# Measuring Shocks to Central Bank Independence using Legal Rulings††thanks:
The authors are grateful to Marek Jarociński for providing many helpful
suggestions and parts of the dataset. Huber and Pfarrhofer gratefully
acknowledge financial support from the Austrian Science Fund (FWF, grant no.
ZK 35) and from the Oesterreichische Nationalbank (OeNB, grant no. 18304).

Stefan Griller Vienna University of Economics and Business Florian Huber
University of Salzburg Michael Pfarrhofer University of Salzburg

We investigate the consequences of legal rulings on the conduct of monetary
policy. Several unconventional monetary policy measures of the European
Central Bank have come under scrutiny before national courts and the European
Court of Justice. These lawsuits have the potential to severely impact the
scope and flexibility of central bank policies, and central bank independence
in a wide sense, with important consequences for the real and financial
economy. Since the number of relevant legal challenges is small, we develop an
econometric approach that searches for minimum variance regimes which we use
to isolate and measure the effects of these events. Our results suggest that
legal rulings addressing central bank policies have a powerful effect on
financial markets. Expansionary shocks ease financial conditions along various
dimensions, and inflation swap reactions suggest inflationary pressures with
stronger effects in the short term.

JEL: E43, E44, E58, K00.

KEYWORDS: Local projection, euro area, financial markets, mixture model,

factor model, identification through heteroskedasticity

##  1 Introduction

The advantages of and challenges to an independent central bank have been
emphasized in a broad body of academic papers over the last 30 years (see,
e.g., Alesina, 1988; Cukierman _et al._ , 1992; Cukierman, 1994; Alesina and
Summers, 1993; Fischer, 1995; Crowe and Meade, 2007; Walsh, 2011; Binder,
2021). In many economies, legislators have been improving the respective legal
architectures and procedures to make central banks less reliant on day-to-day
policy making and to free them from undue political pressures and influence.
While measuring the precise economic consequences of such changes is
difficult, there exists some evidence that related shifts in central bank
policies in the United States (US) and the United Kingdom (UK) have led to
declining levels of inflation, decreasing inflation volatility and the
stabilization of inflation expectations (Cukierman and Meltzer, 1986; Blinder
_et al._ , 2008).

During the global financial crisis and the Covid-19 pandemic, central banks
such as the US Federal Reserve and the Bank of England, however, actively
cooperated with governments to fight these economic downturns. Such
coordinated action did not harm central bank independence due to the legal
structure of the Bank of England and the Federal Reserve System. In the euro
area, coordinated actions also took place, but with member states exhibiting
heterogeneous macroeconomic and, most importantly, debt dynamics. The question
whether the European Central Bank (ECB) is legally allowed to intervene
frequently in this regard arose in the media and policy discussions.

The independence of the ECB constitutes a main pillar of the design of the
European Monetary Union (EMU). In the last decade, ECB expertise and capacity
have been challenged by several severe economic crises which called for
unconventional monetary policy actions (see Hartmann and Smets, 2018, for a
review). These actions involved buying government debt securities in
unprecedented amounts and this raised the question whether the ECB is
exceeding its mandate. As a result, selected ECB programs have come under
legal scrutiny. These lawsuits on the (il)legality of the ECBs unconventional
monetary policy measures could, in the worst case, severely limit the scope of
the ECB’s options during crises and thus also its independence and credibility
in the context of timely and effective policy making. This argument gives rise
to our interpretation of the outcome of such a ruling as a central bank
independence shock.111For a recent study that uses Twitter data to identify
central bank independence shocks, see Bianchi _et al._ (2021).

We take a closer look at three preliminary rulings handed down by the European
Court of Justice (CJEU) at the request of highest national courts, and at the
related decisions of these courts. We scrutinise the Pringle Case which
concerned the legality of establishing the European Stability Mechanism (ESM)
which is a member states’ measure, and was brought up by the Irish Supreme
Court; further, we assess the Gauweiler and the Weiss Case, both raised by the
German Federal Constitutional Court (GFCC), and both concerning non-
traditional ECB measures which allegedly were outside the ECB’s mandate.

In this paper, we aim to exploit variation surrounding the announcements of
these rulings to identify a central bank independence shock and evaluate its
effect on a broad panel of crucial financial indicators. We achieve
identification by assuming that the variance of the endogenous variables is
higher on ruling days than on so-called non-event days.222This feature of
exploiting heteroskedasticity for shock identification relates our paper to a
broad literature on using time-varying volatilities in macroeconomic and
financial data to pin down structural shocks statistically for enabling
economically meaningful interpretation (see, e.g., Sentana and Fiorentini,
2001; Rigobon, 2003; Lewis, 2017, 2021; Bertsche and Braun, 2022; Brunnermeier
_et al._ , 2021). Our approach is closely related to the factor model proposed
in Gürkaynak _et al._ (2020). Since the number of events (i.e., days with news
about rulings) relative to the non-event days is very small, we propose a
mixture model that endogenously selects the non-event days. This mixture model
assumes that the panel of time series features a common volatility process
moving through a set of regimes (see also Carriero _et al._ , 2016). The non-
event set is then defined to comprise of observations with minimum variance,
and thus captures the notion that if the common variance is minimal, we can
safely regard this regime as a non-event regime (i.e., a regime without
meaningful events that could jeopardize our identification strategy).

We consider a panel of time series including euro area-wide, country-level and
international information. Among several key variables of interest, we control
for cross-country movements in 10-year yields by including government bond
yields relative to German Bunds for several euro area member states. These
series are complemented by financial indicators such as measures of systemic
stress, short-term interest rates, corporate bond spreads, as well as stock
market and exchange rate information. Inflation expectations are captured by
including inflation swaps at different maturities, and oil and gold prices are
included as important non-domestic variables. To estimate the effects of
independence shocks on this panel of series we compute local projections
within a unified econometric framework and thus obtain a dynamic picture on
the effects of these rulings.

Our results can be summarized as follows. A positive independence shock (i.e.,
no impairment in the variety of policy actions available to the central bank)
leads to lower spreads between German 10-year Bunds and 10-year government
bond yields across selected euro area member states. Considering the cross-
sectional responses reveals appreciable heterogeneity with respect to the
dynamic responses. Countries such as Greece, Spain and Italy display more
pronounced reactions whereas countries located to the euro area core seem to
be somewhat less affected. A similar pattern occurs for measures of financial
stress across euro area member states. On aggregate, expansionary shocks ease
financial conditions, yield gains on stock markets and result in modest
increases in short-term inflation expectations.

The paper proceeds as follows. Section 2 provides the legal background and a
brief summary of the three major legal cases which we use to form our set of
events. Section 3 introduces the empirical framework used to estimate the
effects of the rulings in our panel of time series. Section 4 presents the
main empirical results of the paper. The final section summarizes and
concludes the paper.

##  2 Controversies on delimiting monetary and economic policy, and resulting
litigation

###  2.1 Exposition of the fundamental legal dimensions

ECB independence is a cornerstone of the EMU design – it is triple guaranteed
by primary law: by Articles 130 and 282(3) Treaty on the Functioning of the
European Union (TFEU), and Article 7 Protocol Number 4 on the Statute of the
European System of Central Banks and of the European Central Bank. Moreover,
ECB independence is bolstered by flanking provisions, among those the
prohibition on monetary financing (Article 123 TFEU), and the appointment
mechanism as well as the not renewable eight-years term of office for members
of the Executive Board. The underlying rationale is that delivering on the
ECB’s primary objective, price stability (Article 127 TFEU), would be made
more difficult if not endangered by external, not the least political
influence of any kind.

ECB independence is guaranteed by the Treaties and thus strongly protected
under European Union (EU) law not only against undue influence on ECB organs
but also against encroachments by member states’ and other EU institutions’
activities. Any violation can be brought before the CJEU (Articles 260 and 263
TFEU). For “monetary policy for the Member States whose currency is the euro”
is an exclusive competence of the EU, with the ECB as the EU institution
entrusted to take care of; “only the Union may legislate and adopt legally
binding acts, the Member States being able to do so themselves only if so
empowered by the Union or for the implementation of Union acts” (Articles 2(1)
and 3(1)(c) TFEU).

