How sensitive are average derivatives?

Tilburg University

How sensitive are average derivatives?

Härdle, W.K.; Tsybakov, A.B.

Publication date:

1994

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Härdle, W. K., & Tsybakov, A. B. (1994). How sensitive are average derivatives? (Reprint Series). CentER for

Economic Research.

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for

~a,s, ~`~~"n III~nIIIIIIIIIIIIIIIIIIVIIIIIQIIINIhllllllll

How Sensitive are Average

Derivatives?

by

Wolfgang H~rdle and

A. B. Tsybakov

Reprinted from Journal of Econometrics,

Vol. 58, 1993, North-Holland

,J~,~~~

Reprint Series

CENTER FOR ECONOMIC RESEARCH

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Hebrew University of Jerusalem S[ockholm School of Economics

Address : P.O. Box 90153, 5000 LE Tilburg, The Netherlands

How Sensitive are Average

Derivatives?

by

Wolfgang Hárdle and

A. B. Tsybakov

í;-

_ii

J~l~~'~ a~Li-'.~..tc.Í ' ~ ~;i~P..1 . ~

~~

TiLái~h~à

Reprinted from Journal of Econometrics,

Vol. 58, 1993, North-Holland

Journal of Economelrics 58 (19931 31--48. North-Holland

How sensitive are average derivatives?~`

Wo(fgang Hárdle

Tilburg UrrioerxitJ~, 5000 LE Tilburg, The Netherlnnd.r

Unirersité Cnrholique de Louuain. B-l348 Louuain-lo-Neuue, Belgiunr

A.B. Tsybakov

Tilburg Unirersify~, 5000 LE Tilburg, The Nclherlonds

Average derivatives are the mean slopes of regression functions. In practice Ihey are estimated via a nonparametric smoothing technique. Every smoothing method needs a calibration parameter that determincs the finite sample performance. In this paper we use the kernel estimation method and develop a formula for lhe bandwidth that describes the sensitivity of the average derivative estimalor. One can determine an oplimal smoothing parameter from this formula which tries out to undersmooth the density ofthe regression variable.

1. Average derivatives in discrete choice analysis

The average derivative is the mean of the slope of a regression function. In

a regression setting Y- m(X ) -F- e with regression curve m: R" -~ R, the average

derivative is the mean gradient Ex(m'(X)), or, more generally, the weighted

mean gradient

á - Ex(m'(X)w(X)),

(i.l)

where ní (x) is the gradient

am

am

m'(x) -

aX ,...

ax )

e R",

t e

x,, ..., xa are components of the vector x, w(x) is some weight function, and

Er is the expectation with respect to the (marginal) X-distribution.

Correspondence ro: Wolfgang Hfirdle, fnstitut fiir Statistik und dkonometrie, FB Wirtschaflswissen-schaften, Humboldt-Universitiit zu Berlin, D-1020 Berlin, Germany.

'Work of the second author was financially supported by the Department of Econometrics, Tilburg University. The Netherlands.

32 _{II'. HnrJh rr~ul ,~.B. T~rhuknr. Hnu~ sensi~ire nrr nceraRe rlerirniiuaa?}

The average derivative ó is interesting in the context of discrete choice

analysis, where in the case of binary choice we want to infer on the

function

P( Y- I I X-.Y) - Ill(X),

from observations {(X;, Y;)};: ~, X; e R', Y; e {0, 1}. A pure nonparametric

approach to estimation of m(x) is possible [see, for example, the recent

mono-graphs by Muller(1988), Eubank ( 1988), Wahba ( 1990), and Híirdle(1990)]. It is

well-known though that this approach is not costless: the precision of the

estimator is exponentially decreasing as the dimension d increases. In order to

avoid this difTiculty one could of course fall back into pure parametric models

for m(.r).

-One such model would be

pl(X) - G(Yr~), _(1.2)

where G, the link function, is of known form, e.g., G-~ would postu)ate

a Probit model.

A model comprising the advantages and simplicity of (1.2) and the flexibility

of a nonparametric smoothing approach is a single-index model,

!II(.Y) - lIl-YTl'), (1.3)

with an unknown link function y and index xT(i.

