European Economic Review 145 (2022) 104120

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European Economic Review
journal homepage: www.elsevier.com/locate/eer

Excess shocks can limit the economic interpretation✩
Adrian Pagan a , Tim Robinson b ,∗ab

School of Economics, University of Sydney, Australia
Melbourne Institute: Applied Economic and Social Research, University of Melbourne, Australia

ARTICLEJEL classification:
E37
C52
Keywords:
Shock recovery
Kalman filter

INFO

ABSTRACT
When a model has more shocks than observed variables (excess shocks) the Estimated Model(EM) shock innovations will be correlated. These correlations limit the usefulness of varianceand variable decompositions, as the latter use the EM shock innovations. Furthermore, anysuch correlations limit the ability to directly test the assumptions made about the Assumed
Model (AM) shocks. These need to be correct in order to interpret impulse responses. A partial
interpretation of the data may be possible if some AM shocks can be recovered. An approach todetermining which shocks can be recovered when using either the current and past or all data
is presented, unifying the existing methods for assessing recovery. It is applicable to a wide
range of macroeconomic models.

1. Introduction
Modern macroeconomics interprets the macroeconomy using a model and its structural shocks. Shocks in models are given namesthat have economic meanings. These names could stem from theory, as in Dynamic Stochastic General Equilibrium (DSGE) models,
but may also just be provided based on a particular structural equation, as with Structural VARs (SVARs) models. The dynamics ofthe economy are then studied through the impulse responses to perturbations in the shocks, and an understanding of past economic
developments is obtained by decomposing the observed data into the contributions of the shocks. This paper discusses a common
situation in which the ability to interpret the macroeconomy using structural shocks can be limited. We explain why these limitations
arise and provide applied economists with a simple approach to determining which shocks are affected. This approach can be applied
after estimation of a wide variety of structural macroeconomic models, including DSGE and SVARs, with or without Time-Varying
Parameters (TVP). The approach can also be applied to other macroeconomic models which involve shocks, but which have lessstructural interpretation, such as Unobserved Components (UC) and some Factor Models (FMs).
The situation which can limit the ability to interpret the economy is when the number of shocks in the model exceeds the numberof observed variables. We term this situation one of excess shocks; alternatively (Forni et al., 2019) call such systems ‘‘short’’ . The
presence of excess shocks is widespread. For UC and FMs it is invariably the case. A common example with structural models is when
DSGE models incorporate measurement error. However, there are many other possible sources, such as indeterminacies (resulting insunspots), stochastically time-varying coefficients, and Markov switching. Excess shocks are common in macroeconomic structuralmodeling; for example, they occur in around one quarter of the estimated DSGE models of the U.S. economy in the Macroeconomic✩ Previously circulated as ‘‘Too Many Shocks Spoil the Interpretation’’. Some of the material appeared in an earlier working paper, Pagan and Robinson(2019). Our thanks to two anonymous referees, the Associate Editor, Ryan Chahrour, Marco Del Negro, Andrew Harvey, George Kapetanios, Mariano Kulish,
James Morley, Dan Rees, Guido Turnip and participants at the 2021 International Association of Applied Econometrics Conference and Econometrics Society
Australasian Meeting for extremely useful and challenging comments which have greatly improved the paper. Our thanks to Qazi Haque and Giovanni Nicolò
for supplying data. This work was supported by ARC Grant DP160102654.
∗ Correspondence to: Level 5, Building 105, The University of Melbourne, Parkville, Victoria, 3010, Australia.
E-mail addresses: adrian.pagan@outlook.com (A. Pagan), tim.robinson@unimelb.edu.au (T. Robinson).

https://doi.org/10.1016/j.euroecorev.2022.104120
Received 25 June 2021; Received in revised form 1 March 2022; Accepted 2 March 2022
Available online 14 April 2022
0014-2921/© 2022 The Authors.
Published by Elsevier B.V. This is an open(http://creativecommons.org/licenses/by/4.0/).

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licenseEuropean Economic Review 145 (2022) 104120

A. Pagan and T. Robinson

Model Database.1 This includes models published in leading journals, such as the American Economic Review, the Economic Journal
and the Review of Economics and Statistics.2 Excess shocks are present in the large-scale DSGE models developed at central banks andother policy institutions, such as RAMSES II (Sveriges Riksbank; Adolfson et al., 2013) and the Multi-Sector Model (Reserve Bank
of Australia; Rees et al., 2016).
As part of giving these shocks names assumptions are made about their nature. There are parallels with regression analysis. In a
regression we make assumptions about the error term — for example, that it is uncorrelated and homoskedastic. In the same way
many DSGE models have shocks, 𝑢𝑡 , that are assumed to follow a first-order autoregressive process of the form 𝑢𝑡 = 𝜌𝑢𝑡−1 + 𝜀𝑡 . In
time-series analysis the 𝜀𝑡 are called innovations as they are assumed to be unpredictable using the past history of 𝑢𝑡 , i.e. the 𝜀𝑡 areuncorrelated with 𝜀𝑡−𝑗 (𝑗 > 0). Usually there are a number of shocks in a DGSE model, each of which is driven by innovations, and
it is assumed that they are uncorrelated with each other. We will label these shocks in the structural model as Assumed Model (AM)shocks. In the case of DSGE models the names of the shocks come from theory, and so they are often said to be ‘‘theoretical shocks’’.
There are two reasons why assumptions are made about the nature of the structural shocks in DSGE models. One is to perform
estimation of the model’s parameters, exactly like a regression. But there is a second reason which relates to interpretation. Assumingthat the innovations driving the structural shocks are uncorrelated with each other allows one innovation to be changed and thereby
for the effects of that perturbation to be traced out by holding constant the innovations in other shocks, i.e. it produces a ceterisparibus assumption. If instead these innovations to the structural shocks are correlated then the one being perturbed will have aneffect on the other shocks, making interpretation of the impulse response functions problematic. Indeed, Ramey (2016) says that
one of the characteristics a shock should have is to be ‘‘ ...uncorrelated with other exogenous shocks; otherwise we cannot identify
the unique causal effects of one exogenous shock relative to another...’’ (p. 75).
A model can be used together with data to produce the Estimated Model (EM) shocks. In a regression they are called residuals. The
residuals need not satisfy the properties investigators assumed about the error term of the regression. As is well known such a failure
can have consequences for the subsequent inferences drawn from the regression, and so many procedures have been developed for
detecting these failures. Exactly the same situation should occur in structural macroeconomic models, such as DSGEs and SVARs. Ina regression it is straightforward; one uses the residuals to check the assumptions made about the errors. But in the case of structuralmodels with excess shocks it is not so simple, and we detail why in the paper. Yet it needs to be done.
To illustrate these issues in this introduction we consider the recent work of Farmer and Nicolò (2021) (FN). We return to amore detailed analysis of it later in the paper. FN estimate a model developed in Farmer and Nicolò (2018) for several countries.
They assess its fit relative to some other popular models, as well as commenting on the role of shocks in explaining cross-country
divergences in economic performance. We will focus on their estimates for the U.S. The FN model is a New Keynesian (NK) modelwith inflation, output and an interest rate driven by supply, demand and monetary shocks. But it lacks a Phillips curve, and instead
the system is closed with beliefs about nominal GDP growth. That can lead to static and dynamic indeterminacies. As a consequence
there will be a sunspot solution which can be represented as relating to the three ‘‘fundamental’’ shocks above and a fourth shock,which we will refer to as animal spirits. Consequently, there are three observed variables and four shocks in this model, resulting
in excess shocks. As one of the conclusions of FN is that differences in shocks (and the responses of central banks to them) explain
divergences in economic performance, one would like to check if these can be recovered in order to assess their properties.
A first complication in checking the properties of the AM shocks arises from the fact that with excess shocks the EM shock
innovations must be correlated, even when this is not the case for AM shock innovations. This means that one cannot simply inferproperties of the AM shock innovations from the EM shocks, as one would do with the residuals from a regression.
A second complication is that the commonly used interpretation procedures of variance and variable decompositions use the EM
shocks. In the presence of excess shocks the EM shock innovations are always correlated and these decompositions do not apply.
For example, in FN’s work the correlation between the estimated supply and monetary shock innovations is .34; this is despite theassumption that the fundamental AM shock innovations are maintained to be uncorrelated. Thus one could not use a variancedecomposition to find which shock is particularly important in driving the business cycle.
We are not the first to draw attention to the fact that there are consequences of having excess shocks. Ravenna (2007, p. 2051),
in the context of understanding the relationship between DSGE and SVARs, stated ‘‘...it will not be possible to map 𝑦𝑡 into a higherdimension vector of orthogonal shocks’’. Forni et al. (2019, p. 226) point out that their recovered (estimated) shock innovations
would be ‘‘linearly dependent’’, i.e. they would be correlated, even if the AM shock innovations are not. Plagborg-Moller and Wolf
(2020) and Chahrour and Jurado (2022) show that excess shocks will mean that not all of the AM shocks are recoverable. Indeed,
Chahrour and Jurado (2022, p.8) say ‘‘...a necessary condition for all the structural shocks to be recoverable is that there be at least
as many observable variables as disturbances...’’. So the consensus of the existing literature is that the EM and AM shock innovations
are different when there is an excess of shocks, and this needs to be accounted for.
In this paper we add to the existing literature by developing an approach to determine the consequences of there being excess
shocks. It is intended to help applied researchers know along what dimensions any of their models which feature excess shocks canbe used to interpret the macroeconomy, and when they cannot. This is done in three stages. First, we ask which of the AM shock
innovations can be recovered from the data, that is, from the EM shock innovations. This simply involves asking whether the AM
shock innovations are the same as the EM ones. In order to avoid secondary issues such as sample size we take the parameters of

1 Version 3.1.14. The Macroeconomic Model Database is an outcome of the Macroeconomic Model Comparison Initiative. In turn this is a product of the
Hoover Institution at Stanford University and the Institute for Monetary and Financial Stability at the Goethe University Frankfurt. For further information see
https://www.macromodelbase.com/. When multiple variants of an individual model are included we use only one to avoid double counting.
2 See Christiano et al. (2014), Chen et al. (2012) and Ireland (2011).

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the assumed model as known. In a regression this would make the assumed error and the residuals the same. Applying our approachto the FN model we find that the supply and sunspot shock innovations cannot be recovered.
Second, for those AM shock innovations that can be recovered we ask whether they exhibit the assumptions made about themby looking at their estimated counterparts. In the FN model it was assumed that monetary policy shock innovations had zero serial
correlation. As these can be recovered from the EM shock innovations the properties of that AM shock’s innovations can be assessed
directly; they are found to be heavily serially correlated and so are not truly innovations.3
Third, we provide a way of assessing the properties of those shocks that cannot be recovered, utilizing indirect inference. It
proceeds by determining the correlation in the EM shock innovations that the presence of excess shocks induces when there is nonein the AM shock innovations. This is then compared to the actual correlations in the EM shock innovations. These should not differ
if the assumptions made about the model shocks were valid.
This article is structured as follows. Section 2 makes the basic point that, when the number of shocks equals the number ofobservables, one can generally treat the EM shocks from the model as the AM shocks, i.e. the latter are recoverable.4 Section 3
then turns to determining whether it is possible to recover some of the AM shock innovations where there is an excess of them.
The existing methods for assessing this are discussed, which highlights that different types of information can be used for recovery;
Forni et al. (2019) use current and past information, which aligns with the traditional concept of invertibility, whereas Chahrour
and Jurado (2022) additionally use future information.
An intuitive way to assess if an assumed shock innovation can be recovered is to ask if the variance (𝑃 ) of the divergence between
it and its estimated counterpart is zero. We show in Sections 3.1 and 3.2 that 𝑃 can be computed using the steady-state Kalman filterand smoother. There will be two 𝑃 𝑠, depending on which of the two information sets mentioned above are used.5 Our approach iseasy to implement and unifies the existing literature.
The Kalman filter is also useful for understanding why there is a divergence between AM and EM shock innovations when there
is an excess. The filter enables the computation of an estimate of the one-step ahead prediction errors of the observed variables, 𝜂𝑡 .
When there are no excess shocks the number of 𝜂𝑡 is the same as the number of assumed shocks, but with excess shocks there areless, and therefore knowledge of 𝜂𝑡 alone cannot be used to recover all the AM shock innovations.6
In Section 3.3 we demonstrate our state-space approach with a simple one variable and two shock UC model. Such models areoften used to generate filters to recover quantities that are functions of the shocks, such as an output gap, a credit gap or a NAIRU.7
The properties of the estimated and assumed gaps may differ considerably. This is known in the literature; we demonstrate that itarises due to excess shocks. Section 3.4 looks at the problems that correlated innovations create for impulse response computations.
Generally, there is no unique response.
Our approach for assessing the properties of the non-recoverable AM shocks uses indirect inference and is presented in
Section 3.5. We then consider in Section 3.5.1 an influential model where the source of excess shocks is the introduction of a sunspot
due to dynamic indeterminacy, namely Lubik and Schorfheide (LS) (2004). It is found that the shock termed a sunspot cannot berecovered. Applying our indirect inference approach points to the assumed sunspot innovation being significantly correlated withthe assumed monetary and supply shock innovations. This is followed in Section 3.5.2 by further analysis of Farmer and Nicolò
(2021), which we have used throughout the introduction.
Section 3.6 provides a demonstration of other ways in which excess shocks can impact on the ability to interpret themacroeconomy. Two examples are used; one is the risk paper of Christiano et al. (2014), where excess shocks come from ‘‘news’’,
and the other a simple TVP model. It is found in the former that the variance of interest rate spreads in the data is almost a thirdof what their model predicts. This is due to the fact that the EM shock innovations are correlated. Variance decompositions about
the importance of risk shocks therefore are not reliable.
Section 4 closely examines a New-Keynesian model, Ireland (2011). We use it to demonstrate our method and some strategies thatapplied researchers can use to deal with excess shocks. This model was chosen as it uses standard techniques that are representativeof much of the DSGE literature. It has four AM shocks but only three observed variables, and therefore has excess shocks. There
are no sunspots or news shocks, as was the case in our earlier examples. We find that two of the four AM shock innovations areclose to being recoverable. Consequently, assuming that they are, we can assess the validity of the assumptions made about them.
Ultimately, these shock innovations are found to be uncorrelated but autocorrelated, so the ability to compute impulse responses is
to some extent limited. The section also deals with some issues raised by referees concerning factor models and the assumption wemade that parameters were known when, in practice, they must often be estimated.
In summary, the AM structural shock innovations provide information about the nature of the model but they do not interpretthe data unless they equal the EM shock innovations, and this cannot be the case for all of them when there are excess shocks.
We believe that it is important for applied researchers to be cognizant of the resulting limitations of their models when making
interpretations in this context. In particular, excess shocks create problems in two ways. First, the correlation they induce in the EM

