ABSTRACT
A l n T J(l)(Xit) f3Hs = --T LLYit ·""A-(6)
n icoo] t=l f(Xit)
where T
/(x) = (nTh)·l ti:K (~~:x) t=[ {-c]
is a kemel density estimator and fOl is its first derivative. The details on the asymptotic properties of the above estimators and the choice
of kernel and h can be found in Hardie (1990) and Pagan and lJllah (1996). Specifically we note that, under the smoothness conditions on the kernel K, E\nit \2+11 < oo for some 8 > 0, and assuming h ---7 0, nhq+Z.< ---7 oo and nhq+~ ---7 0 as n ---7 oo:
where .s represents the sth derivative and a~(x) = V(u I x); for s = 0, ,n<0l(x) = m(x), and for s = l, ,nOl(x) = {J(x). Jn practice a~(x) can be replaced by its consistent estimator a~(x) = (t~rK(x)tnr)- 1 t~rK(x)~, where~ is the vector of the squared nonparametric residuals llJ1 = (Yit - m(xi1))
2 • We can then use (7) to calculate the confidence intervals. The conditions for the asymptotic normality of fJR and !Jns are very similar to the conditions of the asymptotic normality of the pointwise {J(x). It follows from Rilstone (1991) and Hardie and Stoker (1989) that, as n-+ 00,
(8)
where
] + V(fl(x)) (9)
In practice, for the confidence intervals and the hypothesis testing, an estimator of E can be obtained by replacing a~(x), ;<1l(x), and fl(x) by their kernel estimators a~(x), j(ll(x), and {J(x) and then taking the sample averages. Then for q = 1,
The second component of E is essentially the variance of the limiting distribution of the sample average Ln Lr f3(xi1)/nT, and it will be zero if the true m(x) is linear in x. We also observe that the asymptotic variance does not depend on the window width h, kernel K, and number of regressors q. This is quite different compared to the pointwise asymptotic variance result in (7)0
Now we turn to the small sample behavior of the bias and mean-square error (MSE) of the above estimators. First, considering m(x) and taking its expectation conditional on x;1 we get
E(m(x) I Xir) = (t'K(x)t)- 1t'K(x)m* where t = tnr and m* = [m(xu), o. o, m(xnr)]. Further
(10)
(11)
where Q(x) = K(x)'E1K(x) and 'E1 is a diagonal matrix with the diagonal elements a~(X;t) = E(u~ I x;t). If E1 =a~!, Q(x) = a~K 2 (x)o In practice a~(x) ean be consistently estimated by a~(x)o If we expand m* by the Taylor series and consider n to be large then it can be shown that (Ruppert and Wand 1994, Pagan and Ullah 1996), up to 0(h2 ),
(12)
and, up to O(ljnhq),
and
h 2
[ ( ((f(ll(x)) 2
ES(x)- {3(x) = ~ tLz mHl(x)- 2 {3(x) - : .... 2 .f(x)
- - -m (x)---f(2l(x)) (3J j(ll(x) J] j(x) f(x)
Essentially, (J 4) is the derivative of(] 2) with respect to x.
