Comparison of methods for constructing joint confidence bands for impulse response functions
Introduction
In vector autoregressive (VAR) analyses, impulse responses are commonly used for investigating the effects of shocks on the system. In practice, because the impulse responses are functions of the VAR parameters, they have to be estimated. Estimation uncertainty is usually indicated by showing confidence intervals around the individual impulse response coefficients. Asymptotic, bootstrap and Bayesian methods are typically used for setting up such intervals (see, e.g., Lütkepohl, 2005).
Despite this practice of reporting the estimation uncertainty for individual impulse response coefficients, economists are often interested in the response of a certain variable to a specific shock over a longer propagation horizon. For example, in a standard real business cycle (RBC) model, a technology shock is expected to increase hours worked in the long-run, that is, for a number of future periods (see, e.g., Galí, 1999, for an empirical investigation of this issue). Similarly, a contractionary monetary policy shock is expected to reduce the price level and bring down inflation (e.g., Uhlig, 2005). When responses over several periods are of interest, it is desirable to have confidence bands for impulse response functions rather than confidence intervals for individual impulse response coefficients.
If individual confidence intervals for a given confidence level are constructed around the impulse response coefficients for each response horizon separately, there is no guarantee that the overall coverage level for all impulse responses of one variable will correspond to the prespecified confidence level. In other words, the probability of the band containing the true impulse response function of a specific variable will generally not be if the confidence band is constructed as the union of individual confidence intervals. Hence, it is desirable to construct confidence bands with an overall prespecified coverage probability. A range of suitable methods are reviewed in this study and a new proposal is considered. A simulation experiment is used to compare the methods, and recommendations for applied work are given. Our criteria for assessing the bands are the coverage level and the width of the confidence band. While different measures of width are conceivable, in this study we calculate it as the sum of the widths of all individual intervals.
This is not the first study to consider the problem of constructing confidence bands for impulse responses. For example, Inoue and Kilian (2013) and Sims and Zha (1999) propose methods based on Bayesian principles. In this study, we will remain within a classical framework where one could use, for example, the Bonferroni inequality for constructing confidence bands with a joint coverage level at least as large as the desired one. The drawback of this method is that it may deliver very conservative bands that provide much larger coverages than desired, and consequently, are unnecessarily wide. Therefore, we propose a strategy for reducing the bands by adjusting the Bonferroni bands. Another proposal was made by Jordà (2009). He constructed the bands on the basis of so-called Scheffé bounds. Unfortunately, though, the underlying inequalities are only approximate, and may fail to deliver correct coverage levels even under ideal conditions, as was argued convincingly by Wolf and Wunderli (2012) in the context of constructing joint forecast bands. In the context of constructing confidence bands for impulse responses, the simulation evidence from Kilian and Kim (2009) points in the same direction. Yet another approach was proposed by Staszewska (2007), who used numerical search methods to find the smallest possible confidence bands for a given coverage level. The disadvantage of this is that it requires a rather substantial computational effort. Moreover, no general results are available to show that the desired coverage level is actually obtained at least asymptotically. All of these proposals will be compared in a simulation experiment.
In the present study, we consider bands for the impulse response functions of individual variables; that is, we consider confidence bands for the response of an individual variable to a specific shock. This approach is in line with the bands proposed and investigated in most of the related literature (e.g., Staszewska, 2007). In contrast, Inoue and Kilian (2013) point out that it may be appropriate to consider the full uncertainty in all impulse response functions jointly. Although some of the methods discussed below can be extended in that direction, we focus on bands for individual impulse response functions because they may be more relevant from a practical point of view.
Bands with given coverage levels are also of interest in computing forecast paths over a number of horizons. The construction of bands around path forecasts has been considered, for instance, by Jordà and Marcellino (2010), Staszewska-Bystrova (2011), Staszewska-Bystrova and Winker (2013) and Wolf and Wunderli (2012). Since impulse responses are conditional forecasts, there is an obvious relationship with the forecast literature, which we will draw on by adapting the method proposed by Wolf and Wunderli (2012) to our framework of constructing confidence bands around impulse responses. The difference between this and the literature on path forecasts is that there are two components of uncertainty attached to forecasts of specific variables, even if the data generation process (DGP) is known apart from its parameters: the intrinsic uncertainty from the DGP and the estimation uncertainty obtained from using estimated instead of true parameters. In contrast, since impulse responses are conditional forecasts that consider only the marginal effect of a specific shock for a given process, only the estimation uncertainty is relevant in the context of evaluating the impulse responses if the correct model is used. Of course, in practice, there is the usual uncertainty about the DGP in both types of analysis. In any case, our results are also of interest for constructing bands around multiple-horizon forecasts for specific variables, although we focus on the impulse response context.
