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FORECASTING IN FINANCIAL MARKET USINGMARKOV REGIME SWITCHING ANDPRINCIPAL COMPONENT ANALYSIS.
NOP SOPIPAN
September 13, 2012
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
Research ObjectivesMethodology
Scope and limitationsResearch Methodology
Descriptive of model
1 Motivation
2 Research Objectives
3 Methodology
4 Scope and limitations
5 Research MethodologyHeteroskedasticity ModelSpurious high persistence problemMulticollinearily
6 Descriptive of model
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
Research ObjectivesMethodology
Scope and limitationsResearch Methodology
Descriptive of model
In Thai
การพยากรณในตลาดทางการเงน ดวย
วธการวเคราะหองคประกอบหลก และ วธสบเปลยนสถานะมารคอฟ
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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The Beginning
Prof.Dr.Pairote Sa-ayatham Prof.Dr.Bhusana Premanode
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Characteristics of returns
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Property in Financial returns
(Weakly) Stationary process.
Look liked White noise process
The model assume random walk model.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Forecast Financial Return
Let Pt denote the series of the financial price at time t andrtt>0 be the logarithmic return (in percent) on the financialprice at time t be a sequence of random variables on a probabilityspace(Ω,F,P) , i.e.
rt = 100 · ln(Pt
Pt−1) (1)
Let rt be explain in mean equation, i.e.
rt = µt + εt (2)
with mean and variance:
µt = E (rt |Ft−1), ht = Var(rt |Ft−1) = E [(rt − µt)2|Ft−1]
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Forecasted inFinancial Return
Mehmet A.(2008), Marcucci J. (2005) assume mean equationin returns is constant
Easton and Faff (1994), and Kyimaz and Berument (2001)assume mean equation in returns with a one week delay intothe regression model
Supot C.(2003)assume mean equation in returns withAutoregressive process.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Construct Model
The constant mean equation cannot be forecast accuratelydue to the inaccuracy of the financial data, since the financialreturns depend concurrently and dynamically on manyeconomic and financial variables.
The fact that the return has a statistically significantautocorrelation indicates that lagged returns might be usefulin predicting future returns.
We consider add some explanatory variables and stationaryAutoregressive Moving-average order p and q (ARMA (p,q)).Then we add them into a mean equation in order to increaseaccuracy.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Construct Model
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
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Scope and limitationsResearch Methodology
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Research Objectives
The objectives of the research are the following :
1 Construct model of equation (3) for forecasting of financialreturn.
2 Solve multicollinearity problem in explanatory variables ofequation (3) by using Principal Component Analysis (PCA).
3 Construct heteroskedasticity model for forecasting of volatilitywith GARCH-type model and Markov Regime SwitchingGARCH model.
4 Compare and evaluate the performance of models betweenconstant mean equation and model of equation (3) with lossfunctions.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
Research ObjectivesMethodology
Scope and limitationsResearch Methodology
Descriptive of model
Research Objectives
The objectives of the research are the following :
1 Construct model of equation (3) for forecasting of financialreturn.
2 Solve multicollinearity problem in explanatory variables ofequation (3) by using Principal Component Analysis (PCA).
3 Construct heteroskedasticity model for forecasting of volatilitywith GARCH-type model and Markov Regime SwitchingGARCH model.
4 Compare and evaluate the performance of models betweenconstant mean equation and model of equation (3) with lossfunctions.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
Research ObjectivesMethodology
Scope and limitationsResearch Methodology
Descriptive of model
Research Objectives
The objectives of the research are the following :
1 Construct model of equation (3) for forecasting of financialreturn.
2 Solve multicollinearity problem in explanatory variables ofequation (3) by using Principal Component Analysis (PCA).
3 Construct heteroskedasticity model for forecasting of volatilitywith GARCH-type model and Markov Regime SwitchingGARCH model.
