RDP 2025-03: Fast Posterior Sampling in Tightly Identified SVARs Using ‘Soft’ Sign Restrictions 1. Introduction

Sign-restricted structural vector autoregressions (SVARs) are used extensively by applied macroeconomists to estimate the effects of macroeconomic shocks. Researchers making use of sign restrictions typically work with the SVAR's ‘orthogonal reduced-form’ parameterisation and conduct Bayesian inference under a uniform prior on the orthonormal (or ‘rotation’) matrix, Q.^[1] Posterior sampling in this setting is almost universally implemented using an accept-reject sampler; proposal draws are obtained from the posterior distribution of the reduced-form parameters and from a uniform distribution for Q, and are rejected if they do not satisfy the sign restrictions (e.g. Uhlig 2005; Rubio-Ramírez, Waggoner and Zha 2010; Arias, Rubio-Ramírez and Waggoner 2018).

Accept-reject sampling can be computationally burdensome when draws satisfy the sign restrictions with low probability, which occurs when the identifying restrictions substantially truncate the identified set for Q – that is, when identification is ‘tight’.^[2] For example, Kilian and Murphy (2014) identify the effects of oil market shocks using sign restrictions on impulse responses augmented with bounds on the price elasticity of oil supply. Baumeister and Hamilton (2024) observe that 5 million draws of the parameters yield only 16 draws satisfying the identifying restrictions. In applications like this, obtaining a sufficiently large number of draws satisfying the identifying restrictions requires a very large number of proposal draws. The computational bottleneck associated with accept-reject sampling under tight identification is likely to become increasingly prominent, owing to a trend towards using richer sets of identifying restrictions (e.g. Inoue and Kilian 2024).

This paper develops an approach to sampling Q that mitigates the inefficiencies associated with brute force accept-reject sampling. The key feature of our approach is an initial sampling step based on ‘soft’ sign restrictions. This step yields draws of Q that do not necessarily satisfy the imposed sign restrictions, but allows the use of computationally efficient algorithms for sampling. Our approach involves: 1) specifying a target distribution that smoothly penalises parameter values that violate (or are close to violating) the sign restrictions; 2) using a Markov chain Monte Carlo (MCMC) algorithm – the slice sampler (e.g. Neal 2003) – to draw from this target distribution; and 3) applying an importance-sampling step to obtain draws from the desired distribution, which we take to be the uniform distribution over the identified set for Q given the (hard) sign restrictions. We emphasise that we do not consider conducting inference under the relaxed set of sign restrictions; the soft sign restrictions that we employ are only used to facilitate sampling from the posterior given the ‘hard’ sign restrictions. For an approach to conducting Bayesian inference under ‘non-dogmatic’ sign restrictions, see Baumeister and Hamilton (2018, 2019).

Our approach can lead to efficiency gains for two reasons. First, because we sample from a smooth density that tends to assign higher probability to parameter regions within the identified set, the sampler tends to move from its initial point towards the identified set. Second, once the sampler has moved to a parameter region within the identified set, it tends to stay there. The accept-reject sampler does not feature either of these behaviours, because candidate draws are independent.

Existing approaches to uniform sampling of Q under sign restrictions can in some cases be more efficient than accept-reject sampling, but are only applicable under particular classes of restrictions. Amir-Ahmadi and Drautzburg (2021) develop a Gibbs sampler that draws from a uniform distribution over the identified set for Q. However, the sampler is not applicable when the sign restrictions constrain all columns of Q (e.g. when there are restrictions on the impulse responses to all shocks) or when each restriction does not linearly constrain a single column of Q. Read (2022) extends this sampler to allow for zero restrictions.

Chan, Matthes and Yu (2025) leverage the fact that the uniform distribution for Q is preserved when permuting and/or flipping the signs of its columns, building on this idea to rapidly generate a large number of draws that satisfy restrictions on impact impulse responses. Their approach could be combined with a secondary accept-reject step to impose other restrictions. Of course, if the additional restrictions substantially tighten the identified set, an accept-reject step could be burdensome. Their approach is likely to be particularly helpful when the number of variables included in the SVAR is large; intuitively, permuting columns can be useful for generating many uniformly distributed draws of Q when there are many possible permutations. The gains from doing this are likely to be smaller in the settings that we consider, where the number of variables is relatively small.^[3]

In contrast with existing approaches, our sampler can be directly applied when the identifying restrictions – potentially nonlinearly – constrain all columns of Q, and can thus handle a wide variety of identifying restrictions. These include restrictions on: impulse responses (e.g. Uhlig 2005); structural coefficients (e.g. Arias, Caldara and Rubio-Ramírez 2019); the magnitudes of structural elasticities (e.g. Kilian and Murphy 2012); long-run cumulative impulse responses (e.g. Furlanetto et al 2025); and forecast error variance decompositions (e.g. Volpicella 2022). They are also applicable under: shape or ‘ranking’ restrictions (e.g. Amir-Ahmadi and Drautzburg 2021); ‘narrative restrictions’ (e.g. Antolín-Díaz and Rubio-Ramírez 2018; Giacomini, Kitagawa and Read 2023); and restrictions on the relationship between ‘proxies’ and structural shocks (e.g. Arias, Rubio-Ramírez and Waggoner 2021; Giacomini, Kitagawa and Read 2022b; Braun and Brüggemann 2023).

