基于LSTAR模型的非线性协整检验

大量的经济理论和实践都表明,宏观经济时间序列经常会出现非平稳和非线性特征,因而在统计分析时,需要进行非线性协整检验。基于逻辑平滑转换自回归(LSTAR)模型将传统的线性协整表述方法拓展为非线性形式,构造实用的检验程序及合适的统计量,利用软件R进行蒙特卡洛模拟给出非线性协整检验统计量的临界值,并通过实际数据分析购买力平价动态系统的非线性协整关系,说明方法的有效性。     

An investigation of lag identification tools for vector nonlinear time series

Exploratory methods for determining appropriate lagged vsrlables in a vector nonlinear time series model are investigated. The first is a multivariate extension of the R statistic considered by Granger and Lin (1994), which is based on an estimate of the mutual information criterion. The second method uses Kendall's ρ and partial ρ statistics for lag determination. The methods provide nonlinear analogues of the autocorrelation and partial autocorrelation matrices for a vector time series. Simulation studies indicate that the R statistic reliabiy identifies appropriate lagged nonlinear moving average terms in a vector time series, while Kendall's ρ and partial ρ statistics have some power in identifying appropirate lagged nonlinear moving average and autoregressive terms, respectively, when the nonlinear relationship between lagged variables is monotonic. For illustration, the methods are applied to set of annual temperature and tree ring measurements at Campito Mountain In California.     

Nonlinear regression models with general distortion measurement errors

This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with both multiplicative and additive distortion measurement errors. We propose conditional variance and conditional mean calibration estimation methods for the unobserved variables, then a nonlinear least squares estimator is proposed. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the estimator and test statistic are established. Lastly, a residual-based empirical process test statistic marked by proper functions of the regressors is proposed for the model checking problem. We further suggest a bootstrap procedure to calculate critical values. Simulation studies demonstrate the performance of the proposed procedure and a real example is analysed to illustrate its practical usage.     

非线性协整的秩检验方法及其响应面函数研究

 本文对非线性协整关系的秩检验方法进行了系统的梳理,运用Monte Carlo模拟给出了不同样本容量的各个秩检验统计量的临界值,并进一步探讨了其响应面函数,给出了各个秩检验统计量临界值的近似计算公式。对中国上证综指与主要发达国家股指关系的秩协整检验表明,与传统线性协整Johansen检验相比,秩协整检验能够检测到更多的线性和非线性协整关系。     

Estimation and inference for varying coefficient partially nonlinear errors-in-variables models

In this article, we study the varying coefficient partially nonlinear model with measurement errors in the nonparametric part. A local corrected profile nonlinear least-square estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. Further, a generalized likelihood ratio (GLR) statistic is proposed to test whether the varying coefficients are constant. The asymptotic null distribution of the statistic is obtained and a residual-based bootstrap procedure is employed to compute the p-value of the statistic. Some simulations are conducted to evaluate the performance of the proposed methods. The results show that the estimating and testing procedures work well in finite samples.     

Wald检验不一致性的微分几何解释及改进

Wald检验对于等价的零假设中不同形式的表达式在有限样本的情况下缺乏一致性,而从微分几何的角度来解释这一现象,并发现由于Wald统计量是一个混杂的不恰当的几何量,从而对不同的含参数的等价表达式不具有一致性。同时还展示了芬斯拉（Finsler）测地统计量如何能较为简便的计算出来、它在线性回归模型中的非线性约束条件下如何应用以及两者在什么情况下保持一致,并提出了一种解决Wald检验不一致性的思路。     

平滑转换自回归模型的单位根检验问题研究

 内容提要：针对非线性模型的单位根检验中存在的问题,本文认为非线性模型的单位根检验不应该在AR模型中进行,而应该在非线性模型中进行。以LSTAR(1)模型为例,本文给出了在其中进行单位根检验的统计量及其临界值。用蒙特卡洛试验证实,本文提出的单位根检验统计量的功效明显高于DF单位根检验,只有当非平稳特征十分明显时,DF检验才能检测出其中的单位根,因此,在非线性模型中进行单位根检验是必要的。     

Limit theorems for intermediate and central order statistics under nonlinear normalization

In this paper the work of Pancheva (1984) for extreme order statistics under nonlinear normalization is extended to order statistics with variable ranks. Two new results are proved. The first is that under nonlinear normalization, the nondegenerate type (family of types) of the distribution functions with two finite growth points is a possible weak limit of any central order statistic with regular rank sequence. The second result is that the possible nondegenerate weak limits of any central order statistic with regular rank under the traditionally linear normalization and under the power normalization are the same. Finally, the class of all possible weak limits for lower and upper intermediate order statistics is derived under power normalization from the corresponding weak limits of extremes under power normalization.     

