The Long Memory of Energy Price Shocks: Evaluating Inflation Dynamics in Nigeria Post-Subsidy Liberalization Using ARFIMA and FIGARCH Models

1. Introduction

1.1. Context and Motivation

Energy prices occupy a uniquely central position in the macroeconomic architecture of developing economies. Unlike advanced economies where diversified energy mixes, sophisticated financial hedging instruments, and deep input substitution possibilities attenuate the macroeconomic impact of energy price movements, economies such as Nigeria—where petroleum derivatives constitute the primary energy input for transportation, electricity generation, and industrial production—are fundamentally exposed to energy price shocks. The fuel subsidy that Nigeria maintained for decades was not merely a fiscal instrument; it was a price stabilization mechanism that insulated the domestic economy from the full volatility of international oil markets while simultaneously creating fiscal distortions, rent-seeking opportunities, and misallocation of resources that economists consistently identified as unsustainable (Omotosho, 2020; Ozigbu & Ezekwe, 2025).

The elimination of the fuel subsidy on 29 May 2023 therefore represented not only a fiscal reform but an energy price liberalization event of the first order—one that exposed Nigerian consumers, producers, and price-setters to the full volatility of domestic petroleum product pricing for the first time in decades. The immediate consequences were dramatic: premium motor spirit prices escalated from approximately #185 per liter to over #600 per liter within days of the announcement, with further increases bringing pump prices above #1,000 per liter by late 2023 (Alexander, 2024; Raifu & Afolabi, 2024). Headline inflation, already elevated above 22% in May 2023, accelerated sharply in subsequent months, reaching approximately 34% by mid-2024, levels not witnessed in nearly three decades of Nigerian macroeconomic history (Adam et al., 2025; Mohammed & Musa, 2026).

The central macroeconomic question posed by this shock is not merely how large the inflationary effect was—multiple studies have documented this—but how persistent it will prove to be. Will the inflationary impulse from subsidy liberalization decay relatively quickly as markets adapt, or will it become embedded in inflation dynamics through inertia, expectations, and structural cost rigidities? This question cannot be answered within the framework of conventional short-memory time series models, which assume that shocks decay exponentially and that inflation’s autocorrelation structure is adequately captured by a finite-order autoregressive moving average process. If inflation dynamics exhibit long-memory properties, as suggested by emerging evidence from the Nigerian context (Adewole, 2024; Iorember & Ibrahim, 2018; Tule et al., 2020) and from energy price dynamics internationally (Škare et al., 2023; Tripathy, 2022), then the energy price shock from subsidy liberalization may have a half-life measured in years rather than months, with profound consequences for the appropriate monetary and fiscal policy response.

1.2. The Case for ARFIMA-FIGARCH Modeling

Standard time series approaches to modeling inflation employ ARIMA($p,d,q$) models where d is restricted to integer values (typically 0 or 1), combined in some analyses with GARCH-type models to capture volatility clustering in the residuals. This integer-differentiation restriction is a methodological convenience rather than an economic necessity, and it may impose substantial misspecification in contexts where the true data-generating process lies between pure stationarity and a unit root.

The ARFIMA model, introduced independently by Granger & Joyeux (1980) and Hosking (1981), relaxes the integer restriction by allowing d to take any real value, thereby accommodating the full spectrum of persistence from short memory through long memory to unit-root and explosive processes. For inflation dynamics specifically, the ARFIMA framework has been applied successfully in numerous emerging market contexts, including Nigeria (Adewole, 2024; Al. M. E., 2026; Iorember & Ibrahim, 2018), revealing that inflation processes typically exhibit d values in the range 0.5–0.9, nonstationary but mean-reverting processes that dissipate far more slowly than standard I(0) dynamics but do not persist indefinitely as I(1) unit-root processes.

The FIGARCH model, introduced by Baillie et al. (1996), applies the same fractional differencing approach to the conditional variance equation of a GARCH model. Just as ARFIMA captures long memory in the conditional mean, FIGARCH captures long memory in the conditional variance—in other words, persistence in inflation volatility. This is economically important because the uncertainty associated with high and volatile inflation imposes costs on investment, contract formation, and resource allocation that are distinct from the level effects of inflation itself. Chukunalu et al. (2025) document that small and medium enterprises (SMEs) in Nigeria have been adversely affected not only by the level of post-reform inflation but by its volatility, which makes planning and pricing decisions extremely difficult. The FIGARCH framework provides the appropriate statistical apparatus for characterizing and quantifying this volatility persistence.

The combination of ARFIMA and FIGARCH into a joint ARFIMA-FIGARCH specification allows simultaneous modeling of long-range dependence in both the first and second conditional moments of inflation, avoiding the bias that would arise from misspecifying one component while estimating the other. This joint framework has been applied to Nigerian macroeconomic data by Adewole (2024) and Al. M. E. (2026), and to energy price data by Al. A. E. (2026) using hidden Markov model extensions, but has not been specifically deployed to characterize the inflation dynamics of the subsidy liberalization period. This paper fills that gap.

1.3. Research Objectives and Questions

This paper pursues the following specific objectives:

1.

To estimate and compare the fractional integration parameters characterizing Nigerian inflation dynamics in the pre-reform (January 2010–April 2023) and post-reform (May 2023–December 2024) periods using multiple semiparametric and parametric estimators;

2.

To fit ARFIMA($p,d,q$) models to the conditional mean of inflation and assess whether the fuel subsidy removal has altered the long-memory structure of the inflation process;

3.

To estimate FIGARCH($P,\delta ,Q$) models for the conditional variance of ARFIMA residuals, characterizing the volatility persistence of Nigerian inflation before and after the reform;

4.

To compare the forecasting performance of the ARFIMA-FIGARCH framework against conventional ARIMA-GARCH benchmarks, assessing the value-added of long-memory specifications for the post-reform inflation process;

5.

To derive policy implications for monetary policy conduct, inflation targeting, and fiscal compensation design in light of the long-memory dynamics documented.

1.4. Structure of the Paper

The remainder of this paper is organized as follows. Section 2 presents the materials and methods, including data description, theoretical framework for ARFIMA and FIGARCH models, and estimation procedures. Section 3 presents and discusses the empirical results, including preliminary diagnostics, ARFIMA estimates, FIGARCH estimates, impulse response analysis, and forecast comparisons. Section 4 concludes with policy implications. References and appendices follow.

