Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

Time Varying Quantile Lasso

Lenka Zbonakova
Wolfgang Karl Härdle
Weining Wang

Abstract:
In the present paper we study the dynamics of penalization parameter lambda of the least absolute shrinkage and selection operator (Lasso) method proposed by Tibshirani (1996) and extended into quantile regression context by Li and Zhu (2008). The dynamic behaviour of the parameter can be observed when the model is assumed to vary over time and therefore the fitting is performed with the use of moving windows. The proposal of investigating time series of and its dependency on model characteristics was brought into focus by Hardle et al. (2016), which was a foundation of FinancialRiskMeter (http://frm.wiwi.hu-berlin.de). Following the ideas behind the two aforementioned projects, we use the derivation of the formula for the penalization parameter lambda
as a result of the optimization problem. This reveals three possible effects driving lambda;
variance of the error term, correlation structure of the covariates and number of nonzero
coefficients of the model. Our aim is to disentangle these three effect and investigate
their relationship with the tuning parameter lambda, which is conducted by a simulation
study. After dealing with the theoretical impact of the three model characteristics on lambda,
empirical application is performed and the idea of implementing the parameter into a
systemic risk measure is presented. The codes used to obtain the results included in this
work are available on http://quantlet.de/d3/ia/.

Keywords:
Lasso, quantile regression, systemic risk, high dimensions, penalization parameter

JEL Classification:
C21, G01, G20, G32

Specification Testing in Nonparametric Instrumental Quantile Regression

Christoph Breunig

Abstract:
There are many environments in econometrics which require nonseparable modeling of a structural disturbance. In a nonseparable model, key conditions are validity of instrumental variables and monotonicity of the model in a scalar unobservable. Under these conditions the nonseparable model is equivalent to an instrumental quantile regression model. A failure of the key conditions, however, makes instrumental quantile regression potentially inconsistent. This paper develops a methodology for testing the hypothesis whether the instrumental quantile regression model is correctly specified. Our test statistic is asymptotically normally distributed under correct specification and consistent against any alternative model. In addition, test statistics to justify model simplification are established. Finite sample properties are examined in a Monte Carlo study and an empirical illustration.

Keywords:
Nonparametric quantile regression, instrumental variable, specification test, local alternative, nonlinear inverse problem

JEL Classification:
C12, C14

A first econometric analysis of the CRIX family

Shi Chen
Cathy Yi-Hsuan Chen
Wolfgang Karl Härdle
TM Lee
Bobby Ong

Abstract:
The CRIX (CRyptocurrency IndeX) has been constructed based on approximately 30 cryptos and captures high coverage of available market capitalisation. The CRIX index family covers a range of cryptos based on different liquidity rules and various model selection criteria. Details of ECRIX (Exact CRIX), EFCRIX (Exact Full CRIX) and also intraday CRIX movements may be found on the webpage of hu.berlin/crix. In order to price contingent claims one needs to first understand the dynamics of these indices. Here we provide a first econometric analysis of the CRIX family within a time-series framework. The key steps of our analysis include model selection, estimation and testing. Linear dependence is removed by an ARIMA model, the diagnostic checking resulted in an ARIMA(2,0,2) model for the available sample period from Aug 1st, 2014 to April 6th, 2016. The model residuals showed the well known phenomenon of volatility clustering. Therefore a further refinement lead us to an ARIMA(2,0,2)-t-GARCH(1,1) process. This specification conveniently takes care of fat-tail properties that are typical for financial markets. The multivariate GARCH models are implemented on the CRIX index family to explore the interaction.

Keywords:

JEL Classification:
C51, C52, G10

Forecasting Limit Order Book Liquidity Supply-Demand Curves with Functional
AutoRegressive Dynamics

Ying Chen
Wee Song Chua
Wolfgang K. Härdle

Abstract:
Limit order book contains comprehensive information of liquidity on bid and ask sides. We propose a Vector Functional AutoRegressive (VFAR) model to describe the dynamics of the limit order book and demand curves and utilize the fitted model to predict the joint evolution of the liquidity demand and supply curves. In the VFAR framework, we derive a closed-form maximum likelihood estimator under sieves and provide the asymptotic consistency of the estimator. In application to limit order book records of 12 stocks in NASDAQ
traded from 2 Jan 2015 to 6 Mar 2015, it shows the VAR model presents a strong predictability in liquidity curves, with R2 values as high as 98.5 percent for
insample estimation and 98.2 percent in out-of-sample forecast experiments. It produces accurate 5-; 25- and 50-minute forecasts, with root mean squared error as low as 0.09 to 0.58 and mean absolute percentage error as low as 0.3 to 4.5 percent.

