RDP 2022-04: The Unit-effect Normalisation in Set-identified Structural Vector Autoregressions 6. Ruling Out Unboundedness Using Alternative Restrictions

In the context of estimating the effects of monetary policy, the identified set for the impulse responses to a 100 basis point shock may be unbounded when the identified set for the impact response of the federal funds rate to a standard deviation monetary policy shock includes zero. Imposing sign, zero or narrative restrictions of the types considered in Section 5 can indirectly rule this possibility out. This section discusses alternative restrictions that could potentially be used to rule out the possibility that the monetary policy shock has no impact effect on the federal funds rate. Although this discussion is framed in the context of monetary policy, it also applies more generally to other settings.

6.1 Direct bounds on impulse responses

One possibility is to directly restrict the impact response of the federal funds rate to a standard deviation monetary policy shock so that it is greater than some (strictly positive) number (i.e. ${\eta }_{1,1,0}\ge \lambda ,$ where $\lambda >0$ is a specified scalar). However, it seems difficult to justify such restrictions on the basis of economic theory – what is the smallest plausible impact effect of a ‘standard deviation’ monetary policy shock on the federal funds rate? Restrictions of this type could potentially be justified on the basis of prior estimates (e.g. from other SVARs or from estimated DSGE models), but these prior estimates may themselves be based on assumptions that lack credibility. Alternatively, one could impose bounds on the responses of variables to a unit shock such that unbounded impulse responses are ruled out by assumption (e.g. $\left|{\tilde{\eta }}_{2,1,0}\right|\le \lambda$ ).^[22] However, it seems similarly difficult to come up with hard bounds on the responses of variables to a 100 basis point shock without these bounds being somewhat arbitrary. Moreover, in either case, when identified sets are unbounded in the absence of such restrictions, inferences may be highly sensitive to changes in the imposed bounds.

To illustrate, return to the bivariate example of Section 3 and consider additionally imposing that ${\eta }_{1,1,0}\ge \lambda$ for some $\lambda >0$. When ${\sigma }_{21}\ge 0$ and $0<\lambda \le {\sigma }_{11}$, the identified set for ${\tilde{\eta }}_{2,1,0}$ is ^[23]

The additional restriction therefore results in the identified set being bounded; in the absence of this restriction (or in the limit as $\lambda$ approaches zero from above), the identified set is $\left(-\infty ,0\right].$ However, the lower bound of the identified set is sensitive to the choice of $\lambda ,$ particularly when $\lambda$ is small; the derivative of the lower bound tends to $\infty$ as $\lambda$ approaches zero from above. Setting $\lambda$ to some small positive number to rule out an unbounded identified set for ${\tilde{\eta }}_{2,1,0}$ may therefore yield an identified set that is highly sensitive to the choice of $\lambda$.

6.2 Bounds on the forecast error variance decomposition

Rather than directly restricting the impact effect of the monetary policy shock on the federal funds rate, one could instead consider restricting the one-step-ahead forecast error variance decomposition (FEVD) of the federal funds rate with respect to the monetary policy shock. This is the contribution of the monetary policy shock to the one-step-ahead forecast error variance (FEV) of the federal funds rate. Such restrictions may indirectly rule out the possibility that the monetary policy shock has no impact effect on the federal funds rate; intuitively, a strictly positive lower bound on the contribution of the monetary policy shock to the one-step-ahead FEV of the federal funds rate means that the impact effect of the shock itself must be strictly positive.

More formally, the horizon- h FEVD of the i th variable with respect to the j th shock is

The impact effect of the j th shock on the i th variable $\left({c^{\prime }}_{i0}\left(\varphi \right)q_j={e^{\prime }}_{i,n}{\Sigma }_{tr}{q}_j\right)$ is zero if and only if $FEV{D}_{i,j,0}\left(\varphi ,Q\right)=0,$ so bounding $FEV{D}_{i,j,0}\left(\varphi ,Q\right)$ away from zero indirectly bounds the impact response away from zero. However, as in the case where the normalising impulse response is directly bounded away from zero, the identified set obtained under some small lower bound on the FEVD will also be sensitive to the choice of this lower bound when the identified set is unbounded in the absence of this restriction (see Appendix A.5 for an analysis of this case in the context of the bivariate model). Volpicella (2022) proposes imposing bounds on the FEVD, where the bounds are elicited from a range of estimated DSGE models. However, if the assumptions underlying the DSGE models that are used to elicit these bounds lack credibility, the derived bounds on the FEVDs will also lack credibility.

Footnotes

These types of restrictions are sometimes referred to as ‘elasticity restrictions’ (as in Kilian and Murphy (2012)). [22]

If $\lambda >{\sigma }_{11}$, the identified set is empty. See Appendix A.4 for details about this example. [23]