Empirical Financial Economics

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Empirical Financial Economics

• 2. The Efficient Markets Hypothesis - Generalized
  Method of Moments

Stephen Brown NYU Stern School of Business UNSW
PhD Seminar, June 19-21 2006
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Random Walk Hypothesis

• Random Walk hypothesis a special case of EMH
• Overidentification of model
• Provides a test of model (variance ratio
  criterion)
• Allows for estimation of parameters (GMM
  paradigm)

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Variance ratio tests
using sample quantities
The variance ratio is
asymptotically Normal
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Overlapping observations
Non-overlapping observations
ln(pt)
t
tT
Overlapping observations
ln(pt)
unbiassed estimators
Variance ratio is asymptotically Normal
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Random walk model and GMM
aggregate into moment conditions
and express as three observations of a nonlinear
regression model
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Generalized method of moment estimators

• Choose to minimize . is
  referred to as the optimal weighting matrix,
  equal to the inverse covariance matrix of
• Estimators are asymptotically Normal and
  efficient
• Minimand is distributed as Chi-square with d.f.
  number of overidentifying information
• Methods of obtaining
• 1. Set (Ordinary Least Squares).
  Estimate model. Set (Generalized
  Least Squares). Reestimate .
• 2. Use analytic methods to infer

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GMM and the Efficient Market Hypothesis
1 asset and 1 instrument
1 equation and k unknowns
m assets and 1 instrument
m equations and gtk unknowns
m assets and n instrument
mxn equations and gtk unknowns
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Autocovariances and cross autocovariances
xt
yt
t
tk
t-k
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Cross autocovariances are not symmetrical!
Autocovariances are given by
Cross autocovariances are given by
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Cross autocovariances and the weighting function
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Assuming stationarity
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Apply this to cross covariances
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A simple expression for the inverse weighting
matrix
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Some applications of GMM

• Fixed income securities
• Construct moments of returns based on
  distribution of it?
• Estimate ? by comparing to sample moments
• Derivative securities
• Construct moments of returns by simulating PDE
  given ?
• Estimate ? by comparing to sample moments
• Asset pricing with time-varying risk premia