Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

LASSO-Driven Inference in Time and Space


Victor Chernozhukov
Wolfgang K. Härdle
Chen Huang
Weining Wang


Abstract
We consider the estimation and inference in a system of high-dimensional regression equations
allowing for temporal and cross-sectional dependency in covariates and error processes, covering
rather general forms of weak dependence. A sequence of large-scale regressions with LASSO is
applied to reduce the dimensionality, and an overall penalty level is carefully chosen by a block
multiplier bootstrap procedure to account for multiplicity of the equations and dependencies in the
data. Correspondingly, oracle properties with a jointly selected tuning parameter are derived. We
further provide high-quality de-biased simultaneous inference on the many target parameters of
the system. We provide bootstrap consistency results of the test procedure, which are based on a
general Bahadur representation for the Z-estimators with dependent data. Simulations demonstrate
good performance of the proposed inference procedure. Finally, we apply the method to quantify
spillover effects of textual sentiment indices in a financial market and to test the connectedness
among sectors.


Keywords:
LASSO, time series, simultaneous inference, system of equations, Z-estimation, Bahadur representation, martingale decomposition

JEL classification:
C12, C22, C51, C53

Learning from Errors: The case of monetary and fiscal policy regimes

Andreas Tryphonides


Abstract
The New Keynesian theory of inflation determination has been under scrutiny
due to identification issues, which rather have to do with the mechanism of inflation determination
at its core (i.e. Cochrane (2011)). Moreover, similar identification problems
arise in the case of fiscal inflation (see for example Leeper and Leith (2016), Leeper and
Li (2017) and Leeper and Walker (2012)). This paper makes a positive contribution.
We argue that statements about observational equivalence stem from referring to the
equilibrium path, while this should not be our primary source of identifying restrictions.
Moreover, policy identification (or lack thereof) relies on assumptions on the underlying
shock structure, which is unobservable. We instead extract shocks using heterogeneous
uncertain restrictions and external datasets, that is, we learn from errors. We are then
able to recover deep and policy parameters irrespective of the prevailing equilibrium. We
provide time varying evidence on the efficacy of policy in stabilizing the US economy and
on the time varying plausibility of Ricardian versus non-Ricardian price determination.
Results are work in progress.


Keywords:
Monetary and fiscal policy, Price Determination, Identification, Learning from errors

JEL classification:
C11, C13, E62, E63

Textual Sentiment, Option Characteristics, and Stock Return Predictability

Cathy Yi-Hsuan Chen
Matthias R. Fengler
Wolfgang Karl Härdle
Yanchu Liu


Abstract
We distill sentiment from a huge assortment of NASDAQ news articles by means of machine
learning methods and examine its predictive power in single-stock option markets and equity
markets. We provide evidence that single-stock options react to contemporaneous sentiment.
Next, examining return predictability, we discover that while option variables indeed predict
stock returns, sentiment variables add further informational content. In fact, both in a
regression and a trading context, option variables orthogonalized to public and sentimental
news are even more informative predictors of stock returns. Distinguishing further between
overnight and trading-time news, we find the first to be more informative. From a statistical
topic model, we uncover that this is attributable to the differing thematic coverage of the
alternate archives. Finally, we show that sentiment disagreement commands a strong positive
risk premium above and beyond market volatility and that lagged returns predict future
returns in concentrated sentiment environments.


Keywords:
investor disagreement; option markets; overnight information; stock return
predictability; textual sentiment; topic model; trading-time information;

JEL classification:
C58, G12, G14, G41

Bootstrap Confidence Sets for Spectral Projectors of Sample Covariance

A. Naumov
V. Spokoiny
V. Ulyanovk


Abstract
Let X1, . . . ,Xn be i.i.d. sample in Rp with zero mean and the
covariance matrix . The problem of recovering the projector onto
an eigenspace of from these observations naturally arises in many
applications. Recent technique from [9] helps to study the asymp-
totic distribution of the distance in the Frobenius norm kPr - bP
rk2
between the true projector Pr on the subspace of the rth eigenvalue
and its empirical counterpart bP
r in terms of the effective rank of .
This paper offers a bootstrap procedure for building sharp confidence
sets for the true projector Pr from the given data. This procedure
does not rely on the asymptotic distribution of kPr - bP
rk2 and its
moments. It could be applied for small or moderate sample size n and
large dimension p. The main result states the validity of the proposed
procedure for finite samples with an explicit error bound for the er-
ror of bootstrap approximation. This bound involves some new sharp
results on Gaussian comparison and Gaussian anti-concentration in
high-dimensional spaces. Numeric results confirm a good performance
of the method in realistic examples.


