Exchange rate volatility modeling: ARIMAX-FIGARCH with wavelet-based outlier detection for USD/PEN

Abstract

Introduction:

Exchange rate volatility in partially dollarized emerging economies poses persistent challenges for financial risk management and monetary policy. Short-memory GARCH specifications provide inferior fit relative to long-memory alternatives under managed floating regimes with recurrent political disruptions.

Methods:

This study proposes a two-stage framework: a wavelet-dummy operator maps the return series into a binary indicator of atypical observations incorporated as an exogenous regressor in an ARIMAX(2,0) model; filtered residuals serve as input for five conditional variance specifications-ARCH(1), GARCH(1,1), GJR-GARCH(1,1,1), EGARCH(1,1,1), and FIGARCH(1,d,1)-applied to 2,803 daily log-returns of the USD/PEN (January 2015-March 2026).

Results:

The operator identifies 36 atypical observations concentrated in the 2020-2023 sub-period. FIGARCH(1,d,1) dominates under the Akaike Information Criterion with d = 0.553 (p < 0.001) under Skewed Student-t innovations, extending long-memory evidence for USD/PEN conditional variance to 2015-2026 and resolving prior ambiguity in the Peruvian Forex literature. Ljung-Box tests confirm white noise in both standardized residuals (p = 0.4846) and squared residuals (p = 0.9921). Predictive evaluation across three validation windows yields Theil's U between 0.93 and 1.02, indicating mean-forecast accuracy comparable to the random-walk benchmark, while expanding-window volatility forecasts consistently outperform the constant-variance benchmark.

Discussion:

These findings demonstrate that wavelet-based outlier detection combined with long-memory GARCH estimation substantially improves volatility modeling for emerging-market currencies under managed floating regimes, institutional fragility and partial dollarization.

1 Introduction

Exchange rate volatility in partially dollarized emerging economies constitutes a persistent source of financial risk with direct implications for corporate treasury management, currency hedging, and the conduct of monetary policy [1, 2]. The interaction between financial dollarization and exchange rate uncertainty is particularly consequential: when households and firms hold liabilities denominated in foreign currency as a form of business-cycle insurance against domestic income fluctuation [3], abrupt movements in the nominal exchange rate translate directly into balance-sheet stress, potential credit contractions, and an amplification of external shock transmission to the domestic financial system [4]. In economies such as Peru, where foreign-currency liabilities coexist with active central bank intervention and recurrent episodes of political and institutional instability, the precise quantification of exchange rate conditional variance becomes an operational priority [5].

The Peruvian sol (PEN) operates under a managed floating regime conducted by the Central Reserve Bank of Peru (BCRP), which intervenes in the foreign exchange market to mitigate excessive fluctuations while allowing the exchange rate to respond to fundamental economic forces. As shown in [4], using Bayesian estimation of a New Keynesian DSGE model for the Peruvian economy, the best-performing monetary policy specification combines a Taylor rule that does not directly respond to exchange rate movements with an active foreign exchange intervention rule. This configuration is consistent with the BCRP's inflation-targeting framework augmented by financial risk management considerations. Notably, the responsiveness of foreign exchange interventions to exchange rate fluctuations increased almost nine fold between the pre-inflation-targeting and inflation-targeting periods, reflecting the growing importance of managing financial dollarization risks as Peru deepened its integration into global capital markets.

Between 2015 and 2026, the USD/PEN series has been subjected to a sequence of shocks of both external and domestic origin: the reversal of the commodity price cycle that intensified from 2015 onward, the political instability associated with successive presidential vacancies and congressional dissolutions between 2016 and 2021, and the COVID-19 pandemic, which generated unprecedented volatility in March–April 2020. Recent evidence confirms that uncertainty shocks systematically weaken emerging market currencies and tighten financial conditions across Latin America, including Peru [6]. The modified ICSS algorithm applied to daily USD/PEN data identified eight statistically significant breakpoints [7] in the unconditional variance process, documenting that the full-sample GARCH(1,1) parameter sum α + β ≈ 0.999 substantially overstates true volatility persistence once structural breaks are accommodated within each subsample—a finding consistent with the broader literature establishing that unmodeled variance breaks generate spurious evidence of integrated or near-integrated volatility [8, 9]. Likewise, [10] demonstrate, within a Markov-Switching Vector Autoregressive framework, that the Peruvian sol is the most volatile and externally sensitive currency among the four members of the Mercado Integrado Latinoamericano (MILA), exhibiting a statistically significant response to US Monetary Policy Uncertainty in both the high- and low-volatility regimes—the only MILA currency for which this dual-regime sensitivity is detected. The return series of exchange rates in these economies exhibit well-documented statistical regularities: conditional heteroscedasticity, volatility clustering, heavy tails, and asymmetric responses to shocks [11, 12]. These features have driven the development of a broad family of conditional variance models whose capacity to capture financial volatility dynamics has been extensively validated in the literature [11–15].

A recurring methodological challenge in this context is the presence of localized extreme innovations—associated with electoral events, institutional crises, or macroeconomic disruptions—that are not generated by the persistent dynamics of the conditional variance but by episodic shocks of an idiosyncratic nature. Evidence from wavelet-based analyses of emerging market currencies confirms that interactions between financial stress and exchange rate uncertainty become significantly more pronounced during political events than during economic developments [16]. The USD/PEN series illustrates this phenomenon clearly: between 2015 and 2026, episodes such as the 2021 electoral process, the political instability of 2022–2023, and the COVID-19 pandemic generated extreme returns that fall outside the series' habitual distribution, as evidenced by the pronounced leptokurtosis documented for daily USD/PEN returns [7]. When these events are not treated explicitly, they distort the estimators of the mean equation and contaminate the variance equation, compromising both the goodness of fit and the diagnostic validity of the residuals. Indeed, the residuals of fitted GARCH-type models are known to exhibit excess kurtosis precisely because of the unaddressed presence of outliers in the returns, which bias parameter estimates and reduce forecast reliability [17].

