# Uncertainty, volatility and the persistence norms of financial time series

Simon Rudkin Corresponding Author. Full Address: Economics Department, School
of Social Sciences, Swansea University, Bay Campus, Swansea, SA1 8EN, United
Kingdom<EMAIL_ADDRESS>Dioscuri Centre in Topological Data
Analysis, Polish Academy of Sciences, Poland  Wanling Qiu Full Address:
Accounting and Finance Department, School of Management, Swansea University,
Bay Campus, Swansea, SA1 8EN, United Kingdom<EMAIL_ADDRESS>Accounting and Finance Department, Swansea University, United Kingdom  Paweł
Dłotko Full Address: Dioscuri Centre in Topological Data Anlaysis, Mathematics
Institute, Polish Academy of Sciences, Warsaw, 01-2000, Poland.
<EMAIL_ADDRESS>. P.Dłotko acknowledges support from the Dioscuri
program initiated by the Max Planck Society, jointly managed by the National
Science Centre (Poland), and mutually funded by the Polish Ministry of Science
and Higher Education and the German Federal Ministry of Education and
Research. Dioscuri Centre in Topological Data Analysis, Polish Academy of
Sciences, Poland

(September 2021)

###### Abstract

Norms of Persistent Homology introduced in topological data analysis are seen
as indicators of system instability, analogous to the changing predictability
that is captured in financial market uncertainty indexes. This paper
demonstrates norms from the financial markets are significant in explaining
financial uncertainty, whilst macroeconomic uncertainty is only explainable by
market volatility. Meanwhile, volatility is insignificant in the determination
of norms when uncertainty enters the regression. Persistence norms therefore
have potential as a further tool in asset pricing, and also as a means of
capturing signals from financial time series beyond volatility.

Keywords: Topological Data Analysis; Uncertainty; Volatility; Predictability;
Time Series

##  1 Introduction

Since Gidea and Katz, (2018) identified the power of topological data analysis
(TDA) persistence norms to warn about financial crashes, there has been strong
interest in how the tools of TDA can be brought to finance and economics (Goel
et al.,, 2020; Bakas and Triantafyllou,, 2020; Gidea et al.,, 2020; Majumdar
and Laha,, 2020). TDA is introduced in the seminal work of Carlsson, (2009),
and is widely developed for, the natural sciences (Bubenik and Dłotko,, 2017;
Bubenik,, 2020). Initial potential for financial market dynamics of the type
described by Hsieh, (1991) comes from the applications in the detection of
phase transitions and system instability (Stolz et al.,, 2017; Smith et al.,,
2021). Promising advancements on correlation and volatility show how each
alter the topology of the time series, but that neither correlation or
volatility alone can explain persistence norms (Aromi et al.,, 2021; Leaverton
et al.,, 2020; Katz and Biem,, 2021). Focus in this literature has been on the
strength of the volatility-norm relationship. We expand that focus to consider
financial market dynamics, asking how persistence norms link to forward
looking measures of uncertainty

Jurado et al., (2015) constructs uncertainty indexes for financial (FIN),
macroeconomic (MAC) and real activity (REA) based upon the predictability of
relevant series under each heading111We provide a discussion of the Jurado et
al., (2015) uncertainty indexes within the supplementary material. The
interested reader is directed to the Jurado et al., (2015) paper, and
associated supplementary material, for full details on index construction.. Of
these, the MAC index sees most adoption in asset pricing (Bali et al.,, 2017,
2021). FIN has a natural association with the topological norms we derive from
the financial markets, as our measure stems from the returns on four major US
stock indices. This paper therefore focuses primarily on FIN and MAC.
Regression of persistence norms on FIN yields an insignificant constant term,
suggesting that FIN can span persistence norms. Further, adding volatility
does not remove statistical significance from FIN. MAC is only significantly
linked to volatility. Conversely, neither norms nor volatility span
uncertainty222There are many combinations of uncertainty and model parameters
that yield qualitatively similar results that are not reported in this paper
for brevity. All results, including those for REA, for horizons of 3 months
and 12 months, and for the economic elements of the uncertainty indexes, are
available on request from the authors..

This paper focuses on contemporaneous relationships between volatility,
uncertainty and persistence norms in recognition of the way these series are
used in the asset pricing literature (Ang et al.,, 2006; Bali et al.,, 2017,
2021). Our empirical contribution is the demonstration that norms are better
explained by FIN than volatility, but that norms and volatility alone do not
directly explain uncertainty. In evaluating the links between Jurado et al.,
(2015) and persistence norms for the first time, this paper furthers the
understanding of how TDA can enhance financial modelling333Code to allow the
recreation of results in this paper is available from the website of Simon
Rudkin at https://sites.google.com/view/simonrudkin/tda-research.

##  2 Persistence Norms

Consider a point cloud $X=\\{x_{1},\ldots,x_{n}\\}$ where each point
$\\{x_{i}\\}_{i=1}^{k}$ have $k$ coordinates. Given $X$, an _abstract
simplicial complex_ $\mathcal{K}$, being a family of subsets of $X$, will be
constructed. Each subset consisting of points $x_{l_{0}},\ldots,x_{l_{k}}$
will constitute a $k-$dimensional simplex. Such a simplex is equipped with a
_filtration value_ being equal to maximal distance between each pair of points
among $\\{x_{l_{0}},\ldots,x_{l_{k}}\\}$. This distance will be referred to as
a _filtration of the simplex_.

Clearly, when each subsets of $X$ constitute a simplex, the size of the
obtained data structure is $2^{n}$ and therefore not tractable
computationally. Consequently, in what follows, only the simplices of a
filtration value bounded by a predefined constant $r$ (and often of a bounded
cardinality) will be considered in a restricted simplicial complex
$\mathcal{K}_{r}$. Note that when a simplex constituted in points
$\\{x_{l_{0}},\ldots,x_{l_{k}}\\}$ belongs to $\mathcal{K}_{r}$, each simplex
supported in a subset of $\\{x_{l_{0}},\ldots,x_{l_{k}}\\}$ will also belong
to $\mathcal{K}_{r}$.

An obtained simplicial complex $\mathcal{K}_{r}$ is called a _Rips complex_ of
$X$. Let us consider the simplices of $\mathcal{K}_{r}$ in order of their
filtration. Concentrating on connected components, we can observe creation of
new connected components, and merging of existing ones as the simplices of
higher and higher filtration values are considered. The filtration value on
which the components connect will be called _birth_ , while the filtration
value at which the connected components connect to another component is the
_death_ of the original connected component. However, the multi-scale approach
presented here is not restricted to connected components. When considering
higher dimensional topological features like holes and voids, analogously each
feature can be equipped with a birth and a death filtration value. Note that
some of the features will not die in $\mathcal{K}_{r}$, in which case, they
are said to have an infinite length. This assignment is a basis of _persistent
homology_. For more formal, yet intuitive introduction to the subject please
consider Carlsson, (2009)

The life-time of a feature is calculated as the death minus the birth value.
Note that in what follows, we only consider features of a finite lifetime.
Life information can be combined into $L_{1}$ being the sum of the lives of
the features, whilst the $L_{2}$ norm is the square root of the sum of the
squared lives of the features. That is:

$\displaystyle L_{11}=\sum_{f=1}^{F_{0}}d_{f}-b_{f}$ (1) $\displaystyle
L_{12}=\sqrt{\sum_{f=1}^{F_{1}}(d_{f}-b_{f})^{2}}$ (2)

where feature $f$ is born at radius $b_{f}$ and dies at radius $d_{f}$. For
dimension as $z$, $z=0,1$, $F_{z}$ is the total number of features identified.
Note that so called 1-persistent landscape norms used in Gidea and Katz,
(2018) are simply $L_{11}$ up to a constant re-scaling, hence we will focus on
them in the further work.

Influences on the norm are the distances between points within the cloud and
the number of points in the cloud. For comparability all of the clouds in this
paper contain the similar numbers of points meaning any difference may be
attributed to the distance between points. Distances are impacted by the
correlation between axes and the variance within the underlying distributions
that generate the cloud. Volatility and the correlation between market index
returns capture those distributional features in this paper.

##  3 Data

Monthly uncertainty data is obtained from the website of Sydney
Ludvigsson444Downloaded 12th September 2021 from
https://www.sydneyludvigson.com/data-and-appendixes.. Daily index data is
downloaded through Yahoo! Finance with daily log returns for day $t$, $R_{t}$,
constructed from adjusted closing prices, $p_{t}$, using
$R_{t}=\ln{p_{t}}-\ln{p_{t-1}}$. Monthly returns may then be aggregated across
all days in the month. Using the daily data we construct point clouds of
length 1 month, 3 months, 6 months and 12 months and compute the persistence
norms there upon. Following Gidea and Katz, (2018) a cloud is produced using
log returns on the S&P 500, Dow Jones Industrial Average, NASDAQ and Russell
2000. The geometric mean of the standard deviation of returns on the four
indices provides the measure of volatility555The shape of the point cloud is
dependent on the variance-covariance matrix of the input series but is not
order dependent. Further each index is weighted equally in the construction of
the persistence norms. Therefore the average volatility is seen as a
parsimonious measure to apply in this paper. See Aromi et al., (2021) for a
discussion of the role of the variance-covariance matrix and empirical
examples using the four index cloud considered here.. Allowing for
availability of stock index returns and uncertainty measures, our sample runs
from 1st January 1993 to 28th June 2021.

