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

2. The Lee and Mykland Statistic
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
April 22, 2019
The Lee and Mykland Statistic

3. The Lee and Mykland Statistic
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
April 22, 2019
The Lee and Mykland Statistic

4. The Lee and Mykland Statistic
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.
April 22, 2019
The Lee and Mykland Statistic

5. The Lee and Mykland Statistic
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 ε > 0, there exists a finite constant Me such that P(abs(Xn)) < ε eventually…It also satisfies the stochastic volatility plus finite activity jump semi-martingale class in Barndorff-nielsen and Sehphard (2004) and the reference therein. p. 7
April 22, 2019
The Lee and Mykland Statistic

6. Optimal Choice of Window
Optimal Window Size K: The optimal choice for window size is studied by simulation…It turns out that it is optimal for K to be the smallest integer in the condition set, K = Op(Δtª) s.t. -1 < a < .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
April 22, 2019
The Lee and Mykland Statistic

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!
April 22, 2019
The Lee and Mykland Statistic

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
April 22, 2019
The Lee and Mykland Statistic

9. The Lee and Mykland Statistic
SPY Level Plot
April 22, 2019
The Lee and Mykland Statistic

10. The Lee and Mykland Statistic
SPY 3D Plot 1
April 22, 2019
The Lee and Mykland Statistic

11. The Lee and Mykland Statistic
SPY 3D Plot 2 & 3
April 22, 2019
The Lee and Mykland Statistic

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
8:30am: 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
≈2:15pm: Federal Open Market Committee announcements
Chicago Mercantile Exchange opens at 8:20am 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 = βhst + εt; st is surprise component compared to market expectations
Vt,h = αh + γh*abs(st) + εt; again st is the surprise component and v is volume
April 22, 2019
The Lee and Mykland Statistic

13. The Lee and Mykland Statistic
PEP Level Plot
April 22, 2019
The Lee and Mykland Statistic

14. The Lee and Mykland Statistic
PEP 3D Plot 1
April 22, 2019
The Lee and Mykland Statistic

15. The Lee and Mykland Statistic
PEP 3D Plots 2 & 3
April 22, 2019
The Lee and Mykland Statistic

16. The Lee and Mykland Statistic
BMY & KO
April 22, 2019
The Lee and Mykland Statistic

17. The Lee and Mykland Statistic
Sampling Frequency
April 22, 2019
The Lee and Mykland Statistic

18. The Lee and Mykland Statistic
Sampling Frequency
April 22, 2019
The Lee and Mykland Statistic

19. The Lee and Mykland Statistic
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
Sampling Frequency
Matches
# Flagged Jumps
5 minutes 28
1031
17.5 minutes 17
559
385 minutes 3 37
April 22, 2019
The Lee and Mykland Statistic