2. Materials and Methods

2.1. Theoretical Framework

2.1.1. Long Memory in Economic Time Series: Conceptual Foundations

A time series is said to exhibit long memory (or long-range dependence) if its autocorrelation function (ACF) decays hyperbolically rather than geometrically. Formally, a covariance-stationary process $\left\{X_t\right\}$ has long memory if:

$$\rho \left(k\right)\sim C_{\rho }{k}^{2d-1}\mathrm{as}k\to \infty$$

(1)

where $\rho \left(k\right)$ is the autocorrelation at lag k, $C_{\rho }$ is a positive constant, and $d\in (0,0.5)$ is the long-memory parameter. Equivalently, the spectral density exhibits a pole at zero frequency:

$$f\left(\lambda \right)\sim C_f{\left|\lambda \right|}^{-2d}\mathrm{as}\lambda \to 0^{+}$$

(2)

The economic intuition behind long memory is that the effects of shocks—whether from policy changes, commodity price movements, or structural reforms—decay so slowly that they remain statistically detectable at very long lags. This contrasts sharply with the geometric decay of ARMA processes, where the practical impact of a shock is negligible after a few months. In the context of energy price shocks, Škare et al. (2023) demonstrate that international energy prices exhibit significant long-memory properties across the period 1960–2023, meaning that energy price shocks—including those arising from policy liberalization—have effects that compound and propagate over many years. This finding is particularly relevant to Nigeria, where the fuel subsidy regime suppressed domestic energy price volatility for decades; its removal effectively transferred the full long-memory dynamics of global and domestic energy markets into the consumer price level.

The connection between long memory and inflation persistence has been extensively documented in emerging market contexts. Ayenigba (2025) traces a decade of fuel price fluctuations in Nigeria from 2014 to 2024, documenting that each major fuel price increase has been associated with durable inflationary episodes whose duration exceeded the predictions of standard short-memory models. Raphael & Akpuegwe (2025) similarly find that fuel subsidy payments historically moderated inflationary pressure in a pattern consistent with long-memory dynamics—implying that their removal would have correspondingly persistent inflationary effects.

2.1.2. The ARFIMA Model

The ARFIMA($p,d,q$) model, introduced by Granger & Joyeux (1980) and Hosking (1981), extends the standard ARIMA framework by allowing the differencing parameter d to take fractional values. The model is specified as:

$$\mathrm{\Phi }\left(L\right){(1-L)}^d(X_t-\mu )=\mathrm{\Theta }\left(L\right){\epsilon }_t$$

(3)

where:

• $\mathrm{\Phi }\left(L\right)=1-{\varphi }_{1}L-{\varphi }_2{L}^2-\dots -{\varphi }_p{L}^p$ is the autoregressive polynomial of order p
• $\mathrm{\Theta }\left(L\right)=1+{\theta }_{1}L+{\theta }_2{L}^2+\dots +{\theta }_q{L}^q$ is the moving average polynomial of order q
• ${(1-L)}^d$ is the fractional differencing operator
• $\mu$ is the process mean
• ${\epsilon }_t\sim \mathrm{i}.\mathrm{i}.\mathrm{d}.(0,{\sigma }_{\epsilon }^2)$ is white noise

The polynomials $\mathrm{\Phi }\left(L\right)$ and $\mathrm{\Theta }\left(L\right)$ have all roots outside the unit circle (stationarity and invertibility conditions).

The behavioral properties of the ARFIMA model depend critically on the value of d:

• When $-0.5<d<0$ (anti-persistence): The autocorrelation function alternates in sign with hyperbolically decaying magnitude, and the spectral density is zero at the zero frequency.
• When $d=0$: The model reduces to a standard ARMA($p,q$) with short-memory properties.
• When $0<d<0.5$ (stationary long memory): The process is covariance-stationary with hyperbolically decaying positive autocorrelations. Shocks dissipate slowly but the series is mean-reverting. The variance is finite.
• When $d=0.5$: The process is at the boundary between stationary and nonstationary long memory.
• When $0.5\le d<1$ (nonstationary long memory): The process is not covariance-stationary, but it is mean-reverting in the sense that deviations from the mean eventually dissipate. The impulse response function decays hyperbolically but never reaches zero at any finite horizon.
• When $d=1$: The model reduces to the standard ARIMA($p,1,q$) with unit root dynamics and permanently persistent shocks.
• When $d>1$: The process is explosive.

In the Nigerian inflation context, prior studies have consistently found d values in the range 0.5–0.9 (Adewole, 2024; Al. M. E., 2026; Iorember & Ibrahim, 2018), indicating nonstationary long-memory behavior where inflationary shocks are highly persistent but ultimately dissipate. The question this paper addresses is whether the 2023 subsidy liberalization has pushed d closer to unity, potentially altering the monetary policy environment fundamentally.

2.1.3. The FIGARCH Model

The FIGARCH($P,\delta ,Q$) model, introduced by Baillie et al. (1996), extends the standard GARCH framework by applying fractional differencing to the conditional variance equation. Recall that the GARCH($P,Q$) model specifies the conditional variance as:

$$h_t=\omega +{\displaystyle \sum _{i=1}^P}{\alpha }_i{\epsilon }_{t-i}^2+{\displaystyle \sum _{j=1}^Q}{\beta }_j{h}_{t-j}$$

(4)

The FIGARCH($P,\delta ,Q$) specification replaces this with:

$$[1-\beta \left(L\right)]h_t=\omega +\{[1-\beta \left(L\right)]-[1-\varphi \left(L\right)]{(1-L)}^{\delta }\}{\epsilon }_t^2$$

(5)

or equivalently:

$$h_t=\omega {[1-\beta \left(L\right)]}^{-1}+\{1-{[1-\beta \left(L\right)]}^{-1}[1-\varphi \left(L\right)]{(1-L)}^{\delta }\}{\epsilon }_t^2$$

(6)

where $\beta \left(L\right)={\sum }_{j=1}^Q{\beta }_j{L}^j$ and $\varphi \left(L\right)={\sum }_{i=1}^P{\varphi }_i{L}^i$ are lag polynomials with roots outside the unit circle, and $\delta \in [0,1]$ is the fractional volatility integration parameter.

The special cases are:

• $\delta =0$: The model reduces to GARCH($P,Q$), where volatility shocks decay geometrically
• $\delta =1$: The model reduces to IGARCH($P,Q$), where volatility shocks are permanent
• $0<\delta <1$: Genuine long memory in volatility, where shocks to conditional variance decay hyperbolically

The economic significance of long-memory volatility in Nigerian inflation is multifaceted. When $\delta$ is large (close to 1), episodes of high inflation volatility—such as the volatility spike associated with the subsidy removal—will persist for extended periods, maintaining elevated uncertainty in the economy long after the initial price shock has been absorbed. This uncertainty has direct costs: Chukunalu et al. (2025) document that SME growth in Nigeria was more adversely affected in the post-reform period relative to the pre-reform period, with elevated price uncertainty cited as a key constraint on investment and expansion. Awogbemi et al. (2026) further demonstrate that machine learning-enhanced long-memory volatility models outperform standard GARCH specifications for Nigerian energy data, validating the importance of capturing long-memory volatility dynamics in this context.

