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# lec18 - 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 . GMM 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe Novemeber 8, 2007 Lecture 18 GMM A dynamic stochastic general equilibrium (DSGE) model can be estimated in three ways: 1. GMM 2. MLE 3. Bayesian Today we will talk about GMM GMM We have data z t , parameter θ , and moment condition Eg ( z t , θ ) = 0. This moment condition identifies θ . Many problems can be formulated in this way. Examples • OLS : we have E ( y t | x t ) = x t β . We can write this as a conditional (on x ) moment condition, E ( y t- x t β | x t ) = 0 or an undconditional moment condition, E [ x t ( y t- x t β )] = 0 • IV : y t = x t β + e t E ( e t | z t ) =0 This gives the moment condition E [ z t ( y t- x t β )] = 0 • Euler Equation : This was the application for which Hansen developed GMM. Suppose we have CRRA utility, u ( c ) = c 1- γ- 1 . 1- γ The first order condition from utility maximization gives us the Euler equation, E t " β c t +1 c ¶- γ R t +1 t- 1 # = This moment condition can be used to estimate γ and β . Estimation Take the moment condition and replace it with its sample analog: 1 Eg ( z t , θ ) ≈ T T X g ( z t , θ ) = g T ( θ ) t =1 Estimation 2 θ ˆ Our estimate, will be such that g ˆ T ( θ ) ≈ 0. Let’s suppose θ is k × 1 and g T ( θ ) is n × 1. If n < k , then θ is not identified. If n = k (and dg θ ˆ dθ ( ) is full rank), then we are just identified, and we can find...
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lec18 - MIT OpenCourseWare http/ocw.mit.edu 14.384 Time...

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