asymptotic properties of the nonparametric part in partial linear

ASYMPTOTIC PROPERTIES OF THE NONPARAMETRIC PART IN PARTIAL LINEAR HETEROSCEDASTIC REGRESSION MODELS Hua Liang, Wolfgang Hardle and Axel Werwatz  Abstract

This paper considers estimation of the unknown function g () in the partial linear regression model Yi = XiT + g (Ti) + "i with heteroscedastic errors. We rst construct a class of estimates gn of g and prove that, under appropriate conditions, gn is weak, mean square error consistent. Rates of convergence and asymptotic normality for the estimator gn are also established.

Key Words and Phrases:Key words and phrases: Asymptotic normality, consistency,

heteroscedasticity, kernel estimation, rates of convergence, partial linear model, semiparametric models.

1 INTRODUCTION Semiparametric models combine the exibility of nonparametric modeling with structural parametric components. One such model that has received a lot of attention in the literature is the semiparametric partial linear regression model

Yi = XiT + g(Ti) + "i; i = 1; : : :; n

(1)

Hua Liang is Associate Professor of Statistics, Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, China. Wolfgang Hardle is Professor of Econometrics, Axel Werwatz is Dr. of Economics. Both of latter are at the Institut fur Statistik und O konometrie, Humboldt-Universitat zu Berlin, D-10178 Berlin, Germany. This research was supported by Sonderforschungsbereich 373 \Quanti kation und Simulation O konomischer Prozesse" . The rst author was supported by Alexander von Humboldt Foundation. The authors would like to thank Mr. Knut Bartels for his valuable comments which greatly improved the presentation of this paper. 

1

where X and T are (possibly) multidimensional regressors, a vector of unknown parameters, g() an unknown smooth function and " an error term with mean zero conditional on X and T: Well-known applications in the econometrics literature that can be put in the form of (1) are the human capital earnings function (Willis (1986)) and the wage curve (Blanch ower and Oswald (1994)). In both cases, log-earnings of an individual are related to personal characteristics (sex, marital status) and measures of a person's human capital like schooling and labor market experience. Economic theory suggests a non-linear relationship between log-earnings and labor market experience, which therefore plays the role of the variable T in (1). The wage curve is obtained by including the local unemployment rate as an additional regressor, with a possibly non-linear in uence. Rendtel and Schwarze (1995), for instance, estimate g() as a function of the local unemployment rate using smoothing-splines and nd a U-shaped relationship. Under various assumptions, several authors have considered estimation of in (1) at a parametric rate. Chen (1988), Heckman (1986), Robinson (1988), Schick (1996) and Speckp man (1988) constructed n?consistent estimators of . Cuzick (1992a) studied ecient estimation of when the error density is known. Ecient estimation when the error distribution is of an unknown form is treated in Cuzick (1992b) and Schick (1993). In this paper, we will instead focus on deriving the asymptotic properties of an estimator of the unknown function g(). We consider its consistency, weak convergence rate and asymptotic normality. We will derive these results for a speci c version of (1) with nonstochastic regressors, heteroscedastic errors and T univariate. The remainder of this paper is organized as follows. In the following section we will describe methods for estimating and g(). We prove consistency and asymptotic normality of the estimator of g() in sections 3 and 4. We illustrate the usefulness of the estimator and the relevance of the asymptotic distribution results for applied work by a small-scale Monte Carlo study and an empirical illustration in the nal section of the paper.

2 THE ESTIMATOR Speci cally, we consider estimation of g() (and ) in the following partial linear, semiparametric regression model:

