第10章-序列相关性课件

,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,第,10,章序列相关性,Serial Correlation/Autocorrelation,Main Contents,What is Serial correlation(Autocorrelation)?,The consequences of serial correlation,How to detect the serial correlation?,Corrections for serial correlation,What is Serial correlation(Autocorrelation)?,The assumption that errors corresponding to different observations are uncorrelated often breaks down in,time-series,studies.,When the error terms from different(usually adjacent)time periods are correlated,we say that the error term is,serially correlated.,That is,Cov(,u,i,u,j,),0,i.e.E(,u,i,u,j,)0 for,i,j.,Patterns of serial correlation,Reasons of serial correlation,Inertia or sluggishness,Model specification errors(omitted variables),What is Serial correlation(Autocorrelation)?,In this chapter,we only deal with the problem of,first-order serial correlation,in which errors in one time period are correlated directly with errors in the ensuing period.For example,u,t,=,r,u,t-,1,+v,t,Second-order serial correlation will be,u,t,=,r,1,u,t-,1,+,r,2,u,t-,2,+,v,t,The consequences of serial correlation(Autocorrelation),OLS estimators will be still unbiased and consistent.,take the simple regression as an example,Y=,b,0,+b,1,X+u,We know the OLS estimator of,b,1,is,The consequences of serial correlation(Autocorrelation),The R,2,and adj-R,2,are still consistent if the time series is stationary(thats,r,1).Or else,for non-stationary time series,the R,2,and adj-R,2,may be invalid.,The consequences of serial correlation(Autocorrelation),OLS estimators will not be efficient.The variance of OLS estimators will be biased.,The consequences of serial correlation(Autocorrelation),t-statistics and F-statistic will be misleading when there are serial correlation in error terms,u,t,.,The variance and standard error of the predicted value will be invalid.,How to detect the serial correlation?,Time-sequence plot,Runs test,Durbin-Watson test,Time sequence plot,Example:Real wages and productivity(Example 10-1),Runs test,First,get the sign of the residuals,e,t,for example,(-)(+)(-)(+)(-),that is,there are 9 negative signs,followed by 8 positive signs and so on.,The same signs in the parentheses are called a,run,.,Let N is the number of observations,and N,1,is the number of positive signs of the residuals,and N,2,is the number of negative signs.And,k,is the number of runs.,Runs test,Swed and Eisenhart give us a table of critical values.,H,0,:the residual,e,is stochastic,that is,there is no serial correlation.,How to test?If the number of run in your model is less than or equal the critical value n1(table A-6a),and larger than or equal to the critical value n2(A-6b),then we can reject the null hypothesis,H,0,means there exists serial correlation.,Runs test(example),If the signs of the residual is (-)(+)(-)(+)(-),9 8 4 2 3,Then,N,1,=8+2=10,N,2,=9+4+3=16,N=26,k=5,then the critical value at 5%significance is 8 and 19.So,if the runs in our model,8 or,19,we should reject the null hypothesis H,0,.,The number of runs in our model is 58,so we reject the H,0,which mean there is serial correlation in our model.,Durbin-Watson Test,Durbin and Watson put forward an d statistic(DW).,In most software,d-value will be provided with R,2,adj-R,2,(Eviews),in STATA,using command,tsset year/*to describe the data is time series*/,estat dwatson/*must using after reg*/,dwstat/*the out of dated command*/,Durbin-Watson Test,There must be a intercept term in the regression model;,It only can be used to detect the first order serial correlation.That is,u,t,=,r,u,t-1,+v,t,-1,r,1.,There is no lagged dependent variable as explanatory variable.,C,t,=b,0,+b,1,Y,t,+b,2,C,t-1,+,u,t,Durbin-Watson Test,We can rewrite the Durbin-Watson d-stat as,r,d-value,-1,4,0,2,1,0,Durbin-Watson Test,If the Durbin-Watson d-stat lies in(d,u,4-d,u,),there is no serial correlation.,If d4-d,L,there are positive and negative serial correlation respectively.,If d,L,dd,U,or 4-d,U,d4-d,L,then we cant detect the serial correlation.,0,d,L,d,U,2,4-d,U,4-d,L,4,Reject H,0,Positive serial correlation,Accept H,0,there is no serial correlation.,Reject H,0,Negative serial correlation,Can not identify.,Can not identify.,Durbin-Watson Test:Procedure,First regress,Y,on,X,s,and get the residuals,e,t,.,Calculate the DW d-stat.May be given by software.,Given the number of observations,n,and the number of explanatory variables,k,check the critical value,d,L,and d,U,.,Using the rule to judge whether there is serial correlation.,Real wages and productivity:DW test,Table 10_1.txt,insheet using“table 10_1.txt”,clear,tsset year,reg rwage product,dwstat or estat,dwa,tson,d=0.2137 n=44 k=1,d,L,(44,1)=1.475 d,U,(44,1)=1.566,d Z,0.05,=1.645,reject H,0,.,Stata command:estat durbinalt,Corrections for serial correlation:Generalized differencing,Y,t,=,b,0,+b,1,X,t,+u,t,(1),If there is first-order serial correlation,that is,u,t,is,AR(1)process.,i.e.,u,t,=,r,u,t-1,+v,t,-1,r,1.,Then