RDP 2022-09: Estimating the Effects of Monetary Policy in Australia Using Sign-restricted Structural Vector Autoregressions Appendix A: Algorithms for Inference

This appendix describes the algorithms used to conduct inference in the body of the paper. I first outline a generic algorithm. I then provide details on how the different steps of the algorithm are implemented in practice, since implementation differs across different types of identifying restrictions.

Algorithm A1 (robust Bayesian inference). Assume the parameter of interest is ${\eta }_{i,j,h}\left(\varphi ,Q\right)$.

For k = 1,...,K:

Step 1: Draw ${\varphi }_k$ from ${\pi }_{\varphi |Y.}$.

Step 2: Check whether the identified set $\text{𝒬}\left({\varphi }_k|S\right)$ is empty. If so, return to Step 1. If not, proceed to Step 3.

Step 3: Compute $\ell \left({\varphi }_k\right)=\underset{Q\in 𝒬\left({\varphi }_k|S\right)}{\min }{\eta }_{i,j,h}\left({\varphi }_k,Q\right)andu\left({\varphi }_k\right)=\underset{Q\in 𝒬\left({\varphi }_k|S\right)}{\max }{\eta }_{i,j,h}\left({\varphi }_k,Q\right).$

Given the output of Algorithm A1, the set of posterior means is approximated by the interval

Define $d\left(\eta ,\varphi \right)=\max \left\{\left|\eta -\ell \left(\varphi \right)\right|,\left|\eta -u\left(\varphi \right)\right|\right\}$, and let $z_{\tau }\left(\eta \right)$ be the sample $\tau$ th quantile of $d\left(\eta ,{\varphi }_k\right).$ An approximated shortest robust credible interval for $\eta$ (with credibility $\tau$) is an interval centred at $\arg {\min }_{\eta }{z}_{\tau }\left(\eta \right)$ with radius ${\min }_{\eta }{z}_{\tau }\left(\eta \right).$ If D is some hypothesis about ${\eta }_{i,j,h}$ (i.e. that the parameter lies within some specified interval), the posterior lower probability assigned to the hypothesis can be approximated by the posterior probability that the identified set is contained entirely within the interval D ($\text{i}\text{.e}\text{.}{K}^{-1}{\Sigma }_{k=1}^{K}1\left({\eta }_{i,j,h}\left({\varphi }_k|S\right)\subset D\right)$, where 1(·) is the indicator function). The posterior upper probability can be approximated by the posterior probability that the identified set intersects the interval $D\left(\text{i}\text{.e}\text{.}{K}^{-1}{\Sigma }_{k=1}^{K}1\left({\eta }_{i,j,h}\left({\varphi }_k|S\right)\cap D\ne \varnothing \right)\right).$

How Steps 2 and 3 are implemented depends on the nature of the identifying restrictions imposed; in particular, whether the identifying restrictions constrain a single column of Q or multiple columns. I discuss these steps below.

A.1 Determining whether the identified set is empty

Step 2 of Algorithm A1 requires determining whether the identified set for Q, $𝒬\left(\varphi |S\right)$, is empty. There are different algorithms available to do this. The applicability and efficiency of the different algorithms depends on the type of sign restrictions imposed.

When the sign restrictions constrain a single column of Q only, which is the case under Restrictions (1)–(5), I use the algorithm proposed in Giacomini, Kitagawa and Volpicella (2022). This algorithm relies on the fact that any non-empty identified set for q₁ must contain a vertex on the unit sphere in ${ℝ}^n$ where n–1 restrictions are active (i.e. binding). The algorithm proceeds by considering different combinations of n–1 active sign restrictions, computing a unit-length vector that satisfies these restrictions with equality and checking whether this vector (or its negative) satisfies the remaining sign restrictions. In practice, one can compute the vector satisfying the active sign restrictions by computing an orthonormal basis for the null space of $\tilde{S}\left(\varphi \right)$, where $\tilde{S}\left(\varphi \right)$ is an (n–1)×n matrix containing a selection of n–1 of the sign restrictions in $S\left(\varphi \right)$. If a vector satisfying the restrictions is found, the identified set is non-empty. If no such vector is found after considering all possible combinations of restrictions, the identified set is empty.

When the sign restrictions constrain multiple columns of Q, which is the case under Restriction (6), I use the rejection-sampling approach described in Giacomini and Kitagawa (2021). This involves drawing Q from a uniform distribution over the space of orthonormal matrices, normalising the draw so that the diagonal elements of A₀ are non-negative, and checking whether the draw satisfies the sign restrictions. If, after a very large number of draws, no draw of Q satisfies the sign restrictions, the identified set is approximated as being empty. A problem with this approach is that it may misclassify the identified set as being empty when it is in fact non-empty. The chance of this happening increases when the sign restrictions substantially truncate the identified set given the sign normalisations. In practice, I obtain 100,000 draws of Q that do not satisfy the sign restrictions before approximating the identified set as empty.

