## B1 - Dynamic factor models

Statistical modelling of time series in high dimensions and in nonstationary situations requires a well-tuned composition of tools for dimension reduction, adaptive selection techniques and numerical tractability. A prominent example being Functional Principal Components Analysis (FPCA) approximating smooth random functions by time varying lower dimensional objects. Numerous applications exist for economic, finance, energy and weather data, but also for extremely high dimensional FRMI signals of neuroeconomics, see e.g. Majer, Härdle, Heekeren and Mohr (2014). Todays data structures in these application fields, though, demand different theoretical constructs: networks and tails. Whereas FPCA is an elegant tool to describe the variation around a theoretical mean of curves, it is the term "mean“ that makes it inutile for tail event variations that are elements of e.g. risk management and regulation. One rather needs to think about FDATES - Functional Data Analysis for Tail Event Structures. Also dynamic networks occurring in e.g. systemic risk, tweet systems and blogs require new mathematical languages and frameworks for their analysis, see Chen, Härdle, Okhrin (2017, JoE submitted).

In financial risk management, extreme tail event curves of asset returns indicate the lower bound of the potential loss, which is of interest for both the risk manager and market regulator. At the same time, this is a high dimensional problem as there are often several hundreds or thousands of asset returns to be considered that might accumulate into systemic risk. In weather and temperature analysis: high and small tail events give the range of possible temperature variation, which is useful for crop growth or to study climate change. In Energy Markets, as an example, consider the impact of wind-sensor measurements on the spot market for electricity. The wind sensors on the turbines form a network which produces a continuous stream of data. Only by aggregating sensor data can an accurate estimate of the risk of a thunderstorm, a tail event, be computed. Such a tail event leads to an immediate emergency halt of the individual wind turbines affected; the local shortage of energy supply triggers rising prices on the spot market, mostly via automated trades. Eventually, volatility of energy prices can evoke adaptations of the infrastructure and even consumer behaviour on the meta scale, see Härdle and Melzer (2017).

FPCA describes linear dependence and works well if there are no outliers. However, knowing linear dependence does not lead to the knowledge of the dynamics in the tails or lower and/or upper bounds. Moreover, for non-Gaussian and highly asymmetric (skewed) data, the methods based on covariance structure are highly corrupted if no correction is made. A customized FDATES may be based on a representation by Härdle et al. (2016): Tail Event Curve (TEC) analysis may be constructed from eigenvectors of an appropriately weighted covariance matrix. This line of research opens up a new statistical technique to describe variations of high dimensional objects in tail event situations.

**Coordination**

**Wolfgang Karl Härdle**: His main interests are non- and semiparametrics statistics and econometrics. His research includes work in nonparametric modelling, local adaptive models, reduction techniques, stationary models, quantile regression.

**Zongwu Ca**i: His main interests are Econometrics, Quantitative Finance, Nonlinear Time Series. His research includes work in Theoretical and applied econometrics, quantitative finance and risk management, nonparametric curve estimation problems, nonlinear and nonstationary time series, panel data analysis.

**Weining Wang**: Her research interest are on financial econometrics and statistics. In particular, her research includes topics like non and semiparametric statistics, network models, high dimensional time series analysis, spatial temporal copula models, etc.

**Muyi Li**: Her research interests are Time series analysis, high dimensional financial econometrics, model specification tests and diagnostic checks.

**Brenda López-Cabrera**: Her research interests are portfolio optimization, empirical and computational finance and applications within the field of statistical analysis of insurance, finance and energy. She concentrates on economic risk of natural hazards and focuses on Catastrophe Bonds, Weather and Energy Markets.

**Yu Ren**: His main interests are Financial Econometrics, Applied Econometrics, Econometric Theory. His research includes work on testing of stochastic discount factor and China finance.

**Yingxing Li**: Her research interests are about mathematical statistics. In particular, she concentrates on research topics including non and semiparametric regression, longitudinal and functional data analysis, dimension reduction, and model specification tests.

