Testing for Common Trends

Cointegrated multiple time series share at least one common trend. Two tests are developed for the number of common stochastic trends (i.e., for the order of cointegration) ina multiple time series with and without drift. Both tests involve the roots of the ordinary least squares coefficient matrix obtained by regressing the series onto its first lag. Critical values for the tests are tabulated, and their power is examined in a Monte Carlo study. Economic time series are often modeled as having a unit root in their autoregressive r presentation, or (equivalently) as containing a stochastic trend. But both casual observation and economic theory suggest hat many series might contain the same stochastic trends o that they are cointegrated. Ifeach of n series is integrated of order 1 but can be jointly characterized by k < n stochastic trends, then the vector representation of these series has k unit roots and n k distinct stationary linear combinations. Our proposed tests can be viewed alternatively as tests of the number of common trends, linearly independent cointegrating vectors, or autoregressive unit roots of the vector process. Both of the proposed tests are asymptotically similar. The first est (qf) is developed under the assumption that certain components of the process have a finite-order vector autoregressive (VAR) representation, and the nuisance parameters are handled by estimating this VAR. The second test (q,) entails computing the eigenvalues of a corrected sample first-order autocorrelation matrix, where the correction is essentially a sum of the autocovariance matrices. Previous researchers have found that U.S. postwar interest rates, taken individually, appear to be integrated of order 1. In addition, the theory of the term structure implies that yields on similar assets of different maturities will be cointegrated. Applying these tests to postwar U.S. data on the federal funds rate and the threeand twelve-month reasury bill rates provides upport for this prediction: The three interest rates appear to be cointegrated.