Our approach could also be used to draw more efficiently from posterior distributions corresponding to non-uniform priors for Q. For example, the posterior sampler in Bruns and Piffer (2023) involves sampling from a uniform normal-inverse-Wishart posterior and using an importance sampler to draw from the posterior under a different prior, including priors specified directly over structural parameters, as in Baumeister and Hamilton (2015). This approach involves using accept-reject sampling to obtain draws of Q satisfying sign restrictions. Our algorithm could be embedded within their sampler to improve efficiency when identification is tight.

The inefficiency of accept-reject sampling can also be a bottleneck when using prior-robust Bayesian methods (e.g. Giacomini and Kitagawa 2021a; Giacomini, Kitagawa and Read forthcoming). In practice, implementing these methods requires calculating the bounds of the identified set for each parameter of interest (e.g. an impulse response) at every draw of the reduced-form parameters. One way to do this, as suggested in Giacomini and Kitagawa (2021a), is by obtaining many draws from a uniform distribution over the identified set for Q and computing the minimum and maximum of the parameter of interest over the draws. A large number of draws may be required to approximate identified sets with a high degree of accuracy (e.g. Montiel Olea and Nesbit 2021). Again, obtaining these draws via accept-reject sampling can be cumbersome when identification is tight.

The idea of ‘softening’ or smoothing out restrictions has been applied elsewhere. In the context of sampling, Souris, Bhattacharya and Pati (2019) develop a Gibbs sampler to draw from a smooth approximation of a truncated multivariate normal distribution subject to linear constraints. Our proposal also involves sampling from a smooth approximation of a truncated normal distribution, but the general set of constraints that we consider may be nonlinear. In the field of operations research, kernel smoothing methods have been employed when computing derivatives of simulated outcomes under processes where there are discontinuities in the sample path (e.g. Liu and Hong 2011; Bruins et al 2018).

We illustrate the utility of our approach in two main settings. First, we use a simple bivariate model to illustrate our approach and explore its efficiency relative to accept-reject sampling. We show that our approach performs favourably when the identified set for Q is assigned small measure under the uniform prior, which occurs when identification is tight. We also demonstrate that the approach continues to effectively sample from the target distribution when the identified set is made up of disconnected parameter regions.

Second, we revisit the model of the global oil market in Antolín-Díaz and Rubio-Ramírez (2018), which builds on Kilian (2009) and Kilian and Murphy (2012).^[4] This model imposes a rich set of sign, elasticity and narrative restrictions, which simultaneously and nonlinearly constrain all columns of Q. We show that our approach is roughly an order of magnitude more efficient than accept-reject sampling in this application. Using the same model, we also illustrate the utility of our approach in conducting prior-robust Bayesian inference, which would be extremely computationally burdensome when implemented via accept-reject sampling. We find that inferences about the effects of shocks in the oil market obtained under this rich set of identifying restrictions are broadly robust to the choice of conditional prior for Q. Importantly, we argue that this apparent robustness is unlikely to be an artefact of numerical approximation error given the large number of draws used when approximating identified sets. We briefly outline an additional empirical application – a model of US monetary policy from Antolín-Díaz and Rubio-Ramírez (2018) – which demonstrates that the favourable performance of our sampler persists in a larger model.

The remainder of the paper is structured as follows. Section 2 describes the SVAR, outlines the identifying restrictions that can be imposed via our algorithm, and briefly explains standard and robust Bayesian approaches to inference in this setting. Section 3 describes accept-reject sampling and introduces our sampler based on soft sign restrictions. Section 4 illustrates our approach and explores its efficiency in a simple example. Section 5 applies the approach in an empirical setting. Section 6 concludes. The appendices contain proofs and additional details related to the empirical applications.

Notation. We use the following notation throughout the paper. e_i,n is the ith column of the n×n identity matrix, I_n. 0_n×m is an n×m matrix of zeros. For a n×m matrix X, vec(X) is the vectorisation operator, which stacks the elements of X into an nm×1 vector. If X is n×n, vech(X) is the half-vectorisation, which stacks the elements lying on or below the diagonal into an n(n+1)/2×1 vector. $𝟙$(.) is the indicator function.

Footnotes

See the references in Baumeister and Hamilton (2018) for many such examples. There is ongoing debate about the appropriateness of this prior (e.g. Inoue and Kilian 2024; Arias, Rubio-Ramírez and Waggoner 2025). For frequentist approaches to inference in this setting, see Gafarov, Meier and Montiel Olea (2018) or Granziera, Moon and Schorfheide (2018). [1]

The identified set is the set of observationally equivalent parameter values, which are parameter values sharing the same value of the likelihood. [2]

Hou (2024) proposes an MCMC algorithm for posterior sampling under (potentially overidentifying) linear equality and inequality restrictions on impact impulse responses. The algorithm can incorporate inequality restrictions on other parameters, though this requires additional accept-reject steps. The posterior corresponds to an independent Gaussian prior over the columns of the impact impulse-response matrix, rather than the uniform prior for Q considered here. [3]

Variants of this model have been widely studied elsewhere (e.g. Baumeister and Peersman 2013; Lütkepohl and Netšunajev 2014; Baumeister and Hamilton 2019; Bacchiocchi et al 2024; Carriero, Marcellino and Tornese 2024; Hoesch, Lee and Mesters 2024). [4]