3 There is a semantic issue here. We should really refer to ‘‘the estimate of the random variable assumed to be an innovation’’, rather than ‘‘EM shock
innovations’’, as the latter may not be innovation at all, i.e. it may have serial correlation.
4 Of course it has to be that the model is not mis-specified and there are no identification issues.
5 In order to perform these operations the model must be placed into a State-Space Form (SSF), which is very common for DSGE and UC models. For the
FN model illustration we used smoothed shocks, which is typical for DSGE models.
6 This is distinct from parameter estimation. Using the Kalman filter, the likelihood for the model is expressed in terms of 𝜂 alone, and consequently as long𝑡
as the parameters are identified, their estimation is not an issue.
7 More complex UC models, including the Hodrick–Prescott filter, are assessed in an appendix. The conclusions are the same.

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shock innovations makes it difficult to examine the assumptions made about the AM shock innovations. This paper addresses howthat difficulty can be overcome. We provide an approach by which the properties of the AM shock innovations can be assessed fora wide variety of models, either directly and indirectly. Second, we demonstrate that the correlation in the EM shock innovations
limits the usefulness of variance and variable decompositions.
2. Recovering the AM shocks when there is no excess
For simplicity, in order to consider the nature of the shocks when there is no excess, let us first assume that the solution for amodel is a SVAR in 𝑛 observable variables 𝑦𝑡 and the 𝑛 𝐴𝑀 shock innovations 𝜀𝑡
𝐴0 𝑦𝑡 = 𝐴1 𝑦𝑡−1 + 𝜀𝑡 ,where 𝐴0 and 𝐴1 are matrices. We assume that these matrices are known. The assumption that the parameters are known meansthat the filtered 𝐸𝑀 innovations, 𝐸𝑡 𝜀𝑡 ≡ E(𝜀𝑡 |𝑦1 ...𝑦𝑡 ) are 𝐸𝑡 𝜀𝑡 = 𝐴0 𝑦𝑡 − 𝐴1 𝑦𝑡−1 , and these coincide with the realization of the AM shock
innovations 𝜀𝑡 . This enables us to use the 𝐸𝑀 innovations to test the assumptions made about the 𝐴𝑀 shock innovations. If instead
the parameters were not known, then these two sets of shocks would not necessarily coincide as, for example, the model may be
mis-specified. Throughout this paper we work with known parameters so as to focus upon the impacts of excess shocks alone.
We further assume that 𝐴−1
exists so that the Moving-Average (MA) representation is0
𝑦𝑡 = 𝐶0 𝜀𝑡 + 𝐶1 𝜀𝑡−1 + ⋯ ≡ 𝐶(𝐿)𝜀𝑡 .

(1)

There is an underlying VAR process for 𝑦𝑡 of the form
𝑦𝑡 = 𝐵1 𝑦𝑡−1 + 𝑒𝑡 ,, 𝐶𝑗 ≡ 𝐵1 𝐶𝑗−1 (𝑗 > 1).8 Suppose the AM shock innovations 𝜀𝑡 are assumed to be uncorrelated with covariance matrix
and 𝐶0 ≡ 𝐴−1
0
𝐼𝑛 . Then 𝐶0𝑖𝑗 is the contemporaneous impulse response of the 𝑖’th variable to the 𝑗’th AM shock.
Whether the 𝐴𝑀 shock innovations are uncorrelated is an assumption which we wish to evaluate. If the parameters of the SVAR
were known there are no excess shocks and the 𝐸𝑀 shock innovations will be the 𝐴𝑀 shock innovations. In practice, however theparameters are estimated and, as the SVAR is exactly identified, the EM shock innovations will be uncorrelated by construction,
regardless of the nature of the 𝐴𝑀 shock innovations. This is just like a regression where the residuals must be orthogonal to thecovariates. Because of this variance decompositions can be done with the 𝐸𝑀 shocks.
Most DSGE models can be represented as a SVAR, in which case 𝐴0 and 𝐴1 are functions of the deep parameters of the DSGE
model. However, the SVAR implied by a DSGE model is rarely exactly identified; see Liu et al. (2018). One example of a source ofoveridentifying restrictions in DSGE models is the common assumption of rational expectations. Thus the 𝐸𝑀 shock innovations fromDSGE models do not have zero correlation after their parameters are estimated and so, when there are no excess shocks, assumptions
made about the 𝐴𝑀 shocks can be potentially checked using the 𝐸𝑀 shocks.
As mentioned in the introduction, our analysis distinguishes between two types of EM shocks. These differ in the information sets
used for their conditioning. If, as above, information up to time 𝑡 is used, then we would have filtered estimates 𝐸𝑡 𝜀𝑡 ≡ E(𝜀𝑡 |𝑦1 ...𝑦𝑡 ),while the smoothed estimates are 𝐸𝑇 𝜀𝑡 ≡ E(𝜀𝑡 |𝑦1 ...𝑦𝑇 ), and use all the data (i.e. including future observations). If there are no excess
shocks 𝐸𝑡 𝜀𝑡 = 𝐴0 𝑦𝑡 − 𝐴1 𝑦𝑡−1 = 𝐸𝑇 𝜀𝑡 , i.e. the 𝐴𝑀 and 𝐸𝑀 shocks are the same.
3. Recovering the AM shocks if there is an excess
As not all models used in macroeconometrics have a SVAR structure, we need to allow for these more general cases. This will
be done with a State Space Form (SSF). The SSF we adopt follows Nimark (2015) and is:𝑧𝑡 = 𝐷1 𝜓𝑡 + 𝐷2 𝜓𝑡−1 + 𝑅𝜀𝑡(2)𝜓𝑡 = 𝑀𝜓𝑡−1 + 𝐶𝜀𝑡 .

(3)Here Eqs. (2) and (3) are the observation and state equations respectively, 𝑧𝑡 are the 𝑛 × 1 observed variables, 𝜓𝑡 the 𝑝 × 1 core model
variables, and 𝜀𝑡 the 𝑚 × 1 vector of shock innovations. The latter are assumed to be 𝑁(0, 𝐼𝑚 ). By ‘‘core’’ model variables we mean
those that cannot be substituted out. DSGE models, for example, often have variables that can be substituted out using identities
included in the model. It must be that 𝑝 ≥ 𝑛 and with excess shocks 𝑚 > 𝑛. Also, to reiterate, frequently in DSGE models the shocks
are assumed autocorrelated; we are therefore focusing on their innovations and to study them we will assume they are placed as
the last elements in 𝜓𝑡 .9 In most cases 𝑝 > 𝑚. 𝐷1 , 𝐷2 , 𝑅 and 𝐶 are matrices of parameters.
We now turn to the case where there may be excess shocks. In many instances macroeconometric models will include variables
that are not observed; some are latent. One example of this occurring is when auxiliary shocks like measurement error are added;
each observed variable will have an unobserved counterpart. Another example is when the DSGE includes stock variables that are

8
9

The reduced-form 𝑒𝑡 are sometimes referred to as innovations in the literature. They are combinations of the structural shock innovations 𝜀𝑡 .
This will generally mean that 𝑅 = 0.
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A. Pagan and T. Robinson

treated as unobserved, such as the capital stock. These are easily handled by using the SSF. Assessing shock recoverability throughthe SSF is also useful as it readily accommodates 𝐼(1) variables.
If the parameters are known we can clearly find the impulse responses to the AM shock innovations 𝜀𝑡 . This is mostly what
researchers do. But for these responses to be relevant it must be the case that the assumptions about 𝜀𝑡 being uncorrelated (and also
not serially correlated) are correct, and that is why we want to check these assumptions. However, the only innovations available
to perform an assessment are those recovered from the data – the 𝐸𝑀 ones – and so, ideally, we will want those to align with the
𝐴𝑀 shock innovations, as was the case in the previous section.

3.1. Tools of recovery — the Kalman filter and smoother
The Kalman filter can be applied to the SSF with known parameters to obtain the filtered EM shock innovations.10 Nimark (2015)shows that the filtered estimate of 𝜓𝑡 , 𝐸𝑡 𝜓𝑡 , given the data available to time 𝑡, and the system in Eqs. (2) and (3), evolves as(4)𝐸𝑡 𝜓𝑡 = 𝛷𝑡 𝐸𝑡−1 𝜓𝑡−1 + 𝐾𝑡 𝑧𝑡
𝐾𝑡 = [𝑀𝑃𝑡−1|𝑡−1 𝛹 ′ + 𝐶𝐶 ′ 𝐷1′ + 𝐶𝑅′ ]𝐹𝑡−1
′

(5)

′

(6)

𝐹𝑡 = 𝛹 𝑃𝑡−1|𝑡−1 𝛹 + 𝛬𝛬

𝑃𝑡|𝑡 = 𝑃𝑡|𝑡−1 − 𝐾𝑡 [𝛹 𝑃𝑡−1|𝑡−1 𝛹′

𝑃𝑡+1|𝑡 = 𝑀𝑃𝑡|𝑡 𝑀 + 𝐶𝐶′′

+ 𝛬𝛬 ]𝐾𝑡′

(7)

′

(8)

𝛹 ≡ 𝐷1 𝑀 + 𝐷2 , 𝛬 ≡ 𝐷1 𝐶 + 𝑅, 𝛷𝑡 ≡ 𝑀 − 𝐾𝑡 𝛹 ,(9)where 𝐾𝑡 is the gain of the Kalman filter, 𝑃𝑡|𝑡 the filtered variance of 𝐸𝑡 𝜓𝑡 − 𝜓𝑡 and 𝑃𝑡+1|𝑡 provides the one-step-ahead predictor of thelatter. As we observed above, to recover the filtered shock innovations the 𝐴𝑀 shock innovations are made elements of 𝜓𝑡 . In some
applications, this is naturally the case; in others, such as DSGE models, they must be added to the endogenous state variables. Thiscan be done by simply appending identities to the state equations (Eq. (2)), and padding the matrices 𝐷1 and 𝐷2 in the observation
equation, Eq. (1), with appropriate zeros, so as to be conformable with the expanded 𝜓𝑡 .
What happens if we try to recover the 𝐴𝑀 shocks using all the information? Kurz (2018, Equation 4.11) shows that the smoothedstates, 𝜓𝑡|𝑇 , are obtained from the recursion(10)

