# Adaptive local VAR for dynamic economic policy uncertainty spillover

Niels Gillmann
ifo Institute Dresden and Technische Universität Dresden
and
Ostap Okhrin
Chair of Econometrics and Statistics, Technische Universität Dresden

###### Abstract

The availability of data on economic uncertainty sparked a lot of interest in
models that can timely quantify episodes of international spillovers of
uncertainty. This challenging task involves trading off estimation accuracy
for more timely quantification. This paper develops a local vector
autoregressive model (VAR) that allows for adaptive estimation of the time-
varying multivariate dependency. Under local, we mean that for each point in
time, we simultaneously estimate the longest interval on which the model is
constant with the model parameters.

The simulation study shows that the model can handle one or multiple sudden
breaks as well as a smooth break in the data. The empirical application is
done using monthly Economic Policy Uncertainty data. The local model
highlights that the empirical data primarily consists of long homogeneous
episodes, interrupted by a small number of heterogeneous ones, that correspond
to crises. Based on this observation, we create a crisis index, which reflects
the homogeneity of the sample over time. Furthermore, the local model shows
superiority against the rolling window estimation.

Keywords: adaptive local estimation, connectedness, local homogeneity,
multivariate time series, vector autoregression.
JEL classification: C32, C53, E3.

##  1 Introduction

Currently, there is a series of events with international consequences which
increase economic uncertainty around the globe. They are the onset of the
corona pandemic in early 2020, followed by the attack of Russia on Ukraine in
2022. Diebold and Yilmaz (2014) stressed the importance of a timely
quantification of the international spillover of current events. In line with
this background, our research question is as follows:

“How can we measure international spillovers of current events in a timely
manner?”

To measure current events, we rely on a dataset created by Baker et al. (2016)
with monthly data on Economic Policy Uncertainty (EPU), which is shown to
react to current events. More recently, EPU has also been used by the European
Central Bank to quantify the increasing uncertainty around the corona
pandemic, see Gieseck et al. (2020). There are many other options for
measuring the uncertainty that current events create, see Bloom (2014) for an
excellent overview. To mention a few: Bachmann et al. (2013) based on the
disagreement of firm survey participants, Jo and Sekkel (2017) on forecast
errors of professional forecasters, Jurado et al. (2015) on forecast error
variance in a large set of financial and economic variables, and Creal and Wu
(2017) on interest rate data. Despite the many alternatives, EPU is well
suited for our research question since it becomes available with a publication
lag of just one month which is much faster than data from official statistics.
Additionally, it covers not only developed countries but also countries, where
other data is more difficult to obtain. This is also why we prefer EPU to
stock market data which also have little publication lag and react to current
events but are only available in relatively wealthy countries with an
established stock market.

Various researchers have already used EPU data to quantify uncertainty
spillovers across countries. One of the first examples is Klößner and Sekkel
(2014), which employs a spillover method for six countries. They find that
spillovers account for one-fourth of variation in EPU across countries and
that spillovers change over time. But most papers investigating spillovers,
focus on the average spillovers over time instead of the dynamics and
quantification of spillovers from recent events, e.g. Clausen et al. (2019),
Liow et al. (2018), Luk et al. (2020), Tzika and Fountas (2021). With our
research question, we aim to fill the gap in the literature and investigate
the temporal dynamics of spillover.

For quantifying international spillovers, we use an updated version of the DY-
spillover method also used in Klößner and Sekkel (2014). This method, created
by Diebold and Yilmaz (2014), is based on Forecast Error Variance
Decompositions (FEVD) of a Vector Autoregressive (VAR) model. It is widely
used to quantify spillovers (see, for example, Demirer et al. (2018) and
Dungey et al. (2019)), and has several advantages that make it particularly
appealing for our research question. First, it provides an intuitive way to
measure spillovers by linking spillover to the question: “How much of country
$A$’s future uncertainty is due to the current situation in country $B$?”.
Second, it avoids additional theoretical knowledge for estimation and offers a
simple quantitative measurement of spillovers. Third, the framework can easily
be adopted into a dynamic setup. Despite the alternative ways of measuring
spillovers, such as Barigozzi and Brownlees (2019), Engle and Kelly (2012) and
Adrian and Brunnermeier (2016), the DY-spillover method is the preferred
method for our application for its simplicity and interpretability.

Usually, a dynamic setup is adopted through rolling window estimation.
However, rolling window sizes are often chosen subjectively and can easily
drive the results. Furthermore, too short windows result in large variance,
while too long ones result in large bias. Therefore, we propose a data-driven
approach to identify meaningful window sizes for dynamic estimation.

Our approach for quantifying international spillovers in a timely manner is
based on the literature on local parametric estimation, introduced by Čížek et
al. (2009) and Spokoiny (2009) through local univariate parametric time series
models. Later methodological contributions include Chen and Spokoiny (2015),
Spokoiny et al. (2013). The local estimation has been applied successfully to
many topics, including temperature risk (Härdle et al. (016b)), crop yields
(Shen et al. (2018)), financial risk management (Fengler and Okhrin (2016)),
electricity price (Chen and Li (2017)), and financial (Härdle et al. (2015))
forecasting.

We adapted the framework by Čížek et al. (2009) to the multivariate time
series context using finite-sample likelihood ratio tests to test for
homogeneous intervals. The works most related to us, which apply the
likelihood testing procedure to univariate time series to identify local
intervals instead of change points, are Chen et al. (2010), Niu et al. (2017),
and Chen et al. (2013). The local estimation approach is particularly well
suited to answer our research question since it is a natural extension of the
already established fixed rolling windows. Additionally, this approach allows
us to obtain estimation results for our sample’s most recent data points,
thereby allowing for timely quantification of the spillover of current events.

Our contribution is threefold: 1. We extend the local parametric estimation
approach from a univariate autoregressive to a VAR setting. 2. We confirm the
successful extension by a set of Monte Carlo simulations by testing if local
VAR models can identify homogeneous intervals correctly, even in the presence
of structural breaks of various types. 3. In the empirical application to the
measurement of EPU spillover, it turns out that spillover of EPU is
homogeneous over long episodes, interrupted only by a few major crises, which
are the GFC, the European debt crisis, and the trade war. These findings are
used to create a _crisis indicator_ , highlighting when the sample becomes
heterogeneous. Furthermore, total dynamic spillover estimated locally tends to
be the same as that estimated by a big rolling window. Only during times of
crisis the local approach results in total dynamic spillover that corresponds
to a small rolling window size.

The paper proceeds as follows: chapter 2 introduces the EPU data, chapter 3
describes the spillover measure, chapter 4 outlines the local estimation
procedure in the VAR context, chapter 5 contains an extensive simulation
exercise, chapter 6 illustrates the empirical application to EPU, and chapter
7 concludes. Some specific simulation scenarios are put in the Appendix.

##  2 Economic Policy Uncertainty

To capture recent events and their potential connections across countries, we
use monthly EPU data, Baker et al. (2016), freely available at
`policyuncertainty.com`. The EPU indices are based on newspaper articles
classified as related to EPU if they contain specific economic, policy, and
uncertainty keywords. The methodology has been used to measure uncertainty in
many countries like Australia (Moore (2017)), Chile (Cerda et al. (2018)), and
Sweden (Armelius et al. (2017)). Researchers employed this variable to measure
uncertainty from current events and estimate its impact on the economy
(Ghirelli et al. (2021), Prüser and Schlösser (2020b)). EPU is also commonly
used to investigate spillover effects across countries (Caggiano et al.
(2020), Nilavongse et al. (2021), Stockhammar and Österholm (2016)). By now,
indices for more than 23 countries are available, with further countries
continuously added by groups of researchers who followed the Baker et al.
(2016) methodology.