Independence, however, does not exempt the ECB from the EU legal order. The
ECB is subject to legal scrutiny by the CJEU. The latter emphasised early on
that independence “does not have the consequence of separating” the ECB from
every rule of EU law (CJEU, Case C-11/00, §135). First and before anything
else, the ECB must act within the limits of the powers conferred on it
(Article 13(2) Treaty on European Union, TEU), which is to “conduct the
monetary policy of the Union” (Article 282(1) TFEU). Acting outside that
mandate, or more generally, violating EU law, would render ECB measures
voidable by the CJEU.

Both aspects – the member states’ respect for the ECB’s powers and, inversely,
the ECB’s respect for its own limits which protect the member states’ economic
(including fiscal) policy powers which they did not transfer to but merely
“coordinate” within the Union (Article 5(1) TFEU) – became controversial
during and in the aftermath of the economic and financial crisis starting in
$2008$. The controversy led to several high-profile judgments clarifying the
limits between European economic and monetary policy. The cases will be
briefly sketched out below (for a more detailed analysis, see Griller, 2021).
All three of them had been filed with the CJEU by highest national courts as
preliminary ruling procedures (Article 267 TFEU), all of them putting critical
questions concerning the legality of measures taken. Only the second and third
judgment (Gauweiler and Weiss) directly concerned (unconventional) ECB
measures, the first (Pringle) was on a measure taken by the member states.
However, all three of them dealt with the scope of monetary policy and thus
ECB independence, and can therefore, if only in a wide sense regarding
Pringle, be addressed as possible independence shocks.333The ECB’s Outright
Monetary Transactions (OMT) program is conditional on a memorandum of
understanding between a member state and the EMS, so the Pringle case had the
potential to nullify the OMT, thus directly impacting the ECB’s toolkit. Table
1 provides a list and further details.444We have verified that the dates of
the rulings do not systematically coincide with other important news. In
particular, we have cross-checked with the dates of the following press
releases: Eurostat releases of euro area flash CPI, core CPI, produces prices,
M3 money growth, trade balance, unemployment rate, business and consumer
surveys, national accounts, industrial production, retail sales; Markit euro
area PMI indices; ECB monetary policy announcements, speeches by ECB
Presidents; Federal Reserve Bank monetary policy announcements (scheduled and
unscheduled FOMC press releases); US initial jobless claims and non-farm
payrolls. For most releases the overlap is on less than three dates, and often
on zero dates.

The outcomes had been far from obvious. The CJEU’s judgments included a
certain margin of discretion. Indeed, all these judgments have been
controversial before, during and in the aftermath of litigation, both at CJEU
and national level. To put it differently, ex ante, these lawsuits had the
potential of ousting crisis intervention measures as illegal. As a result,
these measures would have had to be terminated. That would have caused a
serious challenge to both the member states’ and the ECB’s efforts to contain
the respective crises and to counteract their adverse effects. Thus, stock
market and interest rate reactions to these decisions are specifically
pertinent when it comes to measuring the economic relevance of court
decisions.

The reactions reflect the mentioned uncertainty and possible expectations or
suspicions of a different outcome. If the court decision as such comes as a
surprise to the markets, which in our cases is conceivable mainly for the
national court decision to request the preliminary ruling by the CJEU (see
below), it is this unexpected event which may be considered as a shock that is
provoking market reactions. Identifying the details of these deviating
expectations and their influence on the resulting dimension of market
reactions – including the important differences between different litigation
stages – goes beyond the scope of our study and might be worthwhile future
research. However, this would necessitate digging both into the economic
weight of the matter and the quality of ex ante market expectations vis-à-vis
the respective judgments, posing a challenge to any research design.

Preliminary ruling procedures under Article 267 TFEU normally provide at least
four, and sometimes even more, prominent stages which are worthwhile
examining. First, the decision of the national court to interrupt its
proceedings and ask the CJEU for an answer to a difficult question of
interpretation or on the validity of secondary EU law. Second, the (non-
binding) Opinion of the CJEU’s Advocate General (AG) whereby the latter
proposes how the CJEU should decide. Third, the requested answer of the CJEU
given in its judgment which is binding under EU law. Fourth, the judgment or
decision of the national court based on the answer given by the CJEU.

Table 1: Ruling dates and description

|  Decision |  Publication |  Number |  Court |  Description
---|---|---|---|---|---
Case C-370 (Pringle), ECLI:EU:C:2012:756
|  2012-07-31 |  2012-07-31 |  [2012] IESC 47 |  SC, Ireland |  Request for preliminary ruling
|  2012-10-26 |  2012-10-26 |  C-317/12 |  CJEU, AG |  Opinion of the AG
|  2012-11-27 |  2012-11-27 |  C-317/12 |  CJEU |  Judgement
Case C-62/14 (Gauweiler), ECLI:EU:C:2015:400
|  2014-01-14 |  2014-02-07 |  2 BvR 2728/13 |  GFCC |  Request for preliminary ruling
|  2015-01-14 |  2015-01-14 |  C62/14 |  CJEU, AG |  Opinion of the AG
|  2015-06-16 |  2015-06-16 |  C62/14 |  CJEU |  Judgement
|  2016-06-21 |  2016-06-21 |  2 BvR 2728/13 |  GFCC |  Judgement
|  |  |  2 BvR 2729/13 |  |
|  |  |  2 BvR 2730/13 |  |
|  |  |  2 BvR 2731/13 |  |
|  |  |  2 BvE 13/13 |  |
Case C-493/17 (Weiss), ECLI:EU:C:2018:1000
|  2017-07-18 |  2017-08-15 |  2 BvR 859/15 |  GFCC |  Request for preliminary ruling
|  |  |  2 BvR 1651/15 |  |
|  |  |  2 BvR 2006/15 |  |
|  |  |  2 BvR 980/16 |  |
|  2018-10-04 |  2018-10-04 |  C-493/17 |  CJEU, AG |  Opinion of the AG
|  2018-12-11 |  2018-12-11 |  C-493/17 |  CJEU |  Judgement
|  2020-05-05 |  2020-05-05 |  2 BvR 859/15 |  GFCC |  Judgement
|  |  |  2 BvR 1651/15 |  |
|  |  |  2 BvR 2006/15 |  |
|  |  |  2 BvR 980/16 |  |

  * Notes: Abbreviations are Advocate General (AG), Court of Justice of the European Union (CJEU), European Commission (EC), German Federal Constitutional Court (GFCC), Supreme Court (SC). In our empirical work, we consider publication dates in case they differ from the decision dates for the timing of the shocks. Publication dates are those of relevance, since this is when the public is informed about the respective decision and judges are prohibited by law from disclosing their rulings prematurely.

###  2.2 The Pringle case: Challenging the European Stability Mechanism

In this case, the legality of the Treaty Establishing the European Stability
Mechanism (ESM Treaty) which is a treaty concluded under international law by
all EMU member states was under scrutiny. Operational from 2012, the ESM is
(primarily) entitled to grant loans to euro area member states in economic
difficulties, conditional on performing a macroeconomic adjustment program.
The overall condition is that such support is necessary to safeguard the
financial stability of the euro area as a whole and of ESM members. The ESM’s
maximum lending capacity is EUR $500$ billion, based on EUR $80.5$ billion
paid-in capital by its member states, and EUR $724.3$ billion of committed
callable capital.

The reason for the Irish challenge was, not the least, an effort to avoid the
financial burden connected with ESM membership. It was claimed that Ireland
(pars pro toto of all member states), by ratifying the Treaty, would violate
several provisions of the TEU and the TFEU on EMU and would directly affect
the respective exclusive EU competence. Most and before all, there was the
alleged violation of the so-called no bailout clause (Article 125 TFEU).

Both arguments were rejected by the CJEU (Case C-370/12, Pringle). The first
was done based on the Court’s definition, for the first time in such clarity,
of monetary policy by its objectives – mainly price stability – rather than
its instruments. The purpose of the ESM, as the Court stressed, is not to
maintain price stability, but rather meeting the financial requirements of
members facing severe financing problems: “Even if the activities of the ESM
might influence the rate of inflation, such an influence would constitute only
the indirect consequence of the economic policy measures adopted” (§97).
Consequently, the CJEU found EU competences to be respected.

