What is an autoregressive (AR) model in time series analysis?

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AcadiFi · Accepted Answer

An autoregressive model of order p, denoted AR(p), expresses the current value of a time series as a linear function of its own past p values plus a random error. Mathematically, x_t equals c plus phi_1 times x_{t-1} plus phi_2 times x_{t-2} plus and so on plus phi_p times x_{t-p} plus epsilon_t.

The AR model captures serial dependence: today's value is partially predictable from yesterday's. An AR(1) model is the simplest and most exam-relevant: x_t equals c plus phi times x_{t-1} plus epsilon_t.

For the model to be covariance stationary, the sum of the phi coefficients must be less than one in absolute value (technically, roots of the characteristic polynomial must lie outside the unit circle). If not, the series has a unit root and behaves like a random walk — covered in a separate question.

Example: analyst Tamsyn Okonkwo at Grayhaven Macro regresses monthly log returns of Stavros Mining Ltd on a one-month lag. The estimated AR(1) is r_t equals 0.002 plus 0.18 times r_{t-1} plus epsilon_t, significant at the 5 percent level. The slight positive autocorrelation suggests momentum.

To diagnose an AR model, examine the autocorrelation function (ACF) for slow decay and the partial autocorrelation function (PACF) — a clean cutoff after lag p in the PACF suggests an AR(p). The Breusch-Godfrey or Durbin-Watson tests check for remaining autocorrelation in residuals, which if present indicates the chosen lag order is insufficient.

What is an autoregressive (AR) model in time series analysis?

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