Testing Serial Correlation And Heteroscedasticity In Partial Linear Models

Testing serial correlation and heteroscedasticity in residuals (errors) for regressionmodels are important practice in econometric data analysis. We generally assume thatthe errorsε_i's are mutually independent and homoscedastic in regression models. Iftheε_i's are not independent, i.e. E(ε_iε_j)≠0, i≠j, we say that the models are seriallycorrelated. If the error variances are not equal, i.e. Var(ε_i)=σ_i~2, i=1,…, n, we saythat the models are heteroscedastic. We request the fitted residuals are white noisefor a good fitted models, that is to say the residuals don't include any information ofthe models. So the assumptions of independent and homoscedasticity are some basicassumptions. Under those assumptions, we can do some ordinary statistic inferencesuch as parameter estimate, hypothesis test and further to forecast. Violating thoseassumptions can lead to many problems. For example, when the errors are seriallycorrelated, we will face with the following problems:1. The parameter estimate is not efficient.2. The significant test for variate is meaningless. The same to other tests.3. The forecast for the models is inefficient.4. Maybe omit some important explanatory variables or the functional form ismisspecified.When the models are heteroscedastic, we can face with the same problems of serialcorrelation. So it is important to diagnose heteroscedasticity and serial correlationbefore statistic inference.Empirical likelihood, proposed by Owen(1988, 1990), is a nonparametric method of inference with sampling properties similar to those of bootstrap. The empiricallikelihood ratio has a limiting chi-squared distribution, leading to hypothesis test andconfidence interval. Compared with other classic or modern statistic methods, empiricallikelihood has many prominent merits. For example, the empirical likelihood ratioconfidence region is range preserving and transformation respecting, and the shapeand orientation of the resulting confidence regions are determined entirely by the data.Also, empirical likelihood can increase accuracy of coverage through the use of auxiliaryinformation, need not to estimate the covariance matrix, and is easy to implement.Indeed, the empirical likelihood has aroused many statlstician's interest, and beenapplied to many fields and statistic models.Partial linear regression model, also called as semiparametric regression model,first introduced by Engle et al(1986) to analyze the relationship between temperatureand electric demand, is an important statistic model since 1980's. Partial linear regres-sion models have received great attention and been applied to many fields includingindustry, agriculture, economics, medicine and finance since it born.In many applications of regression analysis, because of the nature of the measurement mechanism, the independent variables may not be observed directly, but withsome errors instead. So the measurement error (errors-in-variables) models are somewhat more practical than the ordinary regression model. However, because of themeasurement error problem, the ordinary least square method is inefficient, this inturn enlarges the difficulty of research. So to research the measurement error modelsis more challenging than the ordinary regression models. In this paper, we investigatethe serial correlation test and heteroscedastic test in partial linear models includingmeasurement error models and obtain some satisfactory results. Now, let us generallyintroduce our works as follows.One of the main results of the paper is firstly introduced empirical likelihood ratiotest to test serial correlation and heteroscedasticity in regression models including linearmeasurement errors model, partial linear model and partial linear measurement errormodel. The second result is that we extend the method of Li & Hsiao(1998) to testserial correlation in measurement error models. We get the asymptotic distribution of all the test statistics under the null hypothesis. We also investigate the finite sampleproperties of our statistics. Our tests are distribution-free and overcome the defaultof score test which depends on the error's distribution. The third result is that weintroduce empirical likelihood to partial linear measurement model. Liang et al (1999)use the normal approximation based method to investigate the same problem, however,because they cannot estimate the covariance matrix of asymptotic distribution, theirmethod cannot be applied directly.