lec19

# lec19 - MIT OpenCourseWare http/ocw.mit.edu 14.384 Time...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 14.384 Time Series Analysis Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Indirect Inference 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe Novemeber 13, 2007 Lecture 19 Simulated MM and Indirect Inference Indirect Inference Suppose we are interested in parameter β ∈ C with true value β . We observe data { x t } T t =1 . We have a model that we can simulate to generate { y j ( β ) } S j =1 . We don’t necessarily believe that our model is the true DGP, but we do think our model can explain some features of the data. These features we want to explain can be written as a function θ ( { x t } ). Let θ ˆ T = θ ( { x t } ) θ ˆ β s = θ ( { y j ( β ) } ) We’ll assume that θ () is an extremum estimator. θ ˆ T = θ ( { x t } ) = arg max Q T ( θ { x t } , θ ) θ ˆ β s = θ ( { y j ( β ) } ) = arg max Q S ( θ { y j ( β ) } , θ ) For example, θ could be simple sample means or moments, or regression coefficients, or more generally β θ ˆ ˆ parameters from some sort of auxillary model. We estimate by matching T to β θ S ˆ ˆ = arg min- ˆ ( β β β θ T θ S ) W ˆ T ( θ T β- θ ˆ S ) This estimator is discussed by Smith (1993) for the case where Q T ( · , θ ) is a pseudo-loglikelihood. Gourierox, Monfort, and Renault (1993) consider a more general setup. We will go through Smith’s setup. Q T ( · , θ ) is a pseudo-loglikelihood: T Q T ( { x t } , θ ) = X log f ( x t , ..., x t- p ; θ ) t = p We call this a pseudo-loglikelihood because we can allow f to be misspecified, ie f need not be the ture density of x t . Assume 1. x t and y t ( β ) ∀ β are stationary and ergodic 2. ∃ a unique β d such that ( x t , ..., x t + p ) = ( y s ( β ) , ..., y s + p ( β )) so θ ( x t ) = θ ( y s ( β )) = θ 3. f is well-behaved (has several continuous, well-bounded derivatives)...
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lec19 - MIT OpenCourseWare http/ocw.mit.edu 14.384 Time...

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