The literature has addressed this challenge from three complementary directions. First, the GARCH family has been extended toward asymmetric specifications—GJR-GARCH [13] and EGARCH [14]—and toward long-memory models such as FIGARCH [15], which captures the hyperbolic dependence of the conditional variance through a fractional differencing parameter d ε (0, 1). The superiority of FIGARCH for exchange rate volatility forecasting has been documented in seminal contributions, and [18] confirm for the EUR/USD pair that FIGARCH(1,d,1) with skewed Student innovations outperforms all short-memory benchmark models—including GARCH, EGARCH, APARCH, GJR-GARCH, and IGARCH—across 1-day, 1-week, and 1-month forecast horizons, and explicitly identify wavelet-based multiresolution analysis as a natural extension of their framework. Long-memory specifications equally dominate in emerging-market contexts: [19] confirm the presence of long memory in the NGN/USD exchange rate (Hurst exponent H = 0.83) and show that the ARFIMA-GARCH hybrid substantially improves both fit and forecast accuracy over stand-alone ARFIMA specifications. For the Peruvian exchange rate market specifically, prior analysis yields only weak and inconclusive evidence of long memory for the 1997–2013 period [20], and suggests that apparent persistence may reflect random level shifts rather than genuine fractional integration [21]—a concern that recent cross-currency evidence corroborates for a broad set of emerging-market currencies [22]. Second, the Discrete Wavelet Transform (DWT) has emerged as a robust tool for detecting extreme observations in financial series [23]. A universal threshold based on the median absolute deviation [24] does not require normality of the original series, while an automated procedure based on the mean absolute deviation of wavelet coefficients [25] is applicable to general time-series processes without prior model specification. More recent work has extended these techniques to financial series with conditional heteroscedasticity: [26] proposed a wavelet-based outlier detection and correction method applied to GARCH model residuals. This framework was generalized [17] by applying the Maximal Overlap Discrete Wavelet Transform (MODWT) to residuals from any fitted volatility model, computing quantile thresholds via Monte Carlo simulation under Normal and Student-t distributions, and distinguishing additive level outliers (ALO) from additive volatility outliers (AVO)—critically removing the power-of-two sample size restriction of the standard DWT and enabling detection of outlier patches in a single pass. Third, several studies have combined wavelet techniques with long-memory GARCH models through spectral estimation strategies aimed at improving estimation of the fractional integration parameter [27], rather than focusing on the explicit detection and treatment of localized shocks through indicator variables.

Although these advances have substantially enriched the available methodological toolkit, the approaches found in the literature differ from the one proposed in this article in a fundamental respect: in those works, wavelets serve a function of frequency analysis, spectral estimation, or correction of the original series—a function that modifies r_t before or during estimation. The formalization of this procedure as an explicit mathematical operator—one that maps the series into the pair (r_t, D_t) and deterministically incorporates extreme shocks into the mean equation of an ARIMAX model whose variance structure includes long-memory specifications—has not been addressed systematically in the exchange rate volatility literature for economies characterized by high institutional fragility and partial dollarization. Moreover, the systematic application of this operator to emerging-market currencies subject to recurrent political disruptions and financial dollarization risks has not been empirically validated, despite the documented evidence that such economies exhibit outlier-contaminated volatility dynamics that standard GARCH specifications fail to capture adequately [17, 26]. The pre-estimation, exogenous design of ȷ directly addresses the concern that apparent long memory in exchange rate volatility may be spurious—generated by localized shocks rather than genuine fractional integration [21, 22]—by removing those shocks before any parametric estimation is performed.

The present study addresses these gaps through three specific contributions. First, it formalizes a wavelet-dummy operator T: {r_t} → ({r_t}, {D_t}) as a deterministic, pre-estimation transformation that resolves the circularity inherent in residual-based outlier detection methods [17, 26]—enabling the explicit separation of persistent volatility dynamics from episodic shocks without altering the distributional structure of the return series. Second, it provides the first systematic comparison of five conditional variance specifications—ARCH(1), GARCH(1,1), GJR-GARCH(1,1,1), EGARCH(1,1,1), and FIGARCH(1,d,1)—applied to 2,803 daily USD/PEN log-returns (January 2015–March 2026), identifying FIGARCH(1,d,1) as the dominant specification (AIC = 3,074.63, d = 0.553, p < 0.001) and resolving the prior ambiguity in the Peruvian Forex literature [20, 21] regarding whether conditional variance long memory is genuine or spurious. Third, it evaluates the predictive performance of the proposed framework across three validation horizons, documenting mean-forecast accuracy comparable to the naïve random-walk benchmark across the three horizons (Theil's U = 0.93–1.02)—consistent with the forecast evaluation metric paradox documented in [28]—and providing operational evidence for the framework's applicability to medium-term FX risk management in partially dollarized emerging economies under managed floating regimes. The remainder of the article is organized as follows: The theoretical framework is presented in Section 2; Section 3 describes the materials and methods; Section 4 reports the results; and Section 5 discusses the implications.

