Testing Constancy in Varying Coefficient Models

2019-01

Working paper. Economics

ISSN 2340-5031

Testing Constancy in Varying Coefficient

Models

Serie disponible en http://hdl.handle.net/10016/11

Web: http://economia.uc3m.es/

Correo electrónico: [email protected]

Creative Commons Reconocimiento-NoComercial- SinObraDerivada 3.0 España

Testing Constancy in Varying Coefficient Models

Miguel A. Delgado2

& Luis A. Arteaga-Molina Universidad Carlos III de Madrid Universidad de Cantabria

Abstract

This article proposes tests for constancy of coefficients in semi-varying coefficients models. The testing procedure resembles in spirit the union-intersection parameter stability tests in time series, where observations are sorted according to the explanatory variable responsible for the coefficients varying. The test can be applied to model specification checks of interactive effects in linear regression models. Because test statistics are not asymptotically pivotal, critical values and p-values are estimated using a bootstrap technique. The finite sample properties of the test are investigated by means of Monte Carlo experiments, where the new proposal is compared to existing tests based on smooth estimates of the unrestricted model. We also report an application to returns of education modeling.

Keywords: Varying coefficient models; Model checks; U-I tests; Concomitants; Interactive effects model checks. JEL Codes: C12, C14, C52.

1_{Research funded by Ministerio Economía y Competitividad (Spain), ECO2017-86675-P & MDM 2014-0431, and by}

Comunidad de Madrid (Spain), MadEco-CM S2015/HUM-3444.

2 _{Corresponding author: e-mail: [email protected]}

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_2?5

X2 , X21, ...., X2k

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$@56= ?6DED 5:D4@?E:?F@FD C68C6DD:@? >@56=D H96C6

β₀z β₀₀ β₀₁zz,

7@C A2C2>6E6C G64E@CDβ₀₀2?5β₀₁,H96C6 E96 5:D4@?E:?F:EJ :D 6IA=2:?65 3J E96 G2C:23=6Z, H9:49 :D E96 EJA:42= 2=E6C?2E:G6 E@ A2C2>6E6C DE23:=:EJ 9JA@E96D:D :? E:>6 D6C:6D 2?2=JD:D H:E9 A2C2>6E6CD 492?8:?8 2E 2? F?<?@H? E:>6 A@:?E E :D ?@E A@DD:3=6 4@?D:DE6?E=J 6DE:>2E:?8 β₀ :? >@56= FD:?8 D>@@E9:?8 32D65 >6E9@5D

+96 8@2= @7 E9:D 2CE:4=6 4@?D:DED @7 E6DE:?8 E92E E96 G2CJ:?8 4@6Q 4:6?ED :? >@56= 2C6 4@?DE2?E :? E96 5:C64E:@? @7 ?@?A2C2>6EC:4 2=E6C?2E:G6D :6 E6DE:?8

H0 V ar β0j 7@C 2== j ,, ..., k1 GD H1 V ar β0j 7@C D@>6 j ,, ..., k1. $@56= 2=D@ ?6DED 2 >@56= H:E9 X2 g0Z , X11g1Z , ..., X1kg kZ , δ0 δ00,δ 01, ...,δ 0k 2?5 k2 k j=0 mj, H96C6δ0j 2C6 F?<?@H?mj×A2C2>6E6C G64E@CD 2?5gj R→Rmj 2C6 <?@H? 7F?4E:@?D

j , ..., k1. ? E9:D 42D6 42? 36 6IAC6DD65 2D

EY_|X, Z X₁β₀Z μ₀Z a.s.

H:E9 ?@?A2C2>6EC:4 β₀ 2?5 A2C2>6EC:4

μ_0· g_0· δ00, g1· δ01, ..., gk· δ0k

7@C D@>6 δ0 δ00,δ01, ...,δ0k

∈Rk_{. +96C67@C6 F?56C E96 >2:?E2:?65 9JA@E96D:D} 2?5 2C6 6BF:G2=6?E E@ @>?:3FD >@56= 4964<:?8 E92E E96 >2C8:?2= 6R64ED @7 X1 2C6 μ₀Z , H9:49 :>A=:6D 2 A2CE:4F=2C A2C2>6E6C:K2E:@? @7 E96 :?E6C24E:G6 6R64ED %665=6DD E@ D2J E92E >@56= :D ?@E :56?E:L23=6 :? >2?J 4:C4F>DE2?46D, 3FE E96 E6DE H6 AC@A@D6 5@6D ?@E ?665 6DE:>2E:?8 E96 >@56= F?56C E96 2=E6C?2E:G6 9JA@E96D:D ? A2CE:4F=2C @FC E6DE :D :? 724E 2 5:C64E:@?2= DA64:L42E:@? E6DE 7@C E96 =:?62C :? A2C2>6E6CD C68C6DD:@? >@56= :? E96 5:C64E:@? @7 2 D6>:G2CJ:?8 4@6Q 4:6?E >@56= E 42? 2=D@ 36 2AA=:65 2D 2? @>?:3FD DA64:L42E:@? E6DE @7 2 D:>A=6 C68C6DD:@? >@56= H:E9 6IA=2?2E@CJ G2C:23=6Z, :6 k1 :? .

"2F6C>2?? 2?5 +FEK 2: 6E 2= 2? 2?5 192?8 2? 6E 2= 2? 2?5 F2?8 (F 2?5 #: 2?5 2: 6E 2= 92G6 4@?D:56C65 E6DE:?8 32D65 @? E96 5:D4C6A2?4J 36EH66? C6DEC:4E65 2?5 F?C6DEC:4E65 DF> @7 DBF2C65 C6D:5F2=D FD:?8 D>@@E9 6DE:>2E6D @7 E96 G2CJ:?8 4@6Q 4:6?ED ? E96D6 AC@A@D2=D D>@@E9 6DE:>2E6D @7β_0j 2C6 ?66565 2?5 96?46 D:EF2E:@?D =:<6 2C6 CF=65 @FE =D@ E96D6 E6DED 2C6 ?@E 2AA=:423=6 H96? E96 >@56= @? E96 2=E6C?2E:G6 :D ?@E :56?E:L65

? E9:D A2A6C H6 252AE 4=2DD:42= A2C2>6E6C DE23:=:EJ E6DED :? E:>6 D6C:6D 68 (F2?5E 96C?@R 2?5 124<D 2EE2492CJJ2 2?5 !@9?D@? :?<=6J

Brown et al., 1975; Sen and Srivastava, 1975; Hawkins, 1977, 1989; Nyblon, 1989; An-drews, 1993; Csörg½o and Hortváth, 1988, 1997; Aue et al., 2008 among many others.)

Given (Yi; Zi; X1i;X2i)n_i=1 i:i:d: as (Y; Z;X1;X2), we interpret (Yi; X1i;X2i)n_i=1 as

sequentially observed with respect to the ordered values of _fZig n

i=1: That is, denote by

Y[i:n];X1[i:n];X2[i:n]

n

i=1 the Z concomitants, or induced order statistics, of

(Yi;X1i;X2i) n

i=1; i.e. Y[i:n];X1[i:n];X2[i:n] = (Yj;X1j;X2j) i¤ Z(n:i) = Zj; where Z(n:1) Z(n:2) ::: Z(n:n) are the ordered statistics of fZign_i=1: We propose to adapt

union-intersection (U-I) type tests in time series to our context. See Hawkins (1989), Andrews (1993), Horváth and Shao (1995) or Csörg½o and Hortváth (1997, Section 3.1.5.) The test consists of comparing ordinary least squares (OLS) estimators ofX1 coe¢ cients

using subsamples Y[i:n];X1[i:n];X2[i:n] j i=1 and Y[i:n];X1[i:n];X2[i:n] n i=j+1 at each j th sample Z quantile.

The rest of the article is organized as follows. Next section discusses and justi…es the testing procedure. Section 3 studies the …nite sample performance of the test in the context of a Monte Carlo experiment. We report comparisons of existing tests for coe¢ cient constancy based on smooth 0 estimates, as well as speci…cation CUSUM type

tests, as proposed by Stute (1997) and Andrews (1997), which are omnibus, i.e. designed to detect any alternative, much broader thanH1 in (1). In Section 4 we apply the testing

procedure to modeling interactive e¤ects ofIQwhen studying education returns. Section 5 is devoted to conclusions. Mathematical proofs can be found in an appendix at the end of the article.