Keywords:
Limit order book, Liquidity risk, multiple functional time series

JEL Classification:
C13, C32, C53

A Mortality Model for Multi-populations: A Semi-Parametric Approach

Lei Fang
Wolfgang K. Härdle
Juhyun Park

Abstract:
Mortality is different across countries, states and regions. Several empirical
research works however reveal that mortality trends exhibit a common pattern and
show similar structures across populations. The key element in analyzing mortality
rate is a time-varying indicator curve. Our main interest lies in validating the
existence of the common trends among these curves, the similar gender differences
and their variability in location among the curves at the national level. Motivated
by the empirical findings, we make the study of estimating and forecasting mortality
rates based on a semi-parametric approach, which is applied to multiple curves with
the shape-related nonlinear variation. This approach allows us to capture the common
features contained in the curve functions and meanwhile provides the possibility to
characterize the nonlinear variation via a few deviation parameters. These parameters
carry an instructive summary of the time-varying curve functions and can be further
used to make a suggestive forecast analysis for countries with barren data sets. In
this research the model is illustrated with mortality rates of Japan and China, and
extended to incorporate more countries. All numerical procedures are transparent and reproduced on www.quantlet.de.

Keywords:
Nonparametric smoothing, Parametric modeling, Common trend, Mortality, Lee-Carter method,
Multi-populations

JEL Classification:
C14, C32, C38, J11, J13

CRIX an Index for blockchain based currencies

Simon Trimborn
Wolfgang Karl Härdle

Abstract:
The S&P500 or DAX30 are important benchmarks for the financial industry. These and other indices describe different compositions of certain segments of the financial markets. For currency markets, the IMF offers the index SDR. Prior to the Euro, the ECU existed, which was an index representing the development of European currencies. It is surprising, though, to see that the common index providers have not mapped emerging e-coins into an index yet because with cryptos like Bitcoin, a new kind of asset of great public interest has arisen. Index providers decide on a fixed number of index constituents which will represent the market segment. It is a huge challenge to set this fixed number and develop the rules to find the constituents, especially since markets change and this has to be taken into account. A method relying on the AIC is proposed to quickly react to market changes and therefore enable us to create an index, referred to as CRIX, for the cryptocurrency market. The codes used to obtain the results in this paper are available via www.quantlet.de

Keywords:
Index construction, model selection, AIC, bitcoin, cryptocurrency, CRIX

JEL Classification:
C51, C52, G10

Academic Ranking Scales in Economics: Prediction and Imputation

Alona Zharova
Andrija Mihoci
Wolfgang Karl Härdle

Abstract:
We address the problem that often hampers decision making in academic institutions – incomplete research profiles. We suggest a framework for collating ranking data of scientists for comparison purposes. As the result of an analysis of the interconnectedness between HB sub-rankings through quantile regression, we propose a HB common score for scholars within the HB community. The cross-ranking dependence analysis of Handelsblatt, Research Papers in Economics and Google Scholar ranking schemes shows that researcher age and field of specialization – mapped onto the JEL classification codes – have a substantial impact on the resulting scores.

Keywords:
ranking, prediction, quantile regression, Handelsblatt, RePEc, Google Scholar

JEL Classification:
C81, C53, C21, M10

Factorisable Sparse Tail Event Curves with Expectiles

Wolfgang K. Härdle
Chen Huang
Shih-Kang Chao

Abstract:
Oberwolfach Report: New Developments in Functional and Highly Multivariate
Statistical Methodology

Keywords:
multivariate functional data, high-dimensional M-estimators, nuclear norm regularizer, factor analysis, expectile regression, fMRI, risk perception

