Take themodel Y m(X)e m(x) 0 1x 2x2 p xp E h X j e i 0, j 0, ...,p

Take themodel Y Æm(X)Åe m(x) Æ ¯0 ů1x ů2x2 Å¢ ¢ ¢Å¯p xp E

h X j e i

Æ 0, j Æ 0, ...,p g (x) Æ

d dx m(x)

with i.i.d. observations (Yi ,Xi ), i Æ 1, ...,n. The order of the polynomial p is known.

(a) How should we interpret the function m(x) given the projection assumption? How should we interpret g (x)? (Briefly)

(b) Describe an estimator bg (x) of g (x).

(c) Find the asymptotic distribution of p

n

¡

bg (x)¡g (x)

¢

as n!1.

(d) Show how to construct an asymptotic 95% confidence interval for g (x) (for a single x).

(e) Assume p Æ 2. Describe how to estimate g (x) imposing the constraint thatm(x) is concave.

(f ) Assume p Æ 2. Describe how to estimate g (x) imposing the constraint that m(u) is increasing on the region u 2 [xL,xU].