RDP 2026-01: Shock-percentile Restrictions for SVARs 1. Introduction

A vast literature uses structural vector autoregressions (SVARs) to identify the effects of structural shocks. The key challenge in doing this is formulating identifying restrictions that are both credible and sufficiently informative to draw economically useful conclusions. In an important contribution, Luvdigson, Ma and Ng (2017, 2021) propose set identifying SVARs using identifying restrictions that include bounds on shocks in selected episodes – what I refer to as ‘shock-magnitude restrictions’. An example is the restriction that the ‘financial uncertainty shock’ in October 1987 – corresponding to the Black Monday stock market crash – was larger than a specified numeric bound (e.g. four standard deviations). In the context of identifying uncertainty shocks, Ludvigson et al (2021) (henceforth, LMN) demonstrate that these restrictions can contribute useful identifying information. A number of papers have adopted shock-magnitude restrictions when identifying uncertainty, and other, shocks.^[1]

This paper makes three main contributions. First, I highlight an important feature of the procedure used to specify shock-magnitude restrictions: bounds on shock realisations are typically elicited by inspecting the distribution of shocks induced by simulating values of the SVAR's structural parameters. I argue that the algorithm used to randomly draw parameters is arbitrary in this setting, which casts doubt on the credibility of these restrictions. Second, I propose a novel type of identifying restriction that is similar in spirit to shock-magnitude restrictions, but that avoids specifying numeric bounds on shocks, which are difficult to credibly elicit. The new ‘shock-percentile restrictions’ impose that the realisation of the shock in a selected episode is more extreme than a specified percentile of the shock's historical distribution; an example is the restriction that the financial uncertainty shock in the Black Monday episode exceeds the 75th percentile of the historical distribution of financial uncertainty shocks. Third, using these restrictions I revisit the relationship between uncertainty and real activity, and estimate the macroeconomic effects of US monetary policy. Both applications demonstrate that shock-percentile restrictions can provide useful identifying information.

Imposing shock-magnitude restrictions, as in LMN, requires specifying a numeric bound on a shock in a selected period. Eliciting such a bound on the basis of a priori or narrative information is difficult; while narrative information may suggest that there was a large shock in some period, it is not necessarily obvious how to translate this into a numeric bound. Instead, LMN propose a simulation-based procedure to elicit a bound. This procedure involves randomly drawing values of the SVAR's parameters using an algorithm that I describe below. The distribution of parameter draws induces a distribution for the structural shocks given the observed data. LMN then take a specified percentile of the shock distribution in the selected period and use this to impose a shock-magnitude restriction. For example, in the application considered by LMN, this procedure implies that the 75th percentile of the distribution of the financial uncertainty shock in the Black Monday episode is around 4.2 standard deviations. LMN use this result to motivate a shock-magnitude restriction in which the financial uncertainty shock in that episode is constrained to exceed 4.2 standard deviations.

Shock-magnitude restrictions can be interpreted as representing the belief that a shock was ‘large’, where the definition of ‘large’ is elicited based on the simulation-based procedure described above. In this procedure, the algorithm used to randomly draw parameter values is from Rubio-Ramírez, Waggoner and Zha (2010) (henceforth RWZ). The algorithm draws orthonormal (or ‘rotation’) matrices from a uniform distribution and is employed widely when estimating Bayesian SVARs (e.g. Arias, Rubio-Ramírez and Waggoner 2018; Baumeister and Hamilton 2018). In the Bayesian inferential setting, this uniform distribution has a well-defined interpretation as a prior or posterior distribution.^[2] However, in the frequentist setting considered in LMN, the distribution induced by this algorithm has no such interpretation; the algorithm is merely a convenient tool for randomly generating draws of the parameters (Fry and Pagan 2011). Consequently, the distribution of the structural shocks induced by this algorithm is arbitrary, which casts doubt on the credibility of shock-magnitude restrictions.^[3]

As an alternative, I propose identifying shocks using novel identifying restrictions that are similar in spirit to shock-magnitude restrictions, but that avoid the difficult problem of specifying numeric bounds on shocks. More specifically, I consider imposing that the shock in a selected episode exceeds a specified percentile of the historical distribution of that shock. These shock-percentile restrictions can be used to impose the belief that a shock was large relative to other realisations of the same shock occurring within the sample. Because these restrictions relate the realisation of a shock to a percentile of its historical distribution, the restrictions have a natural interpretation as reflecting beliefs about how ‘rare’ such a shock is, expressed in probabilistic terms. In contrast, shock-magnitude restrictions have no such probabilistic interpretation in the absence of a parametric distributional assumption about the structural shock (e.g. that it is Gaussian).

