Edgeworth and Moment Approximations: The Case of MM and QML Estimators for the MA(1) Models

Demos, Antonis; Kyriakopoulou, Dimitra

doi:10.1080/03610926.2011.597919

Cited by 5 publications

(24 citation statements)

References 41 publications

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“…The lbf is

θ / (1 + θ^{2})

and thereby the GMR1 estimator is equal to

(1 - \sqrt{1 - 4 β_{n}^{2}}) / (2 β_{n})

when

β_{n} \in (- c, c)

, equal to −1 when

β_{n} \leq - c

and equal to 1 when

β_{n} \geq c

; see Gourieroux et al. () and Demos and Kyriakopoulou (). To evaluate the GMR2 estimator, the

E_{θ} [β_{n}]

is needed.…”

Section: General Frameworkmentioning

confidence: 99%

“…Validity would follow in the context of Theorem along with the validation of the conditions of Gotze and Hipp () (Assumptions 1–4) for the derivatives of

ς_{n}

. For details, see Demos and Kyriakopoulou (). Smoothness follows from the smoothness of the moments appearing as coefficients in the Edgeworth densities due to the smoothness of the process itself, as well as due to dominated convergence.…”

Section: General Frameworkmentioning

confidence: 99%

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A class of indirect inference estimators: higher‐order asymptotics and approximate bias correction

Arvanitis

Demos

2015

The Econometrics Journal

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In this paper, we define a set of indirect inference estimators based on moment approximations of the auxiliary estimators. Their introduction is motivated by reasons of analytical and computational facilitation. Their definition provides an indirect inference framework for some classical bias correction procedures. We derive higher-order asymptotic properties of these estimators. We demonstrate that under our assumption framework, and in the special case of deterministic weighting and affinity of the binding function, these are second-order unbiased. Moreover, their second-order approximate mean square errors do not depend on the cardinality of the Monte Carlo or bootstrap samples that our definition might involve. Consequently, the second-order mean square error of the auxiliary estimator is not altered. We extend this to a class of multistep indirect inference estimators that have zero higher-order bias without increasing the approximate mean square error, up to the same order. Our theoretical results are also validated by three Monte Carlo experiments.

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“…The lbf is

θ / (1 + θ^{2})

and thereby the GMR1 estimator is equal to

(1 - \sqrt{1 - 4 β_{n}^{2}}) / (2 β_{n})

when

β_{n} \in (- c, c)

, equal to −1 when

β_{n} \leq - c

and equal to 1 when

β_{n} \geq c

; see Gourieroux et al. () and Demos and Kyriakopoulou (). To evaluate the GMR2 estimator, the

E_{θ} [β_{n}]

is needed.…”

Section: General Frameworkmentioning

confidence: 99%

“…Validity would follow in the context of Theorem along with the validation of the conditions of Gotze and Hipp () (Assumptions 1–4) for the derivatives of

ς_{n}

Section: General Frameworkmentioning

confidence: 99%

A class of indirect inference estimators: higher‐order asymptotics and approximate bias correction

Arvanitis

Demos

2015

The Econometrics Journal

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“…Corollary provides with a simple application of the previous results, which essentially uses the results of concerning the derivation of the rate and of a Karhunen–Loeve‐type representation of the asymptotic distribution of the MLE in shrinking neighbourhoods of − 1 as well as the results of when θ ∈ ( − 1,1), c = 0 under the assumption of independence and normality for the noise process. Corollary Suppose that Assumption holds and that v t = 1 for all

t MathClass-rel\in double-struckZ

, while

z_{t} ∽ N ((), 0 MathClass-punc, 1)

, then

r_{n MathClass-punc, ϑ MathClass-punc, θ} MathClass-rel= ({, \begin{matrix} falsenonefalsearray \\ arraycenter \sqrt{n} if θ \in (- 1, 1), c = 0 \\ arraycenter \sqrt{n} if θ \notin [- 1, 1], for all c and η_{n} \\ arraycenter n^{1 ∕ 4} if θ = - 1, c \geq 0, η_{n} = n \end{matrix}) MathClass-punc,

and …”