Keywords:


JEL classification:

Construction of Non-asymptotic Confidence Sets in 2 -Wasserstein Space

Johannes Ebert
Vladimir Spokoiny
Alexandra Suvorikova


Abstract
In this paper, we consider a probabilistic setting where the probability measures
are considered to be random objects. We propose a procedure of construction
non-asymptotic confidence sets for empirical barycenters in 2 -Wasserstein space and
develop the idea further to construction of a non-parametric two-sample test that is
then applied to the detection of structural breaks in data with complex geometry. Both
procedures mainly rely on the idea of multiplier bootstrap (Spokoiny and Zhilova [29],
Chernozhukov, Chetverikov and Kato [13]). The main focus lies on probability measures
that have commuting covariance matrices and belong to the same scatter-location
family: we proof the validity of a bootstrap procedure that allows to compute confidence
sets and critical values for a Wasserstein-based two-sample test.


Keywords:
Wasserstein barycenters, hypothesis testing, multiplier bootstrap,
change point detection, confidence sets.

JEL classification:

Bayesian inference for spectral projectors of covariance matrix

Igor Silin
Vladimir Spokoiny


Abstract
Let X1; : : : ;Xn be i.i.d. sample in Rp with zero mean and
the covariance matrix . The classic principal component analysis esti-
mates the projector P
J onto the direct sum of some eigenspaces of
by its empirical counterpart bPJ . Recent papers [20, 23] investigate the
asymptotic distribution of the Frobenius distance between the projectors
k bPJ ??P
J k2 . The problem arises when one tries to build a condence set
for the true projector eectively. We consider the problem from Bayesian
perspective and derive an approximation for the posterior distribution of
the Frobenius distance between projectors. The derived theorems hold true
for non-Gaussian data: the only assumption that we impose is the con-
centration of the sample covariance b
in a vicinity of . The obtained
results are applied to construction of sharp condence sets for the true pro-
jector. Numerical simulations illustrate good performance of the proposed
procedure even on non-Gaussian data in quite challenging regime.


Keywords:
covariance matrix, spectral projector, principal
component analysis, Bernstein { von Mises theorem.

JEL classification:

Toolbox: Gaussian comparison on Eucledian balls

Andzhey Koziuk
Vladimir Spokoiny


Abstract
In the work a characterization of difference of multivariate Gaussian measures is found on the
family of centered Eucledian balls. In particular, it helps to derive (xx see paper).


Keywords:
multivariate Gaussian measure, Kolmogorov distance, Gaussian comparison

JEL classification:

Pointwise adaptation via stagewise aggregation of local estimates for multiclass classification

Nikita Puchkin
Vladimir Spokoiny


Abstract
We consider a problem of multiclass classification, where the
training sample Sn = {(Xi, Yi)}n
i=1 is generated from the model P(Y =
m|X = x) = m(x), 1 6 m 6 M, and 1(x), . . . , M(x) are unknown Lip-
schitz functions. Given a test point X, our goal is to estimate 1(X), . . . ,
M(X). An approach based on nonparametric smoothing uses a localization
technique, i.e. the weight of observation (Xi, Yi) depends on the distance
between Xi and X. However, local estimates strongly depend on localiz-
ing scheme. In our solution we fix several schemes W1, . . . ,WK, compute
corresponding local estimates e(1), . . . , e(K) for each of them and apply an
aggregation procedure. We propose an algorithm, which constructs a con-
vex combination of the estimates e(1), . . . , e(K) such that the aggregated
estimate behaves approximately as well as the best one from the collection
e(1), . . . , e(K). We also study theoretical properties of the procedure, prove
oracle results and establish rates of convergence under mild assumptions.


Keywords:


JEL classification:

Gaussian Process Forecast with multidimensional distributional entries

Francois Bachoc
Alexandra Suvorikova
Jean-Michel Loubes
Vladimir Spokoiny



Abstract
In this work, we propose to define Gaussian Processes indexed by multidimensional distributions.
In the framework where the distributions can be modeled as i.i.d realizations of a measure on
the set of distributions, we prove that the kernel defined as the quadratic distance between the
transportation maps, that transport each distribution to the barycenter of the distributions, provides
a valid covariance function. In this framework, we study the asymptotic properties of this process,
proving micro ergodicity of the parameters.


Keywords:
Gaussian Process, Kernel methods, Wasserstein Distance

JEL classification:

Gaussian Process Forecast with multidimensional distributional entries

Andzhey Koziuk
Vladimir Spokoiny



Abstract
IV regression in the context of a re-sampling is considered in the work. Comparatively, the contribution
in the development is a structural identication in the IV model. The work also contains a
multiplier-bootstrap justication.


