# Efficient Multi-Change Point Analysis to decode Economic Crisis Information
from the S&P500 Mean Market Correlation

Martin Heßler
Institute for Theoretical Physics
University of Münster
48149 Münster, North Rhine-Westphalia, Germany
<EMAIL_ADDRESS>
& Tobias Wand††footnotemark:
Institute for Theoretical Physics
University of Münster
48149 Münster, North Rhine-Westphalia, Germany
<EMAIL_ADDRESS>
& Oliver Kamps
Center for Nonlinear Science
University of Münster
48149 Münster, North Rhine-Westphalia, Germany
<EMAIL_ADDRESS>
Center for Nonlinear Science, University of Münster, 48149 Münster, Germany

###### Abstract

Identifying macroeconomic events that are responsible for dramatic changes of
economy is of particular relevance to understand the overall economic
dynamics. We introduce an open-source available efficient Python
implementation of a Bayesian multi-trend change point analysis which solves
significant memory and computing time limitations to extract crisis
information from a correlation metric. Therefore, we focus on the recently
investigated S&P500 mean market correlation in a period of roughly 20 years
that includes the dot-com bubble, the global financial crisis and the Euro
crisis. The analysis is performed two-fold: first, in retrospect on the whole
dataset and second, in an on-line adaptive manner in pre-crisis segments. The
on-line sensitivity horizon is roughly determined to be 80 up to 100 trading
days after a crisis onset. A detailed comparison to global economic events
supports the interpretation of the mean market correlation as an informative
macroeconomic measure by a rather good agreement of change point distributions
and major crisis events. Furthermore, the results hint to the importance of
the U.S. housing bubble as trigger of the global financial crisis, provide new
evidence for the general reasoning of locally (meta)stable economic states and
could work as a comparative impact rating of specific economic events.

_Keywords_ Bayesian Multi-Change Point Analysis $\cdot$ Linear Trend Segment
Fit $\cdot$ Computationally Efficient Open-Source Python Implementation
$\cdot$ S&P500 $\cdot$ Mean Market Correlation $\cdot$ Economic Crises $\cdot$
Econophysics

##  1 Introduction

Previous work [1, 2, 3] has identified the mean correlation of the S&P500’s
stock time series to capture essential features of the global market movement
by uncovering a strong agreement between the mean correlations, the maximum
eigenvalue of the correlation matrices and the first principal components of
the stock correlations. Following this observation, the one dimensional time
series should contain valuable information about economic dynamics, such as
e.g. bubbles and economic crises. But if the information is stored in the
condensed mean correlations, we should consequently be able to extract that
information in some way from the strongly fluctuating time series. A
demonstration of that would first, lead to new insights into how the
information is encoded in the mean correlation and second, consolidate —
independently from previous works — the significant role of the mean
correlation as a global market observable. In this article, we find that the
quantitatively most probable points of changing mean correlation trends
identified by a Bayesian change point (CP) analysis show rather good
correspondence to the major economic crisis events of the observed time
period, i.e. the Dot-com bubble, the financial and the Euro crisis. A related
approach of a CP analysis is used in a publication of Dehning et al. [4] to
quantify the efficiency of government decisions during the COVID-19 pandemics.
Nevertheless, there are significant differences to our research task: First,
the authors of [4] have access to well-established pandemic models, e.g. the
susceptible-infected-recovered (SIR) model, whereas we have to assume linear
segments with different slopes in absence of better models for the mean
correlation dynamics. Second, Dehning et al. can determine a certain range for
change point positions based on known government actions and incorporate
gradual changes of parameters directly in the model equations of the Monte
Carlo runs. In contrast, we determine the CP probability distributions with
respect to all possible CP configurations based on the given data without
employing Monte Carlo methods.
We demonstrate that the approach lends itself for extracting information about
economic state history, could help to separate endogenous from exogenous
crises and implies that it might be possible in a way to derive precursors for
endogenous events from the mean correlation. Naturally, the approach applied
in this article cannot be interpreted in terms of precursor information, since
it is applied off-line and in retrospect to the whole mean correlation time
series, i.e. it contains data from before and after the major economic events.
This holds also if we change into an on-line perspective in which we
successively add future data and perform the CP analysis on each updated time
series. The CP will only manifest itself if we include a minimum amount of
data that encodes the uprising trend change. However, the on-line sensitivity
horizon of the method to detect trend changes in the investigated time series
lies roughly between $80$ and $100$ trading days which might provide a chance
to identify current events for experts to focus on with respect to possible
new and locally quasi-stationary economic states. Depending on the time
horizon of macroeconomic research it might even provide some evidence for
ongoing changes of the market dynamics.
Nonetheless, the separation of the investigated time period into segments of
almost constant linear trends offers a rather straight-forward interpretation
in terms of temporally stable market periods. Even if it is not possible to
interpret these states as abstract conceptional entities, as it would be
possible e.g. by the clustering approach that is described in Stepanov et al.
[1], the method allows for a more direct evaluation of the data and a closer
connection to real events. It is also robust with respect to intrinsic
parameter choices which is not guaranteed for clustering algorithms and the
pre-assumed number of clusters.
The remainder of this article is structured as follows: Section 2 explains the
data gathering and preprocessing in 2.1 and the Bayesian methodology used for
the change point detection in 2.2. Section 3 shows the ex post identified
change points in 3.1 and an on-line analysis in 3.2. The results are compared
in detail to a time-line of global economic events of the period. Finally, we
discuss and compare our results to the work of Stepanov et al. [1] in section
4.

##  2 Data and Methods

The preprocessing of the data that we analyse is shortly summarized in
subsection 2.1. More details about the CP analysis method can be found in
subsection 2.2.

###  2.1 Data Preparation

The chosen procedure to calculate the mean market correlation of the S&P500
stock index is chosen analogously to the articles [1, 5]. Confined to the
considered time period between 1 January 1992 and 28 December 2012 we filter
for companies that are included in the S&P500 index for at least
$99.5\text{\,}\mathrm{\char 37\relax}$ of the time and interpolate
occasionally missing data. The data is used to compute pairwise correlations
$C_{i,j}$ of the locally normalised returns of the remaining 249 daily-
resolved stock price time series for 5291 trading days. The correlations are
computed on overlapping rolling data windows of size $\tau=42$. Shown by the
principle component analysis in [1], the mean correlation

$\bar{C}=\frac{1}{N}\sum_{\textmd{i,j}}C_{i,j}$ (1)

is in high agreement with the first principle component. In addition, it
describes the greatest part of the variability in the data. Thus, we focus on
the mean market correlation $\bar{C}(t)$ in which the mean correlation of each
window is identified with its 21st time stamp. The corresponding time series
is shown in blue in figure 1. More details about the preprocessing procedure
can be found in Appendix A.

Figure 1: The calculated mean correlation of the S&P500 market. The time
series are the mean values of correlation matrices which were calculated based
on moving windows of length $\tau=42$ days plotted against the 21st time stamp
of the $\tau$-days-interval. The original raw correlation time series is
computed on a sequence of data windows that are successively shifted by one
trading day. The thinned version which contains every 40th data point of the
raw time series, i.e. 132 data points in total, is used to avoid exorbitant
computation times due to the combinatorics of CP configurations involved in
the CP analysis.

