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Analyzing the Lee and Mykland Statistic

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Title: Analyzing the Lee and Mykland Statistic


1
Analyzing the Lee and Mykland Statistic
  • Econ 201FS
  • Duke University
  • April 4, 2007
  • Peter Van Tassel

2
Outline
  • The Paper Jumps in Real-Time Financial Markets
    A New Nonparametric Test and Jump Dynamics
  • Subtleties of the statistic
  • Quantity of flagged jumps, window size, sampling
    frequency
  • Quantity of flagged jumps, window size, time of
    day
  • Comparison to BNS

3
Breakdown of Paper
  • Section 1 Theoretical Model for the Test Its
    Asymptotic Theory
  • Section 2 Misclassifications, failure to detect
    and false positives
  • Section 3 Monte Carlo Simulation
  • Section 4 Implications of Identifying Jump
    Dynamics
  • Section 5 Conclusion

4
The Test Statistic
  • Presented in section 1. Drift term (of order dt)
    is assumed to be negligible compared to the
    diffusion term (of order dt.5) and jump
    component (of order 1). A second statistic is
    discussed in the back of the paper that does not
    assume the drift term is zero.

5
Window Size
  • Following Pollard (2002), we use Op notation
    throughout this paper to mean that, for random
    vectors Xn and non-negative random variable
    dn, Xn Op(dn), if for each e gt 0, there
    exists a finite constant Me such that P(abs(Xn))
    lt e eventuallyIt also satisfies the stochastic
    volatility plus finite activity jump
    semi-martingale class in Barndorff-nielsen and
    Sehphard (2004) and the reference therein. p. 7

6
Optimal Choice of Window
  • Optimal Window Size K The optimal choice for
    window size is studied by simulationIt turns out
    that it is optimal for K to be the smallest
    integer in the condition set, K Op(?tª) s.t. -1
    lt a lt .5, because it gives the lowest mean
    squared error in our simulation. p. 27
  • We suggest the smallest integer K that satisfies
    the necessary condition as an optimal choice for
    K. Our specific recommendation of optimal window
    sizes for 1 week, 1 day, 1 hour, 30 minute, 15
    minute, and 5 minute data are 7, 16, 78, 110,
    156, and 270, respectively. p. 11

7
Monte Carlo Simulation
  • We compare the performance of the three tests in
    terms of probability of global success in
    detecting actual jumps within a given interval
    and probability of global spurious detection of
    jumps in that interval. Our test does not use the
    conventional terms of size and power, but
    introduces the misclassification of jumps in
    Section 2, because the detection criterion for
    our test is based on the distribution of maximums
    of null distribution, which is diferent from the
    usual hypothesis tests and we do not select one
    model over another when we do single test for
    jump detection. In essence, the probability of
    global success in detecting actual jumps is the
    power of the test and the probability of global
    spurious detection of jumps is the size of the
    test. p. 23.
  • Opportunity here for interesting research!

8
Subtleties of the Statistic
  • Z1 (Quantity of Flagged Jumps, Time of Flagged
    Jumps, Window Size)
  • SPY, PEP, KO, BMY Level plots and 3d plots
  • 17.5 minute sampling frequency for all plots!
  • Z2 (Quantity of Flagged Jumps, Sampling
    Frequency, Window Size)
  • SPY 3d plots

9
SPY Level Plot
10
SPY 3D Plot 1
11
SPY 3D Plot 2 3
12
Macroeconomic Data Releases
  • The High-Frequency Effects of US Macroeconomic
    Data Releases on Prices and Trading Activity in
    the Global Interdealer Foreign Exchange Market
    November 2004, The Federal Reserve Board,
    International Finance Discussion Papers, Chaboud,
    Chernenko, Howorka, Krishnasami, Liu, Wright
  • 830am GDP, Nonfarm Payrolls, Business
    inventory, Durable goods orders, Housing Starts,
    Initial Claims, Personal Consumption Expenditure,
    Personal Income, PPI, Retail Sales, Trade Balance
    Data
  • 10am Consumer Confidence, Factory Orders, ISM
    Index, New Existing Home Sales
  • 215pm Federal Open Market Committee
    announcements
  • Chicago Mercantile Exchange opens at 820am and
    closes at 3pm
  • Results
  • 1. Spikes in trading volume around the time of
    planned announcements
  • 2. Systematic relationship between the surprise
    component of the news announcement and the level
    of the exchange rate
  • 3. Macroeconomic announcements are immediately
    followed by higher trading volume and volatility,
    and that volume and volatility remain elevated
    for a period of time
  • Rt,h ßhst et st is surprise component
    compared to market expectations
  • Vt,h ah ?habs(st) et again st is the
    surprise component and v is volume

13
PEP Level Plot
14
PEP 3D Plot 1
15
PEP 3D Plots 2 3
16
BMY KO
17
Sampling Frequency
18
Sampling Frequency
19
BNS Comparison Tripower statistic used in
previous presentations flags 28 jumps for PEP
using 5 minute returns. Some preliminary results
that compare BNS to Lee / Mykland are presented
below.
Lee Mykland Lee Mykland Lee Mykland

Sampling Frequency Matches Flagged Jumps

5 minutes 28 1031
     
17.5 minutes 17 559
     
385 minutes 3 37
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