Keywords:
Gaussian Process, Kernel methods, Wasserstein Distance
JEL classification:

Understanding Latent Group Structure of Cryptocurrencies Market: A Dynamic Network Perspective

Li Guo
Yubo Tao
Wolfgang Karl Hardle



Abstract
In this paper, we study the latent group structure in cryptocurrencies market
by forming a dynamic return inferred network with coin attributions. We develop
a dynamic covariate-assisted spectral clustering method to detect the communities
in dynamic network framework and prove its uniform consistency along the horizons.
Applying our new method, we show the return inferred network structure and
coin attributions, including algorithm and proof types, jointly determine the market
segmentation. Based on the network model, we propose a novel \hard-to-value"
measure using the centrality scores. Further analysis reveals that the group with a
lower centrality score exhibits stronger short-term return reversals. Cross-sectional
return predictability further conrms the economic meanings of our grouping results
and reveal important portfolio management implications.


Keywords:
Community Detection, Dynamic Network, Return Predictability, Behavioural
Bias, Market Segmentation, Bitcoin

Optimal contracts under competition when uncertainty from adverse selection and moral hazard are present

Natalie Packham



Abstract
In a continuous-time setting where a risk-averse agent controls the drift of an output
process driven by a Brownian motion, optimal contracts are linear in the terminal output;
this result is well-known in a setting with moral hazard and – under stronger assumptions
– adverse selection. We show that this result continues to hold when in addition reser-
vation utilities are type-dependent. This type of problem occurs in the study of optimal
compensation problems involving competing principals.


Keywords:
Principal-agent modelling; contract design; stochastic process; stochastic control

A factor-model approach for correlation scenarios and correlation stress-testing

Natalie Packham
Fabian Woebbeking



Abstract
In 2012, JPMorgan accumulated a USD 6.2 billion loss on a credit derivatives portfolio,
the so-called \London Whale", partly as a consequence of de-correlations of non-perfectly
correlated positions that were supposed to hedge each other. Motivated by this case, we
devise a factor model for correlations that allows for scenario-based stress-testing of correlations.
We derive a number of analytical results related to a portfolio of homogeneous
assets. Using the concept of Mahalanobis distance, we show how to identify adverse scenarios
of correlation risk. As an example, we apply the factor-model approach to the \London
Whale" portfolio and determine the value-at-risk impact from correlation changes. Since our
ndings are particularly relevant for large portfolios, where even small correlation changes
can have a large impact, a further application would be to stress-test portfolios of central
counterparties, which are of systemically relevant size.


Keywords:
Correlation stress testing, scenario selection, market risk, "London Whale"

JEL Classication:
C58, G15, G17, G18

Correlation Under Stress In Normal Variance Mixture Models

Michael Kalkbrener
Natalie Packham


Abstract
We investigate correlations of asset returns in stress scenarios where a common risk
factor is truncated. Our analysis is performed in the class of normal variance mixture
(NVM) models, which encompasses many distributions commonly used in nancial
modelling. For the special cases of jointly normally and t-distributed asset returns
we derive closed formulas for the correlation under stress. For the NVM distribution,
we calculate the asymptotic limit of the correlation under stress, which depends on
whether the variables are in the maximum domain of attraction of the Frechet or
Gumbel distribution. It turns out that correlations in heavy-tailed NVM models are
less sensitive to stress than in medium- or light-tailed models. Our analysis sheds light
on the suitability of this model class to serve as a quantitative framework for stress
testing, and as such provides valuable information for risk and capital management
in nancial institutions, where NVM models are frequently used for assessing capital
adequacy. We also demonstrate how our results can be applied for more prudent stress
testing.


Keywords:
Stress testing, risk management, correlation, normal variance mixture distribution, multivariate normal distribution, multivariate t-distribution.

Model risk of contingent claims

Nils Detering
Natalie Packham


Abstract
Paralleling regulatory developments, we devise value-at-risk and expected shortfall type
risk measures for the potential losses arising from using misspecied models when pricing
and hedging contingent claims. Essentially, losses from model risk correspond to losses realized
on a perfectly hedged position. Model uncertainty is expressed by a set of pricing
models, relative to which potential losses are determined. Using market data, a unied
loss distribution is attained by weighing models according to a relative likelihood criterion.
Examples demonstrate the magnitude of model risk and corresponding capital buers necessary
to suciently protect trading book positions against unexpected losses from model
risk.


