RDP 2022-09: Estimating the Effects of Monetary Policy in Australia Using Sign-restricted Structural Vector Autoregressions 2. Framework

This section describes the SVAR framework, outlines the concepts of identifying restrictions and identified sets, and explains the Bayesian approaches to inference used throughout the paper.

I will make use of the following notation. For a matrix X, vec(X) is the vectorisation of X. When X is symmetric, vech(X) is the half-vectorisation of X, which stacks the elements of X that lie on or below the diagonal into a vector. e_i,n is the i th column of the n×n identity matrix, I_n. 0_n×m is an n×m matrix of zeros.

2.1 SVAR, orthogonal reduced form and identified sets

Let y_t be an n×1 vector of endogenous variables following the SVAR( p ) process:

where A₀ is an invertible n×n matrix with positive diagonal elements (a normalisation on the signs of the structural shocks), $x_t={\left(y_{t-1}^{\prime },\mathrm{...},y_{t-p}^{\prime },z_t^{\prime }\right)}^{\prime }$ with z_t a vector of exogenous variables, and $A_{+}=\left(A_1,\mathrm{...},A_p,A_z\right)$ contains the coefficients on the variables in x_t. Conditional on x_t, the structural shocks ${\epsilon }_t$ are normally distributed with zero mean and identity covariance matrix.^[3]

In set-identified SVARs, it is convenient to reparameterise the model into its ‘orthogonal reduced form’ (e.g. Arias, Rubio-Ramírez and Waggoner 2018):

where $B=\left(B_1,\mathrm{...},B_p\right)=A_0^{-1}{A}_{+}$ is the matrix of reduced-form coefficients, ${\Sigma }_{tr}$ is the lower-triangular Cholesky factor of the variance-covariance matrix of the reduced-form VAR innovations, $\Sigma =E\left(u_t{u}_t^{\prime }\right)=A_0^{-1}{\left(A_0^{-1}\right)}^{\prime }$ with u_t = y_t – Bx_t, and Q is an n×n orthonormal matrix (i.e. QQ′ = I_n).

I denote the reduced-form parameters by $\varphi ={\left(\text{vec}{\left(B\right)}^{\prime },\text{vech}{\left({\Sigma }_{tr}\right)}^{\prime }\right)}^{\prime }$ and the space of orthonormal matrices by $O\left(n\right)$. In the absence of identifying restrictions, the structural parameters of the SVAR are set identified, because any $Q\in O\left(n\right)$ is consistent with the joint distribution of the data (summarised by $\varphi$).

The impulse responses to a monetary policy shock are obtained from the coefficients of the vector moving average representation:

where C_h are the reduced-form impulse responses, defined recursively by $C_h={\displaystyle {\sum }_{l=1}^{\min \left\{h,p\right\}}{B}_l{C}_{h-1}}$ for $h\ge 1$ with C₀=I_n. The (i, j) th element of the matrix $C_h{\Sigma }_{tr}Q$ is the horizon-h impulse response of the i th variable to the j th structural shock, denoted by ${\eta }_{i,j,h}\left(\varphi ,Q\right)=c_{ih}^{\prime }\left(\varphi \right)q_j,$ where $c_{ih}^{\prime }\left(\varphi \right)=e_{i,n}^{\prime }{C}_h{\Sigma }_{tr}$ is the i th row of $C_h{\Sigma }_{tr}$ and $q_j=Q{e}_{j,n}$ is the j th column of Q. Given the normalisation that the structural shocks have unit standard deviation, these impulse responses represent responses to standard deviation shocks.

Sign restrictions on impulse responses or structural coefficients (i.e. elements of A₀) can be represented as linear inequality restrictions on a column of Q, where the restrictions are a function of the reduced-form parameters. For example, the sign restriction ${\eta }_{i,j,h}\left(\varphi ,Q\right)=c_{ih}^{\prime }\left(\varphi \right)q_j\ge 0$ is a linear inequality restriction on q_j. Imposing sign restrictions can be viewed as restricting Q to lie in a subspace of $O\left(n\right)$. Let $S\left(\varphi ,Q\right)\ge 0_{s\times 1}$ represent a collection of s sign restrictions (including the sign normalisation $\text{diag}\left(A_0\right)\ge 0_{n\times 1}$).^[4] Given the sign restrictions, the identified set for Q is

The identified set for Q contains observationally equivalent parameter values, which are parameter values corresponding to the same value of the likelihood (Rothenberg 1971). Any values of Q within $𝒬\left(\varphi |S\right)$ are therefore equally consistent with the joint distribution of the data. The identified set for a particular impulse response is the set of values of ${\eta }_{i,j,h}\left(\varphi ,Q\right)$ as Q varies over its identified set:

2.2 Standard and robust Bayesian inference under sign restrictions

The standard approach to conducting Bayesian inference in set-identified SVARs involves specifying a prior for the reduced-form parameters $\varphi$ and a uniform prior for the orthonormal matrix Q (Uhlig 2005; Rubio-Ramírez, Waggoner and Zha 2010; Arias et al 2018).^[5] To draw from the resulting posterior in practice, one samples values of $\varphi$ from its posterior and Q from a uniform distribution over $O\left(n\right)$ and discards draws that violate the sign restrictions. Draws of the impulse responses are obtained by transforming the draws of $\varphi$ and Q via the function ${\eta }_{i,j,h}\left(\varphi ,Q\right),$ and the posterior is summarised using quantities such as the posterior mean and quantiles.