It is well-known that (i in (1.3) can only be identified up to scale [see

Hárdle and Stoker (1989)]: the (weighted) average derivative (ADE) for this

modef is

b- Ex

L

d(XT~) w(X)

J

l~ -)'r'í~,

(1.4)

so we see that we can estimate ~3 (up to scale) if we know how to estimate S and if

~~B is difTerent from zero. A simple examp)e for (1.4) is a linear link function g(' );

then the coefiïcients ~ are multiplied by the slope of g(-) times EX(w(X)). For

general, nonlinear g( .), as in binary choice models, the ~3 coe(Ticients are

multiplied by the average slope

d

!V. Nnrdle mid A.B. T.~rbaknr, Ho~~ s~.nsrrine nre nrernge derivarrnes? 33

We use kernel estimators for the average derivative b since they are

straiglit-forward to implement and easy to understand on an intuitive Ievel. Other

possibilities include splines and

orthogonal series, but to our

know-ledge these techniques have not been employed to estimate average

deriva-tives. The main point in this paper is about the selection of the bandwidth,

the kernel smoolhing parameler, for lhe d-dimensional case. The

one-dímensional case with a focus on estimation of income efïects is treated in

H~rdle, Hart, Marron, and Tsybakov ( 1991). From an asymptotical viewpoint

the choice of bandwidth does not afiect the behavior of ADE estimators.

It influences only the higher-order terms of asymptotic expansions for mean

squared error, not the main term which is of order O(I~n), where n is the

number of observations. In practice though, the choice of the smoothing

parameter is an important issue as has been pointed out by Hsieh and Manski

(1987, p. 55I).

In this paper we consider tlle special choice of weight function: w(x) - j(x), where j(.Y) is the marginal density of X[cL Powell, Stock, and Stoker ( 1989)].

This is motivated by several reasons. First, under such choice of w we avoid tlie

random denominator appearing if rv(x) - I [in fact, for w(x) - I the ADE estimators contain the density estimator in denominator, see Hiirdle and 5toker (1989) for details]. Because of lhe random denominator the necessary asymp-totic expansions hold under somewhat restrictive assumptions on the underly-ing density j [H~rdle and Stoker ( 1989), Hiirdle, Hart, Marron, and Tsybakov (1991)]. Next, for the multi-dimensional case the O(I~n) rate of the mean-squared error is nol attained unless the oscillating higher-order kernels are implemented. This causes a difliculty in trealing the case of w(x) - I: the ADE estimator is not well-defined and it requires some truncation [Híirdle and

Stoker ( 1989)]. The choice of truncation threshold appears to be crucial in this context. This creates an additional problem which could be easily eliminated if

ll'(.Y) - j(X).

In section 2 we quantify the sensitivity of ADE via a second-order expansion

of inean squared error of a kernel estimator for b. Section 3 is devoted to the

proof ot our main theorem. [n the appendix we prove some lemmas.

2. The sensitivity of ADE

Assume tliat independent pairs (X;, Y;), i- 1, ..., n, of random variables,

X; e R", Y; e R', are observed and that they have the same distribution as

(X,Y),XeR",YeR'.

Let the regression function m(.Y) - E( Y~X - x) exist and let X have the

density j(x) wilh respect to Lebesgue measure in R'. Suppose, moreover, that the

regression function rn and the density jare continuously difierentiable and that

34 _{1{'. Hiirdle ain! A.B. T.r~hakor, Nmr .rr~tririve nre nverage deriralioe.r?}

Using partial integration (over the support of X) we get

b - J m'(x)J'(r)dx

- - 2 Jm(x)f'(x)f(x)dx

- -2E(YI~(X)).

wliereJ

'(.r) -(a~~ax,

_{, ..., af~a.rd) and tlie expectation is now taken over the}

joint distribution of (X, Y).

If we knew the marginal density Jwe could estimate S by means of the sum

-(2~ri)~~-r Y;J'(X;) which is obtained if one substitutes the expectation in

(2.l) by the empirical average.

In our approach we do not know the density function. We shall estimate it

from the data via the kernel method. The marginal densityJ(-) is estimated by

n

Jh(-C) - n-~ ~ .~h(x - Xi),

i-l

where .~Yh(u) - h-",~Y'(ul~h, ..-, nd~h) for a multivariate kernel function, e

.X'(u, , . . . , ne) - ]-] K(u~).

u - (u, , . . . , ue) E Re,

(2.3)

i-r

based on a one-dimensional kernel K. The scaling of .7Y~ is through h~ 0, the

bandwidth, or smoothing parameter.

The gradient J'(x) is estimated by

1

_n;-i

where

~n(lt) - h-e-' K' I uj~ n K~u'~,

`h

k:j

h

and K' denotes the derivative of one-dimensional kernel K.

Using (2.4) we can construct an estimate of the average derivative

b~ - - - ~ Y~fí,(X~).

(2.5)

I{'. Nnrdle~ iuul A.R. T~rhakat', Huu srnsr,ioc arr ne~raRe derioariues? 3S

We study the asymptotic mean squared error of b„ under the following

assump-tions:

(A I) The kernel K is bounded, continuously difïerentiable, symmetric with

support [ - I, I]; K'(0) - 0.

(A2) JK(u)du - I, and there exists a positive integer k~ 2 such that Ju~ K(u)

xdu-0, j- I,...,k- I, f ukK(rQdu-dK~O.

