The team, led by Luis A. Gil-Alana of the University of Navarra in Spain together with OlaOluwa S. Yaya of the University of Ibadan in Nigeria, Ramiro Gil-Serrate of Universidad Antonio de Nebrija and Peter O. Ohue of Queens University, set out to answer a deceptively simple question: when a wet or dry month arrives in these drylands, how long does its influence persist? The answer matters enormously for the hundreds of millions of people whose agriculture, livestock and water supplies depend almost entirely on rain-fed systems. If rainfall shocks fade quickly, droughts are transient annoyances; if they persist, a single bad season can cascade into years of hardship.

To quantify persistence, the researchers turned to the econometric concept of fractional integration. In classical time-series analysis, a series is either stationary, meaning shocks die out quickly, or nonstationary, meaning shocks accumulate indefinitely and require full differencing to stabilise. Fractional models allow an intermediate possibility: a differencing parameter d that can take any real value. When d lies between zero and 0.5, the series displays long memory, with autocorrelations that decay hyperbolically slowly rather than exponentially fast, so disturbances echo through the record for a very long time even though the series ultimately reverts to its mean. When d reaches or exceeds 0.5, the series becomes nonstationary and shocks become effectively permanent in statistical terms.

The innovation in this study lies in extending that framework to seasonality. Monthly rainfall is dominated by an annual cycle, and standard seasonal unit root tests such as the HEGY procedure assume that the seasonal pattern can be removed by differencing the data twelve months apart. Preliminary HEGY tests on the African data rejected both non-seasonal and seasonal unit roots, but the authors argue that such tests are too restrictive: forcing d to equal exactly one risks overdifferencing, which can manufacture spurious conclusions. Their seasonal fractional model instead treats the operator (1 minus L to the twelfth power) raised to the power d, where L is the lag operator and d is estimated freely from the data. This allows the seasonal memory of rainfall to be measured on a continuum rather than forced into a binary stationary-or-not verdict.

The data came from the Climatic Research Unit Time Series dataset hosted on the World Bank Climate Change Knowledge Portal, a globally gridded observational product built by interpolating station measurements onto a regular 0.5-degree grid. Each country series contained 1,488 monthly observations. The descriptive statistics alone tell a story of extremes: mean monthly rainfall ranges from a mere 1.8 millimetres in Egypt and 3.4 millimetres in Libya to more than 70 millimetres in Ethiopia and 83 millimetres in South Sudan. Coefficients of variation frequently exceed 100 percent in countries such as Niger, Mauritania and Namibia, and every series is positively skewed, reflecting long stretches of near-zero rainfall punctuated by occasional torrential events.

The headline finding is that all estimated fractional parameters fall strictly between zero and one, confirming genuine fractional integration in every country examined. But the values split the continent into two hydroclimatic camps. In one group, the estimates are low enough to guarantee stationarity: Somalia records the lowest value at 0.25, followed by Algeria, Tunisia and Kenya at 0.27, Morocco and Egypt at 0.32, and Botswana, Libya, South Africa and Namibia in the 0.36 to 0.40 range. In these countries, rainfall deviations from the long-term mean are corrected relatively quickly, and the confidence intervals rule out nonstationarity. At the other extreme, Mauritania (0.54), Niger (0.56), Sudan and South Sudan (0.60), Chad (0.66) and Mali (0.67) all show estimates at or above the 0.5 threshold, meaning rainfall anomalies there are highly enduring and revert only slowly. Ethiopia sits almost exactly on the boundary at 0.49.

The geography of these results is not accidental. The high-persistence countries are concentrated in the Sahelian belt, where rainfall arrives in a single monsoonal wet season governed by the annual migration of the Intertropical Convergence Zone and the West African Monsoon. Rainfall events in these regions occur in coherent seasonal sequences, and prolonged dry seasons with relatively low intra-seasonal variability enhance temporal persistence. By contrast, the low-persistence countries experience climatically noisy regimes. Kenya and Somalia are shaped by two distinct rainy seasons, the so-called long rains and short rains, modulated strongly by the El Niño-Southern Oscillation and the Indian Ocean Dipole, while Algeria and Tunisia receive episodic precipitation from Mediterranean cyclones and mid-latitude weather systems. Multiple, interacting rainfall-generating mechanisms fragment the seasonal signal and weaken long memory.

