Estimation of dynamic linear models in short panels with ... - Cemmap

More documents

Recommendations

Info

Hajivassiliou, V. and Ruud, P. (1994). Classical estimation methods for LDV modelsusing simulation, in Engle, R. F. and McFadden, D. L. (eds.) Handbook ofEconometrics 4, 2383-2441. Amsterdam: North-Holland.Heckman, J. J. (1978). Simple statistical models for discrete panel data developed andapplied to test the hypothesis of true state dependence against the hypothesis ofspurious state dependence, Annales de l’INSEE 30, 227-269.Heckman, J. J. (1981a). Statistical models for discrete panel data, in StructuralAnalysis of Discrete Data with Econometric Applications, Manski, C. F. andMcFadden, D. (eds.), 227-269. Cambridge MA: MIT Press.Heckman, J. J. (1981b). The incidental parameters problem and the problem of initialconditions in estimating a discrete time-discrete data stochastic process, inStructural Analysis of Discrete Data with Econometric Applications, Manski, C.F. and McFadden, D. (eds.), 179-195. Cambridge MA: MIT Press.Van Praag, B. and Ferrer-i-Carbonell, A. (2004). Happiness Quantified. A SatisfactionCalculus Approach. Oxford: Oxford University Press.Wooldridge, J.M. (2000), A framework for estimating dynamic, unobserved effectspanel data models with possible feedback to future explanatory variables,Economics Letters 68, 245-250.- 20 -
*Appendix 1: A simulation approximation to L~ψψOur aim is to estimate the probability limit of the partial Hessiann−1∂2L( ψˆ, ~ τ ) / ∂ψˆ∂ψˆ' . Let V be an asymptotically valid approximation to the*2covariance matrix of ψˆ : for example, we might use−1−1V = −n H(ˆ)θ . Since V reflectsthe variability of ψˆ and is O p (n -1 ), it provides a good metric for the approximation ofderivatives. Let K be the Choleski factor such that V = KK′. Note that K = O p (n -1/2 ).Now generate a sequence of independent pseudo-random N(0, I) vectors v 1 … v M andm 1mconstruct z n ( L ( ψˆ+ Kv , ~ τ ) − L *( ψˆ, ~ τ ))= − µ , where µ is a steplength*n22parameter. Note that z m , and any covariance (with respect to the distribution of v m ) ofz m with powers of v m , are O p (n -1/2 µ ). Expand z m in a Taylor series:z2m= µmm mdpvp+ ∑ 2∑ ∑ Dpqvpvq+ ∑p n p q>p 2npnµµDppm 2 m( v ) + ζ−1where d p and D pq are elements of d = K' ( n ∂L*( ψˆ, ~ τ ) / ψˆ2∂ )−12D = n K ( n ∂ L *( ψˆ, ~ τ ) / ∂ψˆ∂ψˆ') K'2are O p (1). We then have the following results:EEEp(A1)n and, and ζ is a remainder term. Note that d and Dm m µm m( z v ) d + E( v ζ )p=p p(A2)nm m mm m m( z v v ) = D + E( v v ζ ),p ≠ qpq2µnpqm m 2m m( z ( v ) ) 3D+ D + E( v ) ζ )ppµ 2⎡⎤⎢2ppqq ⎥ p2nq≠p= ∑⎣q⎦(A3)(A4)where the expectations are taken with respect to v m .If necessary, the process for generating the variates v m should be appropriatelymtruncated to ensure that L *(ψˆ+ µ Kv , ~nτ2) always exists and that terms of the formv3 *( v v ( ∂ L ∂ψ∂ψ∂ψ| +)E exist for any function ψ + (v) bounded byij/ p q r ψ = ψ ( v )( ψ ˆ , ψˆ+ µ Kv). In practice, this requires ensuring that simulation ofmˆ avoidsψ + µKvregions where the Γ r are non-ordered. A sufficiently small value for µ (for example- 21 -
Page 1 and 2: ESTIMATION OF DYNAMIC LINEAR MODELS
Page 3 and 4: 1 IntroductionIn discrete data mode
Page 6 and 7: imperfect observation: there is a l
Page 8 and 9: and therefore Pr(y it =1, y it-1 =1
Page 10 and 11: y*itt= α y*i0+t−1∑s=0s 1−αs
Page 18: then done sequentially, starting wi
Page 21: 7 ConclusionsWe have considered an
Page 25: Appendix 2: SD model estimatesTable

Estimation of dynamic linear models in short panels with ... - Cemmap

Create successful ePaper yourself

Delete template?

Save as template?