Table 1: Correlations between index return volatility, persistence norms and
uncertainty

| $FIN1$ | $MAC1$ | $REA1$ | $\bar{\sigma}$ | $L_{11}$ | $L_{12}$ |  | $FIN1$ | $MAC1$ | $REA1$ | $\bar{\sigma}$ | $L_{11}$ | $L_{12}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Panel (a) 1 Month Cloud: |  | Panel (b) 3 Month Cloud: |
$\bar{\sigma}$ | 0.743 | 0.585 | 0.413 | 1 |  |  |  | 0.804 | 0.683 | 0.515 | 1 |  |
$L_{11}$ | 0.147 | 0.100 | 0.167 | 0.059 | 1 |  |  | 0.532 | 0.287 | 0.247 | 0.465 | 1 |
$L_{12}$ | 0.185 | 0.117 | 0.174 | 0.113 | 0.967 | 1 |  | 0.602 | 0.379 | 0.319 | 0.586 | 0.940 | 1
| Panel (c) 6 Month Cloud: |  | Panel (d) 12 Month Cloud:
$\bar{\sigma}$ | 0.814 | 0.721 | 0.560 | 1 |  |  |  | 0.789 | 0.697 | 0.547 | 1 |  |
$L_{11}$ | 0.592 | 0.304 | 0.240 | 0.551 | 1 |  |  | 0.597 | 0.288 | 0.188 | 0.617 | 1 |
$L_{12}$ | 0.663 | 0.403 | 0.318 | 0.653 | 0.964 | 1 |  | 0.665 | 0.381 | 0.264 | 0.715 | 0.982 | 1

Notes: Figures report correlations between stated row variables and columns.
$FIN1$, $MAC1$ and $REA1$ are the 1 month ahead financial, macroeconomic and
real uncertainty index measures from Jurado et al., (2015) as downloaded from
the website of Sydney Ludvigsson at https://www.sydneyludvigson.com/data-and-
appendixes. Point clouds are constructed from the daily returns on four stock
indices, the S&P 500, NASDAQ, the Dow Jones Industrial Average and the Russell
2000. Cloud lengths are indicated as the panel names. $\bar{\sigma}$ is the
average volatility from the four stock indices. $L_{11}$ and $L_{12}$ are the
dimension 1 $L_{1}$ and $L_{2}$ persistence norms respectively. Pairwise
correlations within the Jurado et al., (2015) uncertainty index rows are
omitted as they are not affected by cloud length. Sample from 1st January 1993
to 28th June 2021.

Correlations between the Jurado et al., (2015) uncertainty indexes and
volatility are high, being approximately 0.7 for $FIN1$, around 0.6 for $MAC1$
and approximately 0.6 for $REAL$. Correlations between volatility and the
uncertainty indexes are always higher than the corresponding correlations
between uncertainty and the persistence norms. Table 1 shows correlations
between $L_{1}$ norms and $FIN1$ to be below 0.6 in the 3-month and 6-month
clouds, rising to near 1 for the 12-month cloud666We present a discussion of
the correlation between measures of volatility and the persistence norms
across the cloud lengths in the supplementary material.. $L_{2}$ norms have a
correlation of just above 0.6 with $FIN1$. With $MAC1$ we see correlations
below 0.4, and with $REA1$ the correlations are lower at 0.3 or less. Whilst
the evidence suggests uncertainty is picking up much of the volatility in the
cloud, the norms are providing different information about the data. A note of
caution is raised by the strong correlation between volatility and $FIN1$ when
using these two metrics to model persistence norms. When $MAC1$ is used the
correlations with volatility are lower meaning fewer modelling concerns.

Figure 1: Persistence norms, volatility and uncertainty time series plots

|
---|---
(a) $FIN1$ 3 months | (b) $FIN1$ 12 months
|
(a) $MAC1$ 3 months | (b) $MAC1$ 12 months

Notes: Figures plot the financial (FIN) and macroeconomic (MAC) uncertainty
index of Jurado et al., (2015). Each plot uses the 1 month ahead uncertainty
index $FIN1$ and $MAC1$. These indices are plotted as blue lines. On each plot
the average return volatility of the four indexes within the cloud; being the
S&P500, Dow Jones Industrial Average, NASDAQ and the Russell 2000 index.
Average volatility is plotted in black. The dimension 1 $L_{2}$ persistence
norm from these clouds are shown in red. Values are standardised by
subtracting the mean value and dividing by the standard deviation. Size of
window used in the estimation is provided below each plot. Sample from 1st
January 1993 to 28th June 2021.

Plotting the uncertainty, volatility and $L_{1}$ persistence norms series,
Figure 1 shows that whilst using more lags, or considering longer time periods
produces more consistency, these are still volatile series. Choice of $L_{1}$
follows Gidea and Katz, (2018), but the plots with $L_{2}$ included in the
supplementary material are very similar as the high correlation between
$L_{1}$ and $L_{2}$ would suggest. Across the full sample it is difficult to
conclude which series can be used to explain the others777Developing lead-lag
relationships within this data is left to future work, with greater focus
needed on TDA measures from the same series as the uncertainty measures..
Panels (a) and (b) show the persistence norms being high during the “dot-com”
bubble bursting in the early part of this century, a time where uncertainty is
high but volatility is not. Meanwhile for both the Covid-19 pandemic and the
global financial crisis we see volatility and uncertainty go to high
standardised values whilst norms remain comparatively low. Panels (c) and (d)
show closer alignment between $MAC1$ and volatility since $MAC1$ does not peak
in the “dot-com” bubble like $FIN1$. These observations are important for
evaluating the use of uncertainty, and particularly $MAC1$ in the asset
pricing literature (Bali et al.,, 2017, 2021). Consequently the time series
plots show potential for $L_{1}$ norms in financial uncertainty.

##  4 Results

Our results are presented for the dimension one $L_{1}$ norm to keep this work
in line with Gidea and Katz, (2018). Results for the corresponding dimension 1
$L_{2}$ norms are provided in the supplementary material.

###  4.1 Persistence Norms

Regression of persistence norms on the uncertainty indices of Jurado et al.,
(2015) and the average volatility index returns tests whether these financial
market variables are able to span the persistence norms. Formally we estimate
a total of 216 models which are based upon three base regressions888For each
measure of uncertainty there are 4 point clouds to consider. Three uncertainty
indices at three different lags, and with both a standard and economic version
for each of the 9, this all means 72 combinations arise from Model 1. Note
that there can only be one of the 18 possible uncertainty indexes and one of
the four cloud lengths used in any one regression. Likewise we can only use
one of the two norms, either $L_{1}$ or $L_{2}$ in a regression. Each model
therefore comprises one uncertainty index, one norm and/or one volatility.:

$\displaystyle\mathtt{Model1:}$ $\displaystyle
L_{11}=\alpha_{1}+\beta_{11}FIN1+\epsilon$ (3) $\displaystyle\mathtt{Model2:}$
$\displaystyle L_{11}=\alpha_{2}+\beta_{21}\bar{\sigma}+\epsilon$ (4)
$\displaystyle\mathtt{Model3:}$ $\displaystyle
L_{11}=\alpha_{3}+\beta_{31}FIN1+\beta_{32}\sigma+\epsilon$ (5)

Where $L_{11}$ is the dimension one $L_{1}$ persistence norm, $\sigma$ is the
standard deviation of S&P 500 returns and $FIN1$ is the one month ahead
financial uncertainty index of Jurado et al., (2015). In all cases $\epsilon$
is the error term. In all estimations Newey and West, (1987) adjusted standard
errors are employed to account for autocorrelation and heteroskedasticity
inherent in financial time series.

Table 2: Determinants of $L_{1}$ persistence norms

|  |  | Model 1 | Model 2 | Model 3
---|---|---|---|---|---
|  |  | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m
Panel (a): Financial Uncertainty ($FIN1$): |  |  |
| Constant | $\alpha$ | 0.031 | -0.308 | -0.530 | -0.354 | 0.133 | 0.511 | 1.171 | 2.365 | -0.005 | -0.206 | -0.151 | 0.727
|  |  | (0.574) | (1.270) | (0.674) | (0.150) | (7.680) | (6.103) | (3.780) | (2.525) | (0.083) | (0.659) | (0.225) | (0.484)
| Uncertainty | $FIN1$ | 0.132 | 1.385 | 3.156 | 5.992 |  |  |  |  | 0.206 | 1.163 | 2.262 | 2.926
|  |  | (2.012) | (4.504) | (3.091) | (1.830) |  |  |  |  | (2.501) | (2.453) | (2.162) | (1.243)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.015 | 0.378 | 0.991 | 2.281 | -0.029 | 0.086 | 0.370 | 1.431
|  |  |  |  |  |  | (0.952) | (4.275) | (2.961) | (2.069) | (1.351) | (0.885) | (1.558) | (2.756)
| Adjusted $R^{2}$ | 0.022 | 0.283 | 0.350 | 0.356 | 0.004 | 0.216 | 0.304 | 0.381 | 0.027 | 0.287 | 0.364 | 0.413
Panel (b): Macroeconomic Uncertainty ($MAC1$): |  |  |  |  |  |
| Constant | $\alpha$ | 0.058 | 0.174 | 0.654 | 2.083 | 0.133 | 0.511 | 1.171 | 2.365 | 0.058 | 0.629 | 1.948 | 4.378
|  |  | (0.702) | (0.907) | (0.801) | (0.684) | (7.680) | (6.103) | (3.780) | (2.525) | (0.608) | (1.527) | (1.516) | (2.616)
| Uncertainty | $MAC1$ | 0.143 | 1.190 | 2.580 | 4.601 |  |  |  |  | 0.141 | -0.238 | -1.660 | -4.424
|  |  | (1.053) | (3.969) | (1.958) | (0.912) |  |  |  |  | (0.764) | (0.271) | (0.553) | (1.007)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.015 | 0.378 | 0.991 | 2.281 | 0.000 | 0.410 | 1.244 | 2.995
|  |  |  |  |  |  | (0.952) | (4.275) | (2.961) | (2.069) | (0.017) | (2.512) | (1.907) | (1.868)
| Adjusted $R^{2}$ | 0.010 | 0.082 | 0.092 | 0.083 | 0.004 | 0.216 | 0.304 | 0.381 | 0.010 | 0.218 | 0.322 | 0.420