The Court saw Article 125 TFEU respected based on his interpretation that this
provision “is not intended to prohibit either the Union or the Member States
from granting any form of financial assistance whatever to another Member
State” (§130). Article 125 TFEU requires that the member states remain subject
to the logic of the market when entering into debts (§135); “[…] it must be
held that that provision prohibits the Union and the Member States from
granting financial assistance as a result of which the incentive of the
recipient member state to conduct a sound budgetary policy is diminished”
(§136). Financial assistance remains possible as long as the recipient member
state remains responsible for its commitments, and the conditionality prompts
a sound budgetary policy. To some surprise, ESM assistance was found
compatible with these requirements – given that the ESM does not act as a
guarantor of the debts and does not assume them: “On the contrary, such
assistance amounts to the creation of a new debt, owed to the ESM by that
recipient member state, which remains responsible for its commitments to its
creditors in respect of its existing debts” (§139).

Accepting the ESM as a measure not encroaching on the EU’s exclusive monetary
union competence and thus respecting the ECB’s independence was less
spectacular than denying any violation of Article 125 TFEU, the so-called no
bailout clause (see, e.g., Craig, 2013; Ohler, 2013). It was this aspect which
made the case so controversial which could not escape the attention of
financial markets. The danger that a CJEU verdict to the contrary would mean
the end of the ESM had been imminent.

###  2.3 The Gauweiler case: The first challenge to the ECB’s non-traditional
measures before the CJEU

In 2012 the ECB announced its readiness to perform Outright Monetary
Transactions (OMT) – in essence, purchases in secondary sovereign bond markets
aiming at safeguarding an appropriate monetary policy transmission and the
singleness of the monetary policy. The program was considered for euro area
member states under an ESM adjustment program, and subject to conditionality.
The announcement was accompanied by a statement from ECB president Mario
Draghi: “ We will do whatever it takes within our mandate […] to have a single
monetary policy […], to maintain price stability […] and to preserve the
euro.” The program itself never became operative.

Nevertheless, it was challenged before the GFCC. The latter in turn introduced
its first ever preliminary ruling procedure with the CJEU, clearly insinuating
that the ECB’s announcement might be beyond its mandate (ultra vires), for
resulting in an economic policy measure instead of respecting the limits of
monetary policy as required under the treaties.

The CJEU dismissed the argument and upheld the ECB’s announcement as legal. It
confirmed its findings in Pringle that determining monetary policy must be
done by referring “principally to the objectives of that measure.” Instruments
are “also relevant,” only (Case C-62/14, Gauweiler, §46). Safeguarding the
singleness and an appropriate transmission of monetary policy would likely
contribute to the primary objective of price stability (§§47–50). “Indeed, a
monetary policy measure cannot be treated as equivalent to an economic policy
measure merely because it may have indirect effects on the stability of the
euro area […]” (§52). As a result, purchasing government bonds on the
secondary market might be both: economic policy if done by the ESM, but
monetary policy if done by the ECB (§64).

A different outcome would have invalidated the OMT program, resulting in a
prohibition to uphold or renew it. The GFCC accepted, if somehow reluctantly,
the verdict of the CJEU, and consequently dismissed and rejected the
constitutional complaints before it (Case 2 BvR 2728/13 et al., June 21,
2016).

###  2.4 The Weiss case: A legal worst-case scenario turning reality

The second ever preliminary ruling procedure introduced by the GFCC also
concerned an ECB program, equally insinuating that it might be ultra vires. At
stake was the “Public Sector Purchase Programme” (PSPP), the most important
part of the ECB’s non-traditional measures taken to fight the economic crisis,
also labeled quantitative easing (QE). Starting in 2015, the ECB purchased
sovereign bonds in secondary markets up to EUR $80$ billion per month (2016
and 2017) in order to support economic stability and avert deflationary
pressures.

Again, however, the CJEU denied that the ECB had overstepped its mandate and
upheld the program. It did so by reconfirming that determining whether a
measure falls within the area of monetary policy is principally depending upon
the objectives pursued, while the instruments employed are only of secondary
relevance (Case C 493/17, Weiss, §§53ff). Equally it reconfirmed “that a
monetary policy measure cannot be treated as equivalent to an economic policy
measure for the sole reason that it may have indirect effects that can also be
sought in the context of economic policy” (§61). The PSPP was also held to be
proportionate (§71ff) and respecting the prohibition to grant overdraft
facilities or any other type of credit facilities of Article 123 TFEU
(§101ff).

This time, however, the GFCC refused to accept the CJEU’s ruling. In its so-
called PSPP judgment, the GFCC found that the Federal Government of Germany
and the Bundestag violated the complainants’ constitutional rights by failing
to take steps challenging the ECB’s PSPP decision (GFCC, 2 BvR 859/15, May 5,
2020; PSPP judgement). The GFCC declared both the ECB’s decision and the
CJEU’s judgment upholding the decision as ultra vires acts depriving them of
their “binding force in Germany” (§§154, 163 and 165). Such an ultra vires act
of the ECB or the CJEU is “not to be applied in Germany, and has no binding
effect in relation to German constitutional organs, administrative authorities
and courts” (§234). The consequences affect, inter alia, also the Bundesbank:
“[…] Following a transitional period of no more than three months […], the
Bundesbank may thus no longer participate in the implementation and execution
of the PSPP Decision” (§235).

The story does not end here. Shortly after, an order of execution was sought
with the GFCC. The GFCC found the application inadmissible (GFCC, Order of the
Second Senate, 2 BvR 1651/15, 2BvR 2006/15, April 29, 2021). To the surprise
of many, the Court explained at length that the applications were not only
inadmissible but also unfounded. In essence it explained that the Federal
Government and the Bundestag dispose of a wide range of measures to fulfil the
requests and that the measures taken did indeed fulfil the PSPP judgement’s
requirements (§108). It also found that the ECB Governing Council’s assessment
of the proportionality of the PSPP had been sufficient (§100). This can be
seen as the GFCC’s “final green light” for the PSPP. The European Commission,
though, was not satisfied. On June 8 2021, the Commission sent a letter of
formal notice (under Article 258 TFEU) to Germany for breach of fundamental
principles of EU law, because of the GFCC’s PSPP judgment. Further
developments remain to be seen.

This discussion shows that courts have the potential to severely limit the
possibilities and flexibility of the ECB to fight adverse economic or other
events. In the next step, we will develop an econometric model with the goal
to measure the effects associated with these rulings on a large panel of
mostly financial indicators for selected euro area countries.

##  3 Measuring central bank independence shocks

Table 1 provides us with several event dates (“Publication”) which we can use
to analyze the effects of ruling announcements on a set of outcome variables.
Since the number of events is rather low, reliable estimation of the causal
effect of the rulings is difficult and calls for appropriate techniques which
we develop in this section. Our econometric framework builds on the notion
that rulings introduce uncertainty at high (i.e., daily) frequencies and these
increases in volatility (relative to days without significant
economic/political events that might impact financial markets) are used to
identify the causal effect of legal challenges to the ECB’s independence. Our
approach fundamentally builds on and extents the model developed in Gürkaynak
_et al._ (2020).

