predictability of nonlinear trading rules in the US market
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PREDICTABILITY OF NON-LINEAR TRADING RULES IN THE US STOCK MARKETCHONG & LAM 2010
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Overview
Liu Min
Qi Yichen
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
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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)
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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)
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AR Model
General Form
Commonly referred as AR (p) ModelWhere p is the order of autoregressive
part
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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
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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
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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
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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
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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
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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
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Trading rule Buy Sell Buy>0 Sell>0 Buy - Sell
SETAR(1,50) 0.000541 -0.000153 0.5149 0.51402 6.9400E-04
SETAR(1,150) 0.000643 -0.000402 0.5186 0.50373 1.0450E-03
SETAR(1,200) 0.000744 -0.000616 0.5259 0.48824 1.3600E-03
AR(1,50) 0.000477 -0.000064 0.5117 0.51883 5.4100E-04
AR(1,150) 0.000575 -0.000295 0.5163 0.50681 8.7000E-04
AR(1,200) 0.000665 -0.00048 0.5234 0.49415 1.1450E-03
VMA(1,50) 0.000393 0.000056 0.5093 0.52258 3.3700E-04
VMA(1,150) 0.000357 0.000089 0.5086 0.52116 2.6800E-04
VMA(1,200) 0.000361 0.000064 0.5086 0.52114 2.9700E-04
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
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NASDAQ composite index
Trading rule Buy Sell Buy>0 Sell>0 Buy - Sell
SETAR(1,50) 0.001152 -0.000983 0.60513 0.48529 2.1350E-03
SETAR(1,150) 0.001380 -0.001412 0.6105 0.47296 2.7920E-03
SETAR(1,200) 0.001398 -0.001423 0.61117 0.47314 2.8220E-03
AR(1,50) 0.001313 -0.001148 0.61158 0.48062 2.4610E-03
AR(1,150) 0.001486 -0.001683 0.6135 0.4641 3.1680E-03
AR(1,200) 0.001386 -0.001515 0.61281 0.46511 2.9010E-03
VMA(1,50) 0.000985 -0.000671 0.59128 0.51082 1.6570E-03
VMA(1,150) 0.000682 -0.000307 0.58204 0.51629 9.8900E-04
VMA(1,200) 0.000638 -0.000270 0.58112 0.51466 9.0700E-04
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New York Stock Exchange composite index
Trading rule Buy Sell Buy>0 Sell>0 Buy - Sell
SETAR(1,50) 0.000689 -0.000369 0.52200 0.51938 1.0590E-03
SETAR(1,150) 0.000834 -0.000705 0.53328 0.49578 1.5399E-03
SETAR(1,200) 0.000871 -0.000790 0.53297 0.49589 1.6610E-03
AR(1,50) 0.000681 -0.000367 0.52418 0.51547 1.0490E-03
AR(1,150) 0.000793 -0.000656 0.53139 0.49656 1.4490E-03
AR(1,200) 0.000854 -0.000814 0.53186 0.49377 1.6680E-03
VMA(1,50) 0.000421 0.000041 0.51291 0.53529 3.8000E-04
VMA(1,150) 0.000425 -0.000017 0.51711 0.52255 4.4400E-04
VMA(1,200) 0.000395 0.000074 0.51638 0.52294 3.7800E-04
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Standard and Poor 500 index
Trading rule Buy Sell Buy>0 Sell>0 Buy - Sell
SETAR(1,50) 0.000416 -0.000032 0.47687 0.56155 3.8400E-04
SETAR(1,150) 0.000556 -0.000299 0.4797 0.53307 8.5500E-04
SETAR(1,200) 0.000656 -0.000509 0.48306 0.52848 1.1640E-03
AR(1,50) 0.000402 -0.000200 0.4758 0.56015 3.8200E-04
AR(1,150) 0.000543 -0.000297 0.48026 0.53206 8.4000E-04
AR(1,200) 0.000572 -0.000376 0.47907 0.53206 9.4800E-04
VMA(1,50) 0.000296 0.000181 0.47907 0.53371 1.1500E-04
VMA(1,150) 0.000360 0.000510 0.47173 0.57009 3.0900E-04
VMA(1,200) 0.000368 0.000016 0.47778 0.55109 3.5200E-04
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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
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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
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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
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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
Heng Seng index Straits Times IndexCount 5961 5753Mean 0.000368024 0.000233276
Standard Error 0.000232311 0.000172286Standard Deviation 0.017936129 0.013067676
Sample Variance 0.000321705 0.000170764Kurtosis 56.6643583224* 8.484925539*
Skewness -2.448071427* -0.121755714*JB stat 721240.1595* 7225.697523*
ρ(1) 0.018987413 0.087835121ρ(2) -0.022648541 0.036533523ρ(3) 0.057481869 -0.000380175ρ(4) -0.019593225 0.000678288ρ(5) -0.030011843 -0.002237887Q(5) 32.58874337** 52.12378136**
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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
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Parameter estimation
Model:
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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)
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Parameter estimation
Remarks:
1. In our case, d (delay parameter) = 1.
2. Observe that the residual variance only takes on at most n distinct values as γ is varied, we set γ = ΔYt-d, t = 2,…,n.
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Parameter estimation
Thus, the estimate of θ is
Given n observations, we use OLS to obtain the fitted coefficients γ and θ and predict ∆Yt+1 based on ∆Yt.
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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.
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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 α and β change 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.
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Trading strategy
For example, given W = 50 and n = 100, 1. Obtain γ and α0 α1 β0 β1.
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.
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Algorithm in java
The model
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Algorithm in java
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Least squares method
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Algorithm in java
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Actual stock index moving by time HANG SENG INDEX
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SETAR(1, 50) model Predicted stock index moving by time - HANG SENG INDEX
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Actual stock index moving by time- STAITS TIMES INDEX
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SETAR(1, 50) model Predicted stock index moving by time - STRAITS TIMES INDEX
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SETAR(1, 150) model Predicted stock index moving by time - HANG SENG INDEX
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SETAR(1, 200) model Predicted stock index moving by time - HANG SENG INDEX
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Empirical results of implementing the trading strategies on the HANG SENG INDEX
Trading rule
Buy Sell σ(Buy) σ(Sell) Buy>0
Sell>0
Buy-Sell
SETAR(1, 50)
0.023595934
-0.0737 21634
0.133134682
0.433664247
0. 5341
0. 4838
0.0972959
SETAR(1, 150)
0.029190398
-0.1230 09524
0.071332314
0.458647862
0.5221 0.4737 0.1521995
SETAR(1, 200)
0.033381611
-0.1038 11652
0.076752117
0.264928711
0.5168 0.4667 0.137249652
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Empirical results of implementing the trading strategies on the STRAITS TIMES INDEX
Trading rule
Buy Sell σ(Buy) σ(Sell) Buy>0
Sell>0
Buy-Sell
SETAR(1, 50)
0.027662
-0.0311 35
0.090686577
0.274115174
0.5343 0.50227
0.058797
SETAR(1, 150)
0.033272476
-0.2598 1744
0.091718648
1.520024078
0.5325 0.4998 0.2930899
SETAR(1, 200)
0.03338161
-0.20877
0.086332393
1.2745767
0.5239 0.4915 0.2421512
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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).
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Thank You