Some nonparametric tests for constancy of regression relationships over time

Abstract

Let Y₁, Y₂... be a sequence of random variables obeying the law Y_i = β’_i + ε_i, 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 = ... = β_m = β, m ≥ k, and that β̂₀ is an asymptotically normal estimate of β based on Y₁ , ..., Y_m. This study develops nonparametric procedures for testing H₀: = β = β_m+1 = β_m+2 = ….

The proposed tests involve sequences of truncated sequential tests. That is, a function of the residuals Y_m+1 - β̂’₀ x_m+1, …, Y_m+N - β̂’₀ x_m+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.