We chose a set of indices that are based on at least two newspapers from five
countries: Germany (DE), India (IN), Japan (JP), South Korea (KR), and the
United States of America (US). This constellation is a good mix of large
economies from the developed and developing worlds. All chosen countries come
from the original database by Baker et al. (2016). The time-series for the
countries in the final dataset are plotted in Figure 1, and some descriptive
statistics are shown in Table 1. The ADF tests in the table indicate that four
out of five series are not stationary when using twelve lags but, became
stationary with just one lag. This is a powerful indication that the time
series should be modeled locally. Based on data availability, the selected
time frame is from $2003M01$ until $2021M01$.

| Med | SD | Min | Max | Skew | Kurt |  ADF12 |  ADF1 | KPSS | $N$ | Time
---|---|---|---|---|---|---|---|---|---|---|---
DE | 138.53 | 80.91 | 28.43 | 498.06 | 2.60 | 9.51 | 0.22 | 0.01 | 0.10 | 2 | 1993M1-2021M1
IN | 80.10 | 50.27 | 24.94 | 283.69 | -1.49 | 5.82 | 0.42 | 0.01 | 0.01 | 7 | 2003M1-2021M1
JP | 105.02 | 34.52 | 48.37 | 240.24 | -1.50 | 7.51 | 0.28 | 0.01 | 0.08 | 6 | 1990M1-2021M1
KR | 129.70 | 70.87 | 37.31 | 538.18 | 2.74 | 11.68 | 0.05 | 0.01 | 0.10 | 3 | 1990M1-2021M1
US | 116.25 | 69.98 | 44.78 | 503.96 | 2.56 | 11.23 | 0.89 | 0.01 | 0.09 | 10 | 1985M1-2021M1
Table 1: Summary statistics of the EPU variables, with $p$-values for the
Augmented Dickey-Fuller (ADF) and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS)
tests. $N$ is the number of newspapers used for construction. Figure 1:
Monthly data time series of EPU for Germany (DE), India (IN), Japan (JP),
Korean Republic (KR), and the United States of America (US).

The period is shaped by major events which resulted in high uncertainty in
most countries. There was the second Gulf War ($2003M03$) at the beginning,
the Lehman Brothers bankruptcy ($2008M09$) and the Eurozone crisis ($2011M06$)
in the middle, and Brexit ($2016M06$) as well as the Trade War ($2019M07$)
towards the end. COVID-19 causes the highest spikes at the very end of the
sample. The question of time variation in uncertainty transmission is not
straightforward. Angelini et al. (2019) estimate a threshold model based on
macroeconomic volatility as a proxy for uncertainty which finds strongly
increasing impacts of uncertainty during recessions. Caggiano et al. (2020)
find similar results using EPU. Prüser and Schlösser (2020a) estimating a TVP
model, however, finds that uncertainty transmission is more or less stable
over time. With the proposed local model, we are trying to provide more
evidence on the time variation of uncertainty, especially since threshold and
TVP models require assumptions about the time dynamics while local models do
not.

Regarding the general, static transmission of uncertainty, it can be expected
that it easily and quickly spills over between countries (Belke and Osowski
(2019)), especially between those with close trade links (Balli et al.
(2017)). Furthermore, larger economies will likely transmit uncertainty to
smaller economies, as stated by Tzika and Fountas (2021). This is partly due
to the fact revealed by Bloom (2017) that small open economies have a large
probability of being affected by uncertainty shocks of a foreign origin. Also,
developing countries were found by Carrière-Swallow and Céspedes (2013) to be
more affected by external uncertainty. Therefore, we can expect, for example,
the spillover between Germany and the US to be large since they have strong
trade links. At the same time, India will probably not receive and transmit
much uncertainty to the other countries in the sample as its ties to the other
sample countries are weaker. Also, the US, the biggest economy in the sample,
will most likely be the primary transmitter of uncertainty. As a small open
economy, Korea is expected to experience a lot of spillover from abroad. Just
from Figure 1, it can already be seen that Germany, Korea, and the US comove a
lot, Japan and India do so to a lesser extent. This observation is also
supported by the negative skewness and lower kurtosis of the latter two in
Table 1.

Applied to the local model, we expect that EPU will be homogeneous for most of
the covered period resulting in long intervals of homogeneity. Around the
events, as mentioned earlier, the intervals of homogeneity might, however, be
relatively short since these events result in sudden surges of EPU. All in
all, the dynamics the data display seem to be a good test for the local
estimation approach.

##  3 Measuring spillover

Spillover is measured using a framework called connectedness by Diebold and
Yilmaz (2014). The framework is prevalent and has been used in many papers,
e.g., Demirer et al. (2018), Dungey et al. (2019), and Hale and Lopez (2019).
We are aware of alternative methods to measure spillover and numerous
suggestions for improving connectedness, such as Baruník and Křehlík (2018),
Buse and Schienle (2019), Lanne and Nyberg (2016). Since the method is only
used to evaluate the local estimation algorithm in an empirical application,
we decided that using the original methodology is the most suitable as it has
been tested and validated the most.

Consider a series of $d$-dimensional EPU data
$EPU_{t}=(EPU_{1,t},...,EPU_{d,t})^{\top}$ with $t\in T$ being the time
component. It is believed that the size of the share to the $H$-step ahead
forecast error of the variance of EPU in the country $i$ due to innovations
from EPU in the country $j$, denoted by $C_{ij}(H),i,j=1,\ldots,N,i\neq j$,
can be interpreted as the connectedness between them. Here $i\neq j$ is used
to identify relevant variance shares for connectedness. Variance
decompositions are used to calculate the variance shares for all countries in
the system. The approximating model for the variance decomposition is a VAR
model. The traditional VAR relies on orthogonal innovations, whereas
connectedness means correlated innovations. Therefore, we used the generalized
VAR framework by Pesaran and Shin (1998) that allows for correlated
innovations by considering the observed distribution of the errors.

Let the cross-sectional dependency between the elements of the EPU vectors be
modeled with a VAR as
$EPU_{t}=\sum_{s=1}^{p}\phi_{s}EPU_{t-s}+\varepsilon_{t}$, where
$\varepsilon\sim N(0,\Sigma)$. When the VAR fulfills the requirements for
stationarity, with all roots lying in the unit circle and having a modulus
smaller than one, it can be inverted to a Vector Moving-Average (VMA) of
infinite order as $EPU_{t}=\sum_{u=0}^{\infty}A_{u}\varepsilon_{t-u}$, where
the $A_{u}$’s are $(d\times d)$ moving average coefficient matrices which obey
the recursion $A_{u}=\phi_{1}A_{u-1}+\ldots+\phi_{p}A_{u-p}$,
$A_{0}=\mathfrak{I}$ (the identity matrix), and $A_{u}=0$ for $u<0$. Based on
this setup, country $j$’s contribution to the country $i$’s $H$-step ahead
generalized forecast error variance, $C_{ij}(H)$, can be calculated using the
previously defined MA coefficients and covariance matrix. The following
formula results on the generalized formulation by Pesaran and Shin (1998)

$C_{ij}(H)=\frac{\sigma_{jj}^{-1}\sum_{h=0}^{H-1}(e^{\top}_{i}A_{h}\Sigma
e_{j})^{2}}{\sum_{h=0}^{H-1}(e^{\top}_{i}A_{h}\Sigma A_{h}^{\top}e_{j})},$ (1)

where $\sigma_{jj}$ is the standard deviation of the error term for the $j$th
equation in the VMA, and $e_{i}$ is a selection vector with zeros except for
the $i$’th entry. Due to correlated innovations, the forecast error variance
contributions might not add to unity. Following Diebold and Yilmaz (2012) each
element is divided by its corresponding row sums
$\tilde{C}_{ij}(H)=\frac{C_{ij}(H)}{\sum_{j=1}^{N}C_{ij}(H)}$. Having all
$\tilde{C}_{ij}(H)$’s, we obtain a spillover table, with diagonal elements
representing variance shares and off-diagonal ones the cross-variance shares.
Based on the spillovers $\tilde{C}_{ij}(H)$, we can compute total spillover,
which is obtained by the total share of the cross-variances in the system
$S(H)=\frac{\sum_{i=1,i\neq j}^{N}\sum_{j=1,j\neq
i}^{N}\tilde{C}_{ij}(H)}{\sum_{i,j=1}^{N}\tilde{C}_{ij}(H)}\times 100$.

Diebold and Yilmaz (2014) further developed the connectedness methodology by
considering the variance decompositions as networks. They are more
sophisticated than simple networks since the adjacency matrix, which
corresponds to the variance decomposition matrix, now contains values ranging
between $0$ and $1$ instead of containing either 0 or 1. Additionally, the
links are directed so that the link from country $i$ to country $j$ might
differ from the link from $j$ to $i$, meaning that the adjacency matrix is not
symmetric anymore.

