spondence to: Michael Jansson, Department of Economics, University of California, Berkeley, ... de Jong and Davidson ~2000!+ ... + On the other hand, de Jong.
Econometric Theory, 18, 2002, 1449–1459+ Printed in the United States of America+ DOI: 10+10170S0266466602186087
CONSISTENT COVARIANCE MATRIX ESTIMATION FOR LINEAR PROCESSES MI C H A E L JA N S S O N University of California, Berkeley
Consistency of kernel estimators of the long-run covariance matrix of a linear process is established under weak moment and memory conditions+ In addition, it is pointed out that some existing consistency proofs are in error as they stand+
1. INTRODUCTION Suppose $Vt : t $ 1% is a sequence of random n-vectors generated by the linear process `
Vt 5
( Cl et2l + l50
This paper considers estimation of V 5 lim Tr` T 21 ( Tt51 ( Ts51 E~Vt Vs' !, the long-run covariance matrix of Vt + Consistency of kernel estimators of V is established under weak conditions on $Cl : l $ 0% and $et : t [ Z%+ In addition, it is pointed out that some existing consistency proofs are in error as they stand+ 2. RESULTS Let 7{7 denote the Euclidean norm, let 5 signify almost sure equality, and let Et21~{! denote conditional expectation with respect to the s-algebra generated by $es : s # t 2 1%+ The development of formal results proceeds under the following assumptions+ a+s+
A1+
( l50 7Cl 7 , `+ `
A2+ Et21~et ! 5 0, Et21 ~et et' ! 5 In , and $7et 7 2 % is uniformly integrable+ a+s+
a+s+
The assumption Et21 ~et et' ! 5 In implies the conditional homoskedasticity rea+s+ striction Et21 ~Vt Vt ' 2 Et21 ~Vt Vt ' !! 5 C0 C0' but does not impose restrictions on the form of the conditional covariance matrix because C0 5 In is not assumed+ a+s+
I am grateful to Don Andrews ~the co-editor! and two anonymous referees for helpful comments+ Address correspondence to: Michael Jansson, Department of Economics, University of California, Berkeley, 549 Evans Hall #3880, Berkeley, CA 94720-3880, USA; e-mail: mjansson@econ+berkeley+edu+
© 2002 Cambridge University Press
0266-4666002 $9+50
1449
1450
MICHAEL JANSSON
The uniform integrability condition is satisfied whenever et ; i+i+d+~0, In ! or supt[Z E7et 7 r , ` for some r . 2+ The best currently available consistency results for linear processes would appear to be those of Robinson ~1991! and de Jong and Davidson ~2000!+ The moment and memory assumptions of these papers are stronger than A1 and A2 in the leading special case where $et % is independent and identically distributed ~i+i+d+!+ When the bandwidth expansion rates recommended by Andrews ~1991! are employed, Robinson ~1991! requires A1 and supt[Z E7et 7 r , ` for some r . _52 + On the other hand, de Jong and Davidson ~2000! only require two finite moments but do require L 2 -near epoch dependence of size 2 _12 + In the linear process case, this condition implies ` ( l51 l w 7Cl 7 2 , ` for some w . 1, a stronger requirement than A1+ ' 21 3 Defining G 5 lim Tr` T 21 ( Tt52 ( t21 s51 E~Vt Vs ! and S 5 lim Tr` T T ' ' E~V V !, the matrix V can be decomposed as V 5 G 1 G 1 S+ In some ( t51 t t applications in nonstationary time series analysis, the matrix G is of interest in its own right+ Obvious examples include the cointegration procedures of Phillips and Hansen ~1990! and Park ~1992!+ In recognition of this fact, the present paper focuses explicitly on consistent estimation of G+ It is assumed that G is estimated by a kernel estimator of the form T t21
