, and the United Arab Emirates). This expansion has increased the collective share of global GDP. Further details can be found at: \url{https://www.weforum.org/stories/2024/11/brics-summit-geopolitics-bloc-international/}.}, collectively accounting for 37.3% of world GDP (PPP) and a rapidly growing share of global trade o2011growth, hopewell2017brics, SENGUPTA2025953, nasir2018implications, understanding and accurately forecasting spot exchange rates in these economies has become increasingly critical for macroeconomic planning, financial stability, and policy coordination. }

Although emerging economies such as the BRIC nations have become increasingly influential in the global economy, they remain particularly vulnerable to global shocks, often experiencing rapid and severe currency fluctuations than developed nations. This increased susceptibility stems from greater exposure to external shocks, fragile financial markets, and risk of sudden reversals in capital flows, a phenomenon termed “flight-to-quality” bernanke1994financial, calvo2005sudden. In response to these challenges, extensive literature has emphasized the importance of incorporating various forms of uncertainty measures, such as economic policy uncertainty (EPU) and geopolitical risks (GPR), into exchange rate forecasting frameworks kumar2024bayesian, salisu2022exchange. From a theoretical perspective, when domestic uncertainty exceeds foreign uncertainty, domestic investors tend to invest in foreign currency assets, triggering exchange rate movements balcilar2016does. Moreover, economic uncertainties affect expectations about costs and returns, which in turn influence both supply and demand in currency markets benigno2012risk. Motivated by these theoretical foundations, recent empirical studies have focused on modeling the interplay between EPU and exchange rate dynamics. For instance, zhou2020can demonstrated the enhanced forecasting performance of Generalised Autoregressive Conditional Heteroskedasticity - Mixed Data Sampling (GARCH-MIDAS) models incorporating Sino-US EPU in predicting Chinese exchange rate volatility, while benigno2012risk explored how monetary, inflation, and productivity uncertainties influence real exchange rates. Further studies have consistently confirmed the robust predictive power of EPU in emerging markets across short and long horizons colombo2013economic, sin2015economic, juhro2018can, abid2020economic. Alongside EPU, other financial uncertainty indices such as the US Equity Market Volatility (EMV) and US Monetary Policy Uncertainty (MPU) measure distinct aspects of the economic environment that potentially impact currency markets. Empirical evidence from mueller2017exchange indicates that US MPU significantly affects currency risk premia, with emerging market currencies exhibiting greater sensitivity due to “reach for yield” behavior. Additionally, istrefi2018subjective documents an inverse relationship between US MPU and economic activity, where increased US MPU typically prompts dollar appreciation driven by safe-haven demand during financial stress. Collectively, these findings underscore the importance of incorporating multifaceted uncertainty measures to better capture the complex dynamics governing exchange rates in emerging economies.

In addition to uncertainty measures, other external economic factors, such as the relationship between oil prices and exchange rates, form a critical nexus in international finance, particularly for economies with significant oil exposure. For instance, beckmann2020relationship demonstrated that the oil price and exchange rate relationship exhibits time-varying characteristics depending on the nature of the underlying shock driving oil prices. The distinction between oil-exporting and oil-importing countries is fundamental, as highlighted by chen2024dynamic, who found that exchange rate-oil price connectedness intensifies during crisis periods, with stronger transmission channels evident in oil-exporting economies. This finding is especially relevant for BRIC countries, which span the spectrum from major oil exporters (Russia, Brazil) to significant importers (China, India). Alongside oil price dynamics, short-term interest rates play crucial roles in exchange rate movements. In a recent study, andrieș2017relationship examined this relationship in Romania using a structural vector autoregressive approach and uncovered bidirectional causality between interest rates and exchange rates, with the strength of the relationship varying across different time horizons. In a similar direction, saracc2016impact investigated the Turkish economy and found that short-term interest rates significantly impact exchange rate volatility, with the relationship intensifying during periods of economic distress. Their analysis further highlighted that interest rate differentials serve as a more reliable predictor of exchange rate movements than absolute interest rate levels. For BRIC economies, which exhibit varying degrees of capital account openness and monetary policy independence, the interest rate channel represents a critical transmission mechanism through which both domestic and external shocks propagate to exchange rates. Thus, when combined with oil price dynamics and uncertainty measures, they can capture the complex dynamics governing exchange rates in emerging economies.

The complexity of exchange rate dynamics in emerging markets makes forecasting challenging, motivating the development of models beyond standard benchmarks. However, meese1983empirical demonstrated that simple random walk models often outperform more advanced econometric approaches in out-of-sample forecasts. Numerous efforts have been made to improve the accuracy of exchange rate predictions, the existing literature encompasses a wide range of strategies, including Taylor rule-based fundamentals molodtsova2009out, nonlinear methods kilian2003so, and Kalman filter-based models date2025modelling. More recently, with advances in data-driven techniques and the increasing availability of macroeconomic datasets, the adoption of statistical and machine learning techniques for exchange rate forecasting has surged plakandaras2015forecasting. For instance, ngan2013forecasting employed an autoregressive integrated moving average (ARIMA) model to capture the exchange rate dynamics in Vietnam, while karemera2006assessing revealed that an autoregressive fractionally integrated moving average (ARFIMA) framework can outperform the random walk method in forecasting exchange rates for several developed economies. Similarly, pilbeam2015forecasting investigated the effectiveness of the univariate GARCH model in forecasting foreign exchange market volatility, and galeshchuk2016neural explored the use of neural networks in forecasting exchange rates across multiple currencies. More recently, deep learning approaches have been applied to forecast exchange rates in emerging economies abir2024use; however, these algorithms require high-frequency data, limiting their effectiveness under data scarcity, and they struggle to capture the long-memory characteristics of exchange rates. Despite the methodological advances, a majority of these studies have predominantly focused on exchange rates of developed countries and OECD nations, largely overlooking the distinct dynamics of emerging economies and their interactions with macroeconomic drivers. To address this gap, we systematically examine how multiple uncertainty measures, such as global EPU (GEPU), US EMV, and US MPU, alongside oil price growth rates and country-specific short-term interest rate differentials, impact BRIC exchange rate dynamics through causal analysis techniques. Given the nonlinear characteristics of the data, we employ a nonlinear Granger causality test to identify the key drivers. Furthermore, we detect that exchange rate series of the BRIC economies exhibit structural complexities, including long-term memory, nonlinearity, and non-stationary behavior that traditional time series forecasting approaches, such as linear ARIMA or ARFIMA models, fail to capture. Although advanced neural networks can model the inherent nonlinearities in the exchange rate series, they often lack the theoretical foundations required to model long-term memory and non-stationary patterns.