Yi = XiT + g(Ti) + "i; i = 1; : : :; n 2

(2)

where is an unknown p dimensional parameter vector, g() an unknown, smooth function from [0; 1] to IR1, (X1; T1), (X2; T2) : : : are known, nonrandom design points and "1; : : : ; "n are independent mean zero random errors with nonconstant variance. We allow the variance of " to depend on X and T in an arbitrary way. Previous work in a heteroscedastic setting has focused on the nonparametric regression model (i.e. = 0). Muller et al. (1987) proposed an estimate of the variance function by using kernel smoother, and then proved that the estimate is uniformly consistent. Hall and Carroll (1989) considered consistency of estimates of g(): Eubank et al. (1990) proposed trigonometric series type estimators g of g. They investigated asymptotic approximations of the integrated mean squared error and the partial integrated mean squared error of g. The heteroscedastic version of (1) with 6= 0 has been considered in Schick (1996) but he considers weighted least squares estimation of . We focus on nonparametric estimation of g() as a function of T . Suppose we knew : Then we may estimate g() by nonparametric regression of Yi ? XiT (the variation in Yi not accounted for by the linear component XiT ) on Ti: In the literature one can nd various methods for estimating g() nonparametrically, e.g., kernel, nearest neighbor, orthogonal series, piecewise polynomial and smoothing splines. See Hardle (1990) for an extensive discussion of their statistical properties. All these estimators may be written as weighted local averages of the observed values of the dependent variable with the weights depending on the values of the explanatory variables. In our case, we can write (still assuming that is known):

g^(t) =

n

X

i=1

!ni(t)(Yi ? XiT );

(3)

where !ni (t) = !ni (t; T1; T2; : : :; Tn) are weight functions that depend on the observations T1; : : : ; Tn. For instance, a Gasser-Muller-type kernel estimator takes s !ni (t) = h1 K t h? s ds 1  i  n s ?1 n for s0 = 0; sn = 1; si = 21 (T(i) + T(i+1)): Here T(1); : : : ; T(n) are sample order statistics, K () is the kernel function and h denotes the bandwidth. See Remark 5 below for details. Given the estimator g^(t) as de ned in (3) we may estimate by the least squares regression of Z

i





i

Yi = XiT + g^(Ti) + i 3

Yi ?

n

X

j =1

n

!nj (Ti)Yj = Xi ? Yi = e

n

X

j =1 T Xi +  i

T

o

!nj (Ti)Xj + i (4)

f

That is, we estimate by the generalized least squares estimator

LS = (X T X )?1X T Y f

f

f

e

(5)

where X = (X1; : : : ; Xn )T and Y = (Y1; : : :; Yn )T are the presmoothed design and response variables. In the nal step we obtain the feasible estimator of g() by substituting LS for the unknown in (3): f

f

f

e

gn (t) =

e

n

e

X

i=1

!ni (t)(Yi ? XiT LS );

(6)

Further motivation for the estimators de ned in (5) and (6) is given in Speckman (1988) and Gao, et al. (1995). Note though that LS is not an ecient estimator in the sense of asymptotic normality. In the following section we state and prove the weak, mean square error consistency and give the rates of convergence of gn (t) under various assumptions.

3 CONSISTENCY RESULTS All technical preliminaries needed in the proofs of the following results are collected in Appendix 6 as lemmas. For convenience and simplicity, we always let C denote some positive constant not depending on n: We will use the following assumptions. Assumption 1. There exist continuous functions hj () de ned on [0; 1] such that each element of Xi satis es

xij = hj (Ti) + uij 1  i  n; 1  j  p

(7)

where uij is a sequence of real numbers which satisfy limn!1 n1 Pni=1 ui = 0 and n X 1 uiuTi = B nlim !1

(8)

is a positive de nite matrix, and k X 1 u lim sup 1max j m < 1 kn

(9)

n i=1

n!1

an

i=1

i

4

for m = 1; : : :; p

holds for all permutations (j1 ; : : : ; jn) of (1; 2; : : : ; n) where ui = (ui1 ; : : :; uip )T ; an = n1=2 log n:

Assumption 2. n ! (t) ! 1 i=1 ni

(a)

P

(b)

P

(c)

P

as n ! 1;

n j! (t)j  C i=1 ni

for all t and some constant C ;

n j! (t)jI (jt ? T j > a) ! 0 i i=1 ni

as n ! 1 for all a > 0;

(d) supin j!ni (t)j = Of(log n)?1g: Denote nt =

o n ! 2 (t) ?1 : i=1 ni

n P

Assumption 3.