GZ T 5 T 21
((k t52 s51
S D
6t 2 s6 Vt Vs' , bT
where k~{! is a ~measurable! kernel function and $bT : T $ 1% is a sequence of bandwidth parameters+ The corresponding estimator of V is T
VZ T 5 T 21
T
((k t51 s51
S D
6t 2 s6 Vt Vs' , bT
which can be written as GZ T 1 GZ T' 1 SZ T , where SZ T 5 T 21 ( Tt51 Vt Vt ' + Because SZ T rp S under A1 and A2, VZ T is a consistent estimator of V whenever GZ T is a consistent estimator of G+ Consider the following assumptions on k~{! and $bT %+ A3+ ~i! k~0! 5 1, k~{! is continuous at zero and supx$0 6k~ x!6 , `+ ~ii! *@0,`! k~ O x! dx , `, where k~ O x! 5 supy$x 6k~ y!6+
A4+ $bT % # ~0,`! and lim Tr` ~bT21 1 T 2102 bT ! 5 0+ Assumption A3 generalizes Robinson’s ~1991! assumption A2~0! and would appear to be satisfied by any kernel in actual use+1 For instance, it holds for the 15 kernels studied by Ng and Perron ~1996!+ Moreover, it holds for all kernels in the class K 3 of Andrews ~1991! and Andrews and Monahan ~1992! and for all kernels satisfying Assumptions 1 and 3 of Newey and West ~1994!+ Assumption A4 is standard and holds whenever the bandwidth expansion rate coincides
COVARIANCE MATRIX ESTIMATION
1451
with the optimal ~under a mean squared error criterion! rate reported in Andrews ~1991, p+ 830!+ An important implication of A3~ii! is the following lemma+ LEMMA 1+ Suppose k~{! satisfies A3(ii) and suppose $bT % # ~0,`!. Then
( *k i51
T21
lim Tr` sup0,a#au bT21
S D* i abT
,`
for any 0 , au , `. Results similar to Lemma 1 have been stated ~without proof ! by Andrews ~1991, p+ 852!, Hansen ~1992, p+ 970!, and Hall ~2000, Lemma 2!+ As demonstrated by the examples that follow, the assumptions made in the cited papers do not imply lim Tr` bT21 ( T21 i51 6k~i0bT !6 , `+ Therefore, the proofs of Theorem 1 of Andrews ~1991! and Theorems 1–3 of Hansen ~1992! are in error as they stand, as are the proofs of Theorems 2 and 3 of Hall ~2000!+ The sequence $bT21 ( T21 i51 6k~i0bT !6% depends on k~{! through $k~ x! : x [ D%, where D 5 øT$1 ø1#i#T21 $i0bT %+ The set D is countable, so it is possible to have k~ x! 5 1 ∀x [ D under Hansen’s ~1992! Condition ~K! and Hall’s ~2000! Assumption 5+ In particular, if lim Tr` T 21 bT 5 0 it is possible to have
( *k i51
T21
lim Tr` bT21
S D* i bT
5 lim Tr`
T 21 5 `+ bT
Moreover, a kernel can have k~ x! 5 1 ∀x [ N and belong to the class K 1 of Andrews ~1991! and Andrews and Monahan ~1992!+ For any such kernel and any $bT % # N with lim Tr` T 2102 bT 5 0,
(*
T21
lim Tr` bT21
i51
S D*
i k bT
{~T21!0bT }
$
lim Tr` bT21
(
6k~ x!6
x51
5 lim Tr`
{~T 2 1!0bT } 5 `, bT