(A3) The marginal density f(.~) of X is compactly supported and has eontinuous

partial derivatives up to the order k f I on Rd.

(A4) The regression function rn(x) has continuous partial derivatives up to the

order k f I on Rd.

(AS) The conditional variance aZ(x) - var( Y~X - x) is bounded on the

sup-port of f.

(Atí) h-h„-.O,andnZh~tZ-. oo asn--~ oo.

Later ~.~ denotes Euclidean norm when applied to vectors.

Theorem.

WIIPPP

Clnd

Under the as.u,niptiorrs (AI)-(A6),

E(ló~ - álZ) - Q,~~-~

t Qah~k

Q, - 4[E(IÍ(X ) n~'(X)~Z) - IE(Ï(X)nt'(X))IZ

f E(QZ(X)II~(X)I~)],

C„ - J ~.~Y"(u)~2 du - d f(K'(u))Z du ( J K 2(u) drQ"- `,

( - 1 )k d

ak } ~ .t(c)~aX, ax;

SK(-~) - dK ~ :

k!

_{j-~ ak.r f(C)~aY,ar;}

36 {P. H~irdle and A.B. T.~7'h~lkur, Hnn'si~~esinr~t' arr' are~nke rlerirurir~~s?

From the Tlteorent we see tl)at the bandwidtlt ll„ minimizing E(~b„ - b~z) is

given by

h' - It rt-zrrzktd.zl

_„

_o

where

Ito - (Qz(ri

f 2)1'nzk,e.z)

`

2kQ~

J

For h„ - h~ , we have

E(Ib„ - blz) - Q~n-' -i. Crl-4kl(2ktdf2) } O(q-4k112ktda2l) f ~ ~ ,2 ~, I7 ~ 00 , n

where

C-

df2)

}( 2k )

)

Oprirnization oj k.

This in fact is reasonable if one believes that f and rn are

infinitely many times continuously difTerenliable. It follows from (2.6) that the

best rate for mean squared error equals n-' and it is attained if k ~(d f 2)~2.

For example, in one-dimensional case ( d - 1) it sufiices to take k- 2. Then tlte

second term in (2.6) equals Cn-8r', and !t~ is proportional to n-zj7 [cf Híirdle,

Hart, Marron, and Tsybakov ( 1991)].

Assumptions ( AI) and ( A2) entail that the order k of the kernel should be

necessarily even. Thus, the condition for choosing k that guarantees the best rate

of convergence becomes:

k is Ure mininm( euen number such that k ~(d f 2)~2.

Optirrtization oj K.

The factor C depends on the kernel K. Optimizing this

factor in K leads to the minimization problem (in view of the definition of

Qz attd Q;):

min ( J uk K ( u) du)d ~ z( J( K'(u))z du)zk ( J K z(u) du )zrd- t)

N~. Hwdlc nrid A.B. Tsrhakor, Ho~r sensitrce ore auerage derieatines? 37

where ~Ih is the class of kernels satistying ( A l) and (A2). For d- I, this problem

was solved by Mammitzsch ( 1990) who showed that the optimal K is the quartic

kernel

K(u) - i~(I - uz)z 1(~u~ 5 1),

where I(. ) denotes ihe indicator function. If d ~ 2, then k ~ 4, and the optimal

K is, clearly, an oscillating kernel taking positive and negalive values.

3. Proof of the Theorem

Denote

á~-ó"--1 S~ Y-

X.,

ó'-b-- mx

x {~ xdx.

Clearly,

E(~b" - b~Z) - 4E(~á~ - b'~2).

(3.1)

Write the estimator S` as

b~ - ~ ~ pn(Xt) f E,)Íh(X;),

_t,;-,

where s; - Y; - m(X;). Since E(e,~X;) - 0, we have

E(ló~ - b`IZ)

~ Í

- E E;Íh(Xt)

_,t;~,

_n;-t

- ~ m(Xr)ií~(Xt) - b~

1 n 2

f

_{n ; ~,}

E(S,) - b'

where S; - m(X;)Jh(Xf).

38 li'. Hiv~lle anJ A.B. Tsrbukor, Hom sensiliue arc ntr~nge derifwlines?

It follows from (A1) that .~f~~,(0) - 0, and thus

1

Ih(Xf) - - ~, ~I,(Xf -

X;)-It j-, ixj

Thus,

E(Si) - E(Sr) - E(mlXr)Ií,(Xf )) -

n - 1

y,

(3.4)

n

where q - E(m(X,).JF~~,(X, - XZ)). Here we used ( 2.4) and the fact that

.7Y~,(0) - 0 which follows from (A1). Now, (3.2) and (3.4) entail

E(Ib~ - b'12) - Vr f vZ f v,,

where

I

"

}

~ ~ (bf - E(~r))IZ~.

n r-,

v~ - IE(Cf) - b'~2.

