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Advanced topics in Financial Econometrics Bas Werker Tilburg University, SAMSI fellow

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In which we will consider the modern theory of asymptotic statistics à la Hájek/Le Cam, with a special emphasis on financial econometric applications, semiparametric analysis, and rank based inference methods

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Contents 1.Introduction 2.Inference in parametric models 3.Semiparametric analysis for models with i.i.d. observations 4.Semiparametric time series models 5.Rank based statistics 6.Semiparametric efficiency of rank based inference

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Literature Aad W. van der Vaart, Asymptotic Statistics, Cambridge University Press, 1998/2000 Reference (AS-x) is to Chapter x of this book Various papers

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Introduction

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Contents Consistency and asymptotic normality (AS- 2,3) M- and Z-estimators (AS-5) Local alternatives and continguity (AS-6) Local power of tests

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Stochastic convergence (AS-2) Consider a sequence of -dimensional random vectors All random variables are (for fixed sample size) defined on the same implicit probability space

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Weak convergence Convergence of the distributions: for each point where is continuous, we have as Convergence in distribution/law Notation

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Convergence in probability Convergence of the random variables: as, for all Euclidean distance Basic to the notion of consistency of estimators Notation

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Continuous mapping theorem Let be a function which is continuous at each point of a set for which, then

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o and O notation Convenient short-hand notation and calculus means bounded in probability, i.e., for all there exists such that means

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Rules of calculus Convenient rules

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Delta method (AS-3) Suppose that for numbers we have Suppose is differentiable at Then

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Uniform Delta method Suppose that for numbers and vectors Suppose Suppose is continuously differentiable in a neighborhood of Then

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M-estimators Define a statistic (estimator) for observations as a maximizer of

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Z-estimators Define a statistic (estimator) for observations as a solution of Also called Estimating equation Often, but not always, based on M-estimator

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Examples Maximum likelihood (Generalized) Method of Moments Chi-square estimation... all parametric inference

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Consistency Uniform convergence of criterion function leads to consistency of M-estimators Approximate maximization is sufficient Theorem AS 5.7 Uniform convergence of criterion function leads to consistency of Z-estimators

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Asymptotic normality Let us be given a Z-estimator Suppose the Z-criterion satisfies Suppose is differentiable with derivative at the zero of Then, under some additional regularity,

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One-step estimators A technical trick to reduce the conditions for consistency and asymptotic normality of Z estimators significantly Starting from an initial root-n consistent estimator, i.e.,, we consider the solution of the (linear) equation

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Asymptotic normality The previously derived asymptotic expansion/distribution holds now under the sole condition

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Discretization trick The previous condition can be relaxed further by considering an initial discretized estimator, i.e., one which essentially only takes a finite number of possible values Now, we only need, for all non-random, that

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Contiguity (AS-6) To understand the idea, consider a statistical model where we observe one variable from a distribution or We want to test if the distribution is or If and are orthogonal, this testing problem is trivial Orthogonality: disjoint support

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Contiguity - 2 If and have the same support, i.e., are absolutely continuous, the problem is non- trivial (this is the interesting case) Clearly, good tests should in that case be based on the likelihood ratio

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Intermezzo Radon-Nikodym derivatives always refer to the derivative defined for the part where dominates As a consequence, expectations of Radon- Nikodym derivatives may be strictly smaller than one

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Contiguity - definition Contiguity the the asymptotic version of absolute continuity for sequences of probability measures Definition: if Definition: if both and

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Le Cams first lemma The well known equivalence for absolute continuity translates in the obvious way to contiguity (AS Lemma 6.4) The following are equivalent

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Consistency An estimator which is consistent under a (sequence of) probability (measures) is also consistent under a contiguous (sequence of) probability (measures)

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Le Cams third lemma Change of probability measures using contiguous probabilities may be taken to the limit See AS Theorem 6.6 It looks complicated, but is actually quite intuitive

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Local alternatives The idea of contiguity is basic to the construction of local alternatives In a sequence of statistical experiments with identical parameter space, asymptotic tests for versus are trivial Non-trivial is versus

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Example Consider the model where we observe i.i.d. copies of a random variable Denote When are and contiguous? What is the asymptotic distribution of he sample average under ?

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Inference in parametric models

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Contents Local Asymptotic Normality (AS-7) Optimal testing Efficiency of estimators (AS-8) Nuisance parameters and geometry Limits of experiments (AS-9)

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Local Asymptotic Normality (AS-7) Local Asymptotic Normality (LAN) is the formalization of a regular statistical experiment The concept is a refinement of contiguity All standard econometric models are LAN

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LAN - definition A statistical model is identified as a sequence of probability models LAN holds if for each and every sequence

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Remarks is called the central sequence and the equivalent of the derivative of the log- likelihood in classical statistics is the Fisher information The root-n rate can be any other, but this is the usual situation

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Terminology The terminology derives from with a single observation from

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Examples In models with i.i.d. observations, differentiability conditions on the densities lead to LAN This is the so-called differentiability in quadratic mean condition See AS Theorem 7.2 Regression, Probit/Logit, etc...

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Time series examples LAN has also been shown to hold for ARMA (Kreiss, 1987) ARCH (Linton, 1993) GARCH (Drost and Klaassen, 1997)... In all cases with the obvious central sequence

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Optimal testing in LAN experiments Consider a (test) statistic in a LAN experiment that satisfies, under, An asymptotic size (under ) test is easily constructed

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Local power Consider a sequence of alternatives Whats the behavior of under ? Le Cams third lemma: under

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Maximize local power To maximize local power, we need to maximize Hence take the central sequence evaluated at the null as statistic Lagrange multiplier type Use quadratic forms in multidimensional case

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Efficiency (AS-8) We may also formalize the Cramér-Rao lower bound idea Lets first look at the asymptotic counterpart of an unbiased estimator... which requires more than mere consistency

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Regular estimator Consider an estimator for satisfying under How does this estimator behave under ?

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Once more... Le Cams third lemma, under, Which leads to the requirement If not, estimator does not follow local shifts Such an estimator is called regular

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Convolution theorem For any regular estimator we have The idea of regularity can be relaxed to general limiting distributions In that case, we find The latter result explains the name

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Efficient estimator An estimator is therefore called efficient if Note that this estimator is trivially regular

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Minimax theorem Theorem on asymptotic loss of any estimator (regular or not) Only gives a bound for the asymptotic risk, no more distribution information

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Nuisance parameters The Convolution theorem also leads to optimal estimators in case we have both a parametric of interest and a parametric as nuisance parameter In that case we need to consider

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Efficient estimation If one is only interested in estimating, one should consider just the upper part of From the partitioned inverses formula, this is

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The geometry of inference with nuisance parameters Using the intuition that Fisher information matrices are variances of central sequences, we find that the central sequence to use when there are nuisance parameters is the residual of the projection of the central sequence for the parameter of interest on the central sequences of the nuisance parameters

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Limits of experiments (AS-9) The previous ideas can be extended to a general concept of convergence of statistical experiments Crucial is an identical parameter space LAN corresponds to a Guassian shift limit Other limits are possible

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