4 Compare and evaluate the performance of models betweenconstant mean equation and model of equation (3) with lossfunctions.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
Research ObjectivesMethodology
Scope and limitationsResearch Methodology
Descriptive of model
Research Objectives
The objectives of the research are the following :
1 Construct model of equation (3) for forecasting of financialreturn.
2 Solve multicollinearity problem in explanatory variables ofequation (3) by using Principal Component Analysis (PCA).
3 Construct heteroskedasticity model for forecasting of volatilitywith GARCH-type model and Markov Regime SwitchingGARCH model.
4 Compare and evaluate the performance of models betweenconstant mean equation and model of equation (3) with lossfunctions.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
Research ObjectivesMethodology
Scope and limitationsResearch Methodology
Descriptive of model
Research Objectives
The process of research had been following :
1stPaper.N. Sopipan, P. Sattayatham and B. Premanode ,”Forecasting Volatility of Gold Price using Markov RegimeSwitching and Trading Strategy”, Journal of mathematicalfinance.Vol. 2(1), pp 121-131 , 2012.
2ndPaper. P. Sattayatham, N. Sopipan and B. Premanode(2012), ”Forecasting the Stock Exchange of Thailand usingDay of the Week Effect and Markov Regime SwitchingGARCH.”, Journal of Forecasting. (in progress)
Next Step Use PCA to solve multicollinearity problem inexplanatory varibles in mean equation forecast return.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
Research ObjectivesMethodology
Scope and limitationsResearch Methodology
Descriptive of model
Research Objectives
The process of research had been following :
1stPaper.N. Sopipan, P. Sattayatham and B. Premanode ,”Forecasting Volatility of Gold Price using Markov RegimeSwitching and Trading Strategy”, Journal of mathematicalfinance.Vol. 2(1), pp 121-131 , 2012.
2ndPaper. P. Sattayatham, N. Sopipan and B. Premanode(2012), ”Forecasting the Stock Exchange of Thailand usingDay of the Week Effect and Markov Regime SwitchingGARCH.”, Journal of Forecasting. (in progress)
Next Step Use PCA to solve multicollinearity problem inexplanatory varibles in mean equation forecast return.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
Research ObjectivesMethodology
Scope and limitationsResearch Methodology
Descriptive of model
Research Objectives
The process of research had been following :
1stPaper.N. Sopipan, P. Sattayatham and B. Premanode ,”Forecasting Volatility of Gold Price using Markov RegimeSwitching and Trading Strategy”, Journal of mathematicalfinance.Vol. 2(1), pp 121-131 , 2012.
2ndPaper. P. Sattayatham, N. Sopipan and B. Premanode(2012), ”Forecasting the Stock Exchange of Thailand usingDay of the Week Effect and Markov Regime SwitchingGARCH.”, Journal of Forecasting. (in progress)
Next Step Use PCA to solve multicollinearity problem inexplanatory varibles in mean equation forecast return.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
Research ObjectivesMethodology
Scope and limitationsResearch Methodology
Descriptive of model
Scope and limitations
The forecasting in Markov Regime Switching for simplicity,this thesis assume presence of two regimes and order of theGARCH-type is (1,1).
The standardized errors follow student-t, generalized errordistributions (GED) and normal distribution.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
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Scope and limitationsResearch Methodology
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Scope and limitations
The forecasting in Markov Regime Switching for simplicity,this thesis assume presence of two regimes and order of theGARCH-type is (1,1).
The standardized errors follow student-t, generalized errordistributions (GED) and normal distribution.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
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Scope and limitationsResearch Methodology
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Heteroskedasticity
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Heteroskedasticity
The generalized autoregressive conditional heteroskedasticity(GARCH)-type models mainly capture for problem ofheteroskedasticity.