###  2.2 Change Point Identification

Since the mean market correlation time series $\bar{C}(t)$ should contain
valuable information about changing market characteristics, we investigate its
changing trends over a time period of stable and crises states in detail.
Therefore, we use a tool that considers the change points (CPs) of linear
trends over time $t$. The open-source Python package antiCPy [6, 7, 8]
implements the Bayesian analysis of changing linear trends, introduced by von
der Linden et al. [9, 10]. By assuming $m$ CPs in the trend, the time series
data $\bar{C}(t)$ is modelled by linear segments $\phi_{K}$ of different
slopes, formally given by

$\phi_{K}(t|\underline{E},\underline{\bar{C}}_{\rm
ord},\mathcal{I})=\bar{C}_{K}\frac{t_{K+1}-t}{t_{K+1}-t_{K}}+\bar{C}_{K+1}\frac{t-t_{K}}{t_{K+1}-t_{K}},$
(2)

with $t_{K}\leqslant t\leqslant t_{K+1}$ and the CP vector $\underline{E}$
with the entries $E_{K}=t_{K+1}$ for $K=1,2,...,N-2$, whereby $N$ denotes the
size of the correlation time series $\bar{C}(t)$ and the vector
$\underline{\bar{C}}_{\rm ord}$ contains the design ordinates $\bar{C}_{K}$ of
the assumed CP positions, i.e. the positions where consecutive segments are
connected with each other. Note that the first and last allowed position of a
CP is restricted to the second and penultimate time stamp $t$. For each
possible configuration $\underline{E}$ of CPs the corresponding posterior
probability $p(\underline{E}|\bar{C}(t),\underline{t},\mathcal{I})$ is
calculated via Bayes’ theorem

$p(\underline{E}|\bar{C}(t),\underline{t},\mathcal{I})=\frac{p(\underline{E}\mid\underline{t},\mathcal{I})\cdot
p(\bar{C}(t)\mid\underline{t},\underline{E},\mathcal{I})}{p(\bar{C}(t)\mid\underline{t},\mathcal{I})}$
(3)

with the underlying time steps $\underline{t}$ and the background information
$\mathcal{I}$. Following this reasoning, we can derive the most probable CP
configurations quantitatively if we perform the calculations for all feasible
CP combinations, i.e. a number of

$Z_{m}=\binom{N-2}{m}=\frac{(N-2)!}{m!((N-2)-m)!}$ (4)

CP configurations and select the most probable ones. Furthermore, the complete
pdf of CP configurations enables us to calculate the most probable linear
segment fit and confidence bands for each time step by calculating the
averaged sum of the linear segment fits at that time weighted with their
respective probabilities. Doing so, even future values and their confidence
bands can be extrapolated based on the current data information in a formally
consistent manner. In addition, the marginal CP probability density functions
can be computed by averaging the joint probabilities at each time stamp $t$
with respect to its ordinal position in each configuration.
However, performing these calculations proves to be almost prohibitive (cf.
also [10]) if we recall the combinatorial number of CP configurations, e.g.
for the whole time series $\bar{C}(t)$ of $5269$ data points for only three
CPs, i.e.

$Z_{2}=\binom{5267}{3}\approx 2.4\cdot 10^{10}$ (5)

combinations. One idea to solve this issue for long time series with moderate
number of change points is presented by von der Linden et al. [10]: The
approach uses a Monte Carlo sum over a limited number of random CP
configurations to estimate the linear segments. However, with increasing
number of change points even a Monte Carlo sum approach is not feasible
anymore, since the extremely high-dimensional CP configuration space is
sampled too sparsely to derive reliable fit results. Furthermore, the time to
compute the linear segment fits serially explodes up to several decades or
hundreds of years. For this reason, first we decide to thin the mean market
correlation data $\bar{C}(t)$ by storing only every 40th data point and
performing the calculations on this dataset of $132$ points which keep the
trend characteristics of the data very well as visible in figure 1.
Nevertheless, a naive serial implementation of the CP analysis would fail with
regards to the computation time and — maybe even worse — collapses due to
memory limitations which do not allow for storing all possible change point
configurations at the same time. Imagine e.g. the situation of the thinned
time series under the assumption of five CPs: We easily run into problems if
we want to store a matrix of dimensions $(286.243.776\times 7)$ with double
precision. To avoid such issues, we take advantage of the efficient
implementation of the change point analysis in the antiCPy package which
involves parallelisation and an algorithm that constructs on the fly and
savely organised only a suitable subset of CP configurations for each parallel
worker while avoiding any redundancy which would compromise the results. More
details of the practical implementation will be prepared in a follow-up
manuscript.

##  3 Results

In subsection 3.1 we apply the CP analysis to the whole mean market
correlation dataset $\bar{C}(t)$ under the assumption of two up to four CPs.
The results are double-checked by analyses with one and five CPs as documented
in Appendix B. In addition, we perform an on-line adaptive CP analysis in the
three pre-crisis segments in subsection 3.2 to complete the analysis.

###  3.1 Change Point Analysis

Figure 2: The change point (CP) analysis results under the assumption of two,
three and four CPs is shown in (a), (b) and (c), respectively. The $x$-axis
ticks correspond to the 1 January of each year. The major economic events as
the dot-com bubble, the global financial crisis and the Euro sovereign debt
crisis are indicated by the grey shaded areas: The dot-com bubble’s rise
starts around 1 January 1999, reaches its all time high at the dashed vertical
line and ends around the 3 October 2022 with a after- bubble trough. The
Financial crisis starts with the burst of the U.S. housing bubble around 1
January 2007 and culminates with the Lehman Brothers’ bankruptcy on 15
September 2008 which marks the end of gray shaded area. The Euro crisis starts
with the first insolvencies of European banks starting with the 1 January 2009
and culminates in the downgrading of Greece’s sovereign debt rating on 27
April 2010. The shown CP probability densities (PDFs) exhibit consistently
peaks in the areas of major economic disturbance. Furthermore, three of the
most probable CP configurations are indicated by vertical lines as an example
consistently with the CP PDFs and the signed crisis events. The insets zoom
into some peaks of the higher order CP PDFs to illustrate that their profile
also contains valuable information, e.g. for the dot-com bubble in (a) and
(b). At least the first CP PDF in (a) is slightly increased shortly after the
dot-com bubble’s all time high, but in (b) only the inset provides insights
into the dot-com bubble period. The linear segment fit with corresponding
confidence intervals is the PDF-weighted average of the linear segment fits at
that time and is shown in orange to illustrate the agreement between (a-c).
For a detailed discussion of the results we refer to the running text.
Additional results for the assumption of one and five CPs can be found in
Appendix B, figure A.1.

As already mentioned above the one-dimensional mean market correlation
$\bar{C}(t)$ is expected to contain valuable information about the global
market dynamics including major crisis events. However, extracting this
information in a reliable manner is an ambitious task. Based on our assumption
that the linear trends of the market correlation encode some information about
changing dynamics due to crisis events, we perform a detailed CP analysis as
described in subsection 2.2. We compute the probability density functions
(PDFs) of the CPs assuming one up to five trend-CPs hidden in the data to
confirm that our results are stable against these a priori assumptions. Even
the results based on only one assumed CP, which obviously cannot reflect the
complicated trend structure in the data, are roughly consistent with the more
elaborate fits and the same is true for models with more than four CPs. The
interested reader can take a look at the results for one and five CPs in
Appendix B, figure A.1.
The most suitable assumptions should contain at least three CPs, since the
investigated period from 31 January 1992 until 28 December 2012 contains three
important major economic events: the dot-com bubble, the financial and the the
Euro crisis. Therefore, in figure 2 of the main article we show our results
from two up to four CPs in increasing order and compare them to these three
major economic events. Unfortunately, the continuously progressing dynamics of
economy can make it rather difficult to determine exact dates of crisis onset
and decline.
In table LABEL:table:_economic_events we provide a timeline of selected
significant events of the three major economic bubbles and crises and compare
them to the time stamps at which maxima occur in the CP PDFs of figure 2. Some
maxima, especially of the higher order CP PDFs are only visible by appropriate
up-scaling. For example the high peak of the first CP PDF in 2(c) is
accompanied by a very small one in (a). In figure 2(b) it is not found.

