ARIMA Models for Time-Series Forecasting

• ARIMA is a linear time-series forecasting model that applies differencing to achieve stationarity and employs autoregressive and moving average components.
• It follows the Box–Jenkins methodology, emphasizing model identification, maximum likelihood estimation, and diagnostic checks to ensure white-noise residuals.
• Its flexibility supports seasonal adaptations and hybrid extensions with exogenous inputs, making ARIMA a benchmark in diverse domains.

ARIMA, the Auto Regressive Integrated Moving Average model, is a classical linear framework for univariate time-series forecasting in which a nonstationary series is differenced until it becomes approximately stationary and the resulting process is modeled with autoregressive and moving-average terms. In the papers surveyed here, ARIMA is consistently presented through the Box–Jenkins workflow and parameterized as ARIMA$(p,d,q)$, where $p$ is the autoregressive order, $d$ is the differencing order, and $q$ is the moving-average order (Siami-Namini et al., 2018). A standard compact form used in several of these studies is

$\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t,$

with $\phi(B)$ the AR polynomial, $\theta(B)$ the MA polynomial, and $(1-B)^d$ the nonseasonal differencing operator (Perez et al., 2019).

1. Formal structure and notation

ARIMA is usually introduced as a generalization of ARMA that combines three components: autoregression, integration through differencing, and moving average (Siami-Namini et al., 2018). In the stationary case, the component models are written as

$$x_t = c + \sum_{i=1}^{p} \phi_i x_{t-i} + \varepsilon_t$$

for AR$(p)$, and

$p$0

for MA$p$1, with ARMA$p$2 combining both terms (Siami-Namini et al., 2018). ARIMA extends this by applying differencing before the ARMA stage.

Several papers also give equivalent differenced-domain formulations. One representative form is

$p$3

which makes explicit that the modeled object is the differenced series rather than the original level process (Gyima-Adu et al., 11 Feb 2025). Another common non-operator form is

$p$4

used in a temperature-forecasting comparison (Saa et al., 2020).

Seasonal structure is handled by the multiplicative extension

$p$5

where $p$6 are the seasonal AR, differencing, and MA orders and $p$7 is the seasonal period (Siami-Namini et al., 2018). In electricity-price forecasting, the seasonal model is written as

$p$8

with $p$9 for hourly daily seasonality (Zhao et al., 2018).

This formalism situates ARIMA as a linear, lag-structured, stationarity-adjusted forecasting model. Several papers describe it as a standard econometric benchmark, especially for short-horizon or one-step-ahead prediction tasks in which lag interpretability and explicit differencing remain desirable (Siami-Namini et al., 2018).

2. Identification, stationarity, and estimation

The methodological core of ARIMA in these papers is the Box–Jenkins sequence: model identification, parameter estimation, residual diagnostic checking, and forecasting (Saa et al., 2020). Identification begins with stationarity assessment. Time plots, ACF, PACF, the Augmented Dickey–Fuller test, and sometimes KPSS are used to determine whether differencing is required (Perez et al., 2019).

The first-difference operator is repeatedly written as

$d$0

or, in one paper’s notation,

$d$1

with higher-order differencing applied if the once-differenced series remains nonstationary (Perez et al., 2019). Variance stabilization is sometimes added before differencing. The Box–Cox transformation appears in the form

$d$2

and

$d$3

in malaria incidence forecasting, where differencing alone removed the unit root but produced a model with one non-significant coefficient and non-normal residuals, whereas Box–Cox transformation before differencing led to an acceptable ARIMA$d$4 model (Perez et al., 2019).

Order identification is typically tied to ACF and PACF. One common rule in these papers is that PACF is used to determine $d$5 and ACF to determine $d$6, often in combination with information criteria such as AIC or BIC (Perez et al., 2019). In the Ghana insurance-penetration study, auto.arima(..., ic="aic", trace=TRUE) searched a candidate set that included both seasonal and nonseasonal models and selected ARIMA$d$7 because it had the lowest AIC, $d$8 (Gyima-Adu et al., 11 Feb 2025). In the Philippine malaria study, three tentative models were considered after stationarity treatment,

$d$9

and ARIMA$q$0 was chosen because it had the smallest AIC, $q$1 (Perez et al., 2019).

Estimation is usually carried out by maximum likelihood. This is stated explicitly in the malaria paper, the Ghana insurance study, the Noja water-pressure study, and the Italian COVID-19 hospitalization comparison (Perez et al., 2019). A plausible implication is that, across contemporary applications, ARIMA practice is increasingly software-mediated: several studies rely on auto.arima, statsmodels, EViews, or SPSS rather than manual Yule–Walker-style workflows.

