Structural Breaks in Grouped Heterogeneity

Abstract

Generating accurate forecasts in the presence of structural breaks requires careful management of bias-variance tradeoffs. Forecasting panel data under breaks offers the possibility to reduce parameter estimation error without inducing any bias if there exists a regime-specific pattern of grouped heterogeneity. To this end, we develop a new Bayesian methodology to estimate and formally test panel regression models in the presence of multiple breaks and unobserved regime-specific grouped heterogeneity. In an empirical application to forecasting inflation rates across 20 U.S. industries, our method generates significantly more accurate forecasts relative to a range of popular methods.

Keywords:

• Bayesian analysis
• Forecasting
• Grouped heterogeneity
• Inflation modeling
• Panel data
• Structural breaks

Supplementary Materials

The supplementary materials provide additional information. Specifically, Section 1 of the supplementary appendix provides more details of the estimation procedure and Section 2 gives details of various robustness checks conducted in the empirical analysis. Finally, the supplementary appendix formally sets out the relation between the Multinomial and Poisson distributions.

Disclaimer

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.

Notes

Acknowledgments

The suggestions of Torben Andersen, Andrii Babii, Vittorio Bassi, Richard Blundell, Stephane Bonhomme, Fabio Canova, Sid Chib, David Childers, Petros Dellaportas, Marco Del Negro, Frank Diebold, Rob Engle, Ivan Fernandez-Val, Wayne Ferson, Michele Fioretti, Eric Ghysels, Jim Hamilton, Jerry Hausman, Ed Herbst, Kirstin Hubrich, Michael Keane, Dimitris Korobilis, Dennis Kristensen, Blake LeBaron, Simon Lee, Laura Liu, Matteo Luciani, John Maheu, James Mitchell, Roger Moon, Hashem Pesaran, Mikkel Plagborg-Moller, Katia Peneva, Jim Powell, Giorgio Primiceri, Simon Reese, Geert Ridder, Jeremy Rudd, Andres Santos, Lukas Schmid, Allan Timmermann, Maria Tito, Herman van Dijk, Dan Vine, Martin Weidner, Yinchu Zhu, and seminar participants at USC, Lancaster University, UC Riverside, Federal Reserve Board, NBER-NSF SBIES 2019, Barcelona GSE Summer Forum 2019, 2019 Panel Data Prediction conference at USC, European Seminar on Bayesian Econometrics 2019, and Bristol Econometric Study Group 2019 have been helpful. Smith was a Postdoctoral Scholar Research Associate at USC Dornsife INET while working on this project. Any remaining errors are my own.

Notes

1 A handful of frequentist time series approaches to detect breaks include Andrews (1993) and Bai and Perron (1998), while Bayesian approaches include Chib (1998), Pesaran, Pettenuzzo, and Timmermann (2006), and Koop and Potter (2007).

2 Some recent studies on panel forecasting include Liu, Moon, and Schorfheide (2020) and Smith and Timmermann (2017b).

3 A small handful of articles on cross-sectional grouping include Canova (2004), Hamilton and Owyang (2012), and Bonhomme and Manresa (2015).

4 Some studies on breaks in panel models or multivariate time series include Bai, Lumsdaine, and Stock (1998), Qu and Perron (2007), and Baltagi, Feng, and Kao (2016).

5 For a framework that incorporates noncommon breaks—see Smith (2018).

6 The data are described in more detail in Section 4.

7 The petroleum industry has the most unusual inflation series with high volatility and frequent and large outliers.

8 Primiceri (2005) and Smith and Timmermann (2017b) apply the reversible jump algorithm in economic settings.

9 As is conventional in the literature we assume ${\tau }_0=0$ and ${\tau }_{K+1}=T$ for convenience.

10 Note that group membership is fixed within regimes and so if the ith series belongs to the g_k th group then it does so for all time periods within the kth regime, that is, for $t={\tau }_{k-1}+1,\dots ,{\tau }_k$.

11 ¹¹ Appendix A of the supplementary materials explains how this specification of multiple independent Poisson distributions is inferentially equivalent to a specification that uses a single Multinomial distribution.

12 The properties of the natural conjugate Normal-Inverse Gamma prior are well known; see, for example, Chapter 2 in Chan et al. (2019). We derive Equation (9) by multiplying the priors by the likelihood and marginalising $\mathit{\beta }$ and ${\mathit{\sigma }}^{\mathit{2}}$ such that $p(\mathit{y}|\mathit{X},\tau ,\mathit{c})=\int \int p(\mathit{\beta },{\mathit{\sigma }}^{\mathit{2}}|\mathit{y},\mathit{X},\tau ,\mathit{c})d\mathit{\beta }d{\mathit{\sigma }}^{\mathit{2}}$.

13 Section 1 of the supplementary materials explains each step in detail.

14 They also show that finite and Dirichlet process mixtures yield similar inference on the number of groups once the two approaches’ hyper priors are matched.

15 Creal and Kim (2021) develop a new Bayesian factor model with regression tree priors that performs variable selection—using a spike and slab prior—from a large set of characteristics, identifying the set of economy-wide variables that drive aggregate time series return predictability, and forms portfolios based on a clustering technique.

16 They estimate the maximum number of groups by experimenting with different values. Bridge sampling techniques obtain the corresponding marginal likelihoods and the value assigned the highest posterior model probability is selected. Our approach does not require this additional computation.

17 In results not shown, we find that there are efficiency gains from using the time-invariant clustering approach when there are no breaks in the group membership.

18 The simulation study also illustrates the bias-variance improvements derived from our framework. Both the pooled panel breakpoint model and the time-invariant clustering model deliver biased parameter estimates. The heterogeneous panel breakpoint model correctly identifies the break dates and thus delivers unbiased parameter estimates. However, its estimates are imprecise. Our model delivers unbiased parameter estimates that are far more precise than the heterogeneous breakpoint model. To save space, these results are not shown but are available upon request.

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

20 For open economies, a second-order approximation of a welfare-optimizing central bank’s objective function can be specified as a function of the output gap, real exchange rate, and the PPI (Gali and Monacelli 2005; De Paoli 2009). This assumes the central bank should focus on domestic inflation and that the PPI excludes imports, unlike the CPI. For closed economies, in a two-stage production model Huang and Liu (2005) show that it is preferable to use the PPI in a simple monetary policy rule because the PPI includes intermediate goods prices, while the CPI does not. With sticky prices, relative price fluctuations between intermediate and final goods distort labor allocation across production stages. The PPI captures more of this information than the CPI and a welfare-optimizing central bank should incorporate this information.

21 The intercept, lagged dependent variable, and lagged industry-level unemployment rate all have a regime-group-specific coefficient in our predictive regression.

22 A small subset of additional studies on large Bayesian VARs includes Carriero, Kapetanios, and Marcellino (2009), Koop (2013), and Carriero, Clark, and Marcellino (2015).

23 Due to data availability constraints, industrial production is not available for financials, accommodation, transportation, real estate, and hospitals.

24 We apply an x13 filter to seasonally adjust the industry-level and aggregate U.S. inflation series.

25 We assume a uniform prior across these models.

26 Our model’s superior predictive accuracy could derive partially from faster detection of changing group membership in real time. If a model quickly detects changing group membership, it may realize most of the potential gains in predictive accuracy. Following the approach of Smith and Timmermann (2017), we compare how quickly—defined as the lag between the month of the recursive forecasting exercise in which the break was first detected and its full sample estimated posterior date—the two models detect the September/October 2009 break. Our approach detects the break with a lag of just 4 months compared with a lag of nine months for the alternate approach.