Inference on power law spatial trends

Power law or generalized polynomial regressions with unknown real-valued exponents and coefficients, and weakly dependent errors, are considered for observations over time, space or space--time. Consistency and asymptotic normality of nonlinear least-squares estimates of the parameters are established. The joint limit distribution is singular, but can be used as a basis for inference on either exponents or coefficients. We discuss issues of implementation, efficiency, potential for improved estimation and possibilities of extension to more general or alternative trending models to allow for irregularly spaced data or heteroscedastic errors; though it focusses on a particular model to fix ideas, the paper can be viewed as offering machinery useful in developing inference for a variety of models in which power law trends are a component. Indeed, the paper also makes a contribution that is potentially relevant to many other statistical models: Our problem is one of many in which consistency of a vector of parameter estimates (which converge at different rates) cannot be established by the usual techniques for coping with implicitly-defined extremum estimates, but requires a more delicate treatment; we present a generic consistency result.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ349 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Correlation Testing in Time Series, SpatialandCross-Sectional Data

We provide a general class of tests for correlation in time series, spatial, spatiotemporaland cross-sectional data. We motivate our focus by reviewing howcomputational and theoretical difficulties of point estimation mount as one movesfrom regularly-spaced time series data, through forms of irregular spacing, and tospatial data of various kinds. A broad class of computationally simple tests isjustified. These specialize to Lagrange multiplier tests against parametric departuresof various kinds. Their forms are illustrated in case of several models for describingcorrelation in various kinds of data. The initial focus assumes homoscedasticity, butwe also robustify the tests to nonparametric heteroscedasticity.heteroscedasticity, Lagrange multiplier tests.

Efficient Estimation of the SemiparametricSpatial Autoregressive Model

Efficient semiparametric and parametric estimates are developed for aspatial autoregressive model, containing nonstochastic explanatoryvariables and innovations suspected to be non-normal. The main stress ison the case of distribution of unknown, nonparametric, form, where seriesnonparametric estimates of the score function are employed in adaptiveestimates of parameters of interest. These estimates are as efficient asones based on a correct form, in particular they are more efficient thanpseudo-Gaussian maximum likelihood estimates at non-Gaussiandistributions. Two different adaptive estimates are considered. One entails astringent condition on the spatial weight matrix, and is suitable only whenobservations have substantially many "neighbours". The other adaptiveestimate relaxes this requirement, at the expense of alternative conditionsand possible computational expense. A Monte Carlo study of finite sampleperformance is included.Spatial autoregression, Efficient estimation, Adaptive estimation,Simultaneity bias.© The author. All rights reserved. Short sections of text, not to exceed two paragraphs,may be quoted without explicit permission provided that full credit, including © notice, isgiven to the source.

The Distance between Rival Nonstationary Fractional Processes

Asymptotic inference on nonstationary fractional time series models, including cointegrated ones, is proceeding along two routes, determined by alternative definitions of nonstationary processes. We derive bounds for the mean squared error of the difference between (possibly tapered) discrete Fourier transforms under two regimes. We apply the results to deduce limit theory for estimates of memory parameters, including ones for cointegrated errors, with mention also of implications for estimates of cointegrating coefficients.Nonstationary fractional processes, memory parameter estimation, fractional cointegration, rates of convergence.

Developments in the Analysis of Spatial Data

Disregarding spatial dependence can invalidate methods for analyzingcross-sectional and panel data. We discuss ongoing work on developingmethods that allow for, test for, or estimate, spatial dependence. Muchof the stress is on nonparametric and semiparametric methods.

Inference On Nonparametrically Trending Time Series With Fractional Errors

The central limit theorem for nonparametric kernel estimates of a smooth trend,with linearly-generated errors, indicates asymptotic independence andhomoscedasticity across fixed points, irrespective of whether disturbances haveshort memory, long memory, or antipersistence. However, the asymptotic variancedepends on the kernel function in a way that varies across these threecircumstances, and in the latter two involves a double integral that cannotnecessarily be evaluated in closed form. For a particular class of kernels, weobtain analytic formulae. We discuss extensions to more general settings,including ones involving possible cross-sectional or spatial dependence.

ROBUST COVARIANCE MATRIX ESTIMATION: "HAC" Estimates with Long Memory/Antipersistence Correction

Smoothed nonparametric estimates of the spectral density matrix at zero frequency have been widely used in econometric inference, because they can consistently estimate the covariance matrix of a partial sum of a possibly dependent vector process. When elements of the vector process exhibit long memory or antipersistence such estimates are inconsistent. We propose estimates which are still consistent in such circumstances, adapting automatically to memory parameters that can vary across the vector and be unknown.Covariance matrix estimation, long memory, antipersistence correction, "HAC" estimates, vector process, spectral density.

Large-Sample Inference on SpatialDependence

We consider cross-sectional data that exhibit no spatial correla-tion, but are feared to be spatially dependent. We demonstrate that a spatialversion of the stochastic volatility model of financial econometrics, entailing aform of spatial autoregression, can explain such behaviour. The parameters areestimated by pseudo Gaussian maximum likelihood based on log-transformedsquares, and consistency and asymptotic normality are established. Asymptotically valid tests for spatial independence are developed.Spatial dependence, Parameter estimation, Asymptotic theory,Independence testing.

Denis Sargan: some perspectives.

We attempt to present Denis Sargan’s work in some kind of historical perspective, in two ways. First, we discuss some previous members of the Tooke Chair of Economic Science and Statistics, which was founded in 1859 and which Sargan held. Second, we discuss one of his articles “Asymptotic Theory and Large Models” in relation to modern preoccupations with semiparametric econometrics.

Multiple Local Whittle Estimation in StationarySystems

Moving from univariate to bivariate jointly dependent long memory time series introduces a phase parameter (?), at the frequency of principal interest, zero; for shortmemory series ? = 0 automatically. The latter case has also been stressed under longmemory, along with the 'fractional differencing' case ( ) / 2; 2 1 ? = d - d p where 1 2 d , dare the memory parameters of the two series. We develop time domain conditionsunder which these are and are not relevant, and relate the consequent properties ofcross-autocovariances to ones of the (possibly bilateral) moving averagerepresentation which, with martingale difference innovations of arbitrary dimension,is used in asymptotic theory for local Whittle parameter estimates depending on asingle smoothing number. Incorporating also a regression parameter (ß) which, whennon-zero, indicates cointegration, the consistency proof of these implicitly-definedestimates is nonstandard due to the ß estimate converging faster than the others. Wealso establish joint asymptotic normality of the estimates, and indicate how thisoutcome can apply in statistical inference on several questions of interest. Issues ofimplementation are discussed, along with implications of knowing ß and of correct orincorrect specification of ? , and possible extensions to higher-dimensional systemsand nonstationary series.Long memory, phase, cointegration, semiparametricestimation, consistency, asymptotic normality.