Alpha Geek
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Suppose that log( y ) follows a linear model with a linear form of heteroskedasticity. We write this as
log( y ) 5 b 0 1 x b 1 u u | x ~ Normal(0, h ( x )),
so that, conditional on x , u has a normal distribution with mean (and median) zero but with variance h ( x ) that depends on x . Because Med( u | x ) 5 0, equation (9.48) holds: Med( y | x ) 5 exp( b 0 1 x b ). Further, using an extension of the result from Chapter 6, it can be shown that
E( y | x ) 5 exp[ b 0 1 x b 1 h ( x )/2].
Given that h ( x ) can be any positive function, is it possible to conclude ∂E( y | x )/∂ x j is the same sign as b j ?
Suppose h ( x ) 5 0 1 x (and ignore the problem that linear functions are not necessarily always positive). Show that a particular variable, say x1 , can have a negative effect on Med( y | x ) but a positive effect on E( y | x ).
Consider the case covered in Section 6.4, where h ( x ) 5 s 2 . How would you predict y using an estimate of E( y | x )? How would you predict yusing an estimate of Med( y | x )? Which prediction is always larger?