The study also detected statistically significant positive time trends in 10 of the 17 countries, with the steepest increases in Ethiopia and Kenya, equivalent to roughly 0.20 and 0.34 millimetres per year respectively under the white-noise specification. To check robustness, the authors repeated the analysis allowing for weakly dependent errors using the non-parametric Bloomfield approach, which approximates the spectrum of autoregressive moving average processes with far fewer parameters, and by aggregating the monthly data to quarterly series. The broad picture held: under the Bloomfield specification, stationarity was confirmed in the same ten countries plus Ethiopia, with only South Sudan, Chad and Mali remaining nonstationary, while the quarterly analysis pushed estimates slightly higher, leaving nine countries in the nonstationary camp, with Chad reaching 0.89 and Mali 0.84. Across all specifications, six countries, Somalia, Algeria, Kenya, Tunisia, Egypt and Morocco, showed consistent evidence of stationary rainfall.

For drought managers and agricultural planners, the implications are stark. In Mali, Chad and South Sudan, a drought year is not simply a bad year that the climate forgets; the statistical signature suggests its effects linger, compounding food insecurity and water stress over successive seasons. Conversely, in the Mediterranean-influenced north and the dual-season east, rainfall anomalies wash out faster, but the very variability that produces this rapid mean reversion also makes rainfall harder to forecast from one season to the next. The authors note that countries with higher seasonal persistence may require fundamentally different drought-monitoring and agricultural-planning strategies than those where anomalies revert quickly, and the findings underscore sharply differing levels of hydroclimatic resilience across the continent.

The authors are candid about limitations. Country-level averages can mask enormous sub-national diversity, particularly in climatically heterogeneous nations such as Kenya, Ethiopia, South Africa and Sudan, where distinct regional rainfall regimes coexist within national borders. Future work, they suggest, will examine monthly rainfall totals, anomalies relative to the 1991 to 2020 climate baseline, standardised precipitation indices and log-transformed series, as well as non-linear trends built from Chebyshev polynomials, Fourier functions or neural networks, and the possible role of structural breaks in creating spurious long memory. Even with these caveats, the study delivers a rare century-scale, statistically rigorous portrait of rainfall behaviour in some of the world’s most vulnerable drylands, and it offers a quantitative foundation for anticipating which regions will recover from climate shocks and which will carry them forward.

Subject of Research: Seasonal persistence and long-memory dynamics of rainfall in arid and semi-arid African climatic zones

Article Title: Testing seasonal persistence of rainfall in arid and semi-arid african climatic zones using a seasonal fractional model

Article References: Gil-Alana, L. A., Yaya, O. S., Gil-Serrate, R., & Ohue, P. O. (2026). Testing seasonal persistence of rainfall in arid and semi-arid african climatic zones using a seasonal fractional model. Theoretical and Applied Climatology, 157(10), Article 628. https://doi.org/10.1007/s00704-026-06551-x

Image Credits: AI Generated

DOI: 10.1007/s00704-026-06551-x

Keywords: rainfall persistence, fractional integration, long memory, Sahel, drought, arid climates, seasonality, Africa, time series analysis, HEGY test, Intertropical Convergence Zone, hydroclimatology

The team, led by Manuel Monge of Universidad Francisco de Vitoria and Universidad Europea de Valencia in Spain, together with Jorge Aracil Jordá of Universidad Pontificia de Comillas and Juan Infante of Universidad Villanueva, set out to resolve a long-standing ambiguity in applied economics. Although consumer expectations and price levels are among the most intensively tracked macroeconomic indicators in the world, the applied literature offers a bewildering range of descriptions of their time-series behavior. Different studies classify the same series as stationary, trending, or unit-root processes depending on the tests used and the sample chosen, leaving policymakers without a clear picture of how lasting an economic shock to confidence or purchasing power really is.

To cut through that ambiguity, the authors adopted what they describe as a unitary empirical strategy rather than a conventional multivariate one. Each series was examined on its own terms, with inflation and general consumer confidence indicators maintained as context series. The first step was a rigorous classification of the order of integration of each variable using augmented versions of the Dickey-Fuller and Kwiatkowski-Phillips-Schmidt-Shin procedures, two of the workhorse tools of time-series econometrics. By explicitly accounting for deterministic specifications such as intercepts and trends, these tests determine whether a series must be differenced once to become stable, or whether it already fluctuates around a fixed mean.

The results of that first stage were clear. The Index of Consumer Sentiment, the purchasing power of the consumer dollar, and the broader consumer confidence index all emerged as first-order integrated, meaning they are non-stationary in levels and only settle down after being differenced. Inflation, by contrast, proved to be level stationary, fluctuating around a stable long-run value without requiring differencing. The purchasing power series itself was constructed as the inverse of the Consumer Price Index for all items for urban consumers, so it directly measures how much real consumption a dollar can command at any point in time.