JEL Classification:
C38, C55, C61, C91, D87

The German Labor Market Miracle, 2003 -2015: An Assessment

Michael C. Burda

Abstract:
This paper reviews the dramatic and widely noted developments in the German labor market in the past decade and surveys the most plausible reasons for these changes. Alternative hypotheses are compared and contrasted. I argue that the labor market reforms associated with the Agenda 2010 – the Hartz reforms – played a role at least as great as that of increasing flexibility of wage determination and the allocation of hours across workers. Until 2010, the German economic miracle could be accounted for by an expansion of part-time work, which has since been supplanted by a sustained expansion of full-time employment. Supported by wage flexibility in this segment, part-time employment represents an important new margin of flexibility in the German labor market.

Keywords:
German labor market miracle, Hartz reforms, part-time work, wage inequality

JEL Classification:
E24, J21

Leveraged ETF options implied volatility paradox: a statistical study

Wolfgang Karl Härdle
Sergey Nasekin
Zhiwu Hong

Abstract:
In this paper, we study the statistical properties of the moneyness scaling transformation by Leung and Sircar (2015). This transformation adjusts the moneyness coordinate of the implied volatility smile in an attempt to remove the discrepancy between the IV smiles for levered and unlevered ETF options. We construct bootstrap uniform confidence bands which indicate that in a statistical sense there remains a possibility that the implied volatility smiles are still not the same, even after moneyness scaling has been performed. This presents possible arbitrage opportunities on the (L)ETF market which can be exploited by traders. We build possible arbitrage strategies by constructing portfolios with LETF shares and options which possibly have a positive value at the point of creation and non-negative value at the expiration time. An empirical data application shows that there are indeed such opportunities in the market which result in risk-free gains for the investor. A dynamic "trade-with-the-smile" strategy based on a dynamic semiparametric factor model is presented. This strategy utilizes the dynamic structure of implied volatility surface allowing out-of-sample forecasting and information on unleveraged ETF options to construct theoretical one-step-ahead implied volatility surfaces. The codes used to obtain the results in this paper, are available on www.quantlet.de.

Keywords:
exchange-traded funds, options, moneyness scaling, arbitrage, bootstrap, dynamic factor models

JEL Classification:
C00, C14, C50, C58

Specification Testing in Random Coefficient Models

Christoph Breunig
Stefan Hoderlein

Abstract:
In this paper, we suggest and analyze a new class of specification tests
for random coefficient models. These tests allow to assess the validity of
central structural features of the model, in particular linearity in coefficients
and generalizations of this notion like a known nonlinear functional relationship.
They also allow to test for degeneracy of the distribution of a random coefficient,
i.e., whether a coefficient is fixed or random, including whether an associated
variable can be omitted altogether. Our tests are nonparametric in nature, and use
sieve estimators of the characteristic function. We analyze their power against
both global and local alternatives in large samples and through a Monte Carlo simulation study. Finally, we apply our framework to analyze the specification
in a heterogeneous random coefficients consumer demand model.

Keywords:
Nonparametric specification testing, random coefficients, unobserved heterogeneity, sieve minimum distance, characteristic function, consumer demand

JEL Classification:
C12, C14

lCARE - localizing Conditional AutoRegressive Expectiles

Xiu Xu
Andrija Mihoci
Wolfgang Karl Härdle

Abstract:
We account for time-varying parameters in the conditional expectile based value at risk (EVaR) model. EVaR appears more sensitive to the magnitude of portfolio losses compared to the quantile-based Value at Risk (QVaR), nevertheless, by fitting the models over relatively long ad-hoc fixed time intervals, research ignores the potential time-varying parameter properties. Our work focuses on this issue by exploiting the local parametric approach in quantifying tail risk dynamics. By achieving a balance between parameter variability and modelling bias, one can safely fit a parametric expectile model over a stable interval of homogeneity. Empirical evidence at three stock markets from 2005- 2014 shows that the parameter homogeneity interval lengths account for approximately 1-6 months of daily observations. Our method outperforms models with one-year fixed intervals, as well as quantile based candidates while employing a time invariant portfolio protection (TIPP) strategy for the DAX portfolio. The tail risk measure implied by our model finally provides valuable insights for asset allocation and portfolio insurance.