𝐸𝑇 𝜓𝑡 = 𝜓𝑡|𝑇 = 𝜓𝑡|𝑡 + 𝑃𝑡|𝑡 𝜏𝑡 ,′−1
𝜏𝑡 = 𝛹 𝐹𝑡+1
𝜂𝑡+1 + 𝐿′𝑡+1 𝜏𝑡+1 ,(11)

𝐿𝑡 ≡ 𝑀 − 𝐾𝑡 𝛹 ,(12)where 𝜂𝑡+1 is the one-step ahead prediction error of 𝑧𝑡+1 and 𝜏𝑇 = 0.
To capture the closeness of the 𝐴𝑀 innovations 𝜀𝑡 with the smoothed EM innovations 𝐸𝑇 𝜀𝑡 , we naturally use the variance of∗ . Kurz (Equation
their difference. These shocks are a sub-set of the 𝜓𝑡 so we denote the part of 𝑃𝑡|𝑇 corresponding to them as 𝑃𝑡|𝑇
4.13) gives the following expression for computing 𝑃𝑡|𝑇
𝑃𝑡|𝑇 = 𝑃𝑡|𝑡 − 𝑃𝑡|𝑡 𝑁𝑡 𝑃𝑡|𝑡−1
𝑁𝑡 = 𝛹 ′ 𝐹𝑡+1
𝛹 + 𝐿′𝑡+1 𝑁𝑡+1 𝐿𝑡+1 , 𝑁𝑇 = 0.
∗ = 0 then we canEqs. (10) and (11) can be used to investigate the nature of the smoothed shock innovations. Firstly, if 𝑃𝑡|𝑡∗recover the 𝐴𝑀 shock innovations from the filtered shocks. Indeed, we see that if 𝑃𝑡|𝑡 = 0, Eq. (10) shows there will be no difference
between filtered and smoothed shocks and they will both have whatever properties the 𝐴𝑀 shock innovations have been assigned.
∗ ≠ 0, then at 𝑡 = 𝑇 the smoothed and filtered innovations are the same but, when 𝑡 < 𝑇 , they will differ. Moreover
Alternatively, if 𝑃𝑡|𝑡the difference depends on 𝜏𝑡 and, even if the one step predictions (𝜂𝑡 ) are white noise, the fact that the 𝜏𝑡 are accumulated means
there will likely be serial correlation in the smoothed shock innovations. Thus the 𝐸𝑇 𝜀𝑡 will inherit some of the properties that 𝐸𝑡 𝜀𝑡has, but may also have more serial correlation.11

10 This variant of the Kalman filter, due to Nimark (2015), allows for a lag of the state in the measurement equations. This is useful for DSGE models which
include permanent shocks. In that case typically the observed non-stationary variables will be expressed as growth rates while the model variables will have beennormalized by another 𝐼(1) variable to make them 𝐼(0), i.e. they have been stationized. To give a concrete example, if the production function is 𝑋𝑡 = 𝐴𝑡 𝑁𝑡 ,where 𝑋𝑡 is output, 𝐴𝑡 a permanent technology shock and 𝑁𝑡 labor (hours), then the model will be expressed in terms of 𝐼(0) variables by using stationized𝑋
output, 𝐴𝑡 . It is the log of this variable that will appear in the model as 𝜓𝑡 , i.e. 𝜓𝑡 = 𝑥𝑡 − 𝑎𝑡 (lowercase denoting logs). These stationized model variables need to𝑡

be related to observables, which are growth rates for the 𝐼(1) variables, i.e. 𝑧𝑡 = 𝛥𝑥𝑡 . Hence 𝛥𝑥𝑡 = 𝛥𝜓𝑡 + 𝑎𝑡 , and this adds a lagged state 𝜓𝑡−1 to the observation
equation. Alternatively, one could expand the state to include a lag and use the standard Kalman filter recursions.
11 Henceforth, for brevity we will describe the smoothed 𝐸𝑀 shocks as correlated, rather than contemporaneously and serially correlated.
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3.2. Can we recover the AM shocks?
The Kalman filter produces filtered EM shock innovations while the Kalman smoother will give the smoothed ones, 𝐸𝑇 𝜀𝑡 .
Chahrour and Jurado (2022) give an excellent discussion of the history of the literature on whether 𝐴𝑀 shock innovations canbe recovered. They note that early work on this looked at whether the 𝐸𝑀 shocks could perfectly predict the 𝐴𝑀 shocks with theinformation used being just current and past. If this could be done then it was said that the model was invertible and the shocks
were fundamental.12 Chahrour and Jurado (2022) then suggest that, for example, when expectations are involved in a model one
would probably want to use all the data rather than just current and past, i.e. to ask the same question but now using smoothed
𝐸𝑀 shock innovations. This gives rise to their definition of recoverabilty. It is clearly wider than that of invertibility as it uses more
information.13
Definition 2 of recoverability in Chahrour and Jurado (2022) involves finding a smoothed estimate of 𝜀𝑡 , defined as the best
linear estimate of 𝜀𝑡 given all the data. That is what 𝐸𝑇 𝜀𝑡 from the Kalman smoother is. In the proof of Theorem 1 in their appendix∗ = 𝑣𝑎𝑟(𝜀 − 𝐸 𝜀 ), sothey state the condition for recoverability as ‖𝜀𝑡 − 𝐸𝑇 𝜀𝑡 ‖2 equaling zero. The Kalman smoother provides 𝑃𝑡|𝑇𝑡
𝑇 𝑡
∗ = 0, 𝐸 𝜀 satisfies the condition of Chahrour and Jurado’s definition of recoverability. We note that they explain that theif 𝑃𝑡|𝑇𝑇 𝑡condition involves ‘‘....the filtered variance of the disturbance of interest is zero in population’’. So they are essentially working∗ .
with 𝑃𝑡|𝑇∗ , Chahrour and Jurado (2022) express ‖𝜀 − 𝐸 𝜀 ‖2 as functions of theRather than using the Kalman smoother estimate of 𝑃𝑡|𝑇𝑡
𝑇 𝑡
coefficients of the moving average (MA) representation and obtain a condition on the MA coefficients that they then test. Directly∗ from the SSF and checking if it is zero is a straightforward alternative way of capturing this variance.14 Because
computing 𝑃𝑡|𝑇
we will have purely stationary variables in the SSF after any stationizing transformations (which is why Chahrour and Jurado canwork with a MA representation), it is known that there is a steady-state Kalman filter, and so we will ask whether the steady-state∗ is zero.15
counterpart of 𝑃𝑡|𝑇
As an illustration of this approach to assessing whether an AM shock is recoverable, consider the three equation model in
Chahrour and Jurado (2022). They show that only the third shock is recoverable, and this is only when future information is used.
⎡.0243⎤∗ , when their example is placed in a SSF, produces diag(𝑃 ∗ ) = ⎢.9757⎥. Consequently
Computing the diagonal of the steady-state 𝑃𝑡|𝑇
𝑡|𝑇
⎢
⎥
⎣ 0 ⎦
our approach also shows that only the third shock is recoverable. This is what we would expect given the equivalence of thesteady-state 𝑃𝑡|𝑇 with what they focus on.
Returning to working with filtered EM shocks, it will be the covariance matrix of the variable 𝜓𝑡 − 𝐸𝑡 𝜓𝑡 , i.e. the relevant elementsof 𝑃𝑡|𝑡 being zero, that shows recoverability of 𝐴𝑀 shock innovations from filtered 𝐸𝑀 shock innovations. Often this is called‘‘invertibility’’; it is possible for some shocks to be invertible (or fundamental) when the model is not invertible, but not all of them.
Forni et al. (2019) described the situation when there is sufficient information in a SVAR to recover a particular structural shock as
one involving partial fundamentalness, and developed a measure of informational deficiency (𝛿𝑖 ) to check for this. It follows Sims
and Zha (2006) and looks at the unexplained variance of the best linear projection of the shock of interest onto the residuals (inpopulation). That quantity is the variance of 𝜀𝑡 − 𝐸𝑡 (𝜀𝑡 ) from the Kalman filter. Their deficiency index 𝛿𝑖 has to be zero for the 𝑖’thshock if the 𝑖′ th 𝐴𝑀 shock innovation is to be recovered when using only current and past data. In the same way the (𝑖, 𝑖) element∗ in our approach will have to be zero. As an illustration of this we look at Forni et al.’s Example 1. This is a two equationof 𝑃𝑡|𝑡system in an output gap (𝑧1𝑡 ), an interest rate (𝑧2𝑡 ), demand shocks (𝜀1𝑡 ), and monetary shocks (𝜀2𝑡 ) of the form
(1 + 𝛾𝛽𝐿)𝑧1𝑡 = (1 + 𝛾𝐿)𝜀1𝑡 − 𝛽𝜀2𝑡(1 + 𝛾𝛽𝐿)𝑧2𝑡 = 𝛾(1 + 𝛼𝐿)𝜀1𝑡 + 𝜀2𝑡 .

[∗) =Using their parameter values of 𝛼 = 3, 𝛾 = .4, and 𝛽 = 1 we find that diag(𝑃𝑡|𝑡].8889, which is identical to what they give0

∗ is a simple way of computing their deficiency index 𝛿 .16 It is noteworthy that, for this example,for 𝛿𝑖 in their Table 3. So using 𝑃𝑡|𝑡𝑖only the second shock would be recoverable and so not all shocks may be recoverable, even where there is not an excess of them.
But having an excess of shocks does mean that not all shocks can be recovered.

12 Closely related is the concept of informational sufficiency (Forni and Gambetti, 2014); see also Forni et al. (2019). Forni and Gambetti (2014) developed
a test for informational sufficiency in a SVAR using Granger causality and principal components; a related approach for testing for non-fundamentalness wasdeveloped by Canova and Sahneh (2018).
13 Chahrour and Jurado (2022) provide a discussion of the differences between invertibility and recoverability.
14 There may be useful information in the MA representation that is not in 𝑃
𝑡|𝑇 about specification issues so there is an argument for computing that as wellas 𝑃𝑡|𝑇 .
15 A referee commented that a non-invertible MA(1) would mean shocks are not recoverable and then this might be taken as ‘‘evidence of excess shocks whennone exists’’. The check on recoverability is not a test for the existence of excess shocks. One knows if there are excess shocks simply by counting the number ofobservable variables and shocks.
16 Note that in the SSF we compute 𝑃 ∗ from the shocks 𝜀 having unit variance, just as in Forni et al. (2019). The answers they obtain depend on how many𝑡𝑡|𝑡
simulations they do and the .8889 they report comes from 1000 simulations. It might be mentioned that the computed 𝛿𝑖 could be used to find the covarianceof 𝐸𝑇 (𝜀𝑡 ) with 𝜀𝑡 . This may be a more informative way of capturing the magnitude of the discrepancy between the 𝐴𝑀 and 𝐸𝑀 shock innovations.

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To reiterate, in our approach there two different measures of 𝑃 . 𝑃𝑡|𝑡 , obtained from the Kalman filter, assesses whether shocks
are recoverable in the sense of invertibility, namely using past and current information. It allows us to assess if a particular shock
is fundamental, even if the model is not invertible. The second measure, 𝑃𝑡|𝑇 , instead relates to the wider concept of recoverability
in Chahrour and Jurado (2022); it additionally uses future information and therefore can be computed from the Kalman smoother.
Which definition of recoverability would one use? A referee asked this and proposed that we look at a non-invertible MA(1)process of the form
(13)𝑧𝑡 = 𝜀𝑡 + 𝜃𝜀𝑡−1 , 𝜃 > 1,

saying that 𝜀𝑡 is not recoverable. This can be placed in our SSF, Eqs. (2) and (3), by defining 𝜁𝑡 = 𝜀𝑡 , 𝐷1 = 1, 𝐷2 = 𝜃 and 𝐶 = 1.
∗ = .8889, so one cannot recover 𝜀 with current and past information. However 𝑃 ∗ = 0, meaning that
With a value for 𝜃 = 3, 𝑃𝑡|𝑡𝑡𝑡|𝑇
the shock is recoverable using future information. This was shown by Amisano and Giannini (1997). As they note, because the first𝜃order autocorrelation of 𝑧𝑡 is 1+𝜃 2 , for any given sample estimate there are two solutions for 𝜃, and so there two observationallyequivalent MA(1) processes for 𝑧𝑡 , namely Eq. (13) and
1
𝜔 .
(14)
𝜃 𝑡−1
So the question is which shock would be recovered in practice if a 𝑀𝐴(1) is fitted to the data on 𝑧𝑡 ? When the MA parameter isestimated the likelihood has two maxima, and one can get either of 𝜃 or 𝜃1 , depending on starting values. Many existing computer
programs however impose invertibility constraints, so using them one finds 𝜃1 . As Amisano and Giannini (1997) point out, while
the two MA models are observationally equivalent their impulse responses are different and this is due to the different shocks
characterizing them.
Another example of non-invertibility given to us by a referee was that in Leeper et al. (2013), which comes from a DSGE modelwith tax rates (𝜏𝑡 ) and the capital stock (𝑘𝑡 ). The model is solved under an assumption that agents receive news 𝜀𝜏,𝑡 that tells them
the tax rate two periods ahead, and results in the following two equations
𝑧𝑡 = 𝜔𝑡 +𝜏𝑡 = 𝜀𝜏,𝑡−2(15)

𝑘𝑡 = 𝛼𝑘𝑡−1 + 𝜀𝐴,𝑡 − 𝜅(𝜀𝜏,𝑡−1 + 𝜃𝜀𝜏,𝑡 ).