where {{} denotes the integer part of the argument+2 Assumption A3~ii! rules out pathological cases such as these+ The main result of the paper is the following theorem+ THEOREM 2+ Suppose A1–A4 hold. Then GZ T rp G and VZ T rp V. In applications, the vectors $Vt % are often functions of an unknown parameter vector u ~say!, Vt 5 Vt ~u0 !, where u0 denotes the true value of u+ Consider the estimators
1452
MICHAEL JANSSON
S D (( S D
6t 2 s6 Vt ~ uZ T !Vs ~ uZ T ! ', bT
T t21
GZ T ~ uZ T ! 5 T 21
((k t52 s51 T
VZ T ~ uZ T ! 5 T 21
6t 2 s6 Vt ~ uZ T !Vs ~ uZ T ! ', bT
T
k
t51 s51
where uZ T is an estimator of u0 satisfying the following assumption+ A5+ Either ~i! Vt ~u! 5 Vt ~u0 ! 2 ~u 2 u0 !X t , where T 102 ~ uZ T 2 u0 !dT21 5 Op ~1! and max1#t#T 7dT X t 7 5 Op ~1! for some sequence $dT % of nonsingular matrices or ~ii! T 102 ~ uZ T 2 u0 ! 5 Op ~1! and supt$1 E~supu[N 7~]0]u ' !Vt ~u!7 2 ! , ` for some neighborhood N of u0 +
Assumption A5~i! is Condition ~V3! of Hansen ~1992! whereas A5~ii! is equivalent to Assumption B of Andrews ~1991! under A1 and A2+ As in Hansen ~1992!, the following corollary is an immediate consequence of Theorem 2+ COROLLARY 3+ Suppose A1–A5 hold. Then GZ T ~ uZ T ! rp G and VZ T ~ uZ T ! rp V. Sample-dependent bandwidth parameters can also be accommodated+ Let
S D (( S D T t21
GZ T ~ uZ T , bZ T ! 5 T 21
((k t52 s51 T
VZ T ~ uZ T , bZ T ! 5 T 21
T
k
t51 s51
6t 2 s6 Vt ~ uZ T !Vs ~ uZ T ! ', bZ T
6t 2 s6 Vt ~ uZ T !Vs ~ uZ T ! ', bZ T
where $ bZ T : T $ 1% is a sequence of ~possibly! stochastic bandwidth parameters satisfying the following assumption+ A4 '+ bZ T 5 a[ T bT , where a[ T . 0, a[ T 1 a[ T21 5 Op ~1!, and $bT % satisfies A4+ COROLLARY 4+ Suppose A1–A3, A4 ', and A5 hold. Then GZ T ~ uZ T , bZ T ! rp G and VZ T ~ uZ T , bZ T ! rp V. 3. PROOFS Using change of variables, GZ T can be written as T21
GZ T 5
(k i51
S DS i bT
T 21
T2i
( Vj1i Vj'
j51
D
+
The proofs of Theorem 2 and its corollaries are based on this representation, Lemma 1, and the following lemmas+
COVARIANCE MATRIX ESTIMATION
1453
LEMMA 5+ Suppose A1 and A2 hold. Then
*
( @Vj1i Vj' 2 E~Vj1i Vj' !# * # bi cT 1 T 2102 hT ,
T2i
E T 21
0 # i # T 2 1,
j51
where $ bi : i $ 0% and $cT , hT : T $ 1% are nonnegative sequences with ` ( i51 bi , `, lim Tr` cT 5 0, and lim Tr` hT , `. LEMMA 6+ Suppose A1 and A2 hold. Moreover, suppose k~{! satisfies A3(i) and suppose $bT % # ~0,`! with lim Tr` bT21 5 0. Then lim Tr` supa$al 7E @ GZ T ~u0 , abT !# 2 G7 5 0 for any 0 , al , `. Proof of Lemma 1+ Let kO be defined as in A3~ii!+ By monotonicity of k,O
* k S ab D* # kO S ab D # kO S a b D # a b E i
i
i
u
T
T
u
T
T
@~i21!0au bT , i0au bT !
k~ O x! dx
for any 1 # i # T 2 1 and any 0 , a # au + As a consequence,
( *k i51
T21
sup0,a#au bT21
S D* E i abT
# au
# au
@0, ~T21!0au bT !
E
@0,`!
k~ O x! dx
n
k~ O x! dx , `+
Proof of Lemma 5+ Because Vj1i Vj ' 5
S
`
( Cl ej1i2l l50
DS
`
( Cm ej2m m50
`
5
`
D
'
`
' ' Cm' 1 ( ( 1$l Þ m 1 i %Cl ej1i2l ej2m Cm' , ( Cm1i ej2m ej2m m50 l50 m50
where 1${% is the indicator function, it follows that T2i
T 21
( ~Vj1i Vj' 2 E~Vj1i Vj' !!
