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.

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.

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.

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.

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 artices 'Asymptotic Theory and Large Models' in relation to modern preoccupations with semiparametric econometrics.Denis Sargan, Tooke Chair of Economic Science and Statistics, asymptotic theory and large models, semiparametric econometrics.

Modelling Memory of Economic and Financial Time Series

Much time series data are recorded on economic and financial variables. Statistical modelling of such data is now very well developed, and has applications in forecasting. We review a variety of statistical models from the viewpoint of 'memory', or strength of dependence across time, which is a helpful discriminator between different phenomena of interest. Both linear and nonlinear models are discussed.Long memory, short memory, stochastic volatility

Variance-type estimation of long memory

The aggregation procedure when a sample of length N is divided into blocks of length m = o(N), m ® ¥ and observations in each block are replaced by their sample mean, is widely used in statistical inference. Taqqu, Teverovsky and Willinger (1995), Teverovsky and Taqqu (1997) introduced an aggregate variance estimator of the long memory parameter of a stationary sequence with long range dependence and studied its empirial performance. With respect to autovariance structure and marginal distribution, the aggregated series is closer to Gaussian fractional noise than the initial series. However, the variance type estimator based on aggregated data is seriously biased. A refined estimator, which employs least squares regression across varying levels of aggregation, has much smaller bias, permitting derivation of limiting distributional properties of suitably centered estimates, as well as of a minimum mean squared error choice of bandwidth m. The results vary considerably with the actual value of the memory parameter

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

Semiparametric Fractional Cointegration Analysis

Fractional cointegration is viewed from a semiparametric viewpoint as a narrow-band phenomenon at frequency zero. We study a narrow-band frequency domain least squares estimate of the cointegrating vector, and related semiparametric methods of inference for testing the memory of observables and the presence of fractional cointegration. These procedures are employed in analysing empirical macroeconomic series; their usefulness and feasibility in finite samples is supported by results of a Monte Carlo experiment.Semiparametric analysis, fractional cointegration.

Long and Short Memory Conditional Heteroscedasticity in Estimating the Memory Parameter of Levels - (Now published in Econometric Theory, 15 (1999), pp.299-336.)

Semiparametric estimates of long memory seem useful in the analysis of long financial time series because they are consistent under much broader conditions than parametric estimates. However, recent large sample theory for semiparametric estimates forbids conditional heteroscedasticity. We show that a leading semiparametric estimate, the Gaussian or local Whittle one, can be consistent and have the same limiting distribution under conditional heteroscedasticity assumed by Robinson (1995a). Indeed, noting that long memory has been observed in the squares of financial time series, we allow, under regularity conditions, for conditional heteroscedasticity of the general form introduced by Robinson (1991) which may include long memory behaviour for the squares, such as the fractional noise and autoregressive fractionally integrated moving average form, as well as standard short memory ARCH and GARCH specifications.long memory, dynamic conditional heteroscedasticity, semiparametric estimation