Sequential empirical process in autoregressive models with measurement errors

In this paper, we study the weak convergence of the sequential empirical process based on the residuals from autoregressive models with measurement errors. It is shown that the sequential empirical process converges weakly to the sum of a Gaussian process which is the limit of a sequential empirical process of certain p-dependent random variables and an additional term depending on the parameter estimators of the model. As an application, we discuss the change point problem in the distribution of the error process in the autoregressive model. We present the numerical result of a simulation study for an asymptotically distribution-free test.     

Diagnostics for elliptical linear mixed models with first-order autoregressive errors

For longitudinal time series data, linear mixed models that contain both random effects across individuals and first-order autoregressive errors within individuals may be appropriate. Some statistical diagnostics based on the models under a proposed elliptical error structure are developed in this work. It is well known that the class of elliptical distributions offers a more flexible framework for modelling since it contains both light- and heavy-tailed distributions. Iterative procedures for the maximum-likelihood estimates of the model parameters are presented. Score tests for the presence of autocorrelation and the homogeneity of autocorrelation coefficients among individuals are constructed. The properties of test statistics are investigated through Monte Carlo simulations. The local influence method for the models is also given. The analysed results of a real data set illustrate the values of the models and diagnostic statistics.     

Separated hypotheses testing for autoregressive models with non-negative residuals

Normal residual is one of the usual assumptions in autoregressive model but sometimes in practice we are faced with non-negative residuals. In this paper, we have derived modified maximum likelihood estimators of parameters of the residuals and autoregressive coefficient. Also asymptotic distribution of modified maximum likelihood estimators in both stationary and non-stationary models are computed. So that, we can derive asymptotic distribution of unit root, Vuong's and Cox's tests statistics in stationary situation. Using simulation, it shows that Akaike information criterion and Vuong's test work to select the optimal autoregressive model with non-negative residuals. Sometimes Vuong's test select two competing models as equivalent models. These models may be suitable or unsuitable equivalent models. So we consider Cox's test to make inference after model selection. Kolmogorov–Smirnov test confirms our results. Also we have computed tracking interval for competing models to choosing between two close competing models when Vuong's test and Cox's test cannot detect the differences.     

Adaptive estimation of vector autoregressive models with time-varying variance: Application to testing linear causality in mean

Linear vector autoregressive (VAR) models where the innovations could be unconditionally heteroscedastic are considered. The volatility structure is deterministic and quite general, including breaks or trending variances as special cases. In this framework we propose ordinary least squares (OLS), generalized least squares (GLS) and adaptive least squares (ALS) procedures. The GLS estimator requires the knowledge of the time-varying variance structure while in the ALS approach the unknown variance is estimated by kernel smoothing with the outer product of the OLS residual vectors. Different bandwidths for the different cells of the time-varying variance matrix are also allowed. We derive the asymptotic distribution of the proposed estimators for the VAR model coefficients and compare their properties. In particular we show that the ALS estimator is asymptotically equivalent to the infeasible GLS estimator. This asymptotic equivalence is obtained uniformly with respect to the bandwidth(s) in a given range and hence justifies data-driven bandwidth rules. Using these results we build Wald tests for the linear Granger causality in mean which are adapted to VAR processes driven by errors with a nonstationary volatility. It is also shown that the commonly used standard Wald test for the linear Granger causality in mean is potentially unreliable in our framework (incorrect level and lower asymptotic power). Monte Carlo experiments illustrate the use of the different estimation approaches for the analysis of VAR models with time-varying variance innovations.     

Statistical inferences for single-index models with measurement errors

An important factor in house prices is its location. However, measurement errors arise frequently in the process of observing variables such as the latitude and longitude of the house. The single-index models with measurement errors are used to study the relationship between house location and house price. We obtain the estimators by a SIMEX method based on the local linear method and the estimating equation. To test the significance of the index coefficient and the linearity of the link function, we establish the generalized likelihood ratio (GLR) tests for the models. We demonstrate that the asymptotic null distributions of the established GLR tests follow χ2-distributions which are independent of nuisance parameters or functions. Finally, two simulated examples and a real estate valuation data set are given to illustrate the effect of GLR tests.     

