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

Azrak, Rajae; Mélard, Guy

doi:10.1007/s11203-021-09242-8

Cited by 5 publications

(6 citation statements)

References 32 publications

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“…It is based on an exact algorithm for the computation of the Gaussian likelihood [14] and an implementation of a Levenberg-Marquardt nonlinear least-squares algorithm. Under some very general conditions [8,12], it is shown that the quasi-maximum likelihood estimator β converges to the true value of β, and β is asymptotically normal, more precisely…”

Section: The Estimation Methodsmentioning

confidence: 99%

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ARIMA Models with Time-Dependent Coefficients: Official Statistics Examples

Mélard¹

2023

Time Series Analysis - New Insights

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About 25 years ago, effective methods for dealing with time series models that vary with time appeared in the statistical literature. Except in a few cases, they have never been used for economic statistics. In this chapter, we consider autoregressive integrated moving average (ARIMA) models with time-dependent coefficients (tdARIMA) applied to monthly industrial production series. We start with a small-size study with time-dependent integrated autoregressive (tdARI) models on Belgian series compared to standard ARI models with constant coefficients. Then, a second, bigger, illustration is given on 293 U.S. industrial production time series with tdARIMA models. We employ the software package Tramo to obtain linearized series and model specifications and build both the ARIMA models with constant coefficients (cARIMA) and the tdARIMA models, using specialized software. In these tdARIMA models, we use the simplest specification for each coefficient: a simple regression with respect to time. Surprisingly, for a large part of the series, there are statistically significant slopes, indicating that the tdARIMA models fit better the series than the cARIMA models.

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Section: The Estimation Methodsmentioning

confidence: 99%

“…We replace e tÀk , k =0,1, … , q, in Eq. (3) [8,12]. In practice, we used an exponential function of time for g n ðÞ t β ðÞ .…”

Section: The Modelmentioning

confidence: 99%

ARIMA Models with Time-Dependent Coefficients: Official Statistics Examples

Mélard¹

2023

Time Series Analysis - New Insights

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“…and it depends on realizations of (unobservable) noise indicator q t = q t (c). Thus, some of the widely used estimation methods, e.g., conditional last squares (CLS) method [26], as well as conditional maximum likelihood (CML) method [27], cannot be used here. Moreover, according to Theorem 2 and Equations ( 15) and ( 16), it is clear that even some moment-based estimation methods, i.e., the well-known Yule-Walker (YW) estimators [28], cannot simply obtain the above (see Section 6).…”

Section: Parameter Estimation Proceduresmentioning

confidence: 99%

Integer-Valued Split-BREAK Process with a General Family of Innovations and Application to Accident Count Data Modeling

Stojanović,

Bakouch,

Gajtanović

et al. 2024

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This paper presents a novel count time-series model, named integer-valued Split-BREAK process of the first order, abbr. INSB(1) model. This process is examined in terms of its basic stochastic properties, such as stationarity, mean, variance and correlation structure. In addition, the marginal distribution, over-dispersion and zero-inflation properties of the INSB(1) process are also examined. To estimate the unknown parameters of the INSB(1) process, an estimation procedure based on probability generating functions (PGFs) is proposed. For the obtained estimators, their asymptotic properties, as well as the appropriate simulation study, are examined. Finally, the INSB(1) process is applied in the dynamic analysis of some real-world series, namely, the numbers of serious traffic accidents in Serbia and forest fires in Greece.

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“…The AM theory was generalized to vector processes by [2], who treated the case of tdVARMA processes where the model coefficients did not depend on n, and by [15] for the general case, called tdVARMA (n) processes. Additionally, [16] provided a better foundation for the asymptotic theory for array processes, a theorem for a reduction of the order of moments from 8 to slightly more than 4 and tools for obtaining the asymptotic covariance matrix of the estimator. In [17], there was an example of vector tdAR and tdMA models on monthly log returns of IBM stock and the S&P500 index from January 1926 to December 1999, treated first in [18].…”

Section: Let φmentioning

confidence: 99%

“…A comparison is difficult here, but it is interesting to note a less restrictive assumption of the existence of fourth-order moments, not eighth-order as in AM. Note that [16] has removed that requirement for the AM theory. Note that the expression for I in [8], which corresponds to our W in (9), did not involve fourth-order moments since no parameter was involved in the heteroscedasticity.…”

Section: A Comparison With the Theory Of Cyclically Time-dependent Mo...mentioning

confidence: 99%

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

Azrak¹,

Mélard²

2022

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

Cited by 5 publications

References 32 publications

ARIMA Models with Time-Dependent Coefficients: Official Statistics Examples

ARIMA Models with Time-Dependent Coefficients: Official Statistics Examples

Integer-Valued Split-BREAK Process with a General Family of Innovations and Application to Accident Count Data Modeling

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

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