Keywords:
expectiles, tail risk, local parametric approach, risk management

JEL Classification:
C32, C51, G17

Nonparametric Estimation in case of Endogenous Selection

Christoph Breunig
Enno Mammen
Anna Simoni

Abstract:
This paper addresses the problem of estimation of a nonparametric regression function from selectively observed data when selection is endogenous. Our approach relies on independence between covariates and selection conditionally on potential outcomes. Endogeneity of regressors is also allowed for. In both cases, consistent two-step estimation procedures are proposed and their rates of convergence are derived. Also pointwise asymptotic distribution of the estimators is established. In addition, we propose a nonparametric specification test to check the validity of our independence assumption. Finite sample properties are illustrated in a Monte Carlo simulation study and an empirical illustration.

Keywords:
Endogenous selection, instrumental variable, sieve minimum distance, regression estimation, convergence rate, asymptotic normality, hypothesis testing, inverse problem

JEL Classification:
C14, C26

CRIX or evaluating Blockchain based currencies

Simon Trimborn
Wolfgang Karl Härdle

Abstract:
The S&P500 or DAX30 are important benchmarks for the financial industry. These and other
indices describe different compositions of certain segments of the financial markets. For
currency markets, the IMF offers the index SDR. Prior to the Euro, the ECU existed, which
was an index representing the development of European currencies. It is surprising,
though, to see that the common index providers have not mapped emerging e-coins into an
index yet because with cryptos like Bitcoin, a new kind of asset of great public interest
has arisen. Index providers decide on a fixed number of index constituents which will
represent the market segment. It is a huge challenge to set this fixed number and develop
the rules to find the constituents, especially since markets change and this has to be
taken into account. A method relying on the AIC is proposed to quickly react to market
changes and therefore enable us to create an index, referred to as CRIX, for the
cryptocurrency market.

Keywords:
Index construction, CRIX, risk analysis, bitcoin, cryptocurrency

JEL Classification:
C51, C52, G10

Estimating inflation expectation co-movement across countries

Shi Chen
Wolfgang Karl Härdle
Weining Wang

Abstract:
Inflation expectation is an important indicator for policy makers and financial investors. To capture a more accurate real-time estimate of inflation expectation on the basis of financial markets, we propose an arbitrage-free term structure model across different countries. We first estimate inflation expectation by modeling the nominal and the inflation-indexed bond yields jointly for each country. The joint dynamic model for inflation expectation is a cross sectional state space model combined with a GeoCopula model, which accounts for the default risk and the non Gaussian dependency structure over countries. We discover that the extracted common trend for inflation expectation is an important driver for each country of interest. Moreover, the model extracts informative estimates of inflation expectations and will provide good implications for monetary policies.

Keywords:
inflation expectation, arbitrage free, yield curve modelling, inflation risk

JEL Classification:
G12, E43, E31

TERES - Tail Event Risk Expectile based Shortfall

Philipp Gschöpf
Wolfgang Karl Härdle
Andrija Mihoci

Abstract:
A flexible framework for the analysis of tail events is proposed. The framework contains tail moment measures that allow for Expected Shortfall (ES) estimation. Connecting the implied tail thickness of a family of distributions with the quantile and expectile estimation, a platform for risk assessment is provided. ES and implications for tail events under different distributional scenarios are investigated, particularly we discuss the implications of increased tail risk for mixture distributions. Empirical results from the US, German and UK stock markets, as well as for the selected currencies indicate that ES can be successfully estimated on a daily basis using a one-year time horizon across different risk levels.

Keywords:
Expected Shortfall, expectiles, tail risk, risk management, tail events, tail moments

JEL Classification:
C13, C16, G20, G28

Tail Event Driven ASset allocation: evidence from equity and mutual funds’ markets

