{"id":"600f9d2d-cde5-4601-868c-33490c866f38","arxiv_id":"2505.19243","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":17,"one_line_summary":"Fractionally differenced price series, especially with a differencing order estimated from an ARFIMA model, improved LSTM forecasts and portfolio trading metrics compared to logarithmic returns.","lead":"Researchers compared four ways of transforming stock index prices before feeding them into an LSTM neural network: ordinary returns, fractional differencing, and two variants based on long-memory time series models. They found that fractional differencing methods often improved forecast accuracy and, for a portfolio, produced better risk-adjusted trading results than simple returns.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reported accuracy gains for fractional differencing may reflect a smoother, more autocorrelated target rather than genuine memory preservation; no persistence baseline is included.","rationale":"The reader correctly identifies that per-method hyperparameter tuning and the absence of significance testing weaken the causal link between differencing method and performance. I agree those are real issues. However, the more fundamental confound is that the comparison targets have very different time-series properties: fractionally differenced series are far more autocorrelated than log returns, so a trivial persistence forecast in the transformed domain will yield much lower price-level error for fractional methods. The paper never benchmarks against such a baseline, so the reported RMSE/MAE/MAPE improvements (Table 4) and the superior trading metrics (Tables 13–14) could be driven by the predictability of the target transformation itself rather than by memory preservation aiding the LSTM. This does not prove the central claim false, but it means the evidence as presented does not discriminate between the memory-preservation hypothesis and a much simpler autocorrelation artifact. The verdict should remain conditional, with the added requirement of a persistence/AR baseline per transform before the claim can be accepted.","tokens_in":36356,"tokens_out":6804,"duration_ms":49331,"concrete_test":"For each asset and each of the four transforms, compute the persistence forecast y_hat_{t+1} = y_t on the transformed series, invert it to price level exactly as in Section 4.1 step 10, and compute RMSE, MAE, and MAPE on the same test window. Then compare the ratio LSTM_error / persistence_error across the four transforms. If the fractional methods do not show a systematically larger improvement over persistence than log returns do, the headline claim is not supported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central comparison in Table 4 computes RMSE, MAE, and MAPE on price-level forecasts obtained by inverting each method's transformed predictions (Section 4.1, step 10). The four target series have very different autocorrelation structures: log returns are near white noise, while fractionally differenced log prices with d≈0.2–0.5 are highly persistent. An LSTM that simply predicts the current transformed value will therefore produce much smaller price-level errors for fractional methods, even if it learns no genuine predictive signal. The paper reports no persistence (no-change) or AR baseline on the same transformed series, so the accuracy advantage in Table 4 may be an artifact of target smoothness rather than evidence that memory preservation improves forecastability. The same issue contaminates the trading-signal comparison, because a smoother target changes how often the threshold rule in Equations [46]–[47] triggers trades. Hyperparameter tuning per method (Section 4.1, steps 7–8) is a second confound, but the missing baseline is the load-bearing one.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper compares four data transformation methods as inputs to LSTM networks for one-day-ahead forecasting of four stock indices (S&P 500, WIG20, DAX, Nikkei 225) over an out-of-sample period 2021-2023: logarithmic returns, De Prado's (2018) fractional differentiation, fractional differentiation using an ARFIMA(0,d,0)-estimated d, and tempered fractional differentiation using ARTFIMA(0,d,lambda,0) parameters. For each asset and method, the authors estimate d (and lambda), construct features from lagged transformed values and technical indicators, train an LSTM with per-method hyperparameter tuning, invert the forecasts to price levels, and compare RMSE/MAE/MAPE. They also construct Long-Short and Long-Only trading strategies from the forecasts, backtest them with transaction costs, and report risk-adjusted metrics (IR, SR, MD, etc.) for individual assets and an equally weighted portfolio. The central claim is that memory-preserving fractional differentiation