Notes: Regressions of the stated outcome with the other two variables as
explanatory factors. 3m and 6m refers to the number of months of data in the
point cloud over which the persistence norms and volatility are calculated.
Point clouds are the daily returns on the S&P 500, Dow Jones Industrial
Average, NASDAQ and Russell 2000 indexes. Uncertainty indexes are the 1 month
ahead estimates from Jurado et al., (2015) as downloaded from the website of
Sydney Ludvigson. Figures are the estimated coefficients and figures in
parentheses are the associated Newey and West, (1987) adjusted t-statistics.
Model 1 has $L_{11}=\alpha_{1}+\beta_{11}FIN1+\epsilon$. Model 1 has
$L_{11}=\alpha_{2}+\beta_{21}\bar{\sigma}+\epsilon$. Model 3 has
$L_{2}=\alpha_{3}+\beta_{31}FIN1+\beta_{32}\bar{\sigma}+\epsilon$. Models 1, 2
and 3 represent the base specification upon which all regression models are
built. Specifications show $FIN1$ as the one month ahead financial uncertainty
(FIN) index. Where macroeconomic uncertainty (MAC) is used then the equations
for the four models are duly updated to replace $FIN1$ with $MAC1$. In each
case $\bar{\sigma}$ is the average standard deviation of the four index
returns in the window and $\epsilon$ is the error term. Estimation employs
Newey and West, (1987) robust standard errors to control for autocorrelation
and heteroskedasticity inherent in these time series. Sample from 1st January
1993 to 28th June 2021. $n=342$

An immediate inference from Table 3 is that regressions involving the
uncertainty indexes do not have significant constant terms. Further,
coefficients on FIN1 are significant in all but model 3 on the 12-month cloud.
Meanwhile, volatility is not significant when uncertainty is added. The
significant constants in Model 2 confirm that the norm of the four index cloud
is picking up additional information beyond the volatility in the cloud. When
MAC1 is used as the uncertainty measure, there is some significance in model 1
for the 3-month and 6-month clouds, but model 3 has no significant
coefficients.

###  4.2 Uncertainty

Reversing the question we regress the uncertainty indexes on the persistence
norms and volatility.

$\displaystyle\mathtt{Model4:}$ $\displaystyle
FIN1=\alpha_{4}+\beta_{41}L_{11}+\epsilon$ (6) $\displaystyle\mathtt{Model5:}$
$\displaystyle FIN1=\alpha_{5}+\beta_{52}\bar{\sigma}+\epsilon$ (7)
$\displaystyle\mathtt{Model6:}$ $\displaystyle
FIN1=\alpha_{6}+\beta_{61}L_{11}+\beta_{62}\bar{\sigma}+\epsilon$ (8)

Notation follows models 1 to 3.

Table 3: Explaining the uncertainty index

|  |  | Model 4 | Model 5 | Model 6
---|---|---|---|---|---
|  |  | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m
Panel (a): Financial Uncertainty ($FIN1$): |  |  |
| Constant | $\alpha$ | 0.877 | 0.710 | 0.645 | 0.602 | 0.667 | 0.617 | 0.584 | 0.560 | 0.651 | 0.577 | 0.539 | 0.518
|  |  | (10.76) | (17.72) | (11.39) | (6.192) | (20.77) | (16.46) | (15.70) | (8.128) | (20.94) | (16.51) | (14.95) | (6.359)
| Norms | $L_{11}$ | 0.163 | 0.204 | 0.111 | 0.059 |  |  |  |  | 0.115 | 0.078 | 0.038 | 0.018
|  |  | (2.422) | (8.155) | (6.500) | (3.641) |  |  |  |  | (2.445) | (3.431) | (2.149) | (1.001)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.213 | 0.251 | 0.274 | 0.290 | 0.211 | 0.222 | 0.236 | 0.250
|  |  |  |  |  |  | (11.02) | (10.50) | (7.333) | (3.727) | (10.91) | (10.80) | (6.070) | (2.800)
| Adjusted $R^{2}$ | 0.022 | 0.283 | 0.350 | 0.356 | 0.552 | 0.646 | 0.663 | 0.622 | 0.563 | 0.678 | 0.693 | 0.642
Panel (b): Macroeconomic Uncertainty ($MAC1$): |  |  |  |  |  |
| Constant | $\alpha$ | 0.634 | 0.580 | 0.562 | 0.554 | 0.528 | 0.492 | 0.468 | 0.455 | 0.522 | 0.497 | 0.487 | 0.489
|  |  | (8.488) | (25.77) | (17.88) | (10.97) | (31.34) | (14.50) | (9.186) | (7.774) | (29.43) | (13.01) | (9.998) | (9.931)
| Norms | $L_{11}$ | 0.070 | 0.069 | 0.036 | 0.018 |  |  |  |  | 0.046 | -0.009 | -0.016 | -0.014
|  |  | (0.618) | (1.041) | (1.260) | (1.096) |  |  |  |  | (0.725) | (0.274) | (0.649) | (0.981)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.105 | 0.134 | 0.153 | 0.161 | 0.105 | 0.138 | 0.168 | 0.194
|  |  |  |  |  |  | (5.057) | (2.614) | (2.311) | (2.268) | (4.908) | (3.274) | (2.798) | (1.932)
| Adjusted $R^{2}$ | 0.010 | 0.082 | 0.092 | 0.083 | 0.342 | 0.467 | 0.520 | 0.486 | 0.346 | 0.468 | 0.533 | 0.519

Notes: Regressions of the stated outcome with the other two variables as
explanatory factors. 1m, 3m, 6m and 12m refers to the number of months of data
in the point cloud over which the persistence norms and volatility are
calculated. Uncertainty indexes are the 1 month ahead estimates from Jurado et
al., (2015) as downloaded from the website of Sydney Ludvigson. Clouds are
formed from the daily log returns on the S&P 500, Dow Jones Industrial
Average, NASDAQ and Russell 2000 stock market indexes.. Figures are the
estimated coefficients and figures in parentheses are the associated Newey and
West, (1987) adjusted t-statistics. Model 4 has
$FIN1=\alpha_{4}+\beta_{41}L_{12}+\epsilon$. Model 5 has
$FIN1=\alpha_{5}+\beta_{51}\bar{\sigma}+\epsilon$. Model 6 has
$FIN1=\alpha_{6}+\beta_{61}L_{12}+\beta_{62}\bar{\sigma}+\epsilon$.
Specifications show $FIN1$ as the one month ahead financial uncertainty (FIN)
index. Where macroeconomic uncertainty (MAC) is used then the equations for
the four models are duly updated to replace $FIN1$ with $MAC1$. In each case
$\bar{\sigma}$ is the average standard deviation of the returns on the four
indices in the window. $\epsilon$ is the error term. Estimation employs Newey
and West, (1987) robust standard errors to control for autocorrelation and
heteroskedasticity inherent in these time series. Sample from 1st January 1993
to 28th June 2021. $n=342$

Table 3 demonstrates the significance of volatility to the uncertainty index.
$FIN1$ has significant terms on the $L_{11}$ norms in both Model 4 and Model
6. Persistence norms are also picking up an element of uncertainty in the
market over and above that induced by volatility. For $MAC1$ any fit benefit
of adding $L_{1}$ norms, model 6, to the univariate model with just
volatility, model 5, is marginal. In all cases the constants are significant
meaning that there are other features in the uncertainty index that the point
cloud measures do not pick up.

###  4.3 High and Low Uncertainty

A feature of the uncertainty indexes of Jurado et al., (2015) is that they
have very clear peaks. In the case of $FIN1$, Figure 1 showed that there were
three peaks within our sample. $MAC1$ has peaks in the global financial crisis
and now in the Covid-19 period. Therefore much of the variation in these
indexes can be explained by factors that are effective at explaining the
peaks. To explore this further we split the sample into those months where
each uncertainty index is above, or below, its respective median. The 342
months of our sample therefore move into two sets of 171.

