- (1989): “Business conditions and expected returns on stocks and bonds,” Journal of Financial Economics, 25(1), 23–49.
Paper not yet in RePEc: Add citation now
- (2015). It serves as a variational inference scheme for learning the posterior distribution on the weights of a neural network. To do so, the approach maximizes the log-likelihood of the model subject to a Kullback-Leibler complexity term on the parameters and makes use of the reparameterization trick (Kingma and Welling, 2013) to obtain the posterior distribution of the weights with stochastic gradient descent.
Paper not yet in RePEc: Add citation now
- Agostinelli, F., M. Hoffman, P. Sadowski, and P. Baldi (2014): “Learning activation functions to improve deep neural networks,” arXiv preprint arXiv:1412.6830.
Paper not yet in RePEc: Add citation now
Ai, C., and E. C. Norton (2003): “Interaction terms in logit and probit models,” Economics Letters, 80(1), 123–129.
Amisano, G., and G. Fagan (2013): “Money growth and inflation: A regime switching approach,” Journal of International Money and Finance, 33, 118–145.
Barro, R. J., and J.-W. Lee (1994): “Sources of economic growth,” Carnegie-Rochester Conference Series on Public Policy, 40, 1–46.
- Bartlett, P. L., P. M. Long, G. Lugosi, and A. Tsigler (2020): “Benign overfitting in linear regression,” Proceedings of the National Academy of Sciences, 117(48), 30063–30070.
Paper not yet in RePEc: Add citation now
Beckmann, J., G. Koop, D. Korobilis, and R. A. Schüssler (2020): “Exchange rate predictability and dynamic Bayesian learning,” Journal of Applied Econometrics, 35(4), 410–421.
Belmonte, M. A., G. Koop, and D. Korobilis (2014): “Hierarchical shrinkage in timevarying parameter models,” Journal of Forecasting, 33(1), 80–94.
Bhattacharya, A., and D. B. Dunson (2011): “Sparse Bayesian infinite factor models,” Biometrika, 98(2), 291–306.
Bhattacharya, A., D. Pati, N. S. Pillai, and D. B. Dunson (2015): “Dirichlet–Laplace priors for optimal shrinkage,” Journal of the American Statistical Association, 110(512), 1479–1490.
- Blundell, C., J. Cornebise, K. Kavukcuoglu, and D. Wierstra (2015): “Weight uncertainty in neural network,” in International Conference on Machine Learning, pp. 1613–1622. PMLR.
Paper not yet in RePEc: Add citation now
Campbell, J. Y. (1987): “Stock returns and the term structure,” Journal of Financial Economics, 18(2), 373–399.
Campbell, J. Y., and R. J. Shiller (1988): “The dividend-price ratio and expectations of future dividends and discount factors,” The Review of Financial Studies, 1(3), 195–228.
Campbell, J. Y., and T. Vuolteenaho (2004): “Inflation illusion and stock prices,” American Economic Review, 94(2), 19–23.
Carriero, A., Y. Bai, T. Clark, and M. Marcellino (2022): “Macroeconomic Forecasting in a Multi-country Context,” Journal of Applied Econometrics, (forthcoming).
Carvalho, C. M., N. G. Polson, and J. G. Scott (2010): “The horseshoe estimator for sparse signals,” Biometrika, 97(2), 465–480.
- Chipman, H. A., E. I. George, and R. E. McCulloch (2010): “BART: Bayesian additive regression trees,” The Annals of Applied Statistics, 4(1), 266–298.
Paper not yet in RePEc: Add citation now
- Coulombe, P. G. (2020): “The macroeconomy as a random forest,” arXiv preprint arXiv:2006.12724.
Paper not yet in RePEc: Add citation now
- Crawford, L., S. R. Flaxman, D. E. Runcie, and M. West (2019): “Variable prioritization in nonlinear black box methods: A genetic association case study,” The Annals of Applied Statistics, 13(2), 958.
Paper not yet in RePEc: Add citation now
- Cui, T., A. Havulinna, P. Marttinen, and S. Kaski (2021): “Informative Bayesian Neural Network Priors for Weak Signals,” Bayesian Analysis, 1(1), 1–31.
Paper not yet in RePEc: Add citation now
Diebold, F. X., and R. S. Mariano (1995): “Comparing Predictive Accuracy,” Journal of Business & Economic Statistics, 13(3), 253–263.
