Θ θ θ θ θ θ θ θ θ θ 1 2 2 2 1 1 1 2 n 1 1 1

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÷ ÷ ÷ θ θ θ θ θ θ θ θ θ θ 1 2 2 2 1 1 1 2 n 1 1 1 2 2 2 1 2 2 2 1 2 1 2 1 ˆ Asy.Var[ ] ( ' ' ') ˆ Asy.Var n[ ] ˆ Asy.Var n[ ] | logL logL 1 E n logL logL 1 E n V V CV C RV C CVR V V V C = R =
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Part 19: MLE Applications M&T Computations ™  23/29 - - = = - - = = = Σ - Σ = Σ - Σ H g g H g g θ θ θ 1 1 1 N N 1 1 1 i 1 i1 i 1 i1 i1 n n 2 1 1 1 N N 1 1 2 i 1 i2 i 1 i2 i3 n n ˆ First equation: =MLE, ˆ ˆ ˆ ˆ or ˆ ˆ Second equation: =MLE| ˆ ˆ ˆ ˆ or V V = =  Σ ÷ ÷ ∂θ   Σ ÷ ÷  C R θ θ θ θ θ θ θ θ θ θ N 2 2 2 2 1 1 2 2 2 2 1 1 i 1 2 1 N 2 2 2 2 1 1 1 1 1 1 i 1 2 1 ˆ ˆ ˆ ˆ lnf (y | , , , ) lnf (y | , , , ) 1 = ˆ ˆ n ˆ ˆ ˆ lnf (y | , , , ) lnf (y | , ) 1 = ˆ ˆ n x x x x x x x
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Part 19: MLE Applications Example Equation 1: Number of Kids – Poisson Regression n p(yi1| x i1, β )=exp(-λi)λiyi1/yi1! n λi = exp( x i1’ β ) n g i1 = x i1(yi1 – λi) n V 1 = [(1/n)Σ(-λi) x i1 x i1’]-1 ™  24/29
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Part 19: MLE Applications Example - Continued Equation 2: Labor Force Participation – Logit p(yi2| x i2, δ ,α, x i1, β )=exp(di2)/[1+exp(di2)]=Pi2 di2 = (2yi2-1)[ δx i2 + αλi] λi = exp( βx i1) Let z i2 = ( x i2, λi), θ 2 = ( δ , α) di2 = (2yi2-1)[ θ 2 z i2] g i2 = (yi2-Pi2) z i2 V 2 = [(1/n)Σ{-Pi2(1-Pi2)} z i2 z i2’]-1 ™  25/29
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Part 19: MLE Applications Murphy and Topel Correction ™  26/29 [ ] [ ] [ ] [ ] N 1 i 1 i2 i2 i2 i2 i2 i i1 N N 1 i 1 i2 i2 i2 i1 i i1 N (y P ) (y P ) (y P ) (y ) = = = Σ - - αλ = Σ - - λ C z x R z x
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Part 19: MLE Applications Two Step Estimation of LFP Model ? Data transformations. Number of kids, scale income variables Create ; Kids = kl6 + k618 ; income = faminc/10000 ; Wifeinc = ww*whrs/1000 $ ? Equation 1, number of kids. Standard Poisson fertility model. ? Fit equation, collect parameters BETA and covariance matrix V1 ? Then compute fitted values and derivatives Namelist ; X1 = one,wa,we,income,wifeinc$ Poisson ; Lhs = kids ; Rhs = X1 $ Matrix ; Beta = b ; V1 = N*VARB $ Create ; Lambda = Exp(X1'Beta); gi1 = Kids - Lambda $ ? Set up logit labor force participation model ? Compute probit model and collect results. Delta=Coefficients on X2 ? Alpha = coefficient on fitted number of kids. Namelist ; X2 = one,wa,we,ha,he,income ; Z2 = X2,Lambda $ Logit ; Lhs = lfp ; Rhs = Z2 $ Calc ; alpha = b(kreg) ; K2 = Col(X2) $ Matrix ; delta=b(1:K2) ; Theta2 = b ; V2 = N*VARB $ ? Poisson derivative of with respect to beta is (kidsi - lambda)´X1 Create ; di = delta'X2 + alpha*Lambda ; pi2= exp(di)/(1+exp(di)) ; gi2 = LFP - Pi2 ? These are the terms that are used to compute R and C. ; ci = gi2*gi2*alpha*lambda ; ri = gi2*gi1$ MATRIX ; C = 1/n*Z2'[ci]X1 ; R = 1/n*Z2'[ri]X1 ; A = C*V1*C' - R*V1*C' - C*V1*R' ; V2S = V2+V2*A*V2 ; V2s = 1/N*V2S $ ? Compute matrix products and report results Matrix ; Stat(Theta2,V2s,Z2)$ ™  27/29
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Part 19: MLE Applications Estimated Equation 1: E[Kids] +---------------------------------------------+ | Poisson Regression | | Dependent variable KIDS | | Number of observations 753 | | Log likelihood function -1123.627 | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Constant 3.34216852 .24375192 13.711 .0000 WA -.06334700 .00401543 -15.776 .0000 42.5378486 WE -.02572915 .01449538 -1.775 .0759 12.2868526 INCOME .06024922 .02432043 2.477 .0132 2.30805950 WIFEINC -.04922310 .00856067 -5.750 .0000 2.95163126 ™  28/29
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Part 19: MLE Applications Two Step Estimator +---------------------------------------------+ | Multinomial Logit Model | | Dependent variable LFP |
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