Noncommon Breaks

Abstract

We develop a new Bayesian approach to estimate noncommon structural breaks in panel regression models. Any subset of the cross-section may be hit at different times within a break window. Break-specific parameters are learned from the cross-section. They reflect whether (i) breaks hit many or few series and (ii) there is a long or short lag between the first and final series hit by a break. In an empirical application to international stock return predictability, the method generates significantly more accurate forecasts than several benchmarks that yield economically meaningful utility gains for a risk averse investor with power utility.

KEYWORDS:

• Bayesian analysis
• Panel data
• Stock return predictability
• Structural breaks

Acknowledgments

The suggestions of numerous seminar and conference participants have been helpful. Any remaining errors are my own. Smith was a Postdoctoral Scholar Research Associate at USC Dornsife INET while working on this project. The views expressed in this article are those of the author and do not necessarily reflect the views and policies of the Board of Governors or the Federal Reserve System.

Disclosure Statement

The author reports there are no competing interests to declare.

Notes

1 A subset of studies on breaks in panel models or multivariate time series include Bai, Lumsdaine, and Stock (1998), Bai (2010), Baltagi, Feng, and Kao (2016), and Smith (2023).

2 A handful of frequentist time series approaches to detect breaks include Bai and Perron (1998), Bai and Perron (2003), and Elliott and Müller (2006), while Bayesian approaches include Chib (1998), Pesaran, Pettenuzzo, and Timmermann (2006), and Koop and Potter (2007).

3 A more detailed description of the data is provided in Section 4.

4 Allowing any subset to be hit at the same date would involve $2^N$ possibilities which becomes infeasible even for moderate cross-sectional dimensions N. Allowing series to be hit at different times exacerbates the problem even further.

5 We assume $N_0=N$.

6 Following convention, we assume ${\tau }_{0_i}=0$ and ${\tau }_{K_i+1}=T$ for all i.

7 The noncommon break window begins when the first series is hit by the break and ends when the final series is hit by it.

8 Note that ${\Delta }_{\mathrm{\text{max}}}$ is a pre-imposed upper bound on the break window length that the algorithm implicitly sets equal to T–2, allowing the break window to cover the entire sample except for the first and final time periods which, by construction of the algorithm, must represent the two regimes separated by the break. However, the user has the option to pre-specify ${\Delta }_{\mathrm{\text{max}}}<T-2$ if they wish to preclude break windows greater than a certain length. For instance, if the user wishes to preclude break windows beyond, say 24 months using monthly data, they could set ${\Delta }_{\mathrm{\text{max}}}=24$. This may yield computational gains as demonstrated in the simulation study in Section 3. Of course, specifying ${\Delta }_{\mathrm{\text{max}}}$ to be lower than the true length of the break windows will hinder the ability of the algorithm to reveal the true break dynamics.

9 This implies that, for every break, ${\eta }_k=1$ and ${\delta }_k=0$.

10 Our prior assumes regimes last 50 periods, on average, since c = 100 and d = 2. We set $g=h=1$, implying that each series has 0.5 prior probability of being hit by breaks, and e = 3 and f = 1 such that affected series will be hit on average with a prior expected length of three periods. We set $a=b=2$ and ${\sigma }_{\beta }^2=0.04$. We set the variance of the proposal distribution of ${\beta }^{*},{\sigma }_{{\beta }^{*}}^2$, as defined in (A.3) of the supplemental web appendix equal to the variance of the empirical distribution of the output of β values from an initial run of the common breaks model (which is the noncommon breaks methodology with η_k=1 and ${\delta }_k=0$ for all k. The simulation study results are unaffected by doubling the value of ${\sigma }_{{\beta }^{*}}^2$.

11 This threshold is more conservative than the 1.2 threshold proposed by Gelman and Rubin (1992).

12 There are efficiency gains (not shown) from using the common break approach relative to our approach when every break in the DGP is common.

13 Specifying ${\Delta }_{\mathrm{\text{max}}}=15$, for N = 100 and T = 500, the computation time reduces from 42 to 29 min and the Gelman-Rubin statistic and AMSE ratios relative to the two benchmark models are unchanged (not shown).

14 We impose at least one time period between the end of a break window and the start of the next one (${\tau }_{2k+1}>{\tau }_{2k}+1$).

15 Using local currency returns also circumvents the need to develop a risk premium model for exchange rates so we can focus on time-variation in expected returns in equity markets (Solnik 1993).

16 In results not shown, the full sample ex-post estimates are unchanged when we include two observed common factors (sourced from GFD): the excess stock return on a worldwide series and the change in a trade-weighted average of several exchange rates against the U.S. dollar which proxies for the strength of the U.S. dollar. These results suggest that, in our setting, local currency excess returns closely approximate currency-hedged excess returns.

17 The corresponding plots for the remaining four benchmarks are displayed in Figure A2 of the supplemental appendix.

18 To evaluate whether our methodology outperforms the benchmarks consistently across the out-of-sample period, we compute the cumulative sum of squared error differences against each benchmark for each country. Our methodology outperforms the benchmarks for most countries consistently through the sample (results not shown to save space, but available upon request).

19 Since their test assumes a rolling window, not an expanding window, we compute log scores and the corresponding p-values of the Amisano-Giacomoni test based on forecasts generated at each time period t from model parameters estimated using only the most recent 120 monthly observations, that is $(t-119,\dots ,t$).

20 Like Campbell and Thompson (2008), we choose a relative risk aversion coefficient of three. However, the results remain robust when we increase this to 5, the value chosen by Pettenuzzo, Timmermann, and Valkanov (2014), albeit the magnitude of outperformance is a touch smaller.

21 The corresponding plots for the remaining four benchmarks are displayed in the right panels of Figure A2 in the supplemental web appendix.

22 The CER values are generally smaller for a mean-variance investor (not shown), reflecting the ability of our approach to generate additional pockets of predictability that are related to higher moments (Smith 2007; Guidolin and Timmermann 2008). We thank an anonymous referee for this suggestion.

23 Due to data availability constraints we only have 13 and 15 countries when forecasting with the price-earnings ratio and term spread.

24 To save space, these results are not shown but are available upon request.

25 The break date of 1982 roughly coincides with the end of the monetarist policy experiment in the United States (1979 through 1982).