ARIMA-LSTM Hybrid Forecasting

• ARIMA–LSTM hybrid is a forecasting architecture that merges ARIMA’s linear modeling with LSTM’s nonlinear sequence learning to capture complex time-series patterns.
• It is implemented via residual decomposition or feature augmentation, where ARIMA extracts linear trends and LSTM refines the remaining nonlinear structure.
• Empirical results in finance, weather, and demand forecasting show that hybrid models can outperform standalone methods when properly tuned.

Searching arXiv for recent and foundational work on ARIMA–LSTM hybrids to ground the article in cited papers. An ARIMA–LSTM hybrid is a time-series forecasting architecture that combines the linear modeling capacity of autoregressive integrated moving average methods with the nonlinear sequence modeling capacity of long short-term memory networks. Across the literature, the hybrid is typically motivated by the assumption that an observed series contains both linear and nonlinear structure, often expressed as $Z_t = L_t + N_t$, with ARIMA or SARIMA used to model the linear component and LSTM used either to model residual nonlinear structure or to incorporate ARIMA outputs as features in a downstream recurrent predictor (Zhou et al., 2021, Duarte et al., 26 Sep 2025, Stempień et al., 26 May 2025). The design has been used in domains including wind speed forecasting, finance, electricity demand, weather, mortality, and fund net value prediction, with substantial variation in how the two components are coupled, how multi-step prediction is handled, and whether the components are trained disjointly or jointly (Mohapatra et al., 2023, Choi, 2018, Grandon et al., 2023, Rajeev et al., 12 Jan 2026, Aydın et al., 2023).

1. Conceptual definition and scope

The core premise of the ARIMA–LSTM hybrid is that classical linear models and recurrent neural networks solve different subproblems. ARIMA is used to capture linear dependence, differenced stationarity, and in some settings trend or seasonality, while LSTM is used to capture nonlinear temporal dependencies, long-range effects, or residual structure not explained by the linear model (Zhou et al., 2021, Choi, 2018, Duarte et al., 26 Sep 2025). In several formulations, the decomposition is stated explicitly as

$Z_t = L_t + N_t,$

where $L_t$ denotes the linear component and $N_t$ the nonlinear component (Zhou et al., 2021, Duarte et al., 26 Sep 2025).

In residual-learning hybrids, ARIMA is fitted first and residuals are defined as

$e_t = y_t - \hat{y}_t^{(\text{ARIMA})},$

or equivalently

$E_t = Z_t - \hat{L}_t,$

after which an LSTM is trained on the residual sequence and the final forecast is formed additively (Choi, 2018, Duarte et al., 26 Sep 2025). In feature-augmented hybrids, the ARIMA forecast or ARIMA residual is instead treated as an input feature to the LSTM, which then produces the final prediction directly (Stempień et al., 26 May 2025, Kashif et al., 2024). This suggests that “ARIMA–LSTM hybrid” is best understood as a family of hybridization strategies rather than a single canonical architecture.

The literature also distinguishes ARIMA from closely related linear components such as SARIMA and SARIMAX. In meteorological and seasonal settings, the seasonal variant is used to represent cyclical structure more explicitly, but the underlying hybrid logic remains the same: a statistical linear backbone is paired with an LSTM residual learner or nonlinear correction model (Rajeev et al., 12 Jan 2026, Aydın et al., 2023). A plausible implication is that the term “ARIMA–LSTM hybrid” often functions as a generic label for ARIMA-family–LSTM systems.

2. Canonical architectural patterns

The most common architectural pattern is the additive residual hybrid. In the fund net value formulation, the series is decomposed as

$y_t = L_t + N_t,$

with residuals

$e_t = y_t - L_t,$

and the LSTM learns

$$N_t = f(e_{t-1}, e_{t-2}, \dots, e_{t-m}) + \varepsilon_t,$$

leading to the final hybrid forecast

$$\hat{y}_t^{(\text{hybrid})} = \hat{L}_t + \hat{N}_t$$

(Zhou et al., 2021). The same additive logic appears in mortality forecasting, where the hybrid forecast is written as

$Z_t = L_t + N_t,$0

or horizon-wise as

$Z_t = L_t + N_t,$1

(Duarte et al., 26 Sep 2025).

