A note on non‐parametric testing for Gaussian innovations in AR–ARCH models

Neumeyer, Natalie; Selk, Leonie

doi:10.1111/jtsa.12018

Cited by 5 publications

(4 citation statements)

References 22 publications

(47 reference statements)

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“…While asymptotically normal estimators of the error density f (z) have been studied in [1,15] and [9], consistent estimator for error distribution F (z) does not exist for the AR(p) model. On the other hand, such estimator has been proposed for nonparametric regression in [8], and uniformly √ n-consistent estimator of error distribution for the nonparametric AR(1)-ARCH(1) model in [21] and nonparametric regression model in [10] and [14]. It has been used for symmetry testing in parametric nonlinear time series by [3], and in nonparametric regression by [19], as well as a test of parameter constancy in [2].…”

Section: Introduction Consider An Ar(p) Process {Xmentioning

confidence: 99%

“…While Fn (z) is used for estimating F (z), for example, in [2,3,8,10,11,14,[17][18][19][20][21][22][23], it is consistently shown to be less efficient than F n (z), as one referee observes; see also Section 4.1. Our unique innovation is proving that the smooth estimator F (z) based on residuals is asymptotically equivalent to, not less efficient than, the smooth estimator F (z) based on errors.…”

Section: Introduction Consider An Ar(p) Process {Xmentioning

confidence: 99%

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Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band

Wang¹,

Liu²,

Cheng³

et al. 2014

Ann. Statist.

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We propose kernel estimator for the distribution function of unobserved errors in autoregressive time series, based on residuals computed by estimating the autoregressive coefficients with the Yule-Walker method. Under mild assumptions, we establish oracle efficiency of the proposed estimator, that is, it is asymptotically as efficient as the kernel estimator of the distribution function based on the unobserved error sequence itself. Applying the result of Wang, Cheng and Yang [J. Nonparametr. Stat. 25 (2013) 395-407], the proposed estimator is also asymptotically indistinguishable from the empirical distribution function based on the unobserved errors. A smooth simultaneous confidence band (SCB) is then constructed based on the proposed smooth distribution estimator and Kolmogorov distribution. Simulation examples support the asymptotic theory. Introduction. Consider an AR(p) process {Xt=−∞ are i.i.d. noises, called errors, EZ t = 0, EZ 2 t = σ 2 , with probability density function (p.d.f.) f (z) and cumulative distribution function (c.d.f.) F (z) = z −∞ f (u) du. For a positive integer k, the k-step ahead linear predictorX n+k of X n+k , based on a length n + p realization {X t } n t=1−p up to time n, is well studied in Chapters 5 and 9 of [7]. While efficient

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Section: Introduction Consider An Ar(p) Process {Xmentioning

confidence: 99%

Section: Introduction Consider An Ar(p) Process {Xmentioning

confidence: 99%

Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band

Wang¹,

Liu²,

Cheng³

et al. 2014

Ann. Statist.

View full text Add to dashboard Cite

show abstract

“…This assumption can be dropped. Nevertheless, in order to get similar convergence rates to those in (22) required in our proofs, we must adopt the weighting scheme in (12) and strengthen Assumption C. Compare, for example, with the somehow related developments in Neumeyer and Selk (2013) and Selk and Neumeyer (2013) when dealing with a weighted empirical distribution function of the residuals.…”

Section: Discussion and Extensionsmentioning

confidence: 99%

A model specification test for the variance function in nonparametric regression

Pardo–Fernández

Jiménez-Gamero

2018

AStA Adv Stat Anal

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The problem of testing for the parametric form of the conditional variance is considered in a fully nonparametric regression model. A test statistic based on a weighted L 2 -distance between the empirical characteristic functions of residuals constructed under the null hypothesis and under the alternative is proposed and studied theoretically. The null asymptotic distribution of the test statistic is obtained and employed to approximate the critical values. Finite sample properties of the proposed test are numerically investigated in several Monte Carlo experiments. The developed results assume independent data. Their extension to dependent observations is also discussed.is the conditional variance function and ε is the regression error, which is assumed to be independent of X. Note that, by construction, E(ε) = 0 and V ar(ε)=1. The covariate X is one-dimensional and continuous with probability density function (pdf) f and compact support R. The pdf f is assumed to be positive on R. The regression function, the variance function, the distribution of the error and the distribution of the covariate are unknown and no parametric models are assumed for them.Parametric specifications for either the regression function or the variance function are quite attractive among practitioners since they describe the relation between the response and the covariate in a concise way. Because of this reason, there are a number of papers dealing with the parametric modelling of the conditional mean and the conditional variance of Y , given X. This paper proposes a new goodness-of-fit test for the parametric form of the conditional variance. Specifically, on the basis of independent observations from (1), we wish to test the null hypothesiswhere σ 2 (·; θ) represents a parametric model for the conditional variance function, against the general alternative H 1 : H 0 is not true.Several tests for H 0 have been proposed in the specialised literature. Some of them were designed for testing homoscedasticity (see, for example, Liero, 2003, Dette andMarchlewski, 2010); others assume that the regression function has a known parametric form (see, for example, Koul and Song, 2010, Samarakoon and Song, 2011, 2012; others were built for fixed design points (see, for example, Dette and Hetzler, 2009a,b); others detect contiguous alternatives converging to the null at a rate slower that n −1/2 (see, for example, Wang and Zhou, 2007, Samarakoon and Song, 2011, 2012. Although initially the test in Koul and Song (2010) was designed for parametric regression functions, they also provide the theory for a version where the mean function is nonparametrically estimated.The test in Dette et al. (2007) (henceforth DNV) does not possess any of the above cited cons. The methodology that will be proposed in the present paper is, in a certain sense, close to that in DNV. Next, we describe our proposal and the precise meaning of such closeness. Let ε = {Y − m(X)}/σ(X) be the regression error and let ε 0 = {Y − m(X)}/σ(X; θ)

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“…In Neumeyer & Selk (), we apply Theorem (for s = 1) to test for normality of the innovations in model (AR). It is the topic of a future project to apply the theory developed here to test for serial independence of innovations or independence of the current innovation and past observations resp.…”

Section: Concluding Remarks and Outlookmentioning

confidence: 99%

Testing for a Change of the Innovation Distribution in Nonparametric Autoregression: The Sequential Empirical Process Approach

Selk¹,

Neumeyer²

2013

Scandinavian J Statistics

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We consider a nonparametric autoregression model under conditional heteroscedasticity with the aim to test whether the innovation distribution changes in time. To this end, we develop an asymptotic expansion for the sequential empirical process of nonparametrically estimated innovations (residuals). We suggest a Kolmogorov–Smirnov statistic based on the difference of the estimated innovation distributions built from the first ⌊ns⌋and the last n − ⌊ns⌋ residuals, respectively (0 ≤ s ≤ 1). Weak convergence of the underlying stochastic process to a Gaussian process is proved under the null hypothesis of no change point. The result implies that the test is asymptotically distribution‐free. Consistency against fixed alternatives is shown. The small sample performance of the proposed test is investigated in a simulation study and the test is applied to a data example.

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A note on non‐parametric testing for Gaussian innovations in AR–ARCH models

Cited by 5 publications

References 22 publications

Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band

Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band

A model specification test for the variance function in nonparametric regression

Testing for a Change of the Innovation Distribution in Nonparametric Autoregression: The Sequential Empirical Process Approach

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