A distribution free test to detect general dependence between a response variable and a covariate in the presence of heteroscedastic treatment effects

In this paper, we present a test of independence between the response variable, which can be discrete or continuous, and a continuous covariate after adjusting for heteroscedastic treatment effects. The method involves first augmenting each pair of the data for all treatments with a fixed number of nearest neighbours as pseudo‐replicates. Then a test statistic is constructed by taking the difference of two quadratic forms. The statistic is equivalent to the average lagged correlations between the response and nearest neighbour local estimates of the conditional mean of response given the covariate for each treatment group. This approach effectively eliminates the need to estimate the nonlinear regression function. The asymptotic distribution of the proposed test statistic is obtained under the null and local alternatives. Although using a fixed number of nearest neighbours pose significant difficulty in the inference compared to that allowing the number of nearest neighbours to go to infinity, the parametric standardizing rate for our test statistics is obtained. Numerical studies show that the new test procedure has robust power to detect nonlinear dependency in the presence of outliers that might result from highly skewed distributions. The Canadian Journal of Statistics 38: 408–433; 2010 © 2010 Statistical Society of Canada     

On unification of the asymptotic theory of nonlinear econometric models

After reading a few articles in the nonlinear econonetric literature one begins to notice that each discussion follows roughly the same lines as the classical treatment of maximum likelihood estimation. There are some technical problems having to do with simultaneously conditioning on the exogenous variables and subjecting the true parameter to a Pittman drift which prevent the use of the classical methods of proof but the basic impression of similarity is correct . An estimator – be it nonlinear least squares, three – stage nonlinear least squares, or whatever – is the solution of an optimization problem. And the objective function of the optimization problem can be treated as if it were the likelihood to derive the Wald test statistic, the likelihood ratio test statistic , and Rao's efficient score statistic. Their asymptotic null and non – null distributions can be found using arguments fairly similar to the classical maximum likelihood arguments. In this article we exploit these observations and unify much of the nonlinear econometric literature. That which escapes this unificationis that which has an objective function which is not twice continuously differentiable with respect to the parameters – minimum absolute deviations regression for example.

The model which generates the data need not bethe same as the model which was presumed to define the optimization problem. Thus, these results can be used to obtain the asymptotic behavior of inference procedures under specification error We think that this will prove to be the nost useful feature of the paper. For example, it i s not necessary toresortto Monte Carlo simulat ionto determine i f a Translog estimate of an elasticity of sub stitution obtained by nonlinear three-stage least squares is robust against a CES truestate of nature. The asymptotic approximations we give here w ill provide an analytic answer to the question, sufficiently accurate for most purposes.     

A nonparametric hypothesis test for heteroscedasticity

In this paper, a hypothesis test for heteroscedasticity is proposed in a nonparametric regression model. The test statistic, which uses the residuals from a nonparametric fit of the mean function, is based on an adaptation of the well-known Levene's test. Using the recent theory for analysis of variance when the number of factor levels goes to infinity, the asymptotic distribution of the test statistic is established under the null hypothesis of homocedasticity and under local alternatives. Simulations suggest that the proposed test performs well in several situations, especially when the variance is a nonlinear function of the predictor.     

Critical values for testing for a single outlier in a nonlinear regression model

The maximum absolute studentized residual is commonly used for testing for a single outlier in a linear regression model. This test statistic, however, is seldom discussed in a nonlinear regression setting. We simulate the critical values for the tests under various nonlinear models. The associated critical values are found to be very close to one another. Moreover, they are very well approximated using the critical values obtained from F-distributions based on the Bonferroni equations in linear models. The results are promising even in samples of size 6.     

Berry esséen type bounds for nonlinear models

Berry Esséen type bounds for the distribution of the nonlinear least-squares estimator as well as of a certain statistic useful for testing a hypothesis on a subset of the parameter vector are derived.     

