RDP 2026-01: Shock-percentile Restrictions for SVARs 2. Framework

This section describes the SVAR, explains the concept of an identified set and defines shock-magnitude restrictions.

2.1 SVAR

Assume y_t = (y_1t ,... ,y_nt)′ has a reduced-form VAR(p) representation:

where u_t are serially uncorrelated reduced-form innovations with $𝔼\left(u_t\right)=0_{n\times 1}$ and $𝔼\left(u_t{u^{\prime }}_t\right)=\Sigma$.^[5] Let ${\Sigma }_{tr}$ be the lower-triangular Cholesky factor of $\Sigma$ with non-negative diagonal elements, so ${\Sigma }_{tr}{{\Sigma }^{\prime }}_{tr}=\Sigma$. Collect the reduced-form parameters into $\varphi ={\left(\text{vec}{\left(B_1\right)}^{\prime },\mathrm{...},\text{vec}{\left(B_p\right)}^{\prime },\text{vech}{\left({\Sigma }_{tr}\right)}^{\prime }\right)}^{\prime }$.

Assume u_t is related to the structural shocks ${\epsilon }_t={\left({\epsilon }_{1t},\mathrm{...},{\epsilon }_{nt}\right)}^{\prime }$ by

where $𝔼\left({\epsilon }_t\right)=0_{n\times 1},𝔼\left({\epsilon }_t{{\epsilon }^{\prime }}_t\right)=I_n$ and H is invertible. H contains the impact impulse responses of the variables in y_t to the structural shocks. The diagonal elements of H are normalised to be non-negative (a ‘sign normalisation’), so a positive realisation of structural shock i does not decrease variable i on impact. Knowledge of H and the reduced-form parameters $\varphi$ allows impulse responses and forecast error variance decompositions (FEVDs) to be computed at any horizon.^[6]

2.2 Identified sets

In set-identified SVARs, it is common to work with the model's ‘orthogonal reduced-form’ parameterisation (e.g. Arias et al 2018). Let $H={\Sigma }_{tr}Q$, where $Q\in \mathcal{O}\left(n\right)$ is a n×n orthonormal matrix and $\mathcal{O}\left(n\right)$ is the space of all such matrices. In the absence of further identifying restrictions, Q is set identified and, consequently, so is H (e.g. Uhlig 2005; RWZ).

Given a value of $\varphi$ define the ‘unconstrained identified set’ for Q as:

Values of Q in $\mathcal{Q}\left(\varphi \right)$ are observationally equivalent, in the sense that they correspond to the same second moments of the data, which are summarised by $\varphi .\mathcal{Q}\left(\varphi \right)$ induces an unconstrained identified set for any object of interest that is a function of Q. For example, the unconstrained identified set for H is $\left\{H={\Sigma }_{tr}Q:Q\in \mathcal{Q}\left(\varphi \right)\right\}$. Unconstrained identified sets completely characterise the information contained in the data about the objects of interest when imposing only the sign normalisation.

Traditionally, sign or zero restrictions would be imposed on functions of H, shrinking the identified set, with point identification attained if there are sufficiently many zero restrictions. However, as argued in LMN in the context of identifying uncertainty shocks, such restrictions have little theoretical motivation given the ambiguous predictions of different theories of uncertainty.^[7] LMN therefore turn to alternative identifying restrictions. These include ‘event constraints’, which are inequality restrictions on structural shocks in selected episodes. I describe these identifying restrictions in detail below. For now, it suffices to note that these identifying restrictions can be written as inequality restrictions on Q. These restrictions can truncate the identified set for Q, sharpening identification of the parameters of interest, but in general leaving them set identified.

Given a generic set of s identifying restrictions, $S\left(\varphi ,Q\right)\ge 0_{s\times 1}$, the identified set for Q is:

In the discussion below, it will become apparent that event constraints depend on additional objects not captured in the definition of $\varphi$, but I suppress this dependence for simplicity.^[8] The identified set for Q induces identified sets for other parameters, such as impulse responses and FEVDs.

2.3 Shock-magnitude restrictions

The event constraints proposed in LMN restrict the values of the structural shocks in selected historical episodes. Since ${\epsilon }_t={\epsilon }_t\left(\varphi ,Q,u_t\right)=Q^{\prime }{\Sigma }_{tr}^{-1}{u}_t$, the event constraints represent restrictions on Q that depend on the reduced-form innovations in the constrained periods in addition to the reduced-form parameters $\varphi$.

LMN consider different types of event constraints, which I describe in Section 5. The focus of the following discussion is on the event constraints that involve imposing a bound on a shock in a selected period, which I refer to as shock-magnitude restrictions. A shock-magnitude restriction on shock j in period t takes the form

where $k\in ℝ$ is a numeric lower bound (specified by the researcher) and q_j is column j of Q.^[9] An upper bound would be imposed similarly.

An important input into the shock-magnitude restriction is the bound, k. More-extreme values of k make the restriction more likely to bind and thus truncate the identified set for Q, potentially delivering sharper identification of the SVAR's structural parameters.^[10] The next section discusses the procedure proposed in LMN to elicit bounds.

Footnotes

I abstract from the inclusion of a constant for simplicity. [5]

For definitions of impulse responses and FEVDs, see Kilian and Lütkepohl (2017). [6]

Similar arguments have been made in other settings. For example, Ramey (2016) argues that the recursive orderings commonly used when identifying the effects of monetary policy have little theoretical motivation. [7]

When the identifying restrictions include event constraints (or narrative restrictions), the restrictions depend on the realisation of the data via the reduced-form innovations, u_t. This means the standard definition of an identified set as the set of observationally equivalent parameters does not apply. In settings like this, Giacomini et al (2023) introduce a refinement of the identified set – a ‘conditional identified set’ – that describes the data-dependent mapping from reduced-form parameters to the structural object of interest. For ease of exposition, I do not distinguish between conditional identified sets and identified sets. [8]

Given the normalisation that $𝔼\left({\epsilon }_t{{\epsilon }^{\prime }}_t\right)=I_n,k$ is measured in standard deviations. [9]

Ludvigson, Ma and Ng (2017) discuss how shock-magnitude restrictions can provide identifying information. [10]