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									           Faculty of Management and Economics
           Chair of Empirical Economics
           Chair of Empirical Economics and Health Economics



Power Law Distribution in High Frequency Financial
                     Data?
                An Econometric Analysis
                 Lora Todorova and Bodo Vogt


          Otto-von-Guericke-University Magdeburg




                Lora.todorova@med.ovgu.de
            Faculty of Management and Economics
            Chair of Empirical Economics
            Chair of Empirical Economics and Health Economics




Outline
 We test whether the large returns of selected
 German stocks are consistent with:

• The power law hypothesis
• The no-arbitrage condition
• The exponential distribution hypothesis

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               Faculty of Management and Economics
               Chair of Empirical Economics
               Chair of Empirical Economics and Health Economics




Data Description 1
• Time period               1/1999 – 12/1999
• Data type                 High frequency financial
                            data
• Basic quantity            Standardized simple price
                            returns
• Time scale                3 minutes


                                                                   3
              Faculty of Management and Economics
              Chair of Empirical Economics
              Chair of Empirical Economics and Health Economics




Data Description 2
• Exchange:        Germany                     US

• Stocks:          Daimler Chrysler Daimler Chrysler
                   SAP               SAP
                   Deutsche Telekom
                   Volkswagen
                   Bauer, Allianz

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             Faculty of Management and Economics
             Chair of Empirical Economics
             Chair of Empirical Economics and Health Economics




Data Description 3
• Time series      a) including all returns
                   b) including only simultaneous
                       returns
           9:00                 15:30      17:30
Germany:                                            t
                                                                         t
                          US:                                    17:00
                                9:30        11:30



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                      Faculty of Management and Economics
                      Chair of Empirical Economics
                      Chair of Empirical Economics and Health Economics




Implications of the Universality Property
        Ger                              US         Ger/Sim        US/Sim
DT, VW, Allianz, Bayer, DCh., SAP       DCh, SAP    DCh, SAP       DCh, SAP

                                                                



                                    =       =                  =   =
              =




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                     Faculty of Management and Economics
                     Chair of Empirical Economics
                     Chair of Empirical Economics and Health Economics




Implications of the Scale Invariance Property
       GER                           US        GER/Sim            US/Sim
DT, VW, Allianz, Bayer, DCh, SAP   DCh, SAP        DCh, SAP           DCh, SAP

                                                                



                                         =     =              =   =




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                    Faculty of Management and Economics
                    Chair of Empirical Economics
                    Chair of Empirical Economics and Health Economics




Implications of the No-Arbitrage condition
        GER                             US        GER/Sim        US/Sim
DT, VW, Allianz, Bayer, DCX, SAP       DCX, SAP   DCX, SAP       DCX, SAP




                                   =    =                    =   =




“=“ Same probability distribution

                                                                            8
                   Faculty of Management and Economics
                   Chair of Empirical Economics
                   Chair of Empirical Economics and Health Economics




Overview of Existing Literature
•   Gopikrishnan et. al. (1998), (1999)
                                                      (α-1)≈3
•   Lux (1996)                                        US, Germany, India
•   Pan /Sinha (2006), (2008)
•   Matia et. al. (2004)                              Exponential decay of
                                                      Indian stock price returns
•   Barari/ Mitra (2008)
•   Bauke (2007)                                       Least squares vs. Maximum
•   Kinsella/ O`Brien (2009)                           likelihood

•   Goldstein/ Morris/ Yen (2004)                      Kolmogorov-Smirnov
                                                       Goodness-of-fit test
•   Clauset/ Shalizi/Newman (2007)
                                                                               9
               Faculty of Management and Economics
               Chair of Empirical Economics
               Chair of Empirical Economics and Health Economics



Power Law – Estimation and Testing?
Most Frequently Used Techniques
• Estimation of the scaling                  Least squares fitting
  exponent

• Estimation of the lower                    Visualization
  bound                                      techniques

• Testing the power law                      No quantitative
  hypothesis                                 testing
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                Faculty of Management and Economics
                Chair of Empirical Economics
                Chair of Empirical Economics and Health Economics




Most Widely Reported Results
• Power law?                 Yes, with (α-1)≈3

• Is it universal?           Yes, it is found to hold
                             for different time periods
                             and diferent exchanges

• Is it scale-free?          Yes, the same relation is
                             found to hold for
                             1 min ≤ Δt ≤1 month


                                                                    11
               Faculty of Management and Economics
               Chair of Empirical Economics
               Chair of Empirical Economics and Health Economics




Our Approach (Clauset/ Shalizi/ Newman 2007)
• Estimation of the           Maximum likelihood estimation
  scaling exponent

• Estimation of             Kolmogorov-Smirnov Statistics
  the lower bound

• Two levels of testing     a) Validity of the properties of PL
  the power law             b) Kolmogorov-Smirnov Statistics
  hypothesis

                                                                   12
                  Faculty of Management and Economics
                  Chair of Empirical Economics
                  Chair of Empirical Economics and Health Economics




