Detecting Multiple Level Shifts in Bounded Time Series

Abstract

The article proposes a sequential statistical procedure to test for the presence of level shifts affecting bounded time series, regardless of their order of integration. The article shows that bounds are relevant for the statistic that assumes that the time series are integrated of order one. In contrast, they do not affect the limiting distribution of the statistic that is defined for time series that are integrated of order zero. The article proposes a union rejection statistic for bounded processes that does not require information about the order of integration of the stochastic processes. The model specification is general enough to consider the existence of structural breaks that can affect either the level of the time series and/or the bounds that limit its evolution. Monte Carlo simulations indicate that the procedure works well in finite samples. An empirical application that focuses on the Swiss franc against the euro exchange rate evolution illustrates the usefulness of the proposal.

Keywords:

• Bounded processes
• Changing bounds
• Structural breaks

Supplementary Materials

Appendix A. Mathematical appendix. Appendix B. Asymptotic and finite sample (response surfaces) critical values, Monte Carlo results, and empirical illustration results.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 The model can accommodate the cases of stochastic processes that are only limited below—that is, $x_t\in [{\underset{¯}{b}}_t,\infty ]$—or only limited above—that is, $x_t\in [-\infty ,{\overline{b}}_t]$—but also covers the case of unbounded processes—that is, $x_t\in [-\infty ,\infty ]$.

2 Cavaliere and Xu (2014) argue that when a time series is known to be regulated, but the bounds are unknown, it might be possible that “a reasonable range of bounds can be inferred from historical observations and/or from the relevant economic theory” or from economic policy implemented by institutions—for example, specification of target values for some macroeconomic variables.

3 The response surfaces can also be used for the statistics in Harvey, Leybourne, and Taylor (2010) for unbounded time series if an arbitrarily large value for the bounds is specified.

4 For instance, if the initial number of observed bounds for the time series is n = 1, placed at time ${\tau }_1^0$, and we have detected one significant level shift (m = 1, placed at time ${\tilde{T}}_1$), the estimation of $\left[{\underset{¯}{\mathit{c}}}_1,{\overline{\mathit{c}}}_1\right]$ in the second stage will need to distinguish up to three regimes—that is, $t\le {\tilde{T}}_1,{\tilde{T}}_1<t\le {\tau }_1^0$ and $t>{\tau }_1^0$, provided that ${\tilde{T}}_1\ne {\tau }_1^0$ and ${\tilde{T}}_1<{\tau }_1^0$. The same qualitative result is obtained if ${\tilde{T}}_1\ne {\tau }_1^0$ and ${\tilde{T}}_1>{\tau }_1^0$. The exception is found when ${\tilde{T}}_1={\tau }_1^0$ so that the actual number of regimes in the second stage is equivalent to the first stage one. In general, the actual number of regimes in the ith step of the sequential testing strategy is $(n+i)$.

5 Additional empirical size and power simulation results for ${\widehat{S}}_d,d\in \left\{0,1\right\}$, and ${\widehat{S}}_U$ statistics for different combinations of $\overline{c}$ and w parameters are provided in the appendix—see Table B.7 for the empirical size and Tables B.8–B.10 for the empirical power with w = 0.15, and Tables B.11–B.13 for the empirical power with w = 0.3.

6 Since 2020 the Euro Overnight Index Average (EONIA) has been replaced by the Euro Short-Term Rate (ESTER) as the European interbank market benchmark.

7 In the third period we have considered the official limit 1.2 instead of the minimum, 1.0968; the results do not change.

8 The original institutional boundaries set in May 2009, September 2011, and January 2015 are (1.671, 1.524, 1.241, 1.189) for the upper boundary and (1.457, 1.118, 1.2, 0.964) for the lower boundary. The initial value of the nominal exchange rate is 1.611, so that for the first stage of the sequential testing procedure the first pair of demeaned boundaries is $[{\underset{¯}{\widehat{b}}}_1,{\widehat{\overline{b}}}_1]=[-0.155,0.06]$, the second pair is $[{\underset{¯}{\widehat{b}}}_2,{\widehat{\overline{b}}}_2]=[-0.493,-0.087]$, the third pair is $[{\underset{¯}{\widehat{b}}}_3,{\widehat{\overline{b}}}_3]=[-0.411,-0.37]$ and, finally, the fourth set is $[{\underset{¯}{\widehat{b}}}_4,{\widehat{\overline{b}}}_4]=[-0.648,-0.423]$, as presented in Panel C of Table 4. This table illustrates the evolution of the estimated boundaries as new statistically significant level shifts are encountered.

9 The potential breaks for the ${\widehat{S}}_U$ and ${\widehat{S}}_U^{\ast }$ statistics do not necessarily need to coincide since the estimation procedure used by the former statistic relies on the raw time series, whereas the latter uses the time series projected against a constant and the set of dummy variables $D{U}_{j,t}=1(t>{\tau }_j^0),j=1,\dots ,n$.

10 Note that δ = 0 corresponds with the original limits. Beyond $\delta =50$, negative lower boundaries would appear, which is incompatible with the nature of the exchange rate.

11 In order to guarantee the robustness of the results we have considered a second option in which the series is not bounded before May 2009—see Figure B.2—and a third option that removes the boundaries after 2015—see Figure B.3. The results show that as the boundaries relax, the number of breaks detected decreases—see Figures B.4 and B.5 for the second and third options, respectively. This means that the effect of the limits is concentrated in the period of strong intervention and explicit target setting. This episode has been studied in Bykhovskaya and Duffy (2022).

Additional information

Funding

The authors acknowledge the financial support from the grants PID2020-114646RB-C41, PID2020-114646RB-C44 and RED2022-134122-T funded by MCIN/AEI/10.13039/501100011033.