RDP 2022-04: The Unit-effect Normalisation in Set-identified Structural Vector Autoregressions 1. Introduction

Estimating the response of the economy to macroeconomic shocks (such as monetary policy shocks) is difficult, because it requires disentangling the effects of the shock from the effects of other shocks hitting the economy at the same time. Macroeconomists do this by imposing ‘identifying restrictions’, which are assumptions about the structure of the economy. When using structural vector autoregression (SVAR) models, researchers have traditionally imposed identifying restrictions that are sufficient to pin down the responses to the shocks of interest, in which case we say that the responses are ‘point identified’. However, it has become increasingly common to impose arguably weaker sets of identifying restrictions at the expense of only being able to determine a set of possible responses, in which case we say that the effects are ‘set identified’. These weaker sets of restrictions often take the form of restrictions on the signs of impulse responses to shocks (e.g. Uhlig 2005).

Set-identified SVARs are typically estimated under the normalisation that structural shocks have unit standard deviation. The impulse responses that are obtained under this ‘standard deviation normalisation’ consequently represent impulse responses to a standard deviation shock. However, these impulse responses often do not answer the pertinent economic question. For instance, central bankers are interested in answering questions like ‘what are the effects of a 100 basis point increase in the policy rate?’ To answer this question, we need to know what happens following a monetary policy shock that results in a 100 basis point increase in the policy rate. The responses to a ‘unit shock’ – a shock that raises a particular variable by one unit – are therefore naturally more relevant in this setting (and many others). Such responses can be obtained under the ‘unit-effect normalisation’ (Fry and Pagan 2011; Stock and Watson 2016, 2018). In this paper, I explore the extent to which set-identifying restrictions are informative about impulse responses to unit shocks (or ‘unit impulse responses’).

Set-identifying restrictions generate an ‘identified set’ for the impulse responses, which is the set of impulse responses that are consistent with the data given the identifying restrictions. The identified set for a unit impulse response may be unbounded (Baumeister and Hamilton 2015, 2018). This implies that set-identifying restrictions may be extremely uninformative about the effects of a unit shock, which is a point that appears to have been underappreciated in the literature.^[1] A contribution of this paper is to highlight this issue and explain why it arises.

To provide some intuition about why the identified set for a unit impulse response may be unbounded, consider estimating the response of output to a monetary policy shock. The impulse response of output to a 100 basis point monetary policy shock can be defined as the impulse response of output divided by the impact response of the policy rate (the ‘normalising impulse response’), where both impulse responses are with respect to a standard deviation monetary policy shock. The identifying restrictions may admit the possibility that the policy rate does not respond on impact to the shock; in other words, the identified set for the normalising impulse response may include zero. In this case, it may be possible to make the impulse response of output to a 100 basis point shock arbitrarily large in magnitude by considering a sequence of parameters converging to the point where the impact response of the policy rate is zero.^[2]

I discuss how researchers can draw useful inferences about unit impulse responses when the identified set is potentially unbounded, with a focus on the prior robust approach to Bayesian inference proposed in Giacomini and Kitagawa (2021). As elaborated on below, this is a natural approach to conducting Bayesian inference in set-identified models, because it eliminates the problem of posterior sensitivity to the choice of prior that arises in this setting. Section 2 describes the modelling framework and outlines the robust Bayesian approach to inference. The key feature of this approach to inference is that it replaces the prior distribution with a class of priors, which contains all priors that are consistent with the identifying restrictions. In turn, the class of priors generates a class of posteriors. Summarising the class of posteriors requires computing the lower and upper bounds of the identified set for each impulse response. Importantly, whether different summaries of the class of posteriors are bounded will depend on the posterior probability that the identified set is bounded. I therefore argue that it is crucial to understand how often the identified set is unbounded in any given application, since this helps to communicate transparently about the informativeness of the identifying restrictions.