2. TESTING METHOD

De…ne M`j(u) = E X`Xjt1fFZ(Z) ug and Sj(u) = E XjY1fFZ(Z) ug ; j; ` = 1;2;

where FZ is the cdf of Z: Assume that A.1 FZ is continuous. A.2 Rank 8 < : 2 4 M11(u) M12(u) M21(u) M22(1) 3 5 9 = ;=k1+k2+ 1 for all u2[0;1]:

For the sake of exposition assume w.l.o.g. thatZ is uniformly distributed on [0;1]:An U-I test of H0 is based on the sample version of 0(u) = 0

+

0 (u); where 0(u) =

θ₀u ,θ₀+u ,θo₀u , θ0u % !" θ_,θ,θo E Y ₋X₁θ₋X₂θo Zu 2 _E Y ₋X₁θ+₋X₂θo Z>u 2 M1u Su , u_∈,, Mu ⎡ ⎢ ⎢ ⎢ ⎣ M11u k+1 M12u k+1 M11 −M11u M12 −M12u M21u M21 −M21u M22 ⎤ ⎥ ⎥ ⎥ ⎦, m :D 2m×m >2EC:I @7 K6C@6D 2?5 Su S1u ,S1 −S1u , S2 .

&3G:@FD=JV ar β_0jZ 7@C 2==j , ..., k1 :>A=:6D E92Eη0u 7@C 2==u∈,. ? C6=6G2?E 4:C4F>DE2?46D 5:D4FDD65 36=@H 2=D@V ar β_0jZ if f η₀u 7@C 2== u_∈,

#'+& $#(' M11u uM11 , , ( !,.( ()(0 # ) %')!. !#'

"$! # M12u uM12 $' !! u _∈ ,, , ( &*+!#) )$ _EX1X1|Z M11 a.s.#EX1X2|Z M12 a.s.'$'S1u M11 E β0Z Zu , # %%!.# "" # #',( η₀u E β0Z Zu −uEβ0Z u₋u u₋u _Zuβ0Z −Eβ0Z dP $' !! u_∈,_⇔β₀Z _Eβ₀Z a.s. #'+& $#(' δ0 %*' +'.# $2 #) "$!. # $' !! u ∈ ,, η₀u M₁₁1u S1u −M11 −M11u 1S1 −S1u M11 −M11u 1M11 M111u S1u −M111 S1 , # η₀u !! u_∈, _⇔ S1u −M11u M111 S1 !! u∈, ⇔ ZuJZ β0Z −EJZ 1 EJZ β₀Z d_P . ,) JZ _EX1X1|Z . # JZ ( #$#(#*!' (, η₀u !! u_∈, _⇔ β₀Z _EJZ 1_EJZ β₀Z a.s. ⇔ V ar β_0jZ $' !! j , ..., k1.

+96 23@G6 EH@ C6>2C<D D9@H E92E E6DE:?8

H0 η0u 2== u∈, GD H1 η0u D@>6 u∈,.

:D 6BF:G2=6?E E@ :? AFC6 G2CJ:?8 4@6Q 4:6?E C68C6DD:@? >@56=D 2D H6== 2D :? D:EF2E:@?D H96C6 E96 6=6>6?ED :? X1X1 2C6 >62? :?56A6?56?E @7 Z. =D@ C6;64E:?8 H0 :? E96 5:C64E:@? @7 H1 :>A=:6D C6;64E:?8 H0 :? E96 5:C64E:@? @7 H1 7@C >2?J @E96C D6>:G2CJ:?8 4@6Q 4:6?E >@56=D 2D 5:D4FDD65 :? *64E:@? +96 D2>A=6 2?2=@8 @7 :Dˆθnu ˆθ n u ,ˆθ + n u ,ˆθ o n u H:E9 ˆ θnu % !" θ_,θ,θo ⎧ ⎨ ⎩ nu i=1 Y[i:n]−X1[i:n]θ−X2[i:n]θ o 2 n i=1+nu Y[i:n]−X1[i:n]θ + −X_2[i:n]θo 2 ⎫ ⎬ ⎭ Mˆ_n1u Sˆnu , u∈,,

H96C6 _· >62?D D>2==6DE ?62C6DE :?E686C, Sˆnu Sn1u ,Sn1 −Sn1u ,Sn2

, Snju _i=1nuXj[i:n]Y[i:n], j ,, ˆ Mnu ⎡ ⎢ ⎢ ⎢ ⎣ M11nu k+1 M12nu k+1 M11n −M11nu M12n −M12nu M21nu M21n −M21u Mn22 ⎤ ⎥ ⎥ ⎥ ⎦, 2?5 Mn ju n1 nu i=1 X [i:n]Xj[i:n], , j ,. *:>:=2C 6IAC6DD:@?D 42? 36 7@F?5 :? E:>6 D6C:6D A2C2>6E6C DE23:=:EJ E6DE:?8. +9:D DF886DED E6DE DE2E:DE:4D 7@C H0 32D65 @? DF:E23=6 7F?4E:@?2=D @7

η_nu ˆθ_{n −}θˆ+_n u RMˆ_n1u Sˆnu ,

H:E9R Ik+1 −Ik+1 k 2?5Im :D 2m×m :56?E:EJ >2EC:I,H9:49 :D E96 5:R6C6?46 36EH66? &#* 6DE:>2E@CD @7 X1 4@6Q 4:6?ED F?56C H0 FD:?8 DF3D2>A=6D

Y[i:n],X1[i:n],X2[i:n] j i=1 2?5 Y[i:n],X1[i:n],X2[i:n] n i=j+1. %@E:46 E92E ˆ θnu θ0u Mˆ 1 n u Nˆnu , ˆ Nnu Nn1u ,Nn1 −Nn1u ,Nn2u 2?5 Nnju n1 nu i=1 Xj[i:n]U[i:n] j

, 2?5 Ui Yi −X1iβ0Z −X2iδ0, . +96 2DJ>AE@E:4 5:DEC:3FE:@? @7 Nn :D @3E2:?65

2AA=J:?8 C6DF=ED 7@C A2CE:2= DF>D @7 4@?4@>:E2?ED :? 2E92492CJ2 6IE6?565 3J *6? *EFE6 @C 2GJ5@G 2?5 8@C@G 2>@?8 @E96CD 6L?6 Nu N 1u , N1 −N1u , N2u _,H96C6N j 36kj×, j ,,G64E@CD @7

46?E6C65 2FDD:2? AC@46DD6D H:E9_E Nu N_jv _E X X_jU2

Zuv , , j ,. %6IE 2DDF>AE:@? DFQ 46D E@ D9@H H62< 4@?G6C86?46 @7 √nNˆn 2?5 F?:7@C> 4@?G6C86?46

@7 Mˆn

_E XU 2 <_∞.

6?467@CE9 7@C 2?J >2EC:I A, A λAA :D E96 DA64EC2= ?@C> H96C6 λC :D E96 >2I:>F> 6:86?G2=F6 @7 E96 >2EC:I C, 2?5 _→d >62?D 4@?G6C86?46 :? 5:DEC:3FE:@?

@7 C2?5@> G2C:23=6D C2?5@> G64E@CD @C C2?5@> 6=6>6?ED :? 2 *<@C@9@GMD DA246 Da, b, _≤a < b_≤. +)*),%-%)( ((*"# AA # A, √ n N_n1,N_{n2 →}dN1, N2 _:?D,, # ! nu&'$[0,1] Mn j −Mj u a.s., , j ,. +96C67@C6 D:?46 η_nu θ_{0 −}θ+₀ u RMˆ_n1u Nˆnu , F?56C H0 2?5 4@?5:E:@?D :? 'C@A@D:E:@? √ nη_n→d η in D ,− , ∈, , H96C6 ηu d R_M1_{u N}0 u H:E9 N0u N0₁u , N0_{1 −}N0₁u , N0₂u , 2?5 N0

, ,:D 2 G64E@C @7 >62? K6C@ 2FDD:2? AC@46DD6D H:E9E N0 u N0jv E X X_jV2Zuv 0 ju∧v ,H:E9 V Y −X1β¯0−X2¯δ0 F?4@CC6=2E65 H:E9 E96 4@>A@?6?ED @7X₁,X₂ 2?5 β¯₀,¯δ_{0 ∈R}1+k+k_{.+92E :D β}¯

0,¯δ0 2C6 E96 A2C2>6E6CD @7 E96 36DE =:?62C AC65:4E@C @7 Y 8:G6? X1,X2 , 2?5 V U 2D F?56C H0. .62< 4@?G6C86?46 @7√nη_n :?D,:D ?@E A@DD:3=6 6G6? 2DDF>:?8 E92EZ :D :?56A6?56?E @7X

2?5 U 2D D9@H? 3J 9:3:D@G 7@C E96 DE2?52C5 6>A:C:42= AC@46DD D66 DF3D64E:@? :? 26?DD=6C 2?5 *EFE6 7@C 5:D4FDD:@?

