Econometric take home APPS_Part_37

Econometric take home APPS_Part_37 - Chapter 25 Models for...

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Chapter 25 Models for Event Counts and Duration Exercises 1. a. Conditional variance in the ZIP model. The essential ingredients that are needed for this derivation are [* | * 0 , ] 1e x p ( ) i i i Ey y λ >= −− λ x = E i * and 1 *1 * * 1 exp( ) exp( ) 1 exp( ) 1 ii i i Var y y E E V ⎛⎞ λλ λ > = −= ⎜⎟ λ λ λ ⎝⎠ x i i [See, e.g., Winkelmann (2003, pp. 33-34).]. We found the conditional mean in the text to be E[y i |x i ,w i ] = x p i F λ λ = F i E i * To obtain the variance, we will use the variance decomposition, Var[y i |x i ,w i ] = E z [Var[y i |x i ,z]] + Var z [E[y i |x i ,z]]. The expectation of the conditional variance is E z [Var[y i |x i ,z]] = (1 – F i ) × 0 + F i × 1 1 exp( ) exp( ) 1 i i −λ λ − ⎝ ⎠ = F i × E i * × V i * The variance of the conditional mean is (1 – F i ) × 2 0 1 exp( ) i F λ + F i λ 2 x p ( )1e x p i F λ λ = F i (1-F i ) 2 1 exp( ) i i λ λ = F i (1 – F i )E i * 2 . The unconditional variance is thus, F i E i * [V i * + (1 – F i )E i *]. To obtain τ i we divide by the conditional mean, which is F i E i *, so τ i = [V i * + (1 – F i )E i *]. Is this greater than E i *? Not necessarily. The figure below plots F i (1 – F i )E i * 2 for F i = .9 and various values of λ from .1 to about 12. There is a large range over which the function is less than one.
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Econometric take home APPS_Part_37 - Chapter 25 Models for...

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