Central limit theorems and uniform laws of large numbers for arrays of random fields

Over the last decades, spatial-interaction models have been increasingly used in economics. However, the development of a sufficiently general asymptotic theory for nonlinear spatial models has been hampered by a lack of relevant central limit theorems (CLTs), uniform laws of large numbers (ULLNs) and pointwise laws of large numbers (LLNs). These limit theorems form the essential building blocks towards developing the asymptotic theory of M-estimators, including maximum likelihood and generalized method of moments estimators. The paper establishes a CLT, ULLN, and LLN for spatial processes or random fields that should be applicable to a broad range of data processes.     

Weak dependence, models and some applications

The paper is devoted to recall weak dependence conditions from Dedecker et al. (Weak dependence, examples and applications. Lecture Notes in Statistics, vol 190, 2007)’s monograph; the main basic results are recalled here and we go further in some new applications. We develop here several models of weakly dependent processes and random fields. Among them an ARCH(∞) model is considered with statistical applications to ordinary least squares. A last part aims at proving new asymptotic results for weakly dependent random fields. Such applications are indeed the main proof of the interest of this theoretical notion which measures the asymptotic decorrelation of a process.     

Gaussian semiparametric estimation of multivariate fractionally integrated processes

This paper analyzes the semiparametric estimation of multivariate long-range dependent processes. The class of spectral densities considered is motivated by and includes those of multivariate fractionally integrated processes. The paper establishes the consistency of the multivariate Gaussian semiparametric estimator (GSE), which has not been shown in other work, and the asymptotic normality of the GSE estimator. The proposed GSE estimator is shown to have a smaller limiting variance than the two-step GSE estimator studied by Lobato [1999. A semiparametric two-step estimator in a multivariate long memory model. Journal of Econometrics 90, 129–153]. Gaussianity is not assumed in the asymptotic theory. Some simulations confirm the relevance of the asymptotic results in samples of the size used in practical work.     

Blockwise bootstrap wavelet in nonparametric regression model with weakly dependent processes

In this paper, we suggest a blockwise bootstrap wavelet to estimate the regression function in the nonparametric regression models with weakly dependent processes for both designs of fixed and random. We obtain the asymptotic orders of the biases and variances of the estimators and establish the asymptotic normality for a modified version of the estimators. We also introduce a principle to select the length of data block. These results show that the blockwise bootstrap wavelet is valid for general weakly dependent processes such as α-mixing, φ-mixing and ρ-mixing random variables.     

Unit root testing via the stationary bootstrap

A nonparametric, residual-based stationary bootstrap procedure is proposed for unit root testing in a time series. The procedure generates a pseudoseries which mimics the original, but ensures the presence of a unit root. Unlike many others in the literature, the proposed test is valid for a wide class of weakly dependent processes and is not based on parametric assumptions on the data-generating process. Large sample theory is developed and asymptotic validity is shown via a bootstrap functional central limit theorem. The case of a least squares statistic is discussed in detail, including simulations to investigate the procedure's finite sample performance.     

Asymptotic theory for nonparametric regression with spatial data

Nonparametric regression with spatial, or spatio-temporal, data is considered. The conditional mean of a dependent variable, given explanatory ones, is a nonparametric function, while the conditional covariance reflects spatial correlation. Conditional heteroscedasticity is also allowed, as well as non-identically distributed observations. Instead of mixing conditions, a (possibly non-stationary) linear process is assumed for disturbances, allowing for long range, as well as short-range, dependence, while decay in dependence in explanatory variables is described using a measure based on the departure of the joint density from the product of marginal densities. A basic triangular array setting is employed, with the aim of covering various patterns of spatial observation. Sufficient conditions are established for consistency and asymptotic normality of kernel regression estimates. When the cross-sectional dependence is sufficiently mild, the asymptotic variance in the central limit theorem is the same as when observations are independent; otherwise, the rate of convergence is slower. We discuss the application of our conditions to spatial autoregressive models, and models defined on a regular lattice.     

