Econometric take home APPS_Part_36

Econometric take home APPS_Part_36 - 4. Using Theorem 24.5,...

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4. Using Theorem 24.5, we have 1 - Φ ( α z ) = 14/35 = .4, α z = Φ -1 (.6) = .253, λ ( α z ) = .9659, δ ( α z ) = .6886. The two moment equations are based on the mean and variance of y in the observed data, 5.9746 and 9.869, respectively. The equations would be 5.9746 = μ + σ (.7)(.9659) and 9.869 = σ 2 (1 - .7 2 (.6886)). The joint solution is = 3.3651, ˆ μ ˆ σ = 3.8594. 5. The conditional mean function is E[y| x ] = Φ ( β′ x i / σ i ) β′ x i + σ i Φ ( β′ x i / σ i ) using the equation before (24- 12). Suppose that σ i = σ exp( α′ x i ) for the same vector x i . (We’ll relax that assumption shortly.) Now, differentiate this expression with respect to x . We differentiate the two parts, first with respect to β′ x then with respect to σ i . () [|] 1 1 '' ' ' ' '1 ' ' ' ' ' i ii i i iii i i i i i Ey i i i ⎡⎤ ⎛⎞ + + ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ ++ + x xx x x x x x x x x φφ σσ σ σσσ σφ ββ β β β β αα β β β β 1' i i ⎛ ⎞ ⎜ ⎟ x β α After collecting the terms, we obtain E[y i | x i ]/ x i = Φ (a i ) β + σ i φ (a i ) α where a i = β′ x i / σ i . Thus, the marginal effect has two parts. one for β and one for α . Now, if a variable appears in σ i but not in x i , then only the second term appears while if a variable appears only in x i and not in σ i , then only the first term appears in the marginal effect. 6. The transformed log likelihood function is logL = Σ y > 0 (-1/2)[log2 π - log θ 2 + ( θ y - x ′γ ) 2 ] + Σ y=0 log[1- Φ ( x ′γ )] It will be convenient to define a i = x
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This note was uploaded on 11/13/2011 for the course ECE 4105 taught by Professor Dr.fang during the Spring '10 term at University of Florida.

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Econometric take home APPS_Part_36 - 4. Using Theorem 24.5,...

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