实证金融研究Empirical Finance:Analysis of non-stationary processes:Estimating and testing long-run relationships in systems of equations [2]
论文作者:留学生论文论文属性:案例分析 Case Study登出时间:2011-01-31编辑:anterran点击率:8420
论文字数:3214论文编号:org201101311022211571语种:英语 English地区:英国价格:免费论文
附件:20110131102221146.pdf
关键词:Empirical FinanceAnalysisnon-stationary processesEstimating and testinglong-run relationshipssystems of equations
ECMs:*1*1*1*0*tttttvXYYX++++=−−φδδμ()**1**01ttttvYX+−+Δ=Δ−εφδ()()**1**01011ttttttttvYXvXY+−+Δ=Δ+−+Δ=Δ−−εφδεφδBoth equations potentially contain information about the long-run parameters via the error correction term. An efficient information will use this information. Therefore a single equation approach (EG 2 Step) will in general be inefficient.There is an important exception. If one of the adjustment parameters is zero then that equation contains no informationabout the long-run parameters. In that case the corresponding variable(e.g., X) is weakly exogenous for the long-run parameters–its equation can be ignored in estimatingthe long-run parameters.Warwick Business School 7
Problems with EG 2 Step
Based on these problems with EG 2 Step, we would like an alternative estimator which
1.Can identify multiple cointegratingrelationships.
2.Is an efficient estimator of the long-run parameters.
This will lead us to look at the Johansen Systems Estimator shortly.
But before that we need to look briefly at VAR models…Warwick Business School 8
From single to multiple equations: VARA popular time series (a-theoretical) model for systems of variables is a VECTOR AUTOREGRESSION (VAR)A VAR model provides a simple framework for estimating the dynamics of a system of endogenous variables. Specifically: •All variables are treated as endogenous(each has its own equation) –noassumptions of exogeneityare made. •Since only lagged variables appear on the RHS there is no endogenous regressorproblem in estimation –applying OLS equation by equation is a consistent (and efficient) estimator of the unrestricted coefficients (the A matrices). •VARs are used widely in forecasting systems of variables (analogous to using AR models in univariatetime series) and testing causality.tptpttvyAyAy+++=−−...11y is an n×1 vector of variablesn×nmatrices of AR parametersIID vector of disturbancesusually assumed to beGaussian for estimation.The model may also include constant and trend terms.Warwick Business School 9
Application of VAR models: testing Granger causality in a stationaryVAR (Brooks 6.14).X is said to Granger causeY if lagged values of X affect Y. A VAR can be used to test this. For example: The null that X does notGranger cause Y is given by: Similarly the null that Y does notGranger cause X is given by:The F tests are only valid in a stationary VAR(Y and X~I(0)). –In a nonstationaryVAR the parameters have non-standard distributions.–In that case need to test causality in either: •A VAR in the I(0) differences of Y and X (if the variables are not cointegrated) or •A VECM (if the variables are cointegrated–see below). tptptptptttptptptpttvXXYYXvXXYYY221212121111111111+++++=++++++=−−−−−−−−δδααδδααKKKK0:1110===pHδδKTest hypothesis with an F-test0:2210===pHααKBivariateVAR(p) modelTest hypothesis with an F-testWarwick Business School 10
Application of VAR models: cointegrationin a non-stationary VAR
The Granger Representation Theorem generalizes to systems of I(1) variables. If the system is cointegratedthen there exists a: VECTOR ERROR CORRECTION MODEL (VECM)
()()()()tttttttttttttttvyyyIvyIAAyIAvyAyIAyvyAyAy+Π+ΔΓ=−+−++Δ−=++−=Δ⇒++=−−−−−−−−−211212211122112211 RHS) on the A add and(subtract ()()IAAIAAvyyyypiitptptptt&
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