Empirical Saddlepoint Approximations for Statistical Inference

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Empirical Saddlepoint Approximations
forStatistical Inference
Fallaw Sowell Tepper School of Business Carnegie
Mellon University September 2006
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Basic Issues

• More accurate statistical inference for better
  decisions.

• Most important in nonlinear models and/or
  situations
• where there is only a small amount of data.

• Computational intensive procedure to give an
  improved
• approximations to the sampling distributions.

• Converges to the familiar asymptotic normal
  distribution
• in large samples.

• Allows multiple modes, asymmetric distribution,
• non-normal tail behavior, relative convergence.

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Basic Intuition

• The traditional asymptotic Normal approximations
• use local information about the objective
  function
• at one point to create an approximation over
  the
• entire parameter space.

• The empirical saddlepoint approximation uses
• information about the global shape of the
  objective
• function to create an approximation over the
  entire
• parameter space.

• Instead of one linear approximation to the
  objective
• function, combine a continuum of linear
  approximations.

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Traditional Approach (MLE)
FOCs
Perform a linear approximation.
Apply a central limit theorem
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Traditional Approach (GMM)
FOCs
Perform a linear approximation.
Apply a central limit theorem
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Traditional Approach Intuition
1. A multivariate normal distribution with the
mean at the location of the extreme value and
covariance given by the convexity at the extreme
value.
2. Linear approximation to the FOCs.
3. Equivalent to a quadratic approximation to the
objective function, an elliptic paraboloid.
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Hall and Horowitz (Econometrica,96)
Look at some sampling distributions, various
values of N.
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Saddlepoint Approximation Intuition
-Measures of convexity at
-Local minima associated with
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Steps to use this in practice
1. GMM moments to create estimation equations.
2. Determine the local minima.
3. Over a grid of values calculate the
convexity.
(Solve m equations in m unknowns.)
4. Numerically integrate to obtain marginal
densities.
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Rate of convergence

• Convergence is relative.

relative
absolute

• Relative implies

• Better tail approximation compared to
• asymptotic and bootstrap approximations.

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Saddlepoint versus Bootstrap