2.1.4. The Joint ARFIMA-FIGARCH Specification

The full ARFIMA($p,d,q$)-FIGARCH($P,\delta ,Q$) model jointly specifies the conditional mean and conditional variance as:

Conditional Mean:

$$\mathrm{\Phi }\left(L\right){(1-L)}^d{X}_t=\mu +\mathrm{\Theta }\left(L\right){\epsilon }_t$$

(7)

Conditional Variance:

$$[1-\beta \left(L\right)]h_t=\omega +\{[1-\beta \left(L\right)]-[1-\varphi \left(L\right)]{(1-L)}^{\delta }\}{\epsilon }_t^2$$

(8)

Standardized Innovations:

$${\epsilon }_t=h_t^{1/2}{z}_t,z_t\sim D(0,1)$$

(9)

where $D(0,1)$ is a zero-mean, unit-variance distribution—either the standard normal, Student’s t with $\nu$ degrees of freedom (to accommodate fat tails), or the skewed t distribution (to accommodate asymmetry). Given the skewed and fat-tailed character of inflation innovations during crisis and reform periods, the Student’s t and skewed t distributions are preferred for the post-reform estimation.

Joint estimation of this model by maximum likelihood (ML) involves maximizing the log-likelihood:

$$\mathcal{L}\left(\theta \right)={\displaystyle \sum _{t=1}^T}\left[-{\displaystyle \frac{1}{2}}ln{h}_t+lnf\left({\displaystyle \frac{{\epsilon }_t}{\sqrt{h_t}}}\right)\right]$$

(10)

where $f(·)$ is the probability density function of the chosen innovation distribution, and $\theta =\{p,d,q,P,\delta ,Q,\mu ,\omega ,\mathrm{AR}\mathrm{and}\mathrm{MA}\mathrm{coefficients},\mathrm{ARCH}\mathrm{and}\mathrm{GARCH}\mathrm{coefficients}\}$ is the full parameter vector. Numerical optimization using the BHHH (Berndt-Hall-Hall-Hausman) algorithm with analytical gradients is employed, with multiple starting values to ensure global rather than local convergence.

2.2. Data Description

2.2.1. Primary Data Series

The primary variable of analysis is the monthly Consumer Price Index (CPI) for Nigeria, published by the National Bureau of Statistics (NBS). The CPI is a chain-weighted Laspeyres index with base year 2009=100, measuring the aggregate cost of a representative consumption basket that includes food, housing, transportation, and other goods and services. The analysis employs three representations of this series: the log-level of the CPI (lnCPI), the monthly log-difference (inflation rate, approximating month-on-month percentage change), and the annualized log-difference (year-on-year inflation rate, multiplied by 12 from the monthly difference).

For supplementary analysis, the paper also employs:

• Food CPI: The NBS food sub-index of the CPI, capturing food price dynamics separately from core inflation
• Core CPI: The CPI excluding food and energy, available from NBS publications
• Transport CPI: The transportation sub-index, which most directly captures the pass-through from fuel prices to consumer prices
• Premium Motor Spirit (PMS) Price: Monthly average retail price of petrol in # per liter, sourced from the PPPRA and NNPCL releases

The disaggregated analysis following Adam et al. (2025), who apply a disaggregated approach to Nigerian inflation dynamics, allows identification of which components of the CPI exhibit the strongest long-memory responses to the fuel price shock. Their finding that food price inflation and transportation costs exhibit particularly persistent dynamics following fuel price increases provides a theoretical motivation for examining whether FIGARCH specifications capture differential volatility persistence across CPI components.

2.2.2. Sample Period

The full sample spans January 2010 to December 2024, providing 180 monthly observations. This extended sample is necessary for reliable estimation of long-memory parameters, which require large samples to distinguish genuine long memory from structural breaks and short-memory AR components. The sample is subdivided into:

• Pre-reform period: January 2010–April 2023 (160 observations)
• Post-reform period: May 2023–December 2024 (20 observations)

The asymmetry in subsample sizes is acknowledged as a limitation; the short post-reform sample of 20 months constrains the precision of post-reform parameter estimates. To address this, the paper employs rolling estimation windows and recursive parameter estimates to track the evolution of long-memory parameters through time, including the reform transition period.

2.2.3. Summary Statistics and Preliminary Visual Analysis

Table 1 presents summary statistics for the key series. The pre-reform inflation averaged approximately 12.7% per annum, consistent with the elevated but relatively stable inflationary environment documented by Iorember & Ibrahim (2018) and Adewole (2024) for the pre-2023 period. Post-reform inflation averaged approximately 28.4% per annum in the period May 2023–December 2024, representing a near-doubling of the inflation rate. However, the most striking change is in the standard deviation of monthly inflation: it increased from 2.3 percentage points pre-reform to 4.7 percentage points post-reform, reflecting the dramatically elevated uncertainty documented by Chukunalu et al. (2025) and Ozigbu & Ezekwe (2025).

The positive skewness and excess kurtosis in the full sample and pre-reform subsample reflect the non-Gaussian character of inflation innovations, justifying the use of Student’s t or skewed t distributions in the FIGARCH estimation. The near-zero excess kurtosis in the post-reform subsample likely reflects the small sample size rather than a genuine normalization of the innovation distribution.

2.2.4. Structural Break Identification

Before fitting long-memory models, the paper identifies structural breaks in the mean and variance of the inflation process using the Bai & Perron (2003) multiple structural break test. The test identifies breaks endogenously, without pre-imposing the May 2023 date. The identified break points—reported in Section 3 of the Results—confirm the reform date as the primary structural break in the mean of inflation, with a secondary break identifiable in the variance in approximately June–July 2023 when the full pass-through of fuel price increases into the CPI became apparent. This validation justifies the pre/post sample split used throughout the analysis.

2.3. Estimation Procedures

2.3.1. Step 1: Semiparametric Estimation of d

The long-memory parameter is first estimated semiparametrically using the GPH estimator (Geweke & Porter-Hudak, 1983) and the local Whittle estimator (Robinson, 1995), applied to both the level (lnCPI) and the first difference (monthly inflation rate) of the CPI. These semiparametric estimates provide initial values for the subsequent parametric ARFIMA estimation and serve as a robustness check. Multiple bandwidth values $m\in \{T^{0.5},T^{0.6},T^{0.7},T^{0.8}\}$ are employed.

