RDP 2026-01: Shock-percentile Restrictions for SVARs Appendix C: Additional Details about Empirical Exercises

This appendix provides additional details related to the empirical applications. Appendix C.1 describes the numerical procedures used to approximate identified sets in both applications. I then revisit two exercises from LMN using the shock-percentile restrictions: Appendix C.2 examines whether recursive identification schemes are consistent with the data; and Appendix C.3 examines whether the structural shocks are non-Gaussian.

C.1 Numerical implementation

To approximate identified sets in Section 5, I obtain draws of Q from (an approximation of) the uniform distribution over its identified set, compute the parameters of interest, and take the minimum and maximum of the parameter over the draws of Q.^[42] Given a finite number of draws, this strategy will generate identified sets that are too narrow, with the approximation error vanishing as the number of draws increases. As discussed in Section B.1, Montiel Olea and Nesbit (2021) provide results about the number of draws required to guarantee a given degree of approximation error. Based on the upper bound in their Theorem 3, approximating an identified set of dimension d = 3 × 3 × 61 = 549 with misclassification error less than $\epsilon =0.01$ and probability at least $1-\delta =0.99$ requires $L\approx 300,000$ draws from inside the identified set.

Under the event and external variable constraints considered in LMN, the identified set for Q is substantially truncated relative to the unconstrained identified set. Obtaining a sufficient number of draws satisfying the restrictions via accept-reject sampling is consequently extremely computationally burdensome. I therefore use the sampler based on ‘soft sign restrictions’ proposed in Read and Zhu (2025), which can be more computationally efficient than accept-reject sampling in cases like this where identification is ‘tight’.

The sampler uses Markov chain Monte Carlo methods (specifically, the slice sampler) to obtain draws of Q from a target distribution that is similar to the uniform distribution but that smoothly penalises parameter values that violate the identifying restrictions. This yields some draws that violate the restrictions, and draws that satisfy the restrictions are not uniformly distributed. An importance-sampling step can be used to discard the draws that violate the restrictions and resample the remaining draws so that they better approximate the target uniform distribution. However, since the objective here is only to approximate the bounds of the identified set, it suffices to discard the draws that violate the identifying restrictions, leaving only draws from inside the identified set, and the resampling step is unnecessary. Otherwise, the implementation of the sampler follows Read and Zhu (2025).^[43]

To obtain L effective draws from inside the identified set, it is necessary to obtain more than L draws from the smoothed target density, because some draws will not satisfy the identifying restrictions. For each set of identifying restrictions, I therefore gross up L by an initial estimate of the effective sample size for the sampler. For example, under the restrictions from LMN, the effective sample size is around 89 per cent, so I obtain L/0.89 draws, which yields approximately L draws from inside the identified set.

To implement the robust Bayesesian approach to inference in the empirical application in Section 6, I use the same approach to approximating identified sets at each draw of the reduced-form parameters. Based on the upper bound from Theorem 3 of Montiel Olea and Nesbit (2021), approximating an identified set of dimension d = 2 × 61 = 122 with misclassification error less than $\epsilon =0.01$ and probability at least $1-\delta =0.99$ requires $L\approx 13,000$ draws of Q at each draw of the reduced-form parameters.^[44] The results are based on 1,000 draws of the reduced-form parameters.

C.2 Recursive identification

Much of the empirical literature on the macroeconomic effects of uncertainty relies on recursive identifying schemes to estimate the effects of uncertainty shocks.^[45] LMN demonstrate that recursive structures are inconsistent with their results. This section examines whether this remains the case when replacing the shock-magnitude restrictions with shock-percentile restrictions.