To address these methodological limitations, we propose the Neural AutoRegressive Fractionally Integrated Moving Average (NARFIMA) model, an ensemble approach that integrates the strengths of classical statistical methods with advanced machine learning techniques. The mechanism of the proposed NARFIMA model operates through a structured two-stage procedure. In the initial phase, a linear ARFIMA model is fitted with lagged observations of the exchange rate series, auxiliary uncertainty measures, and macroeconomic drivers to produce in-sample residuals. These ARFIMA model residuals capture the unexplained variations in the exchange rate dataset after accounting for linear long-memory effects and key economic drivers. In the subsequent step, the residuals are combined with the exchange rate time series data, uncertainty measures, and macroeconomic drivers, and are then modeled using an autoregressive neural network architecture to capture nonlinear dependencies and underlying structural patterns. This ensemble framework can capture long-term memory, nonlinear dependencies, and complex interactions between exchange rates and multiple economic drivers. Furthermore, the asymptotic stationarity and geometric ergodicity conditions obtained through Markov chain analysis confirm the theoretical robustness of the NARFIMA model; we have empirically verified that the assumptions necessary for these results are satisfied. These theoretical properties have several economic implications and policy relevance, as central banks and financial institutions rely on forecasting models that are asymptotically stable, interpretable, and reliable. Empirical evaluations conducted on the BRIC economies' exchange rates demonstrate that the NARFIMA model substantially enhances predictive accuracy and robustness, surpassing state-of-the-art statistical and machine learning forecasting frameworks. Its superior performance in capturing complex nonlinear interactions between exchange rate and causal drivers while modeling the long-term memory and non-stationary patterns of the endogenous variable underscores the practical relevance and reliability of our proposed approach. Overall, the integration of theoretical rigor with empirical validation positions NARFIMA as an effective forecasting technique, particularly suited for analyzing the complex exchange rate dynamics under varying policy uncertainties and critical macroeconomic shocks. By capturing both long-term memory and complex nonlinear dynamics between uncertainty measures, short-term interest rate differentials, oil shocks, and exchange rates, our approach provides a comprehensive tool that can deliver valuable insights for policymakers and researchers operating in an increasingly uncertain global economic environment. To further enhance its practical relevance, we perform uncertainty quantification using conformal prediction, which can be integrated with the NARFIMA framework and is vital for policy applications.

{The remainder of this paper is structured as follows. Section (ref) offers a detailed description of the data characteristics. Section (ref) introduces the proposed NARFIMA methodology, detailing its architecture and theoretical properties, including asymptotic stationarity of the NARFIMA model. In Section (ref), we present the causality analysis, empirical validation of theoretical conditions, and performance evaluation comparing the forecasting accuracy of the NARFIMA model with sixteen benchmark methods across six forecast horizons. Robustness checks and statistical significance tests supplement the evaluation. Uncertainty quantification through conformal prediction and ablation study is presented in Section (ref). Policy implications arising from our findings, relevant for central banks and international investors, are discussed in Section (ref). Finally, Section (ref) concludes this study and outlines future research directions.}

Data Characteristics

In this study, we analyze monthly spot exchange rates and several macroeconomic covariates for the BRIC economies from January 1997 (1997-01) to October 2023 (2023-10). Our analysis employs a rolling window approach with six forecast horizons: short-term (1- and 3-month-ahead), semi-long-term (6- and 12-month-ahead), and long-term (24- and 48-month-ahead) for each country. For short-term forecasting, with 1-month-ahead and 3-month-ahead horizons, the training periods span from 1997-01 to 2023-09 and 1997-01 to 2023-07, while the test horizons are 2023-10 and 2023-08 to 2023-10, respectively. In the case of semi-long-term forecasting, which includes 6-month-ahead and 12-month-ahead horizons, the training periods extend from 1997-01 to 2023-04 and 1997-01 to 2022-10, with corresponding test horizons from 2023-05 to 2023-10 and 2022-11 to 2023-10, respectively. Finally, for long-term forecasting with 24-month-ahead and 48-month-ahead horizons, the training period ranges from 1997-01 to 2021-10 and 1997-01 to 2019-10, while the test horizons cover 2021-11 to 2023-10 and 2019-11 to 2023-10, respectively. The target variable, spot exchange rate of BRIC countries, is obtained from the Federal Reserve Economic Data (FRED) repository\footnote{\url{https://fred.stlouisfed.org/}.}. Monthly exchange rate values are calculated as the averages of daily exchange rates based on noon buying rates in New York City for foreign currency cable transfers and are seasonally unadjusted. Table (ref) presents the time series plots of exchange rate dynamics, along with their autocorrelation functions (ACF) plots for the BRIC nations. We observe that all BRIC currencies, except the Chinese yuan, have depreciated against the USD over time. China maintained a currency peg until 2005, and even after shifting away from the peg, its central bank continued active intervention in the exchange rate market. In contrast, the other BRIC nations follow a floating exchange rate system. Additionally, a sharp spike in exchange rates is evident during the global recession, affecting all BRIC countries except China. On the other hand, the ACF plots for the BRIC nations indicate that the autocorrelation decays slowly and stays above the significance bounds for some lags, providing weaker evidence of long memory. Furthermore, to investigate possible structural breakpoints in the exchange rate series, we employ the ordinary least squares (OLS)-based CUSUM test, which helps account for regime shifts ploberger1992cusum. The test examines the cumulative sum of recursive residuals to identify any deviations from the model’s stability, indicating potential structural breakpoints. Our implementation of the OLS-based CUSUM test shows that no statistically significant breakpoints were found in the exchange rate series for BRIC countries, as depicted in Table (ref). The absence of significant breakpoints supports the conclusion that the exchange rate series has been relatively stable during the observed period, suggesting that regime shifts are unlikely to affect the forecasting models.

Among the economic drivers, we consider four widely used news-based uncertainty measures\footnote{The historical dataset for all uncertainty indicators is obtained from \url{https://www.policyuncertainty.com/}.}, namely GEPU, US EMV, US MPU, and GPR. An overview of their definitions and construction is provided in the Appendix (ref). Beyond uncertainty indices, we incorporate macroeconomic indicators such as oil prices, interest rates, and inflation. We consider the global price of West Texas Intermediate (WTI) crude oil (USD per barrel) and stabilize its variability by computing the oil price growth rates. The resultant series helps to capture the effect of oil price fluctuations on exchange rate movements over time. Additionally, we include short-term interest rates, which reflect the cost of borrowing for a duration under 24 hours between financial institutions or for government securities. These rates, measured in percentages for the BRIC economies and the US, serve as a key indicator of short-term financial system liquidity, responding to central bank interventions and market fluctuations. To assess relative monetary policy stance and capital flow dynamics, we compute the short-term interest rate differential (IRD) between the US and each BRIC economy. This country-specific short-term IRD significantly influences exchange rates, as higher BRIC interest rates tend to attract capital inflows, strengthening the domestic currency, while lower rates can lead to depreciation. We also consider CPI (Consumer Price Index) inflation rates for both the US and the BRIC economies, which measure price changes for a fixed basket of goods and services. The CPI inflation differential, reflecting the gap between US and country-specific CPI inflation rates, influences exchange rate movements, as higher domestic inflation indicates currency depreciation and vice versa. All macroeconomic indicators used in this analysis are collected from the FRED repository.

We compute the summary statistics and analyze several global characteristics of the exchange rate datasets and auxiliary variables. Notably, we focus on seven key time series features: skewness, kurtosis, nonlinearity, long-range dependence, seasonality, stationarity, and outlier detection hyndman2018forecasting. We employ Tsay's and Keenan's one-degree tests to assess nonlinearity, while long-range dependence is examined using the Hurst exponent. Unlike the ACF plots, which show finite-sample correlations, the Hurst exponent captures global scaling. Additionally, The Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test is used to evaluate stationarity, while Ollech and Webel's test is applied to detect seasonal patterns. The statistical characteristics of these datasets, summarized in Appendix (ref), reveal that most of the macroeconomic time series are non-stationary, except the oil price growth rate series, the CPI inflation series for Brazil and India, and the GPR series for Brazil and Russia. Most of the series do not exhibit seasonality, except for the GEPU and US EMV series. Additionally, nonlinear patterns are present in the majority of the dataset. The Hurst exponent values greater than 0.5 suggest persistent long-range dependence in exchange rate dynamics and all economic indicators. Moreover, to detect the presence of outliers in the dataset, we apply the Bonferroni outlier test using studentized residuals weisberg1982residuals. Our analysis indicates that all series, except for the exchange rate series of Russia, exhibit significant outliers.

