A Heisenberg Bound for Stationary Time Series

Heisenberg's principle of indeterminacy is applied to stationary time series models. The position and velocity of a forecast are defined and are shown to be imperfectly correlated. Then a first-order autoregression is used to illustrate the trade-off between precision of position and precision of velocity. A counterpart of Planck's constant is identified, and the Heisenberg bound is derived for several autoregressive moving- average models. The time-energy version of the Heisenberg principle is discussed in the context of a stationary model in continuous time.