SPSS command: SPSS AREG
Regression with autocorrelated errors (AREG)
When a regression is run on time series, the errors are often correlated from one period to the next, which makes ordinary standard errors too small.
Free · No account · Files stay on your device · Updated 18 October 2026
What it tells you
AREG estimates the first-order autocorrelation rho together with the coefficients by the Prais-Winsten or Cochrane-Orcutt method, as SPSS AREG does, and reports tests you can trust.
- How strongly the errors are autocorrelated (rho), with its standard error and significance.
- The regression coefficients with standard errors corrected for the AR(1) errors.
- How rho, Durbin-Watson and the mean squared error changed over the iterations, starting from ordinary least squares.
- R, R², adjusted R² and the standard error of the estimate for the final transformed regression.
When to use it
Use it when
- You regress a monthly outcome (sales, awareness) on drivers (price, spend) and the Durbin-Watson is well below 2.
- You need valid significance tests for a driver in time-series data, not just a forecast.
- You want to reproduce an SPSS AREG result by Prais-Winsten (keeps the first observation) or Cochrane-Orcutt (drops it).
Use something else when
- The errors follow a higher-order or seasonal pattern: AREG corrects only AR(1).
- The data are cross-sectional, not ordered in time: use ordinary linear regression.
- You need forecasts beyond the data: AREG here reports fitted values and intervals for observed periods only.
How to run it in Crosstabs
- 01
Open Autoregression
In Statistics mode, choose Autoregression (AREG) in the Forecasting group.
- 02
Define the time axis
Order the rows in time. Every time point in the analysis span needs the dependent and all independent series.
- 03
Choose the series
Pick the Dependent series and Independent series 1; use Add an independent series for up to six predictors.
- 04
Choose the Method
Set Method to Prais-Winsten or Cochrane-Orcutt and decide whether to Include constant in model.
- 05
Run
Click Run. Iteration stops when every parameter changes by less than 0.001, or after 10 iterations.
Worked example
Computed by the Crosstabs engine from the synthetic data below when this page was built — not typed in.
Data
48 synthetic months of unit sales and average price. Sales fall 3 units per unit of price, and the errors follow an AR(1) process with ρ = 0.7, generated with a fixed seed.
| Iteration | Rho (AR1) | Durbin-Watson | Mean Squared Error |
|---|---|---|---|
| 0 | 0.000 | 0.414 | 4.190 |
| 1 | 0.764 | 1.944 | 1.594 |
| 2 | 0.775 | 1.970 | 1.590 |
| 3 | 0.776 | 1.972 | 1.590 |
Converged: every parameter changed by less than 0.001.
| R | R Square | Adjusted R Square | Std. Error of the Estimate | Durbin-Watson |
|---|---|---|---|---|
| 0.943 | 0.889 | 0.886 | 1.261 | 1.972 |
| Item | B | Std. Error | t | Approx. Sig. |
|---|---|---|---|---|
| Rho (AR1) | 0.776 | 0.094 | 8.247 | <.001 |
| Constant | 76.459 | 3.354 | 22.798 | <.001 |
| price | −2.716 | 0.326 | −8.337 | <.001 |
Prais-Winsten estimation; residual df 45.
How to read it
Ordinary least squares (iteration 0, ρ = 0) leaves a Durbin-Watson of 0.41, a sign of positively autocorrelated errors. Prais-Winsten estimates ρ = 0.776 (p < .001) and, after 3 iterations, a price effect of −2.72 units (SE 0.33, p < .001). The transformed residuals have Durbin-Watson 1.97, so the AR(1) correction has absorbed the serial correlation.
Output tables explained
- Iteration History
- Rho, Durbin-Watson and mean squared error at each iteration; iteration 0 is ordinary least squares, so its Durbin-Watson shows how much autocorrelation there was.
- Model Fit Summary
- R, R Square, Adjusted R Square, standard error of the estimate and Durbin-Watson of the final transformed regression.
- Parameter Estimates
- Rho (AR1) and each coefficient with B, standard error, t and approximate significance, plus the residual df.
Assumptions and notes
- The regression errors follow a stationary first-order autoregressive process (|rho| < 1).
- The predictors are not lagged values of the dependent series and are not themselves driven by its errors.
- Time points are complete and equally spaced; AREG stops if a value is missing inside the span.
How the results are validated
Prais-Winsten and Cochrane-Orcutt iterations are checked against an independent Python implementation of the SPSS manual: the rho at each iteration, final rho and its standard error, coefficients, mean squared error and Durbin-Watson agree to 1e-10, coefficient standard errors to 1e-9, with the same residual df. Cochrane-Orcutt coefficients and standard errors also match statsmodels GLSAR. Validation evidence
Questions
Prais-Winsten or Cochrane-Orcutt?
Prais-Winsten keeps the first observation through a rescaling, so it is the better choice for short series and the SPSS default. Cochrane-Orcutt drops the first observation; results are close on long series.
How do I know I need AREG instead of plain regression?
Look at the Durbin-Watson in iteration 0: values well below 2 signal positive autocorrelation. If rho in the Parameter Estimates is significant, the plain regression standard errors would have been misleading.
Why is R² lower than in ordinary regression?
R² is computed on the final transformed (quasi-differenced) regression, which removes the shared drift that inflates R² in trending data.
Can I add lagged predictors?
Yes: create the lag with Create time series, download it, add it to your file and select it as an independent series. Cross-correlations help choose the lag.
Related procedures
- Linear RegressionREGRESSION
- Cross-CorrelationsCCF
- ARIMAARIMA
- Autocorrelations (ACF and PACF)ACF
- Forecast Goodness of FitFIT
- Free SPSS alternative: full coverage list
Run Autoregression (AREG) 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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