Asymptotically Valid Bootstrap Inference for Proxy SVARs

Abstract

Proxy structural vector autoregressions identify structural shocks in vector autoregressions with external variables that are correlated with the structural shocks of interest but uncorrelated with all other structural shocks. We provide asymptotic theory for this identification approach under mild α-mixing conditions that cover a large class of uncorrelated, but possibly dependent innovation processes. We prove consistency of a residual-based moving block bootstrap (MBB) for inference on statistics such as impulse response functions and forecast error variance decompositions. The MBB serves as the basis for constructing confidence intervals when the proxy variables are strongly correlated with the structural shocks of interest. For the case of one proxy variable used to identify one structural shock, we show that the MBB can be used to construct confidence sets for normalized impulse responses that are valid regardless of proxy strength based on the inversion of the Anderson and Rubin statistic suggested by Montiel Olea, Stock, and Watson.

Keywords:

• a-mixing
• Anderson–Rubin confidence set
• External instruments
• Proxy variables
• Residual-based moving block bootstrap
• Weak proxy

Supplementary Materials

We provide an online appendix with additional theoretical detail, all proofs, and additional simulation results. We also provide replication files, coded in Python, for the simulations in the article and in the online appendix.

Acknowledgments

Some results in this article originally circulated in a working article titled, “Proxy SVARs: Asymptotic Theory, Bootstrap Inference, and the Effects of Income Tax Changes in the United States.” We are grateful to Todd E. Clark, Lutz Kilian, Helmut Lütkepohl, José Luis Montiel Olea, Johannes Pfeifer, Carsten Trenkler, and participants at the IAAE 2016 Annual Conference and the CFE 2018 conference for helpful comments and conversations.

Notes

1 “Proxy variable” and “external instrument” are different names for the same variable in this literature. SVARs identified with these variables have been called “proxy SVARs” or “external instrument SVARs.” To simplify communication, we use “proxy variable” and “proxy SVAR” going forward. However, it should be understood that we could equivalently have used “external instrument” or “external instrument SVARs.”

2 Throughout the article, “IRF” refers to structural impulse response functions from orthogonalized economic shocks. It does not refer to forecast error impulse response functions as in (Lütkepohl 2005, p. 52).

3 In principle, all dependence concepts and regularity conditions that are sufficiently strong to allow for the derivation of meaningful asymptotic theory can be employed in place of mixing-type conditions. For instance, classical linearity conditions, but also more modern approaches such as, for example, physical dependence or weak dependence could be used.

4 See Davis and Mikosch (2009) for the probabilistic properties of stochastic volatility models.

5 See Stock, Wright, and Yogo (2002) and Andrews, Stock, and Sun (2019) for surveys on weak IVs.

6 Either standard percentile or Hall’s percentile intervals can be used for confidence intervals. We focus on standard percentile intervals and discuss Hall’s percentile intervals in the supplemental appendix.

7 MSW note that the quadratic form of FEVDs prevents the application of the AR method to FEVDs.

8 Section 4.3 of MSW notes that AR confidence sets can be extended for when r > 1 proxy variables identify r structural shocks but that this extension is inefficient.

9 See Equation (8) in Mertens and Ravn (2013) and our discussion in Section 2.4.

10 The possibility of unbounded confidence sets is not unique to us. Rather, it is a general feature of AR confidence sets. See Andrews, Stock, and Sun (2019, sec. 5.1) and MSW, sec. 4.

11 Assumption 2.1(i) might be restrictive in applications where highly persistent (vector) autoregressions, including integrated and nearly-integrated DGPs, are estimated from the data. In such cases, lag-augmentation as originally proposed by Dolado and Lütkepohl (1996) and Toda and Yamamoto (1995) and investigated recently in Inoue and Kilian (2020) might be helpful to increase the (finite sample) coverage accuracy of impulse response confidence intervals. In simulations in our supplemental appendix, lag augmentation increases coverage accuracy and can also increase the length of confidence intervals.

12 As examples, see Christiano, Eichenbaum, and Evans (1999), Mertens and Ravn (2013), and MSW.

13 We use ${\mathbb{E}}_t$ to denote the conditional expectation given $\{y_s,s\le t\}$ as in Lütkepohl (2005, sec. 2.2.2).

