APA 7th Edition Template

An ARMA(p, q) model was fitted to the [series name] series (n = ___). Order
selection by [BIC / AIC] yielded ARMA(__, __). The fitted coefficients were
φ̂₁ = ___ (SE = ___), …, θ̂₁ = ___ (SE = ___). The model produced
AIC = ___, BIC = ___, and residual variance σ̂² = ___. Residual diagnostics
(Ljung–Box Q(10) = ___, p = ___) were [consistent with / inconsistent with]
white noise, suggesting the model is [adequate / requires revision].

Methods Section Template

Univariate time-series modelling followed the Box–Jenkins methodology. Stationarity was assessed with the Augmented Dickey–Fuller test. Candidate ARMA(p, q) models with p, q ∈ {0, 1, 2, 3} were estimated by conditional maximum likelihood, and the final order was selected by minimum BIC. Adequacy was confirmed by inspecting the ACF and PACF of residuals and the Ljung–Box Q-statistic at lag 10.

Reporting Rules

• Always report (p, q) explicitly — never just "ARMA model".
• Quote AIC and BIC; readers from different traditions will look for one or the other.
• Report Ljung–Box Q(m), df, and p-value as a residual-adequacy check.
• State whether a constant / mean term was included in the fit.
• Express forecasts with their 95% prediction intervals — never the point forecast alone.

The free ARMA model calculator on this page is appropriate when you want to model a univariate, stationary time series whose dependence on its own past can be captured by a finite number of AR and MA terms. It answers the question: what linear, parsimonious process best describes the autocorrelation in my data?

Decision Checklist

• Your series is stationary (or near-stationary after de-trending) — confirmed by ADF / KPSS
• There is no obvious seasonality at a fixed period
• You have at least 50 observations (100+ ideal)
• The relationship between y_t and its lags looks linear
• Do NOT use if the series has a strong unit root → difference and use ARIMA
• Do NOT use if you see periodic peaks at lag 12, 24, 7, etc. → use SARIMA
• Do NOT use if you have multiple interacting series → use VAR
• Do NOT use if variance changes over time (volatility clusters) → use GARCH

Real-World Examples

1. Finance — daily log-returns. Equity log-returns are typically near-stationary with weak autocorrelation; an ARMA(1,1) often captures the short-run dependence and is a baseline before adding GARCH.
2. Hydrology — daily river-flow residuals. After removing seasonality, the residual flow series often follows a low-order ARMA, used for short-term flood forecasting.
3. Psychology — reaction-time deviations. Trial-by-trial residuals from a baseline model frequently exhibit AR(1) structure that ARMA can model and remove.
4. Ecology — nightly camera-trap detections. Detection counts at a single station after removing weekly periodicity often follow a stationary ARMA process useful for activity-pattern modelling.
5. Quality control — process measurements. Manufacturing measurements at a stable line are often modelled as ARMA to detect drift outside control limits.

Sample Size Guidance

Minimum n ≥ 50 for ARMA(1,1); aim for n ≥ 100 when (p + q) ≥ 3. Small n inflates SEs and makes order selection unstable.

1. Enter your data — paste comma-separated values, upload a CSV/Excel, or use a sample dataset.
2. Pick a sample dataset — five built-in series cover finance, hydrology, psychology, economics, and ecology.
3. Edit the group name — the label appears in charts and the APA paragraph.
4. Set p and q — start with (1, 1) if unsure, or choose Auto-BIC to let the tool search (p, q) ∈ {0..3}.
5. Set α and forecast horizon — α governs prediction-interval coverage; horizon h sets how far ahead to forecast.
6. Run the analysis — click "Fit ARMA Model"; results render in under 1 second.
7. Read the summary cards — p, q, AIC, BIC, σ̂², and Ljung–Box at a glance.
8. Inspect the four charts — series + forecast, residual ACF, residual PACF, and residual scatter.
9. Read the detailed interpretation and conclusion — these auto-fill with your numbers.
10. Export — Download Doc (.txt) or Download PDF (print-to-PDF) for your report.