Some nonparametric tests for constancy of regression relationships over time

Abstract

Let Y₁, Y₂... be a sequence of random variables obeying the law Yi = β’i + εi, where β₁, β₂, ... is a sequence of unknown k-dimensional regression vectors; x₁, x₂, ... is a sequence of known k-dimensional regressor vectors; and ε₁ , ε₂, ... is a sequence of independent and identically distributed random variables. Assume that β₁ = ... = βm = β, m ≥ k, and that β̂₀ is an asymptotically normal estimate of β based on Y₁ , ..., Ym. This study develops nonparametric procedures for testing H₀: = β = βm+1 = βm+2 = …. The proposed tests involve sequences of truncated sequential tests. That is, a function of the residuals Ym+1 - β̂’₀ xm+1, …, Ym+N - β̂’₀ xm+N is examined for a shift in the model. If no shift is indicated all m+N observations are pooled and a new estimate of β, β̂₁, is formed. The next N residuals are then examined for a shift. The procedure continues until a.shift is indicated. Brownian motion results are used to obtain approximate critical values when the function of the residuals is: the cumulative sum of the signs of the residuals; the sequential Wilcoxon scores; the ordinary cumulative sums of residuals. Exact results are obtained for the cumulative sum of signs procedure when testing for a shift in median. Asymptotic relative efficiency results are also obtained.