Wolfgang Karl Härdle
David Lee Kuo Chuen
Sergey Nasekin
Xinwen Ni
Alla Petukhina

Abstract:
Classical asset allocation methods have assumed that the distribution of asset returns is smooth, well behaved with stable statistical moments over time. The distribution is assumed to have constant moments with e.g., Gaussian distribution that can be conveniently parameterised by the first two moments. However, with market volatility increasing over time and after recent crises, asset allocators have cast doubts on the usefulness of such static methods that registered large drawdown of the portfolio. Others have suggested dynamic or synthetic strategies as alternatives, which have proven to be costly to implement. The authors propose and apply a method that focuses on the left tail of the distribution and does not require the knowledge of the entire distribution, and may be less costly to implement. The recently introduced TEDAS - Tail Event Driven ASset allocation approach determines the dependence between assets at tail measures. TEDAS uses adaptive Lasso based quantile regression in order to determine an active set of portfolio elements with negative non-zero coefficients. Based on these active risk factors, an adjustment for intertemporal dependency is made. The authors extend TEDAS methodology to three gestalts differing in allocation weights’ determination: a Cornish-Fisher Value-at-Risk minimization,
Markowitz diversification rule and naive equal weighting. TEDAS strategies significantly outperform other widely used allocation approaches on two asset markets: German equity and Global mutual funds.

Keywords:
adaptive lasso, portfolio optimisation, quantile regression, Value-at-Risk,
tail events

JEL Classification:
C00, C14, C50, C58

Copula-Based Factor Model for Credit Risk Analysis

Lu, Meng-Jou
Chen, Cathy Yi-Hsuan
Härdle, Karl Wolfgang

Abstract:
A standard quantitative method to access credit risk employs a factor model based on joint
multi-variate normal distribution properties. By extending a one-factor Gaussian copula
model to make a more accurate default forecast, this paper proposes to incorporate a
state-dependent recovery rate into the conditional factor loading, and model them by
sharing a unique common factor. The common factor governs the default rate and recovery
rate simultaneously and creates their association implicitly. In accordance with Basel III, this paper shows that the tendency of default is more governed by systematic
risk rather than idiosyncratic risk during a hectic period. Among the models considered,
the one with random factor loading and a state-dependent recovery rate turns out to be
the most superior on the default prediction.

Keywords:
Factor Model, Conditional Factor Loading, State-Dependent Recovery Rate

JEL Classification:
C38, C53, F34, G11, G17

An Adaptive Approach to Forecasting Three Key Macroeconomic Variables for Transitional China

Linlin Niu
Xiu Xu
Ying Chen

Abstract:
We propose the use of a local autoregressive (LAR) model for adaptive estimation and forecasting of three of China’s key macroeconomic variables: GDP growth, inflation and the 7-day interbank lending rate. The approach takes into account possible structural changes in the data-generating process to select a local homogeneous interval for model estimation, and is particularly well-suited to a transition economy experiencing ongoing shifts in policy and structural adjustment. Our results indicate that the proposed method outperforms alternative models and forecast methods, especially for forecast horizons of 3 to 12 months. Our 1-quarter ahead adaptive forecasts even match the performance of the well-known CMRC Langrun survey forecast. The selected homogeneous intervals indicate gradual changes in growth of industrial production driven by constant evolution of the real economy in China, as well as abrupt changes in interest rate and inflation dynamics that capture monetary policy shifts.

Keywords:
Chinese economy, local parametric models, forecasting

JEL Classification:
E43, E47

Stochastic Population Analysis: A Functional Data Approach

Lei Fang
Wolfgang K. Härdle

Abstract:
Based on the Lee-Carter (LC) model, the benchmark in population forecasting, a variety of extensions and modifications are proposed in this paper. We investigate one of the extensions, the Hyndman-Ullah (HU) method and apply it to Asian demographic data sets: China, Japan and Taiwan. It combines ideas of functional principal component analysis (fPCA), nonparametric smoothing and time series analysis. Based on this stochastic approach, the demographic characteristics and trends in different Asian regions are calculated and compared. We illustrate that China and Japan exhibited a similar demographic trend in the past decade. We also compared the HU method with the LC model. The HU method can explain more variation of the demographic dynamics when we have data of high quality, however, it also encounters problems and performs similarly as the LC model when we deal with limited and scarce data sets, such as Chinese data sets due to the substandard quality of the data and the population policy.

Keywords:
Functional principal component analysis; Nonparametric smoothing; Mortality forecasting; Fertility forecasting; Asian demography; Lee-Carter model, Hyndman-Ullah method

JEL Classification:
C14, C32, C38, J11, J13