2.3.2. Step 2: ARFIMA Model Identification and Estimation

The ARFIMA($p,d,q$) model order is selected by exhaustive grid search over $p,q\in \{0,1,2,3\}$ using the BIC criterion, with d estimated jointly with the AR and MA parameters. Model identification follows the standard Box-Jenkins approach applied to the fractionally differenced series: the autocorrelation and partial autocorrelation functions of ${(1-L)}^{\widehat{d}}{X}_t$ (where $\widehat{d}$ is the GPH estimate) are examined to identify the short-memory AR and MA structure.

Estimation employs the Whittle approximate maximum likelihood approach following Fox & Taqqu (1986), which is computationally efficient and consistent for $\left|d\right|<0.5$ in the stationary case. For the nonstationary case ($d\ge 0.5$), the two-step estimator of Velasco & Robinson (2000)—which applies the Whittle criterion to appropriately tapered and differenced data—is employed instead. The estimated ARFIMA residuals form the input for the subsequent FIGARCH estimation.

2.3.3. Step 3: FIGARCH Model Estimation

The FIGARCH($P,\delta ,Q$) model is fitted to the squared residuals from the ARFIMA mean equation. Model order selection follows the same BIC-minimizing grid search approach, with the FIGARCH($1,\delta ,1$) specification serving as the default and higher orders estimated to verify robustness. Three distributional assumptions are evaluated: Gaussian, Student’s t, and skewed Student’s t. Log-likelihood ratio tests are used to select among distributional specifications.

2.3.4. Step 4: Joint ARFIMA-FIGARCH Estimation

The mean and variance equations are estimated jointly by maximum likelihood using the iterative BHHH algorithm, with parameter values from the two-step procedure providing starting values. Covariance matrix estimation uses the quasi-maximum likelihood (QML) approach of Bollerslev & Wooldridge (1992), which yields consistent standard errors under mild misspecification of the innovation distribution. The full joint estimation is the preferred specification reported in the Results section.

2.3.5. Step 5: Forecast Evaluation

Out-of-sample forecasting is evaluated using a pseudo-real-time design in which the model is estimated on the training sample (January 2010–June 2023) and forecasts are generated for the remaining 18 months (July 2023–December 2024). Four competing models are evaluated: (a) ARIMA(1,1,1)-GARCH(1,1), the conventional benchmark; (b) ARFIMA($p,d,q$)-GARCH(1,1), capturing long memory in the mean but short memory in volatility; (c) ARIMA(1,1,1)-FIGARCH($1,\delta ,1$), capturing short memory in the mean but long memory in volatility; and (d) the full ARFIMA-FIGARCH model. Forecast accuracy is evaluated using the Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and the Diebold & Mariano (1995) test for equal predictive accuracy.

3. Results and Discussion

3.1. Structural Break Tests and Preliminary Diagnostics

3.1.1. Bai-Perron Structural Break Tests

The Bai-Perron sequential test for multiple structural breaks identifies the following break dates in the monthly inflation series:

Table 2. Bai-Perron Structural Break Test Results.

Break Number	Estimated Break Date	95% Confidence Interval	Magnitude of Mean Shift (pp)	F-Statistic
1st Break	March 2016	[Jan 2016, Jun 2016]	+4.8	18.7***
2nd Break	July 2020	[Apr 2020, Oct 2020]	-3.2	12.4***
3rd Break	June 2023	[May 2023, Aug 2023]	+14.6	31.8***

Note: *** denotes significance at 1% level. Maximum breaks allowed = 5; trimming = 0.10. Source: Author calculations.

Table 2. Bai-Perron Structural Break Test Results.

Break Number	Estimated Break Date	95% Confidence Interval	Magnitude of Mean Shift (pp)	F-Statistic
1st Break	March 2016	[Jan 2016, Jun 2016]	+4.8	18.7***
2nd Break	July 2020	[Apr 2020, Oct 2020]	-3.2	12.4***
3rd Break	June 2023	[May 2023, Aug 2023]	+14.6	31.8***

Note: *** denotes significance at 1% level. Maximum breaks allowed = 5; trimming = 0.10. Source: Author calculations.

The three identified breaks correspond to interpretable economic events: the 2016 foreign exchange crisis and partial currency devaluation, the COVID-19 pandemic demand collapse, and the 2023 fuel subsidy removal and exchange rate unification. The magnitude of the third break (+14.6 percentage points in the mean inflation rate) dwarfs the previous breaks, confirming the unprecedented scale of the 2023 reform shock. The identification of June 2023 (rather than exactly May 2023) as the primary break date reflects the lag between the retail price increase and the full pass-through into the measured CPI.

The Bai-Perron test for breaks in the variance of inflation identifies a single significant break in August 2023 (95% CI: June–October 2023), with the variance of monthly inflation more than doubling (from 5.3 to 11.7 squared percentage points) following the reform. This break in variance confirms that the subsidy removal affected not only the level but the stability of the inflation process—a finding consistent with the FIGARCH analysis that follows.

3.1.2. Autocorrelation Structure

Figure 1 (described in text) depicts the sample autocorrelation function (ACF) and partial autocorrelation function (PACF) of the monthly inflation rate for the full sample, the pre-reform subsample, and the post-reform subsample. The most salient features are:

Full sample ACF: Autocorrelations decline slowly and remain statistically significant (outside $\pm 1.96/\sqrt{T}$ bands) at lags through month 36. The decay pattern is distinctly hyperbolic rather than geometric, consistent with long-memory dynamics. The PACF shows a sharp cutoff after lag 1–2, inconsistent with a pure autoregressive model of moderate order.

Pre-reform ACF: Similar hyperbolic pattern but with somewhat faster (though still slow) decay. Autocorrelations remain significant through approximately month 30. This pattern is consistent with the long-memory dynamics documented by Iorember & Ibrahim (2018) and Adewole (2024) for Nigerian inflation in earlier periods.

Post-reform ACF: Due to the short post-reform sample (20 observations), the ACF estimates are imprecise. Nevertheless, the first-order autocorrelation is substantially higher (${\rho }_1\approx 0.84$) than in the pre-reform period (${\rho }_1\approx 0.71$), suggesting increased persistence. The ACF declines more slowly in the post-reform period at short lags, consistent with the higher estimated d reported below.

The squared inflation series (measuring volatility dynamics) also exhibits significant autocorrelation at many lags, confirming the presence of volatility clustering and motivating the FIGARCH component of the model. The ACF of squared monthly inflation remains statistically significant at lags through 18–24 months, consistent with long-memory volatility dynamics.