####  3.1.1 Dot-Com Bubble

First, let us consider the dot-com bubble: We highlight in grey approximately
the time interval in which the NASDAQ Composite Index starts to grow
relatively fast from 1 January 1999 up to 1 October 2002, the day at which the
NASDAQ Composite Index fell roughly to its lowest value after the bubble’s
all-time high marked by the black dotted vertical line on 3 October 2000. The
trough of 1 October 2002 is accompanied by a more general stock market
downturn in 2002 which started around the 11 September 2001 terrorist attacks
[11]. The most interesting observation for the dot-com bubble is that around
its onset and decline the marginal CP PDFs of the second (cf. figure 2(a)),
second and third (b) or third and fourth (c) CP are pronounced as visible in
the insets. For orientation in the insets, the beginning and end of the dot-
com interval are indicated by black and grey solid vertical lines,
respectively. In figure 2(a) and (c) the CP PDFs of the first (a) and first
and second CP (c), respectively, also reach local maxima shortly after the
bubble’s all-time high. These are plausible results, since the dot-com bubble
was of speculative nature which should become visible in the studied
correlations of the S&P500 stock index returns without notably delay. For
completeness, we mention the 14 May 1999 as an important economic date
independent from the dot-com bubble. On this day a sharp rise in consumer
prices in the U.S. led to a drop in the stock market [12]. Even if the width
of the CP PDFs and their pronounced profile at the end of the bubble tend to
support mostly a connection with the continuous strong rise of the NASDAQ
Composite Index, of course our approach cannot decide definitely whether
events like the 14 May 1999 or the aftermath of the Asian financial crisis
[13] around 1997-1998 or Russian financial crisis [14] around 1998-1999 also
might influence the CP PDF profiles. Keeping that in mind, the correspondence
of CP positions and onset/decline of the dot-com bubble nevertheless
strengthens the hypothesis that the mean market correlations trends encode
distinct market periods.

####  3.1.2 Financial Crisis

Second, we consider the financial crisis. While the dot-com bubble was mostly
created by speculative investment decisions, less speculatively induced
economic crises like the global financial crisis include a complex cascade of
sub-crisis events and cannot be simplified in that manner. Therefore, we
proceed with indicating their approximate beginning up to their culmination
events. The very pronounced peaks of the first (a), the first and second (b)
and the first and third (c) CP PDFs are located rather exactly in the time
interval from 31 October 2006 up to 3 January 2007 which coincides with the
burst of the U.S. housing bubble, the trigger event for the global financial
crisis of the later years [15]. Nevertheless, the commonly accepted definition
of the beginning of the financial crisis is on 9 August 2007 when the interest
rates for inter-bank financial loans rose sharply [16, 17]. This emphasises
the problem of determining exact dates for crisis events. Our results suggest
that the U.S. housing bubble is clearly notable as a change in the mean market
correlation, whereas the change in interest rates for inter-bank financial
loans is not detected by our CP analysis. This is a rather expected result,
since we consider stock return correlations: The correlations probably capture
financial events like strong price changes of stock bubbles rather accurately,
but their response to real-world events (like the changing interest rates)
might be delayed.
However, to demonstrate the precise coincidence we mark the burst of the U.S.
housing bubble around 1 January 2007 as the beginning of the grey shaded area
for the global financial crisis and end with the Lehman Brothers’ insolvency
on 15 September 2008 [18]. Also for the Lehman Brothers’ bankruptcy at least
the first and second CP PDFs exhibit local maxima regardless of assumed number
of CPs. Note that similar to the dot-com bubble’s key dates there is almost no
delay between the event and the detected trend change, since the Lehman
Brothers’ bankruptcy had an almost immediate effect on the world economy. Only
two weeks later on 29 September 2008 the U.S. stock market collapsed [19].

####  3.1.3 Euro Crisis

Third, we discuss the Euro crisis as the direct consequence of the global
financial crisis. We indicate its earliest start when the first ten European
banks asked for a bailout up to January 2009, represented in the figure by the
1 January 2009, and end with the culmination of the crisis when Greece’s
sovereign debt rating was downgraded on 27 April 2010 by Standard & Poor’s
[20]. The temporal proximity and strong causal connection of the global
financial crisis and the Euro crisis are captured by the width of the local
maxima of the second (a), third (b) and first, second and fourth (c) CP PDFs
in the passage from one crisis to the other. Also the commonly mentioned
culmination event of the downgrading of Greece’s sovereign debt rating is
mirrored by the CP analysis results. Even if the downgrading alone might have
only a moderate influence on the mean market correlation trends in terms of
symbolic manifestation of the crisis, the indicated culmination event is
mirrored by small peaks in the second and first CP PDF in (a) and (b),
respectively, and in the well-pronounced peaks of the first and second CP PDFs
in (c).
Apart from these considerations, the CP PDFs in the cases (a-c) show
pronounced maxima in the second half of 2010 up to December 2011. It seems to
be impossible to deduce origin events for each individual CP PDF peak, since
the time period is full of economic (and political) events including a series
of sovereign debt downgradings of European countries, bailout plans, the Arab
Spring and more, summarised in table LABEL:table:_economic_events. But in
spite of that it seems to be rather probable that the trend changes detected
by the above mentioned CP PDFs are correlated with the turbulent economic
situations at that time.

####  3.1.4 Remarks and Intermediate Conclusion

Since the CP PDFs are computed over the thinned grid of time steps only every
40th trading day is sampled. For the definition of the maxima we examine these
really calculated support points. In cases of modes of a certain width we
yield intervals of higher CP PDF. Of course, the listed events are not
complete and its content should not be considered as causal deduction between
the listed real world events and the CP PDF peaks. It should rather illustrate
that the uncovered times of high trend change PDFs in fact correlate with
highly turbulent times in the economy. The maxima of the CP PDFs are clustered
around events that are connected with the dot-com bubble, the global financial
crisis and the Euro sovereign debt crisis.
All in all, the results support the hypothesis that the trend in the mean
market correlation encodes valuable information about major economic events
that can be uncovered by applying a Bayesian trend CP analysis. As the
similarities of the time stamps for CP PDF peaks in table
LABEL:table:_economic_events illustrate, the results are comparable to a high
extent, regardless of how many change points we assume. For an extended
robustness check compare the results in figure A.1 under the assumption of one
and five CPs in Appendix B. It might be unsurprising that sharp drops and
bursting bubbles like the dot-com bubble or the U.S. housing bubble are often
well-captured by our approach, since we rely on correlations of stock returns
which show these changes almost immediately. Nevertheless, considering the
mean market correlation with a CP analysis enables us to eliminate trend
changes that manifest only in a small subset of stock prices and thus, acts
like a filter to focus only on economic events of global importance.
Furthermore, in some cases the ex post derived CP PDF peaks could help to
identify important economic events or even rate the impact of specific time
periods or isolated economic events. Moreover, the general ansatz can be
adapted to other economic observables or subsets of stock prices, e.g.
national companies, companies of certain sectors or of certain size etc., to
filter for economic events that are of special interest for the considered
subset. However, we emphasise that the analysis is performed on the whole
available data which makes it an ex post analysis that cannot be interpreted
in terms of an ex ante economic risk predictor.
We conclude the CP analysis discussion with some remarks on the consistency of
the results. The blue mean market correlation $\bar{C}(t)$ in figure 2 is
well-fitted by the orange segment fits with corresponding
$3\sigma$-credibility bands (CBs). The credibility bands quantify the
certainty of the given fit based on the surrounding data information. In that
sense the narrow shape of the CBs suggests that the fit performs well for the
underlying mean market correlation data $\bar{C}(t)$. A comparison of the very
similarly shaped fits (a-c) also confirms the robustness of the results.
Finally, we studied the individual most probable CP configurations three of
which are drawn by green dashed, violet dash-dotted and red dotted vertical
lines in figure 2 as an example. In (a-c) the most probable configurations are
clustered around the CP PDF peaks of the global financial crisis and the Euro
sovereign debt crisis. The dot-com bubble is included only a bit later in the
most probable configurations, but consistent with the less pronounced CP PDFs
of that period. However, the first configuration that includes a date between
the 1 January 1999 and the 1 October 2002 is identified to be the 264th of
$8385$ total configurations in (a), the 26th combination of $357760$ in (b)
and the 354th of $11358880$ configurations in (c). This corresponds to the
most probable $3.15\text{\,}\mathrm{\char 37\relax}$,
$7.28\text{\,}\cdot\mathrm{1}\mathrm{0}^{-5}\mathrm{\char 37\relax}$ and
$3.12\text{\,}\cdot\mathrm{1}\mathrm{0}^{-5}\mathrm{\char 37\relax}$
configurations in (a),(b) and (c), respectively, and round out our analysis.

