Autoregression (AR)

Autoregression (AR) is a way of modeling a time series in which each value is explained by the values that came before it. The series is regressed on itself: yesterday's reading, and perhaps the readings before that, are the predictors of today's. An autoregressive model of order p, written AR(p), uses the previous p values.

Autoregression is the simplest model of persistence in data and one of the building blocks of time series econometrics. It forms the AR part of ARIMA models. In markets it is used to describe quantities that drift back toward a normal level, such as interest rate spreads, volatility and the price gap between closely related assets.

The AR(p) model is:

• $y_t$ is the value of the series at time $t$.
• $c$ is a constant.
• $\phi_i$ is the coefficient on the value $i$ periods earlier.
• $\varepsilon_t$ is a random error with zero mean, unrelated to past values.

For the AR(1) case, with one lag, the series has a long-run mean of:

Take $c = 2$ and $\phi_1 = 0.6$. The long-run mean is 2 ÷ 0.4 = 5. If the latest value is 10, the forecast for the next period is 2 + (0.6 × 10) = 8. The forecast after that is 2 + (0.6 × 8) = 6.8. Each step closes 40 percent of the remaining gap to the mean of 5.

In an AR(1) model the size of $\phi_1$ sets the behavior of the series.

• Between 0 and 1. The series is persistent but pulled back toward its mean. The closer the coefficient is to 1, the slower the return.
• Between −1 and 0. The series tends to flip from above the mean to below it and back.
• Equal to 1. There is no pull to any level. The series is a random walk, and shocks last forever.
• Greater than 1 in size. The series is explosive.

The model satisfies stationarity only when the coefficient is smaller than 1 in absolute value. Testing whether it equals 1 is the purpose of the Dickey-Fuller test.

A useful summary for traders is the half-life, the expected time for a deviation from the mean to shrink by half:

With a daily coefficient of 0.9, the half-life is −0.693 ÷ −0.105, or about 6.6 days.

Because the predictors are just earlier values of the same series, an AR model can be estimated by ordinary linear regression. The order p is chosen by looking at the partial autocorrelation function, which shows the direct effect of each lag after allowing for shorter ones, or by an information criterion that balances fit against the number of terms. After fitting, the residuals should show no remaining autocorrelation. Adding lags always improves the fit on past data, so a model with too many invites overfitting.

Returns on liquid assets have autocorrelation close to zero, which is why prices are hard to forecast from their own history. Where a small effect exists it has trading names: negative autocorrelation over days or weeks is short-term reversal, and positive autocorrelation over months is momentum.

Other series are strongly autoregressive. Realized volatility today is a good guide to volatility tomorrow, and autoregressive models of it are standard in risk management. In pairs trading, the spread between two cointegrated prices is commonly fitted as an AR(1) process. The coefficient gives the half-life of mean reversion, which helps set holding periods and decide whether a spread reverts fast enough to be worth trading after costs. Continuous-time interest rate models that pull the rate toward a long-run level are AR(1) processes when observed at fixed intervals.

The basic model has several extensions. Adding past errors gives ARMA and ARIMA. Modeling several series together, each depending on the past of all, gives vector autoregression, the framework behind Granger causality tests. Applying the autoregressive idea to the variance of the errors leads to ARCH and GARCH.

An AR model is linear and assumes fixed coefficients. Estimates near 1 are biased downward in small samples, which makes a series look more mean-reverting than it is. Coefficients fitted on one period can shift in the next, and a relationship that held for years can end without warning. Autoregression describes how a series has behaved; it offers no reason why, so it is best used with an economic explanation for the pattern.