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oxdk_day3b - Advanced Programming in Quantitative Economics...

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Advanced Programming in Quantitative Economics Introduction, structure, and advanced programming techniques 17 – 21 August 2009, Aarhus, Denmark Charles Bos [email protected] VU University Amsterdam Tinbergen Institute Advanced Programming in Quantitative Economics – p. 1
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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
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Optimization and restrictions Take model y = + ǫ, ǫ ∼ N (0 2 ) Parameter vector θ = ( β ) 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
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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
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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. Advanced Programming in Quantitative Economics – p. 5
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Transform: General solution Distinguish θ = ( β ) and θ = ( β , log σ ) . Steps: Get starting values θ Transform to θ Optimize θ
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