2 Theoretical framework

2.1 Conditional heteroscedasticity: from ARCH to GARCH

Financial return series systematically violate the assumption of constant variance. The Autoregressive Conditional Heteroscedasticity (ARCH) model, introduced by [29], was the first framework to formalize this dependence by allowing the conditional variance to be a function of past squared innovations. Let r_t denote the log-return of an exchange rate at time t. The ARCH (q) model is defined by the system shown in Equations 1, 2:

where ω > 0 is the variance intercept and α_i ≥ 0 ensures non-negativity of the conditional variance. Although the ARCH (q) model captures volatility clustering, it requires a large number of lags to represent persistent shocks. This limitation was resolved by [30], who proposed the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model, which adds lagged conditional variances. The GARCH(1,1) specification takes the form:

with constraints ω > 0, α₁ ≥ 0, β₁ ≥ 0, and α₁ + β₁ < 1 for covariance stationarity. The persistence sum α₁ + β₁ measures the decay rate of volatility shocks: values close to unity indicate that innovations have long-lasting effects on the conditional variance [11, 12].

2.2 Asymmetric GARCH specifications: GJR-GARCH and EGARCH

Symmetric GARCH models impose an equal response of conditional variance to positive and negative innovations of the same magnitude, a restriction empirically rejected for most financial series. The leverage effect—whereby negative shocks tend to increase volatility more than positive shocks of equal size—requires asymmetric specifications. Two models have become canonical in this regard. The GJR-GARCH(1,1,1) model [13] augments the variance equation with an indicator function that amplifies the effect of negative innovations:

where I_{t − 1} = 1 if ε_{t − 1} < 0 and 0 otherwise, and γ₁ ≥ 0 is the asymmetry parameter. A statistically significant positive γ₁ constitutes direct evidence of the leverage effect, since the total impact of a negative shock is α₁ + γ₁, while a positive shock contributes only α₁.

The Exponential GARCH (EGARCH) model [14] specifies the logarithm of the conditional variance to guarantee non-negativity without sign constraints on parameters, and captures both magnitude and sign effects of innovations:

where z_{t − 1} = ε_{t − 1}/σ_{t − 1} is the standardized innovation. The α₁ term captures the magnitude effect and γ₁ the sign effect; a negative and significant γ₁ confirms the leverage effect. Because the specification is in logarithms, no sign restriction on α₁ or γ₁ is needed, which confers additional flexibility relative to GJR-GARCH [14].

2.3 Long memory in conditional variance: FIGARCH

A distinctive feature of financial volatility is the slow hyperbolic decay of the autocorrelation function of squared returns, which indicates that shocks to conditional variance are neither short-lived (GARCH) nor permanently persistent (IGARCH). The Fractionally Integrated GARCH (FIGARCH) model [15] accommodates this intermediate persistence through a fractional differencing parameter d ε (0, 1). The FIGARCH(1,d,1) model is expressed as:

where L is the lag operator, v_t = – is the innovation to the conditional variance process, ϕ (L) and β (L) are lag polynomials, and (1–L)^d is the fractional differencing operator with binomial expansion:

The fractional parameter d governs the memory properties: for d = 0 the model reduces to GARCH(1,1); for d = 1 it becomes IGARCH; and for 0 < d < 1 the impact of past innovations decays hyperbolically, defining the long-memory regime. The conditional variance formulation is:

[18] find that FIGARCH(1,d,1) with skewed Student innovations achieves the lowest AIC and best out-of-sample forecasting accuracy for EUR/USD across 1-day, 1-week, and 1-month horizons, with an estimated fractional parameter of d = 0.431, statistically significant at the 1% level. Long-memory specifications equally dominate in emerging-market contexts: [19] confirm long memory in the NGN/USD series (Hurst exponent H = 0.83) and show that the Autoregressive Fractionally Integrated Moving Average (ARFIMA)-GARCH hybrid substantially outperforms short-memory counterparts. For the Peruvian exchange rate market, prior studies report inconclusive results: [20] find only weak evidence of long memory in Forex volatility for the 1997–2013 period, while [21] argue that the apparent persistence may reflect random level shifts rather than genuine fractional integration. These findings motivate the inclusion of FIGARCH as the primary long-memory benchmark in the present study. As shown in Section 4.2, the estimated fractional parameter for the USD/PEN series (d = 0.553) exceeds slightly the value reported by [18] for EUR/USD (d = 0.431), consistent with the higher volatility persistence expected in a partially dollarized emerging-market currency. Long-memory specifications have also been validated for exchange rate volatility in emerging economies using FIGARCH models that account for structural breaks and frictions [31]. Long memory in conditional volatility has been empirically confirmed in frontier equity markets, with FIGARCH under Student-t innovations providing superior fit for the Portuguese PSI-20 index, with a significant fractional differencing parameter d = 0.397 (p < 0.01) [32].

2.4 Wavelet-based outlier detection

The Discrete Wavelet Transform (DWT) decomposes a time series into components localized simultaneously in time and frequency, making it particularly suited for detecting anomalous observations in financial series [23]. Given a series {r_t} of length n, the DWT with mother wavelet ψ produces wavelet coefficients W_j,k at decomposition level j and translation k as defined in Equation 9:

The wavelet coefficients are sensitive to jumps, outliers, and structural discontinuities by virtue of their oscillatory, zero-mean nature: their absolute values are large precisely at time positions where irregular features occur [23]. The choice of the Daubechies db4 filter—with four vanishing moments—suppresses polynomial trends up to degree three and provides localization properties superior to the simpler Haar wavelet, making it appropriate for daily financial return series.

Threshold selection follows the universal median absolute deviation (MAD)-based rule of [24], which estimates the noise standard deviation robustly from the first-level detail coefficients cD₁ without assuming normality:

The detection threshold is then:

The divisor 0.6745 ensures consistency with the normal standard deviation, while the MAD itself provides robustness against the very outliers being detected. The multiplier k = 3 corresponds to a three-sigma threshold [24]. This approach was extended [25] by basing the threshold on the mean absolute deviation of wavelet coefficients for general time-series processes. Wavelet-based detection was subsequently applied to GARCH model residuals [26] operating on standardized residuals after model fitting. This procedure was further generalized [17] using the Maximal Overlap DWT (MODWT), which processes series of any sample size by removing the power-of-two restriction of the standard DWT, defines quantile thresholds through Monte Carlo simulation under Normal and Student-t distributions, and distinguishes additive level outliers (ALO)—affecting only the mean—from additive volatility outliers (AVO), which also contaminate the variance equation and bias GARCH estimates.