14 For example, Mertens and Ravn (2013) used an SVAR with seven variables and identify the effects of two structural shocks: changes in average personal income tax rates and in average corporate income tax rates.

15 Throughout the article, we use subscripts to denote scalar elements, $y_{j,t}$ is the jth element of y_t , and superscripts to denote sub-vectors and matrices, $H^{(2)}$ is a sub-matrix of H.

16 Mertens and Ravn (2013) also imposed $\mathbb{E}(m_t)=0$ and $\mathbb{E}(m_t{y}_{t-j}^{\prime })=0$ for all $j=1,\dots ,p$. They argue that this is not restrictive as one can always regress the proxies on the lags of y_t and keep the residuals as the new proxies. In our earlier working article, Jentsch and Lunsford (2019a), we also used these assumptions. In comparison, Assumption 2.2 now gets along without such additional conditions on the proxies, yielding a different limiting distribution in Theorem 2.1 than was derived in Jentsch and Lunsford (2019a).

17 More generally, we can use ${\Sigma }_{ϵ}=I_K$.

18 Imposing signs directly on H is common in the broader SVAR literature. For example, Christiano, Eichenbaum, and Evans (1999) impose that the diagonal elements of H are positive.

19 More generally, the unit effect normalization imposes that the diagonal elements of H are all 1.

20 In earlier articles, Jentsch and Lunsford (2016, 2019a), we define $\mathit{\phi }=\text{vec}(\Psi H^{(1)\prime })$, which we have changed to $\mathit{\phi }=\text{vec}(H^{(1)}\Psi \prime )$ for this article.

21 Although not stated in their Assumption 2.1, Brüggemann, Jentsch, and Trenkler (2016) already required summability of fourth-order cumulants to assure finiteness of $V^{(2,2)}$ in their Theorem 2.1.

22 We do provide explicit limiting results for IRFs and FEVDs in proxy SVARs for the $r=g=1$ setting based on the Delta method in the supplemental appendix.

23 For example, if $(x_t)$ is iid, we get that $\sqrt{T}(\widehat{\mathit{\alpha }}-\mathit{\alpha })$ is asymptotically uncorrelated with $\sqrt{T}(\widehat{\mathit{\sigma }}-\mathit{\sigma })$ and $\sqrt{T}(\widehat{\mathit{\phi }}-\mathit{\phi })$, respectively, where $\mathit{\alpha }=\text{vec}(A_1,\dots ,A_p)$, which excludes the intercept, ν, from $\mathit{\beta }$, and $\widehat{\mathit{\alpha }}=\text{vec}({\widehat{A}}_1,\dots ,{\widehat{A}}_p)$. This result extends the classical setting considered, for example, in Lütkepohl (2005, prop. 3.6), where estimators for VAR slope coefficients and innovations covariance matrix are asymptotically uncorrelated, to the proxy SVAR setting. Formulas for the iid case are in the supplemental appendix.

24 We do not discuss one standard deviation IRFs or FEVDs here as they are not the focus of MSW.

25 A similar result, based on Theorem 2.1 for the strong proxy case, is in the supplemental appendix.

26 We center the resampled VAR residuals because block resampling implies that the VAR residuals cannot be resampled into any arbitrary order. Use the example of T = 3 and $\mathcal{l}=2$. Then, the set of blocks is $\left\{\right\{{\widehat{u}}_1,{\widehat{u}}_2\},\{{\widehat{u}}_2,{\widehat{u}}_3\left\}\right\}$. Hence, for example, ${\widehat{u}}_3$ can never show up as ${\tilde{u}}_1^{*}$. Thus, even though an intercept in the VAR guarantees ${\sum }_{t=1}^T{\widehat{u}}_t=0$, it is not the case that $E^{*}({\tilde{u}}_1^{*})=0$, and we center ${\tilde{u}}_1^{*}$ by subtracting $({\widehat{u}}_1+{\widehat{u}}_2)/2$. In our working article, Jentsch and Lunsford (2019a), we used the assumption $\mathbb{E}(m_t)=0$ and, hence, also included a centering step for m_t . Because we no longer use $\mathbb{E}(m_t)=0$, this step can be dropped.