j51
`
5
( Cm1i m50 `
1
`
S
(( l50 m50
T2i
T 21
S
' 2 In ! ( ~ej2m ej2m
j51
Cl 1$l Þ m 1 i %T 21
D
T2i
Cm'
' ej1i2l ej2m ( j51
D
Cm'
1454
MICHAEL JANSSON
under A1 and A2+ Using subadditivity of 7{7 and the fact that 7AB7 # 7A7{7B7 for conformable A and B, this expression can be bounded as follows:
*
~Vj1i Vj ' 2 E~Vj1i Vj ' !! * ( j51
T2i
T 21
(* m50 `
#
( (* l50 m50 `
1
' ~ej2m ej2m 2 In ! * 7Cm 77Cm1i 7 ( j51
T2i
T 21 `
' ej1i2l ej2m ( * 7Cl 77Cm7+ j51
T2i
1$l Þ m 1 i %T 21
' ' 2102 Therefore, E7T 21 ( T2i hT , where j51 @Vj1i Vj 2 E~Vj1i Vj !#7 # bi cT 1 T `
bi 5
( 7Cm 77Cm1i 7,
m50
cT 5 sup
m$0 0#i#T21
hT 5
S
*
max
' ~ej2m ej2m 2 In ! * , ( j51
T2i
max E T 21
*
0#i#T21 l, m$0
By A1, `
( bi 5
i51
`
`
((
' ej1i2l ej2m ( * j51
T2i
sup E 1$l Þ m 1 i %T 2102
i51 m50
S( D
2
`
7Cl 7 +
l50
2
`
7Cm 77Cm1i 7 #
DS ( D
7Cm 7
, `+
m50
' 2 In : j $ 1% is a uniformly integrable martingale Each element of $ej2m ej2m difference sequence under A2+ As a consequence, for any « . 0 there is a finite constant l « ~independent of i and m! such that
*
' 2 In ! * # ~T 2 i !102 T 21 l « 1 ~T 2 i !T 21 « ( ~ej2m ej2m
T2i
E T 21
j51
# T 2102 l « 1 «, where the first inequality is obtained by proceeding as in the proof of Hall and Heyde ~1980, Theorem 2+22!+ Therefore, lim Tr` cT # « for any « . 0, so cT r 0+ Finally,
*
T2i
1$l Þ m 1 i %T 2102
' ( ej1i2l ej2m
j51
5 1$l Þ m 1 i %tr
FS
T
*
2
( j 51
DS '
T2i
2102
ej11i2l ej'12m
1
T2i
T 2102
T2i T2i
5 T 21
( ( 1$l Þ m 1 i %ej' 2m ej 2m ej' 1i2l ej 1i2l +
j151 j 251
1
2
1
( ej 1i2l ej' 2m
j 251
2
2
2
DG
COVARIANCE MATRIX ESTIMATION
1455
By A2, E~1$l Þ m 1 i %ej'22m ej12m ej'11i2l ej 21i2l ! 5 n 2 1$ j1 5 j 2 %1$l Þ m 1 i %, because, e+g+, E~ej'22m ej12m ej'11i2l ej 21i2l ! 5 E @Ej 22m ~ej'12m !ej 22m ej'11i2l ej 21i2l # 5 0 when j1 . j 2 and l . m 1 i, whereas ' ' ej2m !ej1i2l ej1i2l # 5 n 2 E~ej'22m ej12m ej'11i2l ej 21i2l ! 5 E @Ej1i2l ~ej2m
when j1 5 j 2 5 j and l . m 1 i+ Therefore,
*
' ( ej1i2l ej2m *
T2i
E 1$l Þ m 1 i %T 2102 #E
S
S*
j51
T2i
1$l Þ m 1 i %T 2102
' ( ej1i2l ej2m
j51
*D 2
((
j151 j 251
D
102
T2i T2i
5 T 21
102
n 2 1$ j1 5 j 2 %1$l Þ m 1 i %
5 @1$l Þ m 1 i %n 2 T 21 ~T 2 i !# 102 # n, where the first inequality uses the Cauchy–Schwarz inequality+ In particular,
S( D
2
`
lim Tr` hT # n{
, `,
7Cl 7
l50
n
as was to be shown+ Proof of Lemma 6+ Under A1 and A2, `
E @ GZ T ~u0 , abT !# 5
( 1$i # T 2 1%k i51
for any a . 0, whereas G 5 of 7{7, 7E @ GZ T ~u0 , abT !# 2 G7
( * 1$i # T 2 1%k i51 I
#
*
S
D
1 sup 6k~ x!6 1 1 { x$0
T2i 2 1 {7E~V11i V1' !7 T
i abT
i abT
`
T2i E~V11i V1' ! T
` ( i51 E~V11i V1' !+ Therefore, by subadditivity
i abT
# max 1$i # T 2 1%k 1#i#I
i abT
S D S D S D
( * 1$i # T 2 1%k i5I11 `
1
S D
*
T2i 2 1 {7E~V11i V1' !7 T
*
I T2i 2 1 { ( 7E~V11i V1' !7 T i51
*
7E~V11i V1' !7 ( i5I11
1456
MICHAEL JANSSON
for any I $ 1+ Because
S
D
`
`
sup 6k~ x!6 1 1 { x$0
( i50 7E~V11i V1' !7 , `,
( 7E~V11i V1' !7 i5I11
can be made arbitrarily small ~under A3~i!! by taking I large enough+ For any given I,
*
S D *S D S* S D *
sup max 1$i # T 2 1%k
a$al 1#i#I
# sup max
a$al 1#i#I
#
sup 0#x#I0~al bT !