There are three variables in the SVAR, so there are 3! = 6 possible recursive orderings. Let

so H_ij is the impact response of variable i to shock j. The six possible recursive orderings are:

At a given value of $\varphi$, we can assess whether a specific recursive ordering is consistent with the data and identifying restrictions by: 1) computing the (point-identified) structural parameters under the recursive ordering; and 2) checking whether the event and external variable constraints are satisfied (i.e. that the identified set is non-empty).^[46] If none of the structural parameters implied by the six recursive orderings are consistent with the event and external variable constraints, recursive structures are inconsistent with the data (given the other identifying restrictions).

At the OLS estimates of $\varphi$, none of the possible recursive orderings are consistent with the event and external variable constraints used in LMN. This continues to be the case when replacing the shock-magnitude restrictions with the shock-percentile restrictions (with $\alpha =0.75$). This suggests that recursive orderings are not supported by the data, conditional on the other identifying restrictions. The results under the shock-percentile restrictions therefore reinforce the challenges associated with relying on recursive assumptions when attempting to tease apart the causal relationship between uncertainty and real activity.^[47]

C.3 Are the shocks non-Gaussian?

Focusing on a particular set of parameters within the identified set, LMN document that the implied structural shocks exhibit features of non-Gaussianity, such as skewness and excess kurtosis. This result may be of interest because researchers often assume that structural shocks are normally distributed when using Bayesian methods to estimate SVARs, and departures from Gaussianity can distort inference (e.g. Petrova 2022). This section examines whether evidence of non-Gaussianity remains under the shock-percentile restrictions.

Rather than focusing on the properties of a single sequence of shocks implied by a particular set of parameters, as in LMN, I compute identified sets for the skewness and kurtosis of each structural shock. Intuitively, each Q in the identified set is associated with a different sequence of shocks, and thus (potentially) a different skewness and kurtosis. I characterise identified sets for the skewness and kurtosis of the shocks by randomly drawing values of Q from the uniform distribution over $\mathcal{Q}\left(\varphi |S\right)$, computing the skewness and kurtosis of the implied shocks, and taking the minimum and maximum of these statistics over the draws of Q.^[48]

Table C1 presents identified sets for the skewness and kurtosis of the structural shocks under the identifying restrictions from LMN and the shock-percentile restrictions with $\alpha =0.75$. If the shocks were Gaussian, we should expect them to have skewness equal to zero and kurtosis equal to three. The identified sets under the LMN restrictions unambiguously point to the presence of non-Gaussian features in the identified shocks; macroeconomic uncertainty shocks are positively skewed, output and financial uncertainty shocks are negatively skewed, and all shocks display excess kurtosis.

While the identified sets under the shock-percentile restrictions are wider than under the LMN restrictions, they continue to suggest that the structural shocks possess some non-Gaussian features. The identified sets for skewness contain zero for the uncertainty shocks and values close to zero for the output shock, so there is no longer clear evidence of skewness in the identified shocks. However, the identified sets unambiguously point to excess kurtosis in all three shocks.

Footnotes

This approach follows Algorithm 2 in Giacomini and Kitagawa (2021a). [42]

The sampler from Read and Zhu (2025) requires specifying a tuning parameter $\text{Δ}$ that determines how strongly parameter values are penalised when they violate the identifying restrictions. I set ${\text{Δ=10}}^{-5}$. [43]

I present identified sets for the responses of two variables (the federal funds rate and output) for horizons h = 0, 1,..., 60. Approximating identified sets for the responses of additional variables with the same degree of accuracy would require more draws of Q. [44]

See Kilian, Plante and Richter (2025) for many examples. [45]

LMN check whether the identified sets for the relevant elements of H exclude zero, which is a sufficient condition for the identified sets to be inconsistent with a recursive ordering. [46]

Kilian et al (2025) demonstrate that this challenge can not be overcome by examining the results obtained under alternative causal orderings, because these do not necessarily bound the true impulse responses when the DGP does not feature a recursive structure. [47]

A similar idea underlies the approach to identification in Andrade et al (2024). Given that higher-order moments are set identified under sign restrictions, they impose inequality constraints on higher-order moments to sharpen identification. [48]