Techniques for Testing the Constancy of Regression Relationships Over Time

Abstract

Summary Methods for studying the stability over time of regression relationships are considered. Recursive residuals, defined to be uncorrelated with zero means and constant variance, are introduced and tests based on the cusum and cusum of squares of recursive residuals are developed. Further techniques based on moving regressions, in which the regression model is fitted from a segment of data which is moved along the series, and on regression models whose coefficients are polynomials in time are studied. The Quandt log‐likelihood ratio statistic is considered. Emphasis is placed on the use of graphical methods. The techniques proposed have been embodied in a comprehensive computer program, timvar. Use of the techniques is illustrated by applying them to three sets of data.

On Testing for a Change-Point in Variance of Normal Distribution

“…The CUSUM of squares test was first proposed by Brown et al (1975) as a test of regression coefficients but the results of Hensen (1991) and Ploberger and Kramer (1992) showed that the CUSUM of square test is essentially a test for the change in variances as described in Section 1 of this paper. In the case of a sequence of normal random variables, the test statistic can be written as,…”

“…The R-test is constructed using the maximum likelihood method, which was initially applied during the late 1950's to late 1970's by Quandt (1958Quandt ( , 1960, Hinkley (1969Hinkley ( , 1970Hinkley ( , 1971, Hawkins (1977Hawkins ( , 1987 and Kim and Siegmund (1989). The C-test is based on CUSUM of square tests due to Page & her associates and later followed by Brown et al (1975) and Ploberger and Kramer (1992). Finally, the LM-test is based on Nyblom's (1989) and Hansen's (1991) Lagrange multiplier method.…”

Preliminary evidence on the determinants of Federal Reserve open market operations

“…The shift is identified to be in 1969 by finding the minimum of the log likelihood ratio (for alternative specifications of the dummy variables). See Brown, Durbin and Evans (1975) for more details about the search for a shift in structure. 8.…”

Sequential moving sums of squares of OLS residuals in parameter stability testing

“…However, such procedures may still indicate situations when it is probably better to apply procedures which are adapted to parameter variability assumptions. Two classes of test procedures of the latter kind are CUSUM tests (Brown et al, 1975) and MOSUM tests (Hackl, 1980). CUSUM as well as MOSUM tests are based on so-called recursive residuals, as ordinary-least-squares (OLS) residuals are expected to cause intractable methodological problems.…”