k
T2i 21 T
T2i 21 T
i abT
5 sup max k a$al 1#i#I
i abT
i abT
*
* * S ab D*D
2 1 1 T 21 i k
i
T
6k~ x! 2 16 1 T 21 I sup 6k~ x!6 x$0
whenever 0 , al , ` and T $ I 1 1+ Lemma 6 now follows because the expression on the last line tends to zero whenever A3~i! holds and n lim Tr` bT21 5 0+ Proof of Theorem 2+ By subadditivity of 7{7, 7 GZ T 2 G7 # 7 GZ T 2 E~ GZ T !7 1 7E~ GZ T ! 2 G7, 7 VZ T 2 V7 # 7 VZ T 2 E~ VZ T !7 1 7E~ VZ T ! 2 V7+ Now, E ~ SZ T ! 5 S, so 7E ~ VZ T ! 2 V7 # 2{7E ~ GZ T ! 2 G7 r 0 by Lemma 6+ Moreover, 7 VZ T 2 E~ VZ T !7 # 2{7 GZ T 2 E~ GZ T !7 1 7 SZ T 2 E~ SZ T !7+ By Lemma 5,
*
@Vj Vj ' 2 E~Vj Vj ' !# * r 0+ ( j51 T
E7 SZ T 2 E~ SZ T !7 5 E T 21
In particular, SZ T 2 E~ SZ T ! 5 op ~1!+ The proof of Theorem 2 can be completed by showing that E7 GZ T 2 E~ GZ T !7 r 0+ By subadditivity of 7{7,
(*
T21
7 GZ T 2 E~ GZ T !7 #
i51
k
S D* * i bT
{ T 21
( @Vj1i Vj' 2 E~Vj1i Vi' !# * +
T2i j51
COVARIANCE MATRIX ESTIMATION
1457
Using Lemmas 1 and 5 and the notation from Lemma 5,
S D* S( D S( D
(*
T21
E7 GZ T 2 E~ GZ T !7 #
i51
k
i bT
*
{E T 21
( @Vj1i Vj' 2 E~Vj1i Vi' !# *
T2i j51
T21
# k~0! O
i51 `
# k~0! O
( *k i51
T21
bi cT 1 T 2102 hT
S
S D* i bT
i51
( *k i51
T21
bi cT 1 ~T 2102 bT hT ! bT21
S D *D i bT
r 0+
n
Proof of Corollaries 3 and 4+ Corollary 3 is a special case ~with a[ T 5 1! of Corollary 4, so it suffices to prove the latter+ Under A4 ', inft$T Pr ~al # a[ t # au ! can be made arbitrarily close to unity for sufficiently large T and some 0 , al # au , `+ Consequently, it suffices to show that for any 0 , al # au , `, sup 7 GZ T ~ uZ T , abT ! 2 G7 5 op * ~1!,
al #a#au
which is easily shown to imply supal #a#au 7 VZ T ~ uZ T , abT ! 2 V7 5 op * ~1!, where op * ~1! denotes convergence to zero in outer probability+3 Now, sup 7 GZ T ~ uZ T , abT ! 2 G7 #
al #a#au
sup 7 GZ T ~ uZ T , abT ! 2 GZ T ~u0 , abT !7
al #a#au
1 1
sup 7 GZ T ~u0 , abT ! 2 E @ GZ T ~u0 , abT !#7
al #a#au
sup 7E @ GZ T ~u0 , abT !# 2 G7,
al #a#au
so the proof can be completed by showing that each term on the right-hand side is op * ~1!+ As in the proofs of Theorems 2 and 3 in Hansen ~1992!, Condition A5 implies that
S
7 GZ T ~ uZ T , abT ! 2 GZ T ~u0 , abT !7 # ~T 2102 bT !{ bT21
( *k i51
T21
S D *D i abT
{QT ,
for any a . 0, where QT is Op ~1! and does not depend on a or bT + Now, T 2102 bT r 0 and lim Tr` supal #a#au bT21 ( T21 i51 6k~i0~abT !!6 , ` for any 0 , al # au , ` ~Lemma 1!, so sup 7 GZ T ~ uZ T , abT ! 2 GZ T ~u0 , abT !7 5 op * ~1!+
al #a#au
1458
MICHAEL JANSSON
Next, by the properties of k~{!, O 7 GZ T ~u0 , abT ! 2 E @ GZ T ~u0 , abT !#7
S D*** ( S D*
( *k i51
T21