(a) supn (nt) sup1in j!ni (t)j < 1; and nt = o(n);

(b) pnt sup1in j!ni (t)j = O(n?=2) for some 1 >  > 0;

(c)

P

n ! 2 (t)E"2 =  2=n + o(1=n ) t 0 t i i=1 ni

for some 02 > 0:

Remark 1. Assumption 1 is a common requirement for proving consistency of in the partial linear model (1). In fact, (7) of Assumption 1 is parallel to the case

hj (Ti) = E (xij jTi) and uij = xij ? E (xij jTi) when (Xi ; Ti) are random variables. (8) is similar to the result of the strong law of large numbers for random errors. (9) is similar to law of the iterated logarithm. More detailed discussions may be found in Speckman (1988) and Gao et al. (1995). Theorem 1. Under Assumptions 1 and 2 E fgn (t)g ! g(t) as n ! 1 at every continuity point of the function g: Proof. Decompose the dierence gn (t) ? g(t) as follows by direct calculation.

gn (t) ? g(t) =

n

X

j =1

!nj (t)fg(Ti) + "i ? XiT (X T X )?1X T g(T ) ? XjT (X T X )?1X T "g ? g(t) f

f

where g(T ) = fg(T1); : : :; g(Tn)gT and g(Ti) = g(Ti) ? It follows that e

e

E fgn (t)g ? g(t) =

e

e

n

nX

i=1

o

!ni (t)g(Ti) ? g(t) ? 5

f

f e

n

X

i=1

f

n ! (T )g (T ) j j =1 nj i

P

f

e

and " just like X: e

!ni (t)XiT (X T X )?1X T g(T ) f

f

f e

f

(10)

The rst term tends to zero by Lemma A.1 (i). By lemmas A.2 and A.1 (i) and CauchySchwarz inequality, we know that every element of (X T X )?1X T g(T ) is o(n?1=2), i.e., n

f e

f

f

o

(X T X )?1X T g (T ) j = o(n?1=2) for j = 1; : : : ; p f

f

f e

It suces to show that every element of n

X

i=1

!ni (t)xij =

P

n

X

i=1

n ! (t)X i i=1 ni

(11)

is O(n1=2). Observe that

!ni (t)fhj (Ti) + uij g

Since hj () is continuous, ni=1 !ni (t)hj (Ti) converges to h(t) on the continuity point of h(t) by the same proof as one for Lemma A.1 (i). Moreover, by Abel's inequality and Assumption 2 (d), P

n

X

i=1

Thus



!ni (t)uij  sup j!ni(t)j 1max kn 1in

n

X

i=1

k

X



uj m = O(n1=2) i

i=1

!ni (t)xij = O(n1=2)

(12)

and we complete the proof of Theorem 1. # Theorem 1 shows that gn (t) is an asymptotically unbiased estimator of g(t) at every continuity point of g(t). The next result, Theorem 2, will demonstrate that gn(t) is also mean square-error consistent. Theorem 2 Assume the conditions of Theorem 1 hold except Assumption 2 (d) which is replaced by supin j!ni (t)j = of(log n)?1 g: Then E fgn(t) ? g (t)g2 ! 0 as n ! 1: Proof. It follows from the Cr ?inequality with r = 2 that n

E fgn (t) ? g(t)g2  CE

X

CE

X

+

i=1 n

i=1

2



!ni (t)g(Ti) ? g(t) + CE

2

n

X

i=1

!ni (t)"i

!ni (t)(X T X )?1X T g(T ) + CE f

f

f e

n

X

i=1

2



2

e

" !ni(t)XiT (X T X )?1X T(13) f

f

f

In the proof of Theorem 1 we obtained that the rst and third terms of (13) converge to zero as n tends to in nity. The second can be shown to be order o(1) by direct calculation. We shall now prove the fourth term also converges to zero. Denote (X T X )?1X T = (ji)pn : f