+96C67@C6 7@C _∈,

Eηu ηv Σ0u∧v RM1u Ω0u∧v M1v R, u, v∈ ,− , H:E9Ω0u E N0u N0u .AA=J:?8 E96 , E6DE:?8 AC:?4:A=6 E9:D DF886DED E6DED 32D65 @? 7F?4E:@?2=D @7 E96 6>A:C:42= AC@46DD

αnu ηnu Σˆ 1 n u ηnu , u∈ ,− . H96C6 ˆ Σnu RMˆ 1 n u Ωˆnu Mˆ 1 n u R, 6DE:>2E6DΣ0u 2?5 ˆ Ωnu ⎡ ⎢ ⎢ ⎢ ⎣ n11u k+1 n12u k+1 n11 −n11u n12 −n12u n21u n21 −n21u n22 ⎤ ⎥ ⎥ ⎥ ⎦ 6DE:>2E6D Ω0u , H:E9 n ju n1 _i=1nuX [i:n]Xj[i:n]V[2i:n], , j ,, 2?5 Vi Yi −

X_1iθ+_{n −} X_2iθo_n 2C6 E96 &#* C6D:5F2=D F?56CH0. DFQ 4:6?E 4@?5:E:@? 7@C 4@?D:D E6?4J @7Ωˆnu :D

_E X 4 <_∞2?5 _E V 4 <_∞.

+9:D 4@?5:E:@? 42? 36 C6=2I65 3J 2DDF>:?8 E92E_EV2

|X, Z _EV2 _σ2 _{a.s. H9:49} :>A=:6D E92EΩ0u σ2Mnu 2?5Σˆnu σ2RMˆ

1

n u R @?D:56C E96 , EJA6 E6DE

ϕ_n !( n+Kjn(1)Kn·αn j n 7@C D>2== ∈ , − K n , H:E9 K < n , 2?5 K k1k2 .+96 EC:>>:?8 A2C2>6E6C :D :?EC@5F465 E@ 2G@:5 3@F?52CJ A@:?ED 2?5 D9@F=5 36 49@D6? 2D 4=@D6 E@ K6C@ 2D A@DD:3=6 :? @C56C E@ 56E64E 2?J A@DD:3=6 4@6Q 4:6?E G2C:2E:@? @? 2== :ED 5@>2:? :?4=F5:?8 E9@D6 4=@D6 E@ E96 3@F?52CJ @H6G6C E@@ D>2== G2=F6D 42? AC@5F46 D6C:@FD D:K6 5:DE@CE:@?D D66 D64E:@? +96 2DJ>AE@E:4 5:DEC:3FE:@? @7 ϕ_n :D 56C:G65 2D 2? :>>65:2E6 4@?D6BF6?46 @7 AC@A@D:E:@? 27E6C D9@H:?8 F?:7@C> 4@?D:DE6?4J @7Σˆn. 6L?6 E96 G64E@C @7 C2?5@> AC@46DD6D

{αu _{}u [,1]} d N0u M1u RΣ₀1u RM1u N0u _u

[,1].

%6IE AC@A@D:E:@? 6DE23=:D96D E96 2DJ>AE@E:4 5:DEC:3FE:@? @7ϕ_n 2D 2 4@?D6BF6?46 @7 'C@A@ D:E:@? 27E6C AC@G:5:?8 4@?D:DE6?4J @7 Σˆnu F?:7@C>=J @?u∈ ,− .

Proposition 2: Assume A1 A4: Under H0; for any small …xed 2 (0;1=2 K=n]; K < n=2 ^ '_{n !}d'₁ d = sup u2[;1 ] 1 (u):

Therefore, a test with signi…cance level is given by the binary random variable ^_{n ( ) = 1}

f'^n >c( )g;where c ( ) is the (1 ) th quantile of'1:

These U-I tests in time series are asymptotically distribution-free under suitable regu-larity conditions, which has a counterpart in our context assuming that

A5 Z is independent of X and U:

Of course, this assumption is not acceptable in practice, but it is worth discussing to illustrate the relation of our proposal with related ones in time series parameter instability testing and the behavior of our test statistic when is too small. Consider the 0 = 0

case for simplicity. Under A:5:, M1j(u) = uM1j(1); 1j(u) = 2 u M1j(1); j = 1;2;

fN₁1(u)g_u2[0;1] d = nM₁₁1=2(1) W0(u) o u2[0;1], W0 is a (1 +k1) 1 vector of independent

Wiener’s processes and

0(u) = 2Rt 2 4 uM11(1) 0 0 (1 u)M11(1) 3 5 1 R = 2 M11(1) u(1 u): (10) Therefore, under A5; '₁ =d sup u2[;1 ] B0(u) u(1 u); (11)

where B0(u) = [W0(u) uW0(1)]t[W0(u) uW0(1)] is the sum of 1 +k1 squared

inde-pendent Brownian bridges: The distribution of '₁ has been tabulated by James et al. (1987) for B0 scalar and di¤erent values of ;and by Andrews (1993) in the general case:

Under A5; one can exploit the information in (10) and, after estimating 2 by ~2_n = n 1Pn

i=1V^ 2

ni;use as test statistic, ~ '(0)_n =n max K j n K ~n j n ; with ~n(u) = ^tn(u) ^ M11n(1)u(1 u) ~2_n ^n(u); u2[0;1];

which resembles the classical U-I tests avoiding any trimming. This statistics, suitably standardized, converges to a extremum distribution, which is proved applying Darling and Erd½os (1956) type results for normalized partial sums. To this end, we need the alternative conditions that replaces A3 and A4 by,

0_E|U_|2+δ<_∞ 2?5 _E X 2+δ <_∞ 7@C D@>6 δ>.

+9:D :>A=:6D F?56C A, E92E _E XU 2+δ < _∞ H9:49 :D DEC@?86C E92? A. +96D6 EJA6 @7 >@>6?E 4@?5:E:@?D H2D AC@A@D65 3J *9@C2< E@ 6IE6?5 2C=:?8 2?5 C5S@D C6DF=E E@ 2==@H =6DD E92? E9C66 >@>6?ED +96D6 42? 36 7FCE96C C6=2I65 FD:?8 :?>29= >@>6?E 4@?5:E:@?. 6?467@CE9 x ₀yx1_ey_{dy , 2?5 E :D 2 C2?}

5@> G2C:23=6 DF49 E92E _PE _≤x ($₋ ($₋x , ax √ #x 2?5 bmx

#x m/ # #x₋ # m/ .+96 4@?G6C86?46 @7 ϕ)(0)_n :D D=@H H9:49 C6DF=ED :? 2 A@@C D:K6 244FC24J 2?5 D@>6 2=E6C?2E:G6D >2J 36 AC676CC65 H96? :D D2E:DL65

.6 42? 2=D@ 4@?D:56C E96 C2>UCG $:D6D EJA6 DE2E:DE:4 ) ϕ(1)_n nK j=K ) αn j n , 2?5 E96 F?H6:89E65 DE2E:DE:4

) ϕ(2)_n !( KjnK jn₋j n α)n j n ,

H9:49 3@E9 92G6 36EE6C D:K6 244FC24J F?56C E92? E96 E6DE 32D65 @?ϕ)(0)_n %6IE AC@A@ D:E:@? AC@G:56D E96 =:>:E:?8 5:DEC:3FE:@? @7ϕ)(_nj), j ,, F?56C H0.

+)*),%-%)( ((*" δ0 , 1# *#' H0, a #n ϕ)(0)_{n −}b1+k #n d →E, ) ϕ(1)_{n →}d 1 0 B0u u₋u du, ) ϕ(2)_{n →}d &'$ u [0,1] B0u .