Neural ARFIMA Model

This section outlines the workflow of the proposed Neural ARFIMA (NARFIMA) model, designed for forecasting exchange rate dynamics in the BRIC economies. The NARFIMA framework integrates the strengths of the AutoRegressive Fractionally Integrated Moving Average with causal exogenous variables (ARFIMAx) to capture long-range dependencies in time series while leveraging a neural network to model complex nonlinear interactions within macroeconomic variables. The proposed model follows a sequential approach, where ARFIMAx is first employed to model exchange rate series based on historical observations and macroeconomic drivers. ARFIMAx captures short-term dependencies through its autoregressive and moving average components, while its fractional differencing mechanism accounts for long-term dependencies in exchange rate dynamics. The inclusion of macroeconomic indicators as exogenous variables further strengthens the model by incorporating external influences, providing a comprehensive representation of the exchange rate dynamics.

Model Formulation

Given $T$ historical observations of exchange rate series $\left\{y_t; t = 1, 2, \ldots, T\right\}$ and $r$ macroeconomic drivers $\left\{X_{j,1}, X_{j,2}, \ldots, X_{j,T}\right\}_{j = 1}^r$, the ARFIMAx model is formulated as

where $\epsilon_t$ is white noise and $\operatorname{B}$ represents the backshift operator, such that $\operatorname{B}y_t = y_{t-1}$ and $\operatorname{B}X_{j,t} = X_{j, t-1}$. The parameters $\tilde{p}$ and $\tilde{q}$ denote the number of autoregressive and moving average terms, respectively, while $d \in \left(0, 0.5\right)$ represents the fractional differencing parameter responsible for capturing long-memory effects granger1980introduction. The coefficients $\left\{\tilde{\phi}, \; \tilde{\theta}, \; \tilde{\pi}, \; \tilde{\mu}\right\}$ correspond to the autoregressive, moving average, exogenous components, and bias term, respectively. The fractional differencing operator, given by $$ \left(1 - \operatorname{B}\right)^d = \sum_{v = 0}^{\infty} \frac{\Gamma\left(v - d\right)\operatorname{B}^v}{\Gamma\left(-d\right)\Gamma\left(v+1\right)}, $$ where $\Gamma\left(\cdot\right)$ denotes the gamma function, ensures that $y_t$ is transformed into a stationary process. By allowing $d$ to take non-integer values, this operator effectively captures long-term memory effects in time series, making it particularly well-suited for exchange rate datasets with slow decaying autocorrelation. Thus, the predictions $\left\{\hat{y}_t^{ARFIMA}\right\}$ generated from the ARFIMAx model by estimating the coefficients in Eqn. (ref) capture the linear trajectory of the exchange rate series and the influence of auxiliary covariates. However, exchange rate series of emerging economies like BRIC often exhibit complex nonlinear structures and dynamic causal interactions, which the linear ARFIMAx model may fail to fully explain. Consequently, the residuals $$e_t = y_t - \hat{y}_t^{ARFIMA}$$ of the ARFIMAx framework, captures the unexplained variations in exchange rate dynamics after accounting for linear long-memory effects. These residuals contain learnable structures with nonlinear dependencies and high-frequency fluctuations that cannot be effectively modeled using traditional parametric approaches. To address these limitations, the NARFIMA$(p,q,k)$ framework integrates a feed-forward neural network to model complex nonlinear patterns.

The neural network component in the NARFIMA model is structured as a single hidden-layer architecture, enabling it to learn intricate nonlinear relationships between lagged values of exchange rate series, ARFIMAx residuals, and economic drivers. This single-layered framework offers a balance between computational efficiency and predictive power, restricting overfitting, and making it suitable for forecasting exchange rates in the presence of both linear and nonlinear dynamics. Due to limited data availability for macroeconomic modeling, highly computational deep learning models often fail to capture the data dynamics. On the contrary, NARFIMA utilizes an artificial neural network structure with only one hidden layer having $k$ neurons; therefore, it does not overfit. The network is designed to receive inputs consisting of $p$ historical exchange rate observations, $q$ lagged values of ARFIMAx residuals, and one lagged value for each of the $r$ macroeconomic covariates. The output of the network is a one-step-ahead forecast of the exchange rate series, which is expressed as: $$ \hat{y}_{t+1} = f\left(y_{t}, y_{t - 1}, \ldots, y_{t - p +1}, e_{t}, e_{t - 1}, \ldots, e_{t - q +1}, X_{1, t}, X_{2, t}, \ldots, X_{r, t}\right); $$ where $t = \max(p,q), \cdots, T$ and $f$ represents the neural network function. By learning from historical data and residuals, the neural network effectively captures both linear and nonlinear relationships between input features and exchange rate dynamics, along with long memory dependence. In the proposed settings, two variants of the network are designed based on the inclusion or exclusion of skip connections. These skip connections allow input features to directly influence the output in addition to passing through the hidden layer, influencing the learning process and impacting model performance. The presence of a skip connection ensures that both linear and nonlinear components of the data are effectively integrated. The skip connection between the input and the output layer preserves the linear dynamics of the exchange rate series while allowing the hidden layer to learn the nonlinear interactions. Additionally, the presence of skip connections enhances training stability by mitigating the vanishing gradient problem and serves as a regularizing mechanism, which reduces overfitting by allowing direct information propagation and preventing unnecessary transformations. Thus, using the single-hidden layer neural network with a skip connection, the one-step-ahead forecasts of the exchange rate series can be generated as:

where $k$ is the number of hidden nodes, $\left\{\tilde{\alpha}_{l} ,\; \beta_{i,l}, \; \gamma_{j,l}, \; \delta_{m,l}\right\}$ are the connection weights between input and hidden layers, $\mu_l$ is the weight vector between hidden and output layers, $\left\{\phi_{u}, \eta_{v}, \zeta_{w}\right\}$ represent skip connection weights, $\mu_0$ is the bias term, and $\sigma\left(\cdot\right)$ is the nonlinear activation function. The network weights are initialized randomly and trained using the gradient descent backpropagation approach rumelhart1986learning. In the variant of the neural network without skip connections, the mechanism follows a similar structure, but the skip connection weights $\left\{\phi_{u}, \eta_{v}, \zeta_{w}\right\}$ are set to zero, removing the skip connection between input and output layers. The above procedure generates a one-step-ahead forecast of the exchange rate series. For the multi-step ahead forecasts, we employ a recursive framework where the input layer is updated with the latest predictions at each step. Alongside point forecasts, the NARFIMA framework can be readily integrated with conformal prediction techniques to produce reliable prediction intervals (see Section (ref)).

