Testing serial correlation in partially linear models with validation data

This article investigates the testing for serial correlation in partially linear models with validation data and applies the empirical likelihood methods to construct serial tests statistics, and then we derive the asymptotic distribution of the test statistics under null hypothesis. Simulation results show that our method performs well.     

Tests for random time effects and spatial error correlation in panel regression models

We consider a panel model with spatial autocorrelation and heterogeneity across time. Various Lagrange multiplier and likelihood ratio test statistics are developed for testing time effects and spatial effects, jointly, marginally or conditionally. Limiting null distributions of the tests are derived. Size and power performances of the proposed tests are compared by a Monte-Carlo experiment.     

Testing a linear relationship in varying coefficient spatial autoregressive models

In this article, we propose a test to check a linear relationship in varying coefficient spatial autoregressive models, in which a residual-based bootstrap procedure is suggested to approximate the null distribution of the resulting test statistic. We conduct simulation studies to assess the performance of the test, including the validity of the bootstrap approximation to the null distribution of the test statistic and the power of the test. The simulation results demonstrate that the residual-based bootstrap procedure gives very accurate estimate of the null distribution of the test statistic and the test is of satisfactory power. Furthermore, a real example is given to demonstrate the application of the proposed test.     

Testing stability in a spatial unilateral autoregressive model

ABSTRACT

Least squares estimator of the stability parameter ? ? |α| + |β| for a spatial unilateral autoregressive process X_{k, ?} = αX_{k ? 1, ?} + βX_{k, ? ? 1} + ?_{k, ?} is investigated and asymptotic normality with a scaling factor n^5/4 is shown in the unstable case ? = 1. The result is in contrast to the unit root case of the AR(p) model X_k = α₁X_{k ? 1} + ??? + α_pX_{k ? p} + ?_k, where the limiting distribution of the least squares estimator of the unit root parameter ? ? α₁ + ??? + α_p is not normal.     

Comparison of predictions by kriging and spatial autoregressive models

In geostatistics, the prediction of unknown quantities at given locations is commonly made by the kriging technique. In addition to the kriging technique for modeling regular lattice spatial data, the spatial autoregressive models can also be used. In this article, the spatial autoregressive model and the kriging technique are introduced. We extend prediction method proposed by Basu and Reinsel for SAR(2,1) model. Then, using a simulation study and real data, we compare prediction accuracy of the spatial autoregressive models with that of the kriging prediction. The results of simulation study show that predictions made by the autoregressive models are good competitor for the kriging method.     

Memory properties and aggregation of spatial autoregressive models

A random field displays long (resp. short) memory when its covariance function is absolutely non-summable (resp. summable), or alternatively when its spectral density (spectrum) is unbounded (resp. bounded) at some frequencies. Drawing on the spectrum approach, this paper characterizes both short and long memory features in the spatial autoregressive model. The data generating process is presented as a sequence of spatial autoregressive micro-relationships. The study elaborates the exact conditions under which short and long memories emerge for micro-relationships and for the aggregated field as well. To study the spectrum of the aggregated field, we develop a new general concept referred to as the ‘root order of a function’. This concept might be usefully applied in studying the convergence of some special integrals. We illustrate our findings with simulation experiments and an empirical application based on Gross Domestic Product data for 100 countries spanning over 1960–2004.     

Maximum likelihood estimation for generalized conditionally autoregressive models of spatial data

Conditionally autoregressive (CAR) models are often used to analyze a spatial process observed over a lattice or a set of irregular regions. The neighborhoods within a CAR model are generally formed deterministically using the inter-distances or boundaries between the regions. To accommodate directional and inherent anisotropy variation, a new class of spatial models is proposed that adaptively determines neighbors based on a bivariate kernel using the distances and angles between the centroid of the regions. The newly proposed model generalizes the usual CAR model in a sense of accounting for adaptively determined weights. Maximum likelihood estimators are derived and simulation studies are presented for the sampling properties of the estimates on the new model, which is compared to the CAR model. Finally the method is illustrated using a data set on the elevated blood lead levels of children under the age of 72 months observed in Virginia in the year of 2000.     