3. Diagnostics, residual structure, and probabilistic forecasts

The standard adequacy condition stated across multiple studies is that residuals should behave like white noise. One paper defines residuals as

$q$2

and notes that, for a satisfactory model, residuals should be uncorrelated and have zero mean (Saa et al., 2020). In the malaria study, residual adequacy was checked with residual plots, Q–Q plots, residual ACF/PACF, and the Ljung–Box statistic

$q$3

with the final ARIMA$q$4 model yielding $q$5 and $q$6, supporting the white-noise interpretation (Perez et al., 2019).

The Ghana study reports a similar residual-validation pattern for ARIMA$q$7: Ljung–Box $q$8, $q$9, residual ACF and PACF within standard error limits, and residual stationarity supported by ADF, PP, and KPSS tests (Gyima-Adu et al., 11 Feb 2025). In the Noja water-supply application, the final ARMAX$\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t,$0 model also passed residual checking, with Durbin–Watson

$\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t,$1

reported as $\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t,$2, effectively indicating no first-order residual autocorrelation (Munoz-Rodriguez et al., 10 Dec 2025).

Forecast assessment in these papers uses a wide range of criteria. Mean Squared Error appears as

$\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t,$3

in temperature forecasting (Saa et al., 2020), while RMSE is written as

$\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t,$4

in economics and finance (Siami-Namini et al., 2018). MAPE, MASE, Theil’s $\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t,$5, and Gaussian-white-noise checks on forecast errors also recur (Perone, 2020).

A notable theoretical extension derives the joint distribution of multistep ARIMA forecast errors rather than only marginal one-step variances. Under Gaussian assumptions, the paper “Joint Probability Distribution of Prediction Errors of ARIMA” shows that the vector

$\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t,$6

is jointly multivariate normal, with covariance obtained through explicit linear transformations of the underlying white noise (Qin et al., 2018). This shifts ARIMA from pointwise forecasting toward horizon-wide probabilistic monitoring.

4. Variants and extensions

The surveyed literature treats ARIMA less as a single fixed model than as a family of related constructions. Seasonal ARIMA is the most classical extension; ARIMA with exogenous regressors appears as SARIMAX, ARMAX, or ARIMAX when external drivers are added (Zhao et al., 2018). In the Noja hydraulic-control study, the final model is not plain ARIMA but

$\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t,$7

with contemporaneous exogenous inputs $\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t,$8 and $\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t,$9, and fitted coefficients $\phi(B)$0 and $\phi(B)$1, reflecting hydraulic head-loss and control effects (Munoz-Rodriguez et al., 10 Dec 2025).

In electricity-price forecasting, the key object is not raw RTLMP but the differential series

$\phi(B)$2

A forecast of this spread is converted back to a real-time price forecast via

$\phi(B)$3

On this transformed target, ARMAX$\phi(B)$4-GARCH$\phi(B)$5 with a weekday/weekend indicator outperformed the paper’s other ARIMA-family models (Zhao et al., 2018). This suggests that, in some domains, exogenous structure and variance modeling are more consequential than the base ARIMA order alone.

Hybridization is a more recent direction. In one algorithmic-investment study, ARIMA is not the final predictor but a feature-generating component whose residuals are passed to an LSTM together with closing price, volatility, and trading volume; the authors interpret ARIMA as capturing short-term linear dependencies while the LSTM captures nonlinear and long-term structure (Kashif et al., 2024). A different hybrid combines ARIMA with a polynomial classifier through

$\phi(B)$6

with $\phi(B)$7 chosen analytically to minimize MSE (Nguyen et al., 11 May 2025).

A still more structural extension is BHT-ARIMA for multiple short time series. There, multi-way delay embedding produces block Hankel tensors, Tucker decomposition compresses them to core tensors, and a tensor-valued ARIMA is applied to the core sequence: $\phi(B)$8 This preserves the $\phi(B)$9 logic while changing the forecasted object from a scalar series to a low-rank tensor state (Shi et al., 2020).

5. Applications across domains

The papers collectively show that ARIMA and ARIMA-family models are used across epidemiology, climate, finance, astronomy, telecommunications, water systems, insurance, electricity, and macro-financial forecasting. Representative applications are summarized below.