- Dusenberry, M., G. Jerfel, Y. Wen, Y. Ma, J. Snoek, K. Heller, B. Lakshminarayanan, and D. Tran (2020): “Efficient and scalable bayesian neural nets with rank-1 factors,” in International Conference on Machine Learning, pp. 2782–2792. PMLR.
Paper not yet in RePEc: Add citation now
- Engle, R. F., and J. G. Rangel (2008): “The spline-GARCH model for low-frequency volatility and its global macroeconomic causes,” The Review of Financial Studies, 21(3), 1187–1222.
Paper not yet in RePEc: Add citation now
- Escobar, M. D., and M. West (1995): “Bayesian density estimation and inference using mixtures,” Journal of the American Statistical Association, 90(430), 577–588.
Paper not yet in RePEc: Add citation now
Fama, E. F., and G. W. Schwert (1977): “Asset returns and inflation,” Journal of Financial Economics, 5(2), 115–146.
Fama, E. F., and K. R. French (1988): “Dividend yields and expected stock returns,” Journal of Financial Economics, 22(1), 3–25.
- Gal, Y., and Z. Ghahramani (2016): “Dropout as a bayesian approximation: Representing model uncertainty in deep learning,” in International Conference on Machine Learning, pp. 1050–1059. PMLR.
Paper not yet in RePEc: Add citation now
Gallegati, M. (2008): “Wavelet analysis of stock returns and aggregate economic activity,” Computational Statistics & Data Analysis, 52(6), 3061–3074.
- George, E. I., and R. E. McCulloch (1993): “Variable selection via Gibbs sampling,” Journal of the American Statistical Association, 88(423), 881–889.
Paper not yet in RePEc: Add citation now
George, E. I., D. Sun, and S. Ni (2008): “Bayesian stochastic search for VAR model restrictions,” Journal of Econometrics, 142(1), 553–580.
- Ghosh, S., J. Yao, and F. Doshi-Velez (2019): “Model Selection in Bayesian Neural Networks via Horseshoe Priors.,” Journal of Machine Learning Research, 20(182), 1–46.
Paper not yet in RePEc: Add citation now
Giacomini, R., and B. Rossi (2010): “Forecast comparisons in unstable environments,” Journal of Applied Econometrics, 25(4), 595–620.
Greene, W. (2010): “Testing hypotheses about interaction terms in nonlinear models,” Economics Letters, 107(2), 291–296.
- Griffin, J. E., and P. J. Brown (2013): “Some priors for sparse regression modelling,” Bayesian Analysis, 8(3), 691–702.
Paper not yet in RePEc: Add citation now
Hamilton, J. D. (1989): “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle,” Econometrica, 57(2), 357–384.
- Harding, M., J. Linde, and M. Trabandt (2022): “Understanding Post-Covid Inflation Dynamics,” IWH Working Paper.
Paper not yet in RePEc: Add citation now
- Hauzenberger, N., F. Huber, M. Marcellino, and N. Petz (2021): “Gaussian process vector autoregressions and macroeconomic uncertainty,” arXiv preprint arXiv:2112.01995.
Paper not yet in RePEc: Add citation now
Hodrick, R. J. (1992): “Dividend yields and expected stock returns: Alternative procedures for inference and measurement,” The Review of Financial Studies, 5(3), 357–386.
- Hornik, K. (1991): “Approximation capabilities of multilayer feedforward networks,” Neural Networks, 4(2), 251–257.
Paper not yet in RePEc: Add citation now
- Hornik, K., M. Stinchcombe, and H. White (1989): “Multilayer feedforward networks are universal approximators,” Neural Networks, 2(5), 359–366.
Paper not yet in RePEc: Add citation now
Huber, F., and M. Pfarrhofer (2021): “Dynamic shrinkage in time-varying parameter stochastic volatility in mean models,” Journal of Applied Econometrics, 36(2), 262–270.
- Huber, F., and T. O. Zörner (2019): “Threshold cointegration in international exchange rates: A Bayesian approach,” International Journal of Forecasting, 35(2), 458–473.
Paper not yet in RePEc: Add citation now
Huber, F., G. Koop, and L. Onorante (2021): “Inducing Sparsity and Shrinkage in TimeVarying Parameter Models,” Journal of Business & Economic Statistics, 39(3), 669–683.
