Testing for the Presence of a Random Walk in Series with Structural Breaks

Abstract

We consider tests for the presence of a random walk component in a stationary or trend stationary time series and extend them to series that contain structural breaks. The locally best invariant (LBI) test is derived and the asymptotic distribution is obtained. Then a modified test statistic is proposed. The advantage of this statistic is that its asymptotic distribution is not dependent on the location of the break point and its form is that of the generalized Cramer–von Mises distribution, with degrees of freedom depending on the number of break points. The performance of this modified test is shown, via some simulation experiments, to be comparable with that of the LBI test. An unconditional test, based on the assumption that there is a single break at an unknown point, is also examined. The use of the tests is illustrated with data on the flow of the Nile and US gross national product.

Testing for (common) stochastic trends in the presence of structural breaks

“…If this is not the case, the existence or not of a break seems better to be established for each series in turn. For a univariate model Busetti and Harvey (2001) have proposed an 'inf-type' statistic for non-stationarity that covers the case of an unknown breakpoint. Alternatively, one could adopt a two-stage strategy, where first the breakpoint is estimated and then this estimate is used to compute our tests.…”

“…This test coincides with the LBI test (11) 1 For other values of the test is, of course, consistent. Furthermore, the simulation results of Busetti and Harvey (2001), relating to the univariate version of the test, show that it suffers only a small loss in power compared with the LBI test.…”

“…The limiting distribution is then Cramér-von Mises with 2N degrees of freedom; see Nyblom (1989) and Busetti and Harvey (2001) for details of the additivity property of Cramér-von Mises random variables.…”

The KPSS test with two structural breaks

“…The way to overcome this drawback consists on proceed to the estimation of the break points instead of assuming them as exogenous. Thus, Hao (1996), Busetti and Harvey (2001) and Lee and Strazicich (2001) apply the minimum functional to the sequence that results from the computation of the KPSS test for all possible break points. The argument that minimizes this sequence is taken as the estimate of the breaking point.…”

Testing in unobserved components models

“…There is no slope in model ð0Ł and so the only break is in the level[ The other models all contain a time trend[ In model ð1Ł there is a structural change in both the level and the slope[ Model ð1aŁ contains a break in the level only while ð1bŁ corresponds to a piecewise linear trend[ LBI test against a random walk Under Gaussianity the LBI "and one!sided LM# test statistics for H 9 ] s 1 h 9 against H A ] s 1 h × 9 in the above models are of the same form as equation "1#[ They will be denoted as h i "l#\ i 0\ 1\ 1a\ 1b\ where the subscript i indicates that the residuals depend on the model\ 0\ 1\ 1a or 1b\ and l denotes that the statistic has been constructed for a speci_c value of the breakpoint location parameter and that its asymptotic distribution depends on it[ As in the previous section\ the limiting distribution can be derived by looking at the asymptotic properties of the process followed by the partial sum of residuals[ This will converge to a limiting process*de_ned on an underlying Wiener process*that will depend on l and collapse to a "second!level# Brownian bridge when l 9 or l 0[ The asymptotic distribution of h i "l# is then the integral on the unit interval of the square of this process[ The form of these distributions is given in Busetti and Harvey "1999#[ The upper tail percentage points of the above asymptotic distributions are reported in Busetti and Harvey "1999# for di}erent values of l[ The _gures for l : 9 or l : 0 correspond to the critical values for the Crame rÐvon Mises distributions given in Table I[ As expected\ the per! centage points*as functions of l*are symmetric around l 0:1\ which is also the minimum for models ð0Ł\ ð1Ł and ð1bŁ[ For models ð0Ł and ð1Ł the asymptotic distributions can be characterized in terms of two independent Crame rÐvon Mises distribution[ To see that this is the case\ _rst notice that we can rewrite the statistic as ðB i "r#Ł 1 dr ¦ "0−l# 1 g 0 9 ðB?…”