predictability of non-linear trading rules in the us stock market chong & Lam 2010

1. predictability of non-linear trading rules in the us stock marketchong & Lam 2010

2. Overview Liu Min QiYichen Zhang Fengtian • Literature review of the paper • A brief introduction of the 3 models used in the paper • The strategies and results • Model selection for our project • Data selection for our project • Statistical tests for data selected • Model regression techniques (parameter estimation and trading strategies for SETAR model) • Algorithm in java • Demonstrate SETAR model (JAVA program) • Results and analysis

3. Literature Review • The paper examines the effect of 3 models • Namely • Variable Moving Average (VMA Model) • Autoregressive Model (AR Model) • Self-Exciting Threshold Autoregressive Model (SETAR Model) • On four major US indices • Dow Jones Industrial Average (DJIA) • NASDAQ composite index • New York Stock Exchange composite index • Standard and Poor 500 index (S&P 500)

4. VMA Model • General Form • VMA(S, L) • Where S represents the short term window • And L represents the long term window • Calculated from the below formula • for a n day moving average • Models used in the paper includes VMA(1, 50), VMA(1, 150) and VMA(1, 200)

5. AR Model • General Form • Commonly referred as AR (p) Model • Where p is the order of autoregressive part

6. AR Model • A linear time series model • For the paper, an AR (1) Model was used • ∆Yt is the natural log difference of the stock index • Where ∆Yt= Yt- Yt-1 • Alphas are the fitted coefficient • εt is the residual error • AR (1) was chosen because the estimated coefficients are significant, suggesting it is good enough for modelling dynamics of return series

7. SETAR Model • General Form: • Usually referred as SETAR(k, p) model • k is the number of regimes • p is the order of the autoregressive part

8. SETAR Model • A non-linear time series model • An extension of AR models • Higher degree of flexibility due to the threshold parameter • Which introduces a regime switching behavior

9. SETAR Model • For the paper, a SETAR(2, 1) model was chosen • ∆Yt is the natural log difference of the stock index • Where ∆Yt= Yt- Yt-1 • Alphas and betas are the fitted coefficient • εt is the residual error • d is the delay factor • γ is the threshold parameter

10. SETAR Model • Why SETAR (2, 1)? • Because (as claimed by paper) • It is simple and has good predictability • Threshold parameter already captures non-linearity • Therefore additional benefit of higher autoregressive order is small • The estimated coefficients are significant based on statistical tests • Suggesting first order model is good enough to describe dynamics of the return series

11. Strategies • For VMA • Pretty straightforward • Buy if MA(S) > MA(L) • Sell if MA(S) < MA(L) • For AR and SETAR • Model fitting is required for every w observations • Buy if > 0 • Sell if < 0

12. Dow Jones Industrial Average index. ‘Buy>0’ and ‘Sell>0’ are the fraction of positive buy and sell returns. Buy , Sell and Buy-Sell columns show the one day conditional mean for buy, sell and buy-sell returns

13. NASDAQ composite index

14. New York Stock Exchange composite index

15. Standard and Poor 500 index

16. Results • Performed using observation window period of: • 50, 150, and 200 days • SETAR performed slightly better than AR for DJIA and S&P 500 • AR performed slightly better in NASDAQ • Both SETAR and AR outperformed VMA

17. Model Selection • Therefore, SETAR model was chosen for our project • Because of the better results obtained from the paper • And also because of its non-linearity • Which gives it flexibility in modelling • However, simulation may be slow due to a need for multi-parameter fitting for each signal

18. Data Selection • We had chosen the HK’s Hang Seng Index and Singapore’s Straits Times Index • Data selection (from yahoo finance) • Hang Seng Index • Daily closing price from 31st Dec 1986 to 31st Dec 2010 • Total 5962 Observations • Straits Times Index • Daily closing price from 31st Dec 1987 to 31st Dec 2010 • Total 5754 Observations

19. Index Statistics • Summary statistics for daily log returns – full sample • JB stat represents the Jarque-Bera test for normality • ρ(i) is the estimated autocorrelation at lag i • Q(5) is the Ljung-Box Q statistic at lag 5 • Numbers marked with * are significant at 1% level

20. Statistical Results • From the values of skewness, kurtosis, and Jarque-Bera statistics • Returns are leptokurtic, skewed, and not normally distributed • Ljung-Box Q statistics at 5th lag significant at 1% • Suggestive of substantial serial correlation in stock returns • Essential for existence of trading-rule profits • These results are consistent with that found in the main paper • Which may be indicative of the model’s efficiency on the Hang Seng Index and Straits Times Index

21. Parameter estimation • Model:

22. Parameter estimation • Use Ordinary Least Square method to find γ and θ. (Refer to Bruce E. Hansen (1997) Inference in TAR Models. Studies in Nonlinear Dynamics & Econometrics, Volume 2, Issue 1)

23. Parameter estimation Remarks: In our case, d (delay parameter) = 1. Observe that the residual variance only takes on at most ndistinct values as γis varied, we set γ = ΔYt-d, t = 2,…,n.

24. Parameter estimation • Thus, the estimate of θ is Given n observations, we use OLS to obtain the fitted coefficients γ and θ andpredict ∆Yt+1 based on ∆Yt.

25. Trading strategy • The SETAR trading strategy is as follows: where W is the observation window and is the conditional expectation of ΔYt+1 based on most recent W observations up to day t.

26. Trading strategy • Remarks: • Just imagine that we use the model to predict the price tomorrow. If the predicted price is higher than today actual price, then we buy. Otherwise, we sell. • The value of αand βchange with the observation window, as we use the most recent w observations. So as we move, we roll the window forward and update the α and β to get the next prediction of ΔY.

27. Trading strategy • For example, given W = 50 and n = 100, • 1. Obtain γ and α0 α1 β0β1. • 2. Obtain ΔYt+1 based on ΔYt and estimated parameters. • 3. Buy if ΔYt+1 > 0. Sell if ΔYt+1 < 0. • 4. Shift the observation window (set t = t+1) and repeat Step 1 to Step 3.

28. Algorithm in java • The model

29. Algorithm in java

30. Least squares method

31. Algorithm in java

32. Actual stock index moving by timeHANG SENG INDEX

33. SETAR(1, 50) model Predicted stock index moving by time - HANG SENG INDEX

34. Actual stock index moving by time- STAITS TIMES INDEX

35. SETAR(1, 50) model Predicted stock index moving by time - STRAITS TIMES INDEX

36. SETAR(1, 150) model Predicted stock index moving by time - HANG SENG INDEX

37. SETAR(1, 200) model Predicted stock index moving by time - HANG SENG INDEX

38. Empirical results of implementing the trading strategies on the HANG SENG INDEX

39. Empirical results of implementing the trading strategies on the STRAITS TIMES INDEX

40. Future work T-Statistics • Mean return of buy periods. • Mean return of sell periods. • Buy- sell return. AR Model • The performance of the nonlinear trading rule (SETAR) is compared with that of the linear model (AR).

41. Thank You