+9:D DF886DED E92E 3642FD6 E96 C2E6 @7 4@?G6C86?46 @7 ϕ_n 492?86D DF556?=J 2E , E6DED 32D65 @? 4C:E:42= G2=F6D @7 E96 2DJ>AE@E:4 2AAC@I:>2E:@? 2C6 6IA64E65 E@ 6I9:3:E A@@C D:K6 244FC24J *66 D:>F=2E:@?D :? D64E:@?

%6IE H6 DEF5J E96 A@H6C @7 E96 E6DE :? E96 5:C64E:@? @7 D6BF6?46D @7 =@42= 2=E6C?2E:G6D @7 E96 7@C> Hn1 βZ β0 τZ √ n a.s.,

7@C 4@?DE2?E β₀ 2?5 2 7F?4E:@? τ _R→R1+k _{DF49 E92E Tu}

E X₁τZ Uu :D 3@F?565 7@C 2== u _∈ , 6L?6 T u T_{u , T −Tu ,} k 2?5 E96 C2?5@> AC@46DD6D, α1u _{u [,1} ] d NT u M1u RΣ₀1u RM1u NT u _{u [,1} ].

In order to study the power of the test underHn1;we need the following extra assumption.

A6 _EkX1 (Z)k<1:

Proposition 4: Assume A1 A4; and A6 for ₂(0;(n 2K)/ 2n]; K < n=2: Under

H1; ^ '_{n !p 1}; (15) and under H1n; ^ '_{n !d} sup u2[;1 ] 1 1(u); (16)

Therefore, the test does not have trivial power in the direction of Hn1 when

sup_{u2[;1 ]} (u)>0 with (u) = Tt(u)M 1(u)Rt ₀1(u)RM 1(u)T(u): Under A5; (u) = T(u) t_M 1 11 (1)T(u) 2 _{u(1 u)} :

This suggests choosing as small as possible in order to give more weight to the extreme values ofZ:

The bootstrapped test statistic is ^ '_n =n sup K+bnc j n K bnc ^_n j n for small 2 0; 1 2 K n ; K < n 2; with ^_n(u) = ^_nt(u)^_n1(u)^_n(u); and ^_n(u) = RM^_n1(u)N^_n(u); where N^_n(u) = N^ t n1(u);N^nt1(1) N^n1t(u);N^nt2(u) t , N^_nj(u) = n 1Pbnuc i=1 Xj[i:n]V^[i:n];

j = 1;2; V^_i = ^V_{i i}; _{f ig}n_i=1 are i:i:d:as ; which satis…es that,

A7 _E( ) = 0,_E 2 = 1 and <₁ a.s

The bootstrap test, justi…ed in next Proposition, is ^n ( ) = 1f'^n >c^n( )g; where

^

c_n( ) = inf_fc:_P (^'_n c) 1 _g and _P is the induced probability of a random variable :

+)*),%-%)( ((*" A₋A, # A $' _∈,n₋K n, K < n/.,#' H1, ! nPξϕ n ≤c Pϕ ≤c 2D 2?5 F?56C H1 E96C6 6I:DED 2 C > DF49 E92E

!

nPξϕ

n > C a.s.

+9:D :>A=:6D E92E E96 2DJ>AE@E:4 A@H6C 7F?4E:@? E2<6D E96 G2=F6 α F?56C H0 2?5 F?56C H1,:6 !nE n α α F?56C H0 2?5 !nE n α F?56C H1. +96 E6DE 42? 2=D@ 36 32D65 @? E96 3@@EDEC2A p₋values p _P

ξϕn ≥ϕn , 2?5 H6

C6;64EH0 2Eα−level @7 D:8?:L42?46 H96? p <α. *:?46 c

nα 2?5 p 2C6 5:Q 4F=E E@ 4@>AFE6 :? AC24E:46 E96J 42? 36 2AAC@I:>2E65 3J

$@?E6 2C=@ 2D 244FC2E6=J 2D 56D:C65 FD:?8 E96 7@==@H:?8 2=8@C:E9> 6?6C2E6 b D6ED @7 C2?5@> ?F>36CD ξ(_ij) n

i=1

, j , ..., b i.i.d. 2Dξ H:E9 b =2C86 @>AFE6 b E6DE DE2E:DE:4D ϕ_{n j}(b), j , ..., b 2Dϕ_n , FD:?8 E96 C2?5@> ?F>36CD :?

AAC@I:>2E6 E96 3@@EDEC2A 4C:E:42= G2=F6Dc_nα 3J

c(b_n)α " c b b j=1 ˆ ϕ_{n j}b<c ≥−α ,

2?5 E96 4@CC6DA@?5:?8 p₋values, p_{, 3J}

p(b) b b j=1 ˆ ϕ_{n j}bϕˆn _.

+96 8C62E6C b, E96 36EE6C E96 3@@EDEC2A 4C:E:42= G2=F6D 2?5 p₋values 2AAC@I:>2E:@? +96 D2>6 3@@EDEC2A 2AAC@I:>2E:@?D 42? 36 A6C7@C>65 7@C E6DED 32D65 @? E6DE DE2E:DE:4D ) ϕ(_nj), j ,,. .6 86?6C2E6 D2>A=6D _{Yi, Zi, X11i, ..., X1ki, X21i, X2ki} n i=1 H:E9 Yi β00Zi k j=1 β_0jZi X1ji k j=1 δ0jX2jiUi, i , ..., n,

with_fZign_i=1 i:i:d:as uniform in[0;1],X`ji =Zi+e`ji; e`ji iidas uniform in[0;1]; `= 1;2, j = 1; :::; k`; and Ui = "iexp( Zi/ 2) p V ar("iexp( Zi/ 2)) ;

with"i iid N(0;1); that is,V ar(Ui) = 1, and governs how severe the heteroskedasticity

is. We generate the random coe¢ cients as

0j(z) = 1 +

f(z)

p

V ar(f(z)); for all j = 0;1; :::; k1; i.e. V ar( 0j(Z)) =

2_{, i.e. governs how serious is the departure}

from the null under the following models,

a)f(z) =z; b) f(z) = [1 + exp( z)] 1;

c)f(z) = sin(2 z); d) f(z) = 1 + 2 1_fz 0:4g:

Model a) is a simple linear model and b) is a nonlinear alternative, almost indistinguish-able for = 1when z ₂ [0;1]; the lower ; the smaller the departure from linearity: We use model b) to check departures form linearity under di¤erent values of : Model c) is harder to …t than a) or b) using smooth methods with moderate sample sizes, and d) is a jump model that cannot be estimated using smoothing methods. We only report results for the 0:4 quantile, but we have also tried other values and the results do not change substantially if the jump is not placed in extreme quantiles. Figure 1 represents 0 for

the di¤erent models and di¤erent values.

FIGURE 1 ABOUT HERE

The simulation study is implemented to provide evidence on the e¤ect of ’s choice on ^n ( ), the accuracy of the bootstrap test, the relative performance of our test with

respect to existing alternatives, and the performance of our test for model checking of in-teractive e¤ects. The Monte Carlo study is based on 1.000 replications and the bootstrap replications are set to 1.000.

Figure 2 provides the percentage of rejections for di¤erent 0_{sfor = 0:05. As expected,} size accuracy is poor when is close to zero. For reasonable values, i.e. bigger that 0.1, the level is close to5%, particularly for the larger sample sizes. On the other hand, under the alternatives, i.e., a), c) and d), the power converges to 1 as ndiverges, independently of the value of . Of course , the power always increases with .

,) &,+ )

? @C56C E@ 4964< E96 =6G6= 244FC24J @7 E96 3@@EDEC2A E6DE H6 4@>A2C6 E96 A6C46?E286 @7 C6;64E:@?D FD:?8 G2=F6D @7 E96 2DJ>AE@E:4 'C@A@D:E:@? 2?5 3@@EDEC2A 'C@A@D:E:@? E6DED H96? Z :D :?56A6?56?E @7 X1 2?5 U FD:?8 E96 E6DE DE2E:DE:4D ϕ)(nj), j ,, :?

