oxdk_day3b - A d v a n c e d P r o g r a m m i n g i n Q u...

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Unformatted text preview: A d v a n c e d P r o g r a m m i n g i n Q u a n t i t a t i v e E c o n o m i c s Introduction, structure, and advanced programming techniques 17 – 21 August 2009, Aarhus, Denmark Charles Bos cbos@feweb.vu.nl VU University Amsterdam Tinbergen Institute Advanced Programming in Quantitative Economics – p. 1 Day 3 - Afternoon 13.00L Optimization II ◦ Restrictions and transformations ◦ Delta method: Covariance estimation ◦ (fixing parameters) 14.30P Implementing covariance estimation ◦ Duration model restricting α and/or β ◦ Covariance of parameters 16.00 End 18.00 Course dinner at ‘Sct. Oluf’ Advanced Programming in Quantitative Economics – p. 2 Optimization and restrictions Take model y = Xβ + ǫ, ǫ ∼ N (0 ,σ 2 ) Parameter vector θ = ( β ′ ,σ ) ′ is clearly restricted, as σ ∈ [0 , ∞ ) or σ 2 ∈ [0 , ∞ ) • Newton-based method (BFGS) doesn’t know about ranges • Alternative optimization (SQP) tends to be slower/worse convergence Hence: First tricks for MaxBFGS . Advanced Programming in Quantitative Economics – p. 3 Transforming parameters Variance parameter positive? Solutions: 1. Use σ 2 as parameter, have AvgLnLiklRegr return 0 when negative σ 2 is found 2. Use σ ≡ | θ k +1 | as parameter, ie forget the sign altogether (doesn’t matter for optimisation, interpret negative σ in outcome as positive value) 3. Transform, optimise θ ∗ k +1 = log σ ∈ ( −∞ , ∞ ) , no trouble for optimisation Last option most common, most robust, neatest. Advanced Programming in Quantitative Economics – p. 4 Transform: Common transformations Constraint θ ∗ θ [0 , ∞ ) log( θ ) exp( θ ∗ ) [0 , 1] log parenleftBig θ 1 − θ parenrightBig exp( θ * ) 1+exp( θ * ) Of course, to get a range of [ L,U ] , use a rescaled [0 , 1] transformation....
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This note was uploaded on 05/28/2010 for the course ECONOMIC FM407 taught by Professor Kimcj during the Spring '04 term at 카이스트, 한국과학기술원.

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oxdk_day3b - A d v a n c e d P r o g r a m m i n g i n Q u...

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