To make these issues clear, I use a bivariate SVAR in which I can analytically characterise identified sets under some sign restrictions on impulse responses (Section 3). I then explain how to verify whether identified sets for the impulse responses to a unit shock may be unbounded in an SVAR of arbitrary dimension identified using both sign and zero restrictions (Section 4). I show that a necessary condition for unboundedness of these identified sets is that zero is included within the identified set for the normalising impulse response. I then provide an easily verifiable sufficient condition under which the identified set for the normalising impulse response includes zero; specifically, if the number of sign and zero restrictions is no greater than the number of variables in the SVAR and the restrictions relate to a single structural shock, the identified set for the normalising impulse response always includes zero. When this sufficient condition is not satisfied (i.e. when there are more restrictions than variables in the SVAR and/or the restrictions relate to multiple shocks), the identified set for the normalising impulse response may or may not include zero. In this case, I explain how to numerically check whether the identified set for the normalising impulse response includes zero. Ultimately, I recommend that researchers report the posterior probability that the normalising impulse response includes zero, since this makes it clear which summaries of the class of posteriors are guaranteed to be bounded.

To illustrate the importance of these issues in practice, I estimate the macroeconomic effects of a 100 basis point shock to the federal funds rate under different combinations of identifying restrictions (Section 5): the sign restrictions on impulse responses to a monetary policy shock proposed in Uhlig (2005); the sign and zero restrictions on the systematic component of monetary policy proposed in Arias, Caldara and Rubio-Ramírez (2019); and the ‘narrative restrictions’ proposed in Antolín-Díaz and Rubio-Ramírez (2018).

Under the restrictions considered in Arias et al (2019), the sufficient condition described above is satisfied, so zero is always included in the identified set for the normalising impulse response; that is, the identifying restrictions never rule out models in which the federal funds rate does not respond on impact to a monetary policy shock. This indicates that identified sets for the impulse responses to a 100 basis point shock may always be unbounded. Numerical approximations of the bounds of the identified set suggest that this is indeed the case. These restrictions are therefore extremely uninformative about the effects of a 100 basis point shock.

Combining the restrictions from Arias et al (2019) with the sign restrictions on impulse responses considered in Uhlig (2005) yields identified sets that are bounded with posterior probability close to, but less than, 100 per cent. Nevertheless, the class of posteriors is consistent with either relatively large decreases or increases in output following a 100 basis point shock, so the identifying restrictions appear to be fairly uninformative about the output response to the shock.

Additionally imposing narrative restrictions on the monetary policy shock (as in Antolín-Díaz and Rubio-Ramírez (2018)) yields identified sets that are bounded at every posterior draw of the reduced-form parameters. The results under this set of restrictions are consistent with the peak effects of monetary policy on output lying towards the smaller end of the range of existing estimates summarised in Ramey (2016).

Finally, I discuss the possibility of using alternative identifying restrictions to ensure that identified sets for unit impulse responses are bounded (Section 6). For instance, I consider directly bounding the normalising impulse response away from zero. However, I argue that it may be difficult to elicit a credible lower bound and inferences may be sensitive to changes in the imposed bound. I conclude that such restrictions are unlikely to be a satisfactory solution.

1.1 Related literature

An extensive literature uses set-identified SVARs to estimate the effects of macroeconomic shocks.^[3] Under the standard approach to Bayesian inference in set-identified SVARs (e.g. Uhlig 2005; Rubio-Ramírez, Waggoner and Zha 2010; Arias, Rubio-Ramírez and Waggoner 2018), it is straightforward to transform from the standard deviation normalisation to the unit-effect normalisation; this transformation simply requires dividing the impulse responses obtained under the standard deviation normalisation by the normalising impulse response.^[4] Repeating this at each draw of the parameters from their posterior distribution generates a posterior distribution for the impulse responses to a unit shock. However, there are well-documented problems with the standard approach to Bayesian inference in set-identified models. In particular, because the model is set identified, the likelihood function is flat with respect to certain parameters. As a consequence, a component of the prior is ‘unrevisable’ in the sense that it is never updated, and posterior inference may be sensitive to the choice of prior (e.g. Poirier 1998).^[5]