(16)

If all one has is 𝑘𝑡 then there are excess shocks and both 𝐴𝑀 shocks cannot be recovered. What happens if 𝜏𝑡 is observed as well?
Can both 𝐴𝑀 shocks be recovered then? Leeper et al. (2013) showed that 𝐴𝑀 shock recovery was not possible using filtered 𝐸𝑀
shocks, due to a lack of invertibility. Indeed using parameter values of 𝛼 = .3, 𝛽 = .99, 𝜏 = .4, 𝜅 = .77; and 𝜃 = 𝛼𝛽(1 − 𝜏) we find that∗ is not zero. However, when future information is used, 𝑃 ∗ = 0. So, in this case one would want to use
when 𝜏𝑡 is observed 𝑃𝑡|𝑡𝑡|𝑇smoothed shocks to recover the 𝐴𝑀 shocks; using it also gave a correct answer for the non-invertible MA discussed earlier.17 In thecontext of DSGE models, while it will depend on the research question, it seems likely that in most instances using the smoothedshocks will be the appropriate choice. Indeed, in the widely used program Dynare it is the smoothed, rather than filtered, shocks
that are available as output.
3.3. Example: A simple unobserved-components model
To illustrate our approach of using the SSF to assess recoverability more explicitly, and the consequences of having excess shocks,
it is useful to consider a simple UC model:
𝑦𝑡 = 𝑦1𝑡 + 𝑦2𝑡
𝑦1𝑡 = 𝜌𝑦1𝑡−1 + 𝜀1𝑡𝑦2𝑡 = 𝜀2𝑡 ,where the innovations 𝜀1𝑡 and 𝜀2𝑡 are assumed to be 𝑛.𝑖.𝑑. (0, 𝐼2 ). When the second variable 𝑦2𝑡 is not observed we will have excess
shocks. With 𝜌 known the model can be written as18
(17)

𝑧𝑡 = (1 − 𝜌𝐿)𝑦𝑡 = 𝜀1𝑡 + 𝜀2𝑡 − 𝜌𝜀2𝑡−1 .

To find[filtered
EM innovations 𝐸𝑡 𝜀1𝑡 and 𝐸𝑡 𝜀2𝑡 from the one observed variable 𝑧𝑡 we express the model in an SSF. Therefore 𝜓𝑡
]
[
]
[
]
𝜀1𝑡
and the matrices involved in the Kalman filter are 𝑀 = 0, 𝐷1 = 1 1 , 𝐷2 = 0 −𝜌 , 𝐶 = 𝐼, and 𝑅 = 0.
𝜀2𝑡Using Eq. (4), the recovered filtered innovations evolve as

equals𝐸𝑡 𝜓𝑡 = −𝐾𝑡 𝐷2 𝐸𝑡−1 𝜓𝑡−1 + 𝐾𝑡 𝑧𝑡 .

17 The referee also commented on this example as showing that conditioning on more variables will lead to ‘‘severe problems’’. But the ‘‘severe problems’’
do not come from having more observed variables. It is the properties of the model which came from the use of the tax equation, Eq. (15), that Leeper et al.
found unacceptable. Those properties do not change with the number of observed variables. All that extra observables do is to expand our ability to recover theshocks. One has to be content with the model before one would ask if the shocks are recoverable.
18 The process for 𝑦 is an ARMA(1,1) of the form (1 − 𝜌𝐿)𝑦 = 𝑒 + 𝛼𝑒 , so three parameters can be estimated — the variance of 𝑒 , 𝜌 and 𝛼. These can be
𝑡
𝑡
𝑡
𝑡−1
𝑡
used to estimate 𝜌 and the variances of the two shocks when they are not known. Consequently the model is identified.

7

European Economic Review 145 (2022) 104120

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Eq. (5) then gives the Kalman filter gain as
𝐾𝑡 = 𝐷1′ [𝐷2 𝑃𝑡−1|𝑡−1 𝐷2′ + 𝐷1 𝐷1′ ]−1 ,and(𝐷2 𝑃𝑡−1|𝑡−1 𝐷2′ + 𝐷1 𝐷1′ )−1 = (𝜌2 𝜎 2 𝑃22,𝑡−1|𝑡−1 + 1 + 𝜎 2 )−1 .
( )
[
𝑐
0
So this is a scalar, 𝑐𝑡 , and it means that 𝐾𝑡 = 𝑡 . Consequently, 𝛷 = −𝐾𝑡 𝐷2 =
𝑐𝑡
0

]
𝑐𝑡 𝜌
, and 𝐸𝑡 𝜓𝑡 evolves as𝑐𝑡 𝜌𝐸𝑡 𝜓1𝑡 = 𝑐𝑡 𝜌𝐸𝑡−1 𝜓2𝑡−1 + 𝑐𝑡 𝑧𝑡(18)

𝐸𝑡 𝜓2𝑡 = 𝑐𝑡 𝜌𝐸𝑡−1 𝜓2𝑡−1 + 𝑐𝑡 𝑧𝑡 ,(19)

giving 𝐸𝑡 𝜓1𝑡 = 𝐸𝑡 𝜓2𝑡 . This implies that the EM shocks – the recovered filtered innovations 𝐸𝑡 𝜀1𝑡 and 𝐸𝑡 𝜀2𝑡 – are the same, unlike
the AM innovations. The presence of excess shocks means we cannot recover all the AM shocks from the EM shocks.19
Nelson (1975) and McDonald and Darroch (1983) pointed out that the one-step prediction errors 𝜂𝑡 will generally be a
combination of all {𝜀𝑘,𝑡−𝑗 }𝑘=1,2 . This implies that both of these shock innovations cannot be recovered. As Eq. (19) shows, the filteredshock 𝐸𝑡 𝜓2𝑡 will be a function of all 𝑧𝑡−𝑗 and so depends upon the histories of the AM innovations 𝜀1𝑡 and 𝜀2𝑡 . Canova and Ferroni(2022) make this same point about the EM shocks, but in the context of a DSGE model whose solution is a Vector AutoRegression
Moving-Average (VARMA) structure, referring to this as ‘‘deformation’’, and saying (p.2) ‘‘ Deformation makes identified shocks
mongrels with little economic interpretation for two reasons. Identified shocks are unlikely to combine structural disturbances ofthe same type...the shocks one can identify will be, in general, linear combinations of current and past structural disturbances’’.20
Is it possible to recover one of the AM shocks with just current and past data? As we have argued, the answer to this resides inthe variance 𝑃𝑡|𝑡 computed by the Kalman filter. To parameterize the model, let 𝜌 = .9. In the limit, using the steady-state Kalman
[
]
.5974
−.4026filter, the steady-state value for 𝑃𝑡|𝑡 is 𝑃 =. The variance of 𝐸𝑡 𝜓𝑗𝑡 − 𝜓𝑗𝑡 is 𝑃𝑗𝑗 ; if it was zero then we could recover−.4026.5974
the AM shocks from the 𝐸𝑀 ones, but this is not the case.
What other differences might we see in the EM shocks that are not in the assumed ones? One is that there can be serial correlationin the 𝐸𝑀‘‘innovations’’.
This is because 𝛷𝑡 = −𝐾𝑡 𝐷2 and it may not be zero.21 When 𝜌 = .9 we find the steady-state value of 𝛷𝑡
[
]
0 .36
is 𝛷 =, so the 𝐸𝑀 innovations follow a VAR process, in contrast to the 𝐴𝑀 innovations which are assumed to have no0 .36
serial correlation. It is the inability to separate the innovations 𝜀1𝑡|𝑡 and 𝜀2𝑡|𝑡 when there is only one observed variable which results
in serial correlation in the recovered shock innovations.
The reason for these different properties can be seen by looking at the one-step prediction error for 𝑧𝑡 , 𝜂𝑡 . From Kurz (2018),
this is a linear combination of all the shocks 𝜀𝑡−𝑗 . Now the log likelihood depends directly on 𝜂𝑡 so that, once we know 𝜂1 , … , 𝜂𝑇 ,we know the likelihood. Because 𝜂𝑡 depends in a linear form on {𝜀𝑘 }𝑡𝑘=1 , when there is an excess of model shocks, i.e. more 𝜀𝑡 thanobservables 𝑦𝑡 , we would need to recover 𝐸𝑡 𝜀𝑡 with a g-inverse.
3.4. Implications of correlated innovations for impulse responses
To us one motivation and appeal of SVAR and DSGE models is that one works with truly exogenous variables; that is what
the shocks are designed to be. This is done so as to remove a problem encountered with regressions in macroeconomics, namelythat regressors can rarely be assumed uncorrelated due to simultaneity. But what can be done if a researcher finds their EM shock
innovations to be correlated? One possible approach is to ‘‘orthogonalize’’ them. Apart from the issue of what economic label one
would attach to such shocks, it needs to be understood that this strategy does not deliver a unique set of impulse responses. Consider
Eq. (3), the state equation, written as
𝜓𝑡 = 𝑀𝜓𝑡−1 + 𝐶𝑅𝑅′ 𝜀𝑡= 𝑀𝜓𝑡−1 + 𝐶𝑅𝑣𝑡 ,where 𝑅𝑐𝑜𝑣(𝜀𝑡 )𝑅′ = 𝐼, and 𝑅 is an orthonormal matrix, i.e. 𝑅′ 𝑅 = 𝐼. A Cholesky factorization of the variance–covariance matrix of
𝜀𝑡 works by choosing one such 𝑅, and this delivers new uncorrelated shock innovations 𝑣𝑡 whose impulse responses are different
to those for 𝜀𝑡 , but with the same fit to the data. To be clear, the Cholesky factorization here is being applied to the innovationsof the 𝐸𝑀 structural shocks, not the reduced-form innovations, in order to produce a new set of shocks that are orthogonal byconstruction. This is distinct from applying the Cholesky factorization to the reduced-form innovations, as is sometimes done toidentify a SVAR. The impulse responses to these new shocks are 𝐶𝑅, so they have changed. However, because the orthonormal
matrix is not unique, this is also true of the impulse responses — there is a set of responses.22 Effectively we obtain a set of impulse

19

In Appendix A we provide an examination of some UC trend-cycle decompositions, including that used to generate the Hodrick–Prescott (1997) filter.
Identified shocks in Canova and Ferroni (2022) are what we term as EM, i.e. compatible with the data.
21 This was also noted by Harvey (1992) for this type of model.
22 This is different to the bounds in Plagborg-Moller and Wolf (2020) which arise from model parameters not being identified. Our identified set arises strictly
due to correlation in shocks, since the model parameters are known. There are also many variance decompositions and not just one. The range of impulse
responses that are found as 𝑅 varies can be very large — see for example Liu et al. (2018).
20