Choice of Parameters

The NARFIMA model consists of primarily four tunable parameters, namely the number of lagged exchange rate observations $(p)$, the number of historical values for ARFIMAx residuals $(q)$, the number of nodes in the hidden layer $(k)$, and the network structure indicating whether a skip connection is included $(skip)$. To determine the optimal values of the parameters, we utilize a time series cross-validation strategy and select $\left(p, q, k\right)$ by minimizing the root mean square error (RMSE) on the validation set ($\mathcal{V}$) as follows: $$ \left(p, q, k\right) = \underset{\left(p, q, k\right)}{\operatorname{arg min}} \sqrt{\frac{1}{|\mathcal{V}|} \sum_{{t'} \in \mathcal{V}} \left(y_{t'} - \hat{y}_{t'} \right)^2}, $$ where $y_{t'}$ and $\hat{y}_{t'}$ are the ground truth and forecasts generated by NARFIMA at time $t' \in \mathcal{V}$, respectively. For all the datasets, we build two different versions of the NARFIMA model, one with skip connections (general case) and one without skip connections. The optimal values of other parameters are identified through temporal cross-validation. To address potential overfitting concerns arising from the model's complexity relative to available data, the NARFIMA framework incorporates several safeguards that ensure robust generalization. The neural network architecture is deliberately constrained to a single hidden layer with a limited number of neurons ($k \leq 5$), preventing excessive parameterization while maintaining sufficient nonlinear modeling capacity. Subsequently, the NARFIMA model is trained with the optimal parameters on the entire training dataset to accurately forecast the exchange rate dynamics. The integration of ARFIMAx, which provides a robust foundation for modeling linear long-term dependencies, with the neural network, which excels in learning complex, nonlinear interactions, allows the NARFIMA architecture to capture both nonlinear and long-range dependencies. This ensures a comprehensive representation of the exchange rate series dynamics by combining long-memory dependencies, macroeconomic influences, and nonlinear fluctuations. In the next subsection, we study the asymptotic stationarity and ergodicity of the proposed NARFIMA model from a nonlinear time series perspective.

Geometric Ergodicity and Asymptotic Stationarity

Stationarity and ergodicity are fundamental for statistical inference in nonlinear time-series analysis. When a process is both stationary and ergodic, a single long realization is sufficient for time averages to recover the data-generating law. Asymptotic stationarity guarantees that distributional features stabilize as time grows, even in the presence of long memory or transient dynamics. Geometric ergodicity of the Markov chain induced by the model’s state ensures exponentially fast convergence to the invariant distribution. Together, these properties justify using NARFIMA for econometric and financial forecasting under long-memory and nonlinear regimes. Building on trapletti1999ergodicity, trapletti2000stationary, chakraborty2020unemployment, which analyze autoregressive neural networks (ARNN) and ARIMA–ARNN models, we develop a unified framework for NARFIMA with skip connections, a strictly more general specification. We present the theory and its economic implications and applications.

We demonstrate the asymptotic properties for the NARFIMA$(1,1,k)$ process with skip connections. However, for simplicity, we omit exogenous variables for establishing the theoretical results. The simple NARFIMA process is given by:

where $\Theta$ denotes the weight vector, $\varepsilon_t$ is a sequence of independently and identically distributed (i.i.d.) random noise, and $f(y_{t-1}, e_{t-1}, \Theta)$ represents an autoregressive neural network with $k$ hidden units and inputs $y_{t-1}$ and the ARFIMA feedback $e_{t-1}$, as defined in Section (ref). The output of the simple NARFIMA$(1,1,k)$ process with an activation function $G$ is as follows:

where $\psi = {(\psi_1,\psi_2)}^\top$ is a weight vector representing the skip connections, the input to hidden layer weight vector is denoted by $\phi = {\left(\phi_{1,1},\ldots,\phi_{k,1},\phi_{1,2},\ldots,\phi_{k,2},\mu_1,\ldots,\mu_k\right)}^\top$, and $\beta = {(\beta_0, \beta_1, \ldots, \beta_k)}^\top$ is the hidden to output layer weight vector. Both $\psi$ and $\beta$ are incorporated into the overall weight vector $\Theta$ of the neural network. The function $g$ represents the nonlinear transformation performed by the hidden layer, combining $\Theta$, $y_{t-1}$, and $e_{t-1}$ through the activation function $G$. The one-step model is given by $$y_t={\psi_1 y_{t-1}+\psi_2 e_{t-1}}+{\beta_0+\sum_{i=1}^k \beta_i G\left(\mu_i+\phi_{i,1} y_{t-1}+\phi_{i,2} e_{t-1}\right)}+\varepsilon_t.$$ Now, we write the NARFIMA process in the state space form as follows:

where $x_t =

, \; S =

, \; \sum =

, \; \Psi =

$, and $F(x_{t-1}) =

$ is the nonlinear part. We say $\{x_t\}$ is a Markov chain with state space $\mathcal{X} \subseteq \mathcal{R}^2$ equipped with Borel $\sigma$-field $\mathcal{B}$ and Lebesgue measure $\lambda$. \\

In order to show the asymptotic stationarity and ergodicity of NARFIMA, we state several key assumptions: Assumption (ref) is the condition for stationarity of the ARFIMA component, where the fractional differencing parameter satisfies $0<d<\frac{1}{2}$ and AR and MA polynomials are invertible granger1980introduction. This ensures ARFIMA residuals are stationary ($\left\{e_t\right\}$ is a stationary process). Assumption (ref) relies on the characteristics of the activation function to be used in the neural network architecture of the NARFIMA process. Assumption (ref) is essential for the stability of the dynamical system since it ensures that the behavior of the nonlinear part in Eqn. (ref) is predictable and does not exhibit erratic changes. This assumption is satisfied by popularly used activation functions, such as sigmoid and tanh, and it is considered in the choice of activation function during the implementation of the NARFIMA model. Additionally, the neural network may have skip connections where output also depends linearly on inputs like previous lag and past residual. Assumptions (ref) and (ref) ensure contraction in the linear part and bound the influence of past states. The innovation $\varepsilon_t$ satisfies Assumption (ref), which are usual restrictions on the noise process. We verify these assumptions empirically in Section (ref) while implementing the NARFIMA model on BRIC exchange rate datasets.

With the assumptions established, we now proceed to prove the asymptotic stationarity and ergodicity of the NARFIMA$(1,1,k)$ process. To establish geometric ergodicity and asymptotic stationarity of the NARFIMA process, we first show that the process is irreducible by proving the forward accessibility of its control system (Lemma (ref) and Lemma (ref)). Then, we construct a suitable Lyapunov function and verify a geometric drift condition, which, together with irreducibility, ensures the existence of a unique invariant distribution to which the process converges geometrically fast, completing the proof. To start with, we initially show the irreducibility of the associated Markov chain $\{x_t\}$. A formal definition of irreducibility is provided below meyn2012markov.

If for each current state $x$, the one-step law $\mathbb{P}(x, \cdot)$ admits a density $p(x, \cdot)$ w.r.t. Lebesgue measure that is strictly positive on every nonempty open set, then the chain is irreducible. From a control theory perspective, irreducibility is closely related to forward accessibility; for details, refer to meyn2012markov. Below is a formal definition of a forward accessible control system.

Since the activation function is nonconstant and bounded and $\psi_1 + \psi_2 \neq 0$ holds, the control model in Eqn. (ref) is forward accessible. Hence, from any state $\left(y_{t-1}, e_{t-1}\right)=(y, e)$, the one-step reachable set contains an open interval (nonempty interior). By iterating the same argument over steps, the reachable set from any initial state has a nonempty interior. The proof of the following lemma is provided in Appendix (ref).