Table 1: Overview about economic events. The table is not claimed to be complete, but serves as chronological orientation. The CP PDF peaks of figure 2 are printed for easy comparison, but should be interpreted carefully, since the analysis does not allow for causal inference. Event Date | Event Description | Source | CP PDF Peak Dates
---|---|---|---
|  |  |  Figure 2(a) |  (b) |  (c)
1994 |  Over the year bond prices fell continuously in consequence of partially unexpected and repeated raise of federal funds rates by the FED. In consequence bonds lost about $1.5 trillion in market value globally. |  [21] |  2 February 1995 to |  |
26 July 1995; |  |  |  |  |
31 March 1995 to |  |  |  |  |
21 September 1995 |  2 February 1995; |  |  |  |
31 March 1995 to |  |  |  |  |
21 September 1995 |  26 July 1995 to |  |  |  |
16 November 1995; |  |  |  |  |
2 February 1995; |  |  |  |  |
26 July 1995 to |  |  |  |  |
16 November 1995 |  |  |  |  |
December 1994 |  Due to devaluation of the peso by the Mexican government and thus anticipating further devaluations, investors rapidly withdrew capital from Mexican investments. In January 1995 the U.S. government coordinates a $40 billion bailout. |  [22] |  2 February 1995 to |  |
26 July 1995; |  |  |  |  |
31 March 1995 to |  |  |  |  |
21 September 1995 |  2 February 1995; |  |  |  |
31 March 1995 to |  |  |  |  |
21 September 1995 |  26 July 1995 to |  |  |  |
16 November 1995; |  |  |  |  |
2 February 1995; |  |  |  |  |
26 July 1995 to |  |  |  |  |
16 November 1995 |  |  |  |  |
1997 - 1998 |  Asian financial crisis in East and Southeast Asia. Multiple origins are discussed. However, long-lasting global contagion stayed away and the markets recovered fast in 1998-1999. |  [13] |  14 May 1999 |  10 October 1997; |
18 March 1999 |  10 October 1997; |  |  |  |
18 March 1999; |  |  |  |  |
14 May 1999 |  |  |  |  |
1998 - 1999 |  Russian financial crisis in which the rubel was devalued. As a result many neighbouring countries experienced also severe crises. |  [14] |  14 May 1999 |  18 March 1999 to |
13 July 1999 |  18 March 1999; |  |  |  |
14 May 1999 |  |  |  |  |
1 January 1999 |  The modest onset of the dotcom-bubble 1998 turns into a rapid rally of the NASDAQ Composite Index. In the literature the long pre-bubble period begins commonly in 1995. |  [23, 24] |  14 May 1999 |  18 March 1999 to |
12 July 1999 |  18 March 1999 |  |  |  |
14 May 1999 |  Stock market drop due to sharp rise of consumer prices to $1.75\text{\,}\mathrm{\char 37\relax}$ in the U.S. |  [12] |  14 May 1999 |  14 May 1999 to |
12 July 1999 |  14 May 1999 |  |  |  |
10 March 2000 |  Based on the NASDAQ Composite Index the Dot-com bubble reaches its all-time high. |  [24, 23] |  16 October 2000 |  22 June 2000; |
9 April 2001 |  kink at |  |  |  |
22 June 2000; |  |  |  |  |
9 April 2001 |  |  |  |  |
11 September 2001 |  In course of a terror attack three airplanes crash into the twin towers of the U.S. World Trade Center and the Pentagon coordinated by the militant Islamist extremist network al-Qaeda. |  [11] |  24 May to |  |
23 July 2002 |  4 October 2001; |  |  |  |
24 May 2002; |  |  |  |  |
23 July 2002 |  4 October 2001; |  |  |  |
30 January 2002; |  |  |  |  |
24 May 2002; |  |  |  |  |
23 July 2002; |  |  |  |  |
18 September 2002 |  |  |  |  |
1 October 2002 |  Based on the NASDAQ Composite Index the Dot-com bubble reaches a new trough after the bubble burst in course of a more general stock downturn since 11 September 2011. |  [23, 24] |  28 July 2005 to |  |
18 November 2005 |  28 October 2003 to |  |  |  |
24 December 2003 |  28 October 2003 to |  |  |  |
24 December 2003 |  |  |  |  |
2002/2003 |  The Venezuelan General Strike massively hinders oil exports among others to the USA. |  [25, 26] |  28 July 2005 to |  |
18 November 2005 |  28 October 2003 to |  |  |  |
24 December 2003 |  28 October 2003 to |  |  |  |
24 December 2003 |  |  |  |  |
2003 |  Approximate beginning of the boom-phase of the U.S. housing bubble. |  [27] |  28 July 2005 to |  |
18 November 2005 |  28 October 2003 to |  |  |  |
24 December 2003 |  28 October 2003 to |  |  |  |
24 December 2003 |  |  |  |  |
2003 - 2016 |  Begin of rising oil prices from under $25/bbl to above $30/bbl. The peak of $147.30/bbl is reached in July 2008, before they return temporarily to $35/bbl in 2009. |  [26, 28] |  28 July 2005 to |  |
18 November 2005 |  28 October 2003 to |  |  |  |
24 December 2003 |  28 October 2003 to |  |  |  |
24 December 2003 |  |  |  |  |
Q4 2006 to Q1 2007 |  Period of most intense burst of the U.S. housing bubble with new lowest price in 2011. |  [15] |  2 November 2006 to |  |
3 January 2007 |  31 October 2006; |  |  |  |
2 January 2007 |  2 November 2006 to |  |  |  |
3 January 2007 |  |  |  |  |
9 August 2007 |  Commonly listed onset of the global financial crisis. On this day the interest rates for inter-bank financial loans rose sharply. |  [17, 16] |  22 August 2007 |  22 August 2007 |  22 August 2007
2008 |  Year of the financial crisis and great recession. The year is characterised by manifold bailouts for banks and stock market crashes. |  [29] |  overall high CP PDFs in 2008 |  overall high CP PDFs in 2008 |  overall high CP PDFs in 2008
January to March 2008 |  The housing prices continue to collapse. The government passes a tax rebate bill on 13 February 2008 and the FED starts bail out programs in the beginning of March. |  [30, 31] |  4 August 2008 |  4 August 2008 |  4 August 2008
30 July 2008 |  U.S. governments passes bailout laws for Fanny Mae and Freddie Mac. |  [32] |  Kink at |  |
4 August 2008 |  4 August 2008 |  4 August 2008 |  |  |
7 September 2008 |  U.S. government’s take over of Fanny Mae Association and Freddie Mac Corporation. |  [33] |  1 October 2008 to |  |
26 November 2008 |  1 October 2008 |  1 October 2008 to |  |  |
26 November 2008 |  |  |  |  |
15 September 2008 |  Lehman Brothers bankruptcy. |  [18] |  1 October 2008 |  1 October 2008 |  1 October 2008
29 September 2008 |  Stock market collapsed when the bailout bill was rejected by the U.S. House of Representatives. |  [19] |  1 October 2008 |  30 September 2008 |  1 October 2008
24 October 2008 |  Many world stock indices lost around $10\text{\,}\mathrm{\char 37\relax}$. |  [34] |  26 November 2008 |  26 November 2008 |  26 November 2008
1 December 2008 |  When the National Bureau of Economic Research officially declared that the U.S. was in a recession since December 2007, the S&P500 lost $8.93\text{\,}\mathrm{\char 37\relax}$ and the financial stocks of the index even $17\text{\,}\mathrm{\char 37\relax}$ based on these news. |  [35] |  high CP PDF after |  |
26 November 2008 |  high CP PDF after |  |  |  |
26 November 2008 |  high CP PDF after |  |  |  |
26 November 2008 |  |  |  |  |
January 2009 |  The first ten European banks ask for bailout programs. |  [36] |  high CP PDF after |  |
26 November 2008 |  high CP PDF after |  |  |  |
26 November 2008 |  high CP PDF after |  |  |  |
26 November 2008 |  |  |  |  |
Early 2010 to mid 2012 |  The Arab Spring. Great anti-government protests in large parts of the Middle East and North Africa. The oil prices rise above $100/bbl. |  [37, 38, 28] |  5 May 2010 to |  |
1 July 2010 |  5 May 2010 to |  |  |  |
1 July 2010 |  5 May 2010 to |  |  |  |
1 July 2010 |  |  |  |  |
27 April 2010 |  Greece’s sovereign debt rating was downgraded by Standard& Poor’s. |  [20] |  5 May 2010 to |  |
1 July 2010 |  5 May 2010 to |  |  |  |
1 July 2010 |  5 May 2010 to |  |  |  |
1 July 2010 |  |  |  |  |
6 May 2010 |  So-called flash crash led to a $9\text{\,}\mathrm{\char 37\relax}$ drop in the Dow Jones Index caused due to high frequency trading. |  [39] |  5 May 2010 to |  |
1 July 2010 |  5 May 2010 to |  |  |  |
1 July 2010 |  5 May 2010 to |  |  |  |
1 July 2010 |  |  |  |  |
8 May 2010 |  Passing drastic bailout plans for Greece in Brussels. A €110 billion package is approved. |  [40, 41] |  5 May 2010 to |  |
1 July 2010 |  5 May 2010 to |  |  |  |
1 July 2010; |  |  |  |  |
28 August 2010 |  5 May 2010 to |  |  |  |
1 July 2010; |  |  |  |  |
1 July to |  |  |  |  |
28 August 2010 |  |  |  |  |
17 May 2010 |  The Euro currency falls into first four-years low. |  [42] |  5 May 2010 to |  |
1 July 2010 |  5 May 2010 to |  |  |  |
1 July 2010; |  |  |  |  |
28 August 2010 |  5 May 2010 to |  |  |  |
1 July 2010; |  |  |  |  |
1 July to |  |  |  |  |
28 August 2010 |  |  |  |  |
Q1/2 2010 |  Several downgradings of debt ratings/bonds of European countries, i.e. Portugal, Spain, Greece. |  [43] |  5 May 2010 to |  |
1 July 2010 |  5 May 2010 to |  |  |  |
1 July 2010; |  |  |  |  |
28 August 2010 |  5 May 2010 to |  |  |  |
1 July 2010; |  |  |  |  |
1 July to |  |  |  |  |
28 August 2010; |  |  |  |  |
3 September 2010 |  |  |  |  |
4 June 2010 |  The Euro currency falls into second four-years low. Major American markets fall more than $3\text{\,}\mathrm{\char 37\relax}$. |  [44] |  5 May 2010 to |  |
1 July 2010 |  5 May 2010 to |  |  |  |
1 July 2010; |  |  |  |  |
28 August 2010 |  5 May 2010 to |  |  |  |
1 July 2010; |  |  |  |  |
1 July to |  |  |  |  |
28 August 2010 |  |  |  |  |
November 2010 |  Bailout request by Ireland. In the end of November Ireland receives €85 billion. |  [45, 46] |  15 April 2011 |  15 April 2011 |  15 April 2011
7 April 2011 |  In the evening of 6 April 2011 Portugal’s government announces that it is the third after Greece’s and Ireland’s governments that will ask for a bailout package. On 17 May 2011 a bailout package of around €78 billion is formally adopted. |  [47] |  15 April 2011 to |  |
14 June 2011 |  15 April 2011 to |  |  |  |
14 June 2011 |  15 May 2011 to |  |  |  |
14 June 2011 |  |  |  |  |
13 June 2011 |  Greece’s credit rankings become worst in the world. |  [48] |  13 June 2011 |  13 June 2011 |  13 June 2011
August 2011 |  Under the Securities Markets Programme the ECB restarts to purchase a significant amount of eurozone sovereign bonds. Spanish and Italian yields breach $6\text{\,}\mathrm{\char 37\relax}$. |  [49, 50] |  6 October 2011 |  9/10 August 2011 |  10 August 2011
|  |  |  |  |
|  |  |  |  |
|  |  |  |  |