Table 4: High and low uncertainty index periods

|  |  | Model 4 | Model 5 | Model 6
---|---|---|---|---|---
|  |  | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m
Panel (a): High Financial Uncertainty ($FIN1$): |  |  |
| Constant | $\alpha$ | 1.054 | 0.963 | 0.943 | 0.945 | 0.877 | 0.836 | 0.821 | 0.822 | 0.867 | 0.808 | 0.792 | 0.806
|  |  | (10.90) | (16.51) | (12.70) | (8.781) | (25.29) | (20.84) | (18.08) | (9.144) | (24.97) | (15.53) | (13.75) | (7.132)
| Norms | $L_{11}$ | 0.044 | 0.084 | 0.042 | 0.019 |  |  |  |  | 0.056 | 0.034 | 0.016 | 0.005
|  |  | (0.516) | (3.754) | (2.633) | (1.176) |  |  |  |  | (1.195) | (1.234) | (0.879) | (0.283)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.128 | 0.151 | 0.158 | 0.157 | 0.128 | 0.143 | 0.148 | 0.149
|  |  |  |  |  |  | (7.316) | (7.887) | (4.930) | (2.144) | (7.230) | (7.106) | (4.141) | (1.740)
|  Adjusted $R^{2}$ | 0.004 | 0.116 | 0.114 | 0.086 | 0.493 | 0.528 | 0.442 | 0.313 | 0.500 | 0.545 | 0.457 | 0.317
Panel (b): Low Financial Uncertainty ($FIN1$): |  |  |
| Constant | $\alpha$ | 0.741 | 0.739 | 0.694 | 0.643 | 0.678 | 0.625 | 0.579 | 0.62 | 0.678 | 0.627 | 0.564 | 0.575
|  |  | (17.05) | (14.28) | (12.44) | (7.811) | (28.10) | (18.77) | (25.68) | (18.85) | (27.95) | (19.42) | (18.90) | (7.302)
| Norms | $L_{11}$ | 0.007 | 0.003 | 0.027 | 0.025 |  |  |  |  | -0.004 | -0.003 | 0.011 | 0.013
|  |  | (0.212) | (0.130) | (1.142) | (1.261) |  |  |  |  | (0.119) | (0.173) | (0.600) | (0.651)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.084 | 0.151 | 0.207 | 0.148 | 0.084 | 0.151 | 0.201 | 0.137
|  |  |  |  |  |  | (3.626) | (4.062) | (8.092) | (4.490) | (3.631) | (4.457) | (7.064) | (3.648)
|  Adjusted $R^{2}$ | 0.000 | 0.000 | 0.046 | 0.092 | 0.142 | 0.257 | 0.392 | 0.348 | 0.142 | 0.257 | 0.399 | 0.374
Panel (c): High Macroeconomic Uncertainty ($MAC1$): |  |  |  |  |  |
| Constant | $\alpha$ | 0.703 | 0.657 | 0.647 | 0.648 | 0.598 | 0.549 | 0.515 | 0.507 | 0.582 | 0.552 | 0.537 | 0.542
|  |  | (8.618) | (23.02) | (16.26) | (10.47) | (24.77) | (16.95) | (11.38) | (7.552) | (28.05) | (14.31) | (13.17) | (9.276)
| Norms | $L_{11}$ | 0.098 | 0.060 | 0.028 | 0.013 |  |  |  |  | 0.103 | -0.006 | -0.018 | -0.016
|  |  | (1.191) | (0.77) | (0.856) | (0.746) |  |  |  |  | (1.935) | (0.119) | (0.652) | (0.891)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.089 | 0.122 | 0.146 | 0.149 | 0.090 | 0.124 | 0.163 | 0.184
|  |  |  |  |  |  | (4.158) | (3.217) | (2.921) | (2.042) | (3.991) | (3.670) | (3.362) | (1.688)
|  Adjusted $R^{2}$ | 0.021 | 0.069 | 0.062 | 0.045 | 0.281 | 0.428 | 0.504 | 0.407 | 0.304 | 0.428 | 0.523 | 0.450
Panel (d): Low Macroeconomic Uncertainty ($MAC1$): |  |  |  |  |  |
| Constant | $\alpha$ | 0.569 | 0.557 | 0.554 | 0.556 | 0.553 | 0.547 | 0.543 | 0.537 | 0.553 | 0.545 | 0.542 | 0.541
|  |  | (54.13) | (39.02) | (26.33) | (28.50) | (57.69) | (40.03) | (40.67) | (30.85) | (57.10) | (34.82) | (31.26) | (32.77)
| Norms | $L_{11}$ | 0.019 | 0.017 | 0.008 | 0.003 |  |  |  |  | 0.002 | 0.005 | 0.001 | -0.002
|  |  | (1.199) | (2.176) | (1.465) | (1.291) |  |  |  |  | (0.189) | (0.653) | (0.148) | (0.453)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.025 | 0.031 | 0.031 | 0.034 | 0.025 | 0.028 | 0.028 | 0.033
|  |  |  |  |  |  | (3.787) | (3.249) | (3.57) | (2.116) | (3.548) | (2.867) | (2.879) | (1.350)
|  Adjusted $R^{2}$ | 0.015 | 0.079 | 0.078 | 0.048 | 0.135 | 0.191 | 0.222 | 0.260 | 0.136 | 0.196 | 0.223 | 0.272

Notes: Regressions of the stated outcome with the other two variables as
explanatory factors. 1m, 3m, 6m and 12m refers to the number of months of data
in the point cloud over which the persistence norms and volatility are
calculated. Panels (a) and (c) cover periods of high financial and
macroeconomic uncertainty respectively. Likewise panels (b) and (d) cover
periods where uncertainty is below the sample median. Uncertainty indexes are
the 1 month ahead estimates from Jurado et al., (2015) as downloaded from the
website of Sydney Ludvigson. Clouds are formed from the daily log returns on
the S&P 500, Dow Jones Industrial Average, NASDAQ and Russell 2000 stock
market indexes. Figures are the estimated coefficients and figures in
parentheses are the associated Newey and West, (1987) adjusted t-statistics.
Model 4 has $FIN1=\alpha_{4}+\beta_{41}L_{11}+\epsilon$. Model 5 has
$FIN1=\alpha_{5}+\beta_{51}\bar{\sigma}+\epsilon$. Model 6 has
$FIN1=\alpha_{6}+\beta_{61}L_{11}+\beta_{62}\bar{\sigma}+\epsilon$.
Specifications show $FIN1$ as the one month ahead financial uncertainty (FIN)
index. Where macroeconomic uncertainty (MAC) is used then the equations for
the four models are duly updated to replace $FIN1$ with $MAC1$. In each case
$\bar{\sigma}$ is the average standard deviation of the returns on the four
indices in the window. $\epsilon$ is the error term. Estimation employs Newey
and West, (1987) robust standard errors to control for autocorrelation and
heteroskedasticity inherent in these time series. The number of observations
in each high and low period is 171. Overall sample from 1st January 1993 to
28th June 2021.

Results in Table 4 derive from the estimation of models 4 to 6 on high, and
low, uncertainty subsamples. In all cases the constant is highly significant
for both FIN and MAC. Persistence norms are significant in high $FIN1$ periods
when the 3-month and 6-month clouds, but this disappears after the addition of
volatility into the model. Volatility is significant in the regressions of
both $FIN1$ and $MAC1$, the significance robust to the addition of persistence
norms to the model. Hence, whilst there is evidence of norms adding power in
explaining uncertainty for the full sample, the split at the median shows the
power to be insufficient in either high or low uncertainty periods.

##  5 Summary

Taking financial markets as complex dynamical systems (Hsieh,, 1991),
persistence norms provide a means to evaluate phase transitions and forewarn
of crashes (Gidea and Katz,, 2018). An intuitive link to uncertainty in the
spirit of Jurado et al., (2015) is made in this paper. Through contemporaneous
regressions amongst the norms, volatility and uncertainty it has been shown
that FIN, not volatility, has the strongest association with persistence
norms. That persistence norms have explanatory power for FIN, but cannot span
uncertainty, positions norms as a potentially useful further metric to add to
the asset pricing arsenal. Although links between MAC and volatility shown in
our results lend interpretation to the observed mispricing of MAC in Bali et
al., (2017), there is only limited significance for persistence norms. Further
development of the norms and volatility from the same dataset as the
uncertainty indexes is a natural next step. Given the demonstrated roles of
uncertainty and volatility in asset pricing, and the fact that volatility
cannot explain the persistence norms, there is strong potential to also
explore TDA signals in asset pricing.

## References

  * Altig et al., (2020) Altig, D., Baker, S., Barrero, J. M., Bloom, N., Bunn, P., Chen, S., Davis, S. J., Leather, J., Meyer, B., Mihaylov, E., et al. (2020).  Economic uncertainty before and during the covid-19 pandemic.  Journal of Public Economics, 191:104274.
  * Ang et al., (2006) Ang, A., Hodrick, R. J., Xing, Y., and Zhang, X. (2006).  The cross-section of volatility and expected returns.  The Journal of Finance, 61(1):259–299.
  * Aromi et al., (2021) Aromi, L. L., Katz, Y. A., and Vives, J. (2021).  Topological features of multivariate distributions: dependency on the covariance matrix.  Communications in Nonlinear Science and Numerical Simulation, page 105996.
  * Bakas and Triantafyllou, (2020) Bakas, D. and Triantafyllou, A. (2020).  Commodity price volatility and the economic uncertainty of pandemics.  Economics Letters, 193:109283.
  * Baker et al., (2016) Baker, S. R., Bloom, N., and Davis, S. J. (2016).  Measuring economic policy uncertainty.  The quarterly journal of economics, 131(4):1593–1636.
  * Bali et al., (2017) Bali, T. G., Brown, S. J., and Tang, Y. (2017).  Is economic uncertainty priced in the cross-section of stock returns?  Journal of Financial Economics, 126(3):471–489.
  * Bali et al., (2021) Bali, T. G., Subrahmanyam, A., and Wen, Q. (2021).  The macroeconomic uncertainty premium in the corporate bond market.  Journal of Financial and Quantitative Analysis, 56(5):1653–1678.
  * Bubenik, (2020) Bubenik, P. (2020).  The persistence landscape and some of its properties.  In Topological Data Analysis, pages 97–117. Springer.
  * Bubenik and Dłotko, (2017) Bubenik, P. and Dłotko, P. (2017).  A persistence landscapes toolbox for topological statistics.  Journal of Symbolic Computation, 78:91–114.
  * Carlsson, (2009) Carlsson, G. (2009).  Topology and data.  Bulletin of the American Mathematical Society, 46(2):255–308.
  * Gidea et al., (2020) Gidea, M., Goldsmith, D., Katz, Y., Roldan, P., and Shmalo, Y. (2020).  Topological recognition of critical transitions in time series of cryptocurrencies.  Physica A: Statistical Mechanics and its Applications, 548:123843.
  * Gidea and Katz, (2018) Gidea, M. and Katz, Y. (2018).  Topological data analysis of financial time series: Landscapes of crashes.  Physica A: Statistical Mechanics and its Applications, 491:820–834.
  * Goel et al., (2020) Goel, A., Pasricha, P., and Mehra, A. (2020).  Topological data analysis in investment decisions.  Expert Systems with Applications, 147:113222.
  * Hsieh, (1991) Hsieh, D. A. (1991).  Chaos and nonlinear dynamics: application to financial markets.  The journal of finance, 46(5):1839–1877.
  * Jurado et al., (2015) Jurado, K., Ludvigson, S. C., and Ng, S. (2015).  Measuring uncertainty.  American Economic Review, 105(3):1177–1216.
  * Katz and Biem, (2021) Katz, Y. A. and Biem, A. (2021).  Time-resolved topological data analysis of market instabilities.  Physica A: Statistical Mechanics and its Applications, 571:125816.
  * Leaverton et al., (2020) Leaverton, L., Katz, Y. A., and Vives, J. (2020).  Some statistical properties of functional norms of persistence topological landscapes.  Available at SSRN 3640139.
  * Lee et al., (2021) Lee, K., Joen, Y., and Kim, M. (2021).  Which uncertainty measures matter for the cross-section of stock returns?  Finance Research Letters, page 102390.
  * Majumdar and Laha, (2020) Majumdar, S. and Laha, A. K. (2020).  Clustering and classification of time series using topological data analysis with applications to finance.  Expert Systems with Applications, 162:113868.
  * Newey and West, (1987) Newey, W. K. and West, K. D. (1987).  A simple, positive semi-definite, heteroskedasticity and autocorrelation.  Econometrica, 55(3):703–708.
  * Smith et al., (2021) Smith, A. D., Dłotko, P., and Zavala, V. M. (2021).  Topological data analysis: concepts, computation, and applications in chemical engineering.  Computers & Chemical Engineering, 146:107202.
  * Stolz et al., (2017) Stolz, B. J., Harrington, H. A., and Porter, M. A. (2017).  Persistent homology of time-dependent functional networks constructed from coupled time series.  Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(4):047410.