E

n

X

i=1



!ni (t)XiT (X T X )?1 X T " f

f

f

2

= E = 6

n

nX

!ni (t)

i=1 p n X n  X X

l=1 i=1 k=1

p n

X X

k=1 l=1

xik kl"l 2



!ni (t)xik kl l2

2

o

f

f

It follows from the arguments for (12) that this equals to O(n) nl=1 kl2 . Since nl=1 kl2 and the elements of the k?th row of (X T X )?1 have the same order O(n?1 ). It follows that P

f

E

n

X

i=1

P

f

2



!ni (t)XiT (X T X )?1X T " = o(1): f

f

f

(14)

Furthermore, we can easily show that

E

n

X

i=1

!ni (t)XiT (X T X )?1 f

f

n

X

k=1

Xk f

n

nX

l=1

!nl (Tk )"k

o

2

= o(1):

(15)

Combining (14) and (15) ensures that the fourth term of (13) is o(1), and thus completes the proof of Theorem 2. # The following result gives the weak convergence rate of gn under stronger assumptions on f!ni (t)g than those given in Assumption 2. Here we list these assumptions. Assumption 2'. The weight functions !ni(t) satisfy: (a) supt j (b) supt

n ! (t) ? 1j = O (n?1=3 log n); P i=1 ni

P

P

n j! (t)jI (jt ? T j > c ) = O(d ); i n n i=1 ni

where dn and cn are n?1=3 log n;

(c) supt max1in j!ni (t)j = OP (n?2=3):

Theorem 3. Assume g() and hj () are Lipschitz continuous of order 1 and Assumptions 1 and 2' hold. Then

gn(t) ? g(t) = OP (n?1=3 log n):

Proof. By Lemma A.1 (ii), n

X

i=1

!ni (t)g(Ti) ? g(t) = OP (n?1=3 log n):

Using Assumption 2' (c) and Chebyshev's inequality we have n

X

i=1

!ni (t)"i = OP (n?1=3 log n):

The similar arguments as that for (11) and (12) yield n

X

i=1

!ni (t)XiT (X T X )?1X T g(T ) = OP (n?1=3 log n): f

f

f e

7

Finally, observe that ni=1 !ni (t)uij = O(1) and then Thus, by the arguments for (14) and (15),

n ! (t)x ij i=1 ni

P

E

n

X

i=1

P

= O(1) for j = 1; : : : ; p:

2

e

!ni(t)XiT (X T X )?1X T " = O(n?1 ): f

f

f

(16)

which entails n

X

i=1

!ni (t)XiT (X T X )?1 X T " = OP (n?1=3 log n): f

f

f

e

#

This completes the proof of Theorem 3. Remark 2. We can conclude from the above arguments that 2=3 log?2 n)E fg (t) ? g (t)g2 < 1: lim sup ( n n n!1

Theorem 4 gives the asymptotic variance of gn(t): Theorem 4. Under Assumptions 1, 2' and 3, ntV arfgn (t)g ! 02 as n ! 1:

Proof.

ntV arfgn (t)g = ntE

n

nX

i=1

?2ntE

2

o

!ni (t)"i + ntE n

nX

i=1

o

!ni (t)"i


n

nX

n

nX

i=1

i=1

o

!ni (t)XiT (X T X )?1X T " f

f

f

e

2

o

!ni (t)XiT (X T X )?1X T " f

f

f

e

The rst term converges to 02: The second term tends to zero by (16), and then the third term also tends to zero by the Cauchy-Schwarz inequality. #

4 ASYMPTOTIC NORMALITY In the nonparametric regression model, Liang (1995) proved asymptotic normality for independent "i's under the mild conditions. In this section, we shall consider the asymptotic normality of gn under the Assumptions 1, 2' and 3. Theorem 5. Assume that "1; "2; : : :; "n are independent random variables with E"i = 0 and inf i i2 > c > 0 for some c : There exists a function G(u) satisfying Z

1 0

uG(u)du < 1

(17)

such that

P (j"ij > u)  G(u); for i = 1; : : :; n and large enough u: 8

(18)

If

max1in !ni2 (t) ! 0 n ! 2 (t) i=1 ni

as n ! 1:

P

(19)

Then

gn (t) ? Egn (t) ?!L N (0; 1) as n ! 1: V arfgn (t)g q

Remark 3. The condition 01 uG(u)du < 1 is to guarantee supi i2 < 1: The proof of Theorem 5. At rst from the proof of Theorem 4, we obtain that R