###  3.2 On-line Change Point Evolution in Pre-Crisis-Segments

Figure 3: Results of a CP analysis on specific subsets of the whole mean
market correlation $\bar{C}(t)$ under the assumption of one CP. The blue pre-
crisis segments start on 31 January 1999 (time series onset), 1 October 2002
(end dot-com bubble) and 15 September 2008 (Lehman Brothers’ bankruptcy) for
the dot-com bubble, the financial crisis and the Euro crisis, respectively,
and end with the dates $t_{0}$ at the crisis onsets on 1 January 1999 (dot-com
bubble), 1 January 2007 (financial crisis) and slightly before the crisis
onset on 1 December 2008 (Euro crisis). The time series are thinned by using
every tenth data point for the dot-com bubble and the financial crisis. The
resulting orange CP PDFs are compared to the black PDFs that are computed on
the red updated time series segments up to dates $t_{1}$, i.e. 1 June 1999
(dot-com bubble), 9 August 2007 (financial crisis) and 1 January 2009 (Euro
crisis). As expected the major crisis onsets are captured by the black CP PDFs
after some in-crisis data is included, whereas the orange CP PDFs may reflect
consequences of economic events confined to the segment interval. Interpreted
from an on-line analysis perspective, the appearance of a pronounced and
narrow peak (cf. black CP PDFs) might help to identify important current
economic events that may influence future economic states more than the ones
detected by the orange smaller an wider CP PDFs. In contrast to the more
pronounced black CP PDF in the former two cases, a higher similarity of the
orange and black CP PDF is observed in the Euro crisis case. This might
indicate the difficulties of isolating the Euro crisis from the financial
crisis as a cascade of events in the end 2008 implies lots of trend changes
with an overall negative mean market correlation trend in the end of 2008. The
green interim CP PDFs are computed on data up to the green dotted vertical
lines at date $\tilde{t}$ and mark the minimum amount of updated data that is
needed to locate the global maximum of the green CP PDF less than $100$
trading days from the defined crisis onset dates. The segment fits with
corresponding confidence bands are shown to guide the eye. More information
can be found in the running text.

