V v v v v v v v v θ θ θ θ θ θ θ θ θ θ θ θ

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v v ... v ... v v ... v ... ... ... ... ... ... ... ... ... ... ... ... v v ... v ... ... ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ
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Part 11: Asymptotic Distribution  Theory Application: CES Function ™  38/42
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Part 11: Asymptotic Distribution  Theory Application: CES Function ™  39/42 1 2 3 4 1 3 2 2 2 1 2 3 4 2 3 2 3 1 2 3 4 3 4 2 3 2 4 2 2 2 3 3 2 2 3 1 2 3 4 exp( ) 0 0 0 0 0 ( ) ( ) 0 1 1 0 ( ) 0 ∂γ ∂γ ∂γ ∂γ ∂β ∂β ∂β ∂β β β ∂δ ∂δ ∂δ ∂δ ∂β ∂β ∂β ∂β β + β β + β = = ∂ν ∂ν ∂ν ∂ν ∂β ∂β ∂β ∂β -β β β + β -β β β β β β β β ∂ρ ∂ρ ∂ρ ∂ρ ∂β ∂β ∂β ∂β G
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Part 11: Asymptotic Distribution  Theory Application: CES Function Using Spanish Dairy Farm Data Create ; x1=1 ; x2=logcows ; x3=logfeed ; x4=-.5*(logcows-logfeed)^2$ Regress ; lhs=logmilk;rh1=x1,x2,x3,x4$ Calc ; b1=b(1);b2=b(2);b3=b(3);b4=b(4) $ Calc ; gamma=exp(b1) ; delta=b2/(b2+b3) ; nu=b2+b3 ; rho=b4*(b2+b3)/(b2*b3)$ Calc ;g11=exp(b1) ;g12=0 ;g13=0 ;g14=0 ;g21=0 ;g22=b3/(b2+b3)^2 ;g23=-b2/(b2+b3)^2 ;g24=0 ;g31=0 ;g32=1 ;g33=1 ;g34=0 ;g41=0 ;g42=-b3*b4/(b2^2*b3) ;g43=-b2*b4/(b2*b3^2) ;g44=(b2+b3)/(b2*b3)$ Matrix ; g=[g11,g12,g13,g14/g21,g22,g23,g24/g31,g32,g33,g34/g41,g42,g43,g44]$ Matrix ; VDelta=G*VARB*G' $ Matrix ; theta=[gamma/delta/nu/rho] ; Stat(theta,vdelta)$ ™  40/42
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Part 11: Asymptotic Distribution  Theory Estimated CES Function --------+-------------------------------------------------------------------- | Standard Prob. 95% Confidence Matrix| Coefficient Error z |z|>Z* Interval --------+-------------------------------------------------------------------- THETA_1| 105981*** 475.2458 223.00 .0000 105049 106912 THETA_2| .56819*** .01286 44.19 .0000 .54299 .59340 THETA_3| 1.06781*** .00864 123.54 .0000 1.05087 1.08475 THETA_4| -.31956** .12857 -2.49 .0129 -.57155 -.06758 --------+-------------------------------------------------------------------- ™  41/42
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Part 11: Asymptotic Distribution  Theory Asymptotics for Least Squares Looking Ahead… Assumptions: Convergence of XX / n (doesn’t require nonstochastic or stochastic X ). Convergence of X’/ n to 0 . Sufficient for consistency. Assumptions: Convergence of (1/ n ) X’ to a normal vector, gives asymptotic normality What about asymptotic efficiency?   42/42
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