##  4 Method

As mentioned before, this paper aims to measure the impact of current events
on international spillover as timely as possible. To quantify spillovers, we
use the DY-connectedness measure relying on VAR models. Though rolling window
techniques are commonly used in the literature to estimate dynamic spillovers,
they are far from optimal. Short windows might work well in the presence of
many events associated with high spillover, but they disregard valuable
information if there are only a few breaks in the data. Long windows will
produce good results in stable times but might incur serious bias when
structural breaks are present in the data. Using VAR models with time-varying
parameters would require extra assumptions about the dynamics in the data.
Hence, we believe that a local estimation approach that allows for time-
varying window size is the best solution to measure dynamic spillovers.
Additionally, local estimation is quickly applicable since it does not require
assumptions on the underlying time dynamics.

###  4.1 Time-varying VAR estimation

Let the temporal and cross-sectional dependency of $EPU_{t}$ be modeled by the
VAR process, which differs from the one introduced in Chapter 3 in the respect
that $\phi$ and $\Sigma$ now change with time $t$

$EPU_{t}=\phi_{0,t}+\sum_{s=1}^{p}\phi_{s,t}EPU_{t-s}+\varepsilon_{t},$ (2)

where $\phi_{0,t}$ and $\phi_{s,t}(s=1,\ldots,p)$ are $(d\times d)$
coefficient matrices and the random noise $\varepsilon_{t}\sim
N(0,\Sigma_{t})$. To simplify notations, let the parameters be summarised in
$\theta_{t}=(\phi_{0,t},\phi_{1,t},\ldots,\phi_{p,t},\Sigma_{t})$. Within VAR
models, parameters driving the temporal dynamics are usually assumed to be
constants. In practice, however, most processes are not constant over time,
and parameters show different behavior during turbulent and calm periods. We
deal with this by allowing parameters to vary over time without making
specific assumptions about the structure of the time variation. The only
assumption we make is that of local homogeneity: For each $\tau\in[1,T]$,
there exists a true unknown local interval of homogeneity $I_{\tau}^{*}=[\tau-
m_{\tau}^{*},\tau]$ over which $\theta_{t}=\theta$ for $t\in I_{\tau}^{*}$.

The assumption of local intervals represents a good balance between the
model’s adaptability and its estimation’s feasibility. Furthermore, local
intervals allow the procedure to handle smooth transitions and sudden jumps of
underlying parameters. Thereby we cover both varying coefficients (Cai et al.
(2000)) and piecewise constant (Bai and Perron (1998)) models. For extensive
details on univariate local parametric models and their theoretical
properties, we refer to Spokoiny (2009), and from here on, we closely follow
the notation of Chen et al. (2010). For each local interval $I_{\tau}$, the
local log-likelihood function is defined as

$\ell(EPU,I_{\tau},\theta)=-\frac{m_{\tau}}{2}\log
2\pi+\frac{m_{\tau}}{2}\log\lvert\Sigma^{-1}\rvert-\frac{1}{2}\sum_{v=\tau-
m_{\tau}+1}^{\tau}\varepsilon_{v}^{\top}\Sigma^{-1}\varepsilon_{v},$ (3)

where $\varepsilon_{v}=EPU_{v}-\phi_{0,v}-\sum_{s=1}^{p}\phi_{s,v-s}EPU_{v-s}$
and all the parameters are collected in
$\theta=(\phi_{0},\phi_{1},\ldots,\phi_{p},\Sigma)$. This results in the
following local maximum likelihood (ML) estimator
$\tilde{\theta}_{\tau}=\underset{\theta\in\Theta}{\mathrm{argmax}}\
\ell(I_{\tau},\theta)$, with $\Theta$ being the parameter space, where for
notational simplicity we denoted $\ell(I_{\tau},\theta)$ for
$\ell(EPU,I_{\tau},\theta)$.

###  4.2 Quality of local estimation

Suppose that for each time point $\tau\in[1,T]$, EPU is driven by a local VAR
process with the true (unknown) parameters $\theta^{*}_{\tau}$ being constant
on the homogeneous interval $I_{\tau}^{*}$. To assess the quality of the local
model with parameters $\tilde{\theta}_{\tau}$, we can measure the deviation
from the model with optimal parameters $\theta^{*}_{\tau}$ using a likelihood
ratio (LR) statistics as

$LR(I_{\tau},\tilde{\theta}_{\tau},\theta^{*}_{\tau})=\ell(I_{\tau},\tilde{\theta}_{\tau})-\ell(I_{\tau},\theta^{*}_{\tau}).$
(4)

There exists a well-established theory for identifying local models with the
LR from Equation (4); e.g., Spokoiny (2009) and Čížek et al. (2009). Polzehl
and Spokoiny (2006) derived a risk bound (RB) which depends on the true
parameter $\theta^{*}_{\tau}$ for the expected deviation (4) and its $r$th-
power transformations with $r>0$ for an iid sequence of Gaussian innovations

$E_{{\theta}^{*}_{\tau}}\lvert
LR(I_{\tau},\tilde{\theta}_{\tau},\theta^{*}_{\tau})\rvert^{r}\leq RB^{r}.$
(5)

The introduced bound is nonasymptotic and can be used for any finite interval
$I_{\tau}$. It allows us to construct confidence intervals for assessing the
quality of estimation, meaning that $\tilde{\theta}_{\tau}$ and the
corresponding LR fulfill the risk bound (5). Hence, the assessment of the
quality of the local estimation is done using the following LR statistics

$\lvert LR(I_{\tau},\tilde{\theta}_{\tau},\theta^{*}_{\tau})\rvert^{r}.$ (6)

In practice, the true parameter $\theta^{*}_{\tau}$ is not known and instead a
hypothetical parameter is used for simulating data and calculating the risk
bound $RB^{r}$. Details on the procedure are given in the next section. For a
series of models with distributions from exponential families, this risk bound
even does not depend on the true parameter, which is, unfortunately, not the
case in our model.

Belomestny and Spokoiny (2007) show that an optimal choice of an interval of
local homogeneity for univariate processes can be obtained via the adaptive
procedure. We concentrate on the construction details for a multivariate VAR
process in the following. A comprehensive simulation study in the next chapter
illustrates the performance of the adaptive procedure in our setting.

###  4.3 Adaptive identification of local intervals of homogeneity

A sequential testing procedure is employed to identify the local homogeneous
intervals of the process. Therefore, we consider a finite set of candidate
intervals $I_{\tau,k}=\\{I_{\tau,1},\ldots,I_{\tau,K}\\}$ with
$I_{\tau,k}=[\tau-m_{k},\tau]$ and ML estimators $\tilde{\theta}_{\tau}^{k}$
with $k=1,\ldots,K$ on each candidate. For each $\tau$, we start with the
shortest possible interval $I_{\tau,1}=[\tau-m_{1},\tau]$, assumed to be
homogeneous. From there on, we extend the interval backward and test whether
parameter $\tilde{\theta}_{\tau}^{1}$ is also well suited on the next bigger
interval $I_{\tau,2}=[\tau-m_{2},\tau]$. If the hypothesis is not rejected, we
consider $I_{\tau,2}$ to be homogeneous and continue extending until the
largest possible interval is reached.

The procedure is performed using the obtained ML estimators
$\tilde{\theta}_{\tau}^{k}$ and the LR statistic (6). The only difference is
that now the true $\theta^{*}_{\tau}$ is replaced with the best-known one –
the adaptive estimator $\hat{\theta}_{\tau}$, which will be determined
sequentially at each backward-looking step and formalized below. The adaptive
estimators will differ for each $\tau$ since it depends on $I_{\tau}^{*}$.
Furthermore, since $I_{\tau,1}$ is assumed to be always homogeneous, the
procedure starts with $\hat{\theta}_{\tau}=\tilde{\theta}_{\tau}^{1}$ and then
continues either until the last interval $I_{\tau,K}$ or where the test
statistic does not exceed the critical value as

$\lvert{LR(I_{\tau,k},\tilde{\theta}^{k}_{\tau},\hat{\theta}_{\tau})}\rvert^{r}\leq\zeta^{r}_{k},\quad
k=2,\ldots,K,$ (7)

with $\zeta^{r}_{k}$ being the critical value at step $k$ and is described in
more detail below. The test statistic measures the difference between the
current local ML estimator $\tilde{\theta}_{\tau}^{k}$ and the adaptive
estimator $\hat{\theta}_{\tau}$ over a possible $k$-th interval of local
homogeneity $I_{\tau,k}$. If the test statistic is small, there is no
significant change in the dynamics, and (7) is not violated. We thus cannot
reject the null of local homogeneity and adopt the new estimator
$\tilde{\theta}_{\tau}^{k}$ as the _adaptive estimator_
$\hat{\theta}_{\tau}=\tilde{\theta}_{\tau}^{k}$. Suppose the test statistic is
bigger than the critical value. In that case, it indicates that the adaptive
estimator $\hat{\theta}_{\tau}$ for the current point in time $\tau$ cannot be
extended further backward and is only valid until $k-1$. Therefore, the
iterative procedure is terminated and $\tilde{\theta}_{\tau}^{k-1}$ is
accepted as the optimal i.e. the _adaptive estimator_ $\hat{\theta}_{\tau}$
for the current ${\tau}$.