Let $\bm{y}_{t}=(y_{1t},\ldots,y_{Mt})^{\prime}$ denote a panel of $M$ daily
variables which, for simplicity, are assumed to have an unconditional mean
equal to zero. We model $\bm{y}_{t}$ using a static factor model with time-
varying factor loadings:

$\bm{y}_{t}=d_{t}\bm{\lambda}f_{t}+\bm{\varepsilon}_{t},\quad\bm{\varepsilon}_{t}\sim\mathcal{N}(\bm{0},\bm{\Sigma}),$
(1)

with $d_{t}\in\\{0,1\\}$ being an indicator which equals one if $t$ is in a
so-called event (or ruling) set $\mathcal{A}_{\text{R}}$ while it equals zero
if $t$ is included in a non-event set $\mathcal{A}_{\text{NE}}$.
$\bm{\lambda}$ denotes an $M$-dimensional vector of factor loadings and
$\bm{\varepsilon}_{t}$ is a Gaussian shock vector with zero mean and a
diagonal variance-covariance matrix
$\bm{\Sigma}=\text{diag}(\sigma_{1}^{2},\dots,\sigma_{M}^{2})$.
$f_{t}\sim\mathcal{N}(0,1)$ is a latent factor which summarizes co-movements
in the yields on ruling dates. The key implication of Eq. (1) is that on
ruling dates, the implied variance-covariance matrix of $\bm{y}_{t}$ is:

$\text{Var}(\bm{y}_{t})=\bm{\lambda}\bm{\lambda}^{\prime}+\bm{\Sigma},$ (2)

whereas on non-event dates, it is simply $\text{Var}(\bm{y}_{t})=\bm{\Sigma}$.
Hence, if $\bm{\lambda}\neq\bm{0}_{M}$, our model implies that the variance of
the $i$th element of $\bm{y}_{t}$, $y_{it}$, on a ruling day is equal to
$\lambda^{2}_{i}+\sigma_{i}^{2}$ and thus exceeds the variance on a non-event
day. The off-diagonal elements of $\text{Var}(\bm{y}_{t})$ on a ruling day are
given by $\lambda_{i}\lambda_{j}$ for $i\neq j$, which results in a full
variance-covariance matrix. By contrast, on non-event days, the shocks in
$\bm{\varepsilon}_{t}$ are independent. In principle, if movements in
$\bm{y}_{t}$ are dominated by legal rulings on a specific ruling date, the
corresponding commonalities given by

$\frac{\lambda^{2}_{i}}{(\lambda_{i}^{2}+\sigma_{i}^{2})}$ (3)

approach unity.

Model estimation can be carried out using maximum likelihood or Bayesian
techniques. In this paper, we follow a Bayesian approach and employ a Gibbs
sampler to simulate from the joint posterior of $\bm{\lambda},f_{t}$ and
$\bm{\Sigma}$. This algorithm, however, relies on knowing the non-event set
$\mathcal{A}_{\text{NE}}$. The next sub-section describes how this set is
formed.

###  3.1 Identifying non-event windows

One important assumption underlying our modeling framework is that we assume
the variance of $\bm{y}_{t}$ to differ between ruling and non-ruling dates.
This is a rather strong assumption. Several important regular events (such as
press conferences after governing council meetings, see, e.g., Altavilla _et
al._ , 2019, or data releases by statistical agencies) or non-regular events
(such as Mario Draghi’s “Whatever it takes” speech) have pronounced effects on
financial variables. Having such observations in the non-event/non-ruling set
would certainly lead to a situation where the assumption of a higher variance
on ruling dates is difficult to maintain. This is the reason why we
meticulously avoided defining non-ruling windows but instead differentiated
between ruling and non-event windows. Moreover, if the number of “event” days
is small to moderate (which is the case for court rulings), simply
discriminating between ruling and non-ruling dates easily masks differences in
variances. This makes identifying the factor (and/or the loadings) difficult
and renders inference imprecise.

In the related literature it is common to avoid corrupting event windows of
interest with other events by specifying tight windows surrounding these
announcements (see, e.g., Kuttner, 2001; Gürkaynak _et al._ , 2005; Gertler
and Karadi, 2015; Jarociński and Karadi, 2020). This paper proposes a
different route which does not rely on using intra-day data and/or precise
timings of the rulings. Let $\mathcal{A}_{\text{NR}}$ denote the set which
includes all days except the ones which feature a ruling decision (i.e., for
all $t\notin\mathcal{A}_{\text{R}}$). We assume that $\bm{y}_{t}$ for
$t\in\mathcal{A}_{\text{NR}}$ evolves according to:

$\bm{y}_{t}\sim\sum_{j=1}^{J}w_{j}\mathcal{N}(\bm{y}_{t}|\bm{0}_{M},\vartheta_{j}^{2}~{}\bm{\Omega})\quad\text{
for }t\in\mathcal{A}_{\text{NR}},$ (4)

with $w_{j}$ denoting a weight with $w_{j}\geq 0$ and $\sum_{j=1}^{J}w_{j}=1$
while $\vartheta_{j}^{2}$ denotes a component-specific variance with
$\vartheta_{1}<\dots<\vartheta_{J}$ and
$\bm{\Omega}=\text{diag}(\omega_{1}^{2},\ldots,\omega_{M}^{2})$ is a known
diagonal matrix which serves to control for differences in the scaling of
$\bm{y}_{t}$.555The ordering restriction on the variances solely refers to the
labeling of the regimes, and is a technical restriction that does not affect
posterior distributions (see Frühwirth-Schnatter, 2006, for details). This
implies that $\bm{\Sigma}=\vartheta_{1}\bm{\Omega}$ in terms of Eq. (1),
featuring solely the observations in the minimum variance regime, i.e., the
set $\mathcal{A}_{\text{NE}}$. In what follows, the diagonal elements of
$\bm{\Omega}$ are set equal to the empirical variances of the different
columns in $\bm{Y}=(\bm{y}_{1},\dots,\bm{y}_{T})^{\prime}$.

It is noteworthy that $\vartheta_{j}$ serves as a common scalar factor which
drives the variances on non-ruling dates. To see this more clearly, Eq. (4)
can be equivalently stated in terms of auxiliary variables $\delta_{t}$ which
select the mixture component:

$\bm{y}_{t}|\delta_{t}=j\sim\mathcal{N}(\bm{y}_{t}|\bm{0}_{M},\vartheta^{2}_{j}~{}\bm{\Omega}),$
(5)

which implies that if $\delta_{t}=j$, the variance of $\bm{y}_{t}$ is
proportional to $\vartheta_{j}^{2}$. Under the ordering restriction that the
variances are increasing, the set
$\mathcal{A}_{\text{NE}}=\\{t:\delta_{t}=1\\}$ collects all time periods where
the variance of the process in Eq. (5) is minimal. Our key assumption is that
on days in the non-event set, no substantial new information is driving
movements in the variables collected in $\bm{y}_{t}$.

This, of course, relies on an appropriate specification of $J$. Too small
values of $J$ could lead to regimes that might still be contaminated by
other/non-ruling based events (and thus feature a too high variance) whereas
too large values of $J$ could translate into regimes with too few observations
in the non-event set. In our case, we use a Bayesian approach in the spirit of
Malsiner-Walli _et al._ (2016) and estimate $J$ by setting it to a large value
(i.e., $J=30$) and then use shrinkage to empty out irrelevant clusters. For
the remaining parameters of the model, we choose adequate weakly informative
priors. Appendix A provides further details on the prior setup and the Markov
chain Monte Carlo sampling algorithm.

###  3.2 Local projections to compute dynamic responses

We use the general model in Eq. (1) specified as a local projection (see
Jordà, 2005) for horizons $h=0,1,\ldots,H$, such that we can interpret the
factor loadings in
$\bm{\lambda}_{h}=(\lambda_{1h},\ldots,\lambda_{Mh})^{\prime}$ directly as the
impulse response function for variable $y_{jt}$ at $h$. Note that our variable
set is comprised mostly of financial variables. We thus assume that their
dynamic evolution is described well with a random walk, and consequently
consider the dependent variable in differences, $\Delta y_{j,t+h}$, similar
to, e.g., Lewis _et al._ (2019); Swanson (2021):666Experiments indicate that
using this type of linear projection rather than including a set of (lagged)
control variables produces very similar results, pointing to the adequacy of
the random walk assumption.

$(y_{j,t+h}-y_{j,t-1})\equiv\Delta
y_{j,t+h}=d_{t}\lambda_{jh}f_{t+h}+\epsilon_{j,t+h}.$

We consider a horizon of up to $H=60$ (corresponding to three months of
workday-weeks). Specifying the model directly as a local projection has the
advantage of taking all sources of uncertainty into account within a unified
econometric framework. By contrast, a two-stage approach, estimating the
factor first and using, e.g., its posterior median to compute dynamic
responses with local projections, may yield incorrect inference and too narrow
credible sets for parameter estimates.