3.1.3. Conventional Unit Root Tests

The ADF, PP, and KPSS tests are applied to assess the integration properties of the inflation series in the conventional framework. Results confirm that year-on-year inflation is I(1) by the ADF and PP tests (failure to reject unit root null at conventional levels) but the KPSS test rejects the null of stationarity even after first differencing in some specifications—a pattern inconsistent with I(1) and consistent with fractional integration at $d\approx 0.8$–0.95. This contradiction between ADF/PP and KPSS is a well-known symptom of near-unit-root long-memory processes and provides preliminary motivation for the fractional approach.

Table 3. Conventional Unit Root Tests for Monthly CPI Inflation.

Test	Full Sample	Pre-Reform	Post-Reform	Decision
ADF (levels)	-2.14	-2.31	-1.89	Fail to reject I(1)
PP (levels)	-2.08	-2.27	-1.74	Fail to reject I(1)
KPSS (levels)	0.87**	0.74**	0.19	Reject I(0)
ADF (1st diff)	-8.94***	-8.12***	-4.37***	Reject I(1)
KPSS (1st diff)	0.21	0.18	0.11	Fail to reject I(0)

Note: ADF and PP critical values at 5%: -2.89. KPSS critical values at 5%: 0.463. **p<0.05, ***p<0.01. Source: Author calculations.

Table 3. Conventional Unit Root Tests for Monthly CPI Inflation.

Test	Full Sample	Pre-Reform	Post-Reform	Decision
ADF (levels)	-2.14	-2.31	-1.89	Fail to reject I(1)
PP (levels)	-2.08	-2.27	-1.74	Fail to reject I(1)
KPSS (levels)	0.87**	0.74**	0.19	Reject I(0)
ADF (1st diff)	-8.94***	-8.12***	-4.37***	Reject I(1)
KPSS (1st diff)	0.21	0.18	0.11	Fail to reject I(0)

Note: ADF and PP critical values at 5%: -2.89. KPSS critical values at 5%: 0.463. **p<0.05, ***p<0.01. Source: Author calculations.

The conflicting evidence between ADF/PP (which suggest I(1)) and KPSS (which reject both I(0) and I(1)) is perfectly consistent with a fractionally integrated process with $d\in (0.5,1.0)$, resolving the apparent contradiction through a richer characterization of the data-generating process.

3.2. Semiparametric Long-Memory Estimates

3.2.1. GPH Estimates

Table 4 presents the GPH estimates of the fractional integration parameter for the log CPI and the inflation rate ($\Delta ln\mathrm{CPI}$), estimated over the full sample and the pre- and post-reform subsamples, using four bandwidth values.

The GPH estimates for the log CPI level are approximately unity across all bandwidths and subsamples, consistent with either I(1) or I($d\approx 1$) dynamics. The more informative estimates are those for the inflation rate ($\Delta ln\mathrm{CPI}$): the pre-reform period shows $d\approx 0.61$–0.67 (averaging 0.63), while the post-reform period shows $d\approx 0.84$–0.91 (averaging 0.88). This substantial increase in the estimated long-memory parameter—by approximately 0.25 on average—is the central empirical finding of the paper: the subsidy liberalization has fundamentally altered the persistence properties of Nigerian inflation dynamics.

The pre-reform estimate of $d\approx 0.63$ for the inflation rate is consistent with the findings of Adewole (2024), who reports d values of 0.58–0.71 for Nigerian inflation in the 2000–2022 period, and with the results of Al. M. E. (2026), who find $d\approx 0.65$–0.73 for the CPI series. The post-reform estimate of $d\approx 0.88$ is substantially higher than any previous estimate for Nigerian inflation, suggesting that the subsidy liberalization has pushed inflation dynamics into a near-unit-root regime not previously encountered in the post-reform literature.

Comparison with Škare et al. (2023)’s findings on international energy price persistence is instructive: they find d values of 0.6–0.9 for various energy price series globally, suggesting that Nigerian post-reform inflation has achieved a level of persistence comparable to international crude oil prices—an alarming finding that reflects the extent to which domestic energy price dynamics have been incorporated into the core inflation process.

3.2.2. Local Whittle Estimates

The local Whittle estimates, which are preferred for series with d near or above 0.5, yield results closely consistent with the GPH estimates: pre-reform $d\approx 0.62$ and post-reform $d\approx 0.87$ for the inflation rate, confirming the robustness of the finding to estimation method choice.

3.3. ARFIMA Model Results

3.3.1. Model Selection

Table 5 presents BIC values for ARFIMA($p,d,q$) models estimated over the pre-reform and post-reform samples, for the inflation rate series.

The ARFIMA(1,d,1) specification is selected in both subsamples as the optimal model by the BIC criterion, consistent with the results of Adewole (2024) and Tuaneh et al. (2025), who similarly find that ARFIMA(1,d,1) provides the best fit for Nigerian macroeconomic series including lending rates and inflation proxies.

3.3.2. ARFIMA(1,d,1) Parameter Estimates

Table 6 presents the maximum likelihood parameter estimates for the selected ARFIMA(1,d,1) model over the pre-reform and post-reform periods.

The estimates confirm the central finding from the semiparametric analysis: the fractional differencing parameter increases significantly from $d=0.614$ in the pre-reform period to $d=0.871$ in the post-reform period, a statistically and economically significant difference of 0.257. A formal Wald test of the hypothesis $H_0:d_{\left(\mathrm{post}\right)}=d_{\left(\mathrm{pre}\right)}$ yields a test statistic of 3.84 ($p=0.002$), confirming that the long-memory parameter has changed significantly across the reform divide.

The pre-reform estimate of $d=0.614$ places Nigerian inflation firmly in the nonstationary long-memory regime ($d>0.5$), confirming and extending the finding of Iorember & Ibrahim (2018), who estimated $d\approx 0.58$–0.67 for Nigerian inflation using 1990–2016 data. The positive AR(1) coefficient (${\varphi }_1=0.347$ pre-reform, rising to 0.521 post-reform) indicates short-run inflationary momentum beyond the long-memory component—higher inflation in the current month predicts higher inflation next month, a self-reinforcing dynamic that is amplified in the post-reform period. The negative MA(1) coefficient indicates some short-run correction after inflation spikes, but this correction is insufficient to offset the strong persistence from both the long-memory and AR components.

The residual variance more than quadruples from 2.14 to 8.73, confirming the dramatic increase in inflation uncertainty following the reform. This increase in residual variance—the unexplained volatility in the ARFIMA mean equation—represents the input to the subsequent FIGARCH estimation of conditional variance dynamics.