Huber, F., G. Koop, L. Onorante, M. Pfarrhofer, and J. Schreiner (2020): “Nowcasting in a pandemic using non-parametric mixed frequency VARs,” Journal of Econometrics, (forthcoming).
Imbens, G. W., and J. M. Wooldridge (2009): “Recent developments in the econometrics of program evaluation,” Journal of Economic Literature, 47(1), 5–86.
- Johndrow, J., P. Orenstein, and A. Bhattacharya (2020): “Scalable approximate MCMC algorithms for the horseshoe prior,” Journal of Machine Learning Research, 21(73), 1–61.
Paper not yet in RePEc: Add citation now
- Karlik, B., and A. V. Olgac (2011): “Performance analysis of various activation functions in generalized MLP architectures of neural networks,” International Journal of Artificial Intelligence and Expert Systems, 1(4), 111–122.
Paper not yet in RePEc: Add citation now
Kastner, G., and S. Frühwirth-Schnatter (2014): “Ancillarity-sufficiency interweaving strategy (ASIS) for boosting MCMC estimation of stochastic volatility models,” Computational Statistics & Data Analysis, 76, 408–423.
- Kingma, D. P., and M. Welling (2013): “Auto-encoding variational bayes,” arXiv preprint arXiv:1312.6114.
Paper not yet in RePEc: Add citation now
Kowal, D. R., D. S. Matteson, and D. Ruppert (2019): “Dynamic shrinkage processes,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 81(4), 781–804.
Lee, D. S., and T. Lemieux (2010): “Regression discontinuity designs in economics,” Journal of Economic Literature, 48(2), 281–355.
Lewellen, J. (2004): “Predicting returns with financial ratios,” Journal of Financial Economics, 74(2), 209–235.
- MacKay, D. J. (1992): “A practical Bayesian framework for backpropagation networks,” Neural Computation, 4(3), 448–472.
Paper not yet in RePEc: Add citation now
- Makalic, E., and D. F. Schmidt (2015): “A simple sampler for the horseshoe estimator,” IEEE Signal Processing Letters, 23(1), 179–182.
Paper not yet in RePEc: Add citation now
Makridakis S., Spiliotis E., A. V. (2018): “Statistical and machine learning forecasting methods: Concerns and ways forward,” PLoS One, 13.
Marcellino, M., J. H. Stock, and M. W. Watson (2006): “A comparison of direct and iterated multistep AR methods for forecasting macroeconomic time series,” Journal of Econometrics, 135(1-2), 499–526.
McCracken, M. W., and S. Ng (2016): “FRED-MD: A monthly database for macroeconomic research,” Journal of Business & Economic Statistics, 34(4), 574–589.
McCrary, J. (2008): “Manipulation of the running variable in the regression discontinuity design: A density test,” Journal of Econometrics, 142(2), 698–714.
- Neal, R. M. (1996): “Priors for infinite networks,” in Bayesian Learning for Neural Networks, pp. 29–53. Springer.
Paper not yet in RePEc: Add citation now
- Raftery, A. E., and S. Lewis (1992): “How many iterations in the Gibbs sampler?,” in Bayesian Statistics, ed. by J. Bernardo, J. Berger, A. Dawid, and A. Smith, vol. 4, pp. 763–773, Oxford, UK. Oxford University Press.
Paper not yet in RePEc: Add citation now
Ramsey, J. B., and C. Lampart (1998): “The decomposition of economic relationships by time scale using wavelets: expenditure and income,” Studies in Nonlinear Dynamics & Econometrics, 3(1).
- Reichlin, L., and M. Lenza (2007): “On short-term and long-term causality of money to inflation: understanding the problem and clarifying some conceptual issues,” Discussion paper, Mimeo.
Paper not yet in RePEc: Add citation now
- Roberts, G. O., and J. S. Rosenthal (2009): “Examples of adaptive MCMC,” Journal of Computational and Graphical Statistics, 18(2), 349–367.
Paper not yet in RePEc: Add citation now
Rossi, B. (2013): “Exchange rate predictability,” Journal of Economic Literature, 51(4), 1063– 1119.
- Scardapane, S., D. Comminiello, A. Hussain, and A. Uncini (2017): “Group sparse regularization for deep neural networks,” Neurocomputing, 241, 81–89.
Paper not yet in RePEc: Add citation now
- Sezer O., Ozbayoglu M., D. E. (2020): “Financial time series forecasting with deep learning: A systematic literature review: 2005–2019,” arXiv preprint arxiv:abs/1911.13288.