2 AFC6 G2CJ:?8 4@6Q 4:6?ED >@56= :6 H:E9 δ0 = 0 +23=6 C6A@CED E96D6 C6DF=ED +96 3@@EDEC2A E6DED 6I9:3:E G6CJ 8@@5 D:K6 244FC24J 7@C E96 E9C66 E6DE DE2E:DE:4D D 6IA64E65 E96 2DJ>AE@E:4 E6DE 32D65 @? ϕ)(0)_n D9@HD BF:E6 A@@C D:K6 AC@A6CE:6D A2CE:4F=2C=J 7@C n D>2== @H6G6C E96 D:K6 244FC24J @7 E96 2DJ>AE@E:4 E6DED 32D65 @?ϕ)(1)_n 2?5 ϕ)(2)_n :D 72:C=J 8@@5 3FE >F49 H@CD6 E92? E96 4@CC6DA@?5:?8 3@@EDEC2A E6DED 2D 6IA64E65

+# &,+ )

%@H H6 A6C7@C> E96 4@>A2C:D@? H:E9 6I:DE:?8 E6DED :? E96 4@?E6IE @7 E96 A2CE=J =:?62C >@56= .6 4@?D:56C E96 @>?:3FD DA64:L42E:@? E6DE AC@A@D65 3J *EFE6 7@C 4@?D:DE6?E E6DE:?8 @7 2?J ?@?A2C2>6EC:4 2=E6C?2E:G6 H9:49 :D 32D65 @? E96 ,*,$ @7 C6D:5F2=D EJA6 AC@46DD ψ_nx n n i=1 Ui k j=1 Xjxj k m=1 Xmxkm, x =x1, ..., xk+k _. +96 ,*,$ E6DE :D 56D:8?65 7@C @>?:3FD C68C6DD:@? >@56= 4964<:?8 :6 :E 56E64ED :? AC:?4:A=6 2?J 56A2CEFC6 7@C> =:?62C:EJ :?4=F5:?8 DA64:L42E:@?D 5:R6C6?E E@ E96 G2CJ:?8 4@6Q 4:6?E >@56= .6 4@?D:56C E96 "@=>@8@C@G*>:C?@G EJA6 DE2E:DE:4

φ_n &'$

x Rkk

√

n ψ_nx .

&FC E6DE :D 5:C64E:@?2= 2?5 :D 6IA64E65 E@ 36 >@C6 A@H6C7F= F?56C H1 .6 2=D@ 4@?D:56C E96 #) EJA6 3@@EDEC2A E6DE @7 2: 6E 2= 7@C E6DE:?8 H0 :? E96 5:C64E:@? @7 H1

Tn RSS0/RSS1 − E92E 4@>A2C6D C6DEC:4E65 2?5 F?C6DEC:4E65 DF> @7 DBF2C65 C6D:5 F2=D #) EJA6 E6DED 2C6 2DJ>AE@E:42==J 5:DEC:3FE:@? 7C66 3J E96 32?5H:5E9 4@?G6C8:?8 E@ K6C@ 2E 2 DF:E23=6 C2E6 2D E96 D2>A=6 D:K6 5:G6C86D D66 2? 2?5 F2?8 @C 2: 6E 2= @H6G6C E6DED 32D65 @? 4C:E:42= G2=F6D 4@CC6DA@?5:?8 E@ E96 2DJ>AE@E:4 5:DEC:3FE:@? 6I9:3:E 2 A@@C D:K6 A6C7@C>2?46 :? L?:E6 D2>A=6D 2: 6E 2= A286 =:?6D 2C8F6 E92E E9:D :D 3642FD6 E96 D6?D:E:G:EJ @7 E96 E6DE E@ 32?5H:5E9 49@:46 2?5

recommend approximating critical values with the assistance of bootstrap. This is why we only report the bootstrap version of Cai et al. (2000)’s test.

In the following set of simulations we consider di¤erent X2 dimensions, k2 = 1;2;3;

= 0:25and = 1:Table 2 provides the percentage of rejections in this simulation study. It shows that, under H1; our directional test works better than the omnibus CUSUM as

k2 increases because of the curse of dimensionality. For instance, whenk2 = 3 and under

model d), our test rejects more than twice than the CUSUM test. The smoothing based test has similar power than ours in all models but the jump model d), due to the poor performance of the Nadaraya-Watson estimator for estimating discontinuous regressions.

TABLE 2 ABOUT HERE

Table 3 reports the percentage of rejections for di¤erent X1 dimensions, k1 = 1;2;3;

= 0:25and = 1. Note that, again, our directional test works better than the omnibus CUSUM as k1 increases. For instance, when k1 = 3 and under model d), the power of

our test is almost twice the CUSUM test. The test using T^n works similarly to ours in

general, but our test performs better when k1 = 3. The smooth test also su¤ers of the

curse of dimensionality; the power decreases as k1 increases. Also, the LM test detects

departures from the null in the direction of jump model d) much less than the other tests, which do not require to estimate the model under the alternative using smoothing.

Under model d) our test also works much better than the LM smoothing based test be-cause of the curse of dimensionality of the Nadaraya-Watson estimator needed to compute

^ Tn.

TABLE 3 ABOUT HERE

In the next set of simulations we apply the test as a regression model check of the linearity hypothesis when k1 = 0; k2 = 1 and X2 = Z. That is, H0 is equivalent to

omnibus speci…cation testing of the simple regression model _E(Y_jZ) = ₀₀+Z 00 a:s.

The resulting test competes with the CUSUM test based on ^n: Since 00 is not

iden-ti…able, tests based on comparing …ts under the null and the alternative, like the LR test using T^n as test statistic, cannot be implemented. We compare our test with the

omnibus speci…cation test, designed to detect more general non-linear alternatives. We consider model b) with di¤erent values in order to check the performance of the test

F?56C D>2== 56A2CEFC6D 7C@> E96 =:?62C:EJ 9JA@E96D:D +23=6 D9@HD E92E @FC E6DE C6;64ED 2=>@DE 5@F3=6 E92? E96 ,*,$ E6DE 7@C 2== ρG2=F6D

+# &,+ )

.6 2=D@ 4@?D:56C E96 E6DE 7@C >@56= 4964<:?8 @7 ?@?=:?62C C68C6DD:@? >@56=D .6 4@?D:56C E6DE:?8 E92E _EY_|Z β₀₀ L₌₁Z δ0 1 a.s. :? E96 5:C64E:@?

EY_|Z β₀₀Z

L

=1

Z δ0 1a.s. H:E9 V arβ00Z ≥a.s.

2?5 β₀₀ F?<?@H? &FC E6DE :D @>?:3FD 7@C E96 ?@?=:?62C DA64:L42E:@? 9JA@E96D:D D:?46 E96 5:C64E:@? @7 :?E6C6DE ?6DED 2?J A@DD:3=6 56A2CEFC6 7C@> E96 ?F== +9:D 4@CC6DA@?5D E@ 2AA=J:?8 @FC E6DE E@ >@56= H:E9gjz zj, j , ..., L.+23=6 C6A@CED C6;64E:@?D 7@C

>@56= 3 H:E9 ρ , H9:49 AC@5F46D 2 D6?D:E:G6 56A2CEFC6 7C@> =:?62C:EJ 7@C 5:R6C6?E L G2=F6D

+# &,+ )

%6IE H6 4@?D:56C E96 A6C7@C>2?46 @7 E96 E6DE 2D 2 DA64:L42E:@? E6DE @7 :?E6C24E:G6 6R64ED :? E96 4@?E6IE @7 >@56= H:E9 k1 > L , g0z 2?5 g1z z. +92E :D @FC E6DE :D :>A=6>6?E65 7@C E6DE:?8 E96 9JA@E96D:D

EY_|X1, Z β00Z δ00Z k j=1 β_0jZ X1jδ0jX1jZ a.s. :? E96 5:C64E:@? EY_|X1, Z β00δ00Z k j=1 β_0jX1j δ0jX1jZ a.s..

+23=6 C6A@CED E96 A6C46?E286 @7 C6;64E:@?D 7@C @FCD 2?5 ,*,$ E6DE :? >@56= 3 H:E9 λ . 5:R6C6?E ρ G2=F6D 2?5 k1 ,,. &FC E6DE A6C7@C>D 36EE6C E92? ,*,$ :? >@DE 42D6D

+# &,+ )

%@H H6 4@?D:56C E6DE:?8 ?@?=:?62C DA64:L42E:@? @7 :?E6C24E:G6 6R64ED :? E96 4@?E6IE @7 >@56= H:E9 k1 > L ,,,, g0z 2?5 gjz zj &FC E6DE :D :>A=6>6?E65

7@C E6DE:?8 E96 9JA@E96D:D

EY_|X1, Z β00Z L =1 Z δ0 1 k j=1 β_0jZ X1j X1j L =1 Z δ0j+L+1 a.s

:? E96 5:C64E:@? EY_|X1, Z β00 L =1 Z δ0 1 k j=1 β_0jX1j X1j L =1 Z δ0j+L+1 a.s.

