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(ui, uj)0, i.e. E(ui, uj) 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, ut=r ut-1+vt Second-order serial correlation will be ut=r1ut-1+r2ut-2+vt,The consequences of serial correlation (Autocorrelation),OLS estimators will be still unbiased and consistent. take the simple regression as an example Y= b0 + b1 X +u We know the OLS estimator of b1 is,The consequences of serial correlation (Autocorrelation),The R2 and adj-R2 are still consistent if the time series is stationary (thats r 1). Or else, for non-stationary time series, the R2 and adj-R2 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 ut. 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, et, 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 N1 is the number of positive signs of the residuals, and N2 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. H0: 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, H0, means there exists serial correlation.,Runs test (example),If the signs of the residual is (-)(+)(-)(+)(-) 9 8 4 2 3 Then, N1=8+2=10, N2=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 H0. The number of runs in our model is 58, so we reject the H0, 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 R2, adj-R2(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, ut=r ut-1+vt, -1r1. There is no lagged dependent variable as explanatory variable. Ct=b0+b1Yt+b2Ct-1+ut,Durbin-Watson Test,We can rewrite the Durbin-Watson d-stat as,Durbin-Watson Test,If the Durbin-Watson d-stat lies in (du, 4- du), there is no serial correlation. If d4-dL, there are positive and negative serial correlation respectively. If dLddU, or 4-dUd4-dL, then we cant detect the serial correlation.,Durbin-Watson Test: Procedure,First regress Y on Xs, and get the residuals et. 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 dL and dU. 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 dwatson d=0.2137 n=44 k=1 dL(44,1)=1.475 dU(44,1)=1.566 ddL, so there exists positive correlation in et.,Durbin-Watson Test: there is a lagged dependent variable,Ct=b0+b1Yt+b2Ct-1+ut Where there is a lagged dependent variable in the model, the Durbin-Watson h test can be used. H0: there is no serial correlation. d=0.8598, h=2.9092 Z0.05=1.645, reject H0. Stata command: estat durbinalt,Corrections for serial correlation: Generalized differencing,Yt=b0+b1Xt+ut (1) If there is first-order serial correlation, that is,