Let us evaluate the terms Vl, VZ, and V,.

Clearly,

J~Z(Xi),

1- f.

-1~, i~ l.

From (3.3) and (3.6) we get

t6'. Hórdle mul A.B. Tst'hnkav, Hou~ srrtsiliue ore average rlerinarives? - ~~J ~~2(xt) E ~ 2

~ .~„(xt -x;) )j(x,)dx,

J-z

- ~~a

J

vz(x) E

~ ~ (,~f~í~(x - XJ),.N'~"n(x - X,))~Í(x)dx.

~:-z

Here and later (~,.) denotes the scalar product. We have

E ~ ~ (.7Y~,(x - XJ), ,~í,(x - X,))

_J

`,, z

39

n

-~ E(I~h(x - x;)IZ) f~ IE(~~,(x - xJ))IZ

(3.s)

J-z J..-z

Js:

- (17 - 1) E(~~h(x - il 1 )~Z)

f (~i - 1)(~~ - 2)IE(~h(x - Xt))IZ.

[t follows from Lemmas 1 and 2 and (3.8) that

E ~ ~ (.~Y'í~(z - X ),~i,(.C - x,)~ `, . z

- (n -1) (~Xj(x)l~-'-Z -~ n3vl, x))

f(n - 1)(~i - 2)Ij'(x) -t- hk Sx(x) f Qt(h, x)~z.

Hence

Vt -

1

1

_J

QZ(x)Ij~(x)IZ

J

v2(x)j~(x)dx

1

~h4

1 i

~o

f0

- t zl,

n~ oo,

(3.9)

rtzh"'z

n

n

40 N'. Hn"rdle nnd A.B. Tsrh~tkno, Hmr .aensifirr are auerage dcri~atioes?

Next,

n

~z - 2~ E((~,,~;)) - IE(~t)Iz - ~zt -~ ~zz - IE(~,)Iz,

_{n ;.~-,}

(3.to)

where

~

1il - Z~ E(Istlz),

_{n ;31}

~zz - Z~ E((St, ~;)).

_{n ;.Ja,}

t:~

We have

E(~Sriz) -'z E Intz(Xf) ~(~á(Xt - Xf),.~n(XI - Xs)) I.

_n

I.sxt

Hence

1

"

l

Vzt - ~~3 E mz(XI) ~ (~í,(Xt - Xf), ~í,(XI - Xz))J.

f.:-z

Using the same argument as in (3.7)-(3.9), we find

V21 - i J

l112~~)~f (-~)~ZI(.Y)dX f n2I1 ~Z CK JnIZ(X)f Z(X)dX

1

f o

nzli tz

J

f O 'n f nz '

n-~ oo.

(3.11)

Consider the term l 2z now. Applying (3.3) we obtain

11'. Ifiudl~ nnrl A.B. Tcrbaknr. Hm~ srn.airiee ure nrernge Arriralit~e.c?

where

dls - E[m(X,)rn(Xz)(-~ í~lXl - XI),.1f í,(Xz - X,))).

Let us treat d,,, separately in the following four cases:

(i)

s - I,

(ii) s~l,l~2,s~1,

(iii) s~! and either 1- 2, s~ I or 1~ 2, s- 1,

(iv) s~l,l-2,s- 1.

In case (i),

(3.13)

In case (ii),

d,.. - E[rn(X, )n,(Xz)(~~i~(XI - XI), ~-í~(XI - Xs))~

- IE(In(X, ).~~n(X, - X,))12 -1912.

In case (iii),

dr. - E[m(X, )n~IXz)(.~h(XI - Xz), ~ n(Xz - X.)))

dcr

- E~(Ip(XI)lll(X2)(~h(XI - X2))..~h(X2 - X D))~ - B2.

lIl CasC (IV), .

dlt - E[,n(X,)nt(Xz)(~í~(XI - Xz), ~h(Xz - XI))7

-

E ! n, X) n, X

JY' X

X

d~r

In (3.16) we used the fact that .X'~, is antisymmetric:

(3.14)

(3.15)

(3.16)

42 li'. Nàrdln ond A.B. T.crhakou, Hmr sensilii,e nre neernge rlericalinn.c?

It follows from (3.12)-(3.16) that

Vz2-Jl-1

n~ C(

n-28 f n2-Sn.f-6

) J(

)19I f 2(n - 2)BZ t B,].

2

(3. I 7)

Now we use Lemma 4 in the appendix, which implies, together with (3.17),

lhat

V22

-1_LJ~j(Y)IJJI~(.Y)I2dX -~IA2(x)~f,(x)~2f(x)dX J

f 1912 -E ~1 - 6~ - n2lle.z fm2(x)I2(x)dx

C"k') C')

From (3.4), (3.10), ( 3.11), and ( 3.I8), we find

V2 VSJ i V22

-n-1

9

- n L Jf

~(x)~rJJ'(x)I2dx - 41912J

By substitution on (A.2) into (3.19), we obtain

12 - ' L

JÍ'(x)~Jn'(x)12dx - 41ó~~2J

n -. ao.