#
T21
#
kO
i51
i abT
i au bT
( @Vj1i Vj' 2 E~Vj1i Vi' !# *
T2i
T 21
T 21
j51
( @Vj1i Vj' 2 E~Vj1i Vi' !# *
T2i j51
for any 0 , al # a # au , `+ By Theorem 2 and its proof, the last line is op ~1!, so sup 7 GZ T ~u0 , abT ! 2 E~ GZ T ~u0 , abT !!7 5 op * ~1!+
al #a#au
Finally, supal #a#au 7E @ GZ T ~u0 , abT !# 2 G7 r 0 by Lemma 6+
n
NOTES 1+ A3 is weaker than Robinson’s ~1991! A2~0! because it is not assumed that *R 6K~l!6dl , `, where K~l! 5 ~2p!21 *R k~ x!exp~ilx! dx+ A well-known example of a kernel satisfying A3 but violating *R 6K~l!6dl , ` is the uniform kernel k~ x! 5 1$6 x6 # 1%, where 1${% is the indicator function+ ` 2+ One kernel k [ K 1 such that k~ x! 5 1 ∀x [ Z is k~{! 5 ( l50 k l ~{!, where, for each l $ 0 and any x [ R,
k l ~ x! 5
5
1 2 ~l 1 1! 2 ~6 x6 2 l !,
if l # 6 x6 # l 1 ~l 1 1!22,
1 2 ~l 1 1! 2 ~l 1 1 2 6 x6!,
if l 1 1 2 ~l 1 1!22 # 6 x6 # l 1 1,
0,
otherwise+
It is easily seen that k [ K 1 + In particular, k is continuous and
E
R
`
6k~ x!6 dx 5 2 (
l50
E
@0,`!
`
p2
l50
3
k l ~ x! dx 5 2 ( ~l 1 1!22 5
, `+
3+ To avoid measurability complications, convergence in outer probability ~rather than convergence in probability! is considered+
REFERENCES Andrews, D+W+K+ ~1991! Heteroskedasticity and autocorrelation consistent covariance matrix estimation+ Econometrica 59, 817–858+ Andrews, D+W+K+ & J+C+ Monahan ~1992! An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator+ Econometrica 60, 953–966+ Hall, A+ ~2000! Covariance matrix estimation and the power of the overidentifying restrictions test+ Econometrica 68, 1517–1527+ Hall, P+ & C+C+ Heyde ~1980! Martingale Limit Theory and Its Application+ New York: Academic Press+ Hansen, B+E+ ~1992! Consistent covariance matrix estimation for dependent heterogeneous processes+ Econometrica 60, 967–972+
COVARIANCE MATRIX ESTIMATION
1459
de Jong, R+M+ & J+ Davidson ~2000! Consistency of kernel estimators of heteroscedastic and autocorrelated covariance matrices+ Econometrica 68, 407– 423+ Newey, W+K+ & K+D+ West ~1994! Automatic lag selection in covariance matrix estimation+ Review of Economic Studies 61, 631– 653+ Ng, S+ & P+ Perron ~1996! The exact error in estimating the spectral density at the origin+ Journal of Time Series Analysis 17, 379– 408+ Park, J+Y+ ~1992! Canonical cointegrating regressions+ Econometrica 60, 119–143+ Phillips, P+C+B+ & B+E+ Hansen ~1990! Statistical inference in instrumental variables regression with I~1! variables+ Review of Economic Studies 57, 99–125+ Robinson, P+M+ ~1991! Automatic frequency domain inference on semiparametric and nonparametric models+ Econometrica 59, 1329–1363+