V arfgn (t)g =

n

X

i=1

n

nX

!ni2 (t)i2 + o

i=1

!ni2 (t)i2

o

Furthermore

gn (t) ? Egn (t) ?

n

X

i=1

!ni (t)"i =

n

X

i=1

!ni (t)XiT (X T X )?1X T " = OP (n?1=2) f

f

f

e

which yields P

n ! (t)X T (X fT f ?1 fT e X ) X " = O (n?1=2n1=2) = o (1) i=1 niq i P P t V arfgn (t)g

It follows that gn (t) ? Egn(t) = V arfgn(t)g q

n ! (t)" i i=1 ni n ! 2 (t) 2 i i=1 ni

P

q P

def

+ oP (1) =

n

X

i=1

ani"i + oP (1);

where ani = p ! !(t2) (t) : Let ani = anii, obviously supi i < 1 due to 01 vH (v)dv < 1: =1 The proof of the theorem immediately follows from the conditions (17){(19) and Lemma A.4. # Remark 4. If "1; : : :; "n are independent identically distributed, then E j"1j2 < 1 and the condition (19) of Theorem 5 can yield the result of Theorem 5. Assumption 2". The weight functions !ni (t) satisfy: Pn i

R

ni

ni

i

(a) Pni=1 !ni (t) ? 1 = o(n?t 1=2 ); (b) Pni=1 j!ni (t)jI (jt ? Ti j > c0n ) = o(d0n ); where c0n and d0n are o(n?t 1=2 ):

9

Theorem 6. Suppose that g(t) is Lipschitz continuous of order 1. Assume the conditions of Theorem 5 hold with the previous Assumption 2" replacing Assumption 2'. Then gn (t) ? g(t) ?!L N (0; 1) as n ! 1: q V arfgn(t)g

Proof. In fact, recall the conclusion of Theorem 5, it suces to show that Egn (t) ? g(t) = pn fEg (t) ? g(t)g + o(1) = o(1): t n V arfgn (t)g q

For c0n = o(n?t 1=2). Note that

jEgn(t) ? g(t)j 

n

X

i=1

j!ni (t)fg(Ti) ? g(t)gjfI (jTi ? tj > c0n ) + I (jTi ? tj  c0n)g +jg(t)j

n

X

i=1

 (g; c0n)
B + 2C

!ni (t) ? 1 n

X

i=1



j!ni (t)jI (jTi ? tj > c0n ) + C

n

X

i=1



!ni (t) ? 1 ;

where C = supt2[0;1] jg(t)j and (g; c0n) = supjt?t0jc0 jg(t) ? g(t0)j: Assumption 2" and the previous arguments yield the conclusion of Theorem 6. # Remark 5. In this remark, we shall give concrete weight functions f!ni (t); i = 1; : : : ; ng which satisfy the assumptions given in the former context, in order to explain the reasonability of the results established in previous sections carefully. Assume s !ni (t) = h1 s K t h? s ds 1  i  n (20) n ?1 n where s0 = 0; sn = 1 and si = 12 (T(i) + T(i+1)); 1  i  n ? 1: hn is a sequence of bandwidth parameters which tends to zero as n ! 1 and K () is a kernel function, which is supported to have compact support and to satisfy n

Z

i





i

Z

supp(K ) = [?1; 1]; sup jK (x)j  C < 1; K (u)du = 1 and K (u) = K (?u) Obviously Assumptions 2(a), (b) and (d) are satis ed for the weight functions given in (20). If Z

jujah?n 1

K (u)du = o(1) 10

Then Assumption 2 (c) hold also. In fact n s !ni(t)I (jTi ? tj > a) = h1 K t h? s dsI (jTi ? tj > a) n i=1 s ?1 n i=1  h1 jT ?sja?max jT ?T j K t h? s ds n ?1 n  h1 jujh?1 (a?max jT ?T j) K (u)du = o(1) ?1 n n

X

X

Z

i





i

Z

Z



i

i

n



i

i

i

Now let us take hn = Cn?1=3 for some C > 0 and suppose Z

jujah?n 1

K (u)du = O(n?1=3 log n)

There exist constants C1; C2 > 0 such that

C1  min jT ? T j  max jT ? T j  C2 n 1in i i?1 1in i i?1 n Then we can take nt = nhn , and Assumptions 3 and 2" hold. Theorems 5 and 6 imply that q

nhn fgn (t) ? g(t)g ?!L N (0; 02) as n ! 1:

This is just the classical conclusion in nonparametric regression estimation.