When testing the procedure with simulations, we realized that for a few
$\tau$, we sometimes obtain implausible interval series like
$I_{\tau,6},I_{\tau+1,1},I_{\tau+2,6}$, corresponding to the intervals of the
lengths $m_{\tau,6}$, $m_{\tau+1,1}$, and $m_{\tau+2,6}$ respectively. For the
real data as $EPU_{t}$, this behavior is hardly empirically interpretable, but
formally this is because the procedure takes each $\tau$ individually. Hence,
we implemented an additional `if-condition` that ensures that the final
intervals $I_{\tau}$ do not allow for sudden unexpected jumps: If
$m_{\tau,k}\neq m_{\tau-1,k}$, then $m_{\tau,k}=m_{\tau,k_{max}}$, where
$k_{max}$ is the index of the interval with the largest LR:
$\max_{k\in[1,K]}\\{LR(I_{\tau,k},\tilde{\theta}^{k}_{\tau},\hat{\theta}_{\tau})\\}$.
It was furthermore verified in the simulations that this additional
restriction does not affect the results but only produces more smooth and
better interpretable changes of the identified intervals of homogeneity.

The whole procedure can be thus summarized as follows

  1. 1.

initialization: Set $\hat{\theta}_{\tau}=\tilde{\theta}_{\tau}^{1}$ and $k=2$

  2. 2.

while
$\lvert{LR(I_{\tau,k},\tilde{\theta}^{k}_{\tau},\hat{\theta}_{\tau})}\rvert^{r}\leq\zeta^{r}_{k}$
and $k\leq K$
do $\hat{\theta}_{\tau}=\tilde{\theta}_{\tau}^{k}$, $k=k+1$.

  3. 3.

set $\hat{m}_{\tau}=m_{\tau,k-1}$ and
$\hat{\theta}_{\tau}=\tilde{\theta}_{\tau}^{k-1}.$

  4. 4.

check if $\hat{m}_{\tau}=m_{{\tau}-1,k}$. If not,
$\hat{m}_{\tau}:=m_{\tau,k_{max}}$, where $k_{max}$ is the index of the
interval with the largest $LR$.

###  4.4 Critical values

The critical values $\zeta_{k}^{r}$ are obtained through Monte Carlo
simulations. There are two main ingredients. The first is the empirical
version with the expectation being replaced by the theoretical risk bound
$RB^{r}$ based on $\theta^{*}_{\tau}$. The second is the empirical counterpart
which is the deviation between the MLE estimator $\tilde{\theta}_{\tau}^{k}$
and the adaptive estimator $\hat{\theta}_{\tau}$ measured by a likelihood
ratio. Additionally,$\frac{k}{K}$ as a normalizing factor to make estimates
based on different $k$ comparable and $\rho$, a tuning factor, to ensure that
the $LR$ and $RB$ match, are needed.

The whole procedure, which is similar to Chen and Spokoiny (2015), Härdle et
al. (2015) and Shen et al. (2018), is summarized in the following algorithm,
where we use the notation $\ell(X_{i},I_{\tau,k},\tilde{\theta}^{k}_{\tau})$
to highlight, that the likelihood is evaluated on the interval $I_{\tau,k}$ of
the sample $X_{i}$ using the parameter $\tilde{\theta}^{k}_{i,\tau}$:

  1. 1.

simulate $N=10^{4}$ homogenous processes $X_{i,t}$, $i=1,\ldots,N$ from a
fixed $\theta^{*}$.

  2. 2.

use (5) to calculate
$\widehat{RB}^{r}_{k}=\frac{1}{N}\sum_{i=1}^{N}\left|\ell(X_{i},I_{\tau,k},\tilde{\theta}^{k}_{i,\tau})-\ell(X_{i},I_{\tau,k},\theta^{*})\right|^{r}$,
for $k=2,\ldots,K$.

  3. 3.

set initial critical values $\zeta_{k}^{r}=\infty$.

  4. 4.

using (6) to get $\hat{\theta}_{i,\tau}(\zeta_{k}^{r})$ for each sample
$i=1,\ldots,N$ select $\zeta_{k}^{r}$ over $k=2,\ldots,K$ by

$\zeta_{k}^{r}=\arg\min_{\zeta}\left|\frac{1}{N}\sum_{i=1}^{N}\left|\ell(X_{i},I_{\tau,k},\tilde{\theta}^{k}_{i\tau})-\ell\\{X_{i},I_{\tau,k},\hat{\theta}_{i,\tau}(\zeta)\\}\right|^{r}-\rho\frac{k}{K}\widehat{RB}^{r}_{k}\right|.$

###  4.5 Selection of $\rho$ and $r$

There are two choices to be made for the calibration of critical values:
$\rho$ and $r$. Keeping $r$ fixed while increasing $\rho$ will lead to smaller
critical values. On the other side, leaving $\rho$ fixed while increasing $r$
will lead to bigger critical values.

Härdle et al. (2015) suggest $r=0.5$ and $\rho=0.5$ in a univariate setting,
while Chen and Netsunajev (2018) recommend $r=0.5$ in a functional AR model.
Since the selection of $\rho$ is often arbitrary, we follow the idea from
Čížek et al. (2009) and determine it by minimizing prediction errors. In
detail, we estimate local models over a grid of values for $\rho$ ranging from
0.01 to 1 (including 0.5). From each set of local models, we predicted from
the estimated VAR the $\widehat{EPU}_{t,\rho}$ for each $\rho$ from the grid.
In the end, we compute the Mean Absolute Percentage Error (MAPE) for each
$\rho$ and select the $\rho$ which corresponds to the smallest MAPE as
$\hat{\rho}=\underset{\rho>0}{\mathrm{argmin}}\frac{1}{T}\sum_{t=1}^{T}\left\lvert\frac{EPU_{t}-\widehat{EPU}_{t,\rho}}{EPU_{t}}\right\rvert$.
The algorithm results in a potentially different $\rho$ and, therefore, a
different critical value for each dataset. This is important because different
datasets might need a higher or a lower break detection sensitivity depending
on the amount of general heterogeneity.

##  5 Simulation

We perform a Monte Carlo study to investigate the performance of the local
estimation procedure. There are three criteria that the procedure should
fulfill. First, it should not detect a break when there is none in the data.
Second, it should detect a break, when there is one. Third, it should not
recover to the maximum length if there is a second break shortly after the
first. Therefore, datasets that include tests for all three conditions are
generated. But before results are presented, the choice of parameters and the
setup has to be described to ensure reproducibility.

###  5.1 Simulation design

We use a finite set of $K+1=7$ candidate intervals based on a geometric grid
$m_{i}=[m_{0}a^{k}]$, with $m_{0}=12$ and $a=1.25$, where the $[x]$ means the
largest integer smaller than $x$, which results in the following set of
interval lengths: $m_{\tau,k}=\\{12,15,19,23,29,37,46\\}$. Note that the first
interval corresponding to a length of 12 is always assumed to be homogenous
and works as a baseline against which to compare the candidate intervals. In
our setting, the smallest interval corresponding to twelve months i.e., one
year seemed a good tradeoff between timeliness and estimation accuracy. The
geometric grid is preferred over the linear grid since it generally yields
better results in the simulation exercise and is also used by most of the
literature (Čížek et al., 2009; Härdle et al., 2015; Spokoiny, 2009).