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responses as there are a range of models that are transformations of the DSGE that we started with, all of which fit the data equally
well. This has come up in the sign restrictions literature and the range may be very large.
To analyze this further, suppose we had the moving-average representation of Eq. (1). Then 𝐶𝑗 can be computed once 𝐶0 and 𝐴1
are known. If the AM innovations 𝜀𝑡 have no serial correlation and are uncorrelated with each other then we can recover 𝜀𝑗𝑡 from𝐸𝑡 𝜀𝑗𝑡 i.e. regressing 𝑧𝑖𝑡 on 𝐸𝑡 𝜀𝑗𝑡 will correctly give 𝐶𝑖𝑗0 . This is because all the other regressors in Eq. (1) will be uncorrelated with
𝜀𝑗𝑡 and so can be omitted. This is not true if the assumptions made about the 𝜀𝑡 are incorrect. Suppose 𝜀𝑗𝑡 is correlated with 𝜀𝑘𝑡 . In
this case omitting 𝜀𝑘𝑡 from the regression will cause a bias in the estimate of 𝐶𝑖𝑗0 . Of course it may be small if the correlation of theregressors is small. But it points to why we would need to check that. Cast in this way we consider the Leeper et al. (2013) modelof Section 3.2. There were two observables and two shocks. The 𝐴𝑀 shocks could not be recovered using the filtered EM shocks,
but could be using the smoothed 𝐸𝑀 shocks. Thus regressing 𝑧𝑖𝑡 on 𝐸𝑇 𝜀𝑗𝑡 will correctly give 𝐶𝑖𝑗0 .
3.5. Assessing the properties of the AM shocks that cannot be recovered with indirect inference
If the 𝐴𝑀 shocks are not recoverable can we still find some information about the correlation properties of the 𝐴𝑀 shock
innovations using the 𝐸𝑀 shocks, as the latter are all we have? One is not seeking information about all observations on the 𝐴𝑀
shock innovations but about parameters of their density, as that is what the correlations depend on, and that is much easier to find.
One approach to doing this is to conduct indirect inference.
Suppose we took the 𝑖’th smoothed 𝐸𝑀 shock innovation 𝜀𝑖𝑡|𝑇 and regress it against 𝜀𝑗𝑡|𝑇 (𝑗 ≠ 𝑖) to get an estimated coefficient̂ The rationale for this is that if the AM shock innovations did have zero correlations then the regression of 𝜀𝑖𝑡 against 𝜀𝑗𝑡 would
𝜃.
give a zero coefficient (in large samples). However, because the 𝐸𝑀 shocks are not the 𝐴𝑀 shocks when there is no recoverability,
we would expect that the 𝜃̂ would not be zero, even in large samples. So we need to ask what the value of 𝜃̂ would be in large
samples when there is a valid zero correlation assumption among the 𝐴𝑀 shock innovations, and to then compare that with what
is seen when the regression uses the 𝐸𝑀 shock innovations. This is an example of indirect inference.
To implement this approach, take the parameter values of a model, say a DSGE model, and simulate a large number ofobservations (𝑁). Then compute 𝐸𝑀 shock innovations using the simulated data. Call these 𝐸𝑀𝑆, and use a superscript 𝑠 to
distinguish them. The regression will now be of 𝜀𝑠𝑖𝑡|𝑁 on 𝜀𝑠𝑗𝑡|𝑇 to get 𝜃 𝑠 . That value will be what we expect to get from using the
𝐸𝑀 shock innovations when the assumptions made about the 𝐴𝑀 shock innovations are correct. So testing if 𝜃̂ = 𝜃 𝑠 is an indirect
test of the assumptions made about the 𝐴𝑀 shock innovations. We use a HAC-adjusted 𝑡 test to guard against any autocorrelation.
Of course one can also use a similar approach to test if the innovations are really that, i.e. to test if they have no serial correlation.
To illustrate this we use the UC model of Section 3.3 and suppose the AM shock innovations are correlated with a correlationof 𝜙. Because the variances of the innovations are unity we can write
𝜀1𝑡 = 𝜙𝜀2𝑡 + 𝑣𝑡 ,with 𝑣𝑎𝑟(𝑣𝑡 ) = 1 − 𝜙2 . Therefore𝑣𝑎𝑟(𝜀1𝑡 ) = 1 = 𝜙2 𝑣𝑎𝑟(𝜀2𝑡 ) − 𝜙2
= 𝑣𝑎𝑟(𝜀2𝑡 ).
In order to simulate 𝑧𝑡 when there is correlation among the AM shock innovations Eq. (17) will become𝑧𝑡 = (1 − 𝜌𝐿)𝑦𝑡 = 𝜙𝜀2𝑡 + 𝑣𝑡 + 𝜀2𝑡 − 𝜌𝜀2𝑡−1= (1 + 𝜙)𝜀2𝑡 + 𝑣𝑡 − 𝜌𝜀2𝑡−1 ,where 𝑣𝑎𝑟(𝑣𝑡 ) = 1 − 𝜙2 . Hence this provides a nested model for a direct test of 𝜙 = 0.
We look at what happens for two cases, namely when {𝜙 = 0, 𝜌 = .6} and {𝜙 = .5, 𝜌 = .6}. In the first case, when there is nocorrelation between the AM shock innovations, the regression of 𝜀1𝑡|𝑇 against 𝜀2|𝑇 (10,000 observations) gives a coefficient of .81.
When 𝜙 = .5 this becomes .55. So the presence of correlation causes a decrease in the correlation of EM shock innovations fromwhat we expect it to be when the AM shocks are uncorrelated.
To test if there is correlation between the 𝐴𝑀 shock innovations we use the 𝐸𝑀 shock innovations computed from the smaller
set of data (200 observations) and get an estimated coefficient of .545. Comparing this to .81 we get a (HAC-adjusted) 𝑡 ratio of−21.30, so there is strong evidence of correlation in the AM shock innovations, as there should be.
3.5.1. Example: Lubik and Schorfheide (2004)
Another example of excess shocks is indeterminacy in a DSGE model leading to sunspots. In an influential study (Lubik andSchorfheide, 2004) (LS) reported this when estimating a simple New-Keynesian (NK) model for the pre-Volcker period. Their NK
model had three observable variables – deviations of output, inflation and interest rates from their steady states – and three basic
𝑑𝑠(‘‘fundamental’’) shocks 𝜀𝑡 - monetary (𝜀𝑅𝑡 ), demand (𝜀𝑡 ) and supply (𝜀𝑡 ). With indeterminacy there is a fourth; a sunspot shock
(𝜀𝑠𝑢
),
and
therefore
excessshocks.
To
look
at
this
case
we
need
to
solve
the model under indeterminacy and we use the approach𝑡of Bianchi and Nicolo (2021).23 The solution from that can be expressed in terms of our SSF(20)

𝑧𝑡 = 𝐷1 𝜁𝑡

23

Their Dynare program for the LS model is available from https://sites.google.com/view/francescobianchi/home/codes.
9European Economic Review 145 (2022) 104120

A. Pagan and T. Robinson(21)𝜁𝑡 = 𝑀𝜁𝑡−1 + 𝐶𝜀𝑡 ,and we use the values for 𝐷1 , 𝑀 and 𝐶 coming from the parameter values in Bianchi and Nicolo’s Dynare code.24)(
𝑠𝑢 ′ , it is found that
Can we recover the AM shock innovations? Arranging these as 𝜀𝑠𝑡 , 𝜀𝑑𝑡 , 𝜀𝑅𝑡 , 𝜀𝑡
⎡ .00
⎢

−.03
∗
𝑃𝑡|𝑇
=⎢
⎢ 0.01
⎢−.01
⎣

−.03
.48
−.12
−.18

.02
−.11
.03
.06

−.01⎤
⎥
−.18⎥
,
.06 ⎥
⎥
.49 ⎦

so the supply and monetary policy shocks are recoverable, or seem nearly so. The demand shock and the sunspot are not.
A noteworthy aspect of LS is that they allow the innovations to the demand and supply 𝐴𝑀 shocks to be correlated; they
estimated it to be .14. The solution to the model remains the same regardless of this correlation, i.e. 𝐷1 , 𝑀 and 𝐶 are invariant toany correlations between the 𝜀𝑡 . It is only in the computation of impulse responses to these shocks that any correlation has to betaken into account. They proceed to then orthogonalize the demand and supply shocks. As we have shown in Section 3.4 this does
not produce unique impulse responses.
As the AM demand and sunspot shock innovations are not recoverable from their 𝐸𝑀 counterparts, in order to assess theirnature we use our indirect inference approach. The matrices below show the correlations in the EM shocks (left-hand side), andthose based on data simulated under the assumption of uncorrelated 𝐴𝑀 shocks (right-hand side). Pairwise testing suggests thatthe sunspot shock is significantly correlated with both the AM monetary and demand shocks (e.g. for the former the HAC-adjusted
𝑡 ratio is −4.31).25
𝐸𝑀 Shock Correlations in Data
⎡ 1
.31
.39
−.29⎤
⎢
⎥
.31
1
.34
.35 ⎥
⎢
;
⎢ .39
.34
1
−.28⎥
⎢−.29 .35 −.28⎥
1 ⎦⎣𝐸𝑀 Shock Correlations Assuming Uncorrelated 𝐴𝑀 Shocks⎡ 1.02−.05 −.04⎤
⎢
⎥
.02
1
.86
.17 ⎥
⎢
.
⎢−.05
.86
1
.33 ⎥
⎢−.04 −.17
⎥
.33
1 ⎦
⎣

3.5.2. Example: The Farmer and Nicolò (2021) beliefs-based model
Farmer and Nicolò (2018) set out a NK system like LS but instead of a Phillips curve being present there is a beliefs function
about nominal GDP growth, 𝑥𝑡 , and the expectations about it are assumed to be(22)

𝐸𝑡 (𝑥𝑡+1 ) = 𝑥𝑡 .

In Farmer and Nicolò (2021) they estimated the model with data from three countries. Using parameter values for their modelin Table 4A and 4B of their paper for the U.S. shows indeterminacy. The solution can be represented as having three fundamental
shocks – monetary 𝑚, demand 𝑑, and supply 𝑠 – as well as an animal spirits shock, 𝑎.26 The demand and supply shocks are assumed
to be AR(1) processes with innovations 𝜀𝑑𝑡 and 𝜀𝑠𝑡 , whereas the monetary and animal spirit shocks are white noise, and therefore𝑎equal their innovations 𝜀𝑚𝑡 and 𝜀𝑡 . There are three observables — inflation 𝜋𝑡 , an interest rate 𝑅𝑡 , and linearly de-trended log GDP
𝑦𝑡 . All these variables are 𝐼(1) in the model and there are two error correction terms 𝑒𝑐1𝑡 = 𝑅𝑡 − 𝜋𝑡 and 𝑒𝑐2𝑡 = (1 − 𝜆)𝜋𝑡 − 𝜇𝑦𝑡 , where
𝜆 and 𝜇 are parameters in the interest rate rule.
The solved FN system can be placed in a SSF, as was done for LS, and this is used to check recovery of the shocks, yielding:
⎡ .35
⎢

−.37
∗
𝑃𝑡|𝑇
=⎢
⎢−.16
⎢ .14
⎣

−.37
.46
.12
−.16

−.16
.12
.14
−.06

.14 ⎤
⎥
−.16⎥
,
−.06⎥
⎥
.06 ⎦

𝑑
where the shock innovations are in the order 𝜀𝑎𝑡 , 𝜀𝑠𝑡 , 𝜀𝑚𝑡 and 𝜀𝑡 . Thus the sunspot and the supply-side shock cannot be recovered.
The serial correlation in the innovations to the 𝐸𝑀 shocks of animal spirits and monetary shocks are substantial, namely .53 and.58 respectively, i.e. they are not really innovations. Turning to the nature of the EM shock innovations, which we have said will
be correlated even if this was not true of the AM shock innovations, we find the correlation between (𝜀𝑎𝑡|𝑇 , 𝜀𝑠𝑡|𝑇 ) is 0.80, whereas(𝜀𝑑𝑡|𝑇 , 𝜀𝑠𝑡|𝑇 ) is −0.07.

3.6. Implications for variance and variable decompositions
To analyze this, suppose that there are only two shocks and we use the identity involving data 𝑧𝑡 = 𝐺(𝐿)𝐸𝑇 𝜀𝑡 , giving
𝑧𝑡 = 𝐺1 (𝐿)𝐸𝑇 𝜀1𝑡 + 𝐺2 (𝐿)𝐸𝑇 𝜀2𝑡

24 Eq. (20) abstracts from the constants in the LS measurement equations. To calculate 𝐷 , 𝑀 and 𝐶 we use simulated observations on 𝑧 and 𝜀 and fit the1𝑡𝑡
identities, Eqs. (20) and (21). The method was used in Liu et al. (2018).
25 This is done by regressing the monetary shock on the sunspot.
26 FN actually work with a sunspot variable that is a linear combination of the four innovations.

