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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 i i i i i i i i i i i i i i i i i i i i E y i i i i i i i i i ⎞ ⎛ = Φ + + ⎟ ⎜ ⎠ ⎝ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ + + + ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ ⎠ ⎝ x x x x x x x x x x x x x φ φ σ σ σ σ σ σ φ φ σ σ σ σ σ σ φ σ σ σ σ β β β β β β β β α α β β β β β β 1 ' i i i i ⎤ ⎛ ⎞ ⎛ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ ⎠ ⎝ x σ σ σ 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
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