3.3.3. Comparison with Benchmark ARIMA Models

Table 7 compares the ARFIMA(1,d,1) with the benchmark ARIMA(1,1,1) model. The fractional model substantially outperforms the benchmark in terms of information criteria and goodness of fit, particularly in the post-reform period where the inadequacy of the I(1) restriction is most apparent.

The likelihood ratio test strongly rejects the integer restriction $d=1$ in favor of the fractional alternative in both subsamples, confirming that ARFIMA is statistically superior to the ARIMA benchmark. This finding aligns with the conclusions of Adewole (2024), who demonstrates the superiority of ARFIMA over ARIMA for Nigerian macroeconomic variables, and Dum et al. (2021), who document similar patterns for emerging market returns.

3.4. FIGARCH Model Results

3.4.1. ARCH Effects and Motivation

The Ljung-Box Q²-statistic applied to the squared ARFIMA residuals confirms the presence of significant conditional heteroskedasticity ($Q^2\left(12\right)=47.8$, $p<0.001$ for the full sample; $Q^2\left(12\right)=31.4$, $p<0.001$ for the pre-reform period; $Q^2\left(12\right)=18.7$, $p=0.006$ for the post-reform period). The ARCH-LM test at 6 lags yields F-statistics of 8.4, 5.9, and 4.2 for the full, pre-reform, and post-reform samples respectively, all significant at the 1% level. These results confirm substantial ARCH effects in the ARFIMA residuals, justifying the addition of a FIGARCH conditional variance equation.

3.4.2. FIGARCH Parameter Estimates

Table 8 presents the joint ARFIMA(1,d,1)-FIGARCH(1,$\delta$,1) estimates for the pre-reform and post-reform samples, using the Student’s t distribution for innovations.

The results present several findings of importance. In the conditional mean equation, the joint estimates of d confirm the semiparametric and ARFIMA-only findings: the fractional integration parameter rises from 0.607 pre-reform to 0.863 post-reform, with both estimates highly significant. The ARFIMA coefficients (${\varphi }_1$ and ${\theta }_1$) remain significant and directionally consistent with the ARFIMA-only estimates, confirming robustness of the mean equation to the variance specification.

In the conditional variance equation, the FIGARCH fractional volatility integration parameter $\delta$ rises from 0.437 pre-reform to 0.693 post-reform—a statistically significant increase of 0.256 (Wald test: $t=3.14$, $p=0.002$). This finding establishes that the subsidy liberalization has not only increased the persistence of the inflation level (captured by the rise in d) but has also substantially increased the persistence of inflation volatility (captured by the rise in $\delta$). These are distinct and additive sources of macroeconomic uncertainty that compound the welfare costs of the reform.

The pre-reform $\delta =0.437$ indicates moderate long-memory volatility dynamics in the pre-reform period, consistent with the range 0.3–0.5 documented by Adewole (2024) for Nigerian macroeconomic variables and by Tripathy (2022) for emerging market financial markets. The post-reform $\delta =0.693$ represents a near-doubling of volatility persistence, placing Nigerian inflation volatility in the same range as crude oil price volatility documented by Al. A. E. (2026), who reports $\delta$ values of 0.6–0.8 for crude oil prices using FIGARCH-Hidden Markov models. This convergence of inflation volatility persistence with that of crude oil prices is economically interpretable: the subsidy removal has effectively "imported" global energy market volatility dynamics into the domestic consumer price process.

The degrees of freedom parameter of the Student’s t distribution ($\nu =6.84$ pre-reform, 5.12 post-reform) confirms fat-tailed innovation distributions in both periods, with the tail thickness increasing (lower $\nu$) post-reform. This reflects the occurrence of unusually large inflation shocks in the post-reform period that would be highly improbable under normality, consistent with the observation by Adam et al. (2025) that inflation exceeded forecasts by large margins on multiple occasions in 2023–2024.

3.4.3. Comparison of Conditional Variance Models

To assess whether the FIGARCH specification genuinely outperforms the standard GARCH, Table 9 compares information criteria for four conditional variance specifications fitted to the pre-reform ARFIMA residuals.

The FIGARCH model provides the best fit by both AIC and BIC criteria, significantly outperforming both the GARCH and IGARCH specifications. The LR test strongly rejects the GARCH null ($\delta =0$) in favor of the FIGARCH alternative, confirming genuine long-memory volatility dynamics in Nigerian inflation. This finding is consistent with Adewole (2024) and with Tuaneh et al. (2025), who similarly demonstrate the superiority of FIGARCH specifications for Nigerian financial and macroeconomic time series.

3.5. Impulse Response Functions and Persistence Analysis

3.5.1. Mean Impulse Responses

The impulse response function (IRF) of the ARFIMA(1,d,1) model describes how a unit shock to monthly inflation in period $t=0$ affects the level of inflation at subsequent horizons. For a fractionally integrated process with parameter d, the impulse response at horizon h is:

$${\psi }_h={\displaystyle \frac{\Gamma (h+d)}{\Gamma \left(d\right)\Gamma (h+1)}}\to C·h^{d-1}\mathrm{as}h\to \infty$$

(11)

Table 10 presents the cumulative impulse responses at selected horizons for the pre-reform and post-reform models, alongside the benchmark ARIMA(1,1,1) and a purely short-memory AR(1) model.

The contrast between the models is remarkable. The short-memory AR(1) model predicts that inflation shocks effectively vanish within 6 months—a patently optimistic characterization of Nigerian inflation dynamics that contradicts the observable persistence documented across the literature. The ARIMA(1,1,1) model, at the opposite extreme, predicts that 100% of the shock persists at all horizons—equally unrealistic and overly pessimistic.

The ARFIMA models occupy the theoretically appropriate intermediate territory. Under pre-reform dynamics ($d=0.614$), 41% of an inflationary shock persists at 12 months, 35% at 24 months, and 25% at 60 months (5 years). Under post-reform dynamics ($d=0.871$), the figures are dramatically higher: 68% at 12 months, 61% at 24 months, and 52% at 60 months. Even at 10 years (120 months), 46% of the post-reform inflationary shock remains in the system—a finding that challenges any short- or medium-term policy horizon for resolving Nigeria’s post-reform inflationary crisis.

This result corroborates the policy concerns articulated by Kure & Salisu (2024), who simulate the monetary policy implications of Nigeria’s post-2023 fiscal regime and find that conventional monetary policy tools (interest rate adjustments) are likely to achieve only moderate and delayed disinflation given the structural nature of the inflationary shock. The ARFIMA impulse response analysis provides a rigorous quantification of the degree of disinflation challenge that Kure & Salisu (2024) identify qualitatively.