GARCH model
The Exponential GARCH (EGARCH) model
GJR-GARCH model
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Heteroskedasticity model
GARCH(1,1):[Bollerslev (1986)]
εt = ηt√
ht , ht = α0 + α1ε2t−1 + β1ht−1 (3)
EGARCH(1,1):[Nelson (1991)]
ln(ht) = α0 + α1|εt−1√ht−1|+ β1ln(ht−1) + ξ
εt−1√ht−1
(4)
GJR-GARCH(1,1):[Glosten, Jagannathan and Runkle(1993)]
ht = α0+α1ε2t−1(1−Iεt−1>0)+β1ht−1+ξε2t−1(Iεt−1>0) (5)
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Heteroskedasticity model
GARCH(1,1):[Bollerslev (1986)]
εt = ηt√
ht , ht = α0 + α1ε2t−1 + β1ht−1 (3)
EGARCH(1,1):[Nelson (1991)]
ln(ht) = α0 + α1|εt−1√ht−1|+ β1ln(ht−1) + ξ
εt−1√ht−1
(4)
GJR-GARCH(1,1):[Glosten, Jagannathan and Runkle(1993)]
ht = α0+α1ε2t−1(1−Iεt−1>0)+β1ht−1+ξε2t−1(Iεt−1>0) (5)
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Heteroskedasticity model
GARCH(1,1):[Bollerslev (1986)]
εt = ηt√
ht , ht = α0 + α1ε2t−1 + β1ht−1 (3)
EGARCH(1,1):[Nelson (1991)]
ln(ht) = α0 + α1|εt−1√ht−1|+ β1ln(ht−1) + ξ
εt−1√ht−1
(4)
GJR-GARCH(1,1):[Glosten, Jagannathan and Runkle(1993)]
ht = α0+α1ε2t−1(1−Iεt−1>0)+β1ht−1+ξε2t−1(Iεt−1>0) (5)
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Spurious high persistence problem
0 200 400 600 800 1000 12000
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Vola
tilit
y
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Review of Markov Regime Swithcing
Hamilton (1989,1990) extended MRS for analyzing structuralbreaks in financial return.
Hamilton and Susmel (1994) and Cai (1994) proposed MRSARCH.
Gray (1996)proposed MRS GARCH.
Klassen (2002) modification of Gray’s MRS GARCH.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Concept of Markov Regime Swithcing
Generalization of the simple dummy variables approach.
Allow regimes (called states) to occur several periods overtime.
In each period t the state is denoted by St .
There can be N possible states: St = 1, . . . ,N.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Markov Regime Swithcing
Probability St equals some particular value j depends on thepast only through the most recent value St−1:PSt = j |St−1 = i , St−2 = k , . . . = PSt = j |St−1 = i =pij .
Process is described as an N-state Markov Chain withtransition probabilitiespiji ,j=1,2,...,N and pi1 + pi2 + . . .+ piN = 1
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Markov Regime Swithcing
Probability St equals some particular value j depends on thepast only through the most recent value St−1:PSt = j |St−1 = i , St−2 = k , . . . = PSt = j |St−1 = i =pij .