Keywords:
Model risk, parameter uncertainty, hedge error, value-at-risk, expected shortfall

JEL Clasification:
G32, G13

Default probabilities and default correlations under stress

Natalie Packham
Michael Kalkbrener
Ludger Overbeck


Abstract
We investigate default probabilities and default correlations of Merton-type credit portfolio
models in stress scenarios where a common risk factor is truncated. The analysis is
performed in the class of elliptical distributions, a family of light-tailed to heavy-tailed distributions
encompassing many distributions commonly found in nancial modelling. It turns
out that the asymptotic limit of default probabilities and default correlations depend on the
max-domain of the elliptical distribution's mixing variable. In case the mixing variable is
regularly varying, default probabilities are strictly smaller than 1 and default correlations
are in (0; 1). Both can be expressed in terms of the Student t-distribution function. In the
rapidly varying case, default probabilities are 1 and default correlations are 0. We compare
our results to the tail dependence function and discuss implications for credit portfolio
modelling.


Keywords:
financial risk management, credit portfolio modelling, stress testing, elliptic distribution, max-domain

MSC classification:
60G70, 91G40

Tail-Risk Protection Trading Strategies

Natalie Packham
Jochen Papenbrock
Peter Schwendner
Fabian Woebbeking


Abstract
Starting from well-known empirical stylised facts of nancial time series, we develop
dynamic portfolio protection trading strategies based on econometric methods. As a criterion
for riskiness we consider the evolution of the value-at-risk spread from a GARCH
model with normal innovations relative to a GARCH model with generalised innovations.
These generalised innovations may for example follow a Student t, a generalised
hyperbolic (GH), an alpha-stable or a Generalised Pareto (GPD) distribution. Our
results indicate that the GPD distribution provides the strongest signals for avoiding
tail risks. This is not surprising as the GPD distribution arises as a limit of tail behaviour
in extreme value theory and therefore is especially suited to deal with tail risks.
Out-of-sample backtests on 11 years of DAX futures data, indicate that the dynamic
tail-risk protection strategy eectively reduces the tail risk while outperforming traditional
portfolio protection strategies. The results are further validated by calculating
the statistical signicance of the results obtained using bootstrap methods. A number of
robustness tests including application to other assets further underline the eectiveness
of the strategy. Finally, by empirically testing for second order stochastic dominance,
we nd that risk averse investors would be willing to pay a positive premium to move
from a static buy-and-hold investment in the DAX future to the tail-risk protection
strategy.

Keywords:
tail-risk protection, portfolio protection, extreme events, tail distributions

JEL Classification:
C15, G11, G17.

Penalized Adaptive Forecasting with Large Information Sets and Structural Changes

Lenka Zbonakova
Xinjue Li
Wolfgang Karl Härdle


Abstract
In the present paper we propose a new method, the Penalized Adaptive
Method (PAM), for a data driven detection of structural changes in sparse linear
models. The method is able to allocate the longest homogeneous intervals over
the data sample and simultaneously choose the most proper variables with the
help of penalized regression models. The method is simple yet exible and can
be safely applied in high-dimensional cases with dierent sources of parameter
changes. Comparing with the adaptive method in linear models, its combination
with dimension reduction yields a method which properly selects signicant
variables and detects structural breaks while steadily reduces the forecast error
in high-dimensional data.

Keywords:
SCAD penalty, propagation-separation, adaptive window choice, multiplier bootstrap

JEL Classification:
C12, C13, C50, E47, G12

On complete convergence in Marcinkiewicz-Zygmund type SLLN for random variables

Anna Kuczmaszewska
Ji Gao YAN


Abstract
We consider a generalization of Baum-Katz theorem for random vari-
ables satisfying some cover conditions. Consequently, we get the result for many
dependent structure, such as END, -mixing, -mixing and -mixing, etc.

Keywords:
Complete convergence; Marcinkiewicz-Zygmund type SLLN; Extended negatively dependent; Mixing dependency; Weakly mean bounded.

JEL Classification:
C00

MSC(2010) Subject Classification:
60F15

Complete Convergence and Complete Moment Convergence for Maximal Weighted Sums of Extended Negatively Dependent Random Variables

Ji Gao YAN


Abstract
In this paper, the complete convergence and complete moment convergence for maximal
weighted sums of extended negatively dependent random variables are investigated. Some su±cient
conditions for the convergence are provided. In addition, the Marcinkiewicz{Zygmund type strong law
of large numbers for weighted sums of extended negatively dependent random variables is obtained.
The results obtained in the article extend the corresponding ones for independent random variables
and some dependent random variables.

Keywords:
Extended negatively dependent, complete convergence, complete moment convergence, maximal weighted sums, strong law of large numbers

JEL Classification:
C00

MR(2010) Subject Classification:
60F15