2.5 The wavelet-dummy operator and the ARIMAX mean equation

The central methodological contribution of this study is the formalization of a pre-processing operator that transforms the raw return series into a structured pair suitable for joint estimation with long-memory volatility models. Let {r_t} be the series of log-returns over t = 1, …, T. Define the wavelet-dummy operator:

where {D_t} is a binary indicator sequence constructed as follows. Apply the DWT with Daubechies db4 filter to {r_t} and extract the first-level detail coefficients cD₁. Compute threshold λ per Equation 11. Then:

The operator 𝒯 defines a deterministic mapping over the return space that preserves the original stochastic process while generating an auxiliary binary sequence with sparse support. This sequence encodes the temporal location of discontinuities, allowing the decomposition of the observed process into a persistent component and a localized shock component without modifying the underlying distribution of returns. Critically, D_t is constructed from the wavelet decomposition of the raw series before any parametric model is estimated, so it is exogenous to the conditional variance process by construction. This design avoids the circularity arising when outliers are identified from contaminated model residuals. Together, these properties produce three identifiable statistical advantages: (i) consistent estimation of the long-memory parameter d by eliminating shock contamination before variance estimation; (ii) valid asymptotic inference on the ARIMAX coefficients—the intervention dummy Dt is treated as a deterministic exogenous regressor following the framework developed in Chapter 8 of [33], while inference for the ARIMAX-FIGARCH specification relies on the quasi-maximum likelihood framework described in Section 3.4; and (iii) a parsimonious specification in which a single binary regressor absorbs the mean impact of 36 extreme events, avoiding the rank deficiency that would result from including individual dummy variables for each atypical observation.

The output of 𝒯 is incorporated into an ARIMAX (p,q) model as a deterministic exogenous regressor in the mean equation:

where μ is the unconditional mean, ϕ_i are the autoregressive coefficients, θ_j are the moving-average coefficients, and β is the coefficient of the wavelet-dummy indicator. The residuals ε_t from Equation 14 serve as input to the conditional variance specifications of Sections 2.1–2.3. The optimal ARIMAX order (p,q) is selected by exhaustive search over p, q ε {0, …, 7} using the Akaike Information Criterion (AIC), with individual coefficient significance verified at the 5% level.

The full estimation pipeline is a two-stage procedure. In the first stage, 𝒯 maps {r_t} into ({r_t}, {D_t}), and the ARIMAX (p,q) model is estimated to produce filtered residuals {ε_t}. In the second stage, the five conditional variance models—ARCH(1), GARCH(1,1), GJR-GARCH(1,1,1), EGARCH(1,1,1), and FIGARCH(1,d,1)—are estimated on {ε_t} and ranked by AIC. Adequacy is verified by Ljung-Box portmanteau tests on both standardized residuals and squared standardized residuals. The preferred specification is the one with the lowest AIC that simultaneously satisfies both diagnostic tests. The complete specification of data sources, software implementation, and estimation parameters is detailed in Section 3.

3 Materials and methods

3.1 Data sources and sample

The empirical analysis uses daily nominal exchange rate data for the US Dollar/Peruvian Sol (USD/PEN) pair, obtained from the Superintendencia de Banca, Seguros y AFP (SBS) of Peru, the official source for exchange rate data in Peru. The series covers the period from 2 January 2015 to 31 March 2026, yielding 2,804 daily price observations and 2,803 log-return observations after first-differencing. The sample was selected to encompass structurally distinct episodes: the commodity price reversal of 2015–2016, the political instability associated with four consecutive presidential administrations between 2016 and 2022, the COVID-19 pandemic (2020–2021), and the post-Castillo political crisis of 2022–2023.

Daily log-returns are computed as in Equation 15:

where P_t denotes the nominal exchange rate at the close of day t. This transformation yields a stationary series expressed in percentage points of continuously compounded daily returns. Table 1 reports descriptive statistics for the USD/PEN log-return series.

Table 2 reports the results of stationarity and ARCH-effect tests applied to the USD/PEN log-return series.

Table 3 reports the Lagrange multiplier test for ARCH effects applied to the log-return series.

3.2 Wavelet-dummy pre-processing

The wavelet-dummy operator 𝒯 defined in Equations 12, 13 is applied to each return series prior to any parametric estimation. The DWT is implemented with a Daubechies db4 mother wavelet at decomposition level j = 1, extracting the first-level detail coefficients cD₁, which capture high-frequency daily variations and are therefore most sensitive to episodic shocks in daily return series. The noise standard deviation is estimated robustly via Equation 10 using the MAD of cD₁, and the detection threshold is set at k = 3 robust standard deviations per Equation 11.

For the USD/PEN series, the wavelet pre-processing yields the parameters reported in Table 4.

The 36 atypical observations (1.28% of the sample) are concentrated pre-dominantly in the 2020–2023 sub-period, coinciding with the onset of the COVID-19 pandemic (March 2020), the electoral process and change of government (April–August 2021), the Russian invasion of Ukraine combined with domestic political instability (February–April 2022), and the post-impeachment crisis following the removal of President Castillo (January 2023). The single largest return in the series (+3.01%) was recorded on 30 July 2021, 2 days after the presidential inauguration of Pedro Castillo. The constructed binary vector D_t enters as an exogenous regressor in the ARIMAX mean equation estimated in the following subsection.

