SPSS command: SPSS ARIMA
Seasonal ARIMA models in your browser
ARIMA models a series through its own past values (AR), past shocks (MA) and differencing (I), with optional seasonal terms.
Free · No account · Files stay on your device · Updated 18 October 2026
What it tells you
Crosstabs runs the SPSS ARIMA procedure by exact maximum likelihood with a Kalman filter, then reports coefficient tests, SPSS-style AIC and SBC, residual autocorrelations and forecasts with t-based limits.
- The estimated AR, MA and seasonal coefficients, with standard errors, t values and approximate significance.
- Whether the residuals are white noise, from the Box-Ljung statistics in Residual Autocorrelations.
- How models compare on log-likelihood, AIC and SBC, and how well they fit on the original scale (R², RMSE, MAPE).
- Forecasts with confidence limits, transformed back when you model the natural log.
When to use it
Use it when
- The ACF and PACF suggest a specific order, e.g. an AR(1) cut-off or a seasonal spike at lag 12.
- You want to reproduce a published or earlier SPSS model such as the airline model ARIMA(0,1,1)(0,1,1)12.
- You need formal tests on the coefficients rather than only a forecast.
Use something else when
- You do not know which order to use: let TSMODEL's Expert Modeler search, or start from the ACF and PACF.
- You need predictors, intervention events or transfer functions: those parts of SPSS ARIMA and TSMODEL are not included.
- The series is very short: after differencing there must be more usable values than parameters plus three.
How to run it in Crosstabs
- 01
Open ARIMA
In Statistics mode, choose ARIMA in the Forecasting group.
- 02
Define the time axis
Map rows to time points and set the Periodicity; seasonal orders use this periodicity as s.
- 03
Set the orders
Choose the Series, then Autoregressive p (0–5), Difference d (0–2), Moving average q (0–5), Seasonal P (0–2), Seasonal D (0–1) and Seasonal Q (0–2).
- 04
Transformation and constant
Set Transformation to None or Natural log, and tick or untick Include constant in model. With differencing the constant acts as a drift.
- 05
Outliers (optional)
Tick Detect outliers automatically and choose the types to search for: Additive outlier and Level shift are ticked by default, as in SPSS; Innovational outlier, Transient change, Seasonal additive, Local trend and Additive patch can be added. Detected outliers are estimated jointly with the ARIMA terms, marked on the chart by type and carried into the forecasts.
- 06
Forecast and run
Enter Periods to forecast and click Run. Estimation runs in a background worker; a note appears if it does not fully converge.
Worked example
Computed by the Crosstabs engine from the synthetic data below when this page was built — not typed in.
Data
60 synthetic monthly values (January 2021 to December 2025) of a weekly-active-users index: an AR(1) process around 50 (φ = 0.5, normal shocks with SD 1 from a fixed seed), with a one-month additive outlier of +6 planted in 2022-06, a two-month additive patch of +5 in 2023-06 and 2023-07, and a level shift of +4 from 2024-09. The model is ARIMA(1,0,0) with a constant, with automatic detection of all seven TSMODEL outlier types.
| Item | Estimate | Std. Error | t | Approx. Sig. |
|---|---|---|---|---|
| AR, lag 1 | 0.372 | 0.120 | 3.099 | .003 |
| Constant | 49.719 | 0.219 | 226.972 | <.001 |
| Additive outlier, 2022-06 | 7.326 | 0.857 | 8.549 | <.001 |
| Additive patch, 2023-06 | 5.798 | 0.916 | 6.330 | <.001 |
| Additive patch, 2023-07 | 6.296 | 0.914 | 6.888 | <.001 |
| Level shift, 2024-09 | 3.733 | 0.404 | 9.230 | <.001 |
Exact maximum likelihood; the outlier effects are estimated jointly with the ARIMA terms.
| Time point | Type | Estimate | Std. Error | t | Approx. Sig. |
|---|---|---|---|---|---|
| 2022-06 | Additive outlier | 7.326 | 0.857 | 8.549 | <.001 |
| 2023-06 | Additive patch (1 of 2) | 5.798 | 0.916 | 6.330 | <.001 |
| 2023-07 | Additive patch (2 of 2) | 6.296 | 0.914 | 6.888 | <.001 |
| 2024-09 | Level shift | 3.733 | 0.404 | 9.230 | <.001 |
Found on the model residuals (Chen and Liu, 1993) with critical value 3.02, then estimated jointly with the ARIMA parameters.
| Item | Value |
|---|---|
| Number of residuals | 60 |
| Residual variance | 0.922 |
| Log-likelihood | −79.763 |
| Akaike's Information Criterion (AIC) | 171.526 |
| Schwarz's Bayesian Criterion (SBC) | 184.093 |
| Time point | Forecast | LCL (95%) | UCL (95%) |
|---|---|---|---|
| 2026-01 | 53.54 | 51.62 | 55.47 |
| 2026-02 | 53.49 | 51.43 | 55.54 |
| 2026-03 | 53.47 | 51.39 | 55.54 |
| 2026-04 | 53.46 | 51.38 | 55.53 |
| 2026-05 | 53.45 | 51.38 | 55.53 |
| 2026-06 | 53.45 | 51.38 | 55.53 |
Exact least-squares forecasts with t(54) limits; the level shift carries into the forecasts.