Adaptive autoregressive priors for area and time structured mortality data

Mortality counts by age and area are relevant to obtaining small area life tables and summary statistics such as life expectancy. A Bayesian approach to small area life tables is proposed here based on the principle of smoothing (or “pooling strength”) over adjacent ages or areas. Several schemes have been suggested to reflect dependence between age categories x or areas i  , such as conditional autoregressive priors based on the principle of local smoothing, determined by adjacency of age groups or spatial proximity. It is argued here that a more flexible approach is to allow a mix of local and global smoothing over age groups and areas, as determined by the data and additional parameters κ∈[0,1]$\kappa \in [0,1]$ and λ∈[0,1]$\lambda \in [0,1]$ for age and area, respectively. An extension is also proposed to reflect the fact that the appropriate mix between local and global smoothing may not be constant across age bands or across the region being studied. For example, local spatial smoothing will not be appropriate if an area is disparate from its neighbours (e.g. in terms of social distance), and so area specific mixing parameters _λi${\lambda }_i$ are introduced. The _λi${\lambda }_i$ may be modelled by logit regression on observed sources of disparity between neighbouring areas. The application considers small area life tables for males over 625 small areas (electoral wards) in London over 2003–2005.     

Testing for residual correlation of any order in the autoregressive process

We are interested in the implications of a linearly autocorrelated driven noise on the asymptotic behavior of the usual least-squares estimator in a stable autoregressive process. We show that the least-squares estimator is not consistent and we suggest a sharp analysis of its almost sure limiting value as well as its asymptotic normality. We also establish the almost sure convergence and the asymptotic normality of the estimated serial correlation parameter of the driven noise. Then, we derive a statistical procedure enabling to test for correlation of any order in the residuals of an autoregressive modelling, giving clearly better results than the commonly used portmanteau tests of Ljung–Box and Box–Pierce, and appearing to outperform the Breusch–Godfrey procedure on small-sized samples.     

Gaussian Markov random field spatial models in GAMLSS

This paper describes the modelling and fitting of Gaussian Markov random field spatial components within a Generalized AdditiveModel for Location, Scale and Shape (GAMLSS) model. This allows modelling of any or all the parameters of the distribution for the response variable using explanatory variables and spatial effects. The response variable distribution is allowed to be a non-exponential family distribution. A new package developed in R to achieve this is presented. We use Gaussian Markov random fields to model the spatial effect in Munich rent data and explore some features and characteristics of the data. The potential of using spatial analysis within GAMLSS is discussed. We argue that the flexibility of parametric distributions, ability to model all the parameters of the distribution and diagnostic tools of GAMLSS provide an ideal environment for modelling spatial features of data.     

GMM inference in spatial autoregressive models

In this study, we investigate the finite sample properties of the optimal generalized method of moments estimator (OGMME) for a spatial econometric model with a first-order spatial autoregressive process in the dependent variable and the disturbance term (for short SARAR(1, 1)). We show that the estimated asymptotic standard errors for spatial autoregressive parameters can be substantially smaller than their empirical counterparts. Hence, we extend the finite sample variance correction methodology of Windmeijer (2005 Windmeijer, F. (2005). A finite sample correction for the variance of linear efficient two-step GMM estimators. Journal of Econometrics 126(1):25–51.[Crossref], [Web of Science ®] , [Google Scholar]) to the OGMME for the SARAR(1, 1) model. Results from simulation studies indicate that the correction method improves the variance estimates in small samples and leads to more accurate inference for the spatial autoregressive parameters. For the same model, we compare the finite sample properties of various test statistics for linear restrictions on autoregressive parameters. These tests include the standard asymptotic Wald test based on various GMMEs, a bootstrapped version of the Wald test, two versions of the C(α) test, the standard Lagrange multiplier (LM) test, the minimum chi-square test (MC), and two versions of the generalized method of moments (GMM) criterion test. Finally, we study the finite sample properties of effects estimators that show how changes in explanatory variables impact the dependent variable.     

Testing for Persistence in the Error Component Model: A One-Sided Approach

This article proposes new simple testing procedures for the joint null hypothesis of absence of persistent effects, in the form of random effects and first-order serial correlation in the error component model. The fact that the presence of random effects is clearly of a one-sided nature, together with the fact that in many empirical applications researchers worry about positive serial correlation leaves room for a power gain that arises from restricting the parameter space under the alternative hypothesis, compared to existing procedures that allow for two-sided alternatives. A Monte Carlo experiment shows that the proposed statistics have good size and power performance in very small samples like those typically used in applied work in panel data. An empirical example illustrates the usefulness of the proposed statistics.     

Empirical likelihood-based serial correlation testing in partially varying coefficient single-index models

ABSTRACT

Partially varying coefficient single-index models (PVCSIM) are a class of semiparametric regression models. One important assumption is that the model error is independently and identically distributed, which may contradict with the reality in many applications. For example, in the economical and financial applications, the observations may be serially correlated over time. Based on the empirical likelihood technique, we propose a procedure for testing the serial correlation of random error in PVCSIM. Under some regular conditions, we show that the proposed empirical likelihood ratio statistic asymptotically follows a standard χ² distribution. We also present some numerical studies to illustrate the performance of our proposed testing procedure.     

