Testing for the Constancy of Parameters over Time

Abstract

Tests are proposed for detecting possible changes in parameters when the observations are obtained sequentially in time. While deriving the tests the alternative one has in mind specifies the parameter process as a martingale. The distribution theory of these tests relies on the large-sample results; that is, only the limiting null distributions are known (except in very special cases). The main tool in establishing these limiting distributions is weak convergence of stochastic processes. Suppose that we have vector-valued observations x 1, …, x n obtained sequentially in time (or ordered in some other linear fashion). Their joint distribution is described by determining the initial distribution for x 1 and the conditional distribution for each x k given the past up to x k–1. Suppose further that these distributions depend on a p-dimensional parameter vector θ. At least locally (i.e., in a short time period) this may be more or less legitimate. In the long run, however, the possibility of some changes in the observation-generating process should be taken into account. Specifically, it is assumed here that those changes occur through a parameter variation in the form of a martingale. The martingale specification has an advantage of covering several types of departure of constancy: for example, a single jump at an unknown time point (the so-called change-point model) or slow random variation (typically random walk). The tests are derived by first finding the locally most powerful test against a martingale-type alternative when the starting value of the parameter process is known. After some simplification a test having a known numerically tractable limiting distribution is developed. When the starting point is unknown an efficient estimate is substituted for it. In addition, the corresponding limiting distribution is established. The proposed tests turn out to be based on cumulative sums of the score function (the derivative of the log-likelihood).