+23=6 C6A@CED E96 A6C46?E286 @7 C6;64E:@?D 7@C 3@E9 @FCD 2?5 ,*,$ E6DED F?56C >@56= 3 H:E9 λ . ρ k1 2?5 5:R6C6?E L G2=F6D &FC E6DE A6C7@C>D 36EE6C :? 86?6C2=

+# &,+ )

.6 4@>A=6>6?E E96 AC6G:@FD $@?E6 2C=@ DEF5J H:E9 2? 2AA=:42E:@? E@ FD:?8 IQ 2D 4@?EC@= @C AC@IJ G2C:23=6 @7 23:=:EJ :? 2 C6EFC?D @7 65F42E:@? >@56= +9:D :D 32D65 @? =24<3FC? 2?5 %6F>2C< H@C< H9:49 :D FD65 :? .@@=5C:586 E6IE3@@< 6I2>A=6 +96 52E2 4@?D:DED @7 @3D6CG2E:@?D 7C@> E96 0@F?8 $6?MD @9@CE %2E:@?2= #@?8:EF5:?2= *FCG6J +96 >2:? @3;64E:G6 4@?D:DED @7 6DE:>2E:?8 E96 >2C8:?2= 6R64E @7 65F42E:@? @? H286D 4@?EC@==:?8 7@C C6=6G2?E 4@G2C:2E6D H9:49 :?4=F56 F?@3D6CG65 23:=:EJ C62D@?23=6 >@56= FD:?8 IQ 2D AC@IJ G2C:23=6 .@@=5C:586 6I2>A=6 :D

LogW AGE β₀₀ β_01·EDU C β_02·IQX₂δ01U, H96C6W AGE 2C6 ,* >@?E9=J 62C?:?8DEDU C :D J62CD @7 65F42E:@?IQ:D :?E6==:86?46 BF@E:6?E AC@IJ @7 23:=:EJ 2?5 X₂ EXP ER, T EN U RE, M ARRIED, SOU T H, U RBAN, BLACK , EXP ER 2C6 J62CD @7 H@C< 6IA6C:6?46 T EN U RE J62CD H:E9 4FCC6?E 6>A=@J6C M ARRIED 2 5F>>J :7 >2CC:65 BLACK 5F>>J :7 3=24< SOU T H 5F>>J :7 =:G6 :? D@FE9U RBAN 5F>>J :7 =:G6 :? FC32? 2C62 *$* 2?5 δ01 δ01, ...,δ06 +96 &#* 6DE:>2E@CD @7β012?5 β02 :? E9:D >@56= 96E6C@D<652DE:4:EJ C@3FDE * :? A2C6?E96D:D 2C6. . 2?5. . ,C6DA64E:G6=J +96 &#* 6D E:>2E@C @7 E96 >2C8:?2= 6R64E @7EDU C β₀₁ :D :?4@?D:DE6?E H96?_EU_|EDU C, IQ,X2 56A6?5D @?EDU C :6 IQ:D ?@E 2 8@@5 AC@IJ 7@C 23:=:EJ 3FE 2=D@ H96? :E @?=J 56A6?5D @? IQ :? 2 ?@?=:?62C 7@C> C62D@?23=6 2=E6C?2E:G6 E@ :D E96 G2CJ:?8 4@6Q 4:6?ED >@56=

LogW AGE β₀₀IQ β₀₁IQ _·EDU CX₂δ0U, H9:49 2==@HD EDU C A2CE:2= 6R64ED E@ 36 2? F?<?@H? 7F?4E:@? @7 IQ. :8FC6 AC@G:56D 6DE:>2E6D @7 β₀₀ 2?5 β₀₁ G2CJ:?8 4@6Q 4:6?ED FD:?8 2: 6E 2= AC@465FC6 H9:49

uses a modi…ed manifold cross-validation criterion for choosing the bandwidth. We also provide OLS estimates of the parametric speci…cation 0j(IQ) =

(1) 0j + (2) 0jIQ+ (3) 0j IQ2; j = 0;1.

FIGURE 3 ABOUT HERE

Thep valuesfor testingH0 :V ar( 0j(IQ)) = 0; j = 1;2versusH1 :V ar( 0j(IQ))>0

some j = 1;2; or H2 :V ar( 00(IQ)) = 0 and V ar( 01(IQ))>0are reported in Table 8,

where we provide thep values.

TABLE 8 ABOUT HERE

We also report the smoothing LR test of Cai et al. (2000). Here the CUSUM test is unable to reject the null hypothesis, but the directional tests reject H0 in the two

directions considered. The p value of our test is the smallest when testing in the direction H1; but the correspondingp value for the smoothing LR test based onT^n is

the smallest in the direction H2:

Next, we apply our test as a model check of the interactive e¤ect of EDU C: The maintained speci…cation is

Log(W AGE) = ( ₀₀(IQ) + 07IQ) + ( 01(IQ) + 08) EDU C +Xt2 0+U; (20)

which is model (19) augmented with the explanatory variables (IQ; EDU C) in the con-stant coe¢ cients terms,i:e:Xt₂ in (19) is substituted by(Xt₂; IQ; EDU C)in (20). Then

H0 is in fact a speci…cation test of the functional form of the varying coe¢ cients in (19).

TABLE 9 ABOUT HERE

In this case, see table 9, we are unable to reject the speci…cation of the interactive e¤ect either with the CUSUM or with our test. We conclude that the speci…cation including

IQ and a simple interactive e¤ect EDU C with IQ cannot be rejected.

5. CONCLUSIONS.

We have proposed a test for constancy of coe¢ cients in semi-varying coe¢ cients mod-els, where the variable responsible for the coe¢ cient varying may depend on the rest of explanatory variables in an unknown form. The test, implemented using bootstrap, is based on comparing the OLS coe¢ cients of subsamples of concomitants to the explana-tory variable in the varying coe¢ cients. The test is justi…ed under fairly weak regularity

conditions, which allow discontinuous random coe¢ cients under the alternative hypothe-sis. Our test forms a basis for speci…cation testing of parametric varying coe¢ cients and, in particular, for testing the functional form of interactive e¤ects. Simulation results have provided evidence of the good performance of our test in …nite samples compared with a CUSUM-type test, designed to omnibus speci…cation testing of linear regression models, and a smooth LM test, designed to test varying coe¢ cients constancy in the direction of smooth alternatives. The CUSUM test, like ours, does not require estimating the model on the alternative, but the LM-type test compares the restricted and unrestricted sum of squared residuals and, hence, requires estimating the nonparametric smooth varying co-e¢ cients. Simulations show that, unlike our test, the two competitors su¤er of the curse of dimensionality. These also show that the LM smooth test exhibit a lack of power, compared with the two competitors, under alternatives with discontinuous varying coef-…cients. We have also included a real data application to model interactive e¤ects of IQ in a returns of education model.

The proposed methodology is applicable to testing constancy of a subset of varying coe¢ cients or a linear combination of them. However, since the model under the null must be estimated, smooth estimation of the unrestricted varying coe¢ cients is necessary. A formal justi…cation of the resulting test is technically demanding, but it seems possible to take advantage of existing asymptotic inference results for varying coe¢ cient models. A relevant extension consists of allowing endogenous explanatory variables using an instrumental variables approach, see e.g. Cai et al. (2017). Extensions to nonlinear and multiple equations estructural systems seems also feasible.

APPENDIX

Proof of Proposition 1. (8) follows from Davidov and Ergorov (2000) Theorem 1. Recall that Z is distributed as an U(0;1): A typical uniformity argument shows that sup_u2[0;1] M~jn Mj (u) = op(1) a:s: with

~ Mjn(u) =n 1 n X i=1 XjiXtji1fZi ug; j = 1;2:

Then (9) follows by noticing thatM^jn(u) = ~Mjn(Zn:bnuc)and thatsupu₂[0;1] Zn:bnuc u = o(1) a:s: by Glivenko-Cantelli theorem.