3.3 ARIMAX mean equation estimation

The mean equation is specified as in Equation 14, with D_t as the exogenous regressor produced by the operator 𝒯. The optimal autoregressive order p and moving-average order q are determined by exhaustive grid search over p, q = {0, 1, …, 7}, minimizing the Akaike Information Criterion (AIC) subject to the constraint that all estimated coefficients are individually significant at the 5% level. This two-criterion selection procedure prevents the retention of spuriously significant lags.

For the USD/PEN series, the search identifies ARIMAX(2,0) as the optimal specification (AIC = 1,967.22; BIC = 1,996.92). Table 5 reports the estimated coefficients.

The filtered residuals ε_t from the ARIMAX(2,0) model constitute the input series for the conditional variance estimation described in the following subsection.

3.4 Conditional variance model estimation

Five conditional variance specifications are estimated on the ARIMAX residuals: ARCH(1), GARCH(1,1), GJR-GARCH(1,1,1), EGARCH(1,1,1), and FIGARCH(1,d,1), as defined in Equations 2–8 of the Theoretical Framework. All models are estimated by Maximum Likelihood (ML) under conditional normality. Although the Jarque–Bera test rejects normality of standardized residuals, the consistency of quasi-maximum likelihood estimators under non-normal innovations justifies this specification for comparative and inferential purposes. To assess the sensitivity of the long-memory estimates to the distributional assumption, the FIGARCH(1,d,1) model is additionally estimated under three alternative innovation distributions: Student-t, Skewed Student-t, and the Generalized Error Distribution (GED). These specifications are selected to capture documented in Table 1. Model comparison across distributions follows the AIC. Model selection follows the AIC, with smaller values indicating better fit adjusted for parsimony. Adequacy of each fitted model is verified by the Ljung-Box portmanteau test [34] applied to the standardized residuals z_t = ε_t/σˆ_t and to their squares , using 10 lags. Non-rejection of the null of no serial correlation in both z_t and simultaneously is required for a model to be considered adequately specified. The ARCH(1) model is retained as a baseline benchmark despite its known limitations in capturing volatility persistence, to quantify the improvement from adding GARCH dynamics.

All estimations are performed in Python 3.12.13 using the arch library (version 8.0.0), which implements MLE for the full GARCH family including FIGARCH via the Baillie–Bollerslev–Mikkelsen parameterization [15]. The fractional differencing parameter d in FIGARCH is estimated jointly with the other parameters, constrained to the interval (0, 1) to ensure the long-memory regime. Replication code and estimated output are deposited at Zenodo (doi: 10.5281/zenodo.19655851).

3.5 Forecasting design and evaluation metrics

Predictive performance is evaluated over three out-of-sample validation windows as an empirical validation of the proposed operator-enhanced volatility framework, assessing whether the separation between persistent dynamics and localized shocks translates into improved forecast accuracy. Each model is estimated once on the corresponding training sample and generates multi-step-ahead (dynamic) forecasts for the full validation horizon without re-estimation. This design replicates the operational setting of a risk manager who calibrates a model at a fixed point in time and projects forward.

Because all five conditional variance specifications share the same ARIMAX(2,0) mean equation, the point forecasts of the return r_t are numerically identical across models. Model differentiation is therefore assessed exclusively through the conditional variance forecasts, the information criteria of Table 6a (Section 4), and the Ljung-Box diagnostics on squared residuals. Forecast accuracy for the mean equation is evaluated using four metrics, defined in Equations 16–19:

where n is the number of observations in the validation window, r_t is the observed log-return, and rˆ_t is the model forecast. The naïve benchmark in Equation 19 is the random walk without drift, which sets rˆ_t = 0 for all horizons. Theil's U < 1 indicates that the model outperforms the naïve benchmark; U > 1 indicates the opposite. The choice of multiple evaluation criteria is deliberate—the forecast evaluation literature documents that different accuracy metrics can yield conflicting model rankings [28], so that reliance on a single measure may produce misleading conclusions about model performance. For this reason, results are reported for all four metrics simultaneously, and no single metric is used as the sole selection criterion.

The MAPE metric is retained for completeness but interpreted with caution: when observed log-returns are close to zero—as is typical for daily exchange rate returns—MAPE values are inflated by near-zero denominators and lose their comparative informativeness. The primary selection metrics are therefore RMSE and Theil's U.

3.6 Use of artificial intelligence tools

During the preparation of this manuscript, the authors used Claude (Anthropic) for language editing and translation support, and SciSpace for bibliographic search assistance. The authors retain full responsibility for the conceptual development, analytical decisions, and content of this publication.

4 Results

4.1 Outlier detection and temporal distribution

The application of the wavelet-dummy operator 𝒯 to the USD/PEN log-return series yields the detection parameters reported in Table 4. With a MAD-based threshold of λ = 0.6943, the procedure identifies 36 atypical observations, representing 1.28% of the total sample of 2,803 daily returns. This proportion is consistent with the theoretical expectation under a three-sigma rule applied to a near-normal residual distribution, and confirms that the operator is not over-detecting noise but capturing genuine episodic disruptions.

Table 7 reports the temporal distribution of the detected outliers across the sample period, grouped by contextual episode.

Figure 1 displays the full USD/PEN log-return series over the sample period, with the 36 atypical observations identified by the operator marked in red. The visual inspection confirms the concentration of outliers in the 2020–2023 sub-period and the relative stability of the series during 2015–2019, interrupted only by the isolated episode of regional emerging market volatility in early 2016.

Figure 1

4.2 Conditional variance model comparison

Table 6a reports the AIC, BIC, and log-likelihood values for the five conditional variance specifications estimated on the ARIMAX(2,0) filtered residuals. The models are ranked from lowest to highest AIC.