(3.I8)

(3.19)

k

n'. Hárdle nnrl A.B. Ts}~hnkor, Noir sensirice are auerngc derivalice.t? 43

tt remains to find the asymptotic expression for V3.

By (A.2) we have

~~ llk

J

_{SAl-r)iGr) nl(z)dx f ~Iz(ll)}

n

q-b'

From (3.5), (3.9), ( 3.20), and ( 3.21) one gets

E(Ib~ - h~`Iz)

- n L J f lx

(.Y)~rn'(X)Iz dx - 41b'~z f

J

az(X)II

~(X)~zÍ(X)dX

J

1

~- nzlíd.z Cx

az(X)Iz(X)dx f

hzk

_J

fO~rlífnz~-Fo

nzli,z~'

rt~oo.

This, in view of (3.1), proves the Theorem.

2

(3.21)

Appendix

Lennnn l. Lel assumpirons (AI)-(A4) be sntisfied. Then

44 IV. Hirrdlc and .4.8. Tsrhnkor, Hu~r sensilive nrp ar~erngc derivalires?

~ahere sup,l~,(h, x)~ - o(hk) ns h~ 0, and

9- r5' - hk

_J

~c

here

I~iz(h)I - o(h4) as Ir ~0.

Proof.

By partial integration,

E(~h(Y-X,))-1idtl

J

.)f"~xlr z~f(z)dz

1

- - h f~'(u)I(x - Uh)du

(A.3)

1

-J

where we used the fact thal .~Y and f are compactly supported. Assumption (A3)

entails that the Taylor expansion is valid, and thus, uniformly in u e supp.~Y~,

a~o~ t ~.t(x)~ar, ax'

f'(x - nrr) - ~ ~r (- I)~'~rr'Ir~'~

_.

a:~a~ src a-

_a

_f(x)~axdax

S Qr (h, x),

(A.4)

where supxl~i,(h, x)~ - o(Irk), h~ 0, and a -(a, ..., ad) is multi-index,

a;10, _{~x~-a, f...~ad} al-a~l...ad1 ,ra-u~~...uë _for

u-(u, , . . , nd) e Rd, and a~'~~ax' - a~'I~aY ~~ . . . .. axá" for x - (x,, . . , .rd) e Rd.

It follows from (A2) that

J

rr'JY(tr)du - 0,

_{0 S Ix~ 5 k- I,}

(A.5)

0,

~x~ - k, card { j: a; ~ 0} ~ I,

tr'JF'(tr)dn

IV. Hitrrllt~ und .i.R. T.ttbnAor, Notr sertsirit~r arr aonrnge deiirarloes? 45

From (A.3)-(A.5) we get

E(.~Y'h(.Y - X, ))

a~s~ } ~.f(.x)~aX, arQ

-- j 1-v) - ki Il"(- I)`

~

.

a:~al-k alal } 1_j(.K),aXdaXa

x

J

u'.~Y(1Qdu-~l,(Il,x)

(A.6)

-- j'(x) - hk SK(x) - 1' 1(Il. X).

which proves (A.1). To get (A.2) note that by (A.I)

CJ - E(Ill(~il).it~hlX1 - X2))

- E(nl(x~t)Es:(~i~(Xt -Xz))

- EOn( Y, )(-j'(X 1) - hkS,;(X1) - ÍItUh X 1)))

- (5 ~` - hk

J

Now, sup,~~i, ( h, x)~ - o(h~), and J m(x)j(x)d.x ~ oo since nl and jare bounded

on the compact support of f. Tliis proves (A.2).

Lemnra 1.

Let assurnplinns ( Al )-(A4) be satisjed. Then

E(I~ti(.x - Xr )I~) - CA j(x)h-"-Z f llallh-x),

ivJrere sup,.~~i(Il,.x)~ - o(Il"d"Z), Il-~0.

Proof. E( ~.~F'~h(.x - X, )I Z) - Itze.z f

I~ `x h Z~ z

j(Z) dZ

1

46 t1'. Hrvdle m~d A.B. T.rrhakov. Nmi~ sensirine are average derinariues?

It follows from (A3) that f is Lipschitz continuous on its support with some

Lipschitz constant Lf. Thus,

Ilrétz jl~'(u)Iz.Í(x - uh) du - li tz.f(x) Cxl S Irét, fI.~f"(u)Iz lul du.

This proves the lemma.