A second pattern is the non-additive feature-augmented hybrid. In financial forecasting, one architecture uses the ARIMA forecast $Z_t = L_t + N_t,$2 as an additional explanatory feature in the LSTM input: $Z_t = L_t + N_t,$3 rather than assuming that the linear and nonlinear components combine additively (Stempień et al., 26 May 2025). A related investment-strategy architecture uses ARIMA residuals as one of several LSTM inputs—together with price, volatility, and volume—while the LSTM remains the final prediction engine (Kashif et al., 2024).

A third pattern is joint state-space optimization. Instead of fitting the statistical model and the recurrent model separately, the hybrid state-space formulation combines an LSTM state vector with an ARMA-family state vector and optimizes them jointly through particle filtering (Aydın et al., 2023). In that formulation, the hybrid prediction is written as

$Z_t = L_t + N_t,$4

where the nonlinear component is the LSTM output and the linear component is the SARIMAX output (Aydın et al., 2023). This suggests a broader taxonomy: additive residual hybrids, feature-augmented hybrids, and jointly optimized state-space hybrids.

3. Statistical and recurrent components

The linear component is generally specified as ARIMA$Z_t = L_t + N_t,$5, or as its seasonal extension. A standard ARIMA representation given in the literature is

$Z_t = L_t + N_t,$6

with lag operator $Z_t = L_t + N_t,$7, differencing order $Z_t = L_t + N_t,$8, autoregressive coefficients $Z_t = L_t + N_t,$9, and moving-average coefficients $L_t$0 (Stempień et al., 26 May 2025). In other expositions the model is written as

$L_t$1

or equivalently

$L_t$2

(Mohapatra et al., 2023, Rajeev et al., 12 Jan 2026). Order selection is commonly described through ACF and PACF inspection combined with AIC or BIC-based model selection (Prajapati et al., 2021, Zhou et al., 2021, Choi, 2018).

The LSTM component is described through the standard gated recurrent equations. A representative formulation is

$L_t$3

$L_t$4

$L_t$5

$L_t$6

$L_t$7

$L_t$8

with $L_t$9, $N_t$0, and $N_t$1 denoting forget, input, and output gates (Prajapati et al., 2021, Mohapatra et al., 2023, Rajeev et al., 12 Jan 2026, Zhou et al., 2021, Duarte et al., 26 Sep 2025). These equations are repeated with minor notational differences across finance, epidemiology, mortality, and weather forecasting papers, indicating a largely standardized recurrent core.

In more elaborate hybrids, the linear and recurrent parts are embedded in a unified state-space model. The LSTM hidden state and cell state are treated as latent states, the ARMA-family coefficients are also treated as state variables, and the observation equation sums the linear and nonlinear predictions (Aydın et al., 2023). This suggests that the distinction between “model combination” and “joint model definition” is central to the methodological evolution of the field.

4. Training regimes and forecast generation

In residual hybrids, the training sequence is usually: fit ARIMA; compute residuals; construct sliding windows over residuals; train LSTM; add ARIMA and LSTM forecasts (Zhou et al., 2021, Choi, 2018). This workflow is used directly in fund net value prediction, where the historical series is first differenced for ARIMA, ARIMA forecasts are computed, residuals are extracted, and a three-layer LSTM is trained on residual windows using a sliding-window online prediction scheme (Zhou et al., 2021).

In sequential feature hybrids, ARIMA outputs are incorporated into the LSTM state representation. One sports forecasting framework constructs an input state matrix $N_t$2, where $N_t$3 is the ARIMA-predicted national team feature matrix and $N_t$4 is an athlete-derived feature matrix, and then feeds this combined representation to the LSTM (Liu et al., 11 Feb 2025). This is explicitly described as sequential rather than parallel, and differs materially from the residual-addition design.

