On the Invariance Principle for Nonstationary Mixingales

McLeish, D. L.

doi:10.1214/aop/1176995772

Cited by 97 publications

(86 citation statements)

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“…The above concept of NED process dated back to Ibragimov (1962) and was further developed by Billingsley (1968), McLeish (1975a,1975b,1977 and Lin (2004). Basically, most of them assumed that f" t g is martingale di¤erence or '-mixing.…”

Section: Introductionmentioning

confidence: 99%

Local Linear Fitting Under Near Epoch Dependence: Uniform Consistency With Convergence Rates

Linton

2012

Econom. Theory

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Local linear fitting is a popular nonparametric method in nonlinear statistical and econometric modelling. Lu and Linton (2007) established the point wise asymptotic distribution (central limit theorem) for the local linear estimator of nonparametric regression function under the condition of near epoch dependence. We further investigate the uniform consistency of this estimator. The uniformly strong and weak consistencies with convergence rates for the local linear fitting are established under mild conditions. Furthermore, general results of uniform convergence rates for nonparametric kernel-based estimators are provided. Applications of our results to conditional variance function estimation and some economic time series models are also discussed. The results of this paper will be of widely potential interest in time series semiparametric modelling. JEL subject classifications: C13, C14, C22.

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Section: Introductionmentioning

confidence: 99%

Local Linear Fitting Under Near Epoch Dependence: Uniform Consistency With Convergence Rates

Linton

2012

Econom. Theory

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“…Using this result we prove a weak invariance principle for a class of dependent random variables, satisfying a Lindeberg-type condition. The weak invariance principle we obtain for (p-mixing sequences shows that the mixing rate used by MCLEISH in Theorem (3.8) of [4] and in Corollary (2.11) of [5], can be improved provided the finite-dimensional distributions converge.…”

Section: A Criterion For Tightness For a Class Of Dependent Random Vamentioning

confidence: 97%

“…Observe that condition (8) improves the mixing rate used by MCLEISH in Theorem (3.8) of [4] and in Corollary 2.11 of [5], the mixing coefficient being at the same time more general. However this corollary does not imply these results of McLeish, since we impose in addition condition (9).…”

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confidence: 99%

A criterion for tightness for a class of dependent random variables

Peligrad

1982

Acta Mathematica Academiae Scientiarum Hungaricae

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1) EX, = 0For each tC [O, 1] put LOYNES [3] proved that if the finite-dimensional distributions of a sequence of martingales converge and if for each time t the variables are uniformly integrable, then weak convergence follows (in either C or D) provided the limiting process satisfies a certain condition; this condition is satisfied by the Wiener process. Using this result we prove a weak invariance principle for a class of dependent random variables, satisfying a Lindeberg-type condition. The weak invariance principle we obtain for (p-mixing sequences shows that the mixing rate used by MCLEISH in Theorem (3.8) of [4] and in Corollary (2.11) of [5], can be improved provided the finite-dimensional distributions converge.Let (X,, i~1) be a sequence of a square integrable random variables on the probability triple (f2, F, P) and put Fm=a(X~; n<=i<-m). For each m=>0, define ~o~,, = sup sup IP(BIA)-P(B)I. , ACr~,BC~',"+~,P(a)~0We denote E(X.IFm) by EmX,, ~q~ by a,. We also assume that i=1 f for all i, E Xi = O(n).where [x] is the greatest integer contained in x. We shall give sufficient conditions for the weak convergence of IV,, in Skorohod's space D=D[0, 1], el.[1], to the standard Brownian motion process on D, denoted by W. For an event A let I a denote its indicator.THEOREM. Let (Xi, i=>1) be a stochastic sequence satisfying (1) and assume that for every e>0, (2) lim 1 ~ EXi2i(rx, l>~a, gZ) = 0 n~ na n i=1 and (3) w.-~ w (i.e. the finite-dimensional distributions converge). Then IV, converges weakly to W.

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“…Based on a stationary process {ε t }, {Y t } and {X t } are defined by From the above definition, we know that the NED process includes the α-mixing process as a special case. The concept of NED process can date back to Ibragimov (1962), and it is further developed by Billingsley (1968), McLeish (1975aMcLeish ( , 1975bMcLeish ( , 1977 and Lin (2004), most of which assume that {ε t } is stationary martingale difference or φ-mixing. In this paper, we study the NED process with respect to a stationary α-mixing process {ε t } (α-mixing dependence is weaker than the φ-mixing).…”

Section: Asymptotic Propertiesmentioning

confidence: 99%

A flexible semiparametric model for time series

Linton

2012

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We consider approximating a multivariate regression function by an a¢ ne combination of one-dimensional conditional component regression functions. The weight parameters involved in the approximation are estimated by least squares on the …rst-stage nonparametric kernel estimates. We establish asymptotic normality for the estimated weights and the regression function in two cases: the number of the covariates is …nite, and the number of the covariates is diverging. As the observations are assumed to be stationary and near epoch dependent, the approach in this paper is applicable to estimation and forecasting issues in time series analysis. Furthermore, the methods and results are augmented by a simulation study and illustrated by application in the analysis of the Australian annual mean temperature anomaly series. We also apply our methods to high frequency volatility forecasting, where we obtain superior results to parametric methods. JEL subject classi…cations: C14, C22.

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On the Invariance Principle for Nonstationary Mixingales

Cited by 97 publications

References 0 publications

Local Linear Fitting Under Near Epoch Dependence: Uniform Consistency With Convergence Rates

Local Linear Fitting Under Near Epoch Dependence: Uniform Consistency With Convergence Rates

A criterion for tightness for a class of dependent random variables

A flexible semiparametric model for time series

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