MLE and KS Statistics – Exact Procedure 1
                                                                                1
                                             ^                n
                                                                    xi (t ) 
• Maximum likelihood estimator –  MLE                1  n   ln         
                                                              t 1 xmin 
                                         ^
• Estimation of the lower bound – x min is the value of xi (t ) that
  monimizes the distance (D) between the empirical data and the
  best-fit power law model for xi (t )  xmin :
        D  max S  xi (t )  x   P  xi (t )  x  , xi (t )  xmin

                       Best-fit power        Empirical
                       law model             data
                                                                                     13
                 Faculty of Management and Economics
                 Chair of Empirical Economics
                 Chair of Empirical Economics and Health Economics




MLE and KS Statistics – Exact Procedure 2
• Testing the power law hypothesis

 Sampling a large number of synthetic data sets from true power
  law distribution;
 Fitting them individually to their best-fit power law model;
 Measuring D for each of these samples;
 Calculating p-value – the fraction of distances of the synthetic
  data sets that are larger than the distance of the empirical data;


                                                                       14
                               Faculty of Management and Economics
                               Chair of Empirical Economics
                               Chair of Empirical Economics and Health Economics




Graphical Representation of the Data 1
                  a) Deutsche Telekom/ Ger                             b) Volkswagen/ Ger
             0                                                  0
            10                                                 10

             -1                                                 -1
            10                                                 10

             -2                                                 -2
Pr(X  x)




                                                   Pr(X  x)
            10                                                 10

             -3                                                 -3
            10                                                 10

             -4                                                 -4
            10                                                 10

             -5                                                 -5
            10 -6       -4     -2     0       2
                                                               10 -7        -5    -3       -1    1
              10      10      10     10      10                  10       10     10       10    10
                               x                                                      x




                                                                                                     15
                                    Faculty of Management and Economics
                                    Chair of Empirical Economics
                                    Chair of Empirical Economics and Health Economics




Graphical Representation of the Data 2
                  c) Allianz/ Ger                                           d) Bayer/ Ger
             0                                                       0
            10                                                      10

             -1                                                      -1
            10                                                      10

             -2                                                      -2
Pr(X  x)




                                                        Pr(X  x)
            10                                                      10

             -3                                                      -3
            10                                                      10

             -4                                                      -4
            10                                                      10

             -5                                                      -5
            10 -5         -3          -1      1
                                                                    10 -5          -3        -1    1
              10        10          10       10                       10         10         10    10
                                x                                                       x




                                                                                                       16
                                   Faculty of Management and Economics
                                   Chair of Empirical Economics
                                   Chair of Empirical Economics and Health Economics




Graphical Representation of the Data 3
                  e) Daimler Chrysler/ Ger                                 f) SAP/ Ger
             0                                                      0
            10                                                     10

             -1                                                     -1
            10                                                     10

             -2                                                     -2
Pr(X  x)




                                                       Pr(X  x)
            10                                                     10

             -3                                                     -3
            10                                                     10

             -4                                                     -4
            10                                                     10

             -5                                                     -5
            10 -5         -3         -1      1
                                                                   10 -5         -3           -1    1
              10        10         10       10                       10         10           10    10
                               x                                                         x




                                                                                                        17
                                   Faculty of Management and Economics
                                   Chair of Empirical Economics
                                   Chair of Empirical Economics and Health Economics




Graphical Representation of the Data 4
                  g) Daimler Chrysler/ US                                  h) SAP/ US
             0                                                      0
            10                                                     10

             -1                                                     -1
            10                                                     10

             -2                                                     -2
Pr(X  x)




                                                       Pr(X  x)
            10                                                     10

             -3                                                     -3
            10                                                     10

             -4                                                     -4
            10                                                     10

             -5                                                     -5
            10 -5         -3         -1      1
                                                                   10 -5         -3          -1    1
              10        10         10       10                       10        10           10    10
                               x                                                        x




                                                                                                       18
                                     Faculty of Management and Economics
                                     Chair of Empirical Economics
                                     Chair of Empirical Economics and Health Economics




Graphical Representation of the Data 5
                   i) Daimler Chrysler/ Sim/ Ger                            j) SAP/ Sim/ Ger
              0                                                        0
            10                                                       10


              -1                                                       -1
            10                                                       10
Pr(X  x)




                                                         Pr(X  x)
              -2                                                       -2
            10                                                       10


              -3                                                       -3
            10                                                       10


              -4                                                       -4
            10 -5           -3            -1        1
                                                                     10 -5            -3        -1    1
              10          10            10        10                   10           10         10    10
                                 x                                                         x




                                                                                                          19
                                     Faculty of Management and Economics
                                     Chair of Empirical Economics
                                     Chair of Empirical Economics and Health Economics




Graphical Representation of the Data 6
                   k) Daimler Chrysler/ Sim/ US                             l) SAP/ Sim/ US
              0                                                        0
            10                                                       10


              -1                                                       -1
            10                                                       10
Pr(X  x)




                                                         Pr(X  x)
              -2                                                       -2
            10                                                       10


              -3                                                       -3
            10                                                       10