Restricted estimation and testing of hypothesis in linear measurement errors models

ABSTRACT

In this article, the linear models with measurement error both in the response and in the covariates are considered. Following Shalabh et al. (2007 Shalabh, Garg, G., Misra, N. (2007). Restricted regression estimation in measurement error models. Comput. Stat. Data Anal. 52:1149–1166.[Crossref], [Web of Science ®] , [Google Scholar], 2009 Shalabh, Garg, G., Misra, N. (2009). Use of prior information in the consistent estimation of regression coefficients in measurement error models. J. Multivariate Anal. 100:1498–1520.[Crossref], [Web of Science ®] , [Google Scholar]), we propose several restricted estimators for the regression coefficients. The consistency and asymptotic normality of the restricted estimators are established. Furthermore, we also discuss the superiority of the restricted estimators to unrestricted estimators under Pitman closeness criterion. We also develop several variance estimators and establish their asymptotic distributions. Wald-type statistics are constructed for testing the linear restrictions. Finally, Monte Carlo simulations are conducted to illustrate the finite-sample properties of the proposed estimators.     

Application of shrinkage estimation in linear regression models with autoregressive errors

In this paper, we consider the shrinkage and penalty estimation procedures in the linear regression model with autoregressive errors of order p when it is conjectured that some of the regression parameters are inactive. We develop the statistical properties of the shrinkage estimation method including asymptotic distributional biases and risks. We show that the shrinkage estimators have a significantly higher relative efficiency than the classical estimator. Furthermore, we consider the two penalty estimators: least absolute shrinkage and selection operator (LASSO) and adaptive LASSO estimators, and numerically compare their relative performance with that of the shrinkage estimators. A Monte Carlo simulation experiment is conducted for different combinations of inactive predictors and the performance of each estimator is evaluated in terms of the simulated mean-squared error. This study shows that the shrinkage estimators are comparable to the penalty estimators when the number of inactive predictors in the model is relatively large. The shrinkage and penalty methods are applied to a real data set to illustrate the usefulness of the procedures in practice.     

Analysis of cointegrated models with measurement errors

We study the asymptotic properties of the reduced-rank estimator of error correction models of vector processes observed with measurement errors. Although it is well known that there is no asymptotic measurement error bias when predictor variables are integrated processes in regression models [Phillips BCB, Durlauf SN. Multiple time series regression with integrated processes. Rev Econom Stud. 1986;53:473–495], we systematically investigate the effects of the measurement errors (in the dependent variables as well as in the predictor variables) on the estimation of not only cointegrating vectors but also the speed of the adjustment matrix. Furthermore, we present the asymptotic properties of the estimators. We also obtain the asymptotic distribution of the likelihood ratio test for the cointegrating ranks. We investigate the effects of the measurement errors on estimation and test through a Monte Carlo simulation study.     

Bayesian testing of restrictions on vector autoregressive models

In this study, we propose a prior on restricted Vector Autoregressive (VAR) models. The prior setting permits efficient Markov Chain Monte Carlo (MCMC) sampling from the posterior of the VAR parameters and estimation of the Bayes factor. Numerical simulations show that when the sample size is small, the Bayes factor is more effective in selecting the correct model than the commonly used Schwarz criterion. We conduct Bayesian hypothesis testing of VAR models on the macroeconomic, state-, and sector-specific effects of employment growth.     

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.     

Objective Bayesian hypothesis testing in regression models with first-order autoregressive residuals

This article considers the objective Bayesian testing in the normal regression models with first-order autoregressive residuals. We propose some solutions based on a Bayesian model selection procedure to this problem where no subjective input is considered. We construct the proper priors for testing the autocorrelation coefficient based on measures of divergence between competing models, which is called the divergence-based (DB) priors and then propose the objective Bayesian decision-theoretic rule, which is called the Bayesian reference criterion (BRC). Finally, we derive the intrinsic test statistic for testing the autocorrelation coefficient. The behavior of the Bayes factor-based DB priors is examined by comparing with the BRC in a simulation study and an example.     

Bayesian modeling of autoregressive partial linear models with scale mixture of normal errors

Normality and independence of error terms are typical assumptions for partial linear models. However, these assumptions may be unrealistic in many fields, such as economics, finance and biostatistics. In this paper, a Bayesian analysis for partial linear model with first-order autoregressive errors belonging to the class of the scale mixtures of normal distributions is studied in detail. The proposed model provides a useful generalization of the symmetrical linear regression model with independent errors, since the distribution of the error term covers both correlated and thick-tailed distributions, and has a convenient hierarchical representation allowing easy implementation of a Markov chain Monte Carlo scheme. In order to examine the robustness of the model against outlying and influential observations, a Bayesian case deletion influence diagnostics based on the Kullback–Leibler (K–L) divergence is presented. The proposed method is applied to monthly and daily returns of two Chilean companies.     

