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1、Some statistics in Econometrics and their developments,Shaoping WangSchool of Economics,Huazhong University of Science and Technology,Wuhan,China,IntroductionClassic tests in Econometrics,More broadly:hypothesis could be set as linear or nonlinear.Likelihood Ratio test:Wald test:LM test:,Three Preva
2、iling Tests in econometrics,Introduction,DW test for first autocorrelationI(1)tests:t statistics and its distribution-DF and ADF distributionWhats difference for the DF and ADF distributionADF test and PP test for I(1)processSome issues,3.1宏观计量,非平稳（单位根I(1)）过程:,3.1宏观计量,非平稳（单位根I(1)）过程:,单位根检验 时间序列yt,yt
3、=yt-1+ut,零假设和备择假设分别是，H0：=1，（yt I(1)H1：1，（yt I(0)）,3.1宏观计量,非平稳（单位根I(1)）过程:,单位根检验 时间序列yt,yt=yt-1+ut,零假设和备择假设分别是，H0：=1，（yt I(1)H1：1，（yt I(0)）,用DF统计量进行单位根检验。t=DF distribution,3.1宏观计量,非平稳（单位根I(1)）过程:,单位根检验 时间序列yt,yt=yt-1+ut,零假设和备择假设分别是，H0：=1，（yt I(1)H1：1，（yt I(0)）,用DF统计量进行单位根检验。t=DF distribution,协整 协整是对非
4、平稳经济变量长期均衡关系的统计描述.非平稳经济变量间存在的均衡关系称作协整关系.定义:如果 X1t,X2t,Xkt I(1)，Zt=X I(0),=(1,2,k)，那么X1t,X2t,Xkt 协整，记为,Xt CI(1,0)，是协整向量.,上图说明:X(t),Y(t)I(1),Z(t)=0.3Y(t)+0.5X(t)I(0),Introduction,Panel unit root test:Survey by Hurlin and Mignon(2004)Assume cross sectional independenceLevin and Lin(1992,1993);Levin,Lin
5、 and Chu(2002);Harris and Tzavalis(1999);Im,Pesaran and Shin(1997,2003);Maddala and Wu(1999);Choi(1999,2001)Assume cross sectional dependenceFlres,Preumont and Szafarz(1995);Tayor and Sarno(1998);Breitung and Das(2004);Bai and Ng(2001,2004);Moon and Perron(2004);Phillips and Sul(2003);Pesaran(2003);
6、Choi(2002);Chang(2002),Introduction,Chang(2002)A NIV estimation Chang test,Performance of Chang test with moderate to high cross sectional dependency,This Paper A Two Step Test Improved the performance,Changs Model,(1):coefficient on the lagged dependent variable:error term which follows the AR(p)pr
7、ocess:(2):lag operator:autoregressive coefficient:some integer that is known and fixed,We are interested in testing for all VS for some,Hypothesis,Model Assumptions,To ensure the AR(p)process in(2)is invertibleAssumption 1:for all andTo restrict the distribution of error termAssumption 2:Denote(1)ar
8、e independent and identically distributed and its distribution is absolutely continuous with respect to Lebesgue measure(2)has mean zero and covariance matrix(3)satisfies for some and has a characteristic function that satisfies for some,NIV Estimation and Chang Test,OLS estimation:Under,the asympto
9、tic distribution of obtained from(3)is asymmetric,and not the usual t-distribution NIV estimation:as instrument for,where is some function satisfying Assumption 3:is regularly integrable and satisfy,Under assumption 1-3,Chang draw the key result:as and are asymptotically uncorrelated regardless of t
10、he cross sectional dependence And the test statistic has a limiting standard normal distribution,NIV Estimation and Chang Test(contd),Findings about Chang(2002),The bigger the N is,the Smaller the correlation coefficient of cross-sectional units becomes.The test statistic does not fully follow the l
11、imiting standard normal distribution when the cross sectional dependence is strong.Chang test perform well in finite samples when when the cross sectional dependence is low.,Our Test:A Two Step Test,Step1:eliminate the cross sectional dependence through the method of principal components.,Step2:appl
12、y Chang test to the treated data,Model Setting,Adopting the DGP in Bai and Ng(2004)to model the cross-sectional dependency by common factor:(3)Where:Ft is the r1 common factor among individuals.Error term has zero mean with covariance matrix,for.,Test procedure:Step one,Under the null hypothesis:The
13、 differenced common component estimator of is times the eigenvectors corresponding to the largest r eigenvalue of the matrix,the estimated loading matrix is given by.,Step one(contd),the data with weak(or no)cross-sectional dependence where can be set as 0.After eliminating the common factor,model(3
14、)can be rewritten as:(4)Model(4)is of the same form as Changs model(1),but with a different error term.,Step two,Denote,where.We have the model with no(weak)cross-sectional dependence(5),Step Two(contd),Denote,The NIV estimator for(5)is:t-ratio of:,is the variance estimator of.Test statistics:.,The
15、Distribution,Theorem 1.Suppose that Assumption 1-3 hold.Under the null hypothesis of panel unit root,we obtain,as,for all and,where denote the correlation coefficient.Theorem 2.Suppose that Assumption 1-3 hold.Under the null hypothesis of panel unit root and as,we obtainExtend Theorem 1 and 2 to pan
16、el data models with individual intercept and/or time trend by de-meaning and/or de-trending schemes,Simulation,DGP with General Cross-sectional dependence The covariance matrix of,DGP with General cross sectional dependence,for size evaluation,for power evaluationThe number of common factor is set a
17、s 1 for eliminating the cross-sectional dependency by method of principal components.,Simulation(contd),DGP with General cross sectional dependence Size,no intercept and no linear trend,The empirical sizes of our test in all cases are fairly close to the nominal sizes(we pick up 5%).The distortions
18、of Chang test are more pronounced when the cross sectional dependence is high(e.g.,0.8).,Simulation(contd),Our test has reasonably good power in all designs and the power increases as N and T increase.The power of BN test is a little lower than the power of our test in some cases.,no intercept and n
19、o linear trend Uniform0.85,0.99.,General cross sectional dependencePower,Simulation(contd),The covariance matrix of,for size evaluation,for power evaluation,Simulation DGP with one common factor,no intercept and no linear trend,The empirical sizes of two tests are almost identical and close to the n
20、ominal sizes,and the powers are high enough.,Simulation(contd):Common factor dependence Size&Power,Idea:remove the cross sectional dependence before applying Chang testMethod:first removing the cross-sectional dependence by method of principal component and then applying Chang test to the treated dataTest Distribution:a limiting standard normal distribution under the null hypothesis of panel unit rootTest Performance:(1)much better than Chang test when the cross sectional dependence is moderate to high;(2)as good as Chang test when the cross sectional dependence is low.,Conclusion,Thank you!,