+))$ )$ +)*),%-%)( 6L?6 Xiu X1iu ,X1i −X1iu ,X2i _{, H:E9} X1iu X1i Ziu, 2?5θ¯0 β¯ 0,β¯ 0,¯δ 0

.+96 C6DF=E :D :>>65:2E6 7C@> 'C@A@D: E:@? 2DDF>:?8 A 2?5 A, 27E6C D9@H:?8 E92E

&'$ K n+u K n ˆ Ωn−Ωˆ 0 n u op , H:E9 ˆ Ω0_nu n n i=1 Xiu X iu V 2 i . ,?56C H0, ˆθn θ¯0Opn1/2 2?5 ˆ Ωnu −Ωˆ 0 nu n nu i=1 X_[i:n]u θˆn −¯θ0 2 X[i:n]X[i:n] − ˆθn −¯θ0 n nu i=1 X[i:n]X[i:n]X[i:n]V[i:n].

J 'C@A@D:E:@? ˆθn −¯θ0 Op n1/2 . +96? 3J A A, 27E6C 2AA=J:?8 W=56CMD

:?6BF2=:EJ &'$K n+u K n ˆ Ωn−Ωˆ 0 n u Opn1 Op n1/2 .

+))$ )$ +)*),%-%)( :CDE ?@E:46 E92E 3642FD6 Z :D :?56A6?56?E @7 U,X E96 4@?4@>:E2?ED U[i:n],X[i:n] n i=1 2C6 ::5 2D U,X . 6L?6 ) ϕ_nn !( K nKα) n _n H:E9 η_nu M₁₁1 N1nu −u N1n u₋u , ) α_nu n_·η_nu M11 u−u σ2 η nu

AA=J:?8 E96 6IE6?D:@? @7 2C=:?8C5S@DME96@C6> 2C=:?8 2?5 C5S@D E@ E96 G64E@C 42D6 2D :? @CGTE9 a #n ϕ)_n−b1+k #n →d E. +96C67@C6 :E DFQ 46D E@ AC@G6 E92E &'$ K nK ) ϕ(0)_{n −}ϕ)_n n op .

+@ E9:D 6?5 LCDE 2AA=J E96 $2C4:?<:6H:4K1J8>F?5 DEC@?8 =2H @7 =2C86 ?F>36CD 9@H 2?5 +6:496C AA E@ 6DE23=:D9 E92E 7@C E96δ > :? A_,

nM11

n − M11 o

2/(2+δ) _{2D 2D}

→ ∞.

n M11 −M11 n −n− M11 o n− 2/(2+δ) 2D 2D _{→ ∞}, n i=1 V_i2 ₋nσ2 o n2/(2+δ) 2D 2D n_{→ ∞}. 6?46 !( K n nM11 − M11n n op n δδ , !( 1 nK n₋ n M11 − M11n −M11 n n op n δδ _. J A.., !( K 1 n M₁₁1_n n Op , !( 1 nK n₋ n M11n −M11n n 1 Op .

=D@ 3J E96 =2H @7 E96 :E6C2E65 =@82C:E9> 7@C A2CE:2= DF>D !( 1 n n √ N1n n Op # #n 2?5 3J 2?5 σ2_n−σ2 n n i=1 V_i2₋σ2 ₋S_n M₁₁1_n Sn op nδ/(2+δ) Op # #n n . %@E:46 E92E n_· α)n−α)n _n n₋ n ηn−η n _n ⎛ ⎝ M11 σ2_o p n δ δ ⎞ ⎠ × n_n− η_n−η_n n , H96C6 n₋ n ηn−η n n n₋ n M₁₁1_n n − n M₁₁1 N1n n − M11n −M11n n 1 − _nn − M 1 11 × N1n −N1n n

n− 2 n3 n M₁₁1_n n nM11 − M11n n M 1 11 n √ N1n n − n− 2 n3 n n₋ M11n −M11n n 1 × n−_n M11 − M11n −M11n n ×M₁₁1 _√ n n₋ N1n −N1n n +96C67@C6 3J n₋ n K!(nK ηn−η n _n ≤ Op ×op nδδ × Op # #n op , H9:49 AC@G6D :?2==J 2?5 7@==@H 3J 2?5 u₋u M1/2 η_nu /σ _{u [0,1] →}d{W0u −uW0 }u [0,1] in D,, 3J 'C@A@D:E:@?

+))$ )$ +)*),%-%)( :CDE ?@E:46 E92E 3J 'C@A@D:E:@? 2?5 F?:7@C>=J :? u_∈ ,₋ ,

) ϕ_n

n →p &'$u1

η₀u Σ₀1u η₀u ,

H9:49 AC@G6D ? @C56C E@ AC@G6 ?@E:46 E92E F?56CH1n

√ n ˆθn−¯θ0 u Mˆ 1 n u ⎡ ⎣ n nu i=1 X[i:n]τZi:n √ nNˆnu ⎤ ⎦, H96C6 F?56C A, &'$ _u1 n1 _i=1nuX[i:n]τZi:n −T u o a.s. FD:?8 D2>6

2C8F>6?ED E@ AC@G6 :? 'C@A@D:E:@? +96? 2AA=J 2?5 E96 4@?E:?F@FD >2AA:?8 +96@C6> E@ 4@>A=6E6 E96 AC@@7

+))$ )$ +)*),%-%)( #6E_Pξ 36 E96 :?5F465 AC@323:=:EJ 5:DEC:3FE:@? @7ξ. E DFQ 46D

E@ D9@H E92E 7@C 2?J c >, Pξϕn ≤c →Pϕ ≤c o a.s., %@E:46 E92E F?:7@C>=J :? u_∈, η_nu RMu o 1Nˆ_nu 2D,

$:>:4<:?8 E96 DEC2E68J @7 AC@@7 :? *EFE6 2= *( 7@C 2 D:>:=2C 3@@EDEC2A DE2 E:DE:4D 7@==@HD 3J D9@H:?8 E92E 4@?5:E:@?2= E@ E96 D2>A=6 √nNˆ_n 4@?G6C86D :? 5:D EC:3FE:@? E@ N 2D :6 7@C 2=>@DE 2== D2>A=6 _{Yi,X1i,X2i, Zi}n_i=1, 3J D9@H:?8 E96

4@?G6C86?46 @7 E96 L?:E6 5:>6?D:@?2= 5:DEC:3FE:@?D L5:D 2?5 E:89E?6DD 6?467@CE9 _Eξ

:D E96 6IA64E2E:@? @A6C2E@C 4@CC6DA@?5:?8 E@ _Pξ. @C L5:D 4@?G6C86?46 LCDE ?@E:46 E92E

7@C u1, u2 ∈,, n_Eξ Nˆ nu1 Nˆ n u2 n n(uu) i=1 X[i:n]X[i:n]V 2 [i:n] n n i=1 XiXiV 2 i _{ZiZnuun} n n i=1 X_i ˆθn −¯θ0 2 XiXiV 2 i _{ZiZnuun} Ω0u1∧u2 o a.s.

D:?46 ˆθn ¯θno a.s., 2?5 2AA=J:?8 E96 2C8F>6?ED 7@C AC@G:?8 FD:?8 2?5

+96? LI:?8 u1, ..., uq, 3J E96 C2>UC.@=5 56G:46 :E DFQ 46D E@ D9@H E92E 7@C 2?J

c <_∞, Pξ √ n q j=1 ajbNˆ nuj ≤c →p P q j=1 ajbNuj ≤c , 7@C 2?J b b1, ..., bk+1 2?5 a =a1, ..., ap .C:E6Wi q_j=1ajbXi ZiZ nuj n √ n q j=1 ajbNˆ nuj √ n n i=1 WiViξi _√ n n i=1 WiViξi. +96? 7@==@HD 3J 4964<:?8 E96 #:?56C36C8MD 4@?5:E:@? Lnδ n n i=1 W_i2 {|WiVˆiξi|δ _n } W_i2V_i2ξ2_id_Pξ→ a.s. 6L?6Wi q j=1ajbXi.*:?46 |ξ|≤τ, Lnδ ≤ κ2 n n i=1 |_W¯_iVˆ_i|δn τ _W2 iV 2 i κ 2 n n i=1 |_W¯_iVi|δn τ _W2 iV 2 i o a.s. o a.s.