Pseudo maximum likelihood estimation of spatial autoregressive models with increasing dimension

Pseudo maximum likelihood estimates are developed for higher-order spatial autoregressive models with increasingly many parameters, including models with spatial lags in the dependent variables both with and without a linear or nonlinear regression component, and regression models with spatial autoregressive disturbances. Consistency and asymptotic normality of the estimates are established. Monte Carlo experiments examine finite-sample behaviour.     

Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances

This study develops a methodology of inference for a widely used Cliff–Ord type spatial model containing spatial lags in the dependent variable, exogenous variables, and the disturbance terms, while allowing for unknown heteroskedasticity in the innovations. We first generalize the GMM estimator suggested in  and  for the spatial autoregressive parameter in the disturbance process. We also define IV estimators for the regression parameters of the model and give results concerning the joint asymptotic distribution of those estimators and the GMM estimator. Much of the theory is kept general to cover a wide range of settings.     

Testing for structural stability in the whole sample

The paper examines a Lagrange Multiplier type test for the constancy of the parameter in general models with dependent data without imposing any artificial choice of the possible location of the break. In order to prove the asymptotic behaviour of the test, we extend a strong approximation result for partial sums of a sequence of random variables. We also present a Monte-Carlo experiment to examine the finite sample performance of the test and how it compares with tests which assume some knowledge of the possible location of the break.     

Statistical estimation of multivariate Ornstein–Uhlenbeck processes and applications to co-integration

Ornstein–Uhlenbeck models are continuous-time processes which have broad applications in finance as, e.g., volatility processes in stochastic volatility models or spread models in spread options and pairs trading. The paper presents a least squares estimator for the model parameter in a multivariate Ornstein–Uhlenbeck model driven by a multivariate regularly varying Lévy process with infinite variance. We show that the estimator is consistent. Moreover, we derive its asymptotic behavior and test statistics. The results are compared to the finite variance case. For the proof we require some new results on multivariate regular variation of products of random vectors and central limit theorems. Furthermore, we embed this model in the setup of a co-integrated model in continuous time.     

The concept of weak exchangeability and its applications

A finite sequence of binary random variables is called a weak exchangeable sequence of order m if the sequence consists of m random vectors such that the elements within each random vector are exchangeable in the usual sense and the different random vectors are dependent. The exact and asymptotic joint distributions of the m-dimensional random vector whose elements include the number of successes in each exchangeable sequence are derived. Potential applications of the concept of weak exchangeability are discussed with illustrative examples.     

Fixed Effects and Random Effects Estimation of Higher-order Spatial Autoregressive Models with Spatial Autoregressive and Heteroscedastic Disturbances

Abstract

This paper develops a unified framework for fixed effects (FE) and random effects (RE) estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroscedasticity of unknown form in the idiosyncratic error component. We derive the moment conditions and optimal weighting matrix without distributional assumptions for a generalized moments (GM) estimation procedure of the spatial autoregressive parameters of the disturbance process and define both an RE and an FE spatial generalized two-stage least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators and derive their joint asymptotic distribution, which is robust to heteroscedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman test of the spatial random against the spatial FE model.     

Large panels with common factors and spatial correlation

This paper considers methods for estimating the slope coefficients in large panel data models that are robust to the presence of various forms of error cross-section dependence. It introduces a general framework where error cross-section dependence may arise because of unobserved common effects and/or error spill-over effects due to spatial or other forms of local dependencies. Initially, this paper focuses on a panel regression model where the idiosyncratic errors are spatially dependent and possibly serially correlated, and derives the asymptotic distributions of the mean group and pooled estimators under heterogeneous and homogeneous slope coefficients, and for these estimators proposes non-parametric variance matrix estimators. The paper then considers the more general case of a panel data model with a multifactor error structure and spatial error correlations. Under this framework, the Common Correlated Effects (CCE) estimator, recently advanced by Pesaran (2006), continues to yield estimates of the slope coefficients that are consistent and asymptotically normal. Small sample properties of the estimators under various patterns of cross-section dependence, including spatial forms, are investigated by Monte Carlo experiments. Results show that the CCE approach works well in the presence of weak and/or strong cross-sectionally correlated errors.     