The remainder of the study is organized as follows. In Section 2, the model setup is presented. Section 3 reviews the methods for constructing joint confidence bands for impulse responses, and a simulation comparison is discussed in Section 4. An illustrative example is given in Section 5, and Section 6 concludes. A number of technical details can be found in the Appendix.
Section snippets
Model setup
A standard reduced-form VAR setup is used, with the variables being generated by a -dimensional VAR process, The are parameter matrices, and the error process is a -dimensional zero mean white noise process with covariance matrix , that is, . The -dimensional intercept vector is the only deterministic term, because such terms are of limited relevance for the following arguments. Adding other
Methods for constructing confidence bands for impulse responses
In empirical studies, VAR parameters are usually estimated using standard methods, with estimates of the impulse responses obtained as (nonlinear) functions of these estimates. The estimated impulse responses are typically plotted with confidence bands. In most cases, these confidence bands are obtained by simply connecting confidence intervals for individual impulse responses. In other words, the joint distribution is ignored when setting up the confidence bands. Such confidence bands are
Monte Carlo comparison with other methods
For our small-sample comparison, we use the DGPs from Kilian (1998), with . This type of DGP is referred to as DGP1.
A second type of DGP (denoted by DGP2) is based on the empirical model described in Section 5. The DGP is a three-dimensional VAR(3), and the errors are assumed to follow a multivariate normal distribution with a covariance matrix as estimated on the basis of the data.
The experiments are
Illustrative example
In this section, the different methods for constructing confidence bands are applied in the context of a structural VAR model which was previously analyzed by Kilian (2009) when investigating the world crude oil market. The dataset is the same as that used by Kilian (2009), and consists of monthly observations for the period 1973:1–2007:12. In other words, we have a sample size of . The variables are the percentage change in global crude oil production, , an index of real economic
Conclusions
Impulse response functions are popular tools in structural vector autoregressive analysis. They are used to investigate the reactions of the variables of a VAR process to specific shocks. Typically, the propagation of a shock is traced over a number of periods, and it is desirable to construct confidence bands for these propagation paths for a prespecified confidence level. A number of proposals for constructing such bands in a classical setting are reviewed, and it is argued that they may not
Acknowledgments
We thank Michael Wolf and two anonymous referees for helpful comments on an earlier version of the paper. The research for this paper was conducted while the first author was Bundesbank Professor at the Freie Universität Berlin. An earlier version of the paper was presented at the DIW Seminar on Macroeconomics and Econometrics, ERCIM 2012, Oviedo, and the Workshop on Multivariate Time Series and Forecasting, Melbourne, February 2013. Support from Deutsche Forschungsgemeinschaft (DFG) through
Helmut Lütkepohl has been Bundesbank Professor at Freie Universität Berlin and Dean of the DIW Graduate Center, Berlin since 2012. He is and has been Associate Editor of a number of journals, such as Econometric Theory, Journal of Econometrics, the Journal of Applied Econometrics, Macroeconomic Dynamics, Empirical Economics and Econometric Reviews. He has published extensively in leading field journals and books, and is author, co-author and editor of a number of books in econometrics and time
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Helmut Lütkepohl has been Bundesbank Professor at Freie Universität Berlin and Dean of the DIW Graduate Center, Berlin since 2012. He is and has been Associate Editor of a number of journals, such as Econometric Theory, Journal of Econometrics, the Journal of Applied Econometrics, Macroeconomic Dynamics, Empirical Economics and Econometric Reviews. He has published extensively in leading field journals and books, and is author, co-author and editor of a number of books in econometrics and time series analysis. For example, he has authored an ‘Introduction to Multiple Time Series Analysis’ (Springer, 1991) and a ‘Handbook of Matrices’ (Wiley, 1996). His current teaching and research interests include methodological issues related to the study of nonstationary, integrated time series, forecasting and structural vector autoregressive modelling.
Anna Staszewska-Bystrova is an Assistant Professor in the Chair of Econometric Models and Forecasts, University of Lodz. She has recently published in journals including the International Journal of Forecasting, Journal of Forecasting and Computational Statistics and Data Analysis. Her research interests are in time series econometrics and simulation and computational methods in economics.
Peter Winker has been professor of statistics and econometrics at the Justus-Liebig- University Giessen since 2006. His research interests focus on computational methods in statistics and econometrics, but also include macroeconomic modeling, experimental design, survey methodology and financial markets. His publications include a substantial number of refereed journal articles. He has also published books including ‘Optimization Heuristics in Econometrics’ (Wiley, 2001), as well as being co-editor of several books such as ‘Computational Methods in Financial Engineering’ (Springer, 2008). He is Managing Editor of the Journal of Economics and Statistics and Associate Editor of several journals including Computational Statistics and Data Analysis and Computational Management Science.