Because Q is set identified under sign restrictions, the likelihood function is flat over $𝒬\left(\varphi |S\right)$ and a component of the prior (specifically, the conditional prior for Q given $\varphi$) is never updated; in other words, the conditional posterior for Q is equal to the conditional prior (Poirier 1998; Baumeister and Hamilton 2015). This raises the concern that posterior inferences may be sensitive to the choice of conditional prior. Given that the conditional prior under the standard approach to Bayesian inference does not necessarily reflect prior information about the parameters, it is desirable that researchers eliminate or quantify the sensitivity of posterior inference to changes in the conditional prior.

I therefore adopt a ‘robust’ (multiple-prior) Bayesian approach to conduct posterior inference in set-identified models (Giacomini and Kitagawa 2021).^[6] In the context of an SVAR, this approach eliminates the posterior sensitivity arising from the choice of the conditional prior. Importantly, this helps to disentangle the information in the posterior that is contributed by the data and the identifying restrictions from the information that is contributed by the conditional prior, and helps to transparently assess the informativeness of the identifying restrictions. The key feature of the approach is that it replaces the conditional prior for Q with the class of all conditional priors that satisfy the identifying restrictions. Combining the class of prior distributions with the posterior distribution for $\varphi ,{\pi }_{\varphi |Y},$ generates a class of posteriors for ${\eta }_{i,j,h}\left(\varphi ,Q\right).$

Giacomini and Kitagawa (2021) suggest summarising the class of posteriors by reporting the ‘set of posterior means’:

where $\ell \left(\varphi \right)=\inf \left\{{\eta }_{i,j,h}\left(\varphi ,Q\right):Q\in Q\left(\varphi |S\right)\right\}$ is the lower bound of the identified set for ${\eta }_{i,j,h}$ and $u\left(\varphi \right)=\sup \left\{{\eta }_{i,j,h}\left(\varphi ,Q\right):Q\in Q\left(\varphi |S\right)\right\}$ is the upper bound. The set of posterior means is an interval containing all posterior means obtainable within the class of posteriors. They also suggest reporting a robust credible region with credibility level $\tau$, which is an interval estimate for ${\eta }_{i,j,h}$ such that the posterior probability assigned to the interval is at least $\tau$ uniformly over the class of posteriors.^[7] Given a particular hypothesis of interest (e.g. that the output response to a monetary policy shock is negative at some horizon), the set of posteriors also generates a set of posterior probabilities for this hypothesis. This set can be summarised by the posterior lower and upper probabilities, which are, respectively, the smallest or largest posterior probabilities assigned to the hypothesis over the class of posteriors. Appendix A describes the numerical algorithms used to compute these quantities.

Footnotes

Bayesian inference under the assumption of homoskedastic, normally distributed structural shocks can be interpreted asymptotically as a limited-information procedure based on the second moments of the data; for discussion of this point, see Plagborg-Møller (2019). [3]

See Stock and Watson (2016) or Kilian and Lütkepohl (2017) for overviews of identification in SVARs. See Giacomini and Kitagawa (2021) for more information about the construction of $S\left(\varphi ,Q\right)$. [4]

This approach to conducting inference in SVARs under sign restrictions is ubiquitous in the empirical literature. An alternative approach is to conduct Bayesian inference under a prior that is specified directly over the structural parameters of the SVAR, as advocated for in Baumeister and Hamilton (2015). There are also frequentist approaches, although these have not (yet) been adopted widely (e.g. Gafarov, Meier and Montiel Olea 2018; Granziera, Moon and Schorfheide 2018). [5]

Giacomini, Kitagawa and Read (2021b) review the literature on robust Bayesian analysis in econometrics, and describe different approaches to conducting robust Bayesian inference in set-identified SVARs. [6]

Giacomini and Kitagawa (2021) provide sufficient conditions under which this approach is valid from a frequentist perspective, in the sense that the set of posterior means is consistent for the true identified set and the robust credible interval has asymptotically correct frequentist coverage of the true identified set. This approach therefore reconciles the asymptotic disagreement between frequentist and Bayesian methods that arises in set-identified models (Moon and Schorfheide 2012), as well as having a well-defined (robust) Bayesian interpretation in finite samples. [7]