How to read it
Searching for all seven outlier types, detection finds the three planted events and nothing else: an additive outlier in 2022-06 of 7.33 points (t = 8.55), an additive patch from 2023-06 to 2023-07 with effects of 5.80 and 6.30 points, and a level shift from 2024-09 of 3.73 points (t = 9.23), against planted sizes of +6, +5 per month and +4. With them in the model, the AR(1) coefficient is 0.372 (p = .003) and the constant, the level before the shift, is 49.72. The residual Box-Ljung Q(12) = 15.10, p = .178, so the residuals look like white noise. The 2026-01 forecast of 53.54 keeps the higher level.
Output tables explained
- Parameter Estimates
- Each AR, MA, seasonal and constant term with estimate, standard error, t and approximate significance. MA terms use the SPSS sign convention, opposite to R and statsmodels.
- Outliers
- With outlier detection on: the period and type of each outlier kept in the model, with its estimated effect, standard error, t and approximate significance, and the critical value used (3 for short series, rising to 4 from 450 points). An additive patch has one row per period, each with its own effect. The footnotes list the types searched for and define each type shown.
- Residual Diagnostics
- Number of residuals and parameters, residual df, adjusted and plain residual sums of squares, residual variance, log-likelihood, AIC and SBC as SPSS ARIMA defines them.
- Model Fit
- TSMODEL-style fit statistics on the original scale: stationary R², R², RMSE, MAPE, MaxAPE, MAE, MaxAE and normalized BIC.
- Forecast
- Exact least-squares forecasts with lower and upper limits from the t distribution with the residual df.
- Residual Autocorrelations
- ACF of the residuals to lag 24 with Box-Ljung statistics whose df subtract the number of AR and MA parameters. Significant values mean the model has missed some structure.
Assumptions and notes
- After differencing, the series is stationary: constant mean and variance and autocorrelation that depends only on the lag.
- The AR part is stationary and the MA part invertible; on the boundary (a unit root) standard errors are withheld.
- Errors are independent and roughly normal with constant variance; a log transformation can stabilise a variance that grows with the level.
- Interior missing values are skipped by the Kalman filter, but the most recent values must be observed to forecast.
How the results are validated
Estimates are compared with statsmodels ARIMA (exact ML, Kalman filter) on ARMA(1,1) with a constant, the log airline model, ARIMA(2,1,0) with a constant and a seasonal AR: the maximised log-likelihood agrees within 1e-3 and is never lower by more than 1e-4, coefficients agree to about three decimals, Hessian standard errors within 2% and forecasts within 1e-3. At fixed coefficients, log-likelihoods match to 7–9 decimals, including a seasonal model and a series with missing values, and forecasts and their standard errors to 1e-6. Outlier detection is checked against R tsoutliers: Chen–Liu statistics agree to 1e-8 for additive, innovational, level-shift, transient and seasonal additive outliers (tsoutliers' seasonal level shift has the TSMODEL seasonal-additive pattern), and the same outliers are found on simulated series. Local-trend and additive-patch statistics are checked to 1e-8 against the same definitions computed in R. On simulated series with planted outliers of every type, the search reports the planted type at the planted time, and the joint fit matches R arima with the outlier regressors (coefficients within 2e-4, log-likelihood within 1e-4). Validation evidence
Questions
Why is my MA coefficient the opposite sign of R or Python?
SPSS writes the MA polynomial as θ(B) = 1 − θ₁B − …, while R and statsmodels use 1 + θ₁B + …. Crosstabs follows SPSS, so flip the sign when comparing.
How do I choose p, d and q?
Difference until the series looks stationary (d, and D for seasonal trends), then read the ACF and PACF of the differenced series: a PACF cut-off suggests AR terms, an ACF cut-off MA terms. Compare candidates on SBC and check the residual Box-Ljung test.
Should I include a constant?
Without differencing the constant is the series mean. With one difference it is a drift, i.e. a linear trend; with two or more differences it is rarely needed.
Does it detect outliers like SPSS?
Yes, for all seven TSMODEL types (additive, innovational, level shift, transient change, seasonal additive, local trend and additive patch), following TSMODEL's automatic outlier detection: candidates are found on the model residuals (Chen and Liu), estimated jointly with the ARIMA terms, and dropped when |t| falls below the critical value. SPSS does not document its critical value, how an additive patch competes with single outliers, or the longest patch it looks for, so Crosstabs uses the Chen and Liu default (3 to 4 by series length), compares a patch's χ² with the same critical value at equal tail probability, requires every period of a patch to reach it, and searches patches of 2 to 4 periods. Detected outliers can differ from SPSS on borderline cases. With innovational outliers ticked, a level shift in the middle of a short series can be reported as an innovational outlier, because the first model, fitted without outliers, takes the shift for persistence (R tsoutliers makes the same choice); the SPSS defaults, additive outliers and level shifts, avoid this.
Why do the forecasts use t limits?
Like SPSS ARIMA, the limits use the t distribution with the residual degrees of freedom, which is slightly wider than normal limits on short series.
Related procedures
- Autocorrelations (ACF and PACF)ACF
- Time Series Forecasting (Expert Modeler)TSMODEL
- Exponential SmoothingEXSMOOTH
- Autoregression (AREG)AREG
- Forecast Goodness of FitFIT
- Free SPSS alternative: full coverage list
Run ARIMA on your own data
Open a .sav, CSV or Excel file. Everything runs in your browser; the file never leaves your device.
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