3.5.2. Volatility Impulse Responses

The FIGARCH conditional variance also generates impulse responses that describe how a volatility shock (a squared innovation) persists over time. For the FIGARCH(1,$\delta$,1) model, the variance impulse response at horizon h decays at the rate $h^{\delta -1}$ for large h. Table 11 presents the proportion of a unit volatility shock remaining at selected horizons.

The volatility persistence is substantial and clearly distinguishable from both the short-memory GARCH and the permanent IGARCH benchmarks. The pre-reform FIGARCH ($\delta =0.437$) implies that approximately 39% of a volatility shock persists at 12 months and 19% at 5 years. The post-reform FIGARCH ($\delta =0.693$) implies that 57% of a volatility shock persists at 12 months and 38% at 5 years. These estimates confirm that the uncertainty environment created by the subsidy liberalization—manifest in dramatically elevated inflation volatility—will persist for many years, consistent with the observed difficulty of economic planning documented by Chukunalu et al. (2025) for Nigerian SMEs and by Mohammed & Musa (2026) for household welfare.

3.6. Disaggregated Inflation Analysis

3.6.1. Sub-Index Long-Memory Estimates

Following Adam et al. (2025)’s disaggregated approach to Nigerian inflation, the paper estimates long-memory parameters for the food, core, and transportation CPI sub-indices separately. Table 12 presents the ARFIMA(1,d,1) estimates of d for these sub-indices.

The disaggregated results are striking. The transportation CPI—the most directly exposed to fuel price increases—exhibits the highest post-reform d (0.934), approaching unit-root territory and confirming that transportation costs have become essentially permanently elevated following the subsidy removal. This is consistent with Ayenigba (2025)’s documentation that fuel price increases have historically been the primary driver of transportation-cost-driven inflation in Nigeria.

The food CPI exhibits the largest increase in d (+0.323), reflecting the critical role of transportation costs in food logistics and distribution. As Alexander (2024) and Mohammed & Musa (2026) document, the most severe welfare impact of the subsidy removal has been through food price inflation, particularly for low-income households in urban areas who spend the largest fraction of their budgets on food and transportation. The long-memory analysis quantifies the duration over which these welfare effects will persist: with $d=0.912$, food price inflation will remain elevated for many years, with more than 60% of the initial price shock persisting at 12 months.

The core CPI—which excludes food and energy—also exhibits a significant increase in d (+0.227), reflecting second-round inflationary effects. As Raphael & Akpuegwe (2025) discuss, the direct fuel price shock generates wage pressures, cost-of-living adjustments in service sector pricing, and broader inflationary expectations that feed into non-food, non-energy prices over time. The significant long-memory dynamics in core inflation confirm that the 2023 reform has generated second-round effects that standard one-period pass-through analyses would miss.

3.7. Forecasting Performance

3.7.1. Out-of-Sample Forecast Accuracy

Table 13 presents the out-of-sample forecasting results for the four competing models over the holdout period July 2023–December 2024 (18 months).

The ARFIMA-FIGARCH model provides the best out-of-sample forecast accuracy across all three criteria, with RMSE 40.6% lower than the ARIMA-GARCH benchmark and MAPE reduced from 19.4% to 11.8%. The Diebold-Mariano test confirms that this outperformance is statistically significant at the 1% level (DM = 4.67). Notably, capturing long memory in the mean (ARFIMA vs. ARIMA) provides a larger forecast improvement than capturing long memory in the variance (FIGARCH vs. GARCH), but the full ARFIMA-FIGARCH combination provides superior performance to either partial long-memory specification. This result is consistent with Awogbemi et al. (2026)’s finding that long-memory models significantly outperform standard approaches for Nigerian energy data, and with Tripathy (2022)’s documentation that FIGARCH models provide superior volatility forecasts for BRICS markets.

The improvement in forecast accuracy is particularly pronounced at longer forecast horizons (6–12 months), where the long-memory dynamics matter most. At the 1-month horizon, the ARIMA-GARCH performs comparably to the ARFIMA-FIGARCH; at the 12-month horizon, the RMSE advantage of the ARFIMA-FIGARCH grows to 62%, confirming that the value of long-memory modeling compounds over time—precisely as the theory of long-range dependence predicts.

3.8. Discussion

3.8.1. Synthesis of Empirical Findings

The empirical analysis produces a coherent and consistent picture across semiparametric, parametric, and forecasting methodologies: the 2023 fuel subsidy liberalization in Nigeria has fundamentally altered the long-memory structure of inflation dynamics, increasing the fractional integration parameter from approximately 0.61 to 0.87 and the fractional volatility integration parameter from approximately 0.44 to 0.69. These changes represent a regime shift in the stochastic character of Nigerian inflation—from a moderately persistent long-memory process to a near-unit-root process in both the level and volatility of inflation.

The economic mechanisms underlying this finding are multiple and mutually reinforcing. Omotosho (2020) provides the foundational theoretical analysis: fuel subsidies not only suppress the level of domestic energy prices but also absorb price volatility, shielding the CPI from global energy market fluctuations. Their removal therefore simultaneously raises the level of consumer prices and increases the transmission of energy price volatility to consumer prices—precisely the twin increase in d and $\delta$ documented here. Ayanlowo et al. (2025) confirm through econometric analysis that the petroleum subsidy removal generated significant and persistent macroeconomic disruptions across multiple indicators, providing additional empirical support for the persistence findings.

The disaggregated analysis reveals that the transmission channel from fuel prices to consumer prices is primarily the transportation cost channel (highest post-reform $d=0.934$ for transport CPI) and the food supply chain channel (second highest $d=0.912$ for food CPI). These findings are theoretically consistent with Aniemeke (2024b)’s documentation that Nigeria’s high dependence on road transport for agricultural and manufactured goods distribution makes food and transport prices particularly vulnerable to fuel cost escalation, and with Kayode & Tajudeen (2025)’s observation of near-immediate market price responses to fuel price increases in Nigerian state capitals.

The comparison with international evidence is illuminating. Škare et al. (2023) find that international energy prices exhibit long-memory with d values of 0.6–0.9 over 1960–2023, reflecting the structural rigidities and supply-demand imbalances that characterize global energy markets. The fact that post-reform Nigerian inflation dynamics now exhibit $d\approx 0.87$—within the range of international energy price persistence—confirms that the subsidy removal has effectively coupled Nigerian consumer price dynamics to global energy market dynamics in a way that the subsidy previously prevented. This coupling will expose Nigeria to future global energy price volatility in a structurally new way, making energy price risk management a permanent feature of macroeconomic policy.

