1 DEPARTMENTOFMATHEMATICSUPPSALAUNIVERSITY EMPIRICAL DATA AND MODELING OF FINANCIAL AND ECONOMIC PROCESSES by Maciej Klimek

2 Bad news from Goldman Sachs

3 Financial theories vs. changing reality OLD, BUT PERSISTENT: The moving target problems:  insufficient sequences of statistical data  “uncertainty principle” = beliefs/practice changing the market Convenience more important than realism (eg CAPM, prevalence of Gaussian distribution, ignoring areas of applicability etc) “Natural science” approach to social phenomena (major weakness of Econophysics) NEW, LARGELY UNEXPLORED: Theoretical background pre-dates the IT-revolution (eg Efficient Market Hypothesis) Globalization of markets vs. theories based on several developed countries (eg new research: Virginie Konlack and Ivivi Mwaniki – comparing stock markets in Kenya and Canada) Complexity of financial instruments obscuring risks (eg subprime mortgages vs. CDO’s and the like)

4 Example: ABN-test Okabe, Matsuura, Klimek 2002

5 Notation Block frame approach – Klimek, Matsuura, Okabe 2007

6 Block frames

7 Basic theorem

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9 Fundamental properties

10 The blueprint algorithm

11 Probability and Hilbert Spaces

12 Hilbert lattices

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14 Basic objects associated with time series:

15 dissipation coefficients

16 MAIN IDEA

17 Instead of analysing a d-dimensional time series X n We use the d(m+1) dimensional time series This is computationally intensive, hence the need for efficient algorithms!

18 Example: Tests of stationarity Given time series data X(n) calculate the sample covariance Use the blueprint algorithm to calculate the alleged fluctuations ν + (n) Normalize: W(n) -1 ν + (n), where W (n) 2 =V (n), W (n) -1 is the Moore-Penrose pseudoinverse of W (n) and Apply a white noise test to the resulting data Original version: Okabe & Nakano 1991 A weak stationarity test:

19 The ABN – test If a stochastic process is strictly stationary and P is a Borel function of k variables, then the process is also strictly stationary Strict stationarity implies weak statinarity Given a time series test for breakdown of weak stationarity a large selection of series constructed through polynomial compositions. These new series are part of the information structure of the original one!

20 Applications: Forecasting “Extended” stationarity analysis Causality tests General adaptive modeling of time series improving on ARCH, GARCH and similar models. Volatility modelling.

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