Table 6b reports the AIC, BIC, log-likelihood, and estimated fractional parameter d for FIGARCH(1,d,1) estimated under four innovation distributions.”

Across all four distributions, the fractional differencing parameter d remains statistically significant at the 1% level, ranging from 0.485 under Normal innovations to 0.553 under Skewed Student-t—the specification with the lowest AIC(949.56). The consistency of d across distributional assumptions provides robust evidence that long memory in USD/PEN conditional variance is a genuine structural feature rather than an artifact of the normality assumption. All four specifications pass the Ljung-Box diagnostic tests on standardized and squared residuals, confirming model adequacy regardless of the innovation distribution. The asymmetry parameter of the Skewed Student-t distribution (λ = −0.0608, p = 0.015) is statistically significant, indicating a mild left-skew in the innovation distribution—consistent with the negative skewness documented for USD/PEN returns in Table 1.

The dominance of FIGARCH(1,d,1) over all short-memory specifications—with the lowest AIC of 3,074.63, an AIC advantage of 25.14 units over EGARCH and 26.13 over GARCH(1,1)—is consistent with the long-memory findings documented for the EUR/USD pair by [18] and for the NGN/USD series by [19], and extends this evidence to the USD/PEN pair. The poor performance of ARCH(1)—with an AIC penalty of 551.92 units relative to FIGARCH—confirms that a single lag of squared innovations is insufficient to capture the persistence structure of USD/PEN conditional variance. The near-identical AIC values of EGARCH(3,099.77), GARCH(1,1) (3,100.76), and GJR-GARCH(3,100.87) suggest that, conditional on the wavelet pre-processing having removed the most extreme observations, the asymmetric effect of negative innovations on variance—captured by EGARCH and GJR-GARCH—provides only marginal additional fit over the symmetric GARCH(1,1). This result is noteworthy: the operator 𝒯 appears to absorb a portion of the asymmetric volatility response that would otherwise require an asymmetric specification.

Figure 2 plots the estimated conditional standard deviation σˆ_t from the FIGARCH(1,d,1) model over the full sample period, overlaid on the absolute log-return series. The estimated fractional differencing parameter is d = 0.553 (p < 0.001), statistically significant at the 1% level, confirming the long-memory regime (0 < d < 1). The sum ϕ + β = 0.6755 < 1 satisfies the covariance stationarity condition, confirming that the process is mean-reverting despite its long-memory dynamics.

Figure 2

Table 8 reports the estimated conditional variance parameters of the FIGARCH(1,d,1) component.

4.3 Residual diagnostics

Table 9 reports the Ljung-Box portmanteau test statistics applied to standardized residuals z_t and squared standardized residuals for each of the five estimated models, using 10 lags. A model is considered adequately specified if it simultaneously passes both tests—that is, fails to reject the null of no serial correlation in both z_t and . Only FIGARCH(1,d,1) satisfies both criteria simultaneously, with p = 0.4846 for standardized residuals and p = 0.9921 for squared residuals, confirming the absence of residual serial correlation and volatility clustering.

Table 10 reports the Jarque-Bera normality test applied to the standardized residuals of each specification.

4.4 Forecasting performance

Table 11 reports the out-of-sample forecast accuracy metrics for the mean equation across the three validation windows (5, 10, and 15 trading days). Because all five conditional variance specifications share the same ARIMAX(2,0) mean equation, the point forecasts of the return r_t are numerically identical across models; the metrics in Table 11 therefore reflect the performance of the ARIMAX(2,0) + 𝒯 framework as a whole, independently of the variance specification chosen.

Table 12 extends the evaluation to the volatility dimension using squared returns () as a proxy for realized variance and reporting MSE-Vol and QLIKE loss functions across the five specifications for the 15-day validation window.

Under both MSE-Vol and QLIKE criteria, ARCH(1) ranks first among all specifications, followed by FIGARCH(1,d,1) under Normal innovations. Notably, FIGARCH(1,d,1)–Normal outperforms the naïve constant-variance benchmark under QLIKE (−0.4729 vs. −0.4377), while FIGARCH(1,d,1)–Skew-t does not. The marginal underperformance of FIGARCH(1,d,1)–Skew-t relative to the naïve benchmark reflects the tension between distributional flexibility and out-of-sample forecast accuracy in short validation windows (n = 15), where additional parameters may not improve predictive performance. EGARCH(1,1,1) does not yield closed-form multi-step conditional variance forecasts and is therefore excluded from this comparison; its in-sample fit metrics are reported in Table 6a.

Under the random-walk benchmark of Equation 19, Theil's U remains close to unity across the three horizons (0.9338 at 15 days, 1.0171 at 10 days, 0.9848 at 5 days), indicating that the ARIMAX (2,0) + T framework achieves mean-forecast accuracy comparable to, though not decisively better than, the naïve benchmark. This is consistent with the well-documented difficulty of outperforming the random walk in exchange rate point forecasting. The framework's predictive value is instead concentrated in the volatility dimension, as shown in Tables 12, 13.

Figure 3 illustrates the dynamic multi-step-ahead forecasts generated by the ARIMAX(2,0) + FIGARCH(1,d,1) model for the 15-day validation window, together with the observed returns and the naïve benchmark. The 95% confidence intervals derived from the estimated conditional variance are shown as shaded bands.

Figure 3

Table 13 reports the expanding-window forecast evaluation using one-step-ahead predictions (h = 1) over 303 out-of-sample observations (January 2025–March 2026).