Baumeister and Hamilton (2015) show that the ‘uniform’ prior that is used in the standard approach to Bayesian inference does not necessarily induce a uniform prior over the parameters that are typically of interest, such as impulse responses. They argue that this prior may therefore drive posterior inference despite not reflecting the researcher's actual prior beliefs. As an alternative, they suggest that researchers should impose a prior directly over the structural parameters (see also Baumeister and Hamilton (2018, 2019)). It remains the case under this approach that a component of the prior will never be updated, so posterior sensitivity to the choice of prior may still be a concern.

To address the problem of posterior sensitivity in set-identified models, Giacomini and Kitagawa (2021) propose conducting Bayesian inference using an approach that is robust to the choice for the unrevisable component of the prior. When applying this approach in a set-identified SVAR, they focus on the impulse responses to standard deviation shocks as the parameters of interest. Giacomini, Kitagawa and Read (2022b) describe an algorithm for conducting robust Bayesian inference in proxy SVARs (i.e. SVARs identified using an external instrument) under the unit-effect normalisation. They note that the identified set may be unbounded, but do not draw out the implications of this issue for conducting inference. Baumeister and Hamilton (2015, 2018) explicitly show that the identified set may be unbounded in a simple bivariate model identified with sign restrictions. My own bivariate example builds on their setting by: 1) explaining that the unbounded identified set arises due to the identifying restrictions not ruling out the possibility that a variable does not respond to its own shock; 2) drawing out additional intuition about this result; and 3) explaining some implications of an unbounded identified set for conducting inference.

Unbounded identified sets for unit impulse responses will also arise in other settings, so some of the results in this paper are applicable more broadly. Existing approaches to frequentist inference in set-identified SVARs focus on impulse responses to standard deviation shocks as the parameters of interest (e.g. Gafarov, Meier and Montiel Olea 2018; Granziera, Moon and Schorfheide 2018). If the maximum likelihood estimator (MLE) of the reduced-form parameters is such that zero is included within the identified set for the normalising impulse response, frequentist estimates of identified sets for impulse responses to a unit shock may be unbounded. Unboundedness of the identified set may also arise when imposing set-identifying restrictions in a local projection framework (Plagborg-Møller and Wolf 2021).

Notation. For a matrix X, vec(X) is the vectorisation of X. When X is symmetric, vech(X) is the half-vectorisation of X, which stacks the elements of X that lie on or below the diagonal into a vector. e_i,n is the i th column of the n×n identity matrix, I_n. 0_n×m is an n×m matrix of zeros.

Footnotes

If the VAR is stable, the identified set for the impulse response to a standard deviation shock is always bounded. The issue of unboundedness is therefore specific to the case where the impulse response of interest is to a unit shock. [1]

I do not make a judgement about the plausibility of the non-response of the policy rate to a monetary policy shock. Rather, I highlight that typical set-identifying restrictions do not necessarily rule out this possibility, and I explore the implications that this has for learning about the impulse responses to unit shocks. Interestingly, conventional macroeconomic theory also does not rule out this non-response; in the textbook New Keynesian model (e.g. Galí 2008), there exists a value for the persistence of the monetary policy shock such that the nominal interest rate does not respond on impact to the shock. [2]

In their online appendix, Baumeister and Hamilton (2018) provide a list of close to 100 studies that estimate set-identified SVARs. [3]

The transformation is not necessarily innocuous in the point-identified setting when conducting frequentist inference. In particular, if the sampling distribution of the estimator for the normalising impulse response has probability mass near zero, the sampling distribution of the estimator for the unit impulse response may be poorly approximated by the usual asymptotic normal distribution; see Stock and Watson (2016) for a heuristic discussion of this point. [4]

The posterior density is the product of the likelihood and the prior density. Conditional on the reduced-form parameters, the likelihood function is flat with respect to the structural parameters, so the posterior will be proportional to the prior. [5]