improves forecasting accuracy and trading performance relative to log returns, with the ARFIMA-based method often performing best and tempered differentiation performing poorly.","tokens_in":36729,"tokens_out":2833,"duration_ms":29717,"significance":"If the claims hold, the paper provides a useful extension of De Prado's (2018) fractional differentiation framework by showing that d estimated by ARFIMA/ARTFIMA models can be plugged into the fixed-width-window weighting scheme, and that such inputs improve LSTM forecasts and trading strategies. The use of four major indices, a clearly described train/validation/test split, out-of-sample evaluation, and transaction-cost-adjusted backtests are strengths. The paper is also candid about the WIG20 anomaly and about the lack of a universally best d. However, the manuscript does not yet establish that the observed improvements are due to memory preservation rather than target smoothness or hyperparameter tuning artifacts, and it provides no statistical significance tests for the forecast or trading comparisons. These gaps are load-bearing for the main hypotheses H1, H2, and H4.","major_comments":[{"comment":"The forecast accuracy comparison is performed on price-level forecasts obtained by inverting each method's transformed predictions (Section 4.1, step 10), but the four target series have very different autocorrelation structures: log returns are near white noise, while fractionally differenced log prices with d≈0.2–0.5 are highly persistent. A trivial no-change (persistence) forecast on the transformed series will produce much smaller price-level errors for the more persistent fractional series, even if the LSTM learns no genuine predictive signal. The paper reports no such persistence baseline, nor an AR(1) or random-walk benchmark applied to the same transformed series, so the RMSE/MAE advantage in Table 4 may reflect target smoothness rather than memory preservation. This issue also affects the trading-signal comparison, because a smoother target changes how often the threshold rule in Equations (46)–(47) triggers trades. Please add persistence and simple linear baselines on the same transformed series, or report error metrics on the transformed scale with appropriate scaling.","section":"§4.2, Table 4"},{"comment":"The LSTM hyperparameters (batch size, epochs, number of layers, number of cells, dropout, recurrent dropout, regularization, learning rate) are tuned separately for every asset and every differencing method. With no fixed architecture, no seed control, and no repeated runs, the performance differences in Tables 4–14 could be artifacts of the tuning process or random weight initialization rather than effects of the differencing method. The paper reports no standard deviations across runs, no sensitivity analysis, and no evidence that the tuning procedure converges to comparable-quality optima across methods. At minimum, please report the selected hyperparameters and seed values, and ideally rerun each configuration several times to show that the ranking is stable.","section":"§4.1, steps 7–8"},{"comment":"The trading performance comparisons are based on single backtest paths over a three-year out-of-sample period, with no statistical significance testing. The portfolio Long-Only ARFIMA result (IR 1.63 vs 0.48 for Buy & Hold, SR 2.99 vs 0.79) looks impressive but could arise from luck given the small number of daily observations and the many configurations compared. Please add Diebold-Mariano or similar tests on the forecast errors, and use block bootstrap or other resampling methods to assess the variability of the trading metrics. Also report the number of trades triggered by each method, since the persistence of the target affects trade frequency and thus transaction-cost impact.","section":"§4.3, Tables 13–14"},{"comment":"For WIG20, De Prado's method with d=0.12 produces the worst forecasts (RMSE 39.43 vs 33.30 for log returns, MAE 30.85 vs 24.32), directly contradicting the blanket statement in the abstract that fractional differentiation improves predictive performance. The text acknowledges this exception, but the conclusion that H1 is 'supported' relies on the other three indices and on the ARFIMA variant for WIG20. Please either soften the general claim to reflect the asset-specific nature of the results, or provide a substantive explanation of why a very small d value can hurt LSTM performance, supported by evidence rather than speculation.","section":"§3.2.1, Table 1; §4.2, Table 4"}],"minor_comments":[{"comment":"The word 'rerurn' in the final paragraph of the conclusions should be 'return'; elsewhere 'ARFTIMA' appears in place of 'ARTFIMA' (e.g., in the introduction's overview of Section II).","section":"Conclusions"},{"comment":"The LSTM equations