Testing for a Unit Root Nonstationarity in Multivariate Autoregressive Time Series

The characteristic equation of a multiple autoregressive time series involves the eigenvalues of a matrix equation which determine if the series is stationary. Suppose one eigenvalue is 1 and the rest are less than 1 in magnitude. We show that ordinary least squares may be used to estimate the matrices involved and that the largest estimated eigenvalue has distributional properties that allow us to test this unit root hypothesis using critical values tabulated by Dickey (1976). See also Fountis (1983). If a single unit root is suspected, a model can be fit whose parameters are constrained to produce an exact unit root. This is the vector analog of differencing in the univariate case. In the fitting process, canonical series can be computed thus extending the work of Box and Tiao (1977) to the unit root case.