Len,mn 3. Let assumptions (A!)-(A4) he satisfied. Tlren

E(rn(X 1)~h(x - X1)) --(!n(x)f(x)) - IIkS1K(x) - l~d(Ir, X), lVll ere

ak}1 (f(x)nl(x)),aX~aXk

S,x(x) - dx (-

1)k ~

;

kr

;` ~

ak},

U(x)rn(x))~ax,ax;

and sup,l~3a(h, x) - o(Irk), h-. 0.

Proof.

The proof is similar to flie proof of (A-I ). In fact, instead of (A.3) we now

have

E(rn(X~

I.XI~,(x - X~)) -

_J

U(-x - uh)m(x - uh))

'.JEr(u)du

-- f[f'(x - ul,) m(x - ulr)

~-fcx - ulr)rr,'

(x - uh)].~Y'(u)du.

Len,ma 4.

Let assumptions (AI)-(A4) be satisfied. 77ren, as h-~ 0,

B, -

_J

~nl'(z)Iz!(x)dx f

J

mz(z)I.Í~(x)IZÍ(x)dx

-F 2

J

n,(x)(rn'(x),f'(x))J'z(x)dx f O(h4),

(A.7)

Bz - - ~nrz(z)II'(x)IZI(x) dx

-

J

m(x)(rrí (x),j'(x))f z(z)dx t O(h"),

(A.8)

I

r

fV. lldrdle mid A.B. T.c}~finkov. Nom sensirrne nre ai~erage deriualives? 47

Proof.

By Lemma 3,

Br -

J

I(x)I E(rn(X r)~í,(Xr - x))IZ dx

- fI(x)I(nr(x)I(x))' ~- h"Srx(x) -t- ~3s(x, h)IZ dx

- JI(r)I(nr(x)I(x))'Iz dx f O(hk),

which gives (A.7).

Using Lemmas 1 and 3 and the antisymme[ric property of .7Yh, we get

Bz

-J

m(x)m(Y)(~í,(x - Y), ~n(Y - z))I(x)I(Y)I(z) dx dy dz

-

J

J

~í,(x - Y)I (-z) m(x) dx, fI(z)~~(Y - z)dz~dy

-

_J

-.I'(Y) - hkS~(Y) - ~r(11,1'))dl'

- -

_J

I(Y)rn(Y)((m(Y)J(Y))'.Í'(Y))dy.

This proves (A.8). To prove ( A.9) note that with ~v -(x - y)~h we have

B~ - ~rrt (x)nt(1')I(x)I(Y)I~n(x - Y)IZ dx dy

1

-- hi.z

nr(Y)I(Y)nr(y f wh)Í(Y ~- wlt)I~í,(w)Iz dydw.

It follows from assumptions ( A3) and ( A4) that j and m are Lipschitz continuous

on the support of f. Hcnce

B~ -- ha:: [ f rnz(Y)Iz(Y)dY J ~~í,(w)Iz dw(1 f O(h))J,

48 {I'. Hnrdle tmd A.B. T.crhnkot~, Hmr sertrilir~ are ooer~Re Jeritxniues?

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No. 90 H. Peters and P. Wakker, lndependence of irrelevant alternatives and revealed group preferences, Ecormmetrica, vol. 59, no. 6, 1991, pp. 1787 - 1801.

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No. 95 A. van den Nouweland, P. Borm and S. Tijs, Allocation rules for hypergrapli

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No. 96 E.J. Bomhoff, Monetary reform in Eastern Europe, European Economic Review, vol. 36, no. 213, 1992, pp. 454 - 458.

No. 97 F. van der Ploeg and A. de Zeeuw, International aspects of pollution control, Ern~ironmental and Resource Economics, vol. 2, no. 2, 1992, pp. 117 - 139. No. 98 P.E.M. Borm and S.H. Tijs, Strategic claim games corresponding to an NTU-game,

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No. 99 A. van Scest and P. Kooreman, Coherency of the indirect translog demand system

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No. 100 Th. ten Raa and E.N. Wolff, Secondary products and the measurement of productivity growth, Regional Science and Urbm: Economics, voL 2l, no. 4, 1991, pp. 581 - 615.

No. 10l P. Kooreman and A. Kapteyn, On the empirical implementation of some game theoretic models of household labor supply, 7T:e Jortnta! ojHurnan Resources, vol. 25, no. 4, 1990, pp. 584 - 598.

No. 102 H. Bester, Bertrand equilibrium in a differentiated duopoly, huernationa! Ecortonric Revierv, vol. 33, no. 2, 1992, pp. 433 - 448.

No. 103 1.A.M. Potters and S.H. Tijs, The nucleolus of a matrix game and other nucleoli, Mathentatics of t7pera[ions Research, vol. l7, no. 1, I992, pp. 164 - 174. No. 104 A. Kapteyn, P. Kooreman and A. van Soest, Quantity rationing and concavity in a

flezible household labor supply model, Review of Economics and Statistics, vol. 72, no. I, 1990, pp. 55 - 62.