5 NUMERICAL EXAMPLES In this section we will illustrate the nite-sample behaviour of the estimator by applying it to real data and by performing a small simulation study. In the introduction we already mentioned the human-capital earnings function as a wellknown econometric application that can be put into the form of a partial linear model. It typically relates the logarithm of earnings to a set of explanatory variables describing an individual's skills, personal characteristics and labour market conditions. Speci cally, we estimate and g() in the model ln Yi = XiT + g(Ti) + "i;

(21)

where X contains two dummy variables indicating the level of secondary schooling a person has completed and T; is a measure of labour market experience (de ned as the number of years spent in the labour market and approximated by subtracting (years of schooling + 6) from a person's age). 11

estimate of g(T) 2.7999992.899999 3 3.099999 3.2

Partial linear fit

10

20

30

40

T

Figure 1: Relationship of log-earnings and labour-market experience Under certain assumptions, the estimate of can be interpreted as the rate of return from obtaining the respective level of secondary schooling. Regarding g(T ), human capital theory suggests a concave form: rapid human capital accumulation in the early stage of one's labor market carrer are associated with rising earnings that peak somewhere during midlife and decline thereafter as hours worked and the incentive to invest in human capital decrease. To allow for concavity, parametric speci cations of the earnings-function typically include T and T 2 in the model and obtain a positive estimate for the coecent of T and a negative estimate for the coecient of T 2. For nonparametric tting, we use a Nadaraya-Watson weight function with quartic kernel (15=16)(1 ? u2)2I (juj  1) and chose the bandwidth using cross-validation. The estimate of g(T ) is depicted in Figure 1. In a sample size that is lower than in most empirical investigations of the human capital earnings function we obtain an estimate that nicely agrees with the concave relationship envisioned by economic theory and often con rmed by parametric model tting. We also conducted a small simulation study to get urther insights into the small-sample 12

0

g(T) and its estimate values 0.5

1

Simulation comparation

0

0.5 T

1

Figure 2: Estimates of the function g(T ) performance of the estimator of g(). We consider the model

Yi = XiT + sin(Ti) + sin(XiT + Ti)"i;

i = 1; : : : ; n = 300

where "i is standard normally distributed and Xi and Ti are sampled from a uniform distribution on [0; 1]: We set = (1; 0:75)T and performed 100 replications of generating samples of size n = 300 and estimating g(): Figure 2 depicts the "true" curve g(T ) = sin(T ) (solid-line) and an average of the 100 estimates of g() (dashed-line). The average estimate nicely captures the shape of g():

6 APPENDIX In this appendix we state some useful lemmas. Lemma A.1. Suppose that Assumption 2 (a)-(c) hold and g() and hj () are continuous. Then (i)



max G (T ) ? 1in j i

n

X

k=1



!nk (Ti)Gj (Tk) = o(1)

13

Furthermore, if g() and hj () are Lipschitz continuous of order 1 and Assumption 2' (a)-(c) and 2 (b) hold. Then

max G (T ) ? 1in j i

(ii)

n

X

k=1



!nk (Ti)Gj (Tk ) = O(cn + dn )

for j = 0; : : : ; p: Where G0 () = g() and Gl () = hl() for l = 1; : : : ; p: Proof. We only present the proof of (ii) for g(). The proofs of other cases and (i) are similar. Observe that n

X

i=1

n

!ni (t)fg(Ti) ? g(t)g =

X

=

X

i=1 n

i=1

+

!ni(t)fg(Ti) ? g(t)g +

n

nX

i=1

o

!ni (t) ? 1 g(t)

