Autoregressive Model (AR)
Autoregressive Model · Also known as: AR model, AR(p) model, autoregression, AR process
An autoregressive model of order p — AR(p) — expresses the current value of a time series as a linear function of its own p most recent past values plus a white-noise error. It is the building block of the Box-Jenkins family of time-series models and is widely used for forecasting stationary economic and financial series.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
+2 more
When to use it
Use the AR model when you have a single stationary time series and want to capture its own-lag dependence for forecasting or description. It is appropriate when the PACF shows a clear cut-off at lag p and the ACF decays gradually. If the series is non-stationary, difference it first or apply a unit-root pre-test; a failing ADF test is a strong contra-indication for applying AR directly. Do not use AR when the series is influenced by other variables (use VAR or ADL instead), or when the moving-average structure dominates (favour ARMA). AR is less suitable for structural interpretation of causal mechanisms — it is primarily a reduced-form forecasting device.
Strengths & limitations
- Simple and computationally fast: estimated by OLS in closed form.
- Directly interpretable lag coefficients showing how strongly past values predict the present.
- Solid theoretical foundation: stationarity conditions and asymptotic inference are well established.
- Building block for richer models — ARMA, ARIMA, VAR, and structural VARs all extend the AR core.
- Forecasting: AR models often outperform more complex alternatives at short horizons in practice.
- Requires stationarity; non-stationary series must be differenced before fitting, potentially losing long-run level information.
- Captures only linear, own-lag dynamics; nonlinear patterns or cross-variable dynamics require different models.
- In finite samples, AIC and BIC can select different orders, introducing model uncertainty not reflected in standard errors.
- Long-lag AR models can be over-parameterised and produce poor out-of-sample forecasts due to estimation noise.
Frequently asked
How do I choose the lag order p?
Inspect the PACF: the PACF of an AR(p) drops to near zero after lag p. Confirm with information criteria — fit AR models for p = 0 to some maximum (e.g., 12 for monthly data) and pick the order minimising AIC or BIC. Then verify that the chosen model's residuals show no remaining autocorrelation via the Ljung-Box test.
What is the difference between AR and ARMA?
An AR(p) model expresses the current value solely through its own past values. An ARMA(p,q) adds a moving-average component that weights past error terms, allowing more parsimonious representation of series whose ACF decays slowly. If the PACF does not cut off sharply, consider ARMA rather than a pure AR.
Can I use AR on a non-stationary series?
No — standard AR inference assumes stationarity. Run an ADF or Phillips-Perron test first. If the series has a unit root, take first differences (or log-differences for growth rates) to achieve stationarity before fitting AR. If the integrated series must be modelled without differencing, use an ARIMA(p,1,0) framework instead.
Is AR the same as OLS regression?
Mechanically yes: estimating an AR(p) by OLS means regressing y_t on its p lagged values, and the algebra is identical to ordinary regression. The difference is conceptual — the regressors are lagged values of the dependent variable itself, so inference requires stationarity and the sample must be large enough for Yule-Walker asymptotics to hold.
When should I use VAR instead of AR?
Use VAR when you have two or more time series that may jointly influence each other. An AR model captures the own-lag dynamics of a single series; VAR generalises this by allowing each variable to depend on the lagged values of all variables in the system, making it essential for studying cross-variable impulse responses and Granger causality.
Sources
- Box, G. E. P., & Jenkins, G. M. (1976). Time Series Analysis: Forecasting and Control (revised ed.). Holden-Day. ISBN: 978-0816211043
- Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press. ISBN: 978-0691042893
How to cite this page
ScholarGate. (2026, June 3). Autoregressive Model. ScholarGate. https://scholargate.app/en/econometrics/autoregressive-model
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- ARIMA modelEconometrics↔ compare
- ARMA modelEconometrics↔ compare
- Augmented Dickey-Fuller unit root testEconometrics↔ compare
- Granger Causality TestEconometrics↔ compare
- Moving Average ModelEconometrics↔ compare
- Vector AutoregressionEconometrics↔ compare