Aggregate density forecasting from disaggregate components using Bayesian VARs

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There is a considerable volume of literature concerned with point forecasting which aims to assess whether producing aggregate forecasts as the sum of the components’ forecasts is better than alternative direct methods, whereas aggregate density forecasting from disaggregate components is still a relatively unexplored field. This paper develops an implementation of the bottom-up approach that is capable of producing well-performing and competitive density forecasts. This is achieved by accounting explicitly for the interaction between components, using Bayesian VARs to estimate the whole multivariate process and produce the aggregate forecasts. An empirical application using CPI and GDP data shows that the method can be used to produce aggregate density forecasts capable of accounting for the events resulting from the crisis. This suggests that it might be particularly useful for forecasting in turbulent times and therefore prove a valuable addition to the forecaster’s toolkit.

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Fig. 1

Source: OECD statistics database

Fig. 2
Fig. 3


  1. 1.

    Examples of these comparisons are Espasa et al. (2002), Benalal et al. (2004), Hubrich (2005) and Giannone et al. (2014) for inflation in the Euro area; Marcellino et al. (2003), Hahn and Skudelny (2008), Burriel (2012) and Esteves (2013) for European GDP growth; and Zellner and Tobias (2000), Perevalov and Maier (2010) and Drechsel and Scheufele (2013) for GDP growth in specific industrialized countries.

  2. 2.

    Actual weights is used to refer to the weights that are actually used to produce the corresponding aggregate by the institution that publishes it.

  3. 3.

    They use univariate autoregressive models to forecast the components.

  4. 4.

    For forecasts beyond one period, predictive simulation is required.

  5. 5.

    Corradi and Swanson (2006) and Rossi and Sekhposyan (2019) present a survey of recent developments.

  6. 6.

    The transformation involves taking the inverse normal cumulative density function transformation of the PITs making it equivalent to a test for uniformity.

  7. 7.

    Scores will suffer both for over-dispersed distributions, because probability mass is too high for infrequent values, and for under-dispersed distributions, because too many observations carry a low probability.

  8. 8.

    It is worth noting that the exercise does not replicate real-time forecasting, given that data revisions are not accounted for.

  9. 9.

    For the UK the production data on the OECD database starts in 1995. The first four years of the sample are obtained by splicing backwards the historical reference tables available from the Office for National Statistics. No inconsistencies arise from the seasonal adjustment given that the aggregates are adjusted indirectly; that is, as the sum of the seasonally adjusted components.

  10. 10.

    Four lags are used for all models.

  11. 11.

    K&K also choose \(\lambda \) and \(\kappa \) empirically over a grid. In this particular exercise, the results are not significantly different from those obtained from setting both parameters to 0.99.

  12. 12.

    Specifically, \(\gamma =e^{i}\) selecting i from \(\{-7, -6, \ldots , -1\}\).

  13. 13.

    That is that the null hypothesis of no calibration failure cannot be rejected at the 5% significance level. The tests are conducted on an individual basis which imply a Bonferroni-corrected (joint) p value of 1.25%.

  14. 14.

    For each series, the homoskedastic models are presented in the top panel and the heteroskedastic models and DMS in the bottom. The aggregate AR is included in both to serve as a reference point.

  15. 15.

    The figures for the fixed-parameter models have been dropped for clarity and because the main findings are present in their time-varying counterparts.

  16. 16.

    As Raftery et al. (2010) point out, using forgetting factors is comparable, but not equivalent, to using rolling windows. The dampening in the former is exponential, while in the latter it does not have to be. In fact, the usual practice is to give equal nonzero weights to the information within the window and zero to the rest.


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I would like to thank two anonymous referees and Andrea Carriero for their very insightful suggestions at different stages of this research. I would also like to thank Antonia Palacios, Linda Craddock and Oliver Cobb for their helpful comments and encouragement. The views expressed in this paper are those of the author and do not necessarily represent those of anyone at the Central Bank of Chile.

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Correspondence to Marcus P. A. Cobb.

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Cobb, M.P.A. Aggregate density forecasting from disaggregate components using Bayesian VARs. Empir Econ 58, 287–312 (2020).

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  • Bottom-up density forecasting
  • Density forecast combination

JEL Classification

  • C32
  • C53
  • E27
  • E37