No. l05 A. Kapteyn and P. Kooreman, Household labor supply: What kind of data can tell us how many decision makers there are?, European Ecortornic Review, vol. 36, no. 213, 1992, pp. 365 - 371.

No. 106 Th. van de Klundert and S. Smulders, Reconstructing growth theory: A survey, De Ecortonrisr, vol. l40, no. 2, 1992, pp. l77 - 203.

No. 107 N. Rankin, [mperfect competition, ezpectations and tlte multiple effects of monetary growth, The Ecorrontic Journal, vol. 102, no. 413, 1992, pp. 743 - 753. No. 108 ]. Greenberg, On the sensitivity of von Neumann and Morgenstern abstract stable

sets: The stable and the individual stable bargaining set, International Jounral oJ Game 7heory, vol. 21, no. 1, 1992, pp. 41 - 55.

No. 110 M. Verbeek and Th. Nijman, Testing for selectivity bias in panel data models, InrernatronalEconanic Ret7e~v, vol. 33, no. 3, 1992, pp. 681 - 703.

No. 1 I 1 Th. Nijman and M. Verbeek, Nonresponse in panel data: The impact on estimates of a life cycle consumption function, Journa! of Applied Ecarometrics, vol. 7, no. 3, 1992, pp. 243 - 257.

No. I 12 I. Bomze and E. van Damme, A dynamical characterization of evolutionarily stable states, Ararals of Operarions Research, vol. 37, 1992, pp. 229 - 244.

No. 113 P.1. Deschatnps, Expectations and intertempora) separability in an empirical model of consumption and investment under uncertainty, Empirical Econornics, vol. 17, no. 3, 1992, pp. 419 - 450.

No. 114 K. Kamiya and D. Talman, Simplicial algoritlun for computing a core eletnent itt a balanced game, Jountal of rlte Operations Research, vol. 34, no. 2, 1991, pp. 222

-228.

No. 115 G. W. imbens, An efficient method of moments estimator for discrete choice models with choice-based sampling, Econanerrica, vol. 60, no. 5, 1992, pp. I187 -1214. No. ! 16 P. Borm, On perfectness concepts for bimatrix games, OR Spektrum, vol. 14, no.

1, 1992, pp. 33 - 42.

No. 117 A.P. Jurg, 1. Garcia Jurado and P.E.M. Borm, On modifications of the concepts of perfect and proper equilibria, OR Spektrum, vol. 14, no. 2, 1992, pp. 85 - 90. No. l 18 P. Borm, H. Keiding, R.P. McC.ean, S. Oortwijn and S. Tijs, The compromise value

for NTU-games, hNernatioaralJorarna! oj Game 7heary, vol. 21, no. 2, 1992, pp. 175 - 189.

No. 119 M. Maschler, J.A.M. Potters and S.H. Tijs, The general nucleolus and the reduced game property, lruerrtatiortal Journal of Cante 77teory, vol. 2l, no. 1, 1992, pp. 85

-l06.

No. 120 K. Wárneryd, Communication, correlation and symmetry in bargaining, Econonrics Letters, vol. 39, no. 3, 1992, pp. 295 - 300.

No. 121 M.R. Baye, D. Kovettock and C.G. de Vries, It takes two to tango: equilibria in a model of sales, Carrtes and Ecorranic Behavior, vol. 4, no. 4, 1992, pp. 493 - 510. No. 122 M. Verbeek, Pseudo panel data, in L. Mátyás and P. Sevestre ( eds.), The EconanetricsofPanelData, Dordrecht: KluwerAcademic Publishers, 1992, pp. 303 - 315.

No. l23 S. van Wijnbergen, Intertemporal speculation, shortages and the political economy of price reform, T7:e Econontic Journal, vol. 102, no. 415, 1992, pp. 1395 - 1406. No. 124 M. Verbeek and Th. Nijman, Incomplete panels and selection bias, in L. Mátyás and

No. 125 _{7.1. Sijben, Monetary policy in a game-theoretic framework, Jahrbr7cher fiir} Natiorralókorromie ~nd Statistik, vol. 210, no. 314, 1992, pp. 233 - 253.

No. 126 H.A.A. Verbon and M.J.M. Verhoeven, Decision making on pension schemes under

rational expectations, Jortrnal of Ecaromics, vol. 56, no. 1, 1992, pp. 71 - 97.

No. l27 _{L. Zou, Ownership structure and efficiency: An incentive mechanism approach,} Jairna! oj Cornparative Ecatomics, vol. 16, no. 3, 1993, pp. 399 - 431.

No. l28 C. Fershtman and A. de Zeeuw, Capital accumulation and entry deterrence: A clarifying note, in G. Feichtinger (ed.), Dynamic Ecorromic Modefs mtd Optimal Contro(, Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1992, pp. 281 - 296.