In the following lemma, we establish the irreducibility of the Markov chain associated with the NARFIMA process. The noise has a strictly positive density, which implies irreducibility. The proof is provided in Appendix (ref).

Once irreducibility is established, we proceed to demonstrate the stationarity of the state-space formulation given in Eqn. (ref). The stationarity of a Markov chain $\{x_t\}$ is closely related to the geometric ergodicity of the underlying process. Heuristically, geometric ergodicity implies that the Markov chain converges to its stationary distribution. A formal definition of geometric ergodicity and asymptotic stationarity is provided below meyn2012markov.

Building on this idea, we extend the analysis to establish the geometric ergodicity of the NARFIMA$(1,1,k)$ process, as formalized in the theorem below. A drift geometric condition with Lyapunov function $V(\cdot)$ is written as meyn2012markov

Combining the geometric drift condition with irreducibility implies geometric ergodicity and thus asymptotic stationarity. The proof is provided in Appendix (ref).

Corollary (ref) follows directly from the proof of Theorem (ref), and it confirms that the NARFIMA process is asymptotically stationary, meaning its distribution stabilizes over time regardless of initialization. This allows us to use long-run statistical properties, ensures the consistency of estimators, and guarantees meaningful long-term forecasting behavior.

Economic Implications and Practical Applications

The theoretical results established for the NARFIMA$(1,1,k)$ process can be extended, under similar conditions, to the more general NARFIMA$(p,q,k)$ process. Although the NARFIMA model captures long memory (frequently observed in exchange rates) dynamics, the process remains asymptotically stationary under the fractional integration parameter $0<d<\frac{1}2{}$ and appropriate constraints on the neural component. The established results on asymptotic stationarity and geometric ergodicity of the NARFIMA model have significant practical relevance in financial and macroeconomic time series analysis. These theoretical properties justify the model’s use in applied forecasting and policy modeling in the following ways:

Experimental Analysis

This section first analyzes the potential drivers of monthly spot exchange rates for BRIC countries by examining historical data and several global and country-specific economic indicators. We investigate the role of GEPU, US EMV, US MPU, oil price growth rates, short-term interest rates, CPI inflation, and GPR index in forecasting exchange rates. This investigation allows us to identify the most relevant predictors to be used as exogenous variables in the proposed NARFIMA approach. Further, we evaluate the effectiveness of the NARFIMA model in forecasting the spot exchange rate of the BRIC economies by comparing its performance against state-of-the-art architectures from various paradigms. To assess its generalizability, we employed a rolling window forecasting approach with six different time horizons of lengths 1 month, 3 months, 6 months, 12 months, 24 months, and 48 months.

Causal Analysis

To assess the causal impact of macroeconomic covariates on the exchange rates of the BRIC economies, we employ a nonlinear Granger causality test. This choice is motivated by the global characteristics of the macroeconomic variables, which predominantly exhibit nonlinear patterns, as identified in Section (ref). The nonlinear Granger causality (GC) test evaluates the temporal predictive causality, determining whether the lagged values of one variable improve the explanatory power of another variable beyond what is provided by its past observations granger1969investigating, hiemstra1994testing. This test specifically assesses the presence of nonlinear causal relationships between the exchange rate and macroeconomic variables. Table (ref) presents the results of the nonlinear GC test across BRIC countries. As presented in the table, the p-value of the nonlinear GC test is below 0.05 for several global covariates, including GEPU, US EMV, US MPU, oil price growth rate, and the country-specific short-term IRD. This indicates a significant nonlinear causation between these macroeconomic drivers and the exchange rates. The causality analysis reveals varying degrees of association between the examined variables and spot exchange rates. Notably, some indicators, such as CPI inflation and GPR, do not demonstrate strong causal relationships with exchange rates across all the BRIC economies. Based on these findings, we refine our selection of auxiliary variables for the forecasting exercise, focusing on GEPU, US EMV, US MPU, oil price growth rate, and the country-specific short-term IRD that show the most consistent and significant associations with exchange rate movements.

Baseline Models

We evaluate the forecasting performance of the proposed NARFIMA model against sixteen baseline statistical and advanced deep learning models, some of which are capable of incorporating auxiliary covariates. Among the statistical frameworks, we consider Naïve, Autoregressive (AR), Autoregressive Integrated Moving Average with exogenous variables (ARIMAx), Autoregressive Fractionally Integrated Moving Average with exogenous variables (ARFIMAx), Exponential Smoothing (ETS), Self-exciting Threshold Autoregressive (SETAR), \textit{Trigonometric Box-Cox ARIMA Trend Seasonal} (TBATS), \textit{Generalized Autoregressive Conditional Heteroscedasticity} (GARCH), and \textit{Bayesian Structural Time Series with exogenous variables} (BSTSx). The deep learning architectures used in this evaluation include \textit{Autoregressive Neural Network with exogenous variables} (ARNNx), \textit{Deep learning-based Autoregressive} (DeepAR), \textit{Neural Basis Expansion Analysis for Time Series with exogenous variables} (NBeatsx), \textit{Neural Hierarchical Interpolation for Time Series with exogenous variables} (NHiTSx), \textit{Decomposition-based Linear model with exogenous variables} (DLinearx), \textit{Normalization-based Linear model with exogenous variables} (NLinearx), and \textit{Time Series Mixer with exogenous variables} (TSMixerx). A detailed description of these baseline models is provided in Appendix (ref).

Evaluation Metrics

To assess the performance of the proposed NARFIMA model and the baseline frameworks in forecasting the exchange rate series of the BRIC nations, we used five popularly used evaluation metrics, namely Mean Absolute Percentage Error (MAPE), Symmetric Mean Absolute Percentage Error (SMAPE), Mean Absolute Error (MAE), Mean Absolute Scaled Error (MASE), and Root Mean Square Error (RMSE). The mathematical formulations of these metrics are provided below: {

} where $y_t$ denotes the ground truth observation at time $t$ with the corresponding forecast $\hat{y}_t$, $T$ indicates the number of training observations, and $h$ is the forecast horizon. By definition, the model with the smallest error metric value is identified as the best-performing model hyndman2018forecasting, panja2023epicasting.