10European Economic Review 145 (2022) 104120

A. Pagan and T. Robinson= 𝑧𝑠1𝑡 + 𝑧𝑠2𝑡 ,[]where 𝐺(𝐿) = 𝐺1 (𝐿) 𝐺2 (𝐿) . If 𝑧𝑠1𝑡 and 𝑧𝑠2𝑡 are uncorrelated, which occurs when the AM innovations are recoverable and found to
be uncorrelated among each other and over time, one can find the contribution to the data variance from each shock. This is not thecase if the 𝐴𝑀 shocks are correlated or there are excess shocks, because the EM shocks are then correlated. Specifically, in thoseinstances 𝑧𝑠1𝑡 and 𝑧𝑠2𝑡 are correlated and so 𝑣𝑎𝑟(𝑧𝑡 ) is not 𝑣𝑎𝑟(𝑧𝑠1𝑡 ) + 𝑣𝑎𝑟(𝑧𝑠2𝑡 ).
Variance decompositions are often presented in applied papers where it is implicitly assumed that 𝑣𝑎𝑟(𝑧𝑡 ) equals 𝑣𝑎𝑟(𝑧𝑠1𝑡 )+𝑣𝑎𝑟(𝑧𝑠2𝑡 ),𝑣𝑎𝑟(𝑧𝑠 )1𝑡
as it is maintained that the contribution to the variance of 𝑧𝑡 of (say) the first shock is 𝑣𝑎𝑟(𝑧𝑠 )+𝑣𝑎𝑟(𝑧𝑠 ) . That is true in fitted exactly1𝑡2𝑡

identified SVARs, since the zero correlation between EM shock innovations is enforced in estimation. But it will not be true ofmodels with excess shocks.
3.6.1. Example: Christiano et al. (2014)
To give one example from the literature where this issue arises we consider the contribution of risk shocks to the spread between
short and long rates in Christiano et al. (2014). In that model the data standard deviation for the spread is .38, while using theirmodel the standard deviation of the component of the spread due to risk shocks alone is almost three times higher (1.085).27 The
only way this can happen is if there is a correlation between the risk and non-risk AM shock innovations. Essentially, it is not adecomposition of the data variance that is presented but the fraction of the synthetic quantity 𝑣𝑎𝑟(𝑧𝑠1𝑡 ) + 𝑣𝑎𝑟(𝑧𝑠2𝑡 ), and that is not𝑣𝑎𝑟(𝑧𝑡 ).
The same issue arises with variable (i.e. historical) decompositions. In these the second shock above is set to zero and thatleaves 𝑧𝑠1𝑡 as the contribution. When we have zero correlation between the innovations 𝜀𝑡 this gives a correct measure. But when
the innovations 𝜀𝑡 are correlated, moving 𝜀2𝑡 to zero changes 𝜀1𝑡 so that the total impact due to a change in 𝜀1𝑡 has a direct effectof 𝐺0,𝑖1 and an indirect effect of 𝐺0,𝑖2 .
3.6.2. Example: A Time-Varying Parameter (TVP) autoregressive process
There are many models where parameters are allowed to be stochastically time varying. Prominent examples are TVP SVARs Primiceri (2005) is an influential reference. Mostly these models involve excess shocks. To see the implications of excess shocks forvariance decompositions in TVP models, consider an AR(1) where the autoregressive coefficient is allowed to be time varying:𝑧𝑡 = 𝛼𝑡 𝑧𝑡−1 + 𝜀1𝑡𝛼𝑡 = 𝜌𝛼𝑡−1 + 𝜀2𝑡 ,with the AM shock innovations 𝜀1𝑡 and 𝜀2𝑡 uncorrelated with unit variances. This can be placed in the SSF above, with the states⎡ 𝛼𝑡 ⎤
𝜁𝑡 = ⎢ 𝜀1𝑡 ⎥. 𝐷1 in this instance is time-varying, with 𝐷1𝑡 = 𝑧𝑡−1 . Note it varies in a known way, since 𝑧𝑡−1 is observed. The Kalman⎢ ⎥⎣𝜀2𝑡 ,⎦
filter and smoother remain the same, except for 𝐷1 being time varying.
The above model was fitted to the inflation data used by Lubik and Schorfheide (2004). The resulting estimated model was𝑧𝑡 = 2.404 + 𝛼𝑡 𝑧𝑡−1 + 1.249𝜀1𝑡𝛼𝑡 = .969𝛼𝑡−1 + .1264𝜀2𝑡 ,showing time variation in the persistence of inflation. Turning to the implications of excess shocks, the smoothed 𝐸𝑀 estimates of
the innovations from the estimated SSF have a correlation of .54. This has important implications for variance decompositions, which
are constructed using the smoothed shocks and are reported in many TVP-SVAR papers. A common methodology is to compute a
variance decomposition at time 𝑡 by setting 𝛼𝑡 = 𝐸𝑇 𝛼𝑡 . With 𝛼𝑡 fixed the impulse responses underlying the variance decomposition
are just computed with respect to 𝜀1𝑡 . Thus, the hypothetical experiment performed involves raising 𝐸𝑇 𝜀1𝑡 . However, since 𝐸𝑇 𝜀1𝑡
and 𝐸𝑇 𝜀2𝑡 are correlated, this would also raise 𝐸𝑇 𝛼𝑡 above what it would be if 𝐸𝑇 𝜀1𝑡 had not been raised.
To assume 𝛼𝑡 = 𝐸𝑇 𝛼𝑡 when constructing variance decompositions one needs to have a zero correlation between 𝐸𝑇 𝜀1𝑡 and 𝐸𝑇 𝜀2𝑡 ,which will not occur due to excess shocks. Essentially it is often ignored that there is an additional shock in the model and it isinstead presumed that the assumption made about the AM shocks being uncorrelated translates to the smoothed shocks being usedin the variance decompositions.
4. Excess shocks in DSGE models: Consequences and strategies
Excess shocks can occur in a wide variety of macroeconomic models. In this section we demonstrate assessing shock recoverability
in a DSGE model, Ireland (2011). This is a very standard DSGE model that has no sunspots etc. To reiterate, for an impulse response
from any model to a particular 𝐴𝑀 shock to be relevant, ideally it must be recoverable, and it needs to be uncorrelated withthe others. To be clear, we are not advocating constructing impulse responses to the EM shocks, as they are correlated when excess
shocks exist. Instead, the application below demonstrates which 𝐴𝑀 shocks can be recovered. After that, testing whether they areuncorrelated can be done for a DSGE model in a straightforward manner. We then turn to two strategies for dealing with excess
shocks, which are demonstrated using the Ireland (2011) model.

27

This is taken from Pagan and Wickens (2022, Table 1). One can see the divergence between the variances in Figure 1 of Christiano et al. (2014).
11

European Economic Review 145 (2022) 104120

A. Pagan and T. Robinson

4.1. The New-Keynesian DSGE model of Ireland (2011)
The methodology used by Ireland (2011) is typical within the DSGE literature. It is assumed that the shocks are driven byuncorrelated innovations, parameters of the model are then estimated, and impulse responses and variance decompositions to theseAM shocks are computed. We treat the estimated parameters as true ones and so assume that the model Ireland estimates is not
mis-specified.
The model consists of three key equations: an IS equation, a NK Phillips Curve, and a Taylor rule. Ireland assumed four
uncorrelated 𝐴𝑀 shocks: preferences 𝑎𝑡 , technology 𝑍𝑡 , cost push 𝑒̂𝑡 and monetary policy 𝜀𝑟𝑡 . Using three observables on inflation,
output growth and the interest rate, he estimated the model and computed impulse responses. There are therefore excess shocks.
The technology shock is permanent, while the remainder are transitory. For completeness, the model equations he used are:𝑎̂𝑡 = 𝜌𝑎 𝑎̂𝑡−1 + 𝜖𝑎𝑡 ,(23)(𝑧 − 𝛽𝛾)(𝑧 − 𝛾)𝜆̂ 𝑡 = 𝛾𝑧𝑦̂𝑡−1 − (𝑧2 + 𝛽𝛾 2 )𝑦̂𝑡 + 𝛽𝛾𝑧E𝑡 (𝑦̂𝑡+1 ) + (𝑧 − 𝛽𝛾𝜌𝑎 )(𝑧 − 𝛾)𝑎̂𝑡 − 𝛾𝑧𝑧̂ 𝑡 ,(24)

𝜆̂ 𝑡 = 𝑟̂𝑡 + E𝑡 (𝜆̂ 𝑡+1 ) − E𝑡 (𝜋̂ 𝑡+1 ),(25)

𝑒̂𝑡 = 𝜌𝑒 𝑒̂𝑡−1 + 𝜀𝑒𝑡 ,(26)

𝑧̂ 𝑡 = 𝜀𝑧𝑡 ,(27)(1 + 𝛽𝛼)𝜋̂ 𝑡 = 𝛼 𝜋̂ 𝑡−1 + 𝛽E𝑡 (𝜋̂ 𝑡+1 ) − 𝜓 𝜆̂ 𝑡 + 𝜓 𝑎̂𝑡 + 𝑒̂𝑡 ,(28)

𝑟̂𝑡 − 𝑟̂𝑡−1 = 𝜌𝜋 𝜋̂ 𝑡 + 𝜌𝑔 𝑔̂𝑡 + 𝜀𝑟𝑡 ,(29)𝑔̂𝑡 = 𝑦̂𝑡 − 𝑦̂𝑡−1 + 𝑧̂ 𝑡 ,(30)

0 = 𝛾𝑧𝑞̂𝑡−1 − (𝑧2 + 𝛽𝛾 2 )𝑞̂𝑡 + 𝛽𝛾𝑧E𝑡 (𝑞̂𝑡+1 ) + 𝛽𝛾(𝑧 − 𝛾)(1 − 𝜌𝑎 )𝑎̂𝑡 − 𝛾𝑧𝑧̂ 𝑡 ,(31)

𝑥̂ 𝑡 = 𝑦̂𝑡 − 𝑞̂𝑡 .

(32)

and

The variables in the model are a Lagrange multiplier coming from the consumer’s budget constraint 𝜆𝑡 ; output 𝑦𝑡 ; the growthrate of technology 𝑧𝑡 ; inflation 𝜋𝑡 ; interest rate 𝑟𝑡 ; output growth 𝑔𝑡 ; the efficient level of output 𝑞𝑡 and the output gap 𝑥𝑡 . Some ofthese variables have been normalized by the non-stationary technology 𝑍𝑡 so as to have a well-defined path to log-linearize about.
̂ denotes a log deviation from steady-state.
The parameters are the autocorrelation coefficients for the respective shock processes (𝜌𝑎 and 𝜌𝑒 ); the steady-state growth rateof technology 𝑧; a discount factor 𝛽; an internal habits intensity 𝛾; the degree of forward-looking behavior in price setting 𝛼; andinterest rate rule parameters on inflation and growth (𝜌𝜋 and 𝜌𝑔 ).
Eqs. (23), (26) and (27) represent three of the model innovations processes and the fourth shock 𝜀𝑟𝑡 is that for monetary policy.
The AM innovations 𝜀𝑟𝑡 , 𝜀𝑧𝑡 etc., are all assumed to be uncorrelated and to have no serial correlation. Eqs. (24) and (25) together
produce the NK IS curve. Eq. (28) is a NK Phillips Curve. Eq. (29) is the interest rate rule. Finally, Eq. (32) is the output gap, with
the natural rate of output defined by Eq. (31).
There is another equation in the model that describes hours (ℎ𝑡 ). This isℎ̂ 𝑡 = 𝑦̂𝑡 = 𝑔̂𝑡 + 𝑦̂𝑡−1 − 𝑧̂ 𝑡

(33)

Because it does not feed back into the earlier equations, and ℎ𝑡 was treated as being not observable, it was ignored, but later in
our work it will become important, as it provides extra information.
4.2. Which AM shocks of Ireland’s model can be recovered and what are their properties?
To examine the implications of excess shocks in this model we define the state vector 𝜓𝑡 in our SSF as consisting of thethree observed variables, the unobserved variable, hours, and the four shock innovations (preference, cost-push, technology andmonetary). Doing so enables us to compute the steady-state 𝑃𝑡|𝑇 and therefore to check which of the AM shocks can be recoveredfrom the data. As excess shocks exist we know not all of them can be recovered.
∗ corresponding to the shocks
Treating Ireland’s parameter estimates as the ‘‘true’’ ones it is found that the diagonal elements of 𝑃𝑡|𝑇
⎡.36⎤
⎢ ⎥
.57
are ⎢ ⎥. Accordingly, only the 𝐴𝑀 technology shock innovation can be recovered, although one might think that the monetary
⎢.00⎥
⎢.07⎥
⎣ ⎦
shock seems quite close to being so. The 𝐴𝑀 preference and cost push shock innovations quite clearly cannot be.
12European Economic Review 145 (2022) 104120

