Asymptotic Properties of Quasi-Maximum Likelihood Estimators for ARMA Models with Time-Dependent Coefficients

2006 ◽  
Vol 9 (3) ◽  
pp. 279-330 ◽  
Author(s):  
Rajae Azrak ◽  
Guy Mélard
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Properties ◽  
Maximum Likelihood Estimators ◽  
Time Dependent ◽  
Arma Models ◽  
Quasi Maximum Likelihood

STRONG CONSISTENCY OF ESTIMATORS FOR MULTIVARIATE ARCH MODELS

1998 ◽  
Vol 14 (1) ◽  
pp. 70-86 ◽  
Author(s):  
Thierry Jeantheau
Keyword(s):  
Maximum Likelihood ◽  
Error Term ◽  
Strong Consistency ◽  
Asymptotic Properties ◽  
Maximum Likelihood Estimators ◽  
Multivariate Garch ◽  
Arch Models ◽  
Multivariate Garch Model ◽  
Quasi Maximum Likelihood ◽  
Derivatives Of

This paper deals with the asymptotic properties of quasi-maximum likelihood estimators for multivariate heteroskedastic models. For a general model, we give conditions under which strong consistency can be obtained; unlike in the current literature, the assumptions on the existence of moments of the error term are weak, and no study of the various derivatives of the likelihood is required. Then, for a particular model, the multivariate GARCH model with constant correlation, we describe the set of parameters where these conditions hold.

scholarly journals Causal Inference for a Population of Causally Connected Units

2014 ◽  
Vol 2 (1) ◽  
pp. 13-74 ◽  
Author(s):  
Mark J. van der Laan
Keyword(s):  
Data Structure ◽  
Maximum Likelihood ◽  
Longitudinal Data ◽  
Maximum Likelihood Estimators ◽  
Time Dependent ◽  
Convergence Theory ◽  
Nuisance Parameters ◽  
Target Parameter ◽  
Targeted Maximum Likelihood ◽  
The Mean

AbstractSuppose that we observe a population of causally connected units. On each unit at each time-point on a grid we observe a set of other units the unit is potentially connected with, and a unit-specific longitudinal data structure consisting of baseline and time-dependent covariates, a time-dependent treatment, and a final outcome of interest. The target quantity of interest is defined as the mean outcome for this group of units if the exposures of the units would be probabilistically assigned according to a known specified mechanism, where the latter is called a stochastic intervention. Causal effects of interest are defined as contrasts of the mean of the unit-specific outcomes under different stochastic interventions one wishes to evaluate. This covers a large range of estimation problems from independent units, independent clusters of units, and a single cluster of units in which each unit has a limited number of connections to other units. The allowed dependence includes treatment allocation in response to data on multiple units and so called causal interference as special cases. We present a few motivating classes of examples, propose a structural causal model, define the desired causal quantities, address the identification of these quantities from the observed data, and define maximum likelihood based estimators based on cross-validation. In particular, we present maximum likelihood based super-learning for this network data. Nonetheless, such smoothed/regularized maximum likelihood estimators are not targeted and will thereby be overly bias w.r.t. the target parameter, and, as a consequence, generally not result in asymptotically normally distributed estimators of the statistical target parameter.To formally develop estimation theory, we focus on the simpler case in which the longitudinal data structure is a point-treatment data structure. We formulate a novel targeted maximum likelihood estimator of this estimand and show that the double robustness of the efficient influence curve implies that the bias of the targeted minimum loss-based estimation (TMLE) will be a second-order term involving squared differences of two nuisance parameters. In particular, the TMLE will be consistent if either one of these nuisance parameters is consistently estimated. Due to the causal dependencies between units, the data set may correspond with the realization of a single experiment, so that establishing a (e.g. normal) limit distribution for the targeted maximum likelihood estimators, and corresponding statistical inference, is a challenging topic. We prove two formal theorems establishing the asymptotic normality using advances in weak-convergence theory. We conclude with a discussion and refer to an accompanying technical report for extensions to general longitudinal data structures.

scholarly journals QML Estimators in Linear Regression Models with Functional Coefficient Autoregressive Processes

2010 ◽  
Vol 2010 ◽  
pp. 1-30 ◽  
Author(s):  
Hongchang Hu
Keyword(s):  
Maximum Likelihood ◽  
Linear Regression ◽  
Regression Models ◽  
Asymptotic Properties ◽  
Asymptotic Distributions ◽  
Autoregressive Processes ◽  
Linear Regression Models ◽  
Unknown Parameters ◽  
Quasi Maximum Likelihood ◽  
Functional Coefficient

This paper studies a linear regression model, whose errors are functional coefficient autoregressive processes. Firstly, the quasi-maximum likelihood (QML) estimators of some unknown parameters are given. Secondly, under general conditions, the asymptotic properties (existence, consistency, and asymptotic distributions) of the QML estimators are investigated. These results extend those of Maller (2003), White (1959), Brockwell and Davis (1987), and so on. Lastly, the validity and feasibility of the method are illuminated by a simulation example and a real example.

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