##  Appendix A Measures of Uncertainty

Jurado et al., (2015) provide three categories of uncertainty index, financial
(FIN), macroeconomic (MAC) and real (REA), each based upon the predictability
of a different set of time series. On an individual series uncertainty may be
understood as the forecast error at a $h$-month horizon given information
available at time $t$. Recognising that the information set $I_{t}$ available
to investors is far wider than a single series it is necessary to use a large
set of series to fully measure uncertainty. Jurado et al., (2015)’s approach
seeks a common component of uncertainty to model from the observed data,
rather than being based upon gathering large volumes of internet or newspaper
data like the economic policy uncertainty index of Baker et al., (2016). From
the perspective of the main paper, the uncertainty indexes are taken as givens
but it remains helpful to recognise the debate that informs their
construction.

Details of the series and processes can be accessed via the paper and via the
website of Sydney Ludvigson. In most asset pricing applications it is the MAC
that is used, with the one-month ahead index (MAC1) being the choice for Bali
et al., (2017, 2021). However all three sets include a one-month, three-month
and twelve-month ahead index. There are also versions of these indices
computed for the subset of series which excludes any health related outcomes.
This is referred to in the data descriptions as “economic”. In the main paper
we focus on MAC1 in line with the asset pricing literature, but also use FIN1
as the indicator derived from the financial markets directly. Here we
demonstrate the tight relationships between the indexes, expanding the
demonstration in Bali et al., (2017) and the exposition in Lee et al., (2021).

Figure A1: Comparison of Uncertainty Measures

---
(a) 1-month ahead
|
(b) Financial Uncertainty (FIN) | (c) Financial Uncertainty (Economic)
|
(d) Macroecnomic Uncertainty (MAC) | (e) Macroeconomic Uncertainty (Economic)
|
(f) Real Uncertainty (REA) | (g) Real Uncertainty (Economic)

Notes: Figures plot uncertainty indexes of Jurado et al., (2015) as obtained
from the website of Sydney Ludvigson https://www.sydneyludvigson.com/macro-
and-financial-uncertainty-indexes. Uncertainty indexes based upon the
financial (FIN), macroeconomic (MAC) and real (REA) uncertainty are shown at
the 1, 3 and 12 month horizon. Panel (a) contrasts the one month ahead
uncertainty index as calculated from the three sets of time series. Panels
(b), (d) and (f) use all available time series, whilst (c), (e) and (g) use
only those classed as “economic”. Full details of construction are provided in
Jurado et al., (2015). Sample period is based upon the availability of data on
the Russell 2000 stock index and runs from 1st January 1993 to 30th June 2021.
All data is monthly.

Figure A1 presents comparisons between the three one-month ahead uncertainty
measures in panel (a), and then the three different time spans for each
uncertainty measure in panels (b) to (g). Immediately it may be seen that the
financial uncertainty measure remains the most volatile, as observed in Jurado
et al., (2015). We see that there are three spikes in the FIN series, 2 in the
MAC and just one in the REA. In every case the one-month ahead is the most
volatile, the 12-month ahead measure being far more persistent through the
time periods.

However, in the Covid-19 period we see that all three uncertainty measures
have very similar behaviour999A further useful commentary on recent
uncertainty is provided in Altig et al., (2020), with a discussion on the
differentials between the Jurado et al., (2015) and other approaches to
capturing uncertainty in the light of Covid-19.. To restrict the effect of
health related variables, the “Economic” series were created. Contrasting the
right hand plot on each row with the total uncertainty we see that there is
very little difference in the series. Following the literature we continue
therefore to use total uncertainty in the main paper. Results using the
economic measure are qualitatively similar and are available on request.

Table A1: Correlation of Uncertainty Measures

| Financial Uncertainty | Macroeconomic Uncertainty | Real Uncertainty
---|---|---|---
| Total | Economic | Total | Economic | Total | Economic
| 1 | 3 | 12 | 1 | 3 | 12 | 1 | 3 | 12 | 1 | 3 | 12 | 1 | 3 | 12 | 1 | 3 | 12
FIN1 | 1 | 0.999 | 0.988 | 0.994 | 0.995 | 0.987 | 0.553 | 0.559 | 0.548 | 0.555 | 0.561 | 0.55 | 0.437 | 0.446 | 0.420 | 0.465 | 0.467 | 0.43
FIN3 | 0.999 | 1 | 0.992 | 0.992 | 0.994 | 0.991 | 0.554 | 0.561 | 0.551 | 0.556 | 0.563 | 0.553 | 0.435 | 0.445 | 0.419 | 0.463 | 0.465 | 0.428
FIN12 | 0.980 | 0.989 | 1 | 0.976 | 0.982 | 0.998 | 0.559 | 0.567 | 0.564 | 0.560 | 0.567 | 0.565 | 0.425 | 0.437 | 0.413 | 0.451 | 0.455 | 0.421
EFIN1 | 0.993 | 0.990 | 0.965 | 1 | 0.999 | 0.982 | 0.528 | 0.534 | 0.523 | 0.530 | 0.536 | 0.525 | 0.408 | 0.418 | 0.390 | 0.437 | 0.439 | 0.400
EFIN3 | 0.994 | 0.994 | 0.977 | 0.998 | 1 | 0.987 | 0.532 | 0.538 | 0.528 | 0.534 | 0.54 | 0.530 | 0.410 | 0.420 | 0.392 | 0.438 | 0.441 | 0.402
EFIN12 | 0.98 | 0.988 | 0.996 | 0.975 | 0.986 | 1 | 0.546 | 0.554 | 0.551 | 0.547 | 0.554 | 0.552 | 0.410 | 0.422 | 0.396 | 0.435 | 0.439 | 0.405
MAC1 | 0.666 | 0.662 | 0.658 | 0.608 | 0.609 | 0.615 | 1 | 0.996 | 0.967 | 0.999 | 0.996 | 0.970 | 0.758 | 0.764 | 0.734 | 0.754 | 0.765 | 0.742
MAC3 | 0.628 | 0.632 | 0.651 | 0.562 | 0.570 | 0.602 | 0.978 | 1 | 0.982 | 0.996 | 0.999 | 0.984 | 0.741 | 0.758 | 0.741 | 0.738 | 0.759 | 0.748
MAC12 | 0.517 | 0.531 | 0.585 | 0.438 | 0.459 | 0.529 | 0.872 | 0.954 | 1 | 0.966 | 0.981 | 0.999 | 0.688 | 0.720 | 0.732 | 0.683 | 0.719 | 0.734
EMAC1 | 0.671 | 0.669 | 0.671 | 0.613 | 0.616 | 0.628 | 0.992 | 0.986 | 0.898 | 1 | 0.996 | 0.969 | 0.758 | 0.765 | 0.735 | 0.758 | 0.768 | 0.744
EMAC3 | 0.639 | 0.642 | 0.660 | 0.574 | 0.583 | 0.613 | 0.974 | 0.998 | 0.950 | 0.99 | 1 | 0.983 | 0.744 | 0.761 | 0.744 | 0.743 | 0.764 | 0.752
EMAC12 | 0.549 | 0.561 | 0.609 | 0.473 | 0.492 | 0.556 | 0.895 | 0.968 | 0.998 | 0.921 | 0.967 | 1 | 0.689 | 0.719 | 0.729 | 0.686 | 0.720 | 0.733
REA1 | 0.520 | 0.524 | 0.545 | 0.433 | 0.443 | 0.480 | 0.897 | 0.924 | 0.909 | 0.887 | 0.906 | 0.905 | 1 | 0.985 | 0.913 | 0.995 | 0.982 | 0.922
REA3 | 0.446 | 0.458 | 0.509 | 0.354 | 0.374 | 0.442 | 0.819 | 0.902 | 0.961 | 0.834 | 0.889 | 0.947 | 0.961 | 1 | 0.959 | 0.982 | 0.995 | 0.964
REA12 | 0.329 | 0.349 | 0.424 | 0.237 | 0.265 | 0.359 | 0.686 | 0.812 | 0.939 | 0.717 | 0.802 | 0.916 | 0.865 | 0.969 | 1 | 0.911 | 0.954 | 0.997
EREA1 | 0.518 | 0.524 | 0.554 | 0.429 | 0.442 | 0.488 | 0.877 | 0.927 | 0.937 | 0.886 | 0.917 | 0.933 | 0.988 | 0.981 | 0.908 | 1 | 0.986 | 0.925
EREA3 | 0.461 | 0.473 | 0.521 | 0.370 | 0.389 | 0.455 | 0.823 | 0.906 | 0.962 | 0.843 | 0.896 | 0.951 | 0.959 | 0.998 | 0.966 | 0.985 | 1 | 0.965
EREA12 | 0.350 | 0.370 | 0.441 | 0.259 | 0.286 | 0.376 | 0.707 | 0.829 | 0.947 | 0.738 | 0.819 | 0.925 | 0.880 | 0.976 | 0.999 | 0.922 | 0.974 | 1

Notes: Figures report correlations between the stated uncertainty indexes of
Jurado et al., (2015). Figures below the diagonal are Pearson correlation
coefficients, whilst those above the diagonal are Spearman’s rank correlation
coefficients. The indexes codes are formed of three letters to indicate the
family, being either financial uncertainty (FIN), macroeconomic uncertainty
(MAC), or real uncertainty (REA). Each code includes a number to indicate the
number of months ahead, being either 1, 3 or 12. Finally for those using the
“economic” series a prefix “E” is added. Sample from 1st January 1993 to 28th
June 2021.