In the CP analysis of the mean market correlation $\bar{C}(t)$ from 31 January
1992 until 28 December 2012 in subsection 3.1, we take a retrospective point
of view to investigate whether the linear trend changes in the mean market
correlation $\bar{C}(t)$ encode information about economic crisis events of
global importance. Since our findings in subsection 3.1 suggest indeed such a
connection, in this subsection we complement the consistency of our results by
changing from a retrospective to an on-line perspective. From the on-line
viewpoint three key questions arise: First, is the time evolution of the CP
PDFs $p(E|\bar{C}(t),\underline{t},\mathcal{I})$ of one CP $E$ consistent with
our expectations if we apply it to currently available data and update these
data whenever new trading days end? Second, how much data of a changing mean
market correlation trend, which potentially marks an important economic
period, has to be included to render the CP analysis sensitive? And third, do
the results change consistently on the stronger timely confined data segments,
i.e. do the CP PDFs mirror in a way the relative impact of the events included
in the sub-intervals? To answer these questions, we assume one CP and compare
the CP analysis results calculated on two slightly differing types of time
series segments for the dot-com bubble, the global financial crisis and the
Euro crisis. The first type is limited exclusively to pre-crisis data of the
mentioned major crisis events and the second type includes a small amount of
mean market correlation data beyond the crises’ onsets at $t_{0}$, i.e. it is
cut after $t_{1}>t_{0}$. The results are presented in figure 3. For the pre-
dot-com bubble segment as well as for the financial crisis data we used every
tenth data point to avoid numerical instabilities due to an overflowing
normalisation factor of the CP PDFs. The short segment of the Euro crisis is
not thinned at all. Analogously to subsection 3.1 we show the accompanying
segment fits of the orange and the black CP PDFs in figure 3 to illustrate the
most probable linear segments each of which might resemble specific economic
periods.

####  3.2.1 Dot-Com Bubble

We start considering the pre-dot-com bubble data in figure 3. The blue segment
is cut from the beginning of the $\bar{C}(t)$ time series on 31 January 1992
up to the approximate onset $t_{0}$ of the rapid increase of the NASDAQ
Composite Index on 1 January 1998. Therefore, it does not contain any dot-com
period data and the CP PDF should accumulate either around the times of
economic events of major impact that are already confined in the data segment
or, in the absence of such events, be less pronounced and probably located
almost randomly due to some noisy outliers in the data (i.e. the probability
to find the mode of the CP PDF exactly in the end period of the data is
relatively low). Note that in any case we have to expect some mode of the CP
PDF $p(E|\bar{C}(t),\underline{t},\mathcal{I})$ regardless of whether there is
a crisis event or not. This traces back to the design of the trend CP analysis
in which we compute the CP PDF $p(E|\bar{C}(t),\underline{t},\mathcal{I})$
conditional on the background information $\mathcal{I}$, i.e. our prior
knowledge/assumption that at least one CP is present in the data. Keeping that
in mind, the CP PDFs will exhibit a flat profile only in the rare situation of
almost no fluctuations of constant trends, i.e. in the limiting case of an
almost unperturbed straight line.
However, the CP PDF $p(E|\bar{C}(t),\underline{t},\mathcal{I})$ indeed
accumulates over the year 1995 coinciding with two important economic events:
the so-called 1994 bond market crisis [21] with sharply decreasing bond prices
over 1994 (and a strong drop in the yield spread of 1-year and 30-year U.S.
Treasury Bonds from roughly $2.83\text{\,}\mathrm{\char 37\relax}$ in 25
January 1994 to $0.54\text{\,}\mathrm{\char 37\relax}$ in 13 December 2004
[51], i.e. an indicator for a heavily flattened yield curve and worse economic
conditions), and the Mexican peso crisis [22] starting in December 1994 which
spread to other emerging markets. Updating the time series segment with new
available data and based on the results of subsection 3.1, we expect that the
CP PDF is shifted consistently to the dot-com bubble onset period in 1999,
once data beyond the onset is included. And in fact, the black CP PDF computed
on the updated red mean market correlation $\bar{C}(t)$ including data up to 1
June 1999 is shifted from 1995 to January 1999 corresponding to the dot-com
bubble CP results in subsection 3.1.
Note that the appearance of the black CP PDF’s peak cannot be interpreted in
terms of a precursor of the upcoming bubble for two reasons. First, we need
some data points beyond the crisis onset and second, from the on-line point of
view the shifted CP PDF might be in turn corrected by newly updated data
points that uncover the detected trend change as a negligible fluctuation in
the long run. Nevertheless, once detected, subsequent inclusion of new
information can also confirm the shifted CP PDF, being the case for the major
events detected in subsection 3.1. In that way the on-line implementation can
be very helpful to focus economic research attempts early onto current time
periods which in terms of impact supersede the last major event — here i.e.
the bond crisis 1994 and the Mexican peso crisis in December 1994 — and have
potentially stronger influence to future economic states. Such a line of
argument might be supported to a certain extent by the clearly more pronounced
and narrow profile of the black CP PDF in Q1 1999 compared to the wider and
less pronounced orange CP PDF of the years 1995, corresponding to a sharper
and more probable trend change in 1999 than in 1995.
Additionally, we determined approximately the minimum amount of data
$\bar{C}(t)$ that are needed to shift the global maximum of the CP PDF to a
date less than $\pm 100$ trading days earlier/later as the bubble’s onset date
defined by the thinned time series date that follows directly on the bubble’s
onset on 1 January 1999, i.e. the 5 January 1999. In detail this procedure
leads to the 30 April 1999 marked by the green dotted vertical line in figure
3. Thus, to make the CP analysis sensitive to the bubble onset we need
approximately raw stock return data up to the 1 June 1999 (i.e. $21$ trading
days later), since the mean market correlation $\bar{C}(t)$ is derived from
windows of $42$ trading days returns and identified with the 21st window date.
The resulting green CP PDF computed on data up to the green dotted vertical
line at date $\tilde{t}$ fits into the previous reasoning as it represents a
snapshot at date $\tilde{t}$ of the time evolution from the orange CP PDF at
$t_{0}$ to the black CP PDF at date $t_{1}$ with $t_{0}<\tilde{t}<t_{1}$. The
old global maximum of the orange CP PDF shrank down to the flat double peaked
relic of the green PDF and the new global maximum in January 1999 has already
manifested itself. By adding new data the green global maximum will eventually
grow to the black CP PDF and the green remnants in 1994/1995 vanish.

####  3.2.2 Financial Crisis

For the financial crisis we observe a very similar behaviour. The segments
start beyond the dot-com bubble’s trough on 1 October 2002. The pre-crisis
segment is truncated with the burst of the U.S. housing bubble as the seed
event of the financial crisis on 1 January 2007, whereas the longer red time
series includes in-crisis data until 9 August 2007 when the interest rates for
inter-bank financial loans rose sharply. Similar to the previously discussed
dot-com bubble segment the orange CP PDF might reflect important events in the
confined context of the segment. Consistently with the coincidence of the CP
PDFs and the U.S. housing bubble’s burst around January 2007, the onset of the
bubble’s pronounced rise began in 2003 [27]. Additionally, rising oil prices
in 2003 from $$20$/bbl to over $$30$/bbl peaking in 2008 with $147/bbl, a
phenomenon referred to as the 2000s energy crisis [26, 28], may have
contributed to the trend change. Also the Venezuelan general strike in
2002/2003 might play a role because it hindered oil exports heavily. In this
time the U.S. imports of oil from Venezuela show a pronounced trough [25, 52].
The ambiguity and plurality of possible events underlines the obvious
limitations of such a CP analysis which is unable to distinguish the cause of
the trend change when the events overlap. However, it might help to reduce the
number of probable events that could be responsible for new economic states to
the area of pronounce CP PDFs.
The green CP PDF is computed with data until 1 March 2007, i.e. the
approximate date from which on the CP PDF is sensitive to the U.S. housing
bubble burst is around 30 March 2007. The strongly accumulated probability
mass in the narrow green global maximum is smoothed out a bit to the black CP
PDF when the data is updated.