No. 129 _{L. Bovenberg and C. Petersen, Public debt and pension policy, Fisca[Studies, vol.} l3, no. 3, 1992, pp. 1- 14.

No. 130 _{R. Gradus and A. de Zeeuw, An employment game between government and firms,} Optima! Coturol Applicatioru cF Methods, vol. 13, no. 1, 1992, pp. 55 - 71. No. 131 _{Th. Nijman and R. Beetsma, Empirical tests of a simple pricing model for sugar}

futures, Arrrrales d'Écononrie et de Statistique, no. 24, 1991, pp. 12l - 131. No. 132 F. Groot, C. Withagen and A. de Zeeuw, Note on tlie open-loop Von Stackelberg

equilibrium in the Cartel versus Fringe model, 7Tte Ecoreomic Joun:al, vol. 102, no. 415, 1992, PP. 1478 - 1484.

No. 133 S. Eijffinger and N. Gruijters, On the effectiveness of daily intervention by the Deutsche Bundesbank and the Federal Reserve System in the U5 dollar - deutsche mark exchange market, in BaltenspergerlSinn (eds), Fxcltange-Rate Regimes and Currency Uniorts, Basingstoke: The Macmillan Press Ltd., 1992, pp. 131 - 156. No. 135 _{A. K. Bera and S. Lee, [nformation matrix test, parameter heterogeneity and ARCH:}

a synthesis, Revietv of Ecortonric Studies, 60, 1993, pp. 229 - 240.

No. 136 H. G. Bloemen and A. Kapteyn, The joint estimation of a non-linear labour supply function and a wage equation using simulated response probabilities, Arrnoles d'Économie et de Statistique, No. 29, 1993, pp. 175 - 205.

No. 137 _{H. Bester, Bargaining versus price competition in markets with quality uncertainty,} 7Tte Anrericon Economic Review, Vol. 83, No. 1, March 1993, pp. 278 - 288. No. 138 _{K. W~rneryd, Anarchy, uncertainty, and the emergence of property rights, Ecortottrics}

and Politics, Vol. 5, No. l, March 1993, pp. 1- 14.

No. 139 _{A. L. Bovenberg and L.H. Goulder, Promoting investment under international capital}

mobility: an intertemporal general equilibrium analysis, 77te Scmtdinavlan Jounta! of

Ecottomics, Vol. 95, No. 2, 1993, pp. 133 - I56.

No. l40 _{S. Eijffinger and E. Schaling, Central bank independence in twelve industrial}

countries, Bmrca Naziortale de( Lavoro Quarterly Review, No. 184, March 1993, pp.

No. 14l _{S. Eijffinger and A. van Rixtel, The Japanese financial system and monetary policy:} a descrip[ive review, Japan and rhe World Econorny, Vol. 4, No. 4, 1992, pp 291-309.

No. 142 A. L. Bovenberg, lnvestment-promoting policies in open economies: the importance of intergenerational and international distributional effects, Journa! of Public Econonucs, Vol. 51, 1993, North Holland, pp. 3-54 .

No. 143 _{A. 0zcam, G. Judge, A Bera and T. Yancey, The risk properties of a pre-test} estimator for Zellner's seemingly unrelated regression model, Journalof Qumuirarive Ecortontics, Vol. 9, No. 1, January 1993, pp. 41-52.

No. l44 F. C. Drost and T. E. Nijman, Temporal aggregation of garch processes, Econometrica, Vol. 6l, No. 4, luly 1993, pp. 909-927.

No. 145 1. J. G. Lemtnen and S.C.W. Eijffinger, The degree of financial integration in the European Community, De Econontisr, Vol. 141, No. 2, 1993, pp. 189-213. No. 146 R. Sarin and P. Wakker, A simple axiomatization of nonadditive expected utility,

Econornerrica, Vol. 60, No. 6, November 1992, pp. 1255-1272.

No. 147 _{S. Muto, On licensing policies in bertrand competition, Games and Economic}

Behaviour, 5, 1993, pp. 257-267.

No. 148 _{M. Verbeek and T. Nijman, Minimum MSE estimation of a regression model with} fixed effects from a series of cross-sections, lournal of Economerrics, 59, 1993, pp. 125-136.

No. 149 R. de Groof and M. van Tuijl, Financial integration and fiscal policy in interdependent two-sector economies with real and nominal wage rigidity, Europemr Journa! of Political Econonry, Vol. 9, 1993, Nonh Holland, pp. 209-232. No. 150 A. van Soest, A. Kapteyn and P. Kooreman, Coherency and regularity of demand

systems with equality and inequality constraints, Jounta! of Econontetrics, Vol. 57, 1993, North- Holland, pp. 161-188.

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Bibliotheek K. U. Brabant