But the study did not stop at the binary distinction between stationary and non-stationary. The authors then estimated a fractional differencing parameter for each series, allowing the order of persistence to lie along a continuum rather than forcing each variable into one of two rigid categories. This fractional integration framework, which has gained traction in macroeconomics over recent decades, captures the idea that many economic processes are neither quickly mean-reverting nor permanently drifting, but something in between: they remember shocks for a very long time, with the effect of any disturbance decaying only slowly, often over years or decades.

The fractional estimates delivered the study’s most intriguing finding. For consumer sentiment, purchasing power, and consumer confidence, the estimated differencing parameters clustered slightly below or slightly above one, placing all three processes in the non-stationary region of the continuum while still implying a capacity for slow recovery toward their means. In practical terms, this means that a collapse in consumer confidence or an erosion of real purchasing power is not necessarily permanent: the series can, in principle, drift back toward their historical average levels, but only gradually, with the imprint of the shock visible in the data for a long time. The consumer price index itself, meanwhile, was not easily differentiated from a unit root, indicating that price levels behave as though shocks to them accumulate without a reliable tendency to reverse.

Perhaps the most counterintuitive result concerns purchasing power. Among the mean-reverting series in the study, purchasing power turned out to be the most mean-reverting of all, a finding the authors note runs contrary to most expectations, which hold that mean-reverting data should exhibit the longest memory. The implication is that the real value of the consumer dollar, despite being buffeted by oil shocks, financial crises, pandemic-era supply disruptions, and the sharp inflation surge of 2021 through 2023, retains a stronger tendency to return toward its long-run level than sentiment or confidence measures do. Each series was specifically characterized through the results, giving economists a persistence fingerprint for every variable examined.

The authors are careful to draw a boundary around what their findings do and do not mean. The study finds no relationship between the estimated sentiment levels and affordability conditions, nor does it assert one. In other words, the statistical memory properties of consumer sentiment and the memory properties of purchasing power are characterized separately, and the paper does not claim that one drives the other or that the degree of persistence in feelings about the economy can be read off from the degree of persistence in prices. That restraint matters, because the temptation to link confidence and affordability causally is strong in both public debate and policy discussions.

Even so, the findings carry real weight for how economists and policymakers interpret the American economic mood. If consumer sentiment behaves as a highly persistent, near-unit-root process, then campaigns or interventions designed to restore confidence after a downturn should not expect rapid normalization; the psychological scar of a recession may persist for years, shaping spending and investment decisions long after objective conditions improve. Conversely, the comparatively stronger mean reversion in purchasing power suggests that the erosion of real wages and savings, while painful, has historically tended to correct itself over time as inflation subsides and nominal incomes adjust, a pattern visible in the data following the inflationary episodes of the early 1980s and the post-pandemic period.

The study also demonstrates the value of fractional integration methods for macroeconomic monitoring more broadly. Traditional unit-root tests force a stark either-or choice that can mislead: a series with a differencing parameter of 0.95 and one with a parameter of 1.05 may both fail standard stationarity tests, yet their long-run behavior differs meaningfully, with the former eventually reverting and the latter drifting without bound. By estimating the differencing parameter directly and reporting it along a continuum, the Spanish research team has provided a more nuanced map of persistence in some of the most consequential numbers in American economic life, from the confidence consumers express in surveys each month to the quiet, cumulative erosion and recovery of the dollar in their pockets. As debates over inflation, affordability, and economic anxiety continue to dominate headlines, the message of this research is that both prices and perceptions move on long timescales, and understanding their memory may be essential to anticipating when, and whether, they will return to normal.

Subject of Research: Long memory and fractional integration in US consumer sentiment and purchasing power, 1978 to 2025

Article Title: Long memory in consumer sentiment and consumer dollar purchasing power in the United States from 1978 to 2025

Article References: Monge, M., Jordá, J. A., & Infante, J. (2026). Long memory in consumer sentiment and consumer dollar purchasing power in the United States from 1978 to 2025. Discover Sustainability. https://doi.org/10.1007/s43621-026-04367-3

Image Credits: AI Generated

DOI: 10.1007/s43621-026-04367-3

Keywords: consumer sentiment, purchasing power, fractional integration, long memory, unit root testing, inflation persistence, consumer confidence, University of Michigan Index of Consumer Sentiment, mean reversion, US macroeconomic time series, Discover Sustainability, time-series econometrics