use a symbol that appears as a garbled font; Equations (1)–(6) are unreadable in the current PDF. Please ensure all math is typeset correctly.","section":"§2.1"},{"comment":"The phrase 'elementwise production operator' should be 'elementwise product operator'.","section":"§2.1, text after Eq. (6)"},{"comment":"The equity-line figures in the annex would be easier to interpret if the y-axis were labeled with the portfolio value or cumulative return, and if the legend distinguished line types in addition to colors for grayscale printing.","section":"Figures 12–21"},{"comment":"The paper states that the ADF critical value is −2.86, but this value depends on the sample size and test specification; please report the exact test regression and critical value used for each index, or at least cite the source of the critical value.","section":"§3.2.1"}],"recommendation":"major_revision","confidential_remarks":"The paper's empirical scope is suitable for a quantitative finance journal, and the extension of De Prado's method using ARFIMA/ARTFIMA parameter estimates is a reasonable contribution. My main concern is that the central comparison is confounded by target persistence and by per-method hyperparameter tuning, and the lack of any statistical testing makes the headline claims difficult to evaluate. These issues are fixable with additional baselines and robustness checks, so I do not recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a straightforward, honest empirical comparison of four differencing methods feeding into LSTMs: log returns, De Prado's fractional differencing, an ARFIMA-d variant, and a tempered ARTFIMA variant. The genuinely new bit is using ARFIMA-estimated d and ARTFIMA (d, lambda) inside De Prado's fixed-width window. That combination does not appear in the cited literature, and the derivation of the tempered recursion (Eq. 36) is clean. Credit where due: the research design is transparent, the hypotheses are stated in advance, the out-of-sample split is genuine, and the authors even report a result (WIG20) where De Prado's method loses to log returns, which suggests they are not cherry-picking.\n\nThe soft spots are real, and the stress-test concern lands. Table 4 compares RMSE/MAE/MAPE on price-level forecasts after inverting each transform. Fractionally differenced series with d around 0.2–0.5 are much smoother and more autocorrelated than log returns. An LSTM that simply predicts the current transformed value will look better on those series. There is no persistence (no-change) or AR baseline on the same transformed series, so the accuracy advantage may be an artifact of target smoothness rather than genuine memory preservation. The same issue contaminates the trading results, because the signal threshold in Equations [46]–[47] will trigger trades less often on a smoother target, and indeed the ARFIMA portfolio strategy shows very low ASD. A second confound is that hyperparameters are tuned separately per method (Section 4.1, steps 7–8), so architecture differences could drive the gap. No standard errors, no Diebold-Mariano test, no multiple-comparison control.\n\nThat said, the direction of the evidence is plausible and the paper is not overclaiming massively—it does reject H3 and partially support H4. The WIG20 exception shows the effect is asset-dependent. But the central claim that fractional differencing improves forecasting performance is not established without a baseline that controls for target persistence.\n\nWho gets value? A reader interested in a clean worked example of applying ARFIMA/ARTFIMA parameters to fractional differencing, and a benchmark for future studies. It deserves a serious referee, but the referee should demand a persistence baseline, significance tests, and either a fixed architecture or a sensitivity analysis across seeds.\n\nRecommendation: send to peer review, but expect major revision before it can support the strong conclusion.","headline":"A competent empirical comparison of fractional differencing variants for LSTM forecasting, but the missing persistence baseline and lack of significance tests make the headline claim fragile.","tokens_in":37273,"tokens_out":1948,"would_cite":false,"duration_ms":22131,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62M10","91B84"],"pacs":[],"model":"deepseek-v4-flash","headline":"Feeding LSTMs fractionally differenced prices, not log returns, improves forecasts and risk-adjusted trading returns.","keywords":["fractional differentiation","LSTM","financial time series forecasting","memory preservation","ARFIMA","ARTFIMA","trading strategy","stock market indices"],"falsifier":"Re-run the four-way comparison holding the LSTM architecture, hyperparameters, and random seeds fixed across all transforms, and check whether the fractionally