Maximum likelihood estimation and Lagrange multiplier tests for panel seemingly unrelated regressions with spatial lag and spatial errors: An application to hedonic housing prices in Paris

This paper proposes maximum likelihood estimators for panel seemingly unrelated regressions with both spatial lag and spatial error components. We study the general case where spatial effects are incorporated via spatial errors terms and via a spatial lag dependent variable and where the heterogeneity in the panel is incorporated via an error component specification. We generalize the approach of Wang and Kockelman (2007) and propose joint and conditional Lagrange multiplier tests for spatial autocorrelation and random effects for this spatial SUR panel model. The small sample performance of the proposed estimators and tests are examined using Monte Carlo experiments. An empirical application to hedonic housing prices in Paris illustrate these methods. The proposed specification uses a system of three SUR equations corresponding to three types of flats within 80 districts of Paris over the period 1990-2003. We test for spatial effects and heterogeneity and find reasonable estimates of the shadow prices for housing characteristics.     

Nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters

We consider nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters. Most of the existing works on asymptotic distributions of a nonparametric/semiparametric estimator or a test statistic are based on some deterministic smoothing parameters, while in practice it is important to use data-driven methods to select the smoothing parameters. In this paper we give a simple sufficient condition that can be used to establish the first order asymptotic equivalence of a nonparametric estimator or a test statistic with stochastic smoothing parameters to those using deterministic smoothing parameters. We also allow for general weakly dependent data.     

Inference on transformed stationary time series

The paper is about an approach for parametric inference on instantaneously transformed stationary processes. The paper discusses the asymptotics of the Whittle estimator of the parameters involved and also provides the explicit expression of the asymptotic covariance matrix which does not necessarily require the innovation Gaussianity assumption. As a specific instantaneous transformation, the paper introduces a new version of the Box–Cox transformation and investigates in detail the vector ARMA processes implemented by that transformation, proposing a computation-intensive procedure for parametric estimation and testing. As a computationally feasible test not relying upon the knowledge of the explicit analytic form of the asymptotic covariance matrix or on the information equality, the paper proposes a Monte Carlo Wald test, providing illustrative simulation and real-data examples.     

Testing for Fractional Integration Versus Short Memory with Structural Breaks*

Although it is commonly accepted that most macroeconomic variables are non‐stationary, it is often difficult to identify the source of the non‐stationarity. Integrated processes and short‐memory models with trending components, possibly affected by structural breaks, imply similar features in the data and, accordingly, are hard to distinguish. The goal of this article is to extend the classical testing framework of I(1) versus I(0) + trends and/or breaks by considering a more general class of models under the null hypothesis: fractionally integrated (FI) processes. The asymptotic properties of the proposed tests are derived and it is shown that they are very well‐behaved in finite samples. An illustration using US inflation data is also provided.     

Local GMM estimation of time series models with conditional moment restrictions

This paper investigates statistical properties of the local generalized method of moments (LGMM) estimator for some time series models defined by conditional moment restrictions. First, we consider Markov processes with possible conditional heteroskedasticity of unknown forms and establish the consistency, asymptotic normality, and semi-parametric efficiency of the LGMM estimator. Second, we undertake a higher-order asymptotic expansion and demonstrate that the LGMM estimator possesses some appealing bias reduction properties for positively autocorrelated processes. Our analysis of the asymptotic expansion of the LGMM estimator reveals an interesting contrast with the OLS estimator that helps to shed light on the nature of the bias correction performed by the LGMM estimator. The practical importance of these findings is evaluated in terms of a bond and option pricing exercise based on a diffusion model for spot interest rate.