Difficult empirical puzzles from finance

Empirical facts from Finance

1. Difficult Empirical Facts from
Finance
Blake LeBaron
International Business School
Brandeis University
www.brandeis.edu/~blebaron
SFI CSSS, 2007
Santa Fe, NM

2. Empirical Facts
Robust over time
Robust over markets

Equity

Bonds

FX

Commodities
Highly significant

3. Empirical Challenges
Two big features

Long memory

Fat tails
Complexity connections:

Length scales (time or space)
Learning connections

Coevolution

4. Finance Facts/puzzles
Price time series

Near martingale behavior

Volatility persistence

Fat tails (leptokurtosis)

Technical trading

Nonlinear features/predictability??
Prices relative to something

Deviations from fundamentals

Equity premium
Trading volume
Microstructure facts

5. Near Martingale Prices
r(t+1) uncorrelated, and difficult to directionally
forecast (including nonlinear)
Many horizons (better short)
Many series (almost any liquid asset)
Theory: EMH (Efficient Market Hypothesis)

€
p
t=logP
t()
E
t(p
t+1)=p
t
log(p
t+1
)=log(p
t
)+r
t+1

6. Volatility Persistence
Return magnitudes persistent
Very persistent!!
History

Mandelbrot

ARCH/GARCH

Stochastic volatility

Realized volatility

€
corr(r
t
2
,r
t−j
2
)>0j>>0
corr(|r
t|,|r
t−j|)>0j>>0
corr(σ
t
2
,σ
t−j
2
)>0j>>0

7. Data Introduction
Dow Jones Industrials

Jan 1897-Sep 2004 (29602 obs)
British Pound

June 1973 - Feb 2006
IBM Daily returns and volume

8. Dow Jones Daily Returns
1897-2004

9. Dow Volatility Persistence

10. Long Memory Stochastic
Volatility

€
r
t
=σ
t
ε
t
ε
t
~N(0,1)
log(σ
t
2
)=f(log(σ
t−j
2
))+u
t
log(σ
t
2
)=a
ie
t−i
i=0
∞
∑

11. Fractionally Integrated
Process (Long Memory)
€
x
t=a
je
t−j
j=0
∞
∑
a
j=
Γ(j+d)
Γ(j+1)Γ(d)
a
j≈
1
Γ(d)
j
d−1
a
j+1/a
j=
(j+1)
d−1
(j)
d−1
→1

12. Autocorrelation Comparisons

€
Standard Models
ρ
k
=α
k
,α=0.97
Long Memory Models
ρ
k=k
2d−1
,d=0.4
ρ
k
=k
−0.2
log(ρ
k)=−0.2log(k)

13. ACF Comparison

14. Dow Versus Long Memory
Volatility Process

15. Daily British Pound Returns: 1973-2006

16. Volatility persistence: Pound versus
Long Memory

17. Long Memory in Volatility
Present in almost all financial series
Best estimates:

Realized volatility

0.35 < d < 0.50
Causes

Adding short memory processes

Regime shifts

Nonlinearities

Other

18. Finance Facts/puzzles
Price time series

Near martingale behavior

Volatility persistence

Fat tails (leptokurtosis)

Technical trading

Nonlinear features/predictability??
Prices relative to something

Deviations from fundamentals

Equity premium
Trading volume
Microstructure facts

19. Fat Tailed Return Distributions
Returns at the < monthly frequency are
not normally distributed

Fat tails

Leptokurtic

Power laws

20. Dow Returns and Gaussian

21. Returns and Student-t(3)

22. Normal Quantile Comparisons
upper = Dow, lower = BP

23. Normal Quantile Comparisons
upper = Dow, lower = long memory (d=0.45)

24. Normal Quantile Comparisons
upper = Dow daily, lower = Dow monthly

25. Return Summary Statistics
1 Day 10 Days100 Days
Dow Kurtosis25.1 13.4 8.7
Long memory
d = 0.45
28.7 13.6 7.5
BP Kurtosis7.6 5.8 5.3
Long memory
d = 0.40
17.2 7.8 5.1

26. Approximate Power-law Tails
Shape Parameter
€
Pr(X>x)≈A|x|
−α
log(Pr(X>x))≈log(A)+−αlog(|x|)
E(|X|
m
)=∞m≥α

27. Dow Tail Probabilities

28. Practical Implications
Higher (>3) moment failure??
If true, problems for variance (not mean)
estimation
Possibly other problems

29. Variance Estimation
Increasing sample sizes

20, 60, 250, 1250, 2500, 5000
5000 length monte-carlo
Record quantiles

(0.01, 0.05, 0.5, 0.95, 0.99)

30. Daily Monte-carlo Series
Gaussian, mean 0, variance 1
Student-t’s, mean 0, variance 1

DF = 3, 4, 5

31. 1 Day Variance Estimate:
Gaussian versus Student-t(3) Returns

32. Fine Sampling Frequency
Use higher frequency data to improve
precision
Doesn’t help for means
Good for variances and covariances

