Analysis of structural break models based on the evolutionary spectrum: Monte Carlo study and application

Abstract

We investigate the instability problem of the covariance structure of time series by combining the non-parametric approach based on the evolutionary spectral density theory of Priestley [Evolutionary spectra and non-stationary processes, J. R. Statist. Soc., 27 (1965), pp. 204–237; Wavelets and time-dependent spectral analysis, J. Time Ser. Anal., 17 (1996), pp. 85–103] and the parametric approach based on linear regression models of Bai and Perron [Estimating and testing linear models with multiple structural changes, Econometrica 66 (1998), pp. 47–78]. A Monte Carlo study is presented to evaluate the performance of some parametric testing and estimation procedures for models characterized by breaks in variance. We attempt to see whether these procedures perform in the same way as models characterized by mean-shifts as investigated by Bai and Perron [Multiple structural change models: a simulation analysis, in: Econometric Theory and Practice: Frontiers of Analysis and Applied Research, D. Corbea, S. Durlauf, and B.E. Hansen, eds., Cambridge University Press, 2006, pp. 212–237]. We also provide an analysis of financial data series, of which the stability of the covariance function is doubtful.

Keywords:

• evolutionary spectrum
• break dates
• size and power
• coverage rates
• selection procedures

JEL Classification :

• C22

Notes

Nouira et al. 20 draw two characteristics a priori contradictory and yet coexistent in the daily returns of exchange rate of euro/US dollar. They indeed show the non-stationarity of the covariance structure of the series and, after the extraction of the unstable variance using the algorithm based on the cumulative sums of squares of Inclan and Tiao 13, the existence of long-memory in the filtered series.

Ben Aïssa et al. 11 adopted this non-parametric approach to propose a test similar to that based on Kolmogorov–Smirnov statistic applied to the evolutionary spectrum to determine the number of changes and their locations in the monthly US inflation series.

The description is local since there is an interval of the frequency domain on which the process {X _t } can be considered as approximately stationary.

For more details on the relations (i) and (ii), and the choice of k and T′, the readers are referred to Priestley 23.

This is because the spectral density is independent of time in each interval where the process is stationary.

From Bai and Perron 8, if the estimation is the sole concern for the study, then the minimal number of observations in each regime can be set to any value greater than one, the number of regressors of the model.

Note that ‘↠’ represents the weak convergence.

In practice, we need to impose some constraints on the distributions of the regressors and the errors across regimes. For more details on the different possible cases, the readers are referred to Bai and Perron 8 10.

Note that the existence of breaks in the variance could be exploited to increase the precision of the break date estimates 8.

The size of a test is the probability that it rejects the null hypothesis when it is true, while the power of a test is the probability that it rejects the null hypothesis when it is false.

We adopt the same choice as Artis et al. 5.

Note that this choice of T′ conforms with the suggestions of Priestley 22 since it satisfies the introduced conditions.

To that effect, a simulation experiment is carried out with and . The obtained results (not reported here, but available upon request from the author) show that the performance of some tests improved, but globally the results remain inadequate.

The results of the simulation experiment carried out for and show that the detection frequencies of the true number of breaks by all the selection procedures improved.

Note that a large shift in the variance entails a large change in the mean of the logarithm of the theoretical evolutionary spectral density in the structural change model given by Equation (10), and vice versa.

In this case, the break date τ₁ takes the value [T ₁/T′]=30.

The mean and the standard deviation over all the simulated samples are the same whatever the distribution of the errors across regimes.

The adopted specification slightly affects the results of the sequential procedure for small values of ϵ especially when the errors are heterogeneous across regimes.

We see that the variance returns to its old value at the second break date. Note that the break dates τ₁ and τ₂ take, respectively, the values [T ₁/T′]=20 and [T ₂/T′]=40.

The error in rejection probability is defined as the difference between the actual rejection frequency under the null hypothesis and the nominal significance level of the test.

The heteroskedasticity and autocorrelation consistent covariance matrix is constructed following Andrews 3 using a quadratic kernel with automatic bandwidth selection based on an AR(1) approximation. We also allow using pre-whitening as suggested by Andrews and Monahan 4 (see 8 for more details).

Unlike the conclusions of Bai and Perron 10, the sequential method is not more powerful than the information criteria for the selection of the number of breaks when we are interested in the instability problem based on the theory of the evolutionary spectrum.

Note that the break dates detected by the information criteria are illustrated on the graph of the series given in Figure 3.