Section: Limit Theory For the Indirect Estimator Under Local Alternatmentioning

confidence: 99%

“…There lies our second motivation. provide with heuristic approximations of the higher‐order asymptotic bias and mean squared error of the estimator at hand, under sequences of local to 1 or − 1 alternatives. These imply that the rate of convergence of the estimator is becoming slower than the standard

\sqrt{n}

in appropriately shrinking neighbourhoods of the boundary cases of non‐invertibility for a semi‐parametric MA(1) model under independence of the noise process.…”

Section: Introductionmentioning

confidence: 99%

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A simple example of an indirect estimator with discontinuous limit theory in the MA(1) model

Arvanitis

2014

Journal Time Series Analysis

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SUMMARYIndirect estimators usually emerge from two‐step optimization procedures. Each step in such a procedure may induce complexities in the asymptotic theory of the estimator. In this note, we are occupied with a simple example in which the estimator defined by the inversion of the binding function has a ‘discontinuous’ limit theory even in cases where the auxiliary one does not. This example lives in the framework of estimation of the MA (1) parameter. The ‘discontinuities’ involve the dependence of the rate of convergence on the parameter, the non‐continuity of the limit distribution w.r.t. the parameter and the estimator's non‐regularity. We are also occupied with a more complex example where the discontinuities occur because of complexities induced in any step of the defining procedure. We present some Monte Carlo evidence on the quality of the approximations from the limit distributions. Copyright © 2014 Wiley Publishing Ltd

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Asymptotic expansions of econometric estimators in time series models

Κυριακοπούλου¹

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Techniques for approximating probability distributions like the Edgeworth expansion have a long history in time series models. The purpose of this thesis is to give a detailed study of the asymptotic properties of the Moving Average (MA) and the Exponential GARCH (EGARCH) models. Extending the results in Sargan (1976) [80] and Tanaka (1984) [87], we derive the asymptotic expansions of the distribution, the bias and the mean squared error of the MM and QML estimators of the first order autocorrelation and the MA parameter for the MA(1) model. It turns out that the asymptotic properties of the estimators depend on whether the mean of the process is known or estimated. A comparison of the moment expansions, either in terms of bias or MSE, reveals that there is not uniform superiority of neither of the estimators, when the mean of the process is estimated. This is also confirmed by simulations. In the zero-mean case, and on theoretical grounds, the QMLEs are superior to the MM ones in both bias and MSE terms. The results are important for bias correction and increasing the efficiency of the estimators. Next, we derive the bias approximations of the ML and QML estimators of the EGARCH(1,1) parameters and we check our theoretical results through simulations. With the approximate bias expressions up to O(1/T), we are then able to correct the bias of all estimators. To this end, a Monte Carlo exercise is conducted and the results are presented and discussed. We conclude that, for given sets of parameters values, the bias correction works satisfactory for all parameters. The results for the bias expressions can be used to formulate the approximate Edgeworth distribution of the estimators. Moreover, the asymptotic properties of EGARCH models are still largely unexplored and are considered difficult tasks (see e.g. Straumann and Mikosch, 2006) [83]. There is still no complete answer to the following questions: under which conditions do EGARCH processes have bounded first and second order variance derivatives? And under which conditions is the expectation of the supremum norm of the second order log-likelihood derivative finite, in a neighborhood around the true parameter value? These questions are important because the existence of such moment bounds permits the establishment of large sample statistical properties, such as the asymptotic normality of the QMLEs.

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Edgeworth and Moment Approximations: The Case of MM and QML Estimators for the MA(1) Models

Cited by 5 publications

References 41 publications

A class of indirect inference estimators: higher‐order asymptotics and approximate bias correction

A class of indirect inference estimators: higher‐order asymptotics and approximate bias correction

A simple example of an indirect estimator with discontinuous limit theory in the MA(1) model

Asymptotic expansions of econometric estimators in time series models

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