Long-horizon forecasting introduces additional design constraints. In the weather residual-learning model, SARIMA is fitted on daily average temperature with exogenous Fourier seasonal encoding, residuals are computed, and a stacked LSTM is trained on 14-day multivariate windows to predict residuals (Rajeev et al., 12 Jan 2026). Because open-loop recursive prediction can diverge, the paper applies a decay factor $N_t$5 to the LSTM residual component after the first 5 forecast days: $N_t$6 (Rajeev et al., 12 Jan 2026). This indicates that hybrid systems sometimes require explicit stabilization mechanisms when the recurrent component is used recursively over long horizons.

Multi-step forecasting strategies are a distinct design axis. In short mortality rate series, the residual LSTM is evaluated under Recursive, Direct, and Multi-Input Multi-Output formulations, while ARIMA remains the linear component (Duarte et al., 26 Sep 2025). The study concludes that the recursive approach leads to better accuracy in residual modeling for mortality rate forecasting, and that the ARIMA-LSTM hybrid using a recursive approach outperforms other models in most cases (Duarte et al., 26 Sep 2025). A plausible implication is that hybrid architecture and multi-step strategy are not separable choices; they interact strongly, especially in short-data regimes.

5. Empirical behavior across application domains

The reported empirical behavior of ARIMA–LSTM hybrids varies sharply across domains, forecast horizons, and hybridization schemes.

In stock return correlation forecasting, an additive residual hybrid is used: ARIMA filters linear tendencies in rolling correlation-coefficient series and passes residuals to an LSTM (Choi, 2018). On the development dataset, the ARIMA–LSTM model achieved MSE 0.1786, RMSE 0.4226, and MAE 0.3420, compared with 0.2954, 0.5435, and 0.4423 for the Constant Correlation model, and 0.4597, 0.6780, and 0.5449 for the Full Historical model (Choi, 2018). Comparable superiority persists on Test1 and Test2, where the hybrid retains the lowest MSE, RMSE, and MAE among the reported baselines (Choi, 2018).

In fund net value prediction, the ARIMA–LSTM hybrid dramatically outperforms the individual components. The reported test-set metrics are MSE 3.61, MAE 1.69, RMSE 1.90 for ARIMA; MSE 0.13, MAE 0.31, RMSE 0.36 for LSTM; and MSE 0.01, MAE 0.09, RMSE 0.12 for the hybrid (Zhou et al., 2021). Because the paper uses a residual superposition design, these gains are attributed to the linear–nonlinear decomposition rather than to a simple ensemble effect (Zhou et al., 2021).

In epidemic forecasting, the relevant paper uses ARIMA–NARNN rather than ARIMA–LSTM, but the residual-learning logic is directly analogous. For India cumulative cases during 6–15 May, the reported values are RMSE 502.30, MAE 459.6, MAPE 0.718 for ARIMA; RMSE 7755.93, MAE 5924.5, MAPE 7.871 for LSTM; and RMSE 437.30, MAE 341.8, MAPE 0.523 for the hybrid ARIMA–NARNN (Prajapati et al., 2021). The paper states that the hybrid combination of ARIMA and NARNN gave the best result among the selected models with a reduced RMSE, which proved to be almost 35.3% better than one of the most prevalent method of time-series prediction (ARIMA) (Prajapati et al., 2021). This suggests that the improvement mechanism often attributed to ARIMA–LSTM hybrids was already visible in earlier ARIMA–ANN residual systems.

In weather forecasting, the residual-learning SARIMA–LSTM model is evaluated over a 293-day fully recursive horizon. The reported values are MAE 2.65 °C and RMSE 3.34 °C for SARIMA-only, MAE 8.01 °C and RMSE 9.32 °C for LSTM-only, and MAE 2.41 °C and RMSE 3.33 °C for the hybrid SARIMA–LSTM (Rajeev et al., 12 Jan 2026). The numerical gains over SARIMA are small, but the paper emphasizes improved handling of nonlinear deviations and extremes over long horizons (Rajeev et al., 12 Jan 2026).