Bias-corrected estimations in varying-coefficient partially nonlinear models with measurement error in the nonparametric part

In this paper, we consider the statistical inference for the varying-coefficient partially nonlinear model with additive measurement errors in the nonparametric part. The local bias-corrected profile nonlinear least-squares estimation procedure for parameter in nonlinear function and nonparametric function is proposed. Then, the asymptotic normality properties of the resulting estimators are established. With the empirical likelihood method, a local bias-corrected empirical log-likelihood ratio statistic for the unknown parameter, and a corrected and residual adjusted empirical log-likelihood ratio for the nonparametric component are constructed. It is shown that the resulting statistics are asymptotically chi-square distribution under some suitable conditions. Some simulations are conducted to evaluate the performance of the proposed methods. The results indicate that the empirical likelihood method is superior to the profile nonlinear least-squares method in terms of the confidence regions of parameter and point-wise confidence intervals of nonparametric function.     

Variance reduction for quantile estimates in simulations via nonlinear controls

Linear controls are a well known simple technique for achieving variance reduction in computer simulation. Unfortunately the effectiveness of a linear control depends upon the correlation between the statistic of interest and the control, which is often low. Since statistics often have a nonlinear relation-ship with the potential control variables, nonlinear controls offer a means for improvement over linear controls. This paper focuses on the use of nonlinear controls for reducing the variance of quantile estimates in simulation. It is shown that one can substantially reduce the analytic effort required to develop a nonlinear control from a quantile estimator by using a strictly monotone transformation to create the nonlinear control. It is also shown that as one increases the sample size for the quantile estimator, the asymptotic multivariate normal distribution of the quantile of interest and the control reduces the effectiveness of the nonlinear control to that of the linear control. However, the data has to be sectioned to obtain an estimate of the variance of the controlled quantile estimate. Graphical methods are suggested for selecting the section size that maximizes the effectiveness of the nonlinear control     

基于分形理论的LSTAR模型检验

针对非线性时间序列的单位根检验和非线性检验中存在的问题,引入分形理论中的Hurst指数进行研究,以LSTAR模型为代表,讨论了相关样本性质。通过蒙特卡洛模拟实验证实,只有当非平稳特征十分明显时,DF检验才能检测出单位根。Hurst指数改善了单位根检验功效,且作为一种非参数检验方法,具有较好的稳健性。     

A rank test for constancy of a location oil scale parameter

X nonlinear rank statistic is proposed to test whether a location or scale parameter of a continuous distribution has remained constant over time. The test is based on frequency domain characteristics of the data and uses the periodogram of appropriate rank scores. The test has good power against alternatives involving multiple abrupt up and down changes     

Two tests for sequential detection of a change-point in a nonlinear model

In this paper, two tests, based on weighted CUSUM of the least squares residuals, are studied to detect in real time a change-point in a nonlinear model. A first test statistic is proposed by extension of a method already used in the literature but for the linear models. It is tested under the null hypothesis, at each sequential observation, that there is no change in the model against a change presence. The asymptotic distribution of the test statistic under the null hypothesis is given and its convergence in probability to infinity is proved when a change occurs. These results will allow to build an asymptotic critical region. Next, in order to decrease the type I error probability, a bootstrapped critical value is proposed and a modified test is studied in a similar way. A generalization of the Hájek–Rényi inequality is established.     

Testing for linearity in generalized linear models using kernel smoothing

A test statistic proposed by Li (1999) for testing the adequacy of heteroscedastic nonlinear regression models using nonparametric kernel smoothers is applied to testing for linearity in generalized linear models. Simulation results for models with centered gamma and inverse Gaussian errors are presented to illustrate the performance of the resulting test compared with log-likelihood ratio tests for specific parametric alternatives. The test is applied to a data set of coronary heart disease status (Hosmer and Lemeshow, (1990).     

Estimation and inference for varying coefficient partially nonlinear models

In this paper, we extend the varying coefficient partially linear model to the varying coefficient partially nonlinear model in which the linear part of the varying coefficient partially linear model is replaced by a nonlinear function of the covariates. A profile nonlinear least squares estimation procedure for the parameter vector and the coefficient function vector of the varying coefficient partially nonlinear model is proposed and the asymptotic properties of the resulting estimators are established. We further propose a generalized likelihood ratio (GLR) test to check whether or not the varying coefficients in the model are constant. The asymptotic null distribution of the GLR statistic is derived and a residual-based bootstrap procedure is also suggested to derive the p-value of the GLR test. Some simulations are conducted to assess the performance of the proposed estimating and testing procedures and the results show that both the procedures perform well in finite samples. Furthermore, a real data example is given to demonstrate the application of the proposed model and its estimating and testing procedures.