A structural Bayesian VAR for model-based fan charts

Abstract

Inflation forecast uncertainty is of importance for a wide range of agents in the economy, central banks in particular. Ways to describe and account for this uncertainty in a consistent manner have received increasing attention of late, in part due to the growing number of inflation-targeting central banks. This article develops a large structural VAR for the Swedish economy and estimates it in a Bayesian framework. The methodology permits not only structural interpretation and analysis but offers a natural way to formalize forecast uncertainty, as the posterior predictive density from the model has the interpretation of a fan chart.

Acknowledgements

I am grateful to Meredith Beechey, Per Jansson, Eric Leeper, Lars Svensson, Mattias Villani, Anders Vredin, seminar participants at Sveriges Riksbank and an anonymous referee for valuable comments and to Mattias Villani for providing Matlab code. A special thanks to Jack Lucchetti for clarifying instructions regarding the issue of identification. Financial support from Jan Wallander's and Tom Hedelius’ foundation is gratefully acknowledged. The views expressed in this article are solely the responsibility of the author and should not be interpreted as reflecting the views of the executive board of Sveriges Riksbank.

Notes

¹ Studies using K-models include Bernanke (1986), Kim and Roubini (2000), Camarero et al. (2002) and Villani and Warne (2003).

² Foreign GDP and CPI have been trade weighted according to the TCW index. The foreign interest rate has been weighted using a subset of the countries included in the TCW index due to missing data for some countries.

³ We are also well aware of the low power of Dickey–Fuller type tests when the investigated series have roots close to–but less than–unity or when they are subject to structural breaks; see e.g. Froot and Rogoff (1995) and Perron (1989).

⁴ Not allowing for time-varying parameters in the model implies–apart from the fact that the model could be mis-specified–that one potential source of uncertainty has been left out. This could clearly affect the fan charts from the model. As pointed out by Cogley et al. (2005) though, constant parameters are a reasonable assumption from a forecasting point of view as long as the forecasting horizon is short–the reason for this being that parameter drift typically is very small. Support for usage of constant parameters can also be found in Doan et al. (1984) who argue that the improvement in forecast performance from using time-varying parameters typically is very small.

⁵ Note that both the foreign central bank and the Riksbank are assumed unable to react to innovations in GDP contemporaneously. This is justified by the substantial publication lag in GDP numbers; see for example Leeper et al. (1996).

⁶ Most of the shocks are standard and straight forward to interpret. Regarding the shocks that could be less obvious to interpretate, the aggregate supply shock corresponds to changes in productivity and/or labour supply, the wage setting shock could for instance reflect changes in bargaining power on the labour market and the price setting shock can correspond to higher input prices or changes in mark-ups.

⁷ This might seem like an overly simplistic way to model the regime shift in Swedish monetary policy which occurred in the early 1990s. However, it appears to have worked well when used previously in the literature; see e.g. Jacobson et al. (2001) and Villani and Warne (2003). Moreover, since several of the countries that have large weights in the TCW index also experienced policy changes around the same period–for example Germany, Norway and the United Kingdom–we also let the dummy variable affect the foreign economy. Changing this assumption, thereby letting the dummy affect the Swedish economy alone, has negligible effects on the results.

⁸ See e.g. Litterman (1986) and Robertson and Tallman (2001).

⁹ See e.g. Villani and Warne (2003).

¹⁰ As the model is invariant to sign switches in the equations in G ₀–see for example Sims and Zha (1999)–we must also employ some kind of normalization to the system. In this article we employ the Waggoner and Zha (2003) normalization which has been shown to have good properties both theoretically and empirically.

¹¹ Results not reported but available upon request.

¹²  is the growth rate in Swedish GDP over the last four quarters. As the VAR is specified in first and not fourth differences for Swedish GDP, the forecasts are therefore first summed up to fourth differences and then evaluated.

¹³ The fact that Sveriges Riksbank also has to use real-time data–whereas the VAR relies on ex post data in the out-of-sample forecast exercise–is a related issue. For a discussion regarding issues on real-time data, see e.g. Croshoure and Stark (2002) and Orphanides and van Norden (2002).

¹⁴ For a discussion on this issue, see e.g. Lawrence et al. (1986) and McNees (1990) and Svensson (2005).

¹⁵ For the model in Equation 5 the unconditional means for the periods 1980Q2 to 1992Q4 and 1993Q1 to 2004Q4 are given by α + Ξ and α, respectively.

¹⁶ Persistence estimates for the different time series are given in Table A2 in Appendix B. Note though that is more persistent than Δy _t due to the overlapping nature of the former series.

¹⁷ Note that for variables specified in first differences, the numbers in Fig. 3 all refer to growth rates over the last four quarters. As was the case for Swedish GDP in the out-of-sample forecast evaluation above, the forecasts are therefore summed up to fourth differences.

¹⁸ See e.g. Diebold et al. (1998) and Diebold et al. (1999) for a discussion on how to evaluate density forecasts.

¹⁹ See e.g. Leeper and Zha (2003).