differenced inputs still improve RMSE, MAE, and MAPE and still beat buy-and-hold; if the ranking shifts with seeds or architecture, the memory-preservation explanation is not established. A complementary test is to simulate ARFIMA series with known fractional orders and verify that the LSTM's advantage over log returns grows with the true $d$.","tokens_in":36135,"feed_emoji":"📈","tokens_out":11497,"duration_ms":84953,"temperature":0.7,"pith_summary":"This paper claims that how prices are transformed before training a forecasting model matters as much as the model itself. The authors compare the standard log-return transform with three variants of fractional differencing — a middle ground between no differentiation and full first-order differencing that removes non-stationarity while retaining more of a series' long-range memory — as inputs to an LSTM network forecasting the S&P 500, WIG20, DAX, and Nikkei 225 over ten years. On the out-of-sample period, the fractionally differenced inputs produce the most accurate one-day-ahead predictions for every index, and trading strategies built on those predictions beat the buy-and-hold benchmark on an equally weighted portfolio, especially on risk-adjusted metrics. If the paper is right, a memory-preserving preprocessing step is a cheap and direct way to improve both forecast accuracy and investment performance.","feed_headline":"Fractional differencing beats log returns in LSTM forecasts","feed_subtitle":"LSTM forecasts gain when prices are fractionally differenced, not log-returned, beating buy-and-hold.","key_machinery":"The load-bearing object is the fractional differencing operator $\\Delta^d=(1-B)^d$, expanded through the binomial series into weights $\\omega_0=1$ and $\\omega_k=-\\omega_{k-1}(d-k+1)/k$ applied to past observations, with the tempered variant $\\omega_k=-\\omega_{k-1}(d-k+1)e^{-\\lambda}/k$ for the ARTFIMA version. A fixed-width window keeps only weights above a threshold $\\tau$, and the minimal order $d$ is the smallest value for which the transformed series passes the augmented Dickey-Fuller test at the 95% level, so the input is stationary while its correlation with the original price level stays near 0.99 (and above 0.9 even for the ARFIMA-based series). These differentiated series, lagged and combined with SMA, RSI, Bollinger Bands, and MACD indicators, enter LSTM memory cells whose gates decide what long-range information to retain. The machinery embodies the stationarity-versus-memory trade-off the comparison rests on: log returns maximize stationarity but erase memory, while fractional orders keep enough of both for the LSTM to exploit.","core_discovery":"The central claim is that fractionally differentiated price series are a better input to an LSTM forecaster than logarithmic returns, because full first-order differencing removes an excessive amount of a series' memory. The differencing order $d$ is estimated three ways — De Prado's minimal value that achieves stationarity per the augmented Dickey-Fuller test, the fractional parameter from an ARFIMA$(0,d,0)$ model, and a tempered variant from an ARTFIMA$(0,d,\\lambda,0)$ model — and each resulting series, augmented with the same technical indicators, feeds a separately tuned LSTM. On out-of-sample data from 2021 to 2023, either the De Prado or the ARFIMA-based transform yields the lowest RMSE, MAE, and MAPE for every index (for example, S&P 500 RMSE of 49.81 versus 65.30 for log returns), while the tempered series resembles log returns and performs no better, leading the authors to reject the tempering hypothesis. The trading results are strongest at the portfolio level, where the ARFIMA-based Long-Only strategy reaches an Information Ratio of 1.63 and a Sortino Ratio of 2.99, against 0.48 and 0.79 for buy-and-hold, with transaction costs included.","pith_inferences":["If the proposed mechanism is right, the advantage should scale with the strength of a series' long memory; a direct test is simulating ARFIMA series with known $d$ and checking that the LSTM's gain over log returns grows as $d$ approaches 0.5.","Because hyperparameters were tuned separately for each method and no random seeds are reported, an essential cross-check is rerunning the comparison with a fixed architecture and multiple seeds to confirm that the transform, not the tuning, drives the ranking.","The portfolio-level outperformance hints that the edge is more visible after diversification across assets; testing the same ordering of transforms on individual stocks, currencies, or cryptocurrencies would show whether the result generalizes.","The failed tempered variant implies the exponential factor $e^{-\\lambda}$ removes exactly the long-range autocovariance the LSTM exploits, a claim that could be tested by sweeping $\\lambda$ on a single asset while holding $d$ fixed and observing whether forecast error rises monotonically with $\\lambda$."],"forward_implications":["Choosing a memory-preserving differencing order is a direct lever on forecast error: for every index, a fractional method beats log returns on RMSE, MAE, and MAPE.","Fractionally differenced forecasts translate into tradeable signals, with the ARFIMA-based Long-Only portfolio strategy beating buy-and-hold on both Information Ratio (1.63 vs 0.48) and Sortino Ratio (2.99 vs 0.79) over 2021-2023.","No single fractional method is universally best: the minimal stationary order from De Prado's method wins for some assets while the larger ARFIMA-estimated order wins for others, so the smallest stationary $d$ is not automatically optimal.","Tempered fractional differencing, as implemented here, removes about as much memory as log returns and offers no advantage, rejecting the hypothesis that tempering helps.","The outperformance survives transaction costs of 0.005%, supporting practical deployment of the strategy rather than an accuracy gain that only exists in the metrics."],"supporting_citations":[{"why":"It supplies the fractional differentiation methodology: the binomial-series weights, the fixed-width window, and the ADF-based choice of the minimal differencing order.","marker":"De Prado (2018)"},{"why":"It is the direct predecessor that showed fractional differentiation improving ANN forecasting on the same four indices, and its hypothesis is what this study extends to LSTMs and trading.","marker":"Gajda and Walasek (2020)"},{"why":"It introduces the ARFIMA long-memory model whose estimated fractional parameter provides the second differencing order.","marker":"Granger and Joyeux (1980)"},{"why":"It establishes fractional differencing theory and the ARFIMA(0,d,0) stationarity conditions used to justify the estimated orders.","marker":"Hosking (1981)"},{"why":"It introduces tempered fractional calculus, the basis of the ARTFIMA-based tempered differencing variant.","marker":"Meerschaert et al. (2015)"},{"why":"It provides ARTFIMA parameter estimation and the artfima package used to obtain the $d$ and $\\lambda$ values applied in tempering.","marker":"Sabzikar et al. (2019)"},{"why":"It sets the LSTM forecasting framework the paper adopts, including the memory-cell equations used to motivate the model choice.","marker":"Fischer and Kraus (2018)"},{"why":"It supplies the Long-Short and Long-Only strategy logic and the risk-adjusted metric framework used in the backtest evaluation.","marker":"Kashif and Ślepaczuk (2025)"}],"fun_headline_variants":["Fractional differencing outperforms log returns in LSTM models","Memory-preserving differencing improves LSTM forecasts","LSTM trading strategies improve with fractional differencing","Fractional differencing beats log returns for LSTM predictions","Fractional differencing: better LSTM forecasts and trading"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The comparison assumes that the per-method hyperparameter tuning (batch size, layers, cells, dropout, regularization, and learning rate) is not what actually decides which differencing transform wins, since no fixed architecture, random seed, or sensitivity analysis is reported to rule tuning effects out.","fun_headline_variants_meta":{"raw":{"variants":["Fractional differencing outperforms log returns in LSTM models","Memory-preserving differencing improves LSTM forecasts","LSTM trading strategies improve with fractional differencing","Fractional differencing beats log returns for LSTM predictions","Fractional differencing: better LSTM forecasts and trading"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000491,"raw_usage":{"total_tokens":2416,"prompt_tokens":951,"completion_tokens":1465,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":1387}},"tokens_in":567,"tokens_out":1465,"duration_ms":10805,"temperature":1.0,"reasoning_tokens":1387,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:17:39.942266+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the four-way comparison holding the LSTM architecture, hyperparameters, and random seeds fixed across all transforms, and check whether the fractionally differenced inputs still improve RMSE, MAE, and MAPE and still beat buy-and-hold; if the ranking shifts with seeds or architecture, the memory-preservation explanation is not established. A complementary test is to simulate ARFIMA series with known fractional orders and verify that the LSTM's advantage over log returns grows with the true $d$.","supporting_citations":[{"cited_title":"M., Sabzikar, F., & Chen, J","cited_arxiv_id":null,"evidence_quote":"It introduces tempered fractional calculus, the basis of the ARTFIMA-based tempered differencing variant."}],"review_version":1}