Tutorial Financial Econometrics/Statistics

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TutorialFinancial Econometrics/Statistics

• 2005 SAMSI program on Financial Mathematics,
  Statistics, and Econometrics

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Goal
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At the index level
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Part I Modeling

• ... in which we see what basic properties of
  stock prices/indices we want to capture

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Contents

• Returns and their (static) properties
• Pricing models
• Time series properties of returns

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Why returns?

• Prices are generally found to be non-stationary
• Makes life difficult (or simpler...)
• Traditional statistics prefers stationary data
• Returns are found to be stationary

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Which returns?

• Two type of returns can be defined
• Discrete compounding
• Continuous compounding

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Discrete compounding

• If you make 10 on half of your money and 5 on
  the other half, you have in total 7.5
• Discrete compounding is additive over portfolio
  formation

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Continuous compounding

• If you made 3 during the first half year and 2
  during the second part of the year, you made
  (exactly) 5 in total
• Continuous compounding is additive over time

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Empirical properties of returns
Mean St.dev. Annualized volatility Skewness Kurtosis Min Max
IBM -0.0 2.46 39.03 -23.51 1124.61 -138 12.4
IBM (corr) 0.0 1.64 26.02 -0.28 15.56 -26.1 12.4
SP 0.0 0.95 15.01 -1.4 39.86 -22.9 8.7
Data period July 1962- December 2004 daily
frequency
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Stylized facts

• Expected returns difficult to assess
• Whats the equity premium?
• Index volatility lt individual stock volatility
• Negative skewness
• Crash risk
• Large kurtosis
• Fat tails (thus EVT analysis?)

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Pricing models

• Finance considers the final value of an asset to
  be known
• as a random variable , that is
• In such a setting, finding the price of an asset
  is equivalent to finding its expected return

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Pricing models 2

• As a result, pricing models model expected
  returns ...
• ... in terms of known quantities or a few almost
  known quantities

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Capital Asset Pricing Model

• One of the best known pricing models
• The theorem/model states

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Black-Scholes

• Also Black-Scholes is a pricing model
• (Exact) contemporaneous relation between asset
  prices/returns

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Time series properties of returns

• Traditionally model fitting exercise without much
  finance
• mostly univariate time series and, thus, less
  scope for tor the traditional cross-sectional
  pricing models
• lately more finance theory is integrated
• Focuses on the dynamics/dependence in returns

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Random walk hypothesis

• Standard paradigm in the 1960-1970
• Prices follow a random walk
• Returns are i.i.d.
• Normality often imposed as well
• Compare Black-Scholes assumptions

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Box-Jenkins analysis
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Linear time series analysis

• Box-Jenkins analysis generally identifies a white
  noise
• This has been taken long as support for the
  random walk hypothesis
• Recent developments
• Some autocorrelation effects in momentum
• Some (linear) predictability
• Largely academic discussion

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Higher moments and risk
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Risk predictability

• There is strong evidence for autocorrelation in
  squared returns
• also holds for other powers
• volatility clustering
• While direction of change is difficult to
  predict, (absolute) size of change is
• risk is predictable

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The ARCH model

• First model to capture this effect
• No mean effects for simplicity
• ARCH in mean

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ARCH properties

• Uncorrelated returns
• martingale difference returns
• Correlated squared returns
• with limited set of possible patterns
• Symmetric distribution if innovations are
  symmetric
• Fat tailed distribution, even if innovations are
  not

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The GARCH model

• Generalized ARCH
• Beware of time indices ...

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GARCH model

• Parsimonious way to describe various correlation
  patterns
• for squared returns
• Higher-order extension trivial
• Math-stat analysis not that trivial
• See inference section later

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Stochastic volatility models

• Use latent volatility process

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Stochastic volatility models

• Also SV models lead to volatility clustering
• Leverage
• Negative innovation correlation means that
  volatility increases and price decreases go
  together
• Negative return/volatility correlation
• (One) structural story default risk

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Continuous time modeling

• Mathematical finance uses continuous time, mainly
  for simplicity
• Compare asymptotic statistics as approximation
  theory
• Empirical finance (at least originally) focused
  on discrete time models

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Consistency

• The volatility clustering and other empirical
  evidence is consistent with appropriate
  continuous time models
• A simple continuous time stochastic volatility
  model

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Approximation theory

• There is a large literature that deals with the
  approximation of continuous time stochastic
  volatility models with discrete time models
• Important applications
• Inference
• Simulation
• Pricing

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Other asset classes

• So far we only discussed stock(indices)
• Stock derivatives can be studied using a
  derivative pricing models
• Financial econometrics also deals with many other
  asset classes
• Term structure (including credit risk)
• Commodities
• Mutual funds
• Energy markets
• ...

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Term structure modeling

• Model a complete curve at a single point in time
• There exist models
• in discrete/continuous time
• descriptive/pricing
• for standard interest rates/derivatives
• ...

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Part 2 Inference
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Contents

• Parametric inference for ARCH-type models
• Rank based inference

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Analogy principle

• The classical approach to estimation is based on
  the analogy principle
• if you want to estimate an expectation, take an
  average
• if you want to estimate a probability, take a
  frequency
• ...