Parameters for testing our algorithm are obtained from fitting two-dimensional
VAR(1) models to EPU111We modified the parameters obtained from the data a bit
since the original ones resulted in a large break which was easy to detect.
The parameters presented here will result in a small break and are, therefore,
challenging. Results for higher dimensions are shown in Appendix A1.. A short
lag is sufficient in our setting because our set of intervals is also
relatively short. The resulting parameters can be found in Table 2. Two points
in the selection of the parameters are worth noting: a) the structural break
will contain changes in both $\phi_{0}$ and $\phi_{1}$; b) changes in the
parameters are small. This is the way the real data typically change.
Theoretical studies usually vary just one single coefficient, but we are
interested in empirical applications where more than one coefficient might
change.

$\theta_{1}$ |  $\phi_{0}=\left[\begin{array}[]{c}29.00\\\ 132.00\end{array}\right]$ $\phi_{1}=\left[\begin{array}[]{cc}0.71&0.08\\\ 0.13&0.08\end{array}\right]$
---|---
$\theta_{2}$ |  $\phi_{0}=\left[\begin{array}[]{c}31.00\\\ 130.00\end{array}\right]$ $\phi_{1}=\left[\begin{array}[]{cc}0.63&0.00\\\ 0.12&0.23\end{array}\right]$
Table 2: Parameters for the simulation study with $d=2$ and $p=1$

The simulation results are divided into three scenarios ranging from “easy” to
“difficult”. Here we present only Scenario 1, and the two other scenarios can
be found in Appendix A1. Scenario 1 contains a single break:
$x_{1},\ldots,x_{84}\sim\theta_{1},x_{85},\ldots,x_{146}\sim\theta_{2}$. The
simulated data are referred to as $x_{t}$, and we generate each dataset 250
times. The notation “$\sim\theta_{1}$” means that observations are generated
from a VAR(1) with parameter $\theta_{1}$. In Appendix A2, we present the
distribution of test statistics for each scenario and compare it with the
corresponding selected critical values. The same exercise is also performed
for different sets of parameters to guarantee robustness, with results
available upon request. Since the growth of the interval of homogeneity for
each $\tau$ goes backward, an initial set of 46 observations corresponding to
the longest interval has to be discarded.

For each scenario, we calculate four different results. First is optimal
$\rho$ and its restriction. Then, results for two fixed $\rho$’s, namely the
most often chosen $\rho$ from the optimal algorithm and a $\rho=0.5$. Finally,
results with the optimal $\rho$ algorithm but without the additional
restriction are also shown.

Results are presented as three combined plots to show the input data and the
algorithm. The setup is as follows: The plot on the top shows the simulated
input data with the break. Its purpose is to serve as an orientation for when
something should happen in the local process. The middle panel shows the
resulting intervals of homogeneity for each $\tau$. Plotted are means (solid
lines) and medians (dashed lines) over 250 repetitions. The plot at the bottom
depicts the values of the likelihood ratio tests for the accepted interval
plus one for each $\tau$ to show whether the algorithm starts to react after
the breaks appear in the input data. Please note that the values might be a
combination of tests for different $k$’s since each repetition might stop at a
distant $k$. The horizontal red line indicates the critical value for the
longest possible interval, $k=6$. A test value below means that the local
interval could have been extended more. There is a dashed line through the top
plot that highlights the point when the parameter set for the input data
changes.

###  5.2 Simulation results

Figure 2 shows the simulation results for Scenario 1 based on 250 repetitions.
There is just one break located in the middle of the observation period. It is
marked by a dashed line to help visualize exactly where the break occurs. The
middle plot shows that all four specifications detect the break. The reaction
to the break is that the window length jumps down to one. After the break, it
slowly increases to maximum length again. The recovery process is
characterized by a step function with six constant regions, which we call
_stairs_. These six stairs correspond to the six intervals. So, while the
stairs for the lower intervals are shorter, the stairs for the higher
intervals are more prolonged, corresponding to the predetermined intervals
$I_{\tau,k},\ k=1,\ldots,K-1$. The number and length of the preselected
intervals can determine the number of stairs and their length.

As expected, the procedure with the optimal $\rho$ performs best. While the
medians are almost identical, with only the median window length for
$\rho=0.5$ being visibly lower than six, there are some differences in the
means. Specification for $\rho=0.5$ results in an average window length of
only four instead of six. On the other side, specification with an optimal
$\rho$ and no restrictions does not result in flat steps for lower window
lengths and instead displays some spikes, particularly for the first and
second steps. The algorithm, without any restrictions, using $\rho=0.088$,
results in the most symmetric distribution of window lengths, as the mean is
closest to the median.

The lower means happen because, over the 250 repetitions, each run will have a
few $\tau$’s, where there is a false alarm, and the algorithm mistakenly
selects a window length of one. Most false alarms appear for specification
$\rho=0.5$, which features the highest value of $\rho$ for this setting. This
large value results in a low critical value, so false alarms happen more often
when the test statistic exceeds the critical value. In the univariate setting,
$\rho=0.5$ was a good choice. Still, in our multivariate setting with small
window sizes, we need higher critical values to prevent false alarms while
detecting true breaks.

The upwards spikes in the stairs of specification with optimal $\rho$ with no
restrictions happen because here, the algorithm jumps down to a small window
length when the break occurs but directly jumps up again in a set of
repetitions so that the mean is upwards biased. In the other specifications,
this upward bias is prevented by using the additional restriction, which keeps
the selected window length low for an adequate time. This is evidence that
applying the additional restriction in the multivariate setting with small
window sizes makes sense.

Figure 2: Simulation results - One break.

Top: Simulated 2dim VAR. The vertical line indicates a break. Mid: Identified
Intervals for: optimal $\rho$ (black), $\rho=0.5$ (blue), $\rho=0.088$
(green), optimal $\rho$ with no restrictions (red). Bottom: Test statistics
for $k+1$ with $\zeta_{6}=3.0$ (horizontal red). Bounds for 5$\%$ and 95$\%$
bounds are depicted in grey.

The likelihood ratio test values in the bottom plot are virtually identical
across specifications, where only $\rho=0.5$ results in slightly higher
values. It is reassuring to see that 95% of test statistics during the
homogeneous part of the sample are below the critical value for the longest
possible interval. Then, as soon as the break in the input data occurs, there
is a slight reaction. However, it takes a bit until the break shows in the
window lengths. This is because the test statistic only becomes large once all
observations inside the smallest interval come from $\theta_{2}$.

##  6 Empirical Application

The goal of this chapter is twofold. First, it should provide evidence that
empirical data are mostly homogeneous over time since crises are rare events.
Second, it should illustrate that longer intervals are more suitable for
measuring dynamic spillover because, during homogeneous periods, there is
little risk of bias when using longer intervals for estimation. Therefore, at
first, the local estimation algorithm identifies the longest homogeneous
interval for each $\tau$ in the sample of EPU for Germany, India, Japan, South
Korea, and the US. Second, these intervals will be used to calculate the
Diebold Yilmaz connectedness measure, a common way to measure spillover,
similar to a rolling window approach. The connectedness measure is detailed in
Chapter 3. The same set of interval lengths
$m_{\tau,k}=\\{12,15,19,23,29,37,46\\}$ as for the simulation in Chapter 5
based on a geometric grid is used for the setup of the local estimation
algorithm. As a benchmark for the second part, rolling windows with windows of
the length $w$ of the size of the shortest ($w=12$) and longest ($w=37$)
interval are additionally applied. Note that the last window ($w=46$) cannot
be selected because of how the local estimation algorithm is designed.
Therefore a length of $w=37$ constitutes the longest possible window length.
Furthermore, since during the estimation of the intervals of homogeneity,
intervals extend backward in time from $\tau$, we need to discard the first
$46$ observations so that the first $\tau$ is $2006M11$. The last one is then
$\tau=2021M01$.

###  6.1 A crisis indicator based on local homogeneity

For all possible pairs of EPU in the five selected countries (DE, IN, JP, KR,
and US), bivariate VAR models are estimated to identify homogeneous intervals.
This strategy is chosen because estimating just one VAR with all five
countries would flatten out the breaks, which only affect a fraction of the
countries. By estimating bivariate VAR models, heterogenous episodes that only
occur in a few country pairs are also considered.