In electricity demand forecasting for Ukraine, the medium-term daily residual is modeled with ARIMA, LSTM, and combined residual hybrids. The reported test-set metrics for the daily component are RMSE 552.8 MW, MAE 430.8 MW, MASE 0.495 for regression-only; RMSE 533.4 MW, MAE 399.6 MW, MASE 0.459 for LM + ARIMA; RMSE 500.6 MW, MAE 374.0 MW, MASE 0.429 for LM + LSTM; and RMSE 504.0 MW, MAE 382.9 MW, MASE 0.440 for LM + ARIMA + LSTM (Grandon et al., 2023). Here the LSTM residual correction improves over ARIMA residual correction, and the explicit ARIMA + LSTM combination does not outperform LM + LSTM (Grandon et al., 2023). This is an important counterexample to the notion that adding both statistical and recurrent residual models always yields the strongest hybrid.

In short mortality rate series, the mean MAPE values across 12 datasets are 1.905% for ARIMA-LSTM (recursive) and 1.994% for ARIMA (recursive), with ARIMA-LSTM ranked first and ARIMA ranked second by mean rank (Duarte et al., 26 Sep 2025). The percentage difference between ARIMA-LSTM and ARIMA is 4.44%, while the percentage differences relative to Lee–Carter, Plat, and the single ML models are much larger (Duarte et al., 26 Sep 2025). This indicates that in short annual series the hybrid gain may be modest but consistent.

6. Domain-specific implementations and methodological extensions

Wind speed forecasting has produced a tri-component extension rather than a pure ARIMA–LSTM system. The hybrid described in the metadata combines ARIMA, Kalman filter and LSTM for estimating wind speed and is reported to work with better accuracy when compared to the existing methods (Mohapatra et al., 2023). Because the paper PDF is unavailable, exact formulas, parameter choices, and numerical results are not available from the source block, but the documented design pattern is a residual hybrid in which ARIMA captures linear structure, Kalman filtering is used for state estimation or smoothing, and LSTM captures nonlinear residual structure (Mohapatra et al., 2023). This suggests that ARIMA–LSTM hybrids are often embedded inside broader cascaded filtering pipelines.

Sports forecasting provides a different extension path, with rich feature engineering layered on top of the hybrid. The architecture is sequential rather than residual-based: ARIMA is trained on country-level feature sequences, embeddings convert categorical data into continuous vectors, PCA reduces the athlete representation to a lower-dimensional latent space, and the LSTM consumes the concatenated state $N_t$7 (Liu et al., 11 Feb 2025). The reported ablation gives RMSE = 0.098 and MAE = 0.072 for the original ARIMA–LSTM hybrid, compared with RMSE = 0.121 and MAE = 0.094 for LSTM without ARIMA (Liu et al., 11 Feb 2025). The same paper augments the hybrid with SHAP for feature importance, KNN for prediction intervals, and the run test for non-randomness analysis (Liu et al., 11 Feb 2025). A plausible implication is that in structured panel or event data, ARIMA–LSTM hybrids may act more as state-construction frameworks than as simple residual learners.

Financial return forecasting has also produced a strong methodological critique of additive residual hybrids. In the systematic comparison of econometric–ML–DL hybrids, two ARIMA–LSTM variants are evaluated: an additive Zhang-style residual hybrid and a non-additive feature-augmented hybrid (Stempień et al., 26 May 2025). For S&P 500 Long-Short, the non-additive LSTM-ARIMA (1) has RMSE 1.2825%, MAE 0.8226%, ARC 10.46%, and IR 0.56, while the additive LSTM-ARIMA (2) has RMSE 1.3114%, MAE 0.8469%, ARC 5.01%, and IR 0.29 (Stempień et al., 26 May 2025). For Bitcoin Long Only, LSTM-ARIMA (1) reaches ARC 38.40%, ASD 53.19%, MD 52.80%, IR 0.72, IR* 0.53, and SR 1.43, whereas LSTM-ARIMA (2) achieves ARC 6.49%, ASD 58.61%, MD 83.39%, IR 0.11, and SR 0.21 (Stempień et al., 26 May 2025). The paper concludes that the most effective hybrid model architecture was achieved by combining the econometric ARIMA model with either SVM or LSTM, under the assumption of a non-additive relationship between the linear and nonlinear components (Stempień et al., 26 May 2025). This directly challenges the widespread assumption that $N_t$8 is always the right hybridization prior.