Implementation of NARFIMA Model

This section describes the implementation and performance of the proposed NARFIMA model in forecasting the exchange rate series of the BRIC economies. To ensure a comprehensive evaluation, NARFIMA is compared with several state-of-the-art forecasting models. The NARFIMA framework is implemented in R statistical software using a two-stage approach. Initially, an ARFIMAx model is fitted using the arfima function from the `forecast' package in R. This step captures the long-memory dependencies of the time series while incorporating significant exogenous variables, including GEPU, US EMV, US MPU, oil price growth rate, and the country-specific short-term IRD. Table (ref) summarizes the estimated parameters for the ARFIMAx model used in this evaluation. In the next stage, the residuals from the ARFIMAx model, along with the training series and exogenous covariates, are modeled using a single hidden-layered feed-forward neural network. The network is built using the nnet function from the `nnet' package in R, where it processes $p$-lagged inputs of the exchange rate series, $q$-lagged residuals from the ARFIMAx model, and one-lagged value from each exogenous variable through $k$ hidden nodes to generate a one-step-ahead forecast of the target series. Multi-step-ahead forecasts are generated recursively from the NARFIMA model. To optimize model parameters, we conduct a time series-based cross-validation over the parameters $(p, q, k)$ within the range 1 - 5, minimizing the RMSE metric. Moreover, the NARFIMA model is implemented with two variations of the feed-forward neural network: one that allows direct connections between the input and the output layers $(skip = \text{TRUE})$ and the other without the direct connections $(skip = \text{FALSE})$. The optimal parameters of the NARFIMA(\(p, q, k, skip\)) model for BRIC countries across different forecast horizons are summarized in Table (ref). To implement the proposed NARFIMA model effectively, it is essential to verify the presence of nonlinearity in the residuals of the ARFIMAx model. To assess the linearity of the residuals, we employ the Terasvirta and BDS tests, which reject the null hypothesis of linearity when the computed p-value falls below 0.05 prabowo2020performance, huang2023nonlinearity. The results of these tests, reported in Appendix (ref), confirm that ARFIMAx residuals exhibit significant nonlinearity. This indicates that the ARFIMAx model successfully captures the linear dependencies while leaving behind a complex nonlinear structure, which is then modeled by the neural network component of NARFIMA.

Validation of Theoretical Results on Empirical Data

We empirically investigate the validity of the theoretical assumptions underlying the NARFIMA model. Assumption (ref) is ensured by the model design, since the hidden layer employs a logistic (sigmoid) activation function, defined as $ \sigma(x)=\frac{1}{1+e^{-x}}$. The function is (i) bounded in $(0,1)$; (ii) $\lim _{x \rightarrow-\infty} \sigma(x)=0, \lim _{x \rightarrow+\infty} \sigma(x)=1$ (asymptotically constant); (iii) $\sigma^{\prime}(x)=\sigma(x)(1-\sigma(x)) \leq \frac{1}{4}$ implies $1 / 4$-Lipschitz and $\sigma_\alpha(x)=\sigma(\alpha x)$ is $\alpha / 4$-Lipschitz. These properties ensure that Assumption (ref) is satisfied in the NARFIMA implementation. We further evaluate Assumptions (ref) and (ref) using BRIC exchange rate data, extending the theoretical framework to the NARFIMA$(p,q,k, skip)$ process. In this setting, Assumption (ref) generalizes to the condition $\sum_{i=1}^{p} \psi_{1,i} + \sum_{j=1}^{q} \psi_{2,j} \neq 0,$ where $\psi_{1,i}$ and $\psi_{2,j}$ denote the autoregressive and residual skip weights, respectively, while Assumption (ref) requires $\Big|\sum_{i=1}^{p} \psi_{1,i}\Big| < 1$. Table (ref) reports the results for the long forecast horizons of 12, 24, and 48 months, for which the optimal parameter configurations in Table (ref) had skip connections ($skip = \text{TRUE}$). Empirically, both Assumptions (ref) and (ref) were satisfied across all countries, confirming the validity of the theoretical framework in practice.

Baseline Comparisons

After implementing the proposed NARFIMA model, we implemented the baseline forecasters and generated exchange rate forecasts of BRIC economies for multiple time horizons. Among the baseline forecasters, the classical time series models ARIMAx, ARFIMAx, ETS, TBATS, and ARNNx are implemented using the `forecast' package in R. The implementation of the Naïve, AR, SETAR, BSTSx, and GARCH models is adopted from `stats', `tsDyn', `bsts', and `tseries' packages in R, respectively. Additionally, the deep learning models DeepAR, NBeatsx, NHiTSx, DLinearx, NLinearx, and TSMixerx are implemented using the `darts' library in Python. The forecasting performance for Brazil, Russia, India, and China is presented in Tables (ref), (ref), (ref), and (ref), respectively. These tables highlight that for most forecasting tasks, the NARFIMA model demonstrates superior performance over the baseline architectures. For short-term forecasting (1-month-ahead horizon), NARFIMA significantly outperforms all baseline models across the BRIC economies, as indicated by all performance metrics. For the 3-month-ahead forecasting horizon, NARFIMA provides the most accurate forecasts for Brazil and China, while ARIMAx performs competitively for Russia and India. In the semi-long-term 6-month-ahead forecasts, the NARFIMA model substantially reduces forecast errors compared to its component frameworks, ARFIMAx and ARNNx, across all BRIC nations and provides the best exchange rate forecasts for India and China. Among the baseline models, ARIMAx and ETS produce comparable forecasts for Brazil and Russia. In the case of 12-month-ahead forecasts of BRIC exchange rates, the NARFIMA model generates the most accurate forecasts for Brazil and Russia, while the Naïve and ARIMAx framework performs better for India and China, respectively. For long-term forecasting (24-month-ahead horizon), the NARFIMA model maintains its performance supremacy across all BRIC countries except India, where NLinearx generates more accurate forecasts. At the 48-month-ahead forecasts, our proposed NARFIMA model consistently outperforms all competing frameworks, highlighting its ability to capture both long-term dependencies and nonlinear patterns in time series data. From a country-specific perspective, NARFIMA delivers the most accurate forecasts in five out of six horizons for Brazil and China. For Russia, it outperforms baseline models in four out of six cases while remaining competitive with ARIMAx and ETS models in the other two horizons. However, for India's exchange rate series, the NARFIMA model performs best in only three forecasting horizons. This is primarily attributed to the linearity of India's exchange rate data, which is better modeled with ARIMAx, Naïve, and NLinearx frameworks. These empirical results confirm that the NARFIMA approach can effectively model volatility, non-stationarity, and nonlinearity in time series data. However, its forecasting performance declines for inherently linear series, where traditional statistical models perform better. Nevertheless, as most macroeconomic variables exhibit nonlinear characteristics franses2000non, NARFIMA remains well-suited for forecasting complex financial time series. On the other hand, deep learning models yield relatively inaccurate exchange rate forecasts compared to both statistical baselines and the NARFIMA model. This is likely due to the low sample size of the dataset, which pose significant challenges in accurately training deep learning architectures. While some baseline models, including ARIMAx and ETS, achieve comparable short-term and semi-long-term performance, their accuracy deteriorates significantly over longer horizons. In contrast, NARFIMA consistently performs well across all time horizons, demonstrating its robustness and generalizability. Additionally, the integration of ARFIMAx feedback residuals with the target series and exogenous variables in NARFIMA proves to be a highly effective technique for capturing long-term dependencies, especially for non-stationary and nonlinear exchange rate series.

Robustness and Statistical Significance Tests

In this section, we assess the robustness of our empirical results by evaluating the performance of different forecasting models based on differences in measurement errors, using multiple comparisons with the best (MCB) test. The model-agnostic MCB test is a nonparametric method that ranks each forecaster based on its performance across multiple datasets koning2005m3. It then identifies the model with the lowest average rank as the best-performing framework and considers the critical distance (CD) of this model as the reference value for comparison. Fig. (ref) presents the MCB test results across different forecasting tasks based on RMSE, MAPE, SMAPE, and MAE metrics. The figure shows that the proposed NARFIMA model achieves the lowest average ranks for RMSE, MAPE, SMAPE, and MAE metrics, making it the `best' performing model, followed by the ARIMAx, BSTSx, and Naïve frameworks in terms of the RMSE metric. The CD of the NARFIMA model (shaded area), serves as the reference value of the MCB test. Since the CD values of all baseline forecasters, except ARIMAx, BSTSx, and Naïve, lie well beyond this reference, their performance differs significantly from that of the best-performing NARFIMA model. Overall, the MCB test highlights the statistical significance of the performance differences, demonstrating the superiority of the NARFIMA model across various datasets and forecast horizons.