####  3.2.3 Euro Crisis

In contrast to the former analyses the Pre-Euro crisis segments tell a
different story due to the fact that the crisis is embedded in turbulent
economic times which make it rather difficult to isolate a precise period of
the Euro crisis. In fact the mean market correlation shows alternating
gradually increasing linear downwards trends interrupted by less pronounced
upwards trends, i.e. an overall downwards trend in the long run. Therefore,
the detected mode of the CP PDF is more sensitive to the chosen time series
period. This is in good agreement to our CP analysis in retrospect (cf.
subsection 3.1) where the CP PDF modes in the appropriate period are also
widened from 2008 until 2010. Anyway, the applicability of the CP analysis is
limited due to the alternating short term trends and the non-isolated nature
of the Euro crisis emerging almost directly after the global financial crisis.
In any case we follow our previous approach for the starting time of the
investigated time series segments, i.e. we start with the culmination event of
the previous crisis, here the Lehman Brothers’ bankruptcy on 15 September
2008. Depending on the end of the segment the CP PDF is shifted consistently
to the right, i.e. the emerging Euro crisis supersedes older events, but
cannot be fixed exactly. It fluctuates depending on the amount of included in-
crisis data which of course add more short term trend information and
compromise the assumption of only one CP more and more. Varying the end date
from our defined onset of the Euro crisis on 1 January 2009 to 1 June 2009
leads to the CP PDF fluctuating roughly between 1 January and 1 April 2009.
However, the onset of the negative long term trend is slightly before 1
January 2009. The corresponding analysis of segments ending on 1 December 2008
and 1 January 2009 is presented in figure 3 and underlines the missing
isolation of the Euro crisis period from the financial crisis and the
interconnected Great Recession of 2008 when the U.S. gross domestic product
fell $4.3\text{\,}\mathrm{\char 37\relax}$ which is the deepest recession
since World War II [53, 54].
In contrast to the former two analysed segments, the orange and black CP PDFs
do not differ a lot in shape suggesting that the related trend changes cannot
be classified in terms of lower or higher impact, but seem to be of similar
nature constructing the downwards trend in the long run. This seems to be
feasible because over 2008 the critical economic conditions led to several
giant rescue plans for banks and a cascade of severe stock market crashes.
From 6 to 10 October 2008 the Dow Jones Industrial Average fell
$18.2\text{\,}\mathrm{\char 37\relax}$, the S&P500 even over
$20\text{\,}\mathrm{\char 37\relax}$ [55]. Some crashes created the need of
interim interruption of the stock markets, e.g. in Indonesia (from 8 to 13
October 2008) due to drop of $10\text{\,}\mathrm{\char 37\relax}$ in one day
or Iceland (9, 10 and 13 October 2008) [56, 57]. On 24 October 2008 again
losses of around $10\text{\,}\mathrm{\char 37\relax}$ were realised in most of
the indices [34]. This as well as the IMF prediction of a worldwide recession
of $-0.3\text{\,}\mathrm{\char 37\relax}$ on 6 November 2008 and accompanying
lowering of target interest rates by the Bank of England and the ECB
contribute most probably to the detected CP of the orange PDF [58]. By
updating the data up to $t_{1}$ finally the black CP PDF emerges slightly
after 1 December 2008 when the National Bureau of Economic Research officially
declared that the U.S. was in a recession since December 2007. In consequence
of these news the S&P500 lost $8.93\text{\,}\mathrm{\char 37\relax}$ and the
financial stock of the index even $17\text{\,}\mathrm{\char 37\relax}$ [35].
Further hints supporting the hypothesis that the event cascade drives the
downwards trend is given by the green interim CP PDF result: As the data is
updated, the green interim CP PDF is first widely smeared out between the old
orange and the new black PDF peak and becomes a bi-modal CP PDF with the peaks
near these two positions. In that way, the time evolution of the CP PDF
resembles the cascade of similarly important events that contribute to the
decline of the mean market correlation in the long run. In principle, also in
the case of further added data the CP PDF travels along the cascade of events
via a bi-modal interim PDF from peak to peak in a similar pattern.
The method is sensitive to the short term trend change if data until 23
December 2008, marked by the green vertical dotted line, is included. This
corresponds to an earliest CP analysis sensitivity from 26 January 2009 on.

