You regress stock A on stock B, obtain a hedge ratio of 1.4, and the residual has a daily AR(1) coefficient of 0.94. What is the half life of the spread, what test must be run before trading it, and what changes if you regress B on A instead?

You regress stock A on stock B, obtain a hedge ratio of 1.4, and the residual has a daily AR(1) coefficient of 0.94. What is the half life of the spread, what test must be run before trading it, and what changes if you regress B on A instead?

Approach: Convert the autoregressive coefficient into a continuous decay rate for the half life, then think about which critical values apply when the hedge ratio was fitted on the same sample.

11.2 days. A residual with an AR(1) coefficient of 0.94 decays at -ln(0.94) = 0.061875 a day, so the half life is ln(2)/0.061875 = 11.2 days and a two standard deviation entry takes about three weeks to give back most of its move. Before trading it, run an augmented Dickey-Fuller test on the residual of the cointegration regression using Engle-Granger critical values, which are wider than the standard ones because the hedge ratio was estimated from the same data and the residual is fitted to look stationary. Regressing B on A does not return 1/1.4 unless the R squared is one, since least squares minimises error in one variable only, so the two orderings give different residuals and can disagree on the test. The Johansen procedure treats both series symmetrically and removes the choice.

Follow-up: How would you hedge the pair if the cointegrating vector itself drifts over the sample?

Key concepts: cointegration test, half life, hedge ratio, augmented dickey-fuller.