Algorithmic investment strategies offer yet another design. In the LSTM-ARIMA investment framework, ARIMA residuals are added to price, volatility, and volume as LSTM inputs; the LSTM then produces final predictions and trading signals (Kashif et al., 2024). The reported modified information ratios show that the hybrid outperforms both ARIMA and LSTM across S&P 500, FTSE 100, and CAC 40, and the ensembled Long-Short LSTM-ARIMA strategy reaches IR** 70.54% (Kashif et al., 2024). This is not a residual-superposition hybrid, but it remains recognizably within the ARIMA–LSTM family because the ARIMA stage contributes structured linear information to the recurrent stage.

7. Debates, misconceptions, and current research directions

A common misconception is that ARIMA–LSTM hybrids are inherently superior to both ARIMA and LSTM. The evidence is more conditional. In the electricity demand case study, LM + LSTM is better than LM + ARIMA + LSTM on the medium-term daily component (Grandon et al., 2023). In mortality forecasting, ARIMA-LSTM improves over ARIMA by 4.44% in mean MAPE rather than by an order of magnitude (Duarte et al., 26 Sep 2025). In long-horizon weather forecasting, the hybrid improves MAE only slightly relative to SARIMA while dramatically outperforming standalone LSTM (Rajeev et al., 12 Jan 2026). These results imply that hybridization must be matched to data length, noise level, horizon, and component design.

A second misconception is that the residual-additive formulation is the canonical or theoretically privileged design. Recent financial evidence argues the opposite in some domains: the Zhang-style additive hybrid can underperform a hybrid that uses the ARIMA forecast as an explanatory feature in a non-additive LSTM mapping (Stempień et al., 26 May 2025). This suggests that the decomposition $N_t$9 is a modeling assumption rather than a fact about the data-generating process.

A third issue concerns training strategy. Most hybrid systems train the statistical and recurrent parts in a disjoint manner, but the state-space learning paper argues that such independent training is sub-optimal and introduces a joint optimization mechanism in which an enhanced recurrent neural network and an ARMA-family time series model are jointly optimized in a single pass via particle filtering (Aydın et al., 2023). This line of work points toward hybrids that are no longer simple pipelines but unified probabilistic dynamical systems.

Several extension directions recur across the literature. These include multivariate inputs with exogenous variables such as temperature, dew point, pressure, or volatility (Rajeev et al., 12 Jan 2026, Kashif et al., 2024); attention mechanisms in sequence-to-sequence residual models (Grandon et al., 2023); long-horizon stabilization through time-decayed residual weighting (Rajeev et al., 12 Jan 2026); uncertainty quantification through KNN intervals (Liu et al., 11 Feb 2025); and hybrid state-space formulations that can replace LSTM with GRU or the ARIMA-family model with ETS, while preserving joint optimization (Aydın et al., 2023).

Taken together, the literature supports a precise but non-monolithic definition. An ARIMA–LSTM hybrid is not merely a two-model ensemble. It is a design principle for combining an ARIMA-family linear inductive bias with LSTM-based nonlinear sequence learning, implemented through residual decomposition, feature augmentation, or joint state-space learning, and evaluated in relation to horizon, data regime, and application structure rather than by a universal performance rule (Zhou et al., 2021, Stempień et al., 26 May 2025, Duarte et al., 26 Sep 2025, Aydın et al., 2023).