!ni(t)fg(Ti) ? g(t)gI (jTi ? tj > cn) n

X

i=1

!ni (t)fg(Ti) ? g(t)gI (jTi ? tj  cn) +

By Assumption 2'(b) and Lipschitz continuity of g() n

X

and

i=1 n

X

i=1

n

nX

i=1

o

!ni (t) ? 1 g(t)

!ni (t)fg(Ti) ? g(t)gI (jTi ? tj > cn ) = O(dn );

(22)

!ni (t)fg(Ti) ? g(t)gI (jTi ? tj  cn) = O(cn ):

(23)

(22)-(23) and Assumption 2 (a) complete the proof of Lemma A.1. Lemma A.2. Under Assumptions 1 and 2'. 1 XT X = B lim n!1 n Proof. Denote hns (Ti) = hs(Ti) ? nk=1 !nk (Ti)xks . It follows from xis = hs (Ti) + uis that the (s; m) element of X T X (s; m = 1; : : : ; p) is f

f

P

f

n

X

i=1

f

xisxim = e

e

n

X

i=1

uisuim +

+ def

=

n X i=1

n

X

i=1

n

X

i=1

hns (Ti)uim

h nm (Ti)uis +

uisuim +

3 X

q=1

n

X

i=1

h ns (Ti)hnm(Ti)

q) R(nsm

The strong law of large number implies that limn!1 1=n ni=1 uiuTi = B and Lemma A.1 (1) means R(3) nsm = o(n), which and Cauchy-Schwarz inequality show that Rnsm = o(n) and R(2) nsm = o(n): This completes the proof of the lemma. P

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The following Lemma is a slight version of Theorem 9.1.1 of Chow and Teicher (1988). We therefore do not give a proof. Lemma A.3. Let nk ; k = 1; : : :; kn ; be independent random variables with Enk = 0; 2 =  2 < 1: Assume that limn!1 k  2 = 1 and max 2 ! 0: Then and Enk 1kk nk nk k=1 nk k  !L N (0; 1) in distribution if and only if k=1 nk P

P

n

n

n

k

n X

k=1

2 I (j j >  ) ! 0 for any  > 0 as n ! 1: Enk nk

Lemma A.4. Let V1; : : :; Vn be independent random variables with EVi = 0 and inf i EVi2 > C > 0 for some constant number C: The function H (v) satisfying 01 vH (v)dv < 1 such R

that

P fjVk j > vg  H (v) for large enough v > 0 and k = 1; : : : ; n:

(24)

Also assume that fani ; i = 1; : : : ; n; n  1g is a sequence real numbers satisfying Pni=1 a2ni = 1: If max1in janij ! 0; then for a0ni = ani=i(V ); n

X

i=1

a0niVi ?!L N (0; 1) as n ! 1:

Proof. Denote nk = a0nk Vk ; k = 1; : : : ; n: We have n

X

k=1

2 I (j j >  ) Enk nk

=

 

n

X

P

n E 2 k=1 nk

= 1: Moreover, it follows that

a0nk 2EVk2I (jank Vk j > )

k=1 n a2 X nk EV 2I ( max ja V j >  ) k 1kn nk k 2  k=1 k (inf 2)?1 sup E fVk2I (1max ja V j > )g: k k kn nk k k

It follows from the condition (24) that sup E fVk2I (1max ja V j > )g ! 0 as n ! 1: kn nk k k

Lemma A.4 is therefore derived from Lemma A.3.

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Schick, A. (1993). On ecient estimation in regression models. Annals of Statistics, 21, 1486-1521. Schick, A. (1996). Weighted least squares estimates in partly linear regression models. Statistics & Probability Letters, 27, 281-287. Speckman, P.(1988). Kernel smoothing in partial linear models. Journal of the Royal Statistical Society, Series B, 50, 413-436. Willis, R. J. (1986). Wage Determinants: A Survey and Reinterpretation of Human Capital Earnings Functions in: Ashenfelter, O. and Layard, R. The Handbook of Labor Economics, Vol.1 North Holland-Elsevier Science Publishers Amsterdam, 1986, pp 525-602.

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