A model selection approach to jointly testing for structural breaks and cointegration with application to the Eurocurrency interest rates market

Throughout the paper, we only consider pure structural break models mainly for concerns of simulation time. Partial break models can be studied without any additional technical difficulty except that restricted MLE is needed to estimate Model (1), which can greatly increase the computing time in dating breaks.

The variant of the search procedure we adopt here is inefficient in the sense that it does not permit recursive calculation of the likelihood function as in Qu and Perron (2007a). The reason is that we allow for non-homogeneous variances across regimes in order to use the simple unrestricted ML estimator for the cointegration model.

We focus on these two classical information criteria as they are still quite popular in empirical studies. However, there are some important developments in this line of literature. For example, Qu and Perron (2007b) argue that traditional AIC and BIC tend to select too parsimonious models and the cointegration tests suffer from substantial size distortions in finite samples. They hence suggest a modified approach that includes the likelihood ratio (LR) test statistic of Johansen (1991) as an additional penalty in calculating the information criteria. Nevertheless, the LR test statistic is not defined for the case of full rank (r = n), whereas the model selection approach we propose here searches over models with all possible cointegration ranks (namely, r = 0, 1,…, n). Therefore, we do not pursue the modified information criteria.

A distinctive feature of the canonical form model (8) is that it is straightforward to extend the bivariate model to higher dimensions by generalizing the scalar coefficients ψ_j to an r-dimension matrix and the scalar 1 to a (n – r)-dimension identity matrix. A similar generalization can be made to the intercept δ_j and the correlation coefficient θ. Intuitively, in this general model specification, w_1t can be understood as a weighted version of the disturbance of the cointegration relation of a hypothetical model that is not explicitly formulated and w_2t is the orthogonal complement that is not stationary.

Critical values of parametric tests for structural breaks depend on the amount of trimming, although the dependence may not be heavy (e.g., Vogelsang 1997). Likewise, the determination of the value of the trimming is a key aspect of the model selection process. However, in our simulation exercises, we do not include it as a part of model selection because of computational burden. All results reported in the study are conditional on the chosen value of trimming (i.e., 0.15). More generally, the determination of the trimming can be guided by a priori information about the process or by using some preliminary nonparametric procedures.

We also consider other values of δ > 0 for the DGP. The unreported results show that the performance of the four criteria is not significantly affected by the magnitude of δ.

In Internet Appendix of our working paper version, we report additional results of Simulation I.

Note that the presence of moving average components in the DGP (except when θ = 0) contradicts our earlier assumption of serially uncorrelated errors so that the likelihood of model (1) is additive. Therefore, we estimate VAR as an approximation of the DGP (9) which is a VARMA. The implication of this approximation is that an information criterion may perform poorly in selecting structural breaks and cointegration rank simply because it performs poorly in selecting appropriate lag orders. It is also possible that an information criterion does not perform well for large (negative) θ because instead of choosing a long lag length to capture the effect of θ, the criterion allows for shorter lags by compensating by splitting up in segments.

Since we consider the τ_N test in Tables 4, 5, and 6, we assume that the DGP is known to have one break at an unknown date in these cases. Therefore, the focus here is on the cointegration test. Furthermore, for Table 4, we use ρ = 1. The DGP is thus a bivariate VAR of two I(1) series that are not cointegrated. Therefore, we only report in Table 4 the cointegration rank test results.

The results for more autocorrelated samples (ρ = 0.7) are not reported here. Consistent with the findings in the earlier tables, HQC, BIC, and PIC are all less powerful for larger ρ, whereas the impact on AIC is small.

Conditional on the choice of the maximum lag length of six, we find that the selected lag orders are mostly in the range of 2 to 4 depending on the information criterion.

Therefore, the date estimates should be read in combination with the performance of the information criteria in selecting correct number of breaks and the cointegration ranks (the numbers in parentheses).

There is one practical concern about the choice of lag order in the presence of structural breaks. As ρ increases from 0.5 to 1 and/or β changes from 0.3 to 0.7 following the break, the optimal lag order of the model may also change across the two regimes. Although allowing for time-varying lag orders is possible with the model selection approach, in the consideration of computing time, throughout the paper we assume that breaks in parameters have no impact on the optimal order of the VARs. Empirically, we find that this does not appear to be a very restrictive assumption for the parameter values we use in the simulation. Based on T = 300, the average of selected lags by BIC is 2.13 for both ρ = 0.5 and ρ = 1 when θ = − 0.5 and η = 0. They are identical 96% of the time. The comparisons are similar for other parameter values and the other two information criteria, AIC and HQC.