The length of the intervals of homogeneity can be treated as an indicator of
structural breaks that happened in the recent past. Moreover, structural
breaks in economic policy are rare events and, in history, were mirrored
through various crises. Based on this logic, we define the _crisis indicator_
CI between countries $i$ and $j$, which reflects the homogeneity of the
multivariate process at each time point $\tau$, as

$CI_{i,j}=1-\frac{\hat{k}_{\tau,i,j}-1}{K-1},$ (8)

where $\hat{k}_{\tau,i,j}$ is the index of the estimated length of the
interval of homogeneity for the VAR(1) model that is modeling EPU for
countries $i$ and $j$ and $K-1$ is the index of the largest possible interval
($K=6$ in our paper). The resulting values fall in the interval $[0,1]$, with
an indication of the crisis via $CI_{i,j}=1$ indicating the shortest interval
length corresponding to $k_{\tau,i,j}=1$, and no crisis with $CI_{i,j}=0$ the
longest one corresponding to $k_{\tau,i,j}=K$. Therefore, a crisis indicator
of zero means no heterogeneity in the sample and the longest possible interval
of homogeneity. As soon as the indicator starts increasing towards one, we
obtain shorter intervals of homogeneity, leading to an increase in
heterogeneity, indicating that some structural change is happening.

Suppose all the models possess the longest homogeneity interval for the given
$\tau$. In this case, the causal effects between EPU for different countries
follow a fixed structure over the given period. This implies that no shocks
were present during this period in economic policy for _all_ the involved
countries. If for some country pairs for a given $\tau$, the homogeneity
interval is short, this implies a structural break in the economic policy
modeling in this pair. Therefore averaging the crisis indicator $CI_{i,j}$
over all the pairs of countries, we obtain the _global crisis indicator_

$\displaystyle CI$ $\displaystyle=$
$\displaystyle\frac{2}{d(d-1)}\sum_{i=1}^{d-1}\sum_{j=i+1}^{d}CI_{i,j}$
$\displaystyle=$
$\displaystyle\frac{K}{K-1}-\frac{2}{d(d-1)(K-1)}\sum_{i=1}^{d-1}\sum_{j=i+1}^{d}\hat{k}_{\tau,i,j},$

with $d$ being the number of countries. Instead of the mean, one can use a
median that neglects the magnitude of the interval indices, thus leading to a
more robust specification.

The CI estimated for EPU is presented in Figure 3. Here, the identified
intervals of a local homogeneity of all ten pairs based on EPU in the five
selected countries are used to compute the global crisis indicator $CI$. It
has long periods of homogeneity where no crisis exists in any country pair.
These episodes usually span several years. The homogeneous episodes are
interrupted by three common crises in nearly all the pairs. The first crisis
lasted less than one year and corresponded to the global financial crisis,
which resulted in a sudden shock to economic policy. Furthermore, it affected
economic policy in countries with less integration into the global financial
market, like India, to a lesser degree. Therefore, the global crisis indicator
does not reach a value of one during this crisis. The second crisis lasted two
years, which is more than double the duration of the first one. It corresponds
to the Eurozone debt crisis, a prolonged episode of uncertainty about economic
policy in the Eurozone. Simultaneously, there was uncertainty about economic
policy in the US due to a debt ceiling debate and the risk of a government
shutdown. Both issues involved discussions over a longer period to resolve
underlying problems. This is reflected in two years when the global crisis
indicator stayed elevated. Still, the crisis indicator again did not reach a
value of one because of India and the same reasoning.

Figure 3: A global crisis indicator based on local homogeneity.

The crisis indicator is obtained by transforming the intervals of local
homogeneity ranging from 1 to 6 into a 0,1 Index. A value of 0 indicates
homogeneity and, therefore, no crisis. A value of 1 indicates the absence of
homogeneity and, thus, a crisis in all pairs.

The last crisis in the sample is different since it does attain the value of
one, meaning that it affected economic policy in all country pairs in the
sample. This crisis can be attributed to two simultaneously occurring events:
the Brexit referendum and the election of Donald Trump as president of the
United States. Both events severely increased the uncertainty about
international cooperation, especially regarding trade agreements. Since India
has historical links to the UK and trades a lot with the US, economic policy
in India was affected by this increase in uncertainty as well, resulting in a
value of one for the global crisis indicator. The crisis also lasted for a
long time since the involved governments of the UK and the US at this time did
not swiftly act to dispel worries about their political agendas. This is in
line with earlier warnings from researchers such as Baker et al. (2016), that
governments should communicate policies transparently and predictably since
they otherwise risk causing high economic policy uncertainty. Furthermore,
Caggiano et al. (2020) found that rising EPU in influential countries such as
the US can trigger large spillovers to smaller countries. Putting that
together with the finding in Davis (2016) that global economic policy
uncertainty peaked during the Brexit referendum further supports our
conclusion that Brexit and the election of Donald Trump resulted in the most
global crisis of economic policy in our sample.

All in all, the proposed global crisis indicator allows a classification of
recent events into periods of crises and non-crises based on the degree of
homogeneity that the data exhibit during the respective events. It helps to
assess how global the specific crises were and also sheds light on how long
each crisis lasted. From the application, economic policy crises last longer
if governments do not clarify their agendas or try to dispel worries about
future policy goals. We also learned that economic policy in developing
countries like India is not that much affected by internal political issues in
the developed world or issues regarding the global financial market. However,
economic policy in developing countries is affected by uncertainty about
international cooperation and trade. Lastly, a minor shortcoming of this
setting with monthly frequency data is that our algorithm can only detect
breaks with a delay. Using monthly frequency will therefore require some
patience to classify recent crises. Daily data might be more suited for
monitoring ongoing crises.

###  6.2 Spillover estimation results

Papers like Angelini et al. (2019); Caggiano et al. (2020) have shown that
uncertainty spillovers from Chapter 3 vary over time. The intervals of
homogeneity identified in the previous section are window lengths of a rolling
window procedure selected in a data-driven way. They feature primarily long
windows, interrupted by a few sequences of short windows during episodes of
crises. This section uses them to estimate bivariate rolling window VARs for
calculating DY spillovers. The top panel of Figure 4 shows the actual EPU
data, and the bottom one shows the spillover for the different window lengths
with LHI standing for Local Homogeneous Intervals. In line with most of the
literature, uncertainty spillovers vary over time. They increase during high
economic policy uncertainty and decrease again when uncertainty is low. This
means that long windows do not, as might be worried by some, smoothen out the
countercyclicality in the data. Instead, the long windows can adequately
capture the data’s features. The data has five peaks marked with vertical
lines, which correspond to: 1. The global financial crisis in October 2008. 2.
The European sovereign debt crisis in August 2011. 3. The US and UK political
struggle from June 2016 onwards. 4. The trade war between the US and China, as
well as the EU in 2019. 5. The outbreak of COVID-19 in 2020. From the previous
section, we know that the third crisis was more global than the first and
second one, affecting all countries in the sample. This did not result in
higher spillover as the spillover during the two previous crises attained the
same level. The third crisis differs from the two other crises, however, in
the time persistence of the elevated spillover level. It seems that a crisis
being more global will lead to a prolonged episode of high spillover instead
of a more pronounced peak.

Figure 4: Spillovers.

Top: EPU data for the five selected countries. Bottom: Total spillover based
on all pairs of 2dim VARs. LHI stands for Local Homogeneous Intervals. Min and
Max RW, respectively, stand for fixed windows of length 12 and 37 months.