Formalising the message from Figure A1, Table A1 presents a full 18x18
correlation matrix for the uncertainty indexes. We denote the economic series
with an E prefix. We see a very clear blocking of both the Pearson
correlations in the lower left triangle, and the Spearman’s ranked correlation
ceofficients above the diagonal. Of the three sets of measure the within
variation for MAC is the highest. Correlation between $MAC1$ and $FIN1$ is
0.666, whilst the rank correlation of the series is little over 0.5. This
difference explains the potential for the persistence norms to provide
additional information to financial uncertainty without necessarily giving
extra explanatory power in regressions with $MAC1$ as the dependent variable.
This is what we find in the main paper.

The brief exposition here is consistent with the discussion in Jurado et al.,
(2015) and the interested reader is referred to the original paper for further
discussion of the relationships between the uncertainty indexes. Whilst
Covid-19 has seen different behaviour from REA, MAC1 shows only a short spike
and FIN is already looking to be beyond its spike at all horizons. There is
undoubtedly an interesting research agenda to understand the rationales for
the observed responses. However, such an exploration of Covid-19 responses is
beyond the focus of this paper.

##  Appendix B Correlation Between Clouds

In the regressions we treat each of our four cloud lengths independently. This
follows since interpretation requires that the volatility and norms be
computed on the same data. However, the results are naturally compared across
the 1-month, 3-month, 6-month and 12-month clouds. In this short appendix we
therefore present the correlation between the three cloud based measures in
Table A2

Table A2: Correlations between cloud based measures

Time | Volatility $\bar{\sigma}$ | Volatility $L_{11}$ | Volatility $L_{12}$
---|---|---|---
| 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m
1m | 1 | 0.857 | 0.779 | 0.715 | 1 | 0.318 | 0.245 | 0.188 | 1 | 0.281 | 0.237 | 0.169
3m | 0.858 | 1 | 0.92 | 0.849 | 0.322 | 1 | 0.777 | 0.661 | 0.362 | 1 | 0.771 | 0.648
6m | 0.723 | 0.893 | 1 | 0.931 | 0.283 | 0.831 | 1 | 0.835 | 0.301 | 0.835 | 1 | 0.830
12m | 0.601 | 0.735 | 0.867 | 1 | 0.272 | 0.733 | 0.879 | 1 | 0.254 | 0.709 | 0.859 | 1

Notes: Correlations of the stated cloud measure between observations from the
four cloud lengths. 1m, 3m, 6m and 12m refer to the number of months of data
in the point cloud over which the persistence norms and volatility are
calculated. Clouds are formed from the daily log returns on the S&P 500, Dow
Jones Industrial Average, NASDAQ and Russell 2000 stock market indexes.
Figures below the diagonal are the Pearson correlation coefficient and figures
above the diagonal are the Spearman’s rank correlation coefficient. Overall
sample from 1st January 1993 to 28th June 2021.

Within the three measures taken from our clouds we can see that the between
lengths correlations are much higher for volatility than they are for the
persistence norms. The correlation between the 1-month cloud volatility and
the 12-month cloud volatility is 0.601, but the corresponding figure for the
$L_{11}$ norms is 0.272 and for the $L_{12}$ norms is 0.254. Between the
1-month and 3-month clouds the correlations are 0.858 for the volatility, but
just 0.322 for $L_{11}$ and 0.362 for $L_{12}$. When we see the differences in
the regression results it is understandable because of these low correlations.
From the construction of persistence norms described in Section 2 of the main
paper, it follows that more points produces a denser cloud and will lead to
features being born, and dying, at lower filtration radii than is the case for
the sparse 1-month cloud.

##  Appendix C Four Market Cloud with $L2$ Norms

In the main paper we follow Gidea and Katz, (2018) to use the $L_{1}$ norms
from the point cloud of index returns. Table 1 shows that the correlation
between the $L_{1}$ and $L_{2}$ norms is high and so we may expect that the
regression results are also similar. Through this appendix we demonstrate that
there is consistency to the inference if using the $L_{2}$ norms.

Figure A2: Persistence norms, volatility and uncertainty time series plots

|
---|---
(a) $FIN1$ 3 months $L_{2}$ | (b) $FIN1$ 12 months $L_{2}$
|
(c) $FIN1$ 3 months $L_{1}$ | (d) $FIN1$ 12 months $L_{1}$
|
(e) $MAC1$ 3 months $L_{2}$ | (f) $MAC1$ 12 months $L_{2}$
|
(g) $MAC1$ 3 months $L_{1}$ | (h) $MAC1$ 12 months $L_{1}$

Notes: Figures plot the financial (FIN) and macroeconomic (MAC) uncertainty
index of Jurado et al., (2015). Each plot uses the 1 month ahead uncertainty
index $FIN1$ and $MAC1$. These indices are plotted as blue lines. On each plot
the average return volatility of the four indexes within the cloud; being the
S&P500, Dow Jones Industrial Average, NASDAQ and the Russell 2000 index.
Average volatility is plotted in black. The dimension one persistence norms
from these clouds are shown in red. Panels (a), (b), (e) and (f) show $L_{2}$
norms, whilst panels (c), (d), (g) and (h) present $L_{1}$ Values are
standardised by subtracting the mean value and dividing by the standard
deviation. Size of window used in the estimation is provided below each plot.
Sample from 1st January 1993 to 28th June 2021.

Figure A2 can be directly compared with Figure 1 of the main paper. To
highlight the similarity below each plot of the $L_{2}$ norm we place the
corresponding $L_{1}$ plot. Thus panels (a), (b), (e) and (f) are the $L_{2}$
plots and (c), (d), (g) and (h) replicate those from the main paper in showing
$L_{1}$ norms. The main contrasts appear in the “dot-com” bubble period
towards the left of the plot. Panels (e) and (g) show this well in the tall
spike in the $L_{2}$ contrasting with a shorter spike in the $L_{1}$. In
Panels (b) and (d) we can see a small first spike in the early part of the
global financial crisis for the $L_{2}$ norm, but this is absent in the
$L_{1}$. Both the $L_{1}$ and $L_{2}$ norms then have a taller spike at the
same time that the volatility reaches its maximum. Through the rest of the
panels behaviour is as similar, as the high correlation between the series
would suggest. Exploring the rationale for these differences would be an
interesting task for further research.

###  C.1 Persistence Norms

Regression of persistence norms on the uncertainty indices of Jurado et al.,
(2015) and the average volatility index returns tests whether these financial
market variables are able to span the persistence norms. Formally we estimate
a total of 288 models which are based upon three base regressions:

$\displaystyle\mathtt{Model12:}$ $\displaystyle
L_{12}=\alpha_{1}+\beta_{11}FIN1+\epsilon$ (9)
$\displaystyle\mathtt{Model22:}$ $\displaystyle
L_{12}=\alpha_{2}+\beta_{21}\bar{\sigma}+\epsilon$ (10)
$\displaystyle\mathtt{Model32:}$ $\displaystyle
L_{12}=\alpha_{3}+\beta_{31}FIN1+\beta_{32}\bar{\sigma}+\epsilon$ (11)

Where $L_{12}$ is the dimension one $L_{2}$ persistence norm, $\sigma$ is the
standard deviation of S&P 500 returns and $FIN1$ is the one month ahead
financial uncertainty index of Jurado et al., (2015). In all cases $\epsilon$
is the error term. In all estimations Newey and West, (1987) adjusted standard
errors are employed to account for autocorrelation and heteroskedasticity
inherent in financial time series. This set up is identical to that in the
main paper, but here we are reporting $L_{2}$ norms instead of $L_{1}$. The
addition of the 2 to the model allows distinguishing with models 1 and 2 in
the main paper. We do not add any additional subscripts to the coefficients
for ease of reading.

As in the main paper Models 12, 22 and 32 are just the building blocks for a
series of estimations that must be performed. Again, for each measure of
uncertainty there are 4 point clouds to consider. Three uncertainty indices at
three different lags, and with both a standard and economic version for each
of the 9, this all means 72 combinations arise from Model 12. Across the three
models the total number of combinations is thus 216. This exposition focuses
on FIN and MAC at the 1 month ahead horizon, but as noted results for other
uncertainty indexes and time horizons are available on request from the
authors.

Table A3: Determinants of $L_{2}$ persistence norms

|  |  | Model 12 | Model 22 | Model 32
---|---|---|---|---|---
|  |  | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m
Panel (a): Financial Uncertainty ($FIN1$): |  |  |
| Constant | $\alpha$ | 0.012 | -0.162 | -0.189 | -0.109 | 0.092 | 0.154 | 0.221 | 0.292 | 0.000 | -0.062 | -0.051 | 0.085
|  |  | (0.303) | (2.112) | (1.531) | (0.428) | (7.129) | (5.528) | (4.491) | (3.347) | (0.008) | (0.620) | (0.377) | (0.499)
| Uncertainty | $FIN1$ | 0.113 | 0.569 | 0.790 | 0.921 |  |  |  |  | 0.139 | 0.350 | 0.465 | 0.369
|  |  | (2.396) | (5.799) | (4.861) | (2.561) |  |  |  |  | (2.543) | (2.372) | (2.135) | (1.340)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.020 | 0.173 | 0.262 | 0.365 | -0.010 | 0.085 | 0.134 | 0.258
|  |  |  |  |  |  | (1.538) | (5.576) | (4.964) | (3.566) | (-0.631) | (2.467) | (2.460) | (4.058)
| Adjusted $R^{2}$ | 0.034 | 0.362 | 0.440 | 0.442 | 0.013 | 0.343 | 0.426 | 0.511 | 0.036 | 0.392 | 0.477 | 0.538
Panel (b): Macroeconomic Uncertainty ($MAC1$): |  |  |  |  |  |
| Constant | $\alpha$ | 0.041 | -0.017 | 0.031 | 0.179 | 0.092 | 0.154 | 0.221 | 0.292 | 0.052 | 0.183 | 0.347 | 0.521
|  |  | (0.747) | (0.314) | (0.294) | (0.439) | (7.129) | (5.528) | (4.491) | (3.347) | (0.802) | (1.484) | (1.770) | (2.854)
| Uncertainty | $MAC1$ | 0.114 | 0.570 | 0.764 | 0.841 |  |  |  |  | 0.075 | -0.059 | -0.269 | -0.504
|  |  | (1.286) | (7.047) | (4.728) | (1.227) |  |  |  |  | (0.604) | (0.226) | (0.547) | (1.100)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.020 | 0.173 | 0.262 | 0.365 | 0.012 | 0.181 | 0.303 | 0.446
|  |  |  |  |  |  | (1.538) | (5.576) | (4.964) | (3.566) | (0.631) | (3.400) | (2.458) | (2.854)
| Adjusted $R^{2}$ | 0.014 | 0.144 | 0.162 | 0.145 | 0.013 | 0.343 | 0.426 | 0.511 | 0.017 | 0.344 | 0.436 | 0.538

