如何在VAR模型中进行联合贝叶斯推断？

How to conduct joint Bayesianinference in VAR models? Andrian Yambolov Disclaimer:This paper should not be reported as representing the views of the European Central Bank(ECB). The views expressed are those of the authors and do not necessarily reflect those of the ECB. Abstract When economic analysis requires simultaneous inference across multiple variablesand time horizons, this paper shows that conventional pointwise quantiles in Bayesianstructural vector autoregressions significantly understate the uncertainty of impulseresponses. The performance of recently proposed joint inference methods, whichproduce noticeably different error band estimates, is evaluated, and calibrationroutines are suggested to ensure that they achieve the intended nominal probabilitycoverage. Two practical applications illustrate the implications of these findings: (i)within a structural vector autoregression, the fiscal multiplier exhibits error bandsthat are 51% to 91% wider than previous estimates, and (ii) a pseudo-out-of-sampleprojection exercise for inflation and gross domestic product shows that joint inferencemethods could effectively summarize uncertainty for forecasts as well.Theseresults underscore the importance of using joint inference methods for more robusteconometric analysis. Keywords:vector autoregressions, impulse responses, forecasts, pointwise inference, joint inferenceJEL Codes:C22, C32, C52 Non-technical summary Standard methods for constructing error bands around impulse response functions considerthem in isolation, neglecting the estimation uncertainty that arises across variables and timehorizons due to the joint nature of the underlying structural parameters. For example, oneapproach to assessing the impact of government expenditure on economic activity—known asthe fiscal multiplier—is to estimate the ratio between the cumulative responses of gross domesticproduct and government expenditure. By convention, practitioners use marginal error bands toaddress this and similar economic questions, which leads to an underestimation of uncertainty. This paper quantifies the extent to which conventional error bands understate uncertainty andfocuses on methods for conducting joint inference to support more robust economic analysis.It conducts a series of simulation experiments using widely adopted Bayesian vector autore-gressions to evaluate the performance of several estimators, discussed by Montiel Olea andPlagborg-Møller (2019) and Inoue and Kilian (2022), that account for the joint uncertainty ofimpulse response functions. In addition, it proposes strategies to improve upon those jointinference methods, which tend to overstate the uncertainty. The policy relevance of the paper is illustrated through two examples. The first draws on awidely used fiscal vector autoregression estimated with U.S. data. It shows that taking intoaccount the joint uncertainty between gross domestic product and government expenditurecan uncover stronger and earlier effects of fiscal policy shocks than conventional approachessuggest. In particular, the results indicate that the fiscal multiplier could exceed one as early asfour quarters before standard estimates would imply. This highlights the potential benefits oftimely and proactive fiscal interventions—especially during periods of economic stress whenswift action is essential. Moreover, as Blanchard and Leigh (2013) emphasize, misestimating thefiscal multiplier can result in substantial forecast errors, particularly during periods of sharpchanges in government expenditure. The second example demonstrates how joint inference methods can be employed to assess theprobability that both the growth rate of gross domestic product and inflation remain withinspecified bounds over the projection horizon. This is especially important for central banks witha dual mandate, but more broadly also for any forecasting exercise which strives to properlymeasure and report economic risks. In practice, the uncertainty around economic projectionsis often adjusted at the discretion of forecasters or policy makers to account for risks thatmacroeconometric models may not fully capture. For instance, if uncertainty around the growthrate of gross domestic product seems to be too low or unbalanced, its error bands may bewidened or skewed. Since economic activity and inflation are linked, the uncertainty aboutthe latter must also be adjusted—something that is often overlooked (see Bernanke, 2024). Themethods developed in this paper provide a practical way to capture this type of estimationuncertainty and improve the reliability of macroeconomic projections. 1Introduction The impulse response function (IRF) is the most widely used tool for analyzing economicquestions within a structural vector autoregression (VAR) framework. Regardless of the estima-tion methodology, whether frequentist or Bayesian, practitioners typically construct pointwise(individual) bands1around the impulse response coefficient