##  4 The effects of central bank independence shocks

In this section we present our empirical findings. The first sub-section
briefly discusses the dataset while Section 4.2 presents features of our model
such as the estimated latent factor, the cluster allocation and the
idiosyncratic variances. Sub-section 4.3 presents the dynamic responses to
central bank independence shocks (we also refer to these interchangeably as
ruling shocks).

###  4.1 Data overview

To understand the impact of the central bank independence shocks we study a
representative set of European, US and international financial variables
observed at a daily frequency. Our sample covers the period from March 2012 to
March 2021.

Among other variables we consider in our discussion of dynamic responses to
ruling shocks is the CISS, which is the Composite Indicator of Systemic Stress
in the financial system (see Hollo _et al._ , 2012). It is a summary measure
aggregating indicators of financial stress in the banking sector, money,
equity, bond and foreign exchange markets. To measure sentiments in the
sovereign debt markets we include the yields on 10-year government bond yields
of the euro area member states. The risk-free yield curve, based on the
overnight index swap (OIS) rates at maturities from 1 month to 2 years
reflects the market expectations regarding ECB policies. European inflation
swaps at maturities 1 to 10 years combine market expectations of inflation and
inflation risk premia. European stock market developments are represented by
the EuroStoxx 50 index accompanied by headline country-indices. Corporate bond
market developments are reflected by the Option-Adjusted Spreads (OAS) of
high-yield, BBB and AA corporate bonds over the sovereign bonds of the same
maturity, constructed by ICE BofAML.

Moreover, the information set also includes several other non-domestic and
international variables such as gold and oil prices, or financial variables
for the US. A full list of variables and additional information are provided
in Appendix B. Series and country-coverage is governed by data availability on
a daily frequency.

###  4.2 Ruling dates from an econometric perspective

Our model yields a posterior distribution over the factors $f_{t}$ for
different values of $h$. These can be interpreted as an unexpected shock to
central bank independence induced by legal events. Since the factor is not
identified up to the sign, economic interpretation needs to be carried out in
light of the estimated factor loadings in $\bm{\lambda}_{h}$. This is what we
do in Section 4.3. In this section, we focus on the quantitative properties of
the factor and assess whether using the mixture model sharpens statistical
inference.

Figure 1: Estimates of the factor $f_{t+h}$ at ruling dates. Upper panel: The
black and blue errorbars (horizontal lines mark 68 and 90 percent posterior
credible sets) are shown alongside the posterior median for a non-mixture and
mixture specification with $J=30$. Lower panel: Horizon-specific 68 and 90
percent posterior credible sets of the factor.

Figure 1 depicts boxplots of the posterior distribution of $f_{t}$ for
$h=0,1,\dots,H$ across the different events we use to identify the shock. To
investigate the effect the mixture model has on the estimated factors, we
include boxplots of a model that naively discriminates between ruling and non-
ruling days (in black) and compare them to our approach which uses a mixture
model to group non-event days into a single cluster (in blue). The figure
suggests that if we do not purge non-ruling days from other events, the
corresponding uncertainty bounds are much wider and include zero in almost all
cases. Two cases in which zero is not included are January 14, 2015
(Gauweiler) and May 5, 2020 (Weiss). In the latter case, the GFCC refused to
accept the CJEU’s ruling and challenged the PSPP of the ECB.

If we estimate $f_{t}$ based on the model discussed in Section 3 we find that
posterior uncertainty decreases appreciably in all cases considered. Comparing
the sign of the posterior medians reveals that both approaches yield similar
findings in terms of the direction of the shock but, in terms of magnitudes,
we find the estimates based on the mixture model being slightly more
pronounced. Again, the strongest reaction of the factor is observed on May 5,
2020.

Next we consider the estimated variance clusters as well as the non-event set.
These are contrasted with the ruling dates and shown in Figure 2 for $h=0$.
The gray shaded areas in the figure refer to the probability that a specific
period is included in the non-event set. The red lines refer to the different
ruling dates while the light blue lines mark important events that moved
financial markets. The upper panel shows the full range of the estimated
volatilities $\theta^{2}_{j}$ over time (i.e., for the time $t$ variance if
$\delta_{t}=j$) whereas the lower panel restricts the range of the graph. For
robustness checks varying the number of mixture components, see Appendix C.

Figure 2: Variance clusters and the non-event set alongside ruling dates for
$h=0$. The solid black line plots $\theta_{j}^{2}$ over time (the solid dark
grey horizontal line marks unit variance). The grey shaded area indicates the
posterior median of a specific period being featured in the minimal-variance
regime (and thus, the non-event set). The red vertical lines are the ruling
dates from Table 1, the light blue vertical lines are other known events that
moved financial markets.

From this figure, it is clearly visible that contrasting ruling dates with
non-ruling dates is insufficient. In this case, other important events and
volatility spikes mask potential effects and lead to imprecise inference on
the effect of the ruling dates. In principle, we find that overall volatility
spikes in the case of several important events and, if these are to dominate
the estimates of the idiosyncratic volatilities, the explanatory power of the
ruling factor is greatly diminished. By contrast, the common idiosyncratic
variance estimated from including only information included in the non-event
and ruling sets is small and the corresponding probabilities that a given day
is located in this minimum variance-set is, in most cases, close to one.

If our approach is successful, the commonalities in Eq. (2) should be sizable
(i.e., close to unity). By contrast, if we do not systematically control for
other events on non-ruling days, the estimate of $\sigma_{i}^{2}$ will be
large and $\lambda_{i}$ relatively small. We investigate this in Figure 3. The
figure depicts the share of variance explained through the common factors on
ruling dates for a model without the mixture setup and one that searches for a
minimum variance regime. These quantities are computed for each forecast
horizon white cells indicate commonalities close to zero whereas red cells
suggest that commonalities approach one.

Figure 3: Share of the variance explained by the factor,
$\lambda_{jh}^{2}/(\lambda_{jh}^{2}+\sigma_{jh}^{2})$, at event dates for the
model with and without the mixture specification.

Considering the left panel of Figure 3 reveals a great deal of white and blue
colored cells. For some quantities (e.g., gold, GS1, OIS 1m) we find that the
factor explains little variation on ruling days (especially for longer
forecast horizons) but a sizable amount of variation is explained in the short
term (with values close to 50%). The opposite is true for several other
variables such as stock markets, spreads vis-á-vis German treasuries, as well
as inflation swaps across different maturities. For these quantities, we find
that the factor explains around 50% of total variation on ruling days after
around 10 to 20 days. Interestingly, when we consider the reactions of
country-specific financial conditions (measured through the CISS),
commonalities reach values close to one for horizons between five to 10 days.
For short-run reactions and longer-run reactions, these shares are
considerably lower. This indicates that, even without controlling for the fact
that the non-ruling set is corrupted by several variance outliers, we find
sizable shares explained through the ruling factor. The fact that CISS
reactions are mostly dominated by ruling shares after a few days implies that
for these horizons, news about rulings dominate movements in financial
conditions but this effect quickly dies out afterwards.

We now turn to the question on how the mixture model impacts commonalities.
The right panel of Figure 3 suggests that for most quantities, we observe
variance shares explained which are close to one. At a general level, we find
that immediate reactions often feature shares which are appreciably below one
but for higher order responses the pattern is quite consistent across
variable-types and countries (with some exceptions being gold in EUR and USD,
GS 1y and 10y, the yield spread between Finland and Germany and the CISS for
Ireland). For most of these it makes sense that the corresponding factor
loadings are close to zero and hence the factor plays a limited role in
explaining time series variation.

Summing up this discussion, we find that even without using the mixture model
some variables display sizable commonalities for several impulse response
horizons. In principle, adding the mixture model and thus purging the non-
event set from other, non-ruling related events which had a pronounced impact
on financial markets, further increases the explanatory power of our approach.