##  4 Discussion and Conclusion

In sections 3.1 and 3.2 we present the results of two change point analyses:
first, in retrospect over the thinned mean market correlation $\bar{C}(t)$
time series and second, in an adaptive approach that updates a pre-crisis
segment continuously as present trading days end. Based on an extensive
literature research we provide a detailed comparison to the overall economic
history of roughly 20 years from 31 January 1992 to 28 December 2012 that
include three major economic crises and check the consistency of our results
against a variation of the method’s intrinsic parameter, i.e. the number of
expected CPs. Applying our computationally efficient implementation of the CP
analysis from one up to five CPs (roughly $3\cdot 10^{8}$ combinations) we
find a weak dependence of the main results of the CP distributions on the
expected number of CPs from two up to five CPs, whereas one CP seems to be a
too simple assumption over 20 years of economic evolution. Given that, we find
consistent and reasonable results from sub-periods up to the full period: The
retrospective analysis of the whole dataset suggests a connection between the
CP PDFs of linear trend changes in the mean market correlation and major
economic events as the dot-com bubble, the global financial crisis and the
Euro crisis. Additionally, lower peaks seem to accumulate in crisis periods of
lower impact. Naturally, the analysis is not feasible to infer on causal
dependencies between trend changes of the mean market correlation and single
economic events, but our studies support the idea of the mean market
correlation (i.e. the market mode) to be an informative measure of
macroeconomic changes as stated in the literature (cf. [2, 3, 1]). We note
that the onset of the global financial crisis is especially reflected by the
CP PDF peaks of the U.S. housing bubble burst and might illustrate the strong
interdependence of the bubble and the global crisis.
Furthermore, the CP PDFs divide the considered period into intervals that
match rather well with the locally stable market state intervals found in an
article of Stepanov et al. [1] by a clustering approach: The authors in [1]
find the first four years to be a rather calm period in cluster states one and
two which is mixed with some intermediate state five around Spring 1996. This
is reflected by weakly pronounced CP PDF peaks at that time. A more notably
transition occurs over the dot-com bubble from 1999 to 2003 into a fully
intermediate transition regime dominated by states three, four, five and six.
The transition is marked by the CP PDFs of the dot-com bubble and by an
increased CP probability in the whole intermediate period up to 2007. Around
this date the authors in Stepanov et al. define states seven and eight to be
economic turbulent crisis states which is in accordance to the high CP PDFs in
the end of 2006 and beginning of 2007 as well as the pronounced local CP PDF
modes in the turbulent period from 2007 to the end of 2012. Interestingly, we
can reconstruct approximately the economic state structure presented in
Stepanov et al. by inferring them directly from the trend changes of the mean
market correlation. Our independent methodological CP approach adds new
evidence to the general reasoning of locally (meta)stable market states. The
existence of quasi-stationary states is further supported by an independent
resilience analysis on the S&P500 mean market correlation time series which
suggests locally stable economic states eventually operating on multiple time
scales [5].
The on-line approach assuming one CP on the pre-crisis segments that are
updated with in-crisis data over time yields further insights into the
methodology’s sensitivity and the relation of CP PDFs and the considered time
period. First, the CP PDF is consistently shifted from left to right in times
of crises’ onsets for the dot-com bubble and the financial crisis. The shift
becomes manifest in our analysis by including data of roughly 80 to 100
trading days after the crisis onset. The character of the Euro crisis seems to
be different from the former two because of the almost direct transition from
the financial crisis, i.e. missing isolation of the events in the time
dimension. The CP PDF is jumping roughly between a stock market crash on 1
December 2008 and April 2009 depending on the time series length. This might
indicate a cascade of non-isolated events in that period. Note that this is
also reflected by the wider modes of the CP PDFs in the retrospective analysis
in section 3.1 between 2008 and 2010. Furthermore, zooming into the pre-crisis
segments instead of analysing the whole $\bar{C}(t)$ time series changes the
resolution of the detected trends in relation to the events included in the
segments. For example, the highest peaks in the retrospective analysis are
around the financial crisis onset and after the Euro crisis’ culmination event
of Greece’s downgrading. More local events as the bond crisis 1994, the
Mexican peso crisis or Asian and Russian financial crisis are not or less
pronounced in relation to the included major events. After exclusion of the
major crisis events by zooming e.g. into the pre-dot-com bubble segment, the
CP PDF accumulates around the bond crisis 1994 and Mexican peso crisis. This
means the CP PDFs might reflect up to a certain degree an economic impact
ranking of the events relative to the most important ones that are included in
the analysis segment. Moreover, such an observation implies that the zoom
changes the event impact resolution of the CP analysis.
Naturally, the on-line approach cannot be seen as a precursor tool for
economic crisis, since it needs a certain minimum amount of in-crisis data and
more important, a recently detected CP might be identified as part of
fluctuations if again new data is available. However, the overall results
imply that the mean market correlation $\bar{C}(t)$ contains information about
changing market dynamics and global crisis events. This gives rise to the idea
of designing leading indicators and precursors based on the market mean
correlation $\bar{C}(t)$. Related scientific attempts of precursor design
based on correlations in general can be already found in the literature [59,
60, 61].
Anyways, although the presented study suggests the mean market correlation to
be a promising economic indicator, it is way too early for final deductions on
the mean market correlation’s role as economic information measure. Future
research on this topic might consider other time periods applying similar
methodologies as the one used here or even find different and adequate
procedures for possibly even causal inference on the relation of mean market
correlation and macroeconomic events.
Furthermore, the CP analysis by design detects almost always a CP apart from
the limiting case of an unperturbed straight line. In that sense, it might be
advisable to assure the detected trend change by involving further analyses.
For example, the computed segment fit can be compared to a straight line via a
Bayesian model comparison [10] or information criteria [62] to infer whether
the number of assumed CPs is a justified condition or not. However, due to the
definition the information criteria might always prefer the straight lines for
time series that include only a few data beyond the CP. Assigning higher
weight to the most recent data might be a possible way to remedy this problem.

## Data and Software Availability

The simulated data and Python codes are available on github via abc under a
GNU General Public License v3.0. The open source python-implementation is
named antiCPy and can be found at https://github.com/MartinHessler/antiCPy
under a GNU General Public License v3.0. The raw data can be downloaded via
[63] and the preprocessed time series is available via [64].

## Author Contributions

Conceptualization, M.H., T.W. and O.K.; methodology/software, M.H.;
validation/interpretation, M.H. and T.W.; formal analysis/investigation, M.H.;
data resources, T.W.; data curation, M.H. and T.W.; writing — original draft
preparation, M.H.; writing — review and editing, M.H., T.H. and O.K.;
visualization, M.H.; supervision, O.K. All authors have read and agreed to the
published version of the manuscript.

## Additional Information

This research received no external funding and the authors declare no conflict
of interest.

## Acknowledgments

M.H. and T.W. thank the Studienstiftung des deutschen Volkes for a scholarship
including financial support.

## Abbreviations

The following abbreviations are used in this manuscript:

CB | Credibility Band
---|---
CP | Change Point
ECB | European Central Bank
FED | U.S. Federal Reserve System
NASDAQ | National Association of Securities Dealers Automated Quotations
U.S. | United States
USA | United States of America
PDF | probability density function
Q1, Q2, … | quarter 1, quarter 2, …
S&P | Standard and Poor’s

## Appendix

##  Appendix A Detailed Data Preparation

Daily stock data from the 500 companies that are included in the S&P500 stock
index which is traded at U.S. stock exchanges, were downloaded via the Python
package yfinance [63]. Confined to the time period between 1 January 1992 and
28 December 2012 which is chosen analogously to [65], we filter for companies
that are included in the S&P500 index for at least $99.5\text{\,}\mathrm{\char
37\relax}$ of the time and linearly interpolate occasionally missing price
data $P_{t}$ via the function interpolate() of the Python package pandas. We
end up with 249 daily-resolved stock price time series $P_{t}$ for 5291
trading days. Analogously to [66] the returns $R_{t}=(P_{t+1}-P_{t})/P_{t}$
are locally normalised as

$r_{t}=\frac{R_{t}-\langle R_{t}\rangle_{n}}{\sqrt{\langle
R^{2}_{t}\rangle_{n}-\langle R_{t}\rangle^{2}_{n}}}$ (6)

with the local mean $\langle\dots\rangle_{n}$ across the $n=13$ (cf. [67])
most recent data points to remedy the impact of sudden changes in the drift of
the time series. The procedure corresponds to a local standard normalisation
transformation with the local mean $\langle R_{t}\rangle_{n}$ and standard
deviation $\sqrt{\langle R^{2}_{t}\rangle_{n}-\langle R_{t}\rangle^{2}_{n}}$.

Figure A.1: Additional results of the CP analysis performed in the main
article. (a,b) CP analysis for one and five CPs. The distributions are in good
agreement with the results for two up to four CPs and underline the
consistency of the results.

Local correlation coefficients

$C_{i,j}=\frac{\langle r^{(i)}_{t}r^{(j)}_{t}\rangle_{\tau}-\langle
r^{(i)}_{t}\rangle_{\tau}\langle
r^{(j)}_{t}\rangle_{\tau}}{\sigma^{(i)}_{\tau}\sigma^{(j)}_{\tau}}$ (7)

are calculated over a time period of $\tau=42$ trading days (cf. [67]) with
the local standard deviations $\sigma^{(i)}_{\tau}$. Shown by the principle
component analysis in [1], the mean correlation

$\bar{C}=\frac{1}{N}\sum_{\textmd{i,j}}C_{i,j}$ (8)

is in high agreement with the first principle component. Furthermore, it
describes the greatest part of the variability in the data. Thus, we focus on
the mean market correlation $\bar{C}(t)$ with each mean correlation calculated
over 42 trading days and identified with the 21st value of each data window.

##  Appendix B Advanced Change Point Analysis

The CP analysis results are complemented by the computations under the
assumption of one and five CPs. In figure A.1 the corresponding PDFs (a,b) are
provided. The overall similarities between the PDF results from one up to five
CPs underline the consistency of the results. To complete the advanced
analysis we performed a search for the earliest occurrence of a CP in the dot-
com bubble period: for the assumption of one and five CPs the seventh and
852th most probable configuration, respectively, includes the dot-com period
for the first time. This corresponds to the most probable
$5.0\text{\,}\mathrm{\char 37\relax}$ and
$3.0\text{\,}\cdot\mathrm{1}\mathrm{0}^{-4}\mathrm{\char 37\relax}$ of
possible CP configurations for the assumption of one or five CPs,
respectively.

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