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 The Lee and Mykland Statistic

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 The Lee and Mykland Statistic

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. The Lee and Mykland Statistic

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 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 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 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! 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 The Lee and Mykland Statistic

9. SPY Level Plot The Lee and Mykland Statistic

10. SPY 3D Plot 1 The Lee and Mykland Statistic

11. SPY 3D Plot 2 & 3 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 The Lee and Mykland Statistic

13. PEP Level Plot The Lee and Mykland Statistic

14. PEP 3D Plot 1 The Lee and Mykland Statistic

15. PEP 3D Plots 2 & 3 The Lee and Mykland Statistic

16. BMY & KO The Lee and Mykland Statistic

17. Sampling Frequency The Lee and Mykland Statistic

18. Sampling Frequency The Lee and Mykland Statistic

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. The Lee and Mykland Statistic