Exploring the Efficiency of the Lee-Mykland Test Statistic

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Exploring the Efficiency of the Lee-Mykland Test
Statistic

• Warren Davis
• October 29, 2007
• Econometrics Luncheon

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Outline

• Barndorff-Nielsen and Shephard Jump Detection
• Discussion of the Lee-Mykland Test
• Change of Statistic
• Simulation Set-Up
• Simulation Results
• Future Directions

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Z-Statistics Test
Davis 3

• To flag jump days, a z-statistic was computed
  using both the Tri-Power Quarticity and
  Quad-Power Quarticity as follows

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Lee and Mykland (2006)

• The paper examines a price series defined as
  follows

Where dJ(t) is a jump-counting process with
intensity ?(t) and Y(t) is the jump size, which
follows a normal distribution.
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Lee and Mykland (cont.)

• The t-statistic is constructed using the
  following ratio of the current return to the
  local volatility.

Where K is the trailing window size
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New Proposed Statistic

• Run regular return-to-volatility ratio
• Set flagged returns to zero
• Construct new local volatility estimate using the
  Realized Variance, which is more efficient if no
  jumps are present

Where q is the number of jumps that have been set
to 0
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Simulated Data
Where the series of returns consisting of a
Brownian motion with non-constant volatility,
interrupted by two jumps terms. These jumps
would be distributed according to a compound
Poisson process. The Poisson terms would be
multiplied by normal distributions that would
have means of opposite signs, and changes in
volatility are defined according to
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Simulated Data (cont.)

• The standard set of parameters used
• A persistence to cause a volatility shock
  half-life of 40-50 days
• Poisson means to generate one jump in every 100
  returns on average.
• At first, .01 significance levels were used for
  both stages

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Sample Volatility Plot
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Persistence vs. Volatility Shocks
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Confusion Matrices
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Results- Varying Significance Levels
.1 Both Stages
.01 Both Stages
1 Both Stages
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Changing the Return-to-Volatility Ratio
Significance Level
5 First Stage
1 First Stage
.1 First Stage
.01 Both Stages
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Effects of Jump Size
1 Jumps
.5 Jumps
1.5 Jumps
2 Jumps
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Effects of Jump Frequency
1 in 50
1 in 100
1 in 10
1 in 25
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Conclusions

• Using a higher significance level for both stages
  flags more jumps correctly, although it
  introduces a higher false negative rate
• The biggest gains in introducing the second stage
  came when jumps were very frequent, most likely
  much too frequent to model real-world markets
• The statistics were only fairly accurate in
  identifying jumps when volatility of volatility
  remained under a certain threshold

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