5 Discussion

This section interprets the empirical results in relation to the three contributions stated in the introduction. Section 5.1 addresses the long-memory vs. spurious persistence debate for USD/PEN conditional variance and contextualizes the FIGARCH(1,d,1) dominance within the emerging-market volatility literature. Section 5.2 discusses the methodological implications of the wavelet-dummy operator 𝒯 and its role in resolving the circularity inherent in residual-based outlier detection. Section 5.3 interprets the forecasting performance and its practical implications for FX risk management. Section 5.4 acknowledges the study's limitations and outlines directions for future research.

5.1 Long memory in USD/PEN conditional variance

The central empirical finding of this study is the identification of statistically significant long memory in the conditional variance of the USD/PEN exchange rate, with an estimated fractional differencing parameter d = 0.553 (p < 0.001). This value situates the USD/PEN firmly within the long-memory regime (0 < d < 1), implying that the impact of past innovations on current volatility decays hyperbolically rather than exponentially. The practical implication is that USD/PEN volatility shocks are substantially more persistent than standard GARCH or EGARCH specifications would suggest, and that risk managers using short-memory models will systematically underestimate the duration of elevated volatility regimes following major shocks [5].

The estimated d = 0.553 exceeds the value of d = 0.431 reported by [18] for the EUR/USD pair, consistent with the expectation that partially dollarized emerging-market currencies exhibit greater volatility persistence than major pairs—a difference attributable to the structural features documented in the introduction: recurrent episodes of political instability and financial dollarization risk that the BCRP's managed floating regime can attenuate but not eliminate. This interpretation is reinforced by [4, 7], whose application of the modified ICSS algorithm to USD/PEN data identified eight structural breakpoints—a feature that, when unaddressed, artificially inflates persistence estimates in short-memory models [8, 9]. The concern that d may reflect spurious rather than genuine long memory [20–22] is directly addressed by the design of 𝒯  removing the 36 localized extreme innovations prior to estimation eliminates the primary source of artificial persistence [8, 9], so that the significance of d = 0.553 after pre-treatment constitutes stronger evidence of genuine long memory than estimates from raw series—a conclusion consistent with [35], who reached the same finding for an emerging-market equity index. The AIC advantage of FIGARCH(1,d,1) over GARCH(1,1) of 26.13 units constitutes decisive evidence in favor of the long-memory specification, consistent with the findings of [19] for the NGN/USD pair—where ARFIMA-GARCH substantially outperformed short-memory alternatives—and extends that evidence to a managed-float South American currency under partial dollarization. This result resonates with the near-universal persistence documented in high-frequency financial returns [11].

The robustness of this conclusion is confirmed by the distributional sensitivity analysis in Table 6b: d remains highly significant across all four innovation distributions—ranging from 0.485 under Normal to 0.553 under Skewed Student-t—and all specifications simultaneously pass Ljung-Box diagnostic on standardized and squared residuals. The Skewed Student-t achieves the lowest AIC with a statistically significant asymmetry parameter (λ = −0.0608, p = 0.015) consistent with the mild negative skewness of USD/PEN returns documented in Table 1. This cross-distributional consistency confirms that long memory in USD/PEN conditional variance is a structural feature rather than an artifact of the normality assumption.

The adequacy of the FIGARCH(1,d,1) specification is further confirmed by the residual diagnostics reported in Section 4.3: Ljung-Box tests on standardized and squared residuals confirm that the model fully absorbs both the serial correlation and the conditional heteroscedasticity of the USD/PEN series. This stands in contrast to all short-memory alternatives, whose squared residuals exhibit significant autocorrelation, reinforcing the conclusion that long memory is a genuine feature of USD/PEN conditional variance dynamics.

5.2 The wavelet-dummy operator: methodological implications

The second principal finding concerns the role of the wavelet-dummy operator 𝒯 in the estimation pipeline. The negative and highly significant coefficient of the binary indicator (β = −0.1421, p < 0.001) confirms that the 36 atypical observations identified by 𝒯 exert a systematically negative effect on the conditional mean of returns. This pattern may reflect the asymmetric nature of currency crises in partially dollarized economies, where sharp depreciations of the sol tend to be more pronounced than appreciations—a feature documented in the broader emerging-market exchange rate literature. While the BCRP's active FX intervention policy [4] may contribute to this asymmetry, the present model does not identify the causal mechanism and this interpretation should be treated as indicative rather than conclusive. By incorporating these observations as deterministic regressors rather than treating them as random innovations, the ARIMAX mean equation produces filtered residuals that are cleaner inputs for the conditional variance stage.

A notable secondary finding is that the inclusion of the operator 𝒯 appears to reduce the marginal value of asymmetric GARCH specifications. The near-identical AIC values of EGARCH(1,1,1) (3,099.77), GARCH(1,1) (3,100.76), and GJR-GARCH(1,1,1) (3,100.87) indicate that, once the most extreme negative returns have been absorbed by the binary regressor D_t, the asymmetric leverage effect—whereby negative shocks amplify volatility disproportionately [13, 14]—provides only marginal additional explanatory power. This is a methodologically relevant result: the operator 𝒯 does not merely control for outliers but may partially reduce the statistical signal that asymmetric specifications would otherwise capture through γ₁–an interpretation that warrants formal testing through direct comparison of γ1 estimates with and without the pre-processing step. Future work should examine whether this pattern holds for other emerging-market currencies with similar institutional fragility profiles.

The design of 𝒯 addresses a fundamental identification challenge in the outlier-GARCH literature. Prior approaches, such as [26], detect outliers from parametric model residuals, creating a circularity: the outlier-contaminated residuals are used to estimate the model whose adequacy is being assessed. By constructing D_t from the wavelet decomposition of the raw return series—prior to any parametric estimation—the operator is exogenous to the conditional variance process by construction. Failure to treat outliers in this pre-estimation stage has been shown to bias GARCH parameter estimates [17] and inflates excess kurtosis in standardized residuals. The Jarque-Bera results in Table 10 are consistent with this interpretation: the combined action of 𝒯 and the FIGARCH long-memory structure captures a larger share of the non-normal features of USD/PEN returns than any of the competing specifications.

