Bayesian modeling and analysis of stochastic volatility in
Bayesian modeling and analysis of stochastic volatility in finance Derrick Hang April 6, 2010 Economics 201 FS
Review from Last Time • Regress for Prices: Possible useful predictors of prices are lost when we take the difference between prices to obtain returns • In general , we expect
Addressing Stationarity Concerns • Classic tests for stationarity of an AR time series (i. e. Dickey. Fuller, Phillips-Peron, etc. ) test the coefficient on the lagged time-series for a unit root • However, these tests assume a CONSTANT coefficient and have LOW POWER • DLM allows for the possibility of “pockets of stationarity” and will reject unit root null at values close to 1 Does stationarity of a model matter if we are looking for short term forecasting?
The Data • Jan 2, 2009 – June 31, 2009 (excluding April 10 th) • Data for market hours only (no weekend) : 9: 35 AM – 3: 55 PM • 5 Minute Data (9778 total points for each dataset) • All prices logged • 10 dependent variable (USD/variable): AUD, CHF, EUR, GBP, JPY, NZD, CAD, NOK, SGD, ZAR • 12 independent variable: 10 lagged forex variables, brent oil futures, comex gold futures • Focus on AUD, GBP, JPY, brent, gold
Back to Basics: Jump Testing • What is the relationship –if any- between jump days and periods of non-unit roots?
Back to Basics: Jump Testing
Back to Basics: Jump Testing
Back to Basics: Jump Testing
Back to Basics: Jump Testing l l l 0. 1% 1% 5% Significance Level = 25 / 127 (2. 23%) Significance Level = 6 / 127 (4. 72%) Significance Level = 1 / 127 (19. 69%)
Back to Basics: Jump Testing l l l 0. 1% Significance Level = 5 / 127 (3. 94%) 1% Significance Level = 11 / 127 (8. 66%) 5% Significance Level = 24 / 127 (18. 90%)
Back to Basics: Jump Testing l l l 0. 1% Significance Level = 3 / 127 (2. 36%) 1% Significance Level = 15/ 127 (11. 81%) 5% Significance Level = 38 / 127 (25. 98%)
Time-Varying Coefficient: Lagged GBP, JPY
Time-Varying Coefficient: Brent, Gold
Time-Varying Coefficient: Lagged AUD
Significant Windows: Lagged AUD (descending order) Start 03 -Jun-2009 09: 40: 00 14 -Jan-2009 09: 35: 00 18 -Mar-2009 10: 50: 00 30 -Mar-2009 09: 35: 00 25 -Jun-2009 09: 40: 00 22 -Jun-2009 09: 35: 00 08 -Apr-2009 09: 35: 00 10 -Mar-2009 10: 25: 00 27 -Jan-2009 09: 35: 00 09 -Feb-2009 09: 50: 00 19 -Feb-2009 11: 00 12 -Jun-2009 14: 25: 00 06 -Mar-2009 13: 35: 00 17 -Apr-2009 14: 15: 00 30 -Jan-2009 09: 35: 00 End 11 -Jun-2009 14: 55: 00 20 -Jan-2009 15: 55: 00 23 -Mar-2009 15: 55: 00 02 -Apr-2009 13: 25: 00 29 -Jun-2009 15: 55: 00 24 -Jun-2009 15: 00 13 -Apr-2009 12: 45: 00 12 -Mar-2009 13: 00 28 -Jan-2009 15: 55: 00 10 -Feb-2009 15: 55: 00 20 -Feb-2009 14: 45: 00 16 -Jun-2009 10: 55: 00 10 -Mar-2009 09: 50: 00 20 -Apr-2009 14: 25: 00 02 -Feb-2009 09: 35: 00
Jump Days: AUD (5% level) 08 -Jan-2009 09: 40: 00 16 -Jan-2009 09: 40: 00 28 -Jan-2009 09: 40: 00 02 -Feb-2009 09: 40: 00 03 -Feb-2009 09: 40: 00 04 -Feb-2009 09: 40: 00 23 -Feb-2009 09: 40: 00 (03 -Mar-2009 09: 40: 00) 12 -Mar-2009 09: 40: 00 23 -Mar-2009 09: 40: 00 31 -Mar-2009 09: 40: 00 01 -Apr-2009 09: 40: 00 13 -Apr-2009 09: 40: 00 15 -Apr-2009 09: 40: 00 16 -Apr-2009 09: 40: 00 01 -May-2009 09: 40: 00 08 -May-2009 09: 40: 00 15 -May-2009 09: 40: 00 20 -May-2009 09: 40: 00 22 -May-2009 09: 40: 00 02 -Jun-2009 09: 40: 00 04 -Jun-2009 09: 40: 00 05 -Jun-2009 09: 40: 00 10 -Jun-2009 09: 40: 00 11 -Jun-2009 09: 40: 00
