Sampling Frequency and Jump Detection

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Sampling Frequency and Jump Detection Mike
Schwert ECON201FS 4/7/08
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• This Weeks Approach and Data
• Counted common jump days between different
  sampling frequencies for Ait-Sahalia Jacod,
  Lee-Mykland and Jiang-Oomen jump tests
• Counted common jump days between different tests
• Summarized effect of sampling frequency on jump
  detection
• Price Data
• GE minute-by-minute 1997 2007 (2670 days)
• ExxonMobil minute-by-minute 1999 2008 (2026
  days)
• ATT minute-by-minute 1997 2008 (2680 days)
• SP 500 every 5 minutes, 1985 2007 (5545 days,
  excluding short days)
• Simulated Data
• Continuous process plus jumps from a tempered
  stable distribution
• Alphas of 0.30, 0.90, 1.50, 1.90
• 78 daily price observations for 2500 days
  (195000 total observations)

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Simulated Data a 0.30
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Simulated Data a 0.90
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Simulated Data a 1.50
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Simulated Data a 1.90
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Contingency Tables BN-S ZQP-max Statistic
a 0.30
a 0.90
a 1.50
a 1.90
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Contingency Tables BN-S ZTP-max Statistic
a 0.30
a 0.90
a 1.50
a 1.90
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• Ait-Sahalia Jacod Test
• Introduced in 2008 article by Yacine Ait-Sahalia
  and Jean Jacod

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Contingency Tables Ait-Sahalia Jacod Test
GE
SP 500
ATT
Exxon Mobil
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Contingency Tables Ait-Sahalia Jacod Test
a 0.30
a 0.90
a 1.90
a 1.50
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• Lee-Mykland Test
• Introduced by Suzanne Lee and Per Mykland in a
  2007 paper
• Allows identification of jump timing, multiple
  jumps in a day

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Daily Contingency Tables Lee-Mykland Test
GE
SP 500
ATT
Exxon Mobil
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Daily Contingency Tables Lee-Mykland Test
a 0.30
a 0.90
a 1.90
a 1.50
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• Microstructure Noise Robust Jiang-Oomen Test
• Similar to Jiang-Oomen Swap Variance test, but
  robust to microstructure noise which often
  contaminates high-frequency data
• NOTE NO SIMULATED RESULTS YET BECAUSE OF
  DIFFICULTIES WITH SEXTICITY CALCULATION

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Microstructure Noise Robust Jiang-Oomen Test
Difference Test
Logarithmic Test
Ratio Test
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Contingency Tables MNR-JO Difference Test
GE
SP 500
ATT
Exxon Mobil
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Contingency Tables MNR-JO Log Test
GE
SP 500
ATT
Exxon Mobil
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Contingency Tables MNR-JO Ratio Test
GE
SP 500
ATT
Exxon Mobil
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• Cross-Test Contingency Tables
• Checked to see if different jump tests find
  jumps on same days
• Tests
• Barndorff-Nielsen Shephard ZQP-max test
• Jiang-Oomen swap variance test
• Microstructure Noise Robust Jiang-Oomen test
• Ait-Sahalia Jacod test
• Lee-Mykland test

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Cross-Test Contingency Tables 5 min sample
GE
SP 500
XOM
a 1.50
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Cross-Test Contingency Tables 10 min sample
GE
SP 500
XOM
a 1.50
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Cross-Test Contingency Tables 15 min sample
GE
SP 500
XOM
a 1.50
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Cross-Test Contingency Tables 20 min sample
GE
SP 500
XOM
a 1.50
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Summary of Tests Sample Robustness

• Denominator is frequency with lower number of
  jumps
• Lee-Mykland test results based on daily findings
  detected jumps are not necessarily at the same
  time during the day

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• Possible Extensions
• Regress Barndorff-Shephard Nielsen ZQP-max
  statistics on changes in daily volume to see if
  high volume days correspond to jump days and
  common jump days between samples
• Examine jump diffusion models other than the
  Poisson process used in most jump detection
  literature