Variable selection and estimation for high-dimensional spatial autoregressive models

Spatial regression models are important tools for many scientific disciplines including economics, business, and social science. In this article, we investigate postmodel selection estimators that apply least squares estimation to the model selected by penalized estimation in high-dimensional regression models with spatial autoregressive errors. We show that by separating the model selection and estimation process, the postmodel selection estimator performs at least as well as the simultaneous variable selection and estimation method in terms of the rate of convergence. Moreover, under perfect model selection, the ℓ₂ rate of convergence is the oracle rate of $\sqrt{s/n}$, compared with the convergence rate of $\text{◂√▸}\sqrt{s\log p/n}$ in the general case. Here, n is the sample size and p, s are the model dimension and number of significant covariates, respectively. We further provide the convergence rate of the estimation error in the form of $\sup$ norm, and ideally the rate can reach as fast as $\text{◂√▸}\sqrt{\log s/n}$.     

Testing for serial correlation in the presence of dynamic heteroscedasticity

Standard serial correlation tests are derived assuming that the disturbances are homoscedastic, but this study shows that asympotic critical values are not accurate when this assumption is violated. Asymptotic critical values for the ARCH(2)-corrected LM, BP and BL tests are valid only when the underlying ARCH process is strictly stationary, whereas Wooldridge's robust LM test has good properties overall. These tests exhibit similar bahaviour even when the underlying process is GARCH (1,1). When the regressors include lagged dependent variables, the rejection frequencies under both the null and alternative hypotheses depend on the coefficientsof the lagged dependent variables and the other model parameters. They appear to be robust across various disturbance distributions under the null hypothesis.     

Testing for serial correlation in the presence of dynamic heteroscedasticity

Standard serial correlation tests are derived assuming that the disturbances are homoscedastic, but this study shows that asympotic critical values are not accurate when this assumption is violated. Asymptotic critical values for the ARCH(2)-corrected LM, BP and BL tests are valid only when the underlying ARCH process is strictly stationary, whereas Wooldridge's robust LM test has good properties overall. These tests exhibit similar bahaviour even when the underlying process is GARCH (1,1). When the regressors include lagged dependent variables, the rejection frequencies under both the null and alternative hypotheses depend on the coefficientsof the lagged dependent variables and the other model parameters. They appear to be robust across various disturbance distributions under the null hypothesis.     

Maximum likelihood estimation for directional conditionally autoregressive models

A spatial process observed over a lattice or a set of irregular regions is usually modeled using a conditionally autoregressive (CAR) model. The neighborhoods within a CAR model are generally formed using only the inter-distances or boundaries between the regions. To accommodate directional spatial variation, a new class of spatial models is proposed using different weights given to neighbors in different directions. The proposed model generalizes the usual CAR model by accounting for spatial anisotropy. Maximum likelihood estimators are derived and shown to be consistent under some regularity conditions. Simulation studies are presented to evaluate the finite sample performance of the new model as compared to the CAR model. Finally, the method is illustrated using a data set on the crime rates of Columbus, OH and on the elevated blood lead levels of children under the age of 72 months observed in Virginia in the year of 2000.     

Simultaneous bootstrap for all three parameters in random coefficient autoregressive models

In this paper we consider autoregressive processes with random coefficients and develop bootstrap approaches that asymptotically work for the distribution of estimated autoregressive parameter as well as for the distribution of estimated variances of the innovation noise and the disturbance noise. We discuss how to obtain approximative residuals of the process and how to separate between the innovation and the disturbance noise in order to be able to extend the classical residual bootstrap for autoregressive processes to the situation considered in this paper. Thereafter, we propose a wild bootstrap procedure as a variation of the residual bootstrap that uses estimated densities of the innovation and the disturbance noise to generate bootstrap replicates of the data generating process. The consistency of the bootstrap approaches is established and their performance is illustrated by a simulation study.     

Serial correlation or random subject effects

In longitudinal data analysis with random subject effects, there is often within subject serial correlation and possibly unequally spaced observations. This serial correlation can be partially confounded with the random between subject effects. In real data, it is often not clear whether there is serial correlation, random subject effects or both. Using inference based on the likelihood function, it is not always possible to identify the correct model, especially in small samples. However, it is important that some effort be made to attempt to find a good model rather than just making assumptions. This often means trying models with random coefficients, with serial correlation, and with both. Model selection criteria such as likelihood ratio tests and Akaike's Information Criterion (AIC) can be used. The problem of modelling serial correlation with unequally spaced observations is addressed. A real data example is presented where there is an apparent heterogeneity of variances, possible serial correlation and between subject random effects. In this example, it turns out that the random subject effects explains both the serial correlation and the variance heterogeneity.