Jump Days: AUD (1% and 0. 1% level) 1% 08 -Jan-2009 09: 40: 00 12 -Mar-2009 09: 40: 00 13 -Apr-2009 09: 40: 00 15 -May-2009 09: 40: 00 22 -May-2009 09: 40: 00 04 -Jun-2009 09: 40: 00
Significant Windows: Lagged AUD, GBP, JPY (descending order) Start 30 -Mar-2009 09: 35: 00 03 -Mar-2009 09: 35: 00 10 -Mar-2009 12: 20: 00 26 -Jun-2009 11: 00 29 -Jan-2009 09: 45: 00 27 -Jan-2009 09: 35: 00 19 -Feb-2009 14: 30: 00 17 -Apr-2009 14: 15: 00 06 -Mar-2009 15: 50: 00 End 01 -Apr-2009 13: 20: 00 05 -Mar-2009 11: 05: 00 12 -Mar-2009 13: 00 29 -Jun-2009 15: 55: 00 30 -Jan-2009 13: 50: 00 28 -Jan-2009 09: 55: 00 20 -Feb-2009 14: 45: 00 20 -Apr-2009 14: 25: 00 09 -Mar-2009 15: 55: 00
Jump Days: AUD (5% level) 08 -Jan-2009 09: 40: 00 16 -Jan-2009 09: 40: 00 28 -Jan-2009 09: 40: 00 02 -Feb-2009 09: 40: 00 03 -Feb-2009 09: 40: 00 04 -Feb-2009 09: 40: 00 23 -Feb-2009 09: 40: 00 03 -Mar-2009 09: 40: 00 12 -Mar-2009 09: 40: 00 23 -Mar-2009 09: 40: 00 31 -Mar-2009 09: 40: 00 01 -Apr-2009 09: 40: 00 13 -Apr-2009 09: 40: 00 15 -Apr-2009 09: 40: 00 16 -Apr-2009 09: 40: 00 01 -May-2009 09: 40: 00 08 -May-2009 09: 40: 00 15 -May-2009 09: 40: 00 20 -May-2009 09: 40: 00 22 -May-2009 09: 40: 00 02 -Jun-2009 09: 40: 00 04 -Jun-2009 09: 40: 00 05 -Jun-2009 09: 40: 00 10 -Jun-2009 09: 40: 00 11 -Jun-2009 09: 40: 00
Jump Days: AUD (1% and 0. 1% level) 1% 08 -Jan-2009 09: 40: 00 12 -Mar-2009 09: 40: 00 13 -Apr-2009 09: 40: 00 15 -May-2009 09: 40: 00 22 -May-2009 09: 40: 00 04 -Jun-2009 09: 40: 00
Significant Windows: Lagged AUD, Brent, Gold (descending order) Start 30 -Mar-2009 09: 35: 00 19 -Mar-2009 11: 10: 00 10 -Mar-2009 10: 25: 00 End 02 -Apr-2009 09: 35: 00 23 -Mar-2009 14: 30: 00 12 -Mar-2009 13: 00
Jump Days: AUD (5% level) 08 -Jan-2009 09: 40: 00 16 -Jan-2009 09: 40: 00 28 -Jan-2009 09: 40: 00 02 -Feb-2009 09: 40: 00 03 -Feb-2009 09: 40: 00 04 -Feb-2009 09: 40: 00 23 -Feb-2009 09: 40: 00 03 -Mar-2009 09: 40: 00 12 -Mar-2009 09: 40: 00 23 -Mar-2009 09: 40: 00 31 -Mar-2009 09: 40: 00 01 -Apr-2009 09: 40: 00 13 -Apr-2009 09: 40: 00 15 -Apr-2009 09: 40: 00 16 -Apr-2009 09: 40: 00 01 -May-2009 09: 40: 00 08 -May-2009 09: 40: 00 15 -May-2009 09: 40: 00 20 -May-2009 09: 40: 00 22 -May-2009 09: 40: 00 02 -Jun-2009 09: 40: 00 04 -Jun-2009 09: 40: 00 05 -Jun-2009 09: 40: 00 10 -Jun-2009 09: 40: 00 11 -Jun-2009 09: 40: 00
Jump Days: AUD (1% and 0. 1% level) 1% 08 -Jan-2009 09: 40: 00 12 -Mar-2009 09: 40: 00 13 -Apr-2009 09: 40: 00 15 -May-2009 09: 40: 00 22 -May-2009 09: 40: 00 04 -Jun-2009 09: 40: 00
Initial Findings • From the data so far, we have seen that the largest windows contain days declared as jump days • Most of the time, windows that contain entire days have a “jump day” inside it • Windows where multiple regressors are significant also contain declared jump days • What does this all mean?
Further Research • Look to see if the relationship between large windows and jump days exist with the other dependent currency datasets • Short term forecasts inside this windows? • Try with different jump tests (other than Mean-adjusted TP) • Fix bugs in code
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