LeadLag Relationship and Structural Change: A Genetic Programming Approach

1
Lead-Lag Relationship and Structural Change A
Genetic Programming Approach

• Donald Lien
• Professor of Economics
• University of Texas - San Antonio
• April 11 2003

2
Background Paper

• Lien, Tse and Zhang Structural Change and
  Lead-Lag Relationship between the Nikkei Spot
  Index and Futures Price A Genetic Programming
  Approach, Quantitative Finance, April, 2003

3
Stock Index and Futures Market

• Spot stock index
• Index futures price
• Example At 4/7/2003, Dow Jones index is 8300.41,
  June Dow Jones Index futures price is 8352, and
  September Dow Jones Index futures price is 8419

4
Lessons from 1987 Market Crash

• Institutional factors Futures market is to be
  blamed?
• Lead-lag relationship between stock and futures
  markets

5
Lead-Lag Relationship

• Causality and lead-lag relationship
• Statistical detection
• Linear causality
• Nonlinear causality

6
Linear Causality

• Stock and futures returns are described by a
  bivariate linear time series model.
• The model is VAR (vector autoregression) with or
  without the EC (error correction) term.
• X leads Y if the past history of X helps provide
  a better description or forecast of Y.

7
Classifications

• X leads Y
• Y leads X
• Neither X leads Y nor Y leads X
• There is a two-way relationship between X and Y
• The effect of error correction term (adjusting to
  the long run equilibrium)

8
Data Description

• Nikkei 225 traded on SGX (formerly SIMEX)
• Daily data from September 26, 1995 to December
  30, 1999
• Total observations 1035

9
Empirical Estimation

• ?St -0.002 0.188 zt-1 0.223 ?St-1 0.217
  ?Ft-1
• (0.615) (3.804) (3.835)
  (3.827)
• ?Ft -0.019 0.069 zt-1 0.044 ?St-1 - 0.089
  ?Ft-1
• (0.643) (1.338) (0.722)
  (1.492)
• ?St log(St/St-1), ?Ft log(Ft/Ft-1), ztv
  log(St/Ft)

10
Interpretations

• At the long run equilibrium, the basis is close
  to zero.
• When z is positive (negative), spot price should
  decrease (increase) and/or futures price should
  increase (decrease) to return z to zero
• Linear causality test indicates Nikkei 225
  futures market leads the stock index

11
Non-Linear Causality

• Consider the residual bivariate time series from
  the linear causality test
• Dimension statistics to examine whether the
  marginal distribution and the conditional
  distribution of the index residual (or the
  futures residual) are the same

12
The Main Idea

• Let Xtm denote the history from time t-m to time
  t and let Xsm denote the history from time s-m to
  time s. Absent from causality,
• Prob d(Xtm, Xsm)
• d(Yt-LyLy, Ys-LyLy)
• Prob d(Xtm, Xsm) e
• Let H denote the the difference of empirical
  estimates of the two probability. Then H is
  asymptotically normally distributed.

13
Test Results

• Futures market leads spot market ( 5 lags and 10
  lags)
• ECM residuals 0.0056, 0.0077
• VAR residuals 0.0045, 0.0086
• Spot market leads futures market ( 5 lags and 10
  lags)
• ECM residuals -0.0342, -0.0027
• VAR residuals -0.0263, 0.0046

14
Interpretations

• All statistics are N(0,1) distributed under the
  null hypothesis of no causality. As a result, we
  cannot reject the null hypothesis
• There is no causality relationship between Nikkei
  225 futures residual and stock index residual

15
Summary for Nikkei 225 Data

• Linear causality test indicates Nikkei 225
  futures market leads the stock index
• Non-linear causality test indicates no
  relationship between Nikkei 225 futures residual
  and stock index residual
• Derivative? Which one?

16
Lessons from 1997 Asian Crisis

• There are two market periods normal period and
  extreme period
• Extreme period behaves substantially different
  from normal period
• We need to examine extreme period by itself

17
Solution I

• Extreme period data and normal period data come
  from the same data generation process. The
  former represents the tail whereas the latter
  the center of the distribution
• Study the behavior of tails
• How to derive more observations on tails?