###  4.3 Impulse response functions from local projections

In this section we discuss the impulse responses to an expansionary ruling
shock. With expansionary we mean a situation in which the courts rejected the
challenges to the ECB’s independence. We normalize the shock such that an
expansionary ruling surprise decreases the average cross-country CISS by one
standard deviation after 10 days.777The majority of rulings in our sample (as
opposed to intermediate legal procedures) is in favor of policies enacted by
the ECB, and thus, expansionary in nature. This is why we choose, for ease of
interpretation, to normalize the shock producing our impulse response
functions to be expansionary. It is worth noting that responses in our model
framework are symmetric with respect to the sign and size of the shock.
Figures 4 to 8 present the responses. The blue shaded areas represent the 68
and 90 percent posterior credible sets and the black lines the posterior
median estimates for up to 60 days after a ruling shock has happened. The red
lines depict the posterior quantiles of the model without the mixture
specification.

Figure 4: Impulse responses for CISS measures across euro area economies. The
solid black line shows the posterior median estimate for the mixture model,
alongside the 68 and 90 percent posterior credible sets in shades of blue. The
red lines mark the posterior median, $16$/$84$th and $5$/$95$th percentile for
the non-mixture specification.

Our prior intuition suggests that an expansionary ruling shock (i.e.,
increased central bank independence) should decrease financial stress. This is
because a decreased likelihood that the ECB is forced to stop its
unconventional monetary policy programs translates into a declining
probability of sharply increasing spreads between core and peripheral bonds,
and provides increased credibility to its accommodative policy stance vis-á-
vis financial market participants, affecting their expectations. Consistent
with our normalization assumption of the shock and our prior reasoning, Figure
4 reports that financial stress as measured by the CISS decreases. When we
focus on the shapes and magnitudes of the responses we observe many
similarities across the majority of countries: immediate effects appear to be
insignificant and after a few days, the CISS starts to decline in a persistent
manner, leveling out at around $-0.1$ points after 60 days. The main
exceptions from this general pattern are Finland, Ireland and the Netherlands.
In these countries, we find that the peak happens much earlier (after around
10 days) and the CISS increases between 10 and 30 days before declining once
more.

The response of the CISS across countries provides valuable insights on the
average effects of legal rulings on financial markets in its own right. But
these findings are also relevant in terms of the real economy, due to the
tight link between financial conditions and economic growth. In particular,
the declines we measure in response to an expansionary ruling shock relate to
reduced macroeconomic downside-risk and increases in real economic activity
(see Hollo _et al._ , 2012; Adrian _et al._ , 2019; Figueres and Jarociński,
2020).

Figure 5: Impulse responses for spreads of domestic government bond yields
relative to Germany. The solid black line shows the posterior median estimate
for the mixture model, alongside the 68 and 90 percent posterior credible sets
in shades of blue. The red lines mark the posterior median, $16$/$84$th and
$5$/$95$th percentile for the non-mixture specification.

Closely related to movements in financial conditions are spreads between
10-year government bond yields of selected euro area member states and
Germany, with the German yields serving as the euro area’s risk-free rate. In
fact, spreads between core and periphery countries in the euro area are a
crucial aspect to be considered by ECB policy makers. This is due to their
impact on the smoothness of the transmission of enacted policies to the member
states, and they have been among the concerns underlying Mario Draghi’s
“Whatever it takes” speech. The corresponding responses are depicted in Figure
5.

Apart from the result that spreads vis-á-vis German treasuries decline for all
countries considered, the single most striking observation is the pronounced
heterogeneity with respect to the magnitudes of the responses. While core
countries such as France, Austria, Belgium or Finland exhibit only modest
decreases in spreads which range from 5 to around 15 basis points (BPs),
spread reactions in the peripheral countries are enormous. For instance, in
the case of Greece we observe that spreads, in response to an expansionary
ruling shock, decline by around 250 BPs after around 40 days (with appreciably
uncertainty surrounding the point estimates). Greece marks the lower bound.
Other countries in that group (such as Italy and Portugal) display
substantially smaller reactions which are, however, still significantly larger
than the ones observed for the core economies. For Portugal, spreads decline
by around 100 BPs after 30 days while Italian spreads decrease by around 60
BPs after around a month. This discussion shows that if the courts reject
lawsuits targeted towards stopping QE, on the one hand, financial market
participants tend to assess Greek, Italian and Portuguese government bonds
more favorably, driving down the yields; on the other hand, financial markets
exhibit less demand for German government bond yields.

The previous discussion has shown that expansionary ruling shocks exert a
powerful effect on the longer end of the yield curve. In Figure 6 we focus on
the reactions at the short-end of the OIS rates with maturities up to 2 years
as well as corporate bond spread reactions. Turning to the OIS responses
reveals only modest reactions for maturities up to 6 months. They become
gradually more pronounced for maturities up to 2 years but with peak reactions
of about 5 BPs, the economic significance remains rather low. These subdued
reactions can be explained by the fact that during our estimation period,
interest rates at the short end of the yield curve have been close to zero and
thus display relatively little variation.

Focusing on the reactions of corporate bond markets across different rating
categories (investment-grade and sub-investment-grade) reveals appreciable
effects of expansionary ruling shocks which increase with decreasing levels of
credit quality. For AA-rated corporate bond spreads the model produces a
decline of around 20 BPs after around 60 days. When we consider BBB-rated
bonds, these effects double in magnitude, peaking at around 40 BPs.
Considering responses of sub-investment grade (high-yield) bond spreads
suggests that these decrease in an even more pronounced manner, with peak
declines of over 130 BPs. This relates to our previous discussion of the CISS
responses and provides a more nuanced picture of the reactions of credit
conditions to ruling shocks.

Figure 6: Impulse responses for short-term interest rates and variables
measuring financial conditions. The solid black line shows the posterior
median estimate for the mixture model, alongside the 68 and 90 percent
posterior credible sets in shades of blue. The red lines mark the posterior
median, $16$/$84$th and $5$/$95$th percentile for the non-mixture
specification.

Next we consider the reactions of stock markets to a ruling shock. Our
intuition suggests that if the ECB is allowed to continue with its
unconventional monetary policy programs, stock markets should profit through
different channels. The asset price/wealth channel (see, e.g., Bernanke and
Kuttner, 2005; Bjørnland and Leitemo, 2009), the risk taking channel (see,
e.g., Borio and Zhu, 2012) and a general decline in the probability that the
ECB has to suddenly stop its unconventional monetary policy operations, with
respect to the economic outlook.

Figure 7: Impulse responses for major stock market indices across euro area
economies. The solid black line shows the posterior median estimate for the
mixture model, alongside the 68 and 90 percent posterior credible sets in
shades of blue. The red lines mark the posterior median, $16$/$84$th and
$5$/$95$th percentile for the non-mixture specification.

When we focus on Figure 7 we find empirical evidence which supports our
hypothesis that expansionary ruling shocks should benefit stock markets.
Across all countries (and for the EuroStoxx 50) we find that stock prices
increase (and persistently do so). The general pattern is extremely similar
across countries: after no substantial short-run reactions equities increase
from day 10 after the shock hit the system. Towards the end of the impulse
response horizon the median increases reach about 10 percent. Notice that
peripheral countries such as Italy, Portugal and Greece display responses
which are observed by slightly more noise. This is driven by the generally
higher volatility of the corresponding time series. These stock market
reactions to ruling shocks are in line with the findings in Rigobon and Sack
(2004) or Bernanke and Kuttner (2005) who document substantial effects of
monetary events on asset prices.

Finally, all results discussed so far deal with reactions of financial
markets. In principle, rejected lawsuits about unconventional monetary
policies should also have powerful effects on inflation and inflation
expectations. Our working hypothesis is that expansionary ruling shocks, by
not constraining the ECB’s policy options, should have inflationary effects –
particularly in light of the unique challenges to monetary policy in the euro
area, and our corresponding sampling period which mainly features inflation
rates below the ECB’s target.888See also the discussion of the impact of
central bank independence on inflation dynamics in Crowe and Meade (2007).
While earlier papers such as Alesina and Summers (1993) or Cukierman _et al._
(1992) suggest an inverse relationship between inflation and central bank
independence in industrial economies, institutional changes and subdued global
inflationary pressures in recent years appear to have reversed this
relationship for the case of the euro area. This conjecture is confirmed when
we consider Figure 8. In this figure, we show the responses of the Euro
inflation swap curve at different maturities. In all these cases, the model
produces statistically and economically significant reactions. In terms of
magnitudes, we find that short-run inflation expectations increase stronger
than longer-run expectations, with increases of about 60 BPs for 1 year and 20
BPs for 10 year swaps.