A. Pagan and T. Robinson

Suppose we decided that, based on the above evidence, the AM technology and monetary shock innovations are recoverable. Thenthey are equal to the equivalent smoothed EM shock innovations. So to look at the correlation between the AM shock innovations
we can compute the correlation between the EM shock innovations. For the technology and monetary policy shocks we find that
it is −.20; generally the monetary 𝐸𝑀 shock innovation is only weakly correlated with all the other EM shocks. Hence this points
to the impulse responses for it being valid. A caveat, however, is that Ireland also assumes that the 𝐴𝑀 shock innovation has noserial correlation while it actually has non-trivial first-order autocorrelation (0.55).
As the preference shock is not recoverable we apply the indirect inference approach. Interestingly, the smoothed 𝐸𝑀 preference
shock innovation is found to not be significantly correlated with the others beyond that induced by excess shocks.
The sizable bivariate correlations that exist amongst some of the EM shock innovations mean that variable and variancedecompositions are not appropriate, regardless of whether the AM shock innovations can be recovered, as these decompositions
use the EM shocks.
4.3. Excess shocks and factor models
There is a literature on factor models in macroeconometrics where there are some common factors and also idiosyncratic shocks
connected to all observables. As a referee noted, it is well known that when the number of observables tends to infinity one canrecover all shocks even though this is a situation of excess shocks — see Bai and Ng (2002). This seems to contradict what we havesaid in previous sections. To appreciate the issues raised here, consider a simple example of such a factor model:𝑧𝑖𝑡 =𝛬𝑖 𝐹𝑡 + 𝜀𝑖𝑡 , 𝑖 = 1, … , 𝑛,

(34)

𝐹𝑡 =

𝜂𝑡 ,

(35)
(36)

⟹ 𝑧𝑖𝑡 = 𝛬𝑖 𝜂𝑡 + 𝜀𝑖𝑡 ,where there are 𝑛 observed variables 𝑧𝑖𝑡 and a factor 𝐹𝑡 . The loadings are 𝛬𝑖 and there are two types of AM shocks - 𝜀𝑖𝑡 and thefactor shock 𝜂𝑡 . There is one more shock than observables and we look at whether shocks are recoverable.
∑∑∑
Consider estimating the factor with some weighted average 𝐹̃𝑡 = 𝛿1 𝑛𝑖=1 𝜔𝑖 𝑧𝑖𝑡 , where 𝑛𝑖=1 𝜔2𝑖 < 𝐷 and 𝛿 = 𝑛𝑖=1 𝜔𝑖 𝛬𝑖 .28 Then𝑛1∑𝐹̃𝑡 = 𝐹𝑡 +𝜔𝜀 .
𝛿 𝑖=1 𝑖 𝑖𝑡

A simple approach is to choose 𝜔𝑖 = 1𝑛 , but there are many other combinations. For example, both the smoothed and filtered shocks
are constructed in this way. Then as 𝑛 → ∞ 𝐹̃𝑡 → 𝐹𝑡 and the factor can be recovered. Using the formula for 𝐹̃𝑡 we then have𝑧𝑖𝑡 = 𝛬𝑖 𝐹̃𝑡 + 𝜀̃𝑖𝑡 .

(37)

Subtracting Eq. (37) from (36) produces
𝛬𝑖 (𝐹̃𝑡 − 𝐹𝑡 ) + (𝜀̃𝑖𝑡 − 𝜀𝑖𝑡 ) = 0.
So 𝜀̃𝑖𝑡 → 𝜀𝑖𝑡 as 𝑛 → ∞. But for any finite 𝑛 there is linear dependence between the estimated shocks. In empirical examples 𝑛 is
finite, so the same issues arise in dynamic factor models as in DSGE models. Consequently, it makes sense to check for AM shock
recovery and the assumptions made about them in the same way as was described earlier.
Often factor models are used in a different way, particularly in the context of a SVAR. The factors are replaced by a weighted
average of a large set of observables, mostly the principal components. A finite number of these are then added to an SVAR which
has some other observable variables. Thus in Bernanke et al. (2005) a few principal components are found from a large set ofobservables. These are then included in a SVAR that also has an interest rate. The monetary shock impulses are subsequently foundby using a recursive ordering. So there are no excess shocks in those type of Factor-Augmented VAR (FAVAR) models.
So what happens in a DSGE model as the number of observables increases? This is context dependent, but we note that Liu
et al. (2018) reviewed the Reserve Bank of Australia’s Multi-Sector Model (Rees et al., 2016). This is a relatively large model, with
17 observed variables and, with measurement error included for most variables, 31 shocks. Correlations amongst some of the 𝐸𝑀
shock innovations were found to be sizable — for example, the correlation between cost-push innovations in the non-traded andimport sectors was 0.78. Sizable correlations also existed between some of the measurement error and structural shock innovations.
Our summary is that one needs to check any DSGE model with excess shocks and factor models with the methods we have shownin order to determine issues when interpreting outcomes.
4.4. Strategies for dealing with excess shocks
To look at this question one needs to know why there are excess shocks and what one wants to learn from the modeling exercise.
Sometimes the situation arises from shocks being added on to a model to improve the fit with the data. Examples of that would

28

We are assuming that the parameters 𝛬𝑖 are known for expository purposes.
13European Economic Review 145 (2022) 104120

A. Pagan and T. Robinson
Table 1
Maximum likelihood estimates of Ireland’s model parameters under different shock and observable specifications.
Parameter

Replication estimate

Excluding preference shock estimate

Adding hours worked estimate

𝛾

0.390
(0.077)0.415(0.046)0.127(0.025)0.980(0.025)
0.000
–

0.942
(0.042)0.419(0.044)0.015(0.013)
0.000
–
0.927
(0.030)0.647(0.054)
0.439
0.050
0.043
(0.022)
0.861
0.043
0.000
–

0.000
–
0.004
(0.001)0.110(0.077)0.001(0.0001)

0.033
0.005
0.009
0.001
0.005
(0.0004)
0.001(0.0001)

𝜌𝜋
𝜌𝑔
𝜌𝑎
𝜌𝑒

Standard deviations of the shocks
𝜎𝑎
𝜎𝑒
𝜎𝑧̂
𝜎𝑟

0.087
(0.102)0.002(0.0002)
0.010(0.002)0.001(0.0001)

Standard errors are shown in parentheses.

be measurement error and time-varying parameters. This is a noble intention but there are costs — one cannot recover all the AM
shock innovations, and the EM shock innovations will be correlated. So, if the argument being advanced from the research does
not necessarily need such information, one might want to question whether their addition incurs more of a cost than a benefit.
However, in other cases they are part of the model, and so there will have to be excess shocks. In such cases we have to recognize
the limitations this imposes on using the model: we would need to ask exactly what can be learned from it, and that requires us to
ask what AM shock innovations are recoverable, and whether the properties assumed about them are correct. More intermediatecases arise where the excess shocks are in the maintained model but, for some reason, not enough observable data is used. Ireland’smodel is an example of that. So a range of situations need to be dealt with, and no one strategy will work in all situations. Hence,
we canvass some possible responses by researchers in the next sub-sections before returning to an assumption we have made in our
analysis that the parameters of models exhibiting excess shocks were known.
4.4.1. Eliminating excess shocks
One possibility is to use more data. Ireland (2011) is an example of a DSGE model where it is straightforward to eliminate excess
shocks by adding in the extra observed variable of hours. The key requirement to do this is that the model has a readily available
counterpart in the data.29 In the event that the excess shocks are measurement error it may be that one can use a summary of thisextra data – such as principal components – to compute items such as impulse responses. Forni et al. (2020) seems to be such an
example in a SVAR. Another alternative would be to remove one of the excess shocks — such as the preference shock. To compare
these strategies we do both here. This does not represent a belief that the preference shocks are unimportant. Because we areusing Ireland’s data it is necessary to first replicate his parameter estimates. Table 1 does this and it shows the implications for theparameter estimates of the DSGE model under the two strategies. It is seen that, relative to his estimates with excess shocks, in bothcases the habits parameter, 𝛾 increases significantly while, when the preference shock is excluded, the cost-push shock becomes
highly persistent.30
Table 2 shows the smoothed 𝐸𝑀 shock innovation correlations. As we have argued earlier the preference 𝐴𝑀 shock cannot
be recovered from its 𝐸𝑀 equivalent, but the other shocks probably can, and there are several sizable correlations among the EM
shock innovations — for example, that between the smoothed cost push (𝜀𝑒 ) and technology growth shocks (𝜀𝑧̂ ) is −0.56. Whenthe preference shock is dropped – so there is no excess now – the evidence is that the three remaining AM shock innovations
seem to be uncorrelated although the innovations are found to have substantial autocorrelations and so are not really innovations.
Consequently, this latter assumption used to compute impulse responses is inappropriate. Allowing for four shocks by using theextra series on hours worked does produce, with the exception of money and cost-push, some moderate correlations between the
𝐸𝑀 (and hence AM ) shocks innovations, but large degrees of serial correlation. Both sets of results point to mis-specification in themodel. Indeed, if we do the same analysis as above but with data simulated from the models – so that there is no mis-specification– and no excess shocks, the estimated correlations and autocorrelations of the assumed innovations are essentially zero.31

29
30
31

Adding hours worked was done by Pagan and Wickens (2022) for a related model, Ireland (2004). The hours worked data are from Kulish et al. (2017).
The standard errors in the replication can differ from Ireland (2011) as he bootstrapped the model, and we do not.
These results are available upon request.
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European Economic Review 145 (2022) 104120

A. Pagan and T. Robinson
Table 2
Properties of the estimated shock innovations.
Replication

Excluding preference shock

Adding hours worked

Shock innovation correlations
Filtered
𝜀𝑎
𝜀𝑒
𝜀𝑧̂
𝜀𝑟

𝜀𝑎
1
0.46
0.38
0.09

𝜀𝑒
0.46
1
−0.64
0.22

𝜀𝑧̂
0.38
−0.64
1
−0.15

𝜀𝑟
0.09
0.22
−0.15
1

𝜀𝑎
–
–
–
–

𝜀𝑒
–
1
0.09
0.12

𝜀𝑧̂
–
0.09
1
0.15

𝜀𝑟
–
0.12
0.15
1

𝜀𝑎
1
−0.27
0.29
0.14

𝜀𝑒
−0.27
1
−0.09
−0.40

𝜀𝑧̂
0.29
−0.09
1
0.05

𝜀𝑟
0.14
−0.40
0.05
1

0.15
1
−0.56
0.00

0.36
−0.56
1
−0.20

0.13
0.00
−0.20
1

–
–
–
–

–
1
0.09
0.12

–
0.09
1
0.15

–
0.12
0.15
1

1
−0.27
0.29
0.14

−0.27
1
−0.09
−0.40

0.29
−0.09
1
0.05

0.14
−0.40
0.05
1

Smoothed
𝜀𝑎
𝜀𝑒
𝜀𝑧̂
𝜀𝑟

1
0.15
0.36
0.13

Shock innovation first-order autocorrelations

𝜀𝑎
𝜀𝑒
𝜀𝑧̂
𝜀𝑟

Filtered

Smoothed

Filtered

Smoothed

Filtered

Smoothed

0.32
−0.14
−0.15
0.55

0.39
−0.53
−0.05
0.55

–
−0.09
−0.46
0.59

–
−0.09
−0.46
0.59

0.02
0.81
−0.03
0.58

0.02
0.81
−0.03
0.58

In Ireland’s model it is important that the extra variable which can be measured was in the model. There may be cases, however,when that is not possible. In such cases one may have to just admit that there will be excess shocks and to ask what that means forany interpretation.
A comment we have received is that there are many more shocks in the economy than in any model. However, there are also
more observed variables, so this does not mean that excess shocks will always be needed. The issue with any model is whether one
can simplify the economy so that we can work with a smaller number of shocks and variables than are actually present in it soas to analyze the particular aspect of interest. If we cannot make this simplification then the shocks we retain will be correlated,making the assumption that they are not more implausible. This is a reason why we would want to check if there is a correlation
between the AM shock innovations. Doing so checks whether we have actually set up a model with enough structure and shocks sothat it can capture the observable variables being used, and therefore whether any particular aspect of interest can be studied in adata-coherent way.
4.4.2. Packages of shocks
In Ireland’s model it was shown that we could recover at most two shocks. Of these the monetary shock innovation shows nocorrelation with the others, while that is not true of the technology shocks. The preference shock and cost-push cannot be recovered.
Reflecting the correlations existing in the EM shock innovations, one way to proceed with analyzing the dynamics of the model might
be to study packages of recovered shocks.
Consequently, the first package would be the monetary shock. Had the cost-push shock been recoverable, a second might betermed a business shock as it would combine the cost-push and technology shocks together. For any given combination we cancompute an impulse response to this business shock. Of course this combination is not unique, but there may be some evidence tosuggest what weights might be applied to construct it, or one could consider a range of weights.
There are limitations to working with packages of correlated shocks, most notably that an economically meaningful name to the
package may not exist. This is likely to be more important in larger models; for example shocks pertaining to different agents inthe model could be correlated.
4.5. The issue of parameter values
In our analysis we have assumed that the model being used had known parameter values. An editorial question was what happens
when you do not know them? Ultimately, questions of recoverability of AM shock innovations will be answered with the parametervalues that are being used by the investigators and these may have been estimated. It could be the case that using different parametervalues alters whether AM shock innovations can be recovered. However, all of the AM shock innovations cannot be recovered when
there are excess shocks. So clearly before starting to look at recovery of AM shock innovations, and testing the assumptions madeabout them, one would want to make every effort to ensure that the model being used is not mis-specified, i.e. the parameter
estimates are appropriate. Applying our material involves assuming that this is the case, and then the question asked is what one
can learn about the AM shock innovations given these parameter values. In the case of Ireland (2011), we assumed the estimatedparameters were the true ones and found, from the recoverable shocks (or those that may nearly be) that those AM shock innovations
did not satisfy some of their assumed properties.
The EM shock are more complex; these will be correlated when there are excess shocks. The indirect inference approach ispresented as a way of determining whether any correlation evident in the innovations is statistically different from that induced by
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A. Pagan and T. Robinson