When a value of Q is obtained satisfying the sign restrictions, it can be treated as a draw from the uniform distribution over $\text{𝒬}\left(\varphi |S\right).$ These draws are used to conduct standard Bayesian inference under the conditionally uniform prior.

When interest is in the impulse responses to a 100 basis point monetary policy shock (Section 4), I check whether $0\in {\eta }_{1,1,0}\left(\varphi |S\right)$ using the approaches described in Read (2022b). Under Restriction (3), the number of sign restrictions is equal to the number of endogenous variables in the VAR, so the sufficient condition in Proposition 4.2 of Read (2022b) applies and $0\in {\eta }_{1,1,0}\left(\varphi |S\right)$ for all values of $\varphi$. Otherwise, the posterior probability that $0\in {\eta }_{1,1,0}\left(\varphi |S\right)$ is approximated by augmenting the set of sign restrictions with the zero restriction ${\eta }_{1,1,0}\left(\varphi ,Q\right)=0$ and numerically checking whether the associated identified set is non-empty at each draw of $\varphi$. Under Restrictions (1), (2), (4) and (5), I use Algorithm 4.1 in Read (2022a), which requires solving a simple linear program. Under Restriction (6), I use the rejection sampling approach from Giacomini and Kitagawa (2021) (i.e. draw a value of Q satisfying ${\eta }_{1,1,0}\left(\varphi ,Q\right)=0$ and check whether it satisfies the remaining sign restrictions).

A.2 Computing the bounds of the identified set

Step 3 of Algorithm A1 requires computing the bounds of the identified set. Again, there are different algorithms available to do this.

When the sign restrictions constrain a single column of Q only, which is the case under Restrictions (1)–(5), I apply the active-set algorithm proposed in Gafarov et al (2018). Given a set of active restrictions, these authors provide analytical expressions (up to sign) for the value function and solution of the optimisation problem that defines $\ell \left(\varphi \right)$ and $u\left(\varphi \right)$. Their algorithm proceeds by computing these quantities at every possible combination of active restrictions and checking whether the corresponding solution satisfies the non-active sign restrictions (i.e. whether the potential solution is feasible). $\ell \left(\varphi \right)$ and $u\left(\varphi \right)$ are then computed as the minimum and maximum, respectively, of the feasible values across the different sets of active restrictions.

The analytical expression for the potential solution of the optimisation problem provided by Gafarov et al (2018) is undefined when the value of the optimisation problem is equal to zero. In this case, the authors note that it is necessary to check whether there exists a vector that satisfies the active restrictions with equality and any non-active sign restrictions, but they do not explain how this should be done. When the number of active restrictions is strictly less than n–1, I apply an algorithm proposed in Read (2022a) to determine whether the identified set is non-empty given a set of sign and zero restrictions. When the number of active restrictions is equal to n–1, I compute an orthonormal basis for the null space of the matrix containing the coefficients of the active restrictions. By the rank-nullity theorem, this null space is one-dimensional (i.e. a vector). I then check whether this vector (or its negative) satisfies the remaining sign restrictions, in which case the corresponding solution is feasible.

When the sign restrictions constrain multiple columns of Q, which is the case under Restriction (6), I compute the bounds of the identified set using the rejection-sampling algorithm proposed in Giacomini and Kitagawa (2021). Specifically, I approximate the bounds of the identified set by obtaining 5,000 draws of Q from a uniform distribution over $\text{𝒬}\left(\varphi |S\right)$ and computing the minimum and maximum of the impulse response over these draws. A drawback of this approach is that the approximated identified set will be too narrow, although – when the identified set is bounded – the approximation error will vanish as the number of draws of Q increases.^[28]

A.3 Approximation errors associated with Restriction (6)

It is possible that the identified set is misclassified as empty at some draws of $\varphi$ under Restriction (6), because the algorithm used to determine whether the identified set is empty is only approximate in the case where multiple columns of Q are restricted. The posterior plausibility obtained under Restriction (6) is therefore a lower bound on the ‘true’ posterior plausibility. This approximation error may also tend to make the reported sets of posterior means and robust credible intervals wider than they actually are, since the algorithm will be more likely to discard reduced-form parameters that yield a ‘narrow’ identified set. On the other hand, at any given value of $\varphi$ that has a non-empty identified set, the algorithm used to approximate the identified set will yield an approximation that is too narrow. It is therefore unclear whether the ‘true’ informativeness of the restrictions – in terms of how much the restrictions tighten the set of posterior means – will be under or overstated. Experiments suggest that the results are fairly insensitive to increases in the number of draws used to check whether the identified set is empty or to approximate the bounds of the identified set.

Footnote

When approximating identified sets for the elements of a d-dimensional vector, Montiel Olea and Nesbit (2021) derive bounds for the number of draws that are required to achieve an approximation error that is at most some value with at least some probability. Based on their upper bound for the number of draws, using 5,000 draws of Q is consistent with an approximation error of at most 9 per cent with probability close to 100 per cent. [28]