During the global financial crisis, spillover jumped up drastically in October
2008. This was driven by the US, where Lehman Brothers declared bankruptcy on
September 15th, triggering the crisis. The European debt crisis was a
prolonged episode of increased uncertainty. The highest spillover in our
sample is recorded in August 2011. Among the countries, it is Germany and
Japan who push up spillover this month. For Germany, it is clear that
discussions about bailouts for other European countries created uncertainty.
Regarding Japan, the government intervened in currency markets to prevent the
yen from rising, which would make Japanese exports less competitive on the
international market. This intervention also seems to have created significant
spillovers of economic policy uncertainty. In June 2016, spillover jumped up.
This is, when the first results of the Brexit referendum became public. It
caused stock markets around the world to fall. In our sample, the event is
most clearly visible in the country pairs of India and Korea, India and the
US, as well as Korea and the US. This is probably because especially India and
the US are closely linked to the UK. During the trade war, spillover is
highest in August 2019 for the country pairs involving the Korean Republic.
This is because Korea announced that it would scrap a military information
agreement with Japan, which reacted to a Japanese decision to tighten high-
tech exports to Korea. This announcement prompted stark criticism from the US
and the international community, spreading uncertainty about Korea’s future
economic policy to the world. The Covid-19 crisis caused the strongest
spillover in March 2020. While spillovers among pairs involving either Japan
or the US were highest, pairs with India generally resulted in very low
spillovers during this month. Japan introduced mandatory quarantine for
travelers from China and Korea in early March. Korea responded by suspending
visas for all Japanese citizens traveling to Korea. Furthermore, Japan
announced in March 2020 that it would postpone the Olympic Games by one year
and implement a range of economic measures to stop the spread of the virus. In
the US, Donald Trump tried to downplay concerns about COVID-19 as long as
possible. But in March 2020, he was forced to announce a national emergency
and had to acknowledge that the US would be heading for a recession.
Therefore, the US transmitted uncertainty because of Trump’s unclear and
hesitant communication. In contrast, Japan transmitted uncertainty due to the
large range of measures, of which the impact on economic development was
unclear at the time of the announcement.

When comparing the different methods for calculating spillover, one
immediately notices that spillovers are much higher when based on small
windows. This is because small windows result in increased variance. The high
spillover, resulting from small windows, does not reflect changes in the data
but the estimation uncertainty of the small windows. The spillover based on
the LHI is similar to the spillover based on long windows, which is more
stable and has lower variance. The proposed method improves the estimation of
spillover compared to rolling windows since it sometimes results in higher
peaks, also staying high for longer periods, than the long windows, meaning
that our local estimation algorithm reacts more to sudden increases in the
data while still maintaining a low variance. This is particularly noticeable
around 2011 during the European sovereign debt crisis and in 2016 during the
US and the UK political struggle. Here, the long windows smoothen out some
variation in EPU, failing to represent time-varying spillover adequately.

The window sizes used in the previous estimation represent our choice based on
ideal windows for monthly frequency. In empirical applications, researchers
used different window sizes. The variance of the window sizes employed in the
literature ranges from 18 months in Yin and Han (2014) to 72 months in Clausen
et al. (2019). Therefore, we use a new grid of windows:
$m_{\tau,k}=\\{18,23,29,36,45,57,72\\}$. Since we now have longer intervals,
we have to discard more data at the start of the sample so that now the
starting month is 2009M01. The results of this exercise are depicted in Figure
5.

Figure 5: Spillovers based on window sizes used in the literature.

Top: EPU data for the five selected countries. Bottom: Total spillover based
on all pairs of 2dim VARs. LHI stands for Local Homogeneous Intervals. Min and
Max RW, respectively, stand for fixed windows of length 18 and 71 months.

The variance between the spillover based on the shortest and longest window
increased compared to the previous figure. Especially the spillover estimated
with the smallest window size seems very large compared to the two other
curves. Interestingly, our algorithm indicates that the longest window is
again preferred for spillover calculation since the spillover based on the LHI
is again very close to the longest window, even though the longest window is
now 71 months instead of just 37 months. The only period were the longest
window size would smoothen out too much variation is again around 2016, when
the spillover based on our method shows more variation and higher spillover.
Based on this finding, one should be careful when selecting window sizes for
rolling window estimation not to choose too small windows. Too small windows
will result in high values of spillover during crises. They are, therefore,
tempting. These results are based more on estimation uncertainty than
increased economic policy uncertainty.

##  7 Conclusion

An existing algorithm for detecting locally homogeneous intervals in a
univariate setting is adapted to the multivariate VAR context. Through a
series of Monte Carlo simulations, the algorithm is shown to perform well in
this multivariate context, even when intervals are small and breaks occur
within short periods from each other or as smooth changes over time. The
algorithm can be applied to many settings by modifying the set of intervals.

In the paper’s last chapter, the algorithm is applied to identify homogeneous
intervals and estimate the spillover of EPU across countries. It turns out
that empirical data are primarily homogeneous, and breaks only occur during a
limited number of common crises. This feature is exploited to create a crisis
indicator, which measures how the homogeneity of the sample changes over time.
The most significant episode of non-homogeneity turned out to be the time
around Brexit and the election of Donald Trump, when all countries experienced
increased uncertainty about economic policy. Even the global financial crisis
had less impact on economic policy uncertainty since it affected developing
countries to a lesser degree. From the spillover estimation part, it does not
seem necessary to have window sizes that vary at each $\tau$ since breaks
occur only during specific periods and for short amounts of time. Most of the
time long window sizes are adequate for quantifying spillovers. This means
that sophisticated assumptions about time dynamics are unnecessary. Small
adaptions to the traditional rolling window approach are sufficient to take
care of potential bias when estimating time dynamics.

Avenues for further research are the application of the algorithm to empirical
settings with different time frequencies of the data, such as quarterly or
weekly data. Furthermore, the crisis indicator is an interesting topic for
further research. It might indicate uncertain times earlier than commonly
available uncertainty indicators since heterogeneous patterns in the data
usually characterize uncertain times.

The authors report there are no competing interests to declare.

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Appendix
Adaptive local VAR for dynamic economic policy uncertainty spillover

##  Appendix A Simulation robustness

This appendix shows simulation results for scenarios two and three as well as
results for $d=4$ for all three scenarios. The set of intervals is the same
for $d=4$, but we do need to use a different set of parameters to estimate
from, and therefore, we also obtain new risk bounds. As in the setting with
$d=2$, we simulate three scenarios ranging from easy to complex. The results
are presented in the following three sections, one for each scenario.

###  A.1 Simulation robustness for Scenario 1: One break

This section only contains results for $d=4$, which are depicted in Figure 6.
In general, the results are very similar to the case when $d=2$. The only
noticeable difference is that there is now a larger heterogeneity in the mean
(solid lines) of the different specifications during the homogeneous part of
the sample. The medians (dashed lines) are still all perfectly detecting the
break.

Figure 6: Simulation robustness $d=4$ \- One break.

Top: Simulated 4dim VAR. The vertical line indicates a break. Mid: Identified
Intervals for: optimal $\rho$ (black), $\rho=0.5$ (blue), $\rho=0.088$
(green), optimal $\rho$ with no restrictions (red). Bottom: Test statistics
for $k+1$ with $\zeta_{6}=4.1$ (horizontal red). Bounds for 5$\%$ and 95$\%$
bounds are depicted in grey.

###  A.2 Simulation robustness: Scenario 2 - Two breaks

The results for Scenario two are depicted in Figure 7. It contains two breaks
that occur within a brief period:
$x_{1},\ldots,x_{84}\sim\theta_{1},x_{85},\ldots,x_{99}\sim\theta_{2},x_{100},\ldots,x_{146}\sim\theta_{1}$.
Since they are barely visible, two dashed lines clarify where the breaks
happen. All four specifications detect both breaks. The behavior before the
first break is very similar to the first scenario; it, however, changes
afterward. Suppose the additional restriction is not applied instead of a
small upward bias. In that case, the algorithm returns to the maximum window
length before jumping down again when the second break happens. This happens
because observations from both sets of parameters mix. Shortly after the first
break, most observations in the tested windows again belong to $\theta_{1}$.
Because of this, specification for optimal $\rho$ with no restrictions will
result in long homogeneous window lengths furthermore too early.

The means of the other three specifications show an inverted u-shape that only
grows to a window length of four before jumping down to one again. The mix of
observations from both sets of parameters causes this smoother behavior.
Nevertheless, the algorithm can still identify the second proper break.

Figure 7: Simulation robustness $d=2$ \- Two breaks.

Top: Simulated 2dim VAR. Vertical lines indicate breaks. Mid: Identified
Intervals for: optimal $\rho$ (black), $\rho=0.5$ (blue), $\rho=0.088$
(green), optimal $\rho$ with no restrictions (red). Bottom: Test statistics
for $k+1$ with $\zeta_{6}=3.0$ (horizontal red). Bounds for 5$\%$ and 95$\%$
bounds are depicted in grey.