excess shocks even with uncorrelated AM shock innovations. However, it is not necessary to know that in order to conclude thatvariance and variable decompositions are inappropriate when excess shocks exist, as they use the EM shock innovations.
Our assumption of treating the parameter values as known is essentially considering the best possible scenario when analyzing
models with excess shocks.
5. Conclusion
Estimating structural shocks is a key aspect of applied macroeconomics, but today often one sees models with more shocks thanobserved variables, i.e. ‘‘excess shocks’’. Models with excess shocks are widespread, and come with a variety of different modelingapproaches. This paper shows both how the shocks in such models can easily be assessed and the difficulties that excess shocks cancreate when interpreting commonly reported items produced from these models.
Excess shocks complicate the interpretation of impulse responses. The investigator typically maintains that the assumed modelshock innovations can be recovered and are uncorrelated. This is done in order to implement the ceteris paribus assumption that
impulse responses are found by varying one shock and keeping the others constant. That the shock innovations are uncorrelated is
an assumption and, as the Gershwins’ song says, ‘‘It Ain’t Necessarily So’’. Because of the importance of this assumption we believe
one should ask if it is contrary to the data. This paper demonstrates how it can be assessed when excess shocks exist.
If excess shocks exist the first problem is to determine whether one can recover the AM shock innovations from the data, while thesecond is whether the recoverable shock innovations are uncorrelated. We describe a method using the Kalman filter and smootherthat enables one to identify which are the recoverable shocks. The results from the Kalman filter coincide with the method suggested
in Forni et al. (2019) for the same purpose in SVARs. Forni et al. focus upon using filtered EM shocks. Our measure extends this toallow for smoothed EM shocks and so provides information about the ability to recover the assumed shocks with the complete data
sample. It agrees with the definition of recoverability in Chahrour and Jurado (2022), unifying these two approaches. Our measureis straightforward to compute as it is an output from the Kalman smoother.
If a shock is not recoverable then its correlation with others cannot be directly assessed. We propose a way it can be checkedusing indirect inference, and this method was demonstrated with the Lubik and Schorfheide (2004) model.
There are three major implications of our analysis for applied macroeconomic research. First, when working with excess shocks
researchers should check which of the assumed shocks can be recovered, and then ask about the validity of the assumptions they
are making about them. Those that cannot be recovered can be assessed using indirect inference. Checking the shock properties
is rarely done in the literature, even when there are no excess shocks. Maintaining that the assumptions hold when they can bechecked is undesirable.
Secondly, while the decision to work with excess shocks is one made by the researchers, they should demonstrate to the reader
the actual properties of the assumed shocks as revealed by the data, so the reader can assess which of the impulse responses presented
are informative. In many ways excess shocks are best avoided if their inclusion is not necessary in order to answer the researchquestion being investigated. For example, one might use more and/or better observed data. In the case of Ireland’s (2011) model,
an additional variable – hours worked – could readily be observed, thereby eliminating excess shocks.
Finally, the third implication of excess shocks relates to the common practice of assessing the importance of various shocks using
variable and variance decompositions of the data. These use the EM shock innovations and not the AM shock innovations. When
there are excess shocks the innovations into the EM shocks are contemporaneously correlated, and may also be serially correlated,and so one cannot perform these decompositions in a unique way. More generally, if it is necessary to work with excess shocks theanalysis should be conducted recognizing the limitations that having an excess of shocks imposes.
In essence, excess shocks can limit our ability to validly interpret many of the common items constructed from a wide range ofmacroeconomic models.
Appendix A. Further applications of assessing shock recoverability

A.1. A permanent/transitory components model
To illustrate the consequences of excess shocks in models with I(1) variables we use a variant of the simple UC model analyzed
previously. Such a model has been used to measure output gaps; see, for example, Orphanides and van Norden (2002). These models
decompose an observed series 𝑦𝑡 as 𝑦𝑡 = 𝑦𝑝𝑡 + 𝑦𝑐𝑡 , where 𝑦𝑝𝑡 is a permanent or ‘‘trend’’ component of 𝑦𝑡 and 𝑦𝑐𝑡 a transitory component,
often called the cycle (or output gap). Assumptions have to be made about how these evolve, and we look at the simplest set:
𝜓1𝑡 = 𝛥𝑦𝑝𝑡 = 𝜀1𝑡
𝜓2𝑡 = 𝑦𝑐𝑡 = 𝜀2𝑡𝑧𝑡 = 𝛥𝑦𝑡 = 𝜀1𝑡 + 𝛥𝜀2𝑡 ,where 𝜀1𝑡 and 𝜀2𝑡 are 𝑛.𝑖.𝑑.(0, 1) and independent of one another.32

32 This model implies that 𝛥𝑦 is a MA(1), 𝛥𝑦 = (1 + 𝛼𝐿)𝑢 , and so there are two parameters that can be estimated from the data − 𝛼 and 𝜎 2 − to give𝑡𝑡𝑡𝑢estimates of the variances of 𝜀1𝑡 and 𝜀2𝑡 . However, we will assume that all of the variance parameters are known so that no issues of identification arise.

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To see the consequences of excess shocks, notice that this is the same model as analyzed in Section 3.4, but now with 𝜌 = 1,
so the conclusions about singularity of the covariance matrix of the filtered EM shock innovations and serial correlation in thesmoothed EM shock innovations still hold.
The perfect correlation between the estimated filtered shocks has economic implications. In this model the cyclical component –the output gap – is 𝜀2𝑡 . Consequently the estimated output gap and innovations to trend growth are perfectly correlated. Therefore,
placing an economic interpretation on trend growth, such as that it reflects increases in aggregate supply, is problematic. This will
occur even if a distinction between the shocks driving the cycle and the trend exists in the assumed shocks.33
This correlation has been noted before. Morley et al. (2003) compared the Beveridge–Nelson (BN) definition of the cycle withwhat one would obtain from a UC model. Drawing on Watson (1986), and using a UC model with uncorrelated innovations, they
noted that the BN decomposition led to a filtered estimate of the cycle, 𝐸𝑡 𝑦𝑐𝑡 , identical to that from the Kalman filter of the UC model,
since it was already known that the BN trend and cycle innovations were perfectly correlated. They also show that, although a BN
decomposition based on an ARIMA model of the observed data implies a different UC model (and trend estimates), the estimatedinnovations to the trend and cycle components of this latter model are perfectly correlated.34 Our contribution is to point out thatthe perfect correlation arises from excess shocks, and so it is a wider issue than just filtering.35 Morley et al. (2003) note that Wallis(1995) made the point that the correlations between the DGP innovations and those evident in the estimated innovations can differ
because of estimation.
A.2. The Hodrick–Prescott filter
The Hodrick–Prescott (HP) filter (Hodrick and Prescott, 1997) is extensively used in applied macroeconomic work and has been
criticized along several dimensions (see, for, example, Hamilton, 2018; Cogley and Nason, 1995 and Fukač and Pagan, 2010). In
this sub-section we demonstrate the inability to recover the hypothesized shocks in determining some of the properties of the HP
filter. To do so we exploit the fact that it can be found from a UC model (see, for example, Harvey and Jaeger, 1993). Doing so
enables our previous analysis to be readily applied to the HP filter. The corresponding UC model is(1 − 𝐿)2 𝑦𝑝𝑡 = 𝜀1𝑡𝑦𝑐𝑡 = 𝜙𝜀2𝑡𝑦𝑡 = 𝑦𝑝𝑡 + 𝑦𝑐𝑡 .
Note the cyclical component is a multiple (𝜙) of the innovation 𝜀2𝑡 . The parameter 𝜙 equals
the HP filter is applied to quarterly data.
The system can be expressed as√
𝜆 and 𝜆 = 1,600 is common when𝑧𝑡 = (1 − 𝐿)2 𝑦𝑡= 𝜀1𝑡 + 𝜙(𝜀2𝑡 − 2𝜀2𝑡−1 + 𝜀2𝑡−2 ).
⎡ 𝜀1𝑡 ⎤
⎡0
Defining 𝜓𝑡 = ⎢ 𝜀2𝑡 ⎥ the SSF has the matrices 𝑀 = ⎢0⎢⎥⎢
⎣𝜀2𝑡−1 ⎦
⎣0

0
0
1

0⎤
⎡1
0⎥ , 𝐶 = ⎢0
⎥
⎢
0⎦
⎣0

0⎤
[
1⎥, 𝐷1 = 1
⎥
0⎦

𝜙

]
[
0 , and 𝐷2 = 0−2𝜙]
𝜙 . This leads

to filtered EM innovations 𝐸𝑡 𝜀2𝑡 = 𝜙𝐸𝑡 𝜀1𝑡 , so the filtered trend innovations and the cycle are perfectly correlated. The AM shocks
are not recoverable with past and current data; the steady-state 𝑃𝑡|𝑡 has non-zero diagonal elements for the one-sided HP filter.
The original HP filter was two sided and therefore generated by smoothed, rather than filtered, shocks. In Section 3.3 it was
shown that the smoothed shocks will have serial correlation even if the filtered shocks do not. That is, even if the hypothesized UC
model used to generate the HP filter is correct there will still be serial correlation in the smoothed cycle innovations. This feature
has often been observed in the literature for estimates of the cycle given by the HP filter and it has been interpreted as implyingthat the assumed UC model is mis-specified. However, the serial correlation in the smoothed cycle innovations arises as the AM shocks
are not recoverable; it occurs even if the HP filter is the correct UC model. The steady-state 𝑃𝑡|𝑇 does not have a zero diagonal.
As an experiment we simulated data using the model above as the DGP with 𝜙 = 40, i.e. 𝜆 = 1600. Consequently the HP filter witha choice of 𝜆 = 1600 is the optimal filter. As predicted, applying it produces estimated innovations that are perfectly correlated,⎡.9995⎤
with 𝐸𝑡 𝜀2𝑡 = 40𝐸𝑡 𝜀1𝑡 . The steady-state 𝑑𝑖𝑎𝑔(𝑃𝑡|𝑡 ) = ⎢.2006⎥. There is little serial correlation in the filtered cycle innovation but⎢⎥⎣.1608⎦⎡.9439⎤large amounts in the smoothed one. The steady-state 𝑑𝑖𝑎𝑔(𝑃𝑡|𝑇 ) = ⎢.0561⎥. As mentioned above, when this serial correlation in the⎢⎥⎣.0561⎦

33

If one uses the smoothed shocks they will still be correlated and are likely to be serially correlated as well.
Anderson et al. (2006) demonstrate how the BN decomposition can be obtained from a ‘‘single source of error’’ state-space model (i.e. where the measurement
and state equations are driven by a common innovation; see Snyder, 1985). This utilizes the perfect correlation between the innovations. Morley (2002) provides
an alternative state-space approach to computing the BN decomposition, which was adopted in Morley et al. (2003).
35 The UC model of Clark (1987) has the same feature.
34

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A. Pagan and T. Robinson

smoothed estimates of the cycle (𝐸𝑇 𝜀2𝑡 ) has been seen in data, it has often been said it shows that the trend has not been correctly
estimated, and so the cyclical component has to be purged of its serial correlation. But here the HP filter’s UC model is the optimalfilter and yet there is a large amount of persistence in the smoothed innovations.
Appendix B. Supplementary data
Supplementary material related to this article can be found online at https://doi.org/10.1016/j.euroecorev.2022.104120.
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