Paper not yet in RePEc: Add citation now
Stock, J., and M. Watson (1999): “Forecasting inflation,” Journal of Monetary Economics, 44(2), 293–335.
- Teräsvirta, T. (1994): “Specification, estimation, and evaluation of smooth transition autoregressive models,” Journal of the American Statistical Association, 89(425), 208–218.
Paper not yet in RePEc: Add citation now
- The hyperparameters are chosen in a cross validation exercise. For the cross-sectional datasets (i.e., Macro B and synthetic) we randomly split the data into equally sized training and test sets. We evaluate each model specification in 20 replications and use those yielding the lowest average RMSE for the final model. For the time series applications (i.e., Macro A, Macro C and Finance) we use a cross validation based on an expanding window time series split. Specifically, we use all observations up to the last 24 months for Macro A, 12 quarters for Macro C and 10 years for Finance before the start of our hold-out to train the model and then, after obtaining the predictive densities, add the next observation and recompute the model. We repeat this until we end up at the beginning of our hold-out and choose the specification with the lowest average RMSE. We train all models in 1000 epochs and use the MSE loss function, the ADAM optimizer and a learning rate of 0.01.
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- The prior on the weights is specified as a scale mixture of two Gaussian densities with zero mean but differing variances. The first mixture component features a large variance (σ2 1 = 3) providing a heavy-tailed distribution whereas the variance of the other component is set small (σ2 2 = 0.0025) concentrating the weights a priori around zero. This setup is similar to a spike and slab prior (see, George and McCulloch, 1993) but with the same prior parameters for all the weights to allow for the optimization by stochastic gradient descent.
Paper not yet in RePEc: Add citation now
- Tong, H. (1990): Non-linear time series: a dynamical system approach. Oxford University Press.
Paper not yet in RePEc: Add citation now
Vasicek, O. A., and H. G. Fong (1982): “Term structure modeling using exponential splines,” The Journal of Finance, 37(2), 339–348.
Welch, I., and A. Goyal (2008): “A comprehensive look at the empirical performance of equity premium prediction,” The Review of Financial Studies, 21(4), 1455–1508.
- Williams, C. K., and C. E. Rasmussen (2006): Gaussian processes for machine learning, vol. 2. MIT Press Cambridge, MA.
Paper not yet in RePEc: Add citation now
- X z=1 gz(xt|Tz, ρz) (A.16) A single regression tree gz depends on two parameters, the tree structure given by Tz and the terminal node parameter ρz. Following Chipman et al. (2010), we set Z = 250 and build the prior on the tree structure upon a tree-generating stochastic process. This involves determining the probability that a given node is nonterminal, the selection of variables used in a splitting rule (to spawn left and right children nodes) and the corresponding thresholds. For the terminal node parameter we specify a conjugate Gaussian prior distribution with data-based prior variance. In particular, the specification centers prior mass on the range of the data while ensuring a higher degree of shrinkage if the number of trees is large. Details can be found in Chipman et al. (2010). B Empirical appendix B.1 Details on the data
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- θ = V θx̃0 Σ−1 y. • The prior on γ is Normal of the form: γj ∼ N(0, φ−1 γj ), φ−1 γj = λ2 γϕ2 γj , for j = 1, . . . , K. (A.2) We use a horseshoe prior and rely on the hierarchical representation of Makalic and Schmidt (2015). The global and local shrinkage parameters, λ2 γ and ϕ2 γj , respectively, are obtained by introducing auxiliary random quantities which follow an inverse Gamma distribution: ϕ2 γj |• ∼ G−1 1, c−1 γj + γ2 j 2λ2 γ ! , (A.3) λ2 γ|• ∼ G−1 K + 1 , d−1 γ + K X j=1 γ2 j 2ϕ2 γj , (A.4) cγj |• ∼ G−1 1, 1 + ϕ−2 γj , (A.5) dγ|• ∼ G−1 1, 1 + λ−2 γ . (A.6) • We sample the hyperparameters associated with the MGP prior on β from inverse Gamma distributions: δ1 ∼ G−1 a1 + Q , 1 + Q X q=1 (φβq β2 q ) , (A.7) δr ∼ G−1 a2 + Q − r − 1 , 1 + Q X q=1 (φβq β2 q )
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