Cross-sectional Correlation Tests For Two-factor Error Models With Time-invariant Regressors

In recent years,most empirical studies in the field of econometrics have adopted panel data modeling.In theoretical research and practical applications,it is usually assumed that the same observation object of the panel data model has a relationship at different observation times,and the cross sections are considered to be independent of each other.Such assumptions are unreasonable,for example,public shocks to cross-sectional data(political changes,financial crises)will lead to cross-sectional dependence,which in turn affect the model's inference methods.Therefore,it is meaningful to test the cross-sectional dependence of the model.The cross-sectional dependence test of the two-way error model is one of the hotspots studied by scholars in recent years.For the two-way error model,the traditional crosssectional dependence test assumes that the regressions is time-varying,but in practice,the time-invariant circumstances are often encountered,such as individual gender,region,religious beliefs,etc.Based on this,this paper generalizes the traditional CD test,so that the test statistic can still well test the cross-sectional dependence problem under the twoway error model with time-invariant regressions.In the case of a small panel,the test statistic asymptotically obeys the standard normal distribution,which can effectively test the cross-sectional dependence of the research object in a specific time.However,in the field of econometrics,the case with large time dimensions(T)also often occurs,such as high-frequency finance,conditional value-at-risk research,etc.Therefore,under the twoway error model with time-invariant regressions of large cross sections(N)and large time dimensions(T),it is deduced that the CD test still asymptotically obeys the standard normal distribution.In order to compare the performance of the test statistics,this paper also generalizes the traditional LMbc test based on the fixed-effect homogeneous panel model,so that it can effectively test the cross-sectional dependence of the two-way error model with time-invariant regressions.In the case of large-panel,the test statistic asymptotically obeys the standard normal distribution.In order to examine the performance of the proposed method in a limited sample,Monte Carlo simulations were used to compare the empirical size and power values of the two test statistics and the LM test.The simulation shows that the LMbc test is more sensitive to the time dimensions(T),and the CD test performs well,consistent with the theoretical propertics.There arc different degrees of cross-sectional dependence in the setting of heterogeneous error terms,and the power functions of the two test statistics show corresponding trends;however,the LM test exhibits a large amount of size distortion.When heterogeneous errors arc set as the factor structure,both the CD test and the LMbc test have sufficient power,even when the factor loadings contain positive and negative values and follow different distributions.When large N and large T,both the CD test and LMbc test have correct size and power;when largcN and short T,the CD test performs well,while the power of LMbc test tends to be normal as T increases.Finally,this paper tests the performance of the CD test and the LMbc test through two real cases.In the case of a small panel:Examining the cross-sectional dependence of a crime model based on panel data of 90 North Carolina counties from 1981 to 1987,the CD test results show that there is regional crime in the counties in the state.In the case of largepanel:Based on the cigarette consumption panel data of 46 states in the United States from 1963 to 1992,a two-way error model with time-invariant regressions was established to judge the correlation of cigarette consumption among states.Numerical results of the CD test and the LMbc test show a strong cross-sectional dependence between cigarette consumption across US states.The cross-sectional dependence test results of the two actual cases are consistent with reality,which lays a foundation for the subsequent model inference.