After the first break, the values of the likelihood ratio test fall below the
threshold so that the window length would be the maximum again. The
restriction prevents the procedure from jumping up, however. This is another
piece of evidence why the additional restriction is beneficial. It keeps
results clear and identified breaks visible.

The downward movements at the end of the sample in the middle and bottom
panels indicate the return to homogeneity. The test statistic falls below the
critical value, and the mean of specification $\rho=0.5$ returns to a window
length of four again.

Results for the second scenario with $d=4$ are depicted in Figure 8. Again,
the results are very similar to the case when $d=2$ except for a bit more
heterogeneity in the means of the different specifications during the
homogeneous part of the sample.

Figure 8: Simulation robustness $d=4$ \- Two breaks.

Top: Simulated 4dim VAR. Vertical lines indicate breaks. Mid: Identified
Intervals for: statistics $\rho$ (black), $\rho=0.5$ (blue), $\rho=0.088$
(green), optimal $\rho$ with no restrictions (red). Bottom: Test statistics
for $k+1$ with $\zeta_{6}=4.1$ (horizontal red). Bounds for 5$\%$ and 95$\%$
bounds are depicted in grey.

###  A.3 Simulation robustness: Scenario 3 - Smooth break

Figure 9 illustrates the simulation results for Scenario three. It contains
one break that is a linear change from $\theta_{1}$ to $\theta_{2}$:
$x_{1},\ldots,x_{96}\sim\theta_{1},x_{97},\ldots,x_{112}\sim\theta=(\frac{16-i}{16}\theta_{1}+\frac{i}{16}\theta_{2})$
with $i\in[1,15],x_{113},\ldots,x_{200}\sim\theta_{2}$. We constructed a tiny
change that is barely visible. Still, the gradual change is detected by all
four specifications. However, the algorithm cannot jump down to the smallest
window since the likelihood ratio is not big enough. Furthermore, while the
medians follow the expected form with stairs at each window length, the mean
increases more smoothly. This is because some repetitions will already result
in a window length that is one higher than the current window length, which
will result in a more smooth line instead of strict stairs. The scenario with
the highest value of $\rho$ can return the smallest window during the break.
However, it will already return smaller windows during the homogeneous part of
the sample. This is because the highest value of $\rho$ will result in the
smallest critical value. Hence, this scenario is best visualizing the trade-
off involved in this procedure. Lower critical values will result in better
detection and a higher false alarm rate. Therefore, we prefer higher critical
values only to detect true breaks and to avoid false alarms, even if this
means that we will not have a big break in the window length.

Figure 9: Simulation robustness $d=2$ \- Smooth break.

Top: Simulated 2dim VAR. Vertical lines indicate breaks. Mid: Identified
Intervals for: optimal $\rho$ (black), $\rho=0.5$ (blue), $\rho=0.088$
(green), optimal $\rho$ with no restrictions (red). Bottom: Test statistics
for $k+1$ with $\zeta_{6}=3.0$ (horizontal red). Bounds for 5$\%$ and 95$\%$
bounds are depicted in grey.

The likelihood ratio test values are different from previous scenarios during
the break. They show a gradual reaction, slowly increasing in value. Only when
the window lengths nearly recovered to their maximum again there are
noticeable spikes in the test values. Despite the minimal reaction of the test
statistic, particularly right after the break occurs, the algorithm still
manages to detect the smooth break.

Results for the third scenario with $d=4$ are depicted in Figure 10. The
results are even more similar to the case when $d=2$ than the other scenarios.

Figure 10: Simulation robustness $d=4$ \- Smooth break.

Top: Simulated 4dim VAR. Vertical lines indicate breaks. Mid: Identified
Intervals for: optimal $\rho$ (black), $\rho=0.5$ (blue), $\rho=0.088$
(green), optimal $\rho$ with no restrictions (red). Bottom: Test statistics
for $k+1$ with $\zeta_{6}=4.1$ (horizontal red). Bounds for 5$\%$ and 95$\%$
bounds are depicted in grey.

##  Appendix B Distribution of test statistics

This appendix illustrates the distribution of test statistics for all three
simulation scenarios when $d=2$. For comparison, the sample is divided into
homogeneous and non-homogeneous parts. Since the data are simulated, we know
precisely which observations to use in which part. The plots are then arranged
in a way that the top panel contains average test statistics for all intervals
across the homogeneous part of the simulated data and the bottom panel for the
non-homogeneous part. The test statistics always represent the question: ”Can
we extend our homogeneous interval from $k-1$ to $k$?” Therefore the number
two on the x-axis represents the test when going from the shortest interval to
the second one. Additionally, each plot contains a dashed horizontal red line
representing the critical value for extending the homogeneous interval from
the first to the second candidate interval. The critical values across
intervals are very similar, so there is no need to draw them for all steps. In
all top panels, the critical value is not visible since the test statistics
are far below it. This is expected since the top panel shows the homogeneous
samples, which should always extend to the longest candidate interval and not
reject any interval length.

###  B.1 Distribution of test statistics for Scenario 1: One break

The first figure in this section corresponds to the results of the first
scenario with just one break. The results for this scenario are shown in the
main body of the paper. For the homogeneous plot, observations for
$\tau\in[1,38]$ are included. This is until the Likelihood ratio test
statistic in the bottom panel of Figure 2 in Chapter 5 reacts. The
heterogenous plot is then based on $\tau\in[39:84]$, which includes all time
points for which the test statistic deviates from the homogeneous values. As
expected, some test statistics exceed the critical value only in the bottom
panel. The increasing pattern in the bottom panel is due to the growing step
length of the algorithm at each interval when recovering from a detected
break.

Figure 11: Test-stats - One break.

Top: Homogeneous subsample containing 38 data points for each distribution
based on $\tau\in[1,38]$. Bottom: Non-homogeneous subsample containing 46 data
points for each distribution based on $\tau\in[39:84]$. The critical value is
3.1 (horizontal red). Results are based on the median of 250 runs. The figure
was created with the function pirateplot() from the package yarr in R.

The clustering below the critical value occurs because it is usually just the
interval covering the break going backward from each respective $\tau$ which
is rejected. For example, when we are already 25 observations away from the
break, the tests for intervals two, three, and four will not exceed the
critical value anymore since they will not contain any breaks anymore. The few
points around the critical value for each step are points where the respective
window does include one or more observations from the structural gap but not
enough to exceed the critical value. This behavior is by design since we want
the algorithm to slowly recover after a structural break instead of jumping
down and up again.

###  B.2 Distribution of test statistics for Scenario 2: Two breaks

The second test statistic figure corresponds to the second scenario, which
contains two breaks within a short distance from each other and is depicted in
Figure 12. The figure looks almost identical to the one for the first
scenario. The only difference is that there is a bit of a mass shift above the
critical value from the highest intervals to the ones in the middle. This is
because this scenario contains more data points with intervals of medium size
and fewer intervals with long length since the recovery process from the first
break is interrupted by the second break. In general, it makes sense that the
figures are very similar since the algorithm also detected the breaks
correctly when they occurred within a short distance. Therefore, our algorithm
can handle structural breaks, even if they occur with high frequency in the
data.

Figure 12: Test-stats - Two breaks.

Top: Homogeneous subsample containing 38 data points for each distribution
based on $\tau\in[1,38]$. Bottom: Non-homogeneous subsample containing 62 data
points for each distribution based on $\tau\in[39:100]$. The critical value is
3.1 (horizontal red). Results are based on the median of 250 runs. The figure
was created with the function pirateplot() from the package yarr in R.

###  B.3 Distribution of test statistics for Scenario 3: Smooth break

The third test statistic figure corresponds to the simulation’s third
scenario, which contains a smooth break and is depicted in Figure 13. For this
scenario, even the test statistics in the bottom part are mostly smaller than
the critical value. In the resulting plot of the third scenario, we can
already see that the algorithm struggles to identify the break in this
setting. Therefore, it is no big surprise that the test statistics, even in
the non-homogeneous sample, are also minimal.

Figure 13: Test-stats - Smooth break.

Top: Homogeneous subsample containing 70 data points for each distribution
based on $\tau\in[1,70]$. Bottom: Non-homogeneous subsample containing 30 data
points for each distribution based on $\tau\in[71:100]$. The critical value is
3.1 (horizontal red). Results are based on the median of 250 runs. The figure
was created with the function pirateplot() from the package yarr in R.