Statistical Inference And Applications Of Several Nonparametric Models Under Locally Stationary Panel Data

Individual effects are divided into fixed effects and random effects accord-ing to whether they are related to explanatory variables.Then,when there is an unknown correlation between the individual effect and the explanatory variable,the effect is a fixed effect.The assumptions used in this paper are that the fixed effect is a set of independent and identically distributed random variables with a mean of 0 and a limited variance on the cross section.When the model contains fixed effects,for the convenience of estimation,recognition conditions are given,and the sum of its features on the cross section is considered to be 0.For con-venience of notation,this condition is called the recognition condition I.Random effects are random variables that are irrelevant to the explanatory variables.They are a set of independent and identically distributed random variables with a mean of 0 and a finite variance on the cross section.With local stationary panel data,the time variability of the model needs to be considered,and it takes time T to go to infinity.First,we discuss a two-variable nonparametric model with covariates(one-dimensional)and time-varying factors.Under two different individual effects,we solve the problem of estimating and testing nonparametric functions.Secondly,when the number of covariates in the model increases,the kernel method will be affected by the”dimensional curse”problem.Then we consider the additive model,whose additive term is a binary time-varying function.The model is firstly solved using a B-spline estimation with a fast calculation speed.However,considering that spline estimators are fast,they do not provide asymptotic distributions.Therefore,Profile kernel estimation is added to the results of B-spline estimation.That is,two-step spline kernel es-timation.Next,consider the special form of additive models,variable coefficient additive models.That is,the binary addition term is separated into a product of a covariant component function and a time-varying coefficient function.The spline approximation is used to estimate the variable coefficient function and the covariate function(additive term function)in order.When the additive function is a linear function,in order to better estimate and identify,a least squares esti-mate with penalty is constructed to improve the accuracy of its estimation.Time-varying features,as an important feature of local stationary data,test whether non-parametric functions in binary nonparametric models and additive models are time-varying,which is also an important content of this paper.Based on the discussion of the above issues,this paper is divided into the following three main parts.The establishment of all non-parametric models will be completed based on local stationary panel data,which will be described here in a unified manner and will not be described again later.Part I:Estimation and Test of Binary Nonparametric Models Under Indi-vidual EffectsUnder the fixed effects and random effects,a binary time-varying non-parametric model is constructed,as suggested by Gao&Li(2013)and Lin et al.(2014).Based on the recognition condition I,the least square kernel estimation method is used to obtain the estimator of the unknown function and the asymptotic nor-mality of the estimator is proved.According to the criterion of minimizing the asymptotic mean square error,the optimal window width with respect to the sam-ple size is determined to be(n T)^-1/6.The effect of numerical simulation window selection on the estimation results.The simulation compares whether the fixed effect is eliminated,and the degree of influence on the estimation result.Under random effects and T tends to infinity,several estimation methods are discussed to prove the convergence properties of these estimators.The effects of profile least squares kernel estimation and local linear kernel estimation methods are compared by numerical simulation.Based on the fixed effect,construct a statistic in the form of L₂to test whether the unknown function is time-varying.The simulation found that the test level of the L₂statistic is close to the true level under a limited sample,and the test power is more sensitive to the strong time-varying function.Finally,using the fixed effect method,empirical analysis of automobile gasoline consumption data in 18 OECD countries,found that the functional relationship between gasoline consumption and gasoline prices has time-varying characteristics.The study found that when the mean value of the individual effects is 0,based on the recognition condition I,using the profile least square kernel estimation,the effect of the effect on the estimation can be transformed into a random conver-gence to 0 Variables,so the profile least square kernel estimators obtained in both cases are consistent.Compared with the results in Pei et al.(2018),our assumptions are more relaxed.Therefore,Bias of the estimator has an additional sample mean of a fixed effect,which will have a certain effect on the estimation effect.But as the sample size increases,this effect becomes weaker.Under random effects,the profile least square kernel estimator is compared with the lo-cal linear kernel estimator,and the profile least square kernel estimator is more accurate under finite samplesPart II:Statistical inference of additive models under fixed effectsUnder the fixed effect,an additive model with a binary variable time-varying function is established.The time interval is divided into several sub-intervals,and the one-dimensional B-spline is used to approximate the binary additive term in the sub-interval.Method eliminates the effects of fixed effects.The spline estima-tor and its convergence rate of each binary addable function are obtained.Using the BIC principle,determine the optimal time subinterval length l_T,and determine the number of points in the spline interval.In order to obtain the asymptotic distri-bution statistics of the additive terms,the two-step spline kernel method is used to estimate,and the asymptotic normality of the kernel estimator is proved.A two-step spline kernel estimator based on each additive term,based on the test method in Part I.Constructing a statistic in the form of L₂proves the asymptotic normality of the test statistic under the null hypothesis and alterna-tive hypothesis.The simulation found that the test of the L₂statistic works well under a limited sample.The proposed method was used to empirically analyze the cigarette consumption data of 46 states in the United States.Based on the additive model,the relationship between the per capita consumption and the lo-cal cigarette price and the lowest price of the neighboring state cigarettes was discussed.The study found that:the one-step B-spline estimation,the estimation effect of this method will be affected by the length of the sub-interval l_T,and the sim-ulation found that the length of the sub-interval l_Tshould not be too large or too short.The theoretical analysis is consistent.The effect of the two-step spline kernel method is not significantly affected by the length of the sub-interval and the number of points in the spline interval.The sample size and window width selection have a significant effect on it.The estimation accuracy increases as the sample size increases.Getting higher.When the range of covariate and function values fluctuates greatly,a smaller window width needs to be selected in kernel estimation.Part III:Estimation and Identification of Additive Models with Variable Coefficients for Fixed EffectsThe three-step method is used to estimate the variable coefficient function and the additive term function in the model.First,the B-spline approximation and mean re-difference method are used to eliminate the fixed effects in the model,and the initial spline values that differ from the additive term function by an unknown constant multiple are obtained,and the convergence rate of the initial statistics of these splines is proved.Secondly,based on these initial values,the model is changed into a variable coefficient model,and the estimation results of the locally linear dummy variable method and the B-spline method are compared,and their convergence rates are proved.Finally,based on the spline estimates of the variable coefficients,the mean-difference and B-spline approximation meth-ods are used to obtain the spline estimators of the additive term function.When the additive function is a linear function,in order to better identify and calculate,based on the SCAD penalty function,a least squares estimate with penalty is constructed.Under panel data with fixed effects,when an additive item satisfies the condition g _k=0,the estimation effect with penalty is better.Based on the BIC criterion,the selection method of the optimal regulator in penalty re-gression was obtained.With a limited sample,the change curve between the ad-justment factor and the BIC value in the penalty estimation is given.Based on the proposed variable coefficient additive model,a four-factor model of stock returns in the US stock market is constructed,and an empirical analysis is performed using a penalty estimation method.It is found that the functions corresponding to the value effect and the momentum effect have obvious non-linear characteristics After analyzing the monotonic change of each factor function,some conclusions consistent with the linear model are obtained.The research finds that under the B-spline method,the estimation of the vari-able coefficient term and the additive term is good.The estimation of the variable coefficient function,comparing the local linear kernel method of the dummy vari-able and the B-spline method,finds that their estimation accuracy is close,but the estimation speed of the B-spline is significantly better than the kernel esti-mation.It is found that when the additivity function is linear,the effect of penalty estimation is significantly better than that of penalty-free estimation.