              -4                                                       -4
            10 -5           -3            -1        1
                                                                     10 -5           -3        -1    1
              10          10            10        10                   10          10         10    10
                                 x                                                        x




                                                                                                         20
               Faculty of Management and Economics
               Chair of Empirical Economics
               Chair of Empirical Economics and Health Economics




Formal Results
Company/Exchange                               α-1
Deutsche Telekom/ Ger                         3.5705
VW/ Ger                                       4.8975
Allianz/ Ger                                  4.1936
Bayer/ Ger                                    3.9897
Daimler Chrysler/ Ger                         3.8407
SAP / Ger                                     4.0758




                                                                   21
                  Faculty of Management and Economics
                  Chair of Empirical Economics
                  Chair of Empirical Economics and Health Economics



Company/Exchange                                   α-1
Deutsche Telekom/ Ger                             3.5705
VW/ Ger                                           4.8975
Allianz/ Ger                                      4.1936
Bayer/ Ger                                        3.9897


Daimler Chrysler/ Ger                             3.8407
                                                             = “univ“
Daimler Chrysler/ US                              4.0088
SAP/ Ger                                          4.0758
                                       “univ“=
SAP/ US                                           4.0765



                                                                        22
                Faculty of Management and Economics
                Chair of Empirical Economics
                Chair of Empirical Economics and Health Economics


Company/Exchange                                α-1
Deutsche Telekom/ Ger                          3.5705
VW/ Ger                                        4.8975
Allianz/ Ger                                   4.1936
Bayer/ Ger                                     3.9897
Daimler Chrysler/ Ger                          3.8407
SAP/ Ger                                       4.0758
Daimler Chrysler/ US                           4.0088
SAP/ US                                        4.0765
Daimler Chrysler/ Sim/ Ger                     5.7534
                                                          = “univ“
Daimler Chrysler/ Sim/ US                      4.0221
SAP/ Sim/ Ger                                  4.361
SAP/ Sim/ US                         “univ“=   4.468
                                                                     23
                Faculty of Management and Economics
                Chair of Empirical Economics
                Chair of Empirical Economics and Health Economics



Company/Exchange                                   α-1
Deutsche Telekom/ Ger                             3.5705
VW/ Ger                                           4.8975
Allianz/ Ger                                      4.1936
Bayer/ Ger                                        3.9897
Daimler Chrysler/ Ger                             3.8407
                                                             = “scl. inv.“
Daimler Chrysler/ Sim/ Ger                        5.7534
Daimler Chrysler/ US                              4.0088
                                  “scl. inv.“=
Daimler Chrysler/ Sim/ US                         4.0221
SAP/ Ger                                          4.0758
                                                           = “scl. inv.“
SAP/ Sim/ Ger                                     4.361
SAP/ US                                           4.0765
SAP/ Sim/ US                       “scl. inv.“=   4.468
                                                                             24
                 Faculty of Management and Economics
                 Chair of Empirical Economics
                 Chair of Empirical Economics and Health Economics




Company/Exchange                   α-1                      p
Deutsche Telekom/ Ger             3.5705                  0.002
VW/ Ger                           4.8975                  0.713
Allianz/ Ger                      4.1936                  0.313
Bayer/ Ger                        3.9897                  0.049
Daimler Chrysler/ Ger             3.8407                    0
Daimler Chrysler/ US              4.0088                  0.115
SAP/ Ger                          4.0758                  0.64
SAP/ US                           4.0765                   0.4
Daimler Chrysler/ Sim/ Ger        5.7534                  0.659
Daimler Chrysler/ Sim/ US         4.0221                  0.435
SAP/ Sim/ Ger                      4.361                  0.081
SAP/ Sim/ US                       4.468                  0.425
                                                                     25
             Faculty of Management and Economics
             Chair of Empirical Economics
             Chair of Empirical Economics and Health Economics




Exponential Distribution and Stock Price Returns
• Density Function
                    1  x/ 
             p( x)  e , x  0,   0
                    

• Chi-square goodness-of-fit test for the tails of
  the sample distributions



                                                                 26
                    Faculty of Management and Economics
                    Chair of Empirical Economics
                    Chair of Empirical Economics and Health Economics




Exponential Distribution Results
Company/Exch.         T*              Company/Exch.                  T*
D. Telekom/ Ger       114.63          Daim. Chr./ US                 115.82
VW/ Ger               116.20          SAP/ US                        113.92
Allianz/ Ger          114.16          Daim. Chr./ Sim/ Ger           131.21
Bayer/ Ger            115.71          SAP/ Sim/ Ger                  138.28
Daim. Chr./ Ger       115.74          Daim. Chr./ Sim/ US            115.61
SAP/ Ger              115.57          SAP/ Sim/ US                   114.96


       The results for all time series are significant at 5% significance level



                                                                                  27
              Faculty of Management and Economics
              Chair of Empirical Economics
              Chair of Empirical Economics and Health Economics




Conclusion
•   Power law
-   KS Statistics                           No
-   Universality                            No
-   Scale Invariance                        No
•   No-arbitrage condition                  No
•   Exponential distribution                Yes

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