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Moment estimation (GMM)

• Consider an ARCH-type model
• We suppose that can be calculated
  on the basis of observations if is known
• Moment condition

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Moment estimation - 2

• The estimator now is taken to solve
• In case of underidentification use instruments
• In case of overidentification minimize
  distance-to-zero

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Likelihood estimation

• In case the density of the innovations is known,
  say it is , one can write down the
  density/likelihood of observed returns
• Estimator maximize this

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Doing the math ...

• Maximizing the log-likelihood boils down to
  solving
• with

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Efficiency consideration

• Which of the above estimators is better?
• Analysis using Hájek-Le Cam theory of asymptotic
  statistics
• Approximate complicated statistical experiment
  with very simple ones
• Something which works well in the approximating
  experiment, will also do well in the original one

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Quasi MLE

• In order for maximum likelihood to work, one
  needs the density of the innovations
• If this is not know, one can guess a density
  (e.g., the normal)
• This is known as
• ML under non-standard conditions (Huber)
• Quasi maximum likelihood
• Pseudo maximum likelihood

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Will it work?

• For ARCH-type models, postulating the Gaussian
  density can be shown to lead to consistent
  estimates
• There is a large theory on when this works or not
• We say for ARCH-type models the Gaussian
  distribution has the QMLE property

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The QMLE pitfall

• One often sees people referring to Gaussian MLE
• Then, they remark that we know financial
  innovations are fat-tailed ...
• ... and they switch to t-distributions
• The t-distribution does not possess the QMLE
  property (but, see later)

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How to deal with SV-models?

• The SV models look the same
• But now, is a latent process and
  hence not observed
• Likelihood estimation still works in principle,
  but unobserved variances have to be integrated out

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Inference for continuous time models

• Continuous time inference can, in theory, be
  based on
• continuous record observations
• discretely sampled observations
• Essentially all known approaches are based on
  approximating discrete time models

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Rank based inference

• ... in which we discuss the main ideas of rank
  based inference

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The statistical model

• Consider a model where somewhere there
• exist i.i.d. random errors
• The observations are
• The parameter of interest is some
• We denote the density of the errors by

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Formal model

• We have an outcome space , with the
  number of observations and the dimension of
• Take standard Borel sigma-fields
• Model for sample size
• Asymptotics refer to

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Example Linear regression

• Linear regression model(with observations
  )
• Innovation density and cdf

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Example ARCH(1)

• Consider the standard ARCH(1) model
• Innovation density and cdf

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Maintained hypothesis

• For given and sample size , the
• innovations can be calculated from
  the
• observations
• For cross-sectional models one may even often
  write
• Latent variable (e.g., SV) models ...

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Innovation ranks

• The ranks are the ranks of the
• innovations
• We also write for the
  ranks
• of the innovations
  based on
• a value for the parameter of interest
• Ranks of observations are generally not very
  useful

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Basic properties

• The distribution
  does
• not depend on nor on
• permutation of
• This is (fortunately) not true for
• at least essentially

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Invariance

• Suppose we generate the innovations as
  transformationwith i.i.d.
  standard uniform
• Now, the ranks are even invariant with
  respect to

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Reconstruction

• For large sample size we have
• and, thus,

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Rank based statistics

• The idea is to apply whatever procedure you have
  that uses innovations on the innovations
  reconstructed from the ranks
• This makes the procedure robust to distributional
  changes
• Efficiency loss due to ?

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Rank based autocorrelations

• Time-series properties can be studied using rank
  based autocorrelations
• These can be interpreted as standard
  autocorrelations
• rank based
• for given reference density and distribution free

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Robustness

• An important property of rank based statistics is
  the distributional invariance
• As a result a rank based estimator is
  consistent for any reference density
• All densities satisfy the QMLE property when
  using rank based inference

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Limiting distribution

• The limiting distribution of depends on
  both the chosen reference density and the
  actual underlying density
• The optimal choice for the reference density is
  the actual density
• How efficient is this estimator?
• Semiparametrically efficient

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Remark

• All procedures are distribution free with respect
  to the innovation density
• They are, clearly, not distribution free with
  respect to the parameter of interest

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Signs and ranks
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Why ranks?

• So far, we have been considering completely
  unrestricted sets of innovation densities
• For this class of densities ranks are maximal
  invariant
• This is crucial for proving semiparametric
  efficiency

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Alternatives

• Alternative specifications may impose
• zero-median innovations
• symmetric innovations
• zero-mean innovations
• This is generally a bad idea ...

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Zero-median innovations

• The maximal invariant now becomes the ranks and
  signs of the innovations
• The ideas remain the same, but for a more precise
  reconstruction
• Split sample of innovations in positive and
  negative part and treat those separately

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But ranks are still ...

• Yes, the ranks are still invariant
• ... and the previous results go through
• But the efficiency bound has now changed and rank
  based procedures are no longer semiparametrically
  efficient
• ... but sign-and-rank based procedures are

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Symmetric innovations

• In the symmetric case, the signed-ranks become
  maximal invariant
• signs of the innovations
• ranks of the absolute values
• The reconstruction now becomes still more precise
  (and efficient)

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Semiparametric efficiency
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General result

• Using the maximal invariant to reconstitute the
  central sequence leads to semiparametrically
  efficient inference
• in the model for which this maximal invariant is
  derived
• In general use

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Proof

• The proof is non-trivial, but some intuition can
  be given using tangent spaces