Domain	Series and model	Representative finding
Epidemiology	Weekly malaria incidence, ARIMA$\theta(B)$0	Selected by lowest AIC $\theta(B)$1 after Box–Cox and differencing (Perez et al., 2019)
Insurance	Quarterly Ghana insurance penetration, ARIMA$\theta(B)$2	Lowest AIC $\theta(B)$3; 13-step-ahead forecasts through 2025 Q4 (Gyima-Adu et al., 11 Feb 2025)
Finance	Bitcoin daily closing price	Short-term prediction works better in stable subperiods; MSE-optimal model for $\theta(B)$4 was ARIMA$\theta(B)$5 (Azari, 2019)
Astronomy	Minute-scale optical turbulence, ARIMA$\theta(B)$6	Forecast PDFs for image metrics were “1.5–2 times thinner” than unconditional PDFs (Kornilov, 2018)
Water systems	CPC pressure in Noja, ARMAX$\theta(B)$7	Validation MAPE $\theta(B)$8, RMSE $\theta(B)$9 m (Munoz-Rodriguez et al., 10 Dec 2025)
Telecommunications	Hourly subscriber throughput	ARIMA reported $(1-B)^d$0 on subscriber data and $(1-B)^d$1 on the subscriber testing set, faster than CNN (Nkongolo, 2024)

Comparative studies, however, do not point in a single direction. In a monthly-temperature study, a convolution-plus-LSTM model achieved MSE $(1-B)^d$2 versus ARIMA’s $(1-B)^d$3 on the final 12-month test window (Saa et al., 2020). In economics and financial series, a fixed ARIMA$(1-B)^d$4 baseline was outperformed by LSTM, with average RMSE reductions of $(1-B)^d$5 for financial series and $(1-B)^d$6 for economic series (Siami-Namini et al., 2018). Yet for daily cumulative COVID-19 cases, ARIMA$(1-B)^d$7 remained broadly comparable to several LSTM variants, with kMAPE as low as $(1-B)^d$8 and kMdSA as low as $(1-B)^d$9 (Barman, 2020). This suggests that ARIMA’s empirical status is domain- and protocol-dependent rather than uniformly superseded.

6. Limitations, misconceptions, and contemporary perspective

A recurrent limitation is linearity. One paper explicitly states that ARIMA is a “linear regression-based forecasting approach” and highlights its difficulty modeling nonlinear relationships (Siami-Namini et al., 2018). Financial examples make this concrete. In Bitcoin forecasting, ARIMA performs reasonably well for short-term, one-day prediction in subperiods “in which the behavior of the time-series is almost unchanged,” but introduces large errors over a three-year span with regime change and cannot capture the sharp fluctuations at the end of 2017 (Azari, 2019).

A second limitation is that differencing alone does not guarantee superiority. In the annual global temperature-anomaly uncertainty study, the original series failed the ADF test, so ARIMA$$x_t = c + \sum_{i=1}^{p} \phi_i x_{t-i} + \varepsilon_t$$0 with $$x_t = c + \sum_{i=1}^{p} \phi_i x_{t-i} + \varepsilon_t$$1 was examined, yet the authors concluded that an uncertainty-derived ARMA$$x_t = c + \sum_{i=1}^{p} \phi_i x_{t-i} + \varepsilon_t$$2 fit better than ARIMA$$x_t = c + \sum_{i=1}^{p} \phi_i x_{t-i} + \varepsilon_t$$3 and ARIMA$$x_t = c + \sum_{i=1}^{p} \phi_i x_{t-i} + \varepsilon_t$$4 on residual diagnostics (Hasan et al., 2023). This directly contradicts the misconception that a nonstationary-looking series should automatically lead to an integrated model.

A third limitation is methodological rather than theoretical: ARIMA performance depends heavily on specification, diagnostics, and the fairness of the baseline. The temperature study introduces ARIMA through standard Box–Jenkins steps but does not report the final $$x_t = c + \sum_{i=1}^{p} \phi_i x_{t-i} + \varepsilon_t$$5, provide ACF/PACF plots, or include residual whiteness results; it also omits a seasonal term despite monthly temperature data (Saa et al., 2020). This suggests that some claims of neural superiority over ARIMA may partly reflect under-specified classical baselines rather than an intrinsic inferiority of the methodology.

At the same time, it would be inaccurate to treat ARIMA as merely obsolete. Across the surveyed work, it remains the standard benchmark classical forecaster, often the best single linear model, and frequently the component that supplies interpretable short-memory structure inside hybrids (Perone, 2020). A plausible implication is that ARIMA’s contemporary role is dual: it remains a practical end model for short-horizon, univariate, stationarity-adjusted forecasting, and it also functions as a modular building block in exogenous, heteroskedastic, tensor, and neural hybrid systems.