Process is described as an N-state Markov Chain withtransition probabilitiespiji ,j=1,2,...,N and pi1 + pi2 + . . .+ piN = 1
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Markov Regime Swithcing
The conditional probability density function for theobservations rt given the state variables St ,St−1 and Ft−1 isf (rt |St , St−1,Ft−1)
The chain rule for conditional probabilities yields then for thejoint probability density function for the variables rt ,St ,St−1given Ft−1 isf (rt ,St ,St−1|Ft−1) = f (rt |St ,St−1,Ft−1)Pr(St ,St−1|Ft−1)
The log-likelihood function to be maximized with respect tothe unknown parameters becomesl(θ) =
∑Tt=1[log(
∑NSt=1
∑NSt−1=1)f (rt ,St ,St−1|Ft−1)]
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Markov Regime Swithcing
The conditional probability density function for theobservations rt given the state variables St ,St−1 and Ft−1 isf (rt |St , St−1,Ft−1)
The chain rule for conditional probabilities yields then for thejoint probability density function for the variables rt , St , St−1given Ft−1 isf (rt ,St ,St−1|Ft−1) = f (rt |St ,St−1,Ft−1)Pr(St ,St−1|Ft−1)
The log-likelihood function to be maximized with respect tothe unknown parameters becomesl(θ) =
∑Tt=1[log(
∑NSt=1
∑NSt−1=1)f (rt ,St ,St−1|Ft−1)]
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
Research ObjectivesMethodology
Scope and limitationsResearch Methodology
Descriptive of model
Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Markov Regime Swithcing
The conditional probability density function for theobservations rt given the state variables St ,St−1 and Ft−1 isf (rt |St , St−1,Ft−1)
The chain rule for conditional probabilities yields then for thejoint probability density function for the variables rt , St , St−1given Ft−1 isf (rt ,St ,St−1|Ft−1) = f (rt |St ,St−1,Ft−1)Pr(St ,St−1|Ft−1)
The log-likelihood function to be maximized with respect tothe unknown parameters becomesl(θ) =
∑Tt=1[log(
∑NSt=1
∑NSt−1=1)f (rt ,St ,St−1|Ft−1)]
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
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Scope and limitationsResearch Methodology
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Markov Regime Swithcing GARCH models
The MRS-GARCH model with only two regimes can be representedas follows.
rt = µt,St + εt = µt,St + ηt√
ht,St (6)
ht,St = α0,St + α1,Stε2t−1 + β1,St ht−1 (7)
where St = 1, 2 , µt,St is the mean and ht,St is the volatility underregime St on Ft−1 .
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Multicollinearily
Multicollinearity is a statistical phenomenon in which two or morepredictor variables in a multiple regression model are highlycorrelated.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Multicollinearily
For the mean equation in (3) i.e.
rt = µt + εt ,
µt = φ0 +k∑
i=1
βiXit +
p∑i=1
φi rt−i −q∑
i=1
θiεt−i
Multicollinearity, or high correlation between the explanatoryvariables in a regression equation. One method for removing suchmulticollinearity and redundant independent variables is principalcomponent analysis (PCA).
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
OutlineMotivation
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Principal Component Analysis
The idea of PCA is to find linear combinations αi such that Zi andZj are uncorrelated for i 6= j and the variances of Zi are as large aspossible. More specifically:
The first principal component of X is the linear combinationZ1 = α
′1X that maximizes Var (Z1) subject to the constraint
α′1α1 = 1.
The second principal component of X is the linearcombination Z2 = α
′2X that maximizes Var (Z2) subject to
the constraint α′2α2 = 1 and Cov(Z2,Z1) = 0.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Principal Component Analysis (Cont.)
The kth principal component of X is the linear combinationZk = α
′kX that maximizes Var (Zk) subject to the constraint
α′kαk = 1 and Cov(Zk ,Zk ′) = 0 for k 6= k ′
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Heteroskedasticity ModelSpurious high persistence problemMulticollinearily
Principal Component Analysis (Cont.)
The new variables from the PCA become ideal to use as predictorsin a regression equation since they optimize spatial patterns andremove possible complications caused by multicollinearity.
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Model of returns
The model of returns (2) as follow:
rt = µt + εt
This thesis use three mean equations of return as follow:(1) Mean equation are constants.
µt = δ.
µt = δS(t).
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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Model of returns
(2) Mean equation as stationary ARMA(p,q) includes day ofthe week effect.
µt = Σ5i=1βiDit + Σp
j=1φj rt−j − Σqk=1θkεt−k ,
whereDit , i = 1, ..., 5 are dummy variables which take on the value of 1 ifthe corresponding return of the five days respectively and 0otherwise,βi , i = 1, ..., 5 are coefficients which represent the average returnfor each day of the week ,φj , θk , i = 1, ..., p, k = 1, ..., q, are coefficients which represent theARMA (p, q).
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.
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THANK YOU FOR ATTENTION
NOP SOPIPAN FORECASTING IN FINANCIAL MARKET USING MARKOV REGIME SWITCHING AND PRINCIPAL COMPONENT ANALYSIS.