Asymtotically Exact Confidence Intervals Of Cusum And Cusumsq Tests: A Numerical Derivation Using Simulation Technique

ASYMPTOTICALLY EXACT CONFIDENCE INTERVALS OF CUSUM AND CUSUMSQ TESTS: A Numerical Derivation Using Simulation Technique Hisashi TANIZAKI Faculty of Economics, Kobe University, Nada-ku, Kobe 657, JAPAN ABSTRACT: In testing a structural change, the approximated confidence intervals are conventionally used for CUSUM and CUSUMSQ tests. This paper numerically derives the asymptotically exact confidence intervals of CUSUM and CUSUMSQ tests. It can be easily extended to nonnormal and/or nonlinear models. KEY WORDS: CUSUM test, CUSUMSQ test, Monte-Carlo simulation, Asymptotically exact confidence interval 1 INTRODUCTION There is a great amount of literature on use of recursive residuals, e.g., Brown, Durbin and Evans (1975), Galpin and Hawkins (1984), Harvey (1989), Johnston (1984), Ploberger (1989), Ploberger, Kr¨amer and Alt (1989), and Westlund and T¨ornkvist (1989). Especially, Brown, Durbin and Evans (1975) described an important application of recursive residuals in testing for structural change over time. The technique is appropriate for time series data, and might be used if one is uncertain about when a structural change might have taken place. We have two tests; cumulative sum (CUSUM) test and cumulative sum of squares (CUSUMSQ) test The null hypothesis is that the coefficient vector β is the same in every period; the alternative is simply that it (or the disturbance variance) is not. The test is quite general in that it does not require a prior specification of when the structural change takes place. However, it is known that the power of the test is rather limited compared to that of the Chow test. The test is frequently criticized on this basis. However, the Chow test is based on a rather definite piece of information, namely, when the structural change takes place. If this is not known or must be estimated, the advantage of the Chow test diminishes considerably (see Greene (1990) and Kr¨amer (1989)). See Galpin and Hawkins (1984) for an application. One of the reasons why the CUSUM test is less powerful is that the confidence interval of the test is approximated. For the CUSUM test statistic, we cannot derive the explicit distribution, and therefore the approximated confidence interval is conventionally used in testing the stability of the coefficient. Although it is known that the CUSUMSQ test statistic is distributed as a beta random variable, the confidence interval of the CUSUMSQ test is also approximated. Therefore, in this paper, an attempt is made to obtain the exact confidence intervals of the CUSUM and CUSUMSQ tests asymptotically using the MonteCarlo simulation technique. Moreover, the power is compared for both the approximated confidence intervals (the confidence intervals conventionally used) and the simulated ones (the confidence intervals proposed in this paper). 1 2 OVERVIEW OF CUSUM AND CUSUMSQ TESTS Consider the following regression model: ut ∼ N (0, σ 2 ), yt = xt β + ut , t = 1, · · · , T, where β is a k × 1 unknown parameter vector. yt is a dependent variable while xt is a 1 × k vector of independent variables. ut is assumed to be normally distributed with mean zero and variance σ 2 . Define Xt and Yt as follows:     y1 x1  y2   x2    Yt =  Xt =   ..  ,  ...  . . xt yt The null hypothesis of no structural change for the model is specified as: H0 : β1 = β2 = · · · = βT = β and σ12 = σ22 = · · · = σT2 = σ 2 , where βt denotes the vector of coefficients in period t and σt2 the disturbance varia