Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge

Exercise 18

Suppose that log(y) follows a linear model with a linear form of heteroskedasticity. We write this as

(i) Given that h(x) can be any positive function, is it possible to conclude ?E(y|x)/?x_j is the same sign as

(ii) Suppose (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).

(iii) Consider the case covered in Section 6.4, where . How would you predict y using an estimate of E(y|x)? How would you predict y using an estimate of Med(y|x)? Which prediction is always larger?

Step 1 of 3

(i)

Theis given as:

Take the partial derivative ofwith respect to an arbitrary element. The result is given as follows:

Now, since exponential functions are always positive, hence.

Thus, given that is positive, the sign of is same as the sign of.