Auto-Regressive Models - Random Walks and Unit Roots

  • This is the case of an AR time series model where the predicted value is expected to equal the previous period plus a random error:

xt = b0 + xt-1 + εt

  • When b0 is not equal to zero, the model is a random walk with a drift, but the key characteristic is a b1 = 1.
  • The expected value of the error is still zero.
  • The mean reverting level for a random walk is not covariance stationary and the technique of first differencing is frequently used to transform an AR model with one time lag variable (AR1) into a model that is covariance stationary.
  • If an AR time series is covariance stationary, then the serial correlations for the lag variables are insignificant or they rapidly drop to zero as the number of time period lags rises.
  • When the lag coefficient is not statistically different from 1, a unit root exists.
  • Dickey-Fuller test = applied to AR1 model to test for a unit root.
  • If a unit root is present, then the model is not covariance stationary; if this is the case, the independent variable must be transformed, so you can re-model.