Notes: Regressions of the stated outcome with the other two variables as
explanatory factors. 3m and 6m refers to the number of months of data in the
point cloud over which the persistence norms and volatility are calculated.
Point clouds are the daily returns on the S&P 500, Dow Jones Industrial
Average, NASDAQ and Russell 2000 indexes. Uncertainty indexes are the 1 month
ahead estimates from Jurado et al., (2015) as downloaded from the website of
Sydney Ludvigson. Figures are the estimated coefficients and figures in
parentheses are the associated Newey and West, (1987) adjusted t-statistics.
Model 12 has $L_{12}=\alpha_{1}+\beta_{12}FIN1+\epsilon$. Model 22 has
$L_{12}=\alpha_{2}+\beta_{21}\bar{\sigma}+\epsilon$. Model 32 has
$L_{2}=\alpha_{3}+\beta_{31}FIN1+\beta_{32}\bar{\sigma}+\epsilon$. Models 12,
22 and 32 represent the base specification upon which all regression models
are built. Specifications show $FIN1$ as the one month ahead financial
uncertainty (FIN) index. Where macroeconomic uncertainty (MAC) is used then
the equations for the four models are duly updated to replace $FIN1$ with
$MAC1$. In each case $\bar{\sigma}$ is the average standard deviation of the
four index returns in the window and $\epsilon$ is the error term. Estimation
employs Newey and West, (1987) robust standard errors to control for
autocorrelation and heteroskedasticity inherent in these time series. Sample
from 1st January 1993 to 28th June 2021.

Table A3 reports estimates from the regressions with Newey and West, (1987)
robust standard errors. Panel (a) shows that $FIN1$ is significant in both
models 12 and 32. Volatility is significant in model 22 and here in model 32.
Herein we see the main difference between $L_{1}$ and $L_{2}$ comes in the
insignificance of $FIN1$ when volatility is included. Caution is urged because
of the high correlation between volatility and $FIN1$, but it may also be
noted that $L_{12}$ has a stronger correlation to $\bar{\sigma}$ than $L_{11}$
in Table 1 of the main paper. Further when using only volatility as the
explanatory variable for the $L_{12}$ norm, the constant term is highly
significant, but this is not the case when uncertainty appears in the
specification. Recalling that the same observation was found for the $L_{1}$
norm we have the consistency between the results of Table A3 and Table 2 of
the main paper.

Further consistency is found in the results of the 1-month cloud, with the
adjusted r-squared being much lower than the longer clouds. As a month only
has 22 trading days on average it follows that the cloud being used is quite
small. At a smaller size the impact of an extreme value is much larger, the
distance between the extremes and the next nearest point in the cloud will
then influence the feature construction in the rips complex. Although the
measures from the 1-month cloud are consistent with those from longer clouds
on average, we are directed to use longer clouds if we wish to explain the
behaviour of the persistence norms.

###  C.2 Uncertainty

Reversing the question we regress the uncertainty indexes on the persistence
norms and volatility.

$\displaystyle\mathtt{Model42:}$ $\displaystyle
FIN1=\alpha_{4}+\beta_{41}L_{12}+\epsilon$ (12)
$\displaystyle\mathtt{Model52:}$ $\displaystyle
FIN1=\alpha_{5}+\beta_{52}\bar{\sigma}+\epsilon$ (13)
$\displaystyle\mathtt{Model62:}$ $\displaystyle
FIN1=\alpha_{6}+\beta_{61}L_{12}+\beta_{62}\bar{\sigma}+\epsilon$ (14)

Here again $FIN1$ is the one month ahead financial uncertainty from Jurado et
al., (2015), $L_{11}$ is the dimension 1 $L_{1}$ persistence norm and
$\bar{\sigma}$ is the mean standard deviation from the four return series in
the cloud. $\epsilon$ is the error term. Models are estimated with Newey and
West, (1987) robust standard errors. Once more we add a 2 to the name of the
model to ensure correct referencing against models 4, 5 and 6 of the main
paper. It must be noted at this point that results from model 52 will be
identical to model 5 in the main paper as uncertainty and volatility are not
affected by the choice of persistence norms.

Table A4: Explaining the uncertainty index

|  |  | Model 42 | Model 52 | Model 62
---|---|---|---|---|---
|  |  | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m
Panel (a): Financial Uncertainty ($FIN1$): |  |  |
| Constant | $\alpha$ | 0.867 | 0.678 | 0.610 | 0.556 | 0.667 | 0.617 | 0.584 | 0.560 | 0.651 | 0.584 | 0.542 | 0.517
|  |  | (13.62) | (18.46) | (11.80) | (6.067) | (20.77) | (16.46) | (15.70) | (8.128) | (21.07) | (16.27) | (15.62) | (6.583)
| Norm | $L_{12}$ | 0.304 | 0.637 | 0.557 | 0.479 |  |  |  |  | 0.168 | 0.212 | 0.192 | 0.149
|  |  | (3.063) | (8.153) | (5.795) | (3.685) |  |  |  |  | (2.474) | (3.395) | (2.467) | (1.252)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.213 | 0.251 | 0.274 | 0.290 | 0.210 | 0.215 | 0.224 | 0.236
|  |  |  |  |  |  | (11.02) | (10.50) | (7.333) | (3.727) | (10.62) | (10.69) | (5.770) | (2.662)
| Adjusted$R^{2}$ | 0.034 | 0.362 | 0.440 | 0.442 | 0.552 | 0.646 | 0.663 | 0.622 | 0.563 | 0.672 | 0.693 | 0.643
Panel (b): Macroeconomic Uncertainty ($MAC1$): |  |  |  |  |  |
| Constant | $\alpha$ | 0.631 | 0.556 | 0.533 | 0.520 | 0.528 | 0.492 | 0.468 | 0.455 | 0.524 | 0.496 | 0.482 | 0.487
|  |  | (10.57) | (14.62) | (8.537) | (8.841) | (31.34) | (14.50) | (9.186) | (7.774) | (29.21) | (12.69) | (10.77) | (9.884)
| Norm | $L_{12}$ | 0.120 | 0.252 | 0.212 | 0.173 |  |  |  |  | 0.053 | -0.021 | -0.063 | -0.109
|  |  | (0.634) | (1.245) | (1.041) | (1.383) |  |  |  |  | (0.565) | (0.225) | (0.792) | (0.949)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.105 | 0.134 | 0.153 | 0.161 | 0.104 | 0.138 | 0.169 | 0.201
|  |  |  |  |  |  | (5.057) | (2.614) | (2.311) | (2.268) | (4.958) | (3.191) | (2.495) | (1.944)
| Adjusted $R^{2}$ | 0.014 | 0.144 | 0.162 | 0.145 | 0.342 | 0.467 | 0.520 | 0.486 | 0.345 | 0.468 | 0.528 | 0.514

Notes: Regressions of the stated outcome with the other two variables as
explanatory factors. 1m, 3m, 6m and 12m refers to the number of months of data
in the point cloud over which the persistence norms and volatility are
calculated. Uncertainty indexes are the 1 month ahead estimates from Jurado et
al., (2015) as downloaded from the website of Sydney Ludvigson. Clouds are
formed from the daily log returns on the S&P 500, Dow Jones Industrial
Average, NASDAQ and Russell 2000 stock market indexes. Figures are the
estimated coefficients and figures in parentheses are the associated Newey and
West, (1987) adjusted t-statistics. Model 42 has
$FIN1=\alpha_{4}+\beta_{41}L_{12}+\epsilon$. Model 52 has
$FIN1=\alpha_{5}+\beta_{51}\bar{\sigma}+\epsilon$. Model 62 has
$FIN1=\alpha_{6}+\beta_{61}L_{12}+\beta_{62}\bar{\sigma}+\epsilon$.
Specifications show $FIN1$ as the one month ahead financial uncertainty (FIN)
index. Where macroeconomic uncertainty (MAC) is used then the equations for
the four models are duly updated to replace $FIN1$ with $MAC1$. In each case
$\bar{\sigma}$ is the average standard deviation of the returns on the four
indices in the window. $\epsilon$ is the error term. Estimation employs Newey
and West, (1987) robust standard errors to control for autocorrelation and
heteroskedasticity inherent in these time series. Sample from 1st January 1993
to 28th June 2021.

Panel (a) of Table A4 shows again that the persistence norm is significant as
an explanatory variable for $FIN1$. However, the constant terms in Model 42
are also highly significant and the adjusted r-squared values are around 0.40.
Volatility has a higher significance and, when combining volatility and norms
in Model 62, it is volatility that has the most significant coefficients.
Almost two-thirds of the variation in FIN1 may be explained by volatility and
persistence norms, a large increase from the just under half that is explained
by volatility alone. Model fits using the $L_{2}$ norm are slightly better
than those obtained the $L_{1}$ norm in the main paper. Importantly the
qualitative messages emerging from panel (a) are very similar.