Figure 8: Impulse responses for inflation swaps. The solid black line shows
the posterior median estimate for the mixture model, alongside the 68 and 90
percent posterior credible sets in shades of blue. The red lines mark the
posterior median, $16$/$84$th and $5$/$95$th percentile for the non-mixture
specification.

Summarising, we find that an expansionary ruling shock eases financial
conditions, lowers spreads between core and periphery euro area member states,
yields modest increases in short-term rates and inflation expectations, and
substantial gains on stock markets. These findings are somewhat reminiscent of
responses to contractionary central bank information shocks (see, e.g.,
Nakamura and Steinsson, 2018; Jarociński and Karadi, 2020). In particular, our
findings relate to “Delphic” monetary policy shocks as described in Andrade
and Ferroni (2021), which correspond to monetary policy announcements that
convey news about future macroeconomic conditions. Given our research design,
this points towards financial market participants categorizing expansionary
ruling shocks as being beneficial to real activity and the economic outlook.

##  5 Conclusions

Legal challenges to the conduct of the ECB’s monetary policy have the
potential to seriously impact policy making by reducing the variety of tools
at the disposal of the central bank. This implies a substantial reduction in
the independence and flexibility of the central bank. In this paper, we
analyze three lawsuits surrounding the ECB’s unconventional monetary policy
programs. The number of corresponding ruling days is small and information on
the precise timing of the announcements of the court decisions is not always
available. Standard tools used in the literature have a difficult time in such
a situation since the non-event days often include significant events which
potentially jeopardize identification of the shock of interest. To cope with
this, we develop an econometric model which discriminates between ruling and
non-event days. With non-event days we refer to days that display very low
levels of time series volatility and can thus safely be regarded as days
without significant economic or political events.

We use this framework to analyze the effects of an expansionary ruling shock
(which can be interpreted as an increase in central bank independence). The
corresponding impulse responses are consistent with our prior expectations.
Financial stress measures decrease and bond spreads relative to German Bunds
decline; short-term interest rates, stock markets and inflation swaps
increase. These effects are mostly driven by adjusting the expectations of
financial market participants to the case where the ECB is left unrestricted
in terms of its policy options, and maintains credibility in its conduct of
monetary policy.

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Appendix

##  Appendix A Bayesian inference and sampling algorithm

###  A.1 Prior setup

We assume independent inverse Gamma priors on the component-specific variances
$\vartheta_{j}^{2}$ in a symmetric fashion for $j=1,\ldots,J$:

$\vartheta_{j}^{2}\sim\mathcal{G}^{-1}(a_{0},b_{0}),$

with $a_{0}=b_{0}=0.1$ establishing a weakly informative setup. Moreover, we
assume independent Gaussian priors centered on zero for the factor loadings:

$\bm{\lambda}\sim\mathcal{N}(\bm{0}_{M},\tau\bm{I}_{M}),$

where $\tau$ governs the tightness of the prior. We again opt for a
comparatively uninformative setup and choose $\tau=1$.

To impose shrinkage on the number of non-empty cluster components, we collect
the component weights in a $J$-dimensional vector
$\bm{w}=(w_{1},\ldots,w_{J})^{\prime}$ and assume a symmetric Dirichlet prior:

$\bm{w}\sim\text{Dir}(c_{0},\ldots,c_{0}).$

The hyperparameter $c_{0}$ governs how superfluous components of the mixture
distributions are treated, and is thus closely related to the degree of
shrinkage on the number of non-empty components. Following Malsiner-Walli _et
al._ (2016), we impose a Gamma prior on this parameter:

$c_{0}\sim\mathcal{G}(d,dJ).$

Note that the expected value of $c_{0}$ is $1/J$ with variance $1/(dJ)^{2}$,
which implies that both the level of shrinkage and tightness of the prior
increases with the maximum number of components $J$. We set $d=10$ in our
empirical application, emptying superfluous components aggressively.

###  A.2 Posterior distributions

####  A.2.1 Sampling the component-specific variances

Let $T_{j}$ denote the number of observations assigned to component $j$. The
component-specific variances, conditional on $\delta_{t}$ for $t=1,\ldots,T$,
take a textbook form and are inverse Gamma distributed:

$\displaystyle\vartheta_{j}^{2}$
$\displaystyle\sim\mathcal{G}^{-1}(a_{j},b_{j}),$ (A.1) $\displaystyle a_{j}$
$\displaystyle=a_{0}+T_{j}/2,$ $\displaystyle b_{j}$
$\displaystyle=b_{0}+\frac{1}{2}\sum_{t:\delta_{t}=j}(\bm{y}_{t}-d_{t}\bm{\lambda}f_{t})^{\prime}(\bm{y}_{t}-d_{t}\bm{\lambda}f_{t}).$

####  A.2.2 Sampling parameters associated with the mixture model

The conditional posterior of the component weights $\bm{w}$ is a Dirichlet
distribution:

$\bm{w}\sim\text{Dir}(c_{1},\ldots,c_{J}),\quad\text{with}\quad
c_{j}=c_{0}+T_{j}.$ (A.2)

The latent component-allocation indicators $\delta_{t}$ are sampled $t$-by-$t$
based on

$\tilde{\xi}_{jt}=\Pr(\delta_{t}=j|\bm{y}_{t};\vartheta_{j}^{2}\bm{\Omega},\bm{w})\propto
w_{j}\mathcal{N}(\bm{y}_{t}|\bm{0}_{M},\vartheta_{j}^{2}\bm{\Omega}).$ (A.3)

The elements of the normalized probability vector
$\bm{\xi}_{t}=(\xi_{1t},\ldots,\xi_{Jt})^{\prime}$ are given by
$\xi_{jt}=\tilde{\xi}_{jt}/\sum_{j=1}^{J}\tilde{\xi}_{jt}$. Recall that the
non-event set is constructed by considering the component variances in
ascending order, such that the first cluster has the lowest variance,
$\vartheta_{1}^{2}$. Observations in the set $\mathcal{A}_{\text{R}}$ are
always assigned to the first cluster.

Combining the likelihood with the prior for the intensity parameter $c_{0}$
using Bayes’ theorem yields $p(c_{0}|\bm{w})\propto p(\bm{w}|c_{0})p(c_{0})$
from which we sample using a Metropolis-Hastings algorithm with a log-Normal
random walk proposal.

Denote the current draw by $c_{0}^{(s)}$ and let $c_{0}^{(\ast)}$ be a
proposal from the candidate density,
$\log\left(c_{0}^{(\ast)}\right)\sim\mathcal{N}\left(\log\left(c_{0}^{(s)}\right),\varsigma^{2}\right)$.
The parameter $\varsigma^{2}$ is a tuning parameter which we scale during half
of the burnin-phase of the algorithm to yield an acceptance probability
between 20 to 40 percent. The logarithm of the acceptance probability based on
the likelihood ratio and the asymmetric candidate density is given by

$\displaystyle\log(\zeta)$
$\displaystyle=\left(c_{0}^{(\ast)}-c_{0}^{(s)}\right)\left(\sum_{j=1}^{J}\sum_{t\in\mathcal{A}_{\text{NR}}}\log\left(\xi_{jt}\right)\right)+\log\left(\Gamma\left(Jc_{0}^{(\ast)}\right)\right)-\log\left(\Gamma\left(Jc_{0}^{(s)}\right)\right)$
(A.4)
$\displaystyle-J\left[\log\left(\Gamma\left(c_{0}^{(\ast)}\right)\right)-\log\left(\Gamma\left(c_{0}^{(s)}\right)\right)\right]+(d-1)\left(\log\left(c_{0}^{(\ast)}\right)-\log\left(c_{0}^{(s)}\right)\right)$
$\displaystyle-
dJ\left(c_{0}^{(\ast)}-c_{0}^{(s)}\right)+\log\left(c_{0}^{(\ast)}\right)-\log\left(c_{0}^{(s)}\right),$

where $\Gamma(\bullet)$ denotes the Gamma function. The draw is accepted with
probability $\min(\zeta,1)$, otherwise, the previous draw is retained.

