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EMPIRICAL DATA AND MODELING OF
FINANCIAL AND ECONOMIC PROCESSES
by
Maciej Klimek
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Bad news from
Goldman Sachs
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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)
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Example: ABN-test
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Okabe, Matsuura, Klimek 2002
Notation
Block frame approach – Klimek, Matsuura, Okabe 2007
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Block frames
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Basic theorem
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Fundamental properties
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The blueprint algorithm
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Probability and Hilbert Spaces
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Hilbert lattices
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Basic objects associated with time series:
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dissipation coefficients
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MAIN IDEA
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Instead of analysing a d-dimensional time series Xn
We use the d(m+1) dimensional time series
Xn


 n

 P1  X 0 , , X n  
 Pn  X , , X  
n
 2 0



 n

 Pm  X 0 , , X n  

 n 0
This is computationally
intensive, hence the
need for efficient
algorithms!
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Example: Tests of stationarity
A weak stationarity test:
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
V ( n )     n  ,   n 
Apply a white noise test to the resulting data
Original version: Okabe & Nakano 1991
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The ABN – test
• If a stochastic process
X n is strictly stationary and P
is a Borel function of k variables, then the process
P  X nk 1,
, Xn 
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!
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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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Contact:
[email protected]
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