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Empirical Saddlepoint Approximations forStatistical Inference Fallaw Sowell Tepper School of Business Carnegie Mellon University September 2006
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.
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.
Traditional Approach (MLE) FOC’s Perform a linear approximation. Apply a central limit theorem
Traditional Approach (GMM) FOC’s Perform a linear approximation. Apply a central limit theorem
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 FOC’s. 3. Equivalent to a quadratic approximation to the objective function, an elliptic paraboloid.
Hall and Horowitz (Econometrica,96) Look at some sampling distributions, various values of N.
Saddlepoint Approximation Intuition -Measures of convexity at -Local minima associated with
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.
Rate of convergence • Convergence is relative. relative absolute • Relative implies • Better tail approximation compared to • asymptotic and bootstrap approximations.