####  A.2.3 Sampling parameters associated with the factor model

The factor loadings are estimated exclusively from observations in the set
$\mathcal{A}_{\text{R}}$, and follow independent Gaussian posterior
distributions for $i=1,\ldots,M$ with moments

$\displaystyle\lambda_{i}$
$\displaystyle\sim\mathcal{N}\left(\overline{\lambda}_{i},\overline{\Lambda}_{i}\right),$
(A.5) $\displaystyle\overline{\Lambda}_{i}$
$\displaystyle=\left(\sum_{t\in\mathcal{A}_{\text{R}}}\left(\frac{f_{t}}{\vartheta_{1}\omega_{i}}\right)^{2}+\tau^{-1}\right)^{-1},$
$\displaystyle\overline{\lambda_{i}}$
$\displaystyle=\overline{\Lambda}_{i}\left(\sum_{t\in\mathcal{A}_{\text{R}}}\frac{y_{it}f_{t}}{\vartheta_{1}^{2}\omega_{i}^{2}}\right).$

The factors are sampled for $t\in\mathcal{A}_{\text{R}}$ from a Gaussian
distribution with moments given by

$\displaystyle f_{t}$
$\displaystyle\sim\mathcal{N}\left(\overline{f}_{t},\overline{F}_{t}\right),$
(A.6) $\displaystyle\overline{F}_{t}$
$\displaystyle=\left(\vartheta_{1}^{-2}\bm{\lambda}^{\prime}\bm{\Omega}^{-1}\bm{\lambda}+1\right)^{-1},$
$\displaystyle\overline{f}_{t}$
$\displaystyle=\overline{F}_{t}\left(\vartheta_{1}^{-2}\bm{\lambda}^{\prime}\bm{\Omega}^{-1}\bm{y}_{t}\right).$

###  A.3 Sampling algorithm

The posterior distributions listed in the previous subsection can be used to
set up a Gibbs sampling algorithm that samples iteratively from these
conditionals. After initializing all variables, our sampler cycles through the
following steps:

  1. 1.

Sample the cluster-specific variances from their inverse Gamma conditional
posterior distribution given by Eq. (A.1).

  2. 2.

Sample the latent cluster-indicators $\delta_{t}$ using the respective
probabilities given by Eq. (A.3).

  3. 3.

Sample the component weights $\bm{w}$ from a Dirichlet distribution with
moments stated in Eq. (A.2). After this step, the variances and all other
parameters associated with the mixture model are permuted such that they are
in ascending order and $j=1$ is the cluster with minimal variance
$\vartheta_{1}^{2}$.

  4. 4.

Sample the intensity parameter $c_{0}$ using the Metropolis-Hastings step with
the acceptance probability given by Eq. (A.4).

  5. 5.

Sample the factor loadings $\bm{\lambda}$ from their Gaussian conditional
posterior distribution with moments given in Eq. (A.5) for $i=1,\ldots,M$.

  6. 6.

Sample the factor $f_{t}$ on a $t$-by-$t$ basis for the ruling set
$\mathcal{A}_{\text{R}}$ from their Gaussian conditional posterior
distribution stated in Eq. (A.6). This step involves checking the sign
restriction of the factor. Pre-multiply $\bm{\lambda}$ and $f_{t}$ by negative
one if the restriction is not fulfilled, otherwise omit this step.

In the empirical application of the paper, we produce a set of $3,000$ draws
from the desired full conditional posterior after disregarding the initial
$2,000$ draws as burnin. To reduce autocorrelation in our MCMC sample, we
consider each third from the retained draws and perform inference based on the
remaining $1,000$ draws.

##  Appendix B Data

Table B.1 provides a list of variables alongside their source and scale. We
obtain data from the statistical data warehouse of the European Central Bank
(ECB SDW, sdw.ecb.europa.eu) and the database of the Federal Reserve Bank of
St. Louis (FRED, fred.stlouisfed.org), Yahoo Finance and Macrobond. If
applicable, maturities are indicated with “m” for months and “yr” for years.
Scale refers to the underlying transformation of the series and how impulse
response functions are to be interpreted: basis points (BPs), in percent (%)
or no transformation (raw). EUR refers to euros and USD denote US dollars.
Country codes are: Austria (AT), Belgium (BE), Germany (DE), Spain (ES),
Finland (FI), France (FR), Greece (GR), Ireland (IE), Italy (IT), the
Netherlands (NL) and Portugal (PT).

Table B.1: List of variables. Label | Information | Scale | Source
---|---|---|---
OIS $D$ | Overnight index swap interest rates | BPs | ECB SDW
|  $D\in\\{$1-m, 3-m, 6-m, 1-yr, 2-yr$\\}$ |  |
Infl. swap $D$ | Inflation swaps (EUR) | BPs |
|  $D\in\\{$1-yr, 2-yr, 3-yr, 4-yr, 5-yr, 10-yr$\\}$ |  |
CISS ($C$) | Composite indicator of systemic stress | raw |
|  $C\in\\{$AT, BE, DE, ES, FI, FR, IE, IT, NL, PT$\\}$ |  |
EuroStoxx 50 | ESTOXX50 index | % |
Hyld OAS (EA) | ICE BofA EA High Yield Option-Adjusted Spread | BPs | FRED
AA OAS (EA) | ICE BofA EA AA Option-Adjusted Spread | BPs |
BBB OAS (EA) | ICE BofA EA BBB Option-Adjusted Spread | BPs |
Hyld OAS (US) | ICE BofA US High Yield Option-Adjusted Spread | BPs |
AA OAS (US) | ICE BofA US AA Option-Adjusted Spread | BPs |
BBB OAS (US) | ICE BofA US BBB Option-Adjusted Spread | BPs |
GS 1y | US 1-yr treasury yield | BPs |
GS 10y | US 10-yr treasury yield | BPs |
BE Infl. 5y (US) | US break even inflation | BPs |
FX USD/EUR | USD/EUR exchange rate | % |
NASDAQ | NASDAQ composite index | % |
VIX | CBOE Volatility Index | % |
Oil prices | Crude oil prices: Brent (Europe) | % |
Gold (USD) | Gold fixing price London Bullion Market (in USD) | % |
Gold (EUR) | Gold fixing price London Bullion Market (in EUR) | % |
Stock market ($C$) |  Stock index for $C$ | % | Yahoo Finance
|  $C\in\\{$AT, BE, DE, ES, FI, FR, GR, IE, IT, NL, PT$\\}$ |  |
|  ATX (AT), BEL 20 (BE), DAX (DE), IBEX 35 (ES) |  |
|  OMXH 25 (FI), CAC 40 (FR), Athex 20 (GR) |  |
|  ISEQ 20 (IE), FTSE MIB (IT), AEX (NL) |  |
|  PSI 20 (PT) |  |
Spread ($C$/DE) |  10-yr government bond yield of $C$ minus DE | BPs | Macrobond
|  $C\in\\{$AT, BE, ES, FI, FR, GR, IE, IT, NL, PT$\\}$ |  |

##  Appendix C Additional empirical results

Figure C.1 reports estimates of the common volatilities for varying numbers of
maximum mixture components $J$ and $h=0$ in the upper panel. The lower panel
indicates the posterior mean estimate of the number of non-empty clusters.
While assuming a larger number of clusters a priori results in a larger
estimate of non-empty clusters for shorter horizons $h$, the estimated
variance paths look very similar. The specific number of a priori clusters $J$
has virtually no impact on the estimated impulse response functions.

Figure C.1: Variance clusters (posterior median for $h=0$) and number of non-
empty components (posterior mean) across different specifications of the
maximum number of components $J$ to determine the non-event set.