Modeling and Forecasting Macroeconomic Downside Risk

Abstract

We model permanent and transitory changes of the predictive density of U.S. GDP growth. A substantial increase in downside risk to U.S. economic growth emerges over the last 30 years, associated with the long-run growth slowdown started in the early 2000s. Conditional skewness moves procyclically, implying negatively skewed predictive densities ahead and during recessions, often anticipated by deteriorating financial conditions. Conversely, positively skewed distributions characterize expansions. The modeling framework ensures robustness to tail events, allows for both dense or sparse predictor designs, and delivers competitive out-of-sample (point, density and tail) forecasts, improving upon standard benchmarks.

Keywords:

• Business cycle
• Downside risk
• Financial conditions
• Score driven
• Skewness

Supplementary Materials

The Supplemental Material reports additional results discussed throughout the paper. Appendix A provides further details on the testing procedure for time-varying conditional asymmetry, as discussed in Section 2. Appendix B offers additional details and derivations of the econometric model. Appendix C highlights the relevance of modeling time-varying asymmetry in the conditional distribution and contrasts our model with the one in Plagborg-Møller et al. (2020). Appendix D provides details on the estimation procedure. Appendix E reports the results of the Monte Carlo experiment, summarized in Section 3, and compares our model to existing ad-hoc approaches to time-varying asymmetry. Appendix F provides details about the data. Appendix G discusses in-sample model fit. Appendix H presents additional detailed results about the forecasting exercise described in Section 5. Appendix I offers additional results about the Sparse model in Section 6. Appendix J collects additional tables and figures.

Acknowledgments

This article is extracted from the first chapter of the Ph.D. thesis of Andrea De Polis at the University of Warwick. We thank three referees, the editor and associate editor, Jesús Fernández-Villaverde for the thoughtful discussion and Domenico Giannone and Elmar Mertens for extensive comments on an earlier draft of the article. We also thank Anastasia Allayioti, Scott Brave, Christian Brownlees, Andrew Butters, Andrea Carriero, Todd Clark, Gabriele Fiorentini, Ana Galvao, Francesco Saverio Gaudio, Gary Koop, André Lucas, Massimiliano Marcellino, Leonardo Melosi, James Mitchell, Mikkel Plagborg-Møller, Bernd Schwaab, Andreas Tryphonides, Fabrizio Venditti, Mark Watson and the participants at various conferences and seminars where the article was presented for valuable feedback and suggestions. We are grateful to the Chicago FED for making the full panel of weighted contribution of the financial indicators available for this work. The views expressed in this manuscript are those of the authors and do not necessarily reflect the views of the Bank of Italy. Any errors and omissions are the sole responsibility of the authors.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 Huber, Koop, and Onorante (2021) note that in this setting sparsity is not an artifact of strong a priori beliefs.

2 Using the Bai and Ng (2005) test over different rolling windows, we often reject the null of symmetry, with significant negative and positive skewness detected over the sample. See Figure A.1 in Appendix A.

3 Differently from the Skew-t distribution of Azzalini and Capitanio (2003), the one of Gómez, Torres, and Bolfarine (2007) has an Information matrix which is always nonsingular, provided $|{\varrho }_t|<1$. For practical purposes, we set ${\varrho }_t=c\tanh ({\delta }_t)$, with c being a constant close but below 1, to ensure ${\varrho }_t\in (-1,1)$. This results in a small change in the Jacobian of the transformation, and is omitted to simplify the exposition.

4 In Section 3.1 we allow the time-varying parameters to feature a permanent and a transitory component.

5 Scaling the gradient by the diagonal of the Information matrix ensures that negative (positive) prediction errors translate into negative (positive) updates of the conditional location and shape, and therefore of the conditional mean. This desirable property is not always guaranteed by the full information matrix.

6 Tapia (1977) and Byrd (1978) show the “superlinear local convergence” property of Newton’s method and its diagonalization for standard optimization problems (see also Dennis and Schnabel 1996, chap. 6).

7 We choose a prior for the learning rate that limits its size, so that it (a) reduces the possibility of overshooting in the direction of the (local) optimum, and (b) assumes conservative views on parameters time variation. See Section 3.2.

8 In addition, our framework allows for non-linearities through the mapping of the predictors into the scores, further down-weighting extreme fluctuations in the data. In Appendix C we highlight that these additional features are important to recover salient features of the distribution of GDP growth.

9 Importantly, Bayesian estimators are not affected by local discontinuities, multiple local minima and flat areas of the likelihood, and they are often much easier to compute, particularly in high-dimensional settings (see, e.g., Tian, Liu, and Wei 2007; Belloni and Chernozhukov 2009).

10 Introducing priors for the coefficients governing the learning rate effectively circumvents the “pile-up” problem, often arising when time-varying parameters feature little variation (see, e.g., Stock and Watson 1998). At the same time these priors are quite conservative, implying that any evidence in favor of the time variation of permanent components reflects strong evidence in the data.

11 Intuitively, the invertibility property ensures that the effect of the initialization vanishes asymptotically and that the filter converges to a unique limit process. We derive the elements of $\frac{\partial s(f,y_t,\theta )}{\partial f{\prime }_t}$ in Appendix B.7.

12 This comes closer to the idea of defining the estimator as the maximand of the likelihood within the invertibility region (see Blasques et al. 2018, sec. 4.2).

13 This is akin to assuming a random walk specification for their law of motion. As an alternative, one could feed predictions for the explanatory variables into the model. The latter approach produces results very similar to the ones reported here.

14 The moment skewness can be computed numerically as $\mathbb{S}\mathit{kew}(y_t|\theta ,{\mathcal{Y}}^{t-1})=\frac{{\int }_{\mathbb{R}}{(y_t-\mathbb{E}[y_t|\theta ,{\mathcal{Y}}^{t-1}])}^{3}p(y_t|\theta ,{\mathcal{Y}}^{t-1})dy}{\mathbb{V}{(y_t|\theta ,{\mathcal{Y}}^{t-1})}^{\frac{3}{2}}},$ where $p(y_t|\theta ,{\mathcal{Y}}^{t-1})$ denotes the conditional density of the Skew-t distribution at time t, and the (conditional) mean and variance are computed as in (10) and (11).

15 Recessions are considered as 3 quarters before and after NBER recession quarters. See Appendix F.

16 Specifically, quantile scores are weighted by ${(1-\alpha )}^2$, where α represent the quantile.

17 For comparability, we follow exactly the procedure of Adrian, Boyarchenko, and Giannone (2019), but re-estimating the model using real-time vintages of GDP growth.

18 In Q1 of 2009, the Skt -4DFI model predicts a negative mean and substantial downside risk, whereas the Gaussian model only predicts a slightly negative growth, with a roughly symmetric assessment of the risk surrounding this prediction; see Figure J2, in Appendix J.

19 As these data are not available in real-time, we assume that at time t the set of predictors corresponds to the quarterly average of the financial indicators from the third week of the previous quarter to the second week of the current quarter. This approach mimics the information set available to the econometrician in real-time, and avoids dealing with overlapping quarters. Once a new indicator enters the model, missing observations are set to 0, while the Euclidean norm required for the sparsification step is computed on the available data (appropriately rescaled to reflect data availability).