Alongside the MCB test, we utilize the Murphy diagram approach to assess the robustness and forecastability of the proposed NARFIMA model. The Murphy diagram technique detects empirical forecast dominance between competing models across a range of scoring functions ehm2016quanttiles. Unlike the MCB method, which focuses on ranking models based on their average forecast accuracy, the Murphy diagram provides a comprehensive evaluation by examining whether a forecasting model consistently outperforms others across different loss functions. The Murphy diagram is constructed by computing the scoring function

where the parameter $\theta \in \mathcal{R}$ controls the shape of the loss function, $y_t$ represents the actual observation, and $\hat{y}_t$ is the point forecast generated by a model at time $t$. To compare the performance of two forecasting frameworks, this distribution-free method computes the average scores for each model as $\mathcal{S}_j(\theta) = 1/h \sum_{t = 1}^h \tilde{s}(\hat{y}_{t,j}, y_t)$, where $\hat{y}_{t,j}$ is the forecast generated by the $j^{th}$ model corresponding to ground truth observation $y_t$ and $h$ is the forecast horizon. The Murphy diagram then plots the extremal scores ($\mathcal{S}_j(\theta)$) for different models across a range of $\theta$ values. The parameter $\theta$, plays a crucial role in evaluating forecast accuracy, as different values emphasize different aspects of prediction error. Smaller values of $\theta$ make the scoring function more sensitive to underpredictions, penalizing models that consistently underestimate actual values, whereas higher values of $\theta$ penalize overpredictions more heavily. This adaptability makes the Murphy diagram a robust tool for comprehensive model evaluation, as it provides insights into how forecasting errors are distributed and whether a model consistently outperforms its competitors under different scoring functions.

To empirically validate the effectiveness of NARFIMA against benchmark models, we utilized the `murphydiagram' package in R to generate the Murphy diagrams. Fig. (ref) presents the Murphy diagrams comparing NARFIMA with the top-performing ARIMAx and BSTSx frameworks, as identified by the RMSE-based MCB test results, for each BRIC nation over a 48-month-ahead forecast horizon. The diagram plots the extremal scores for competing models, where a lower score indicates better model performance. The results reveal distinct patterns across different economies. In the case of Brazil, NARFIMA consistently outperforms both ARIMAx and BSTSx across all scoring functions, establishing its superiority in exchange rate forecasting. For Russia, the model provides similar or more accurate exchange rate forecasts when $\theta$ lies below 72 or above 75, suggesting that its dominance depends on specific error considerations. In India and China, NARFIMA remains competitive; however, sometimes ARIMAx and BSTSx outperform it. Overall, these findings establish NARFIMA as an accurate and reliable forecasting model across diverse economic conditions. The combination of the MCB test and the Murphy diagram provides a robust validation of NARFIMA’s forecasting superiority, confirming that it consistently outperforms benchmark methods across various scoring functions and forecast horizons.

Uncertainty Quantification and Ablation Study

Conformal Prediction

Alongside the point forecasts of the exchange rate dynamics, we quantify the uncertainty associated with the NARFIMA model predictions using conformal prediction intervals. The distribution-free conformal prediction approach converts uncertainty scores to prediction intervals that contain the true outcome vovk2005algorithmic. This model-agnostic method offers several advantages over simulation-based prediction intervals, including computational efficiency, fewer assumptions about the underlying data distribution, and guarantees coverage. In the context of time series setup, the conformal prediction framework utilizes the sequential ordering of the data to generate the prediction interval. Given the training set $\{\left(y_t, \tilde{x}_t\right)_{t=1}^T\}$, where $y_t$ represents the target series and $\tilde{x}_t$ indicates the set of features including lagged values of $y_t$, ARFIMAx residuals, and the exogenous variables, we apply the NARFIMA framework and an uncertainty model ($\widehat{\Psi}$) on $\tilde{x}_t$ to generate a measure of uncertainty. The conformal score $\tilde{S}_{t}$ is then calculated as: $\tilde{S}_{t} = \frac{\left|y_{t} - \operatorname{NARFIMA}\left(\tilde{x}_{t} \right)\right|}{\widehat{\Psi}(\tilde{x}_{t})}.$ Due to the temporal dependencies in the series, the conformal quantiles, computed using a weighted conformal method with a fixed window of size $\tau$ defined as $\omega_{\tilde{t}} = \mathbb{1}\{ \tilde{t} \geq t - \tau\} \;\; \forall \; \tilde{t} < t$, are as follows:

Using these weight-adjusted quantiles, the conformal prediction interval at time step $t$ with $100(1-\alpha)$% confidence is given by $ \left[\operatorname{NARFIMA}\left(\tilde{x}_{t} \right) \pm {\operatorname{CQ}}_{t} \; \widehat{\Psi}\left(\tilde{x}_{t} \right)\right].$ Fig. (ref) represents the 95% conformal prediction intervals of the NARFIMA framework for the long-term 48-month-ahead forecast horizon. The figure depicts the point forecasts of the exchange rate series generated by the NARFIMA model and the two best-performing benchmarks, namely ARIMAx and BSTSx, as identified by RMSE-based MCB test results (Fig. (ref)). From the plots, it is evident that the proposed architecture can better capture the fluctuations in the exchange rate dynamics of the BRIC economies in comparison to the ARIMAx and BSTSx models. Furthermore, the width of conformal prediction intervals for the NARFIMA model varies across different countries, although it can still consistently capture most of the variations in the exchange rate datasets. However, for Brazil and Russia, both point forecasts and prediction intervals of NARFIMA and other leading models occasionally fail to capture the inherent dynamics of the exchange rate series, primarily due to abrupt macroeconomic fluctuations triggered by the COVID-19 pandemic and geopolitical conflicts, respectively. The overall analysis thus offers insights into uncertainties associated with the exchange rate forecasts for the BRIC economies.

Sensitivity Analysis of Residual Selection

This section examines the sensitivity of the NARFIMA model to residual selection. The choice of feedback residuals plays a crucial role in designing the overall forecasting framework, as it directly influences how well long-term dependencies are captured. In the proposed NARFIMA architecture, long-memory modeling is incorporated into the neural network by utilizing the residuals of the ARFIMAx model, along with exchange rate series and macroeconomic covariates. To empirically validate the importance of ARFIMAx residuals, we conduct an iterative forecast evaluation in two ways. First, we replace them with residuals from ARIMAx, BSTSx, and Naïve, the three best-performing benchmark models identified in the MCB plot, while keeping the neural network structure unchanged. This results in three NARFIMA variants, namely Neural ARIMAx (NARIMA), Neural BSTSx (NBSTS), and Neural Naïve (NNaïve), respectively. Second, we examine a residual-free variant of NARFIMA, identical to ARNNx, where the neural network is trained solely on exchange rate series and macroeconomic covariates, removing residual feedback altogether. This empirical evaluation aims to assess how residual selection affects overall forecast accuracy. The MCB plot in Fig. (ref) demonstrates the statistical significance of the performance improvement among NARFIMA and its variants in terms of the RMSE metric. As evident from the plot, NARFIMA achieves the lowest average rank across all BRIC economies and forecast horizons, further validating the superiority of our proposal over its variants. These findings confirm that the forecasting performance of NARFIMA relies on both the neural network and ARFIMAx residuals, which optimizes its modeling capabilities.

