Some asymptotic properties of the least-squaresestimator of the parameters of an AR model of orderp, p ≥ 1, are studied when theroots of the characteristic polynomial of the givenAR model are on or near the unit circle.Specifically, the convergence in distribution isestablished and the corresponding limiting randomvariables are represented in terms of functionals ofsuitable Brownian motions.

Further, the preceding convergence in distribution isstrengthened to that of convergence uniformly overall Borel subsets. It is indicated that the methodemployed for this purpose has the potential of beingapplicable in the wider context of obtainingsuitable asymptotic expansions of the distributionsof leastsquares estimators.

In econometrics, seminonparametric (SNP) estimatorsoriginated in the consumer demand literature. TheFourier flexible form is a well-known example. Theidea is to replace the consumer's indirect utilityfunction with a truncated series expansion and thenuse a parametric procedure, such as nonlinearmultivariate regression, to set a confidenceinterval on an elasticity. More recently, SNPestimators have been used in nonlinear time seriesanalysis. A truncated Hermite expansion with an ARCHleading term is used as the conditional density ofthe process. The method of maximum likelihood isused to fit it to data.

This paper considers the consistency property of sometest statistics based on a time series of data.While the usual consistency criterion is based onkeeping the sampling interval fixed, we let thesampling interval take any equispaced path as thesample size increases to infinity. We consider testsof the null hypotheses of the random walk andrandomness against positive autocorrelation(stationary or explosive). We show that tests of theunit root hypothesis based on the first-ordercorrelation coefficient of the original data areconsistent as long as the span of the data isincreasing. Tests of the same hypothesis based onthe first-order correlation coefficient of thefirst-differenced data are consistent againststationary alternatives only if the span isincreasing at a rate greater thanT½, whereT is the sample size. On theother hand, tests of the randomness hypothesis basedon the first-order correlation coefficient appliedto the original data are consistent as long as thespan is not increasing too fast. We provide MonteCarlo evidence on the power, in finite samples, ofthe tests Studied allowing various combinations ofspan and sampling frequencies. It is found that theconsistency properties summarize well the behaviorof the power in finite samples. The power of testsfor a unit root is more influenced by the span thanthe number of observations while tests of randomnessare more powerful when a small sampling frequency isavailable.

In a regression model with an arbitrary number of errorcomponents, the covariance matrix of thedisturbances has three equivalent representations aslinear combinations of matrices. Furthermore, thisproperty is invariant with respect to powers, matrixaddition, and matrix multiplication. This result isapplied to the derivation and interpretation of theinconsistency of the estimated coefficient varianceswhen the error components structure is improperlyrestricted. This inconsistency is defined as thedifference between the asymptotic variance obtainedwhen the restricted model is correctly specified,and the asymptotic variance obtained when therestricted model is incorrectly specified; when someerror components are improperly omitted, and theremaining variance components are consistentlyestimated, it is always negative. In the case wherethe time component is improperly omitted from thetwo-way model, we show that the difference betweenthe true and estimated coefficient variances is oforder greater than N–1in probability.

The limited dependent variable models with errorshaving log-concave density functions are studiedhere. For such models with normal errors, theasymptotic normality of the maximum likelihoodestimator was established by Amemiya [1]. We show,when the density of the error distribution islog-concave, that the maximum likelihood estimatorexists with arbitrarily large probability for largesample sizes, and is asymptotically normal. Thegeneral theory presented here includes the importantspecial cases of normal, logistic, and extreme valueerror distributions. The main results areestablished under rather weak conditions. It is alsoshown that, under the null hypothesis, theasymptotic distribution of the likelihood ratiostatistic for testing a one-sided alternativehypothesis is a weighted sum of chi-squares.

Alternative definitions of the concentration ellipsoidof a random vector are surveyed, and an extension ofthe concentration ellipsoid of Darmois is suggestedas being the most convenient and natural definition.The advantage of the proposed definition inproviding substantially simplified proofs of resultsin (linear) estimation theory is discussed, and isillustrated by new and short proofs of two keyresults. A not-so-well-known, but elementary,extremal representation of a nonnegative definitequadratic form, together with the correspondingCauchy-Schwarẓ-type inequality, is seen to play acrucial role in these proofs.

This paper is concerned with deriving formulae forhigher order derivatives of exogenous variables foruse in estimating the parameters of an opensecondorder continuous time model with mixed stockand flow data and first and second order derivativesof exogenous variables which are not observable.This should provide the basis for the futureestimation of continuous time models in a range ofapplied areas using the new Gaussian estimationcomputer program developed by Nowman [4].