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Solution II

• Extreme period data and normal period data come
  from different data generation process.
• Parametric modeling change in parameters or
  relationships
• Non-parametric approach

19
Non-Parametric Approach

• Structural change and extreme periods
• Identification of structural changes
• Learning and recognition
• Extension to the multivariate system

20
Genetic Programming

• Generation
• Activity function
• Terminal set
• GP-tree
• Fitness
• Reproduction, mutation, and crossover

21
Learning and Recognition

• Choose 35 samples to learn and 5 samples for
  fitness comparisons
• Choose 26 GP-trees each represent a prediction
  equation
• Choose 10 generations for learning
• Fitness is based on average squared errors of the
  best 10 trees at the end of learning

22
Activity Function

• F - . / sin cos exp log abs
• K xt-1 xt-2 xt-3
• Examples
• (1) xt-2 (2) exp -- xt-1
• (3) / -- xt-3 cos abs -- -- xt-1 xt-2

23
Fitness Measure

• g a GP-tree at the end of learning
• Fit (j) the square of the difference between
  xtj and its prediction based on g.
• Fitness of a tree is the average of Fit (j) over
  j 1,, 5
• Use fitness to choose the next generation

24
New Generation

• A GP-tree with a smaller fitness has a greater
  probability to reproduce
• If not reproduce, a mutation or crossover will
  occur
• A mutation is a new tree
• A crossover substitutes a branch of the tree by a
  branch from another tree

25
Structural Change

• At the end of 10 generations, the average fitness
  over the 10 best trees (out of 100) is calculated
  and denoted by F
• Let Fn and Fn1 denote the average fitness for
  the n and n1 windows. Calculate
• D Fn1 / Fn
• D 1 (or smaller) if no structural change

26
Implementation

• Structural change is detected when D 1.2
• The GP approach is applied to Nikkei spot index,
  Nikkei futures price, and the bivariate system
• For the bivariate system, the prediction is based
  on the spot index whereas the arguments are drawn
  from both spot index and futures price

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Lead-Lag Relationship

• If a structural change occurs in the spot index
  (and the system) before it occurs in the futures
  price, then the spot market leads the futures
  market
• If a structural change occurs in the futures
  price (and the system) before it occurs in the
  spot index, then the futures market leads the
  spot market

28
Contemporaneous Relationship

• When a structural change occurs in both spot
  index and futures price but not in the bivariate
  system, we detect a contemporaneous relationship
  between spot and futures markets.

29
Temporary Imbalance

• When a structural change in either spot or
  futures not accompanied by a structural change in
  the bivariate system (and a structural change in
  the other market), the detected change is likely
  the result of temporary market imbalance

30
Empirical Results

• On 13/11/96, a structural change occurs in the
  futures market. On 20/11/96, a structural change
  occurs in both spot and the bivariate system.
  The changes end at 20/02/97 for both spot and
  futures markets but at 13/02/97 for the system
• Futures leads spot or market imbalance

31
Empirical Results

• On 24/07/97 a structural change occurs in both
  the spot and the system, followed by a structural
  change in the futures market on 31/07/97. The
  changes end at 19/11/97 for both the spot and the
  system and 11/12/97 for the futures market
• Spot leads futures

32
Empirical Results

• On 02/07/98 a structural change occurs in both
  the spot and the system, followed by a structural
  change in the futures market on 16/07/98. The
  changes end at 05/10/98 for the spot market and
  19/10/98 for both the futures market and the
  system
• Spot leads futures

33
Empirical Results

• On 30/03/98 there is a structural change in the
  spot market but no corresponding change in either
  the future market or the bivariate system
• The change ends at 03/06/98

34
Conclusions

• Structural changes for spot market and for the
  system always occur simultaneously suggesting a
  bigger impacts for spot market than the futures
  market
• Spot leads futures more often in recent years