【文章內(nèi)容簡介】
y uncorrelated with constant variance, then the correlation between uit and ui,t1 can be shown to be . ?If uit follows a stable AR(1) model, then uit will be serially correlated. Test Serial Correlation in the FirstDifferenced Equation ? Methods: (AR(1)) ? Zero Assumption: ? Steps: ? First, we estimate () by pooled OLS and obtain the residuals, ? Then, we run the regression again with r?i,t1 as an additional explanatory variable. ? The coefficient on r?i,t1 is an estimate of , and so we can use the usual t statistic on r?i,t1 to test H0: 0. Correct for the AR(1) Serial Correlation ? Unfortunately, standard packages that perform AR(1) corrections for time series regressions will not work. Standard CochraneOrcutt or PraisWinsten methods will treat the observations as if they followed an AR(1) process across i and t。 this makes no sense, as we are assuming the observations are independent across i. ? Corrections to the OLS standard errors that allow arbitrary forms of serial correlation (and heteroskedasticity) can be puted when N is large (and N should be notably larger than T ). ? If there is no serial correlation in the errors, the usual methods for dealing with heteroskedasticity are valid. Chap 14 Advanced Panel Data Methods ?Two Methods for Estimating Unobserved Effects Panel Data Model: ? Fixed Effects Estimation ? Random Effects Estimation Fixed Effects Estimation ? An alternative Methods to eliminate the fixed effects—— Fixed Effects Transformation (Within Transformation): ? ? for each i, average this equation over time: ? Substracting: ? Fixed Effects Estimator (Within Estimator) Fixed Effects Estimator ? Unbiasedness: Under a strict exogeneity assumption on the explanatory variables, the fixed effects estimator is unbiased: roughly, the idiosyncratic error uit should be uncorrelated with each explanatory variable across all time periods. ? The other assumptions needed for a straight OLS analysis to be valid are that the errors uit are homoskedastic and serially uncorrelated (across t) ? the degrees of freedom for the fixed effects estimator: df = NT- N- k= N(T- 1)- k. ? The goodnessoffit: The Rsquared obtained from estimating () is interpreted as the amount of time variation in the yit that is explained by the time variation in the explanatory variables. Other ways of puting Rsquared are possible, one of which we discuss later. Notes on some explanatory in Fixed Effects Estimation ? We cannot include variables such as gender or whether a city is located near a river as any explanatory variable that is constant over time for all i gets swept away by the fixed effects transformation ? Although timeconstant variables cannot be included by themselves in a fixed effects model, they can be interacted with variables that change over time and, in particular, with year dummy variables. ? When we include a full set of year dummies— that is, year dummies for all years but the first— we cannot estimate the effect of any variable whose change across time is constant. Example The Return to Education over Time The Dummy Variable Regress