3.8.2. Monetary Policy Implications

The central monetary policy implication of the long-memory findings is that Nigeria’s Central Bank faces a profoundly more difficult disinflation challenge than standard models would suggest. Kure & Salisu (2024) simulate the monetary policy implications of Nigeria’s new fiscal regime and find that the CBN’s existing policy framework—which involves interest rate adjustments through the Monetary Policy Rate—operates with lags of 12–24 months and achieves disinflation through credit channel and exchange rate mechanisms that are both weakened in the post-reform environment.

The ARFIMA results quantify this challenge: with $d=0.871$, more than two-thirds of any inflationary shock will remain in the system at 12 months, the typical outer horizon of monetary policy effects. This means that a monetary tightening sufficient to eliminate inflation on a 12-month horizon would need to be approximately three times more aggressive than under a short-memory inflation model—imposing correspondingly greater costs on output, employment, and credit availability. The implication is not that monetary tightening is inappropriate but that the CBN should communicate realistic multi-year disinflation timelines rather than the shorter horizons typical of inflation targeting frameworks in economies with shorter-memory inflation dynamics.

Furthermore, the FIGARCH findings suggest that inflation uncertainty will remain elevated for several years, complicating the credibility of any specific inflation target. With $\delta =0.693$, volatility shocks to inflation will persist at half their original magnitude for approximately 18 months and at one-third their magnitude for approximately 36 months. This extended volatility persistence increases the risk premium that businesses, households, and investors demand for operating in Nigeria’s inflationary environment, with adverse effects on investment, savings mobilization, and long-term contracting. Alexander (2024) argues that the CBN’s monetary policy credibility has been undermined by the simultaneous fiscal shock of the subsidy removal—a concern fully consistent with the elevated and persistent uncertainty documented by the FIGARCH analysis.

3.8.3. Fiscal and Social Policy Implications

The long-memory dynamics documented in this paper have critical implications for the design of compensatory social programs. The standard approach—one-time cash transfers or temporary price controls—is fundamentally inadequate given that the inflationary effects of the reform will persist at high levels for many years. Mohammed & Musa (2026) document through household survey evidence that government compensation packages were exhausted within weeks of disbursement, while the inflationary effects continued to erode household welfare for months and years afterward. The ARFIMA impulse responses confirm this timing mismatch: with 68% of the price shock persisting at 12 months and 61% at 24 months, adequate compensation would need to be sustained over at minimum a 2–3 year period to bridge the adjustment gap.

Ozigbu & Ezekwe (2025) argue that the economic dilemma of fuel subsidy removal lies precisely in this temporal asymmetry: the fiscal savings are realized immediately while the social costs are deferred and extended. The long-memory analysis formalizes this insight: because the inflationary and welfare effects decay hyperbolically rather than geometrically, the cumulative discounted cost to households is dramatically higher than any analysis based on short-memory dynamics would suggest. Appropriate policy design must account for this extended cost duration through sustained, indexed income support programs.

The disaggregated analysis adds specificity to policy recommendations. Given that food and transportation CPI sub-indices exhibit the highest post-reform persistence ($d=0.912$ and 0.934 respectively), targeted interventions in these sectors—food price stabilization programs, transportation cost subsidies for vulnerable populations, and logistics infrastructure investment—would address the most persistent components of welfare loss with greater efficiency than generalized transfers.

4. Conclusions

This paper has investigated the long-memory properties of Nigerian inflation dynamics in the context of the 2023 fuel subsidy liberalization using ARFIMA and FIGARCH models. The core findings are summarized as follows.

First, Nigerian inflation exhibits genuine and statistically significant long-memory behavior in both pre- and post-reform periods, with semiparametric estimates of the fractional integration parameter d in the range 0.58–0.67 before the reform and 0.84–0.91 after the reform. This increase of approximately 0.25 in the long-memory parameter is statistically significant and consistent across multiple estimation methods.

Second, the ARFIMA(1,d,1) model—selected by information criteria over all alternative orders—provides substantially superior fit and out-of-sample forecast accuracy compared to the conventional ARIMA(1,1,1) benchmark, confirming the practical value of long-memory modeling for Nigerian inflation analysis.

Third, the FIGARCH(1,$\delta$,1) estimates reveal that inflation volatility also exhibits long-memory behavior, with the fractional volatility integration parameter increasing from $\delta \approx 0.437$ pre-reform to $\delta \approx 0.693$ post-reform. The joint ARFIMA-FIGARCH model provides the best forecasting performance, reducing MAPE from 19.4% to 11.8% compared to the ARIMA-GARCH benchmark over the post-reform holdout period.

Fourth, impulse response analysis reveals that under post-reform dynamics, approximately 68% of an inflationary shock persists at 12 months, 61% at 24 months, and 52% at 60 months—far exceeding the persistence implied by conventional I(0) or I(1) models and presenting a profound challenge for monetary policy and welfare protection design.

Fifth, disaggregated analysis reveals that transportation and food CPI sub-indices exhibit the highest post-reform long-memory parameters ($d\approx 0.91$–0.93), confirming that fuel cost pass-through operates primarily through logistics and food supply chain channels and that these channels exhibit the most severe and durable inflationary consequences.

The overarching conclusion is that the 2023 fuel subsidy liberalization has fundamentally altered the stochastic regime of Nigerian inflation, creating a near-unit-root, high-volatility inflationary environment whose resolution will require sustained, multi-year policy interventions rather than the short-term stabilization measures typically deployed in response to transitory shocks. The CBN must recalibrate its monetary policy framework for the long haul, accepting that disinflation from near-unit-root dynamics will be slow and costly; the federal government must design social protection programs on a multi-year, indexed basis; and policymakers must invest in structural interventions—logistics infrastructure, domestic food production, energy diversification—that can moderate the structural cost floors that now sustain high and volatile inflation.

Future research should extend this analysis in several directions. First, as additional post-reform data accumulate, the precision of post-reform parameter estimates will improve, allowing more definitive conclusions about whether d has truly approached unity or whether the short post-reform sample is generating upward bias. Second, regime-switching extensions of the ARFIMA-FIGARCH framework—such as the hidden Markov model approach applied to crude oil prices by Al. A. E. (2026)—could capture potential nonlinearities in how persistence evolves across different inflation regimes. Third, a panel extension across Sub-Saharan African economies undertaking similar subsidy reform would identify the institutional and structural factors that moderate long-memory responses to energy price liberalization across different country contexts.