Shock-percentile restrictions are identifying restrictions that directly involve the realisations of structural shocks in selected episodes. They therefore fall into the broad class of ‘narrative restrictions’ proposed in Antolín-Díaz and Rubio-Ramírez (2018) and analysed in Giacomini, Kitagawa and Read (2023). Shock-percentile restrictions differ to the specific narrative restrictions proposed in Antolín-Díaz and Rubio-Ramírez (2018), which constrain the signs of structural shocks in selected episodes and/or the relative contributions of shocks to forecast errors (i.e. historical decompositions). While restrictions on the relative magnitudes of structural shocks in different periods have been considered elsewhere, these have typically involved restricting a shock to be larger than the realisation of the shock in all other periods (e.g. Ben Zeev 2018; Giacomini, Kitagawa and Read 2021; Read 2022; Abbate, Eickmeier and Prieto 2023). I demonstrate that much weaker restrictions on the relative size of structural shocks can, in some cases, contribute substantial identifying power. Using a simple model and Monte Carlo exercises, I illustrate the potential for shock-percentile restrictions to sharpen identification. I then apply shock-percentile restrictions in two empirical applications.

First, I revisit the relationship between uncertainty and real activity by replacing the shock-magnitude restrictions in LMN with shock-percentile restrictions. Doing this yields qualitatively similar conclusions about the effects of financial uncertainty shocks on output. In particular, it remains the case that positive financial uncertainty shocks unambiguously lead to a decline in output, and financial uncertainty shocks explain a nontrivial share of output fluctuations beyond short horizons. More generally, however, the shock-percentile restrictions tend to yield ambiguous conclusions about the relationship between uncertainty and real activity. Replacing shock-magnitude restrictions with shock-percentile restrictions in this exercise is therefore an example of the trade-off between ‘credibility’ and ‘certitude’ discussed in Manski (2003, 2011).^[4] Imposing an additional shock-percentile restriction related to the 1998 Russian Financial Crisis better disentangles shocks to macroeconomic and financial uncertainty and yields results that are qualitatively consistent with the results from LMN across several dimensions.

Second, I exploit a shock-percentile restriction to sharpen identification of the macroeconomic effects of US monetary policy, building on a model from Antolín-Díaz and Rubio-Ramírez (2018). I start with their narrative restrictions, which constrain the sign of the monetary policy shock and its contribution to changes in the federal funds rate in selected episodes. I then impose a single shock-percentile restriction reflecting the belief that the monetary policy shock in October 1979 – corresponding to the Volcker disinflation – was large relative to the historical distribution of monetary policy shocks. Despite the original narrative restrictions already providing substantial identifying power, this single shock-percentile restriction materially sharpens identification of the output responses to a monetary policy shock; relative to the original set of restrictions, there is much stronger evidence that output declines following a positive shock, especially at short horizons.

Roadmap. Section 2 describes the SVAR, explains the concept of an identified set and defines shock-magnitude restrictions. Section 3 discusses the procedure used to elicit bounds for shock-magnitude restrictions. Section 4 introduces shock-percentile restrictions, illustrates their potential to contribute identifying power in a simple model and relates them to existing identifying restrictions. Section 5 revisits the relationship between uncertainty and real activity, comparing and contrasting results obtained using shock-magnitude and shock-percentile restrictions. Section 6 exploits a shock-percentile restriction to sharpen identification of the effects of US monetary policy. Section 7 concludes. The appendices contain additional details related to the numerical and empirical exercises.

Notation. e_i,n is column i of the n×n identity matrix, I_n. 0_n×m is an n×m matrix of zeros. For n×m matrix X, vec(X) is the vectorisation of X, which stacks the elements of X into an nm×1 vector. If X is n×n, vech(X) is the half-vectorisation of X, which stacks the elements lying on or below the diagonal into an n(n+1)/2×1 vector. $𝟙(\cdot )$ is the indicator function.

Footnotes

Several papers employ shock-magnitude restrictions to help identify uncertainty shocks, including Caggiano, Castelnuovo and Kima (2020), Caggiano et al (2020), Himounet (2022), Theophilopoulou (2022), Caggiano and Castelnuovo (2023), Carriero, Marcellino and Tornese (2023), Baker, Bloom and Terry (2024) and Kang and Park (2024). Shock-magnitude restrictions have also been used in other settings. For example, they are applied by Ludvigson et al (2017) in models of the global oil market and US monetary policy, and by Ferreira (2022) to disentangle forward guidance and conventional monetary policy shocks. [1]

Arias, Rubio-Ramírez and Waggoner (2025) show that the uniform prior for the orthonormal matrix induces a joint uniform prior over the vector of impulse responses. [2]

Section 3.3 argues that a Bayesian analogue of this bound-elicitation procedure would also be problematic. [3]

Shock-percentile restrictions are not necessarily always weaker than comparable shock-magnitude restrictions. See Section 5.2.3 for further discussion. [4]