LS estimation of periodic autoregressive models with non-Gaussian errors: a simulation study

In this article, the least squares (LS) estimates of the parameters of periodic autoregressive (PAR) models are investigated for various distributions of error terms via Monte-Carlo simulation. Beside the Gaussian distribution, this study covers the exponential, gamma, student-t, and Cauchy distributions. The estimates are compared for various distributions via bias and MSE criterion. The effect of other factors are also examined as the non-constancy of model orders, the non-constancy of the variances of seasonal white noise, the period length, and the length of the time series. The simulation results indicate that this method is in general robust for the estimation of AR parameters with respect to the distribution of error terms and other factors. However, the estimates of those parameters were, in some cases, noticeably poor for Cauchy distribution. It is also noticed that the variances of estimates of white noise variances are highly affected by the degree of skewness of the distribution of error terms.     

Joint detection for functional polynomial regression with autoregressive errors

In this article, we are concerned with detecting the true structure of a functional polynomial regression with autoregressive (AR) errors. The first issue is to detect which orders of the polynomial are significant in functional polynomial regression. The second issue is to detect which orders of the AR process in the AR errors are significant. We propose a shrinkage method to deal with the two problems: polynomial order selection and autoregressive order selection. Simulation studies demonstrate that the new method can identify the true structure. One empirical example is also presented to illustrate the usefulness of our method.     

Best subset selection of autoregressive models with exogenous variables and generalized autoregressive conditional heteroscedasticity errors

Summary.  We develop an efficient way to select the best subset autoregressive model with exogenous variables and generalized autoregressive conditional heteroscedasticity errors. One main feature of our method is to select important autoregressive and exogenous variables, and at the same time to estimate the unknown parameters. The method proposed uses the stochastic search idea. By adopting Markov chain Monte Carlo techniques, we can identify the best subset model from a large of number of possible choices. A simulation experiment shows that the method is very effective. Misspecification in the mean equation can also be detected by our model selection method. In the application to the stock-market data of seven countries, the lagged 1 US return is found to have a strong influence on the other stock-market returns.     

Partial linear single-index models with additive distortion measurement errors

We study partial linear single-index models (PLSiMs) when the response and the covariates in the parametric part are measured with additive distortion measurement errors. These distortions are modeled by unknown functions of a commonly observable confounding variable. We use the semiparametric profile least-squares method to estimate the parameters in the PLSiMs based on the residuals obtained from the distorted variables and confounding variable. We also employ the smoothly clipped absolute deviation penalty (SCAD) to select the relevant variables in the PLSiMs. We show that the resulting SCAD estimators are consistent and possess the oracle property. For the non parametric link function, we construct the simultaneous confidence bands and obtain the asymptotic distribution of the maximum absolute deviation between the estimated link function and the true link function. A simulation study is conducted to evaluate the performance of the proposed methods and a real dataset is analyzed for illustration.     

Variable selection in quantile regression when the models have autoregressive errors

This paper considers a problem of variable selection in quantile regression with autoregressive errors. Recently, Wu and Liu (2009) investigated the oracle properties of the SCAD and adaptive-LASSO penalized quantile regressions under non identical but independent error assumption. We further relax the error assumptions so that the regression model can hold autoregressive errors, and then investigate theoretical properties for our proposed penalized quantile estimators under the relaxed assumption. Optimizing the objective function is often challenging because both quantile loss and penalty functions may be non-differentiable and/or non-concave. We adopt the concept of pseudo data by Oh et al. (2007) to implement a practical algorithm for the quantile estimate. In addition, we discuss the convergence property of the proposed algorithm. The performance of the proposed method is compared with those of the majorization-minimization algorithm (Hunter and Li, 2005) and the difference convex algorithm (Wu and Liu, 2009) through numerical and real examples.     

Robust unit root tests with autoregressive errors

ABSTRACT

This article presents a new test for unit roots based on least absolute deviation estimation specially designed to work for time series with autoregressive errors. The methodology used is a bootstrap scheme based on estimating a model and then the innovations. The resampling part is performed under the null hypothesis and, as it is customary in bootstrap procedures, is automatic and does not rely on the calculation of any nuisance parameter. The validity of the procedure is established and the asymptotic distribution of the statistic proposed is proved to converge to the correct distribution. To analyze the performance of the test for finite samples, a Monte Carlo study is conducted showing a very good behavior in many different situations.