On conditions in central limit theorems for martingale difference arrays

Alj, Abdelkamel; Azrak, Rajae; Mélard, Guy

doi:10.1016/j.econlet.2014.03.008

Cited by 10 publications

(5 citation statements)

References 22 publications

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“…Proof. It is shown in Alj et al (2014) and Gaenssler et al (1978) that the conditional Lindeberg condition follows from the unconditional Lyapunov condition. We will show in the following, that n j=2 E|Y n,j | 4 = o(1) and decompose n j=2 E|Y n,j | 4 = L n,1 + L n,2 + L n,3 + L n,4 , where…”

Section: B Auxiliary Resultsmentioning

confidence: 99%

Testing Exogeneity in the Functional Linear Regression Model

Dorn,

Birke,

Jentsch

2026

STAT SINICA

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We propose a novel test statistic for testing exogeneity in the functional linear regression model. In contrast to Hausman-type tests in finite dimensional linear regression setups, we show that a direct extension to the functional linear regression model is not possible. Instead, we propose a test statistic based on the sum of squared differences of projections of the two estimators used for testing the null hypothesis of exogeneity in the functional linear regression model. We derive asymptotic normality under the null and show consistency under general alternatives. Moreover, we establish bootstrap consistency results for residual-based bootstrap approaches. In simulations, we investigate the finite sample performance of the proposed exogeneity tests and illustrate the superiority of bootstrap-based approaches. In particular, the bootstrap-based results turn out to be much more robust with respect to the choice of the regularization parameter.

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Section: B Auxiliary Resultsmentioning

confidence: 99%

Testing Exogeneity in the Functional Linear Regression Model

Dorn,

Birke,

Jentsch

2026

STAT SINICA

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“…The proof continues as in Theorem 2 and 2' of [1], using a weak law of large numbers for a mixingale array in [53] and referring to Theorems 1 and 1' of [1], which make use of a central limit theorem for a martingale difference array (see [54]), modified with a Lyapunov condition (see [55]).…”

Section: Discussionmentioning

confidence: 99%

Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches

Azrak¹,

Mélard²

2022

Stats

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Autoregressive-moving average (ARMA) models with time-dependent (td) coefficients and marginally heteroscedastic innovations provide a natural alternative to stationary ARMA models. Several theories have been developed in the last 25 years for parametric estimations in that context. In this paper, we focus on time-dependent autoregressive (tdAR) models and consider one of the estimation theories in that case. We also provide an alternative theory for tdAR processes that relies on a -mixing property. We compare these theories to the Dahlhaus theory for locally stationary processes and the Bibi and Francq theory, made essentially for cyclically time-dependent models, with our own theory. Regarding existing theories, there are differences in the basic assumptions (e.g., on derivability with respect to time or with respect to parameters) that are better seen in specific cases such as the tdAR(1) process. There are also differences in terms of asymptotics, as shown by an example. Our opinion is that the field of application can play a role in choosing one of the theories. This paper is completed by simulation results that show that the asymptotic theory can be used even for short series (less than 50 observations).

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“…Proof. It is similar to Alj et al (2014) (except that there k n = n and σ = 1 were assumed) where it is shown that it is a consequence of Brown (1971), more precisely Brown and Eagleson (1971), and of Gaenssler et al (1978).…”

Section: Proofsmentioning

confidence: 91%

“…As shown by Gaenssler et al (1978), the unconditional Lindeberg condition implies the conditional Lindeberg condition. Since Lyapunov conditions are stronger than Lindeberg conditions, Alj et al (2014) have fully justified the replacement mentioned above. t ; t = 1, ..., k n , n ∈ N} be a scalar zeromean square-integrable martingale difference array, and {k n } an increasing integer sequence with {k n } ↑ ∞.…”

Section: Proofsmentioning

confidence: 97%

“…That was done in Theorem 1' of Azrak and Mélard (2006) who have faced a case where almost sure convergence could not be obtained for estimators in ARMA models with time-dependent coefficients that depend also on the length of the series, n. In that context, the series cannot be considered as generated by a stochastic process but well by a triangular array process, see e.g. Alj et al (2014).…”

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confidence: 99%

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Asymptotic properties of conditional least-squares estimators for array time series

Azrak

Mélard

2021

Stat Inference Stoch Process

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The paper provides a kind of Klimko-Nelson theorems alternative in the case of conditional least-squares and Mestimators for array time series, when the assumptions of almost sure convergence cannot be established. We do not assume stationarity nor even local stationarity. In addition, we provide sufficient conditions for two of the assumptions and two theorems for the evaluation of the information matrix in array time series. In addition to timedependent models, illustrations to a threshold model and to a count data model are given.

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On conditions in central limit theorems for martingale difference arrays

Abstract: An alternative central limit theorem for martingale difference arrays is presented. It can be deduced from the literature but it is not stated as such. It can be very useful for statisticians and econometricians. An illustration is given.

Cited by 10 publications

References 22 publications

Testing Exogeneity in the Functional Linear Regression Model

Testing Exogeneity in the Functional Linear Regression Model

Autoregressive Models with Time-Dependent Coefficients—A Comparison between Several Approaches

Asymptotic properties of conditional least-squares estimators for array time series

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