Unstable Diffusion Indexes: With an Application to Bond Risk Premia
| Date | 01 December 2019 |
| Author | Daniele Massacci |
| Published date | 01 December 2019 |
| DOI | http://doi.org/10.1111/obes.12311 |
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©2019 Bank of England. Oxford Bulletin of Economics and Statistics ©2019 The Department of Economics, University of Oxford and John Wiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 81, 6 (2019) 0305–9049
doi: 10.1111/obes.12311
Unstable Diffusion Indexes: With an Application to
Bond Risk Premia*
Daniele Massacci
Bank of England, Threadneedle StreetLondon, EC2R 8AH, UK
(e-mail: daniele.massacci@gmail.com)
Abstract
This paper studies the empirically relevantproblem of estimation and inference in diffusion
index forecasting models with structural instability. Factor model and factor augmented
regression both experience a structural change with different unknown break dates. In
the factor model, we estimate factors and loadings by principal components. We consider
least squares estimation of the factor augmented regression and propose a break test. The
empirical application uncovers instabilities in the linkages between bond risk premia and
macroeconomic factors.
I. Introduction
The diffusion index forecasting model of Stock and Watson (1998, 2002) and Bai and Ng
(2006) is a regression model with observable and latent covariates, wherethe latter are the
common factors in the variables of a large-scale data set. The model is usually estimated
under the assumption of structural stability in the loadings of the factor model and in the
slope coefficients of the factor augmented regression: Connor and Korajczyk (1986, 1988,
1993), Bai and Ng (2002), Stock and Watson (2002) and Bai (2003) deal with linear static
factor models; Forni et al. (2000, 2004), Forniand Lippi (2001) and Forni et al. (2015) study
the linear generalized dynamic factor model; Stock and Watson (2002) and Bai and Ng
(2006) focus on stable factor augmented regressions. Violation of the stability assumption
may affect empirical results. Forexample, out-of-sample forecasts may not be accurate due
JEL Classification numbers: C12, C13, C38, C52, G12.
*This paper was started when the author was Franco Modigliani Research Fellow in Economics and Finance at
the Einaudi Institute for Economics and Finance (EIEF). The views are the author’s and do not necessarily reflect
those of the Bank of England or its policy committees. The author is indebted to Marco Lippi for introducing him
to factor models and for several enlightening conversations. This paper benefits from comments from participants
at the International Association for Applied Econometrics (IAAE) 2016Annual Conference, at the Symposium on
Statistical Penalization Methods and Dimension Reduction Methods for Economic and Financial Analysis at the
University of York, at the Vienna Workshop on High-Dimensional Time Series in Macroeconomics and Finance
2017, and at the seminar series at The University of Nottingham Granger Centre for Time Series Econometrics;
and from conversations with Laura Coroneo, Pasquale Della Corte, Domenico Giannone, Christian Gourieroux,
Refet G¨urkaynak, George Kapetanios,Andrew Meldrum and Hashem Pesaran. Errors and omissions are the author’s
responsibility.The financial support from the Associazione Borsisti Marco Fannoand from UniCredit and Universities
Foundation is gratefully acknowledged.
Unstable diffusion indexes 1377
to instabilities in the factor augmented regression (Stock and Watson, 2002, 2009) or in
the factor model (Giannone, 2007; Banerjee, Marcellino and Masten, 2008). In the latter,
structural breaks enlarge the factor space (Breitung and Eickmeier, 2011; Chen, Dolado
and Gonzalo, 2014) without conveying more information: including additional factors
in the factor augmented regression increases estimation noise and deteriorates forecast
performance (Han and Inoue, 2015).
A number of contributions analyses large dimensional factor models with structural
instabilities. Bates et al. (2013) study the robustness of the principal components estimator
as applied to factor models under neglected instability. Breitung and Eickmeier (2011),
Chen et al. (2014), Han and Inoue (2015), Yamamoto and Tanaka (2015) and Barigozzi
and Trapani (2017) develop statistical tools to detect breaks. Chen (2015), Cheng, Liao
and Schorfheide (2016), Bai, Han and Shi (2017) and Baltagi, Kao and Wang (2017)
focus on estimation under the single break assumption. Baltagi, Kao and Wang (2016),
Ma and Su (2016) and Barigozzi, Cho and Fryzlewicz (2018) allow for multiple breaks.
Massacci (2017) studies large dimensional threshold factor models with regime shifts in
the loadings driven by a covariate. These contributions do not consider the whole diffusion
index forecasting model. Corradi and Swanson (2014) propose a test for the joint null
hypothesis of stable factor model and factor-augmented regression. Wang, Cui and Li
(2015) estimate unstable factor augmented regressions with factors extracted from a linear
model. To the very best of our knowledge, estimation and inference in the set up with
instabilities in the factor model and in the factor augmented regression has not been studied:
we aim at filling this gap.
We start from the single break factor model. Let Nand Tdenote the cross-sectional
and time series dimensions respectively. We estimate the model by least squares by mini-
mizing the sum of squared residuals (Baltagi et al., 2017; Massacci, 2017): the resulting
principal components estimator for factors and loadings has the same convergence rate
CNT =min{√N,√T}as in the linear case (Bai and Ng, 2002). We then turn to the sin-
gle break factor augmented regression. We estimate the parameters by least squares by
replacing the latent factors with their estimates (Bai, 1997; Bai and Ng, 2006): despite the
structural instability, the least squares estimator for the slope coefficients is √Tconsistent
and asymptotically normal provided that √T/N →0asN,T→∞. We then propose a
Lagrange multiplier test for the null hypothesis of stability: we showthat the critical values
provided in Andrews (1993) remain valid also in the presence of latent factors estimated
from an unstable large dimensional model.
Finally,we apply our methodology to bond risk premia (Famaand Bliss, 1987; Cochrane
and Piazzesi, 2005; Ludvigson and Ng, 2009). Common factors extracted from a large set
of macroeconomic series help predicting bond excess returns (Ludvigson and Ng, 2009);
however, the loadings are not stable over time (Breitung and Eickmeier, 2011; Chen et al.,
2014; Cheng et al., 2016). Pricing equations for bond risk premia experience structural
breaks (Smith and Taylor, 2009; Bikbov and Chernov, 2010). We thus use our model
to uncover instabilities in the linkages between bond market risk premia and macroeco-
nomic fundamentals. Depending on the maturity of the bond, we show that: predictive
regressions for bond risk premia are stable for most of the Great Moderation (Joslin,
Priebsch and Singleton, 2014); a break occurred in the early 1980s (Smith and Taylor,
2009).
©2019 Bank of England. Oxford Bulletin of Economics and Statistics ©2019The Depar tment of Economics, Universityof Oxford and John Wiley & Sons Ltd.
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