27 We provide Monte Carlo results for Hall’s percentile intervals in the supplemental appendix. As in Kilian (1999), we find that they are not systematically better than standard percentile intervals for inference on IRFs. However, coverage rates of Hall’s intervals can be closer to target at longer impulse horizons. For FEVDs, coverage rates from Hall’s intervals are often worse than those from standard percentile intervals.

28 With a weak proxy variable, Theorem 4.3(i.a) below shows that the bootstrapped normalized IRF does not asymptotically coincide with the limiting distribution in Theorem 3.2(i.a). Hence, neither standard nor Hall’s percentile intervals are valid with a weak proxy variable.

29 For example, the popular class of generalized autoregressive conditional heteroscedastic (GARCH) processes is geometrically strong mixing under mild assumptions on the conditional distribution such that the summability condition in Equation (28) holds.

30 Because the asymptotic normal distribution is symmetric, we have $-{\widehat{q}}_{g,1-\alpha /2}={\widehat{q}}_{g,\alpha /2}$ and $-{\widehat{q}}_{g,\alpha /2}={\widehat{q}}_{g,1-\alpha /2}$. Hence, using $-{\widehat{q}}_{g,1-\alpha /2}\le \sqrt{T}(s{e}_j^{\prime }{\widehat{\Phi }}_i-{\xi }_g{e}_m^{\prime })\widehat{\mathit{\phi }}\le -{\widehat{q}}_{g,\alpha /2}$ is asymptotically equivalent to using ${\widehat{q}}_{g,\alpha /2}\le \sqrt{T}(s{e}_j^{\prime }{\widehat{\Phi }}_i-{\xi }_g{e}_m^{\prime })\widehat{\mathit{\phi }}\le {\widehat{q}}_{g,1-\alpha /2}$ in step 4 of the algorithm. We use $-{\widehat{q}}_{g,1-\alpha /2}\le \sqrt{T}(s{e}_j^{\prime }{\widehat{\Phi }}_i-{\xi }_g{e}_m^{\prime })\widehat{\mathit{\phi }}\le -{\widehat{q}}_{g,\alpha /2}$ because its small sample features closely align with the MBB percentile intervals in our simulations.

31 We provide simulation results for one standard deviation IRFs in the supplemental appendix.

32 The DGP parameters imply that the unconditional distribution of $\text{ln}\hspace{0.17em}{\sigma }_{i,t}$ is $\mathcal{N}(-0.081,0.081)$. Hence, roughly 95% of ${\sigma }_{i,t}$ realizations fall in the interval $(0.53,1.61)$ so that ${\sigma }_{i,t}$ is roughly 3 times larger at the high end of this range than at the low end. Roughly 99.9% of ${\sigma }_{i,t}$ realizations fall in the interval $(0.36,2.35)$ so that the extreme high realizations of ${\sigma }_{i,t}$ can be roughly 6.5 times bigger than the extreme low realizations.

33 We find that adding a nonzero mean to m_t has no affect on our results. Monte Carlo simulations with censored proxy variables are in the supplemental appendix. Consistent with Jentsch and Lunsford (2019b), we find that the MBB’s coverage rates are very similar with and without censoring.

34 First, we regress m_t on ${\widehat{u}}_t$ as was done in Stock and Watson (2012) and test the null hypothesis that the coefficients on ${\widehat{u}}_t$ are zero. Second, we regress the first element of ${\widehat{u}}_t$ on m_t as in Gertler and Karadi (2015) and test the null hypothesis that the coefficient on m_t is zero. Lunsford (2015) provided further discussion.

35 We find that larger block sizes, such as $\mathcal{l}=8$ or $\mathcal{l}=12$, yield slightly lower coverage rates for the MBB.

36 Consistent with this, we show that coverage rates of the grid MBB AR and MSW confidence sets are more similar with a less persistent DGP or a larger sample size in our supplemental appendix.

Additional information

Funding

Parts of this research were conducted while Carsten Jentsch held a position at the University of Mannheim, where he was financially supported by the German Research Foundation DFG via the Collaborative Research Center SFB 884 “Political Economy of Reforms” (Project B6) and the Baden-Württemberg-Stiftung via the Eliteprogram for Postdocs. The views expressed herein are solely those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Cleveland or the Federal Reserve System.