Panel (b) of Table A4 considers MAC and specifically the one-month ahead
macroeconomic uncertainty index ($MAC1$). Here, as in the main paper, we see
no significance of the persistence norms from clouds of any-length. Because
the volatility is not affected by the choice of norms in the model, results
for model 52 match those from model 5 exactly. Drawing parallels with the
$L_{1}$ norms in the main paper, we may again see that the results are
qualitatively similar. Hence whilst persistence norms add to the fit of FIN
they do not provide additional information in the fit of MAC.

###  C.3 High and Low Uncertainty

As with the main paper, the strong correlation between volatility and
uncertainty means that we do not consider high and low norm periods. Here we
ask whether there is a different role for norms and volatility in periods of
high, or low, uncertainty. To achieve this we split the sample at the median
of the respective uncertainty indexes. As neither the uncertainty index, nor
the volatility, are affected by the choice of norm the results from model 52
are again identical to those from model 5 in the main paper.

Table A5: High and low uncertainty index periods

|  |  | Model 4 | Model 5 | Model 6
---|---|---|---|---|---
|  |  | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m | 1m | 3m | 6m | 12m
Panel (a): High Financial Uncertainty ($FIN1$): |  |  |
| Constant | $\alpha$ | 1.048 | 0.933 | 0.906 | 0.907 | 0.877 | 0.836 | 0.821 | 0.822 | 0.866 | 0.815 | 0.792 | 0.806
|  |  | (12.24) | (21.47) | (16.83) | (9.791) | (25.29) | (20.84) | (18.08) | (9.144) | (25.38) | (15.24) | (14.47) | (7.227)
| Norms | $L_{12}$ | 0.105 | 0.291 | 0.238 | 0.175 |  |  |  |  | 0.087 | 0.086 | 0.081 | 0.038
|  |  | (0.993) | (4.197) | (2.980) | (1.385) |  |  |  |  | (1.246) | (1.05) | (1.121) | (0.352)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.128 | 0.151 | 0.158 | 0.157 | 0.127 | 0.140 | 0.142 | 0.146
|  |  |  |  |  |  | (7.316) | (7.887) | (4.930) | (2.144) | (7.013) | (6.518) | (3.896) | (1.708)
|  Adjusted $R^{2}$ | 0.011 | 0.176 | 0.178 | 0.125 | 0.493 | 0.528 | 0.442 | 0.313 | 0.501 | 0.541 | 0.458 | 0.317
Panel (b): Low Financial Uncertainty ($FIN1$): |  |  |
| Constant | $\alpha$ | 0.739 | 0.724 | 0.663 | 0.588 | 0.678 | 0.625 | 0.579 | 0.620 | 0.678 | 0.625 | 0.558 | 0.568
|  |  | (17.40) | (14.73) | (14.30) | (7.435) | (28.10) | (18.77) | (25.68) | (18.85) | (28.09) | (19.82) | (20.58) | (8.163)
| Norms | $L_{12}$ | 0.026 | 0.069 | 0.200 | 0.275 |  |  |  |  | -0.001 | 0.000 | 0.080 | 0.127
|  |  | (0.570) | (0.971) | (1.984) | (2.345) |  |  |  |  | (0.024) | (0.006) | (0.945) | (0.856)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.084 | 0.151 | 0.207 | 0.148 | 0.084 | 0.151 | 0.193 | 0.125
|  |  |  |  |  |  | (3.626) | (4.062) | (8.092) | (4.490) | (3.584) | (4.117) | (6.217) | (2.787)
|  Adjusted $R^{2}$ | 0.001 | 0.012 | 0.102 | 0.182 | 0.142 | 0.257 | 0.392 | 0.348 | 0.142 | 0.257 | 0.406 | 0.379
Panel (c): High Macroeconomic Uncertainty ($MAC1$): |  |  |  |  |  |
| Constant | $\alpha$ | 0.699 | 0.627 | 0.615 | 0.612 | 0.598 | 0.549 | 0.515 | 0.507 | 0.583 | 0.548 | 0.531 | 0.540
|  |  | (10.34) | (11.54) | (7.829) | (9.758) | (24.77) | (16.95) | (11.38) | (7.552) | (27.65) | (12.97) | (12.99) | (9.157)
| Norms | $L_{12}$ | 0.166 | 0.233 | 0.176 | 0.133 |  |  |  |  | 0.145 | 0.005 | -0.071 | -0.121
|  |  | (1.096) | (0.828) | (0.694) | (0.993) |  |  |  |  | (1.762) | (0.037) | (0.757) | (0.940)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.089 | 0.122 | 0.146 | 0.149 | 0.088 | 0.122 | 0.164 | 0.194
|  |  |  |  |  |  | (4.158) | (3.217) | (2.921) | (2.042) | (3.914) | (3.417) | (3.223) | (1.754)
|  Adjusted $R^{2}$ | 0.027 | 0.133 | 0.123 | 0.091 | 0.281 | 0.428 | 0.504 | 0.407 | 0.302 | 0.428 | 0.516 | 0.446
Panel (d): Low Macroeconomic Uncertainty ($MAC1$): |  |  |  |  |  |
| Constant | $\alpha$ | 0.568 | 0.556 | 0.552 | 0.551 | 0.553 | 0.547 | 0.543 | 0.537 | 0.553 | 0.546 | 0.543 | 0.541
|  |  | (55.27) | (45.45) | (29.97) | (28.49) | (57.69) | (40.03) | (40.67) | (30.85) | (57.13) | (38.22) | (35.09) | (32.75)
| Norms | $L_{12}$ | 0.030 | 0.050 | 0.042 | 0.032 |  |  |  |  | 0.002 | 0.009 | 0.001 | -0.015
|  |  | (1.334) | (2.689) | (1.905) | (1.808) |  |  |  |  | (0.096) | (0.422) | (0.041) | (-0.409)
| Volatility | $\bar{\sigma}$ |  |  |  |  | 0.022 | 0.028 | 0.031 | 0.037 | 0.021 | 0.026 | 0.031 | 0.043
|  |  |  |  |  |  | (3.787) | (3.249) | (3.57) | (2.116) | (3.499) | (2.568) | (2.842) | (1.257)
|  Adjusted $R^{2}$ | 0.018 | 0.092 | 0.092 | 0.072 | 0.135 | 0.191 | 0.222 | 0.260 | 0.135 | 0.192 | 0.222 | 0.269

Notes: Regressions of the stated outcome with the other two variables as
explanatory factors. 1m, 3m, 6m and 12m refers to the number of months of data
in the point cloud over which the persistence norms and volatility are
calculated. Panels (a) and (c) cover periods of high financial and
macroeconomic uncertainty respectively. Likewise panels (b) and (d) cover
periods where uncertainty is below the sample median. Uncertainty indexes are
the 1 month ahead estimates from Jurado et al., (2015) as downloaded from the
website of Sydney Ludvigson. Clouds are formed from the daily log returns on
the S&P 500, Dow Jones Industrial Average, NASDAQ and Russell 2000 stock
market indexes. Figures are the estimated coefficients and figures in
parentheses are the associated Newey and West, (1987) adjusted t-statistics.
Model 42 has $FIN1=\alpha_{4}+\beta_{41}L_{12}+\epsilon$. Model 52 has
$FIN1=\alpha_{5}+\beta_{51}\bar{\sigma}+\epsilon$. Model 62 has
$FIN1=\alpha_{6}+\beta_{61}L_{12}+\beta_{62}\bar{\sigma}+\epsilon$.
Specifications show $FIN1$ as the one month ahead financial uncertainty (FIN)
index. Where macroeconomic uncertainty (MAC) is used then the equations for
the four models are duly updated to replace $FIN1$ with $MAC1$. In each case
$\bar{\sigma}$ is the average standard deviation of the returns on the four
indices in the window. $\epsilon$ is the error term. Estimation employs Newey
and West, (1987) robust standard errors to control for autocorrelation and
heteroskedasticity inherent in these time series. The number of observations
in each high and low period is 171. Overall sample from 1st January 1993 to
28th June 2021.

Panel (a) of Table A5 shows again that persistence norms from the 3-month and
6-month clouds are significant in explaining high periods of FIN. Compared to
the results in the main paper the adjusted r-squared is higher, but the
inference on the role of norms is qualitatively similar. We also see strong
similarity in the significance of the persistence norms after the inclusion of
volatility in the model. Here the change to the r-squared value is negligible.
It remains the case that the constant terms are highly significant in all
models. Panel (b) shows that $L_{2}$ norms give significance in the 6-month
and 12-month clouds but that this does not survive the introduction of
volatility into the model. Recognising the links between volatility and norms,
attributes this result to volatility. Primarily the inference on FIN is that
the results are qualitatively similar.

Turning to $MAC1$ in panel (c) we again see that persistence norms from the
financial markets do not explain contemporaneous uncertainty. The only
coefficient identified with statistical significance for the $L_{2}$ norm is
in the 1-month point cloud and this is a very volatile signal. With typically
22 points in the cloud caution should be extended about interpreting the
significance of the $L_{2}$ norm in model 62 versus the insignificance of the
$L_{1}$ in model 6 of the main paper. Panel (d) shows again that in low
periods of $MAC1$ the persistence norm does pick up a significant coefficient
in longer length clouds. However, this significance in model 42 disappears
when moving to model 62. Again it is volatility that is the main explanatory
variable for $MAC1$.

###  C.4 Summary

This appendix has repeated the analysis from the main paper using the $L_{2}$
persistence norm in place of the $L_{1}$. We have shown that the results are
qualitatively similar as would have been expected from the strong correlations
in Table 1 of the main paper. Some small differences in the series were
observed around the “dot-com” bubble and understanding the reasons for these
may help evaluate the best choice between $L_{1}$ and $L_{2}$ norms in
practical applications. It remains the case that the use of $L_{1}$ after
Gidea and Katz, (2018) is sufficient to provide inference on the links between
persistence norms, volatility and the uncertainty indices of Jurado et al.,
(2015).