Advanced topics in Financial Econometrics

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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