Policy Implications

{Accurate forecasting of spot exchange rates is of immense importance for the central banks, especially within emerging market economies such as BRIC. Exchange rate movements substantially influence macroeconomic stability through multiple channels, including trade competitiveness, inflation dynamics, capital flows, and the effectiveness of monetary policy. Given the varied exchange rate regimes among the BRIC economies, from managed floats to more flexible arrangements, precise forecasting is essential for mitigating external shocks, guiding foreign exchange interventions, and enhancing overall economic resilience. The recent acceleration of de-dollarization efforts in the BRIC economies, which aim to reduce dependence on the USD in trade, reserves, and settlement, has further intensified the complexity of exchange rate dynamics by introducing new drivers alongside traditional macroeconomic determinants. This shift toward greater currency diversification in trade and reserves increases the need for models that can simultaneously capture long-term dollar spillover effects and emerging local currency influences. In particular, the BRIC economies deepen their integration within global trade and financial networks while simultaneously seeking greater monetary sovereignty. From the central banks' standpoint, improved accuracy in exchange rate forecasts enables enhanced management of foreign exchange reserves, smoother market interventions, and more effective transmission of monetary policy. Reliable forecasts also equip policymakers to proactively address capital flow volatility, assess external vulnerabilities accurately, and adjust interest rates strategically to maintain economic stability.

Given these insights, BRIC central banks must adapt their policy frameworks and exchange rate forecasting models to reflect de-dollarization dynamics. As BRIC nations increase trade settlement in local currencies, forecasting models must account for shifting demand patterns for BRIC currencies. The persistence of partial dollarization requires the models to incorporate both dollar-based and local currency-based trade flows, ensuring an accurate representation of exchange rate drivers. While the BRIC economies are reducing their reliance on dollar reserves, empirical data suggest that the USD still dominates global reserves. This means that exchange rate forecasting models must still incorporate US monetary policy spillovers, even as BRIC central banks increase holdings in gold, yuan, and other non-dollar assets. This aligns with empirical evidence underscoring the pivotal role of uncertainty measures in exchange rate forecasting, particularly within emerging markets pursuing de-dollarization. A recent study by abid2020economic confirms that heightened EPU induces short-run and long-run depreciation pressures on emerging market currencies, suggesting that integrating EPU into forecasting models becomes even more critical during transitions away from dollar dependence. Further, christou2018role highlights the predictive power of EPU on exchange rate volatility under extreme market conditions, underscoring its relevance in forecasting strategies during periods of monetary transition. The significant impact of US MPU on global currency volatility, as highlighted by mueller2017exchange, reinforces the necessity for the BRIC economies to incorporate MPU into predictive frameworks, particularly as they attempt to reduce vulnerability to Fed policy shifts through currency diversification. The causal relationships identified in Table (ref) affirm the importance of these uncertainty indicators as causal drivers for BRIC exchange rates.

In this context, the proposed NARFIMA model provides a robust framework capable of dynamically capturing the complex influences of uncertainty measures, country-specific short-term IRD, and oil shocks on exchange rates. The model's capacity to incorporate these key causal drivers is particularly valuable as the BRIC economies navigate the transition toward de-dollarization, where currency valuations become increasingly sensitive to both global uncertainties and structural shifts in international monetary arrangements. Additionally, our proposed NARFIMA model can address nonlinearities, long memories, and structural breaks inherent in de-dollarization processes, equipping central banks with a powerful tool to anticipate exchange rate movements in an environment characterized by evolving currency preferences and settlement mechanisms. Although initiatives such as the BRICS Contingent Reserve Arrangement and expanded currency swap agreements can help mitigate exchange rate volatility, the absence of a unified BRIC currency system means that fluctuations across national currencies continue to affect inter-BRIC trade. Finally, the study underscores the need for policy coordination among BRIC nations in the face of shared vulnerabilities to oil shocks and global uncertainty. A forecasting tool like NARFIMA that incorporates both domestic fundamentals and global risk drivers can serve as a basis for joint monitoring mechanisms, facilitating greater financial cooperation and resilience building within the bloc. }

Conclusion and Discussion

{This paper introduces a Neural ARFIMA model to accurately forecast the spot exchange rate dynamics in BRIC countries by taking into account various uncertainty measures, oil shocks, and country-specific short-term IRD. Our proposed approach effectively combines the memory properties of fractionally integrated processes with the flexible nonlinear mapping capabilities of neural networks, creating a powerful methodological solution for capturing both long-range dependencies and complex nonlinear patterns in exchange rate data. Our empirical results across BRIC economy exchange rates demonstrate that the proposed NARFIMA model consistently outperforms traditional statistical and state-of-the-art deep learning approaches across various forecasting horizons. The conditions for asymptotic stationarity and geometric ergodicity of the NARFIMA process are established by analyzing the asymptotic behavior of the associated Markov chain. Under appropriate parameter constraints on skip connections and activation function characteristics, we prove that the NARFIMA process converges to a unique stationary distribution at a geometric rate, regardless of initial conditions. This mathematical framework provides critical assurance regarding the model's long-term behavior, validating its application to exchange rate forecasting problems where stability and convergence are essential properties. The nonlinear GC test depicted the complex interplay between key macroeconomic drivers and exchange rate series. By focusing on spot exchange rates rather than real effective exchange rates, our framework targets the price directly faced by market participants in trade, investment, and policy operations, ensuring that the forecasts are immediately relevant for day-to-day decision-making in foreign exchange management. The NARFIMA model, therefore, offers significant practical value for central banks and financial institutions engaged in monetary policy formulation and foreign exchange operations, particularly during periods of heightened uncertainty. While our current implementation focuses on the BRIC economies and incorporates several key uncertainty measures, future research could extend the NARFIMA framework to include additional factors such as climate risk indicators and social media-based uncertainty measures derived from platforms like Twitter (X). Furthermore, alternative neural network architectures could be explored within the NARFIMA structure to potentially enhance modeling flexibility and forecasting accuracy. The model could also be adapted to forecast other financial indicators characterized by long memory and nonlinear dynamics, including interest rates, commodity prices, and equity market volatility. Given its robust theoretical foundations and empirical performance, NARFIMA presents a promising direction for advancing our understanding of complex economic relationships in an increasingly interconnected global economy. Although the NARFIMA model is primarily developed for long memory and nonlinear time series problems arising in BRIC exchange rate series, it can also be useful for similar complex temporal data problems arising in epidemiology, demand forecasting, and climatology.}

Competing interests

There are no competing interests to be declared.

Data and Code Availability Statement

The spot exchange rate, along with all macroeconomic indicators used in this analysis, is obtained from the Federal Reserve Economic Data (FRED) repository: \url{https://fred.stlouisfed.org}. Data for all uncertainty indicators are sourced from the Economic Policy Uncertainty website: \url{https://www.policyuncertainty.com}. The code and data necessary to reproduce the results of this study are available at: \url{https://github.com/mad-stat/NARFIMA}.