33. Monthly/daily Volatility Estimates
Gaussian Returns

34. Monthly/daily Volatility Estimates
Student-t(3) Returns

35. Why Should We Care About
Tails?
Large events / risk
Portfolio construction
Robust learning

Variance estimation

Filtering and parameter updating

36. Finance Facts/puzzles
Price time series

Near martingale behavior

Volatility persistence

Fat tails (leptokurtosis)

Technical trading

Nonlinear features/predictability??
Prices relative to something

Deviations from fundamentals

Equity premium
Trading volume
Microstructure facts

37. Technical Trading Patterns
Simple trend following rules have some
predictive abilities
Both equity and fx markets

38. Moving Average Trading
Rules (Simplest)

€
a
t,L=
1
L
p
t−j
j=0
L−1
∑
p
t
≥a
t,L:Buy
<a
t,L
:Sell

⎧

⎨

⎩

39. Simple Rule Test

€
s
t=
1:Buy
−1:Sell

⎧

⎨

⎩
x
t=s
tr
t
x =
1
n
x
t
t=1
n
∑,s
2
=var(x)
z=
x
(s
2
/n)
(1/2)

40. Dow MA lengths
Returns and T-tests
MA length
(days)
Annual Return
(percentage)
T-stat
25 10.9 6.87
50 8.9 5.60
150 7.4 4.62
250 7.5 4.72
350 6.5 4.10

41. BP MA lengths
Returns and T-tests
MA length
(days)
Annual Return
(percentage)
T-stat
25 6.0 3.57
50 6.0 3.59
150 3.1 1.85
250 4.2 2.50
350 3.7 2.16

42. Dow Total Annual Return:
Long/short strategy (150 day MA)

43. Total Annual Return BP
(long/short, 150 day MA)

44. Nonlinear Features
“Leverage” effect
Volume/volatility and autocorrelations
Key question:

Stability and reliability

45. Finance Facts/puzzles
Price time series

Near martingale behavior

Volatility persistence

Fat tails (leptokurtosis)

Technical trading

Nonlinear features/predictability??
Prices relative to something

Deviations from fundamentals

Equity premium
Trading volume
Microstructure facts

46. Fundamentals
Dividend price ratios
Interest rate differentials
Real exchange rates

47. Real S&P Level and Shiller’s Dividend
Discounted Price

48. S&P Dividend Yield

49. Dividend Yield Autocorrelations

50. US Dollar - British Pound
Interest Differential

51. Interest Differential:
Autocorrelation

52. Equity Premium
Equity real return = 7-8% per year
Bond real return = 1%
Spread of 6% difficult to explain
Explanations

Tails and risk estimates

Learning (premium is falling over time)

53. Finance Facts/puzzles
Price time series

Near martingale behavior

Volatility persistence

Fat tails (leptokurtosis)

Technical trading

Nonlinear features/predictability??
Prices relative to something

Deviations from fundamentals

Equity premium
Trading volume
Microstructure facts

54. Trading Volume
Persistence
“Trading Time”

55. IBM Trading Volume

56. Detrended IBM Trading Volume

57. IBM Volume Autocorrelations

58. IBM Volatility/volume
Cross Correlation

59. Mixtures of Distributions
(Clock/calendar time)
Clark(1973, Econometrica)
Ann and Geman, (J of Fin, 2000)
Martens and van Dijk (2006, Erasmus)
Gillemot, Farmer, and Lillo (2005)

“There’s more to volatility than volume”

60. Finance Facts/puzzles
Price time series

Near martingale behavior

Volatility persistence

Fat tails (leptokurtosis)

Technical trading

Nonlinear features/predictability??
Prices relative to something

Deviations from fundamentals

Equity premium
Trading volume
Microstructure facts

61. Microstructure Facts
Seasonalities in spreads and volume
Order flows

Evans and Lyons (2002, Journal of Political
Economy, and others)

Predictability

Lillo and Farmer (2004) (and others)

Long memory

Schulmeister (2006, Financial Research Letters)

Technical trading and order flows

62. More Microstructure Facts
Book matters

Large moves

Osler, “Stop loss orders and price cascades”

Farmer et. al. “What really causes large price
changes?”, SNDE (2004).

Depth/liquidity
Trading time/mixtures of distributions

63. High Frequency Example
EBS: Electronic trading system
$/Euro exchange rates (high frequency
quotes and deals)
12/28/02 - 03/03/06
Clock versus event (deal) time
High/low range volatility estimate

(H-L)/( 0.5(H+L) )

1 Hour / 100 deal windows

64. 1 Hour $/Euro Volatility ACF

65. 1 Hour $/Euro Volatility ACF
Time of Day Effects Removed

66. 100 Event $/Euro Volatility ACF

67. 100 Event $/Euro Volatility ACF
Time of Day Effects Removed

68. Finance Facts/puzzles
Price time series

Near martingale behavior

Volatility persistence

Fat tails (leptokurtosis)

Technical trading

Nonlinear features/predictability??
Prices relative to something

Deviations from fundamentals

Equity premium
Trading volume
Microstructure facts

69. Explanations
Many facts hard to “explain” with
traditional modeling approaches

Fat tails

Volatility persistence

Deviations from fundamentals
Agent-based approaches