5.3 Forecasting performance

The forecasting results merit careful interpretation. Across the three horizons, Theil's U remains close to unity (0.9338 at 15 days, 1.0171 at 10 days, 0.9848 at 5 days), indicating that the mean equation performs on par with the random walk without drift—in line with the well-documented difficulty of beating the random walk in exchange rate point forecasting.

The volatility forecasting evaluation in Table 12 reveals a complementary pattern. Under MSE-Vol and QLIKE criteria, ARCH(1) ranks first, a result consistent with the broader evidence that parsimonious specifications outperform more complex alternatives in out-of-sample volatility forecasting under high structural uncertainty [36]. Nevertheless, FIGARCH(1,d,1) under Normal innovations outperforms the naïve constant-variance benchmark under QLIKE (−0.4729 vs. −0.4377), confirming that long-memory specifications retain informational value for volatility prediction beyond a trivial benchmark. For risk management applications such as Value-at-Risk estimation and capital allocation, the accurate characterization of long-run volatility persistence is more relevant than minimizing one-step-ahead point forecast error. The significant d = 0.553 (p < 0.001) under Skewed Student-t innovations confirms that volatility shocks in USD/PEN are not short-lived—a structural feature that ARCH(1) cannot capture by construction, given its single-lag memory.

This pattern is consistent with the forecast evaluation metric paradox documented by [28], who show that model rankings under different loss functions are not invariant, and that a model can dominate under one metric while being dominated under another.

These findings have direct implications for financial risk management in Peru. The BCRP's FX intervention framework, which [4] show to respond primarily to exchange rate fluctuations rather than levels, requires reliable conditional variance forecasts over medium-term horizons to calibrate intervention thresholds. The consistent outperformance of the volatility forecasts over the constant-variance benchmark (Table 13) suggests that ARIMAX(2,0) + FIGARCH(1,d,1) + 𝒯 is a viable operational tool for this purpose. Similarly, corporate treasury managers in partially dollarized firms face currency exposure over horizons of 2 to 4 weeks, for which reliable conditional variance forecasts are particularly valuable. These practical implications should be interpreted in light of the study's limitations: the multi-step forecasting evaluation covers a single out-of-sample period, the framework is validated exclusively for the USD/PEN pair, and the full model comparison is conducted under conditional normality for four of the five specifications—factors that constrain the generalizability of the findings to other emerging-market currencies and volatility regimes.

The expanding-window evaluation in Table 13 confirms the robustness of these findings under a more rigorous sequential forecasting scheme. Over 303 one-step-ahead predictions, all specifications consistently outperform the naïve constant-variance benchmark under both MSE-Vol and QLIKE. FIGARCH(1,d,1) under Normal innovations is statistically indistinguishable from GARCH(1,1) in MSE-Vol—the difference is 0.0000011—confirming that the long-memory specification does not sacrifice forecasting accuracy relative to short-memory alternatives. Notably, ARCH(1), which ranked first in the static 15-day evaluation, falls to last place under this sequential scheme, suggesting that its earlier performance was horizon-specific rather than a robust general feature.

5.4 Limitations and future research directions

Several limitations of this study should be acknowledged. First, while FIGARCH(1,d,1) is estimated under four innovation distributions in this study—Normal, Student-t, Skewed Student-t, and GED—the remaining four conditional variance specifications (ARCH, GARCH, GJR-GARCH, EGARCH) are estimated exclusively under conditional normality. Extensions to heavy-tailed distributions for the full model comparison would provide a more comprehensive robustness assessment, particularly for Value-at-Risk applications [11, 18].

Second, the wavelet-dummy operator 𝒯 as implemented in this study does not distinguish between additive level outliers (ALO) and additive volatility outliers (AVO) as defined by [17]. The current implementation treats all atypical observations identified by the threshold λ = 0.6943 as ALOs affecting the mean equation. A full AVO treatment—which would additionally include the outlier dates as dummy variables in the conditional variance equation—represents a natural extension that could further reduce residual excess kurtosis.

Third, the study focuses exclusively on the USD/PEN exchange rate. The proposed framework is directly applicable to other MILA currencies—COP/USD, CLP/USD, and MXN/USD—which [10] identify as exhibiting long memory and sensitivity to US monetary policy uncertainty under managed floating regimes. The wavelet-based evidence from [16] suggests that currency stress in emerging economies intensifies during political events across multiple markets simultaneously, reinforcing the potential regional applicability of the proposed framework. A cross-currency comparative analysis would allow formal testing of whether the horizon-dependent forecasting advantage of the 𝒯 + FIGARCH framework generalizes across managed-float emerging-market currencies, and whether the estimated d values vary systematically with institutional fragility and dollarization depth.

Fourth, while the expanding-window evaluation in Table 13 provides sequential evidence of forecasting stability over 303 one-step-ahead predictions, the multi-step evaluation (h = 5, 10, 15 days) is conducted over a single out-of-sample period. A rolling-window approach applied to multi-step horizons would provide additional robustness evidence across different volatility regimes. This is particularly relevant given the structural heterogeneity of the 2015–2026 sample, which encompasses both the low-volatility pre-pandemic period and the high-volatility 2020–2023 sub-period.

Finally, future work could formally test whether d = 0.553 represents true or spurious long memory using the semiparametric methods proposed by [22], which would provide a robustness check complementary to the parametric FIGARCH evidence. Additionally, Bayesian estimation of the FIGARCH parameters would provide posterior distributions for d and allow formal model comparison across the five competing specifications—an avenue that would complement the Maximum Likelihood approach adopted in this study.

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