In Brüggemann, Jentsch & Trenkler (2016), we consider a framework for asymptotically valid inference in stable vector autoregressive (VAR) models when the innovations are uncorrelated, but not independent. This setup is referred to as a weak VAR model. We provide asymptotic theory for weak VARs under strong mixing conditions on the innovations and prove a joint central limit theorem for the LS estimators of VAR coefficients and variance parameters of the innovations. Our results allow for asymptotically correct inference on statistics that depend on both VAR coefficients and variance parameters of the innovations as e.g. structural impulse response functions (IRFs). To identify structural shocks in VARs, proxy structural VARs (proxy SVARs) use external proxy variables that are correlated with the structural shocks of interest, but uncorrelated with other structural shocks. In Jentsch & Lunsford (2016+), we extend the results from weak VARs to weak proxy SVARs and provide asymptotic theory when the VAR innovations and proxy variables are jointly strong mixing.
As inference based on normal approximation is cumbersome due to the complicated limiting variance, bootstrap methods are commonly used. In Brüggemann et al. (2016) we showed that (residual-based) wild and pairwise bootstrap schemes are generally inappropriate for inference on (functions of) the variance parameters of the innovations if the VAR innovations are not independent.
As discussed in Jentsch & Lunsford (2016+), this bootstrap inconsistency result translates directly also to proxy SVARs. Hence, the wild bootstrap as propagated by Mertens & Ravn (2013) to produce confi dence intervals for the IRFs in proxy SVARs is not appropriate and simulations show that their coverage rates for IRFs can be much too low. In contrast, we propose a residual-based moving block bootstrap (MBB) and prove its consistency for inference on statistics that depend jointly on VAR coefficients and on covariances of the VAR innovations and proxy variables. Using the MBB to reestimate confi dence intervals for the IRFs in Mertens & Ravn (2013), we show that inference cannot be made about the eff ects of tax changes on output, labor, or nonresidential investment.
References
Brüggemann, R., Jentsch, C. & Trenkler, C. (2016), 'Inference in vars with conditional heteroskedasticity of unknown form', Journal of Econometrics 191(1), 69-85.
Jentsch, C. & Lunsford, K. G. (2016+), 'Asymptotic theory and bootstrap inference for proxy svars', Technical Report.
Mertens, K. & Ravn, M. O. (2013), 'The dynamic e ffects of personal and corporate income tax changes in the united states', American Economic Review 103(4), 1212-1247.
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