U. Leiden, Feb. 22, 2010 Exercice #2 DFA: Detrended ...pjhdent/Exercise2MA.pdf · U. Leiden, Feb....
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U. U. LeidenLeiden, Feb. 22, 2010, Feb. 22, 2010Exercice Exercice #2#2
DFA: DFA: Detrended Detrended Fluctuation Fluctuation AnalysisAnalysis
Marcel Ausloos (*)7 rue des Chartreux, B-4122 Neupré
(*) previously at GRAPES(*) previously at GRAPES(Group for Research and Applications of Physics in Economy and Sociology)(Group for Research and Applications of Physics in Economy and Sociology)
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MotivationMotivation
in in order order to to understand understand how how strong strong are Longare LongRange Range and and Short Range fluctuationsShort Range fluctuations
to to suggest some investment suggest some investment techniquetechnique
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Get rich Get rich ? ? (à la Ausloos)(à la Ausloos)
obtain the obtain the power power law behavior describing the rmslaw behavior describing the rms..deviation deviation of of the the fluctuationsfluctuations
as a as a function function of of the window the window sizesize see see αα value value and and range of range of windowswindows
calculate the calculate the time time dependence dependence of of αα exponent exponent moving moving a a window with appropriate window with appropriate size size along the along the time axis,time axis,
supposing stationarity supposing stationarity of of the increments the increments !!!!!! invent some investment strategyinvent some investment strategy calculate the returns calculate the returns (gains, of course)(gains, of course)
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DEFINITIONSDEFINITIONS
““VariabilityVariability””
““Log returnsLog returns””
““Normalized variabilityNormalized variability””
DΔt y(t) = y(t+Δt) - y(t)
Z(t,Δt) = [DΔt y(t) - <DΔt y>] /sΔt
Time lag effect ?
L Z(t ,Δt) = log[ y(t+Δt) /y(t) ]
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Standard & Poor 500Standard & Poor 500
5056 data points
http://finance.yahoo.com
(normalized)
Jan. 01,1980- Dec. 31,1999
Daily data
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S&P500 S&P500 PDF returnsPDF returns
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Returns Returns 97-0297-02
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Southern Oscillation IndexSouthern Oscillation Index
P (Darwin) -
P (Tahiti) =
1672 data points
http://www.cpc.ncep.noaa.gov/data/indices/
(normalized)
1886-2006
Monthly data
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Detrended Detrended Fluctuation AnalysisFluctuation Analysis (integrated series or not?)(integrated series or not?)
x=0.56
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DFA DFA webweb
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Detrended Detrended Fluctuation AnalysisFluctuation Analysis
LongRangeCorrelations ?
on SP500increments
Brownianvalues
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SP500 80-99 SP500 80-99 ββ
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SGPSGP
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Detrended Detrended Fluctuation AnalysisFluctuation Analysis
LongRangeCorrelations ?
on SOIincrements
NonBrownianvalues
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Polish Polish group DFA for crashesgroup DFA for crashes
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DAX DFA(q)DAX DFA(q)
+ DFAq
+ DFA q,m
Variants/generalizations
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DFA(q)DFA(q)word length (L); word frequency (F)word length (L); word frequency (F)
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DFA DFA forecasting strategyforecasting strategy If If αα < 0.5< 0.5 AntipersistenceAntipersistence Negative correlationNegative correlation «« very noisyvery noisy »» If If αα <<<< 0.5<<<< 0.5 up, down, up, down, up,up, down, up, down, up,
down, up, down,..down, up, down,.. .... .... probabilisticprobabilistic .. αα = 0 : = 0 : white white noisenoise
If If αα > 0.5> 0.5 PersistencePersistence Positive Positive correlationcorrelation «« very smoothvery smooth »» If If αα >>>> 0.5>>>> 0.5 Up, up, up, up, up, ..., down,Up, up, up, up, up, ..., down,
down, down, ..down, down, .. .... .... probabilisticprobabilistic αα = 1 : = 1 : smooth smooth lineline
αα = 0.5 (or = 0.5 (or ββ = 2) : = 2) : Brownian Brownian motionmotion
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Local DFA: DJIA Jan.84-Aug.97Local DFA: DJIA Jan.84-Aug.97
““locallocal”” (or temporal) (or temporal)DFADFA
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«« LocalLocal » » DFA DFA
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BEF - DEMBEF - DEM ((veryvery) ) antipersistentantipersistent α α = 0.23= 0.23 ...... N.B.N.B.
NLG/BEFNLG/BEF : 0.26: 0.26 DKK/BEF : 0.31DKK/BEF : 0.31 FRF/BEFFRF/BEF : 0.37: 0.37
N.B.N.B. JPY/USDJPY/USD : 0.55: 0.55 USD/CAD : 0.55USD/CAD : 0.55 GBP/USDGBP/USD :: 0.550.55 USD/DEM : 0.55USD/DEM : 0.55
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USD/DEM gainUSD/DEM gain
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GoldGold
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Make Make moneymoney