**Bernd Fitzenberger**: His research interests are microeconometrics (panel data regression, quantile regression) and empirical applications in labour economics and the economics of education. Part of his research focuses on the econometric analysis of wage inequality

**Christoph Breunig**: His research interests are in developing methodology to account for econometric analysis of unobserved individual heterogeneity and endogenous selection.

**Haiqiang Chen**: His main interests are Financial Econometrics, Time Series Econometrics and Financial Economics. His current research includes the estimation and inference for nonlinear nonstionary time series models, quantitative prediction models and high frequency data.

**Ying Fang**: His main interests are Econometrics, Applied Econometrics, Economy of China. His research includes work in nonparametric and semi-parametric method, panel data analysis, and instrumental variable selection.

**Ming Lin**: His main interests are Monte Carlo Methods, self selection, dimension reduction methods. His research includes work in Monte Carlo algorithm, Bayesian statistics, nonparametric statistics.

**Exemplary PhD-Theses**

- Liquidity Supply-Demand Curves with Functional AutoRegressive Dynamics

The limit order book (LOB) contains comprehensive information of liquidity on bid and ask sides. A Vector Functional AutoRegressive (VFAR) model describes the simultaneous dynamics of liquidity supply and demand curves and may be used to forecast the joint evolution of the multiple functional time series. A closed form of a sieve based MLE may be employed and its asymptotic consistency may be developed. The shape of the liquidity curves reflect the liquidity level that the steeper the curves are, the less price impact there is for large orders and hence the more liquidity is supplied or consumed in the market. It is important to understand the dynamics of the LOB and/or liquidity curves, since as one determinant of market quality, liquidity attracts much attention of regulators, market makers and traders. An imbalance in market liquidity creates major challenges not just for market participants but also for the financing structure of the economy in long term.

- DYTEC - Dynamic tail event curves

Today’s financial data, in particular, high-frequency data provides us with huge quantities of information that enables us to calculate and control future risk. In doing so one is primarily concerned about extreme, so called "tail", events and is interested in including their dynamic effects into a modelling step. DYTEC is an extensions of fPCA for the reduction of dimensionality to capture tail event variability, dynamics and their dependence over time. Applications can be found in climatology, where daily data over years about temperature, rainfalls or strength of wind are available. With the resulting factor loadings one may price weather derivatives or develop trading strategies. Technically speaking one needs to develop an analogue of PCA in an asymmetric norm, rather than the classic L2-type orthogonal projection. This will cover both quantiles and expectiles. This concept of an asymmetric check function may be extended to factor models. The complexity of the number of dependent variables, should be reduced by different penalizations, e.g. LASSO or nuclear norm.

- Multilevel fPCA on fMRI data

In neuroeconomics, an important research question is to investigate individuals' brain activity and infer their risk attitude and perception. fMRI, as a non-invasive technique that records brain signal, could provide ample information in the form of a time series consisting of more than 1000 images with roughly 100,000 voxels for each individual. To address the challenges due to the complicated dynamics of such a massive data set, we exploit the multilevel functional principle component analysis to extract both intra- and inter- subject information. To be specific, we employ the fast multivariate Penalized spline techniques to identify the active brain regions and project the high dimensional input onto a low dimensional subspace. We then analyse the relationship between the estimated loadings and individuals' risk attitudes to demonstrate the effectiveness of our methods.

**References**

- Chen CYH, Härdle W, Okhrin Y (2017) Tail event driven networks of SIFIs, SFB 649 Discussion Paper 2017-004 (submitted to JoE).
- Majer P, Mohr PNC, Heekeren H R, Härdle WK (2015) Portfolio Decisions and Brain Reactions via the CEAD method, Psychometrika, DOI: 10.1007/s11336-015-9441-5
- Tran N, Burdejovà P, Osipenko M, Härdle W (2016) Principal Component Analysis in an Asymmetric Norm, SFB 649 Discussion Paper 2016-040 (submitted to J Royal Stat Soc).