An optimal test against a random walk component in a non-orthogonal unobserved components model

In this paper we consider the problem of testing the null hypothesis that a series has a constant level (possibly as part of a more general deterministic mean) against the alternative that the level follows a random walk. This problem has previously been studied by, inter alia, Nyblom and Makelainen (1983) in the context of the orthogonal Gaussian random walk plus noise model. This model postulates that the noise component and the innovations to the random walk are uncorrelated. We generalize their work by deriving the locally best invariant test of a fixed level against a random walk level in the non-orthogonal case. Here the noise and random walk components are contemporaneously correlated with correlation coefficient rho. We demonstrate that the form of the optimal test in this setting is independent of rho; i.e. the test statistic previously derived for the case of rho=0 remains the locally optimal test for all rho. This is a very useful result: it states that the locally optimal test may be achieved without prior knowledge of rho. Moreover, we show that the limiting distribution of the resulting statistic under both the null and local alternatives does not depend on rho, behaving exactly as if rho=0. Finite sample simulations of these effects are provided to illustrate and generalizations to models with dependent errors are considered. Copyright Royal Economic Society, 2002