PREDICTABILITY OF NON-LINEAR TRADING RULES IN THE US STOCK MARKETCHONG & LAM 2010

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

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)

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)

AR Model General Form

Commonly referred as AR (p) ModelWhere p is the order of autoregressive

part

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

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

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

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

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

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

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

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

New York Stock Exchange composite indexTrading 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

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

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

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

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

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**

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

Parameter estimation Model:

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)

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.

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.

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.

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.

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.

Algorithm in java The model

Algorithm in java

Least squares method

Algorithm in java

Actual stock index moving by time HANG SENG INDEX

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

Actual stock index moving by time- STAITS TIMES INDEX

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

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

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

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

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

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).

Thank You