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

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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

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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.

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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

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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

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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!

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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

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SPY Level Plot
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SPY 3D Plot 1
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SPY 3D Plot 2 3
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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

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PEP Level Plot
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PEP 3D Plot 1
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PEP 3D Plots 2 3
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BMY KO
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Sampling Frequency
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Sampling Frequency
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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