FD:?8 2C8F>6?ED :? ? @C56C E@ D9@H E:89E?6DD :E DFQ 46D E@ 4964< :==:?8D=6J +96@C6> 2D :? *( #6>>2 6L?6 α_nbu √nbNˆ_nu _√ n n i=1 bXi_{ZiZ nun} Viξi. .6 >FDE D9@H 2D :? *( #6>>2 E92E 7@C 2?Jb _∈R1+k 2?5 _≤u0 ≤u1 ≤u2 ≤, Eξαnbu1 −αnbu0 2 α_nbu2 ₋α_nbu1 2 _≤CJnu2 −Jnu0 2,

H96C6 C < _∞ :D 2 86?6C:4 4@?DE2?E, Jn >@?@E@?6 a.s., 2?5 Jn →J 2D +96? 2AA=J:?8

#6>>2 @7 *EFE6 LHS _≤ n2 i=j Eξλ2iEξγ2i, λi bXiViξi_{ZnunZiZnun} 2?5 γi b _X iViξi_{ZnunZiZnun}.+96? LHS _≤ n2 i=j bXi 2 bXj 2 V_i2V_j2 {ZnunZiZnun}{ZnunZiZnun} ≤ Jnu2 −Jnu0 2 , H96C6 Jnu n n i=1 bXi 2 V_i2 {ZiZnun} :D >@?@E@?6 2?5 Jnu →Ju E bX 2V2Zu a.s. F?:7@C>=J :?u.

9>25 #66=292?@? * #: ( Q 4:6?E 6DE:>2E:@? @7 2 D6>:A2C2>6EC:4 A2CE:2==J =:?62C G2CJ:?8 4@6Q 4:6?E >@56= ?? *E2E P

?5C6HD ." 4@?5:E:@?2= "@=>@8@C@G E6DE 4@?@>6EC:42 P ?5C6HD ." +6DED 7@C A2C2>6E6C :?DE23:=:EJ 2?5 DECF4EFC2= 492?86 H:E9 F?

<?@H? 492?86 A@:?E 4@?@>6EC:42 P

F6 @CGTE9 # "@<@DK<2 ' *E6:?63249 ! $@?:E@C:?8 D9:7ED :? >62? DJ>AE@E:4 ?@C>2=:EJ @7 DE@AA:?8 E:>6D +6DE P

92EE2492CJ2 '" @?G6C86?46 @7 D2>A=6 A2E9D @7 ?@C>2=:K65 DF>D @7 :?5F465 @C56C DE2E:DE:4D ?? *E2E P

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]  ] D] ] ]  ] E] H] ]  ] F] VLQ] ]  ] G] _^]` :8FC6 %'(#))$# $ η₀ $' 3'#) "$!( ,# λ !* *'+ λ . %*'%! *'+ # λ . ' *'+   +  +D  +F  +G   +  +D  +F  +G :8FC6 %'(#))$# $ n α $' 3'#) "$!( ,# λ . ' *'+ # λ . !* *'+

,4  ,4 ,4 ,4  ,4 ,4 :8FC6 %'(#))$# $ β₀₀IQ # β₀₁IQ $' ) ()")( $ ) +'.# $2 #)( *(# '#!( ,) %!*# #,) ' *'+ # ()")( $ ) %'")'/)$# %*'%! *'+

1% 5% 10% k1 0 1 2 3 0 1 2 3 0 1 2 3 ~ '(0)n (bootstrap) 50 0.2 0.3 0.2 0.5 2.5 2.5 2.8 2.4 5.7 6.1 6.4 5.9 100 0.5 0.5 0.7 0.6 3.5 3.2 3.1 2.4 8.5 6.6 6.0 5.8 200 1.2 1.1 0.5 0.4 4.4 4.2 3.6 2.1 8.4 7.9 6.3 4.4 500 0.7 0.7 0.7 1.1 4.1 3.8 3.6 4.5 9.1 7.7 8.4 8.4 ~ '(1)n (bootstrap) 50 0.6 0.7 0.1 0.1 4.1 4.1 3.2 3.4 8.7 9.6 8.0 8.3 100 1.2 1.0 0.7 0.5 4.6 4.5 3.9 3.6 9.5 9.4 9.1 8.5 200 1.2 1.2 0.7 0.7 5.3 4.3 5.2 3.9 9.7 10.5 9.2 7.8 500 1.0 0.9 0.6 1.2 4.7 4.1 4.5 5.1 11.1 9.0 8.7 9.5 ~ '(2)n (bootstrap) 50 0.7 1.0 0.3 1.3 4.5 5.1 5.2 4.6 9.2 11.6 9.9 11.3 100 1.0 1.0 0.7 0.6 5.1 4.5 4.5 4.4 11.0 9.4 9.2 9.6 200 1.2 1.0 0.7 0.7 5.2 5.1 4.9 2.6 10.3 10.8 9.2 8.4 500 1.2 1.0 1.2 1.5 4.7 5.4 5.7 5.9 9.7 10.0 9.4 10.2 ~ '(0)n (asymptotic) 50 0.0 0.1 0.5 3.6 1.7 2.4 6.7 23.3 5.9 8.2 18.1 43.9 100 0.0 0.0 0.1 1.0 1.3 1.1 3.9 9.1 5.3 5.9 10.3 21.8 200 0.0 0.0 0.0 0.4 1.5 1.4 2.5 4.6 4.4 4.6 5.9 13.4 500 0.0 0.0 0.0 0.0 1.5 1.0 2.4 3.3 4.3 3.9 6.2 10.3 ~ '(1)n (asymptotic) 50 0.5 0.1 0.0 0.0 2.9 2.7 1.9 1.0 6.8 6.4 5.1 3.3 100 1.0 0.7 0.6 0.1 4.9 3.8 3.7 2.4 10.8 8.5 7.9 6.7 200 1.3 1.2 0.5 0.6 4.6 5.3 4.0 4.1 8.5 10.3 8.8 7.4 500 0.8 1.1 0.8 0.4 4.9 4.5 4.9 4.3 9.5 9.4 9.5 8.7 ~ '(2)n (asymptotic) 50 0.2 0.1 0.1 0.0 2.0 1.5 1.2 1.4 4.9 4.0 4.1 3.7 100 0.3 0.2 0.4 0.1 3.3 2.5 2.6 4.6 7.8 5.5 5.0 4.6 200 0.7 0.7 0.4 0.3 4.1 3.5 3.2 1.6 8.2 7.2 6.4 4.1 500 0.7 0.7 0.7 0.8 4.4 3.9 4.0 4.7 8.1 8.3 7.8 8.1

Table 1. Percentage of times H0 was rejected ( k2 = 0 and = 0)

Model H0 H1 :a H1 :c H1 :d k2 1 2 3 1 2 3 1 2 3 1 2 3 ^ '_n0:02 50 3.3 4.4 4.6 11.5 9.5 7.1 13.3 12.5 11.4 15.0 11.2 10.3 100 4.0 5.0 4.6 26.6 15.9 12.4 25.9 23.0 21.2 30.0 20.5 19.8 200 4.5 4.3 3.6 49.1 31.4 22.4 56.0 45.6 40.6 60.9 45.4 38.4 ^ n 50 4.5 4.4 4.6 12.7 9.4 4.8 14.0 8.1 6.4 14.4 7.9 6.3 100 4.6 5.0 5.4 26.8 10.9 7.8 27.8 16.5 9.9 28.4 14.4 8.5 200 4.4 4.7 4.1 48.1 20.9 11.7 57.0 34.6 18.2 56.9 30.5 15.0 ^ Tn 50 4.7 4.9 6.6 15.8 9.0 7.6 15.0 13.7 7.4 13.2 10.3 9.3 100 3.8 4.0 6.2 32.1 21.6 12.1 31.9 29.0 18.7 29.5 21.4 18.8 200 4.9 5.1 4.2 57.7 40.4 28.3 62.4 55.3 45.5 56.8 41.5 33.6 Table 2. Percentage of times H0 was rejected, 5% of signi…cance ( k1 = 0, = 0:25 and

= 1)

Model H0 H1 :a H1 :c H1 :d k1 1 2 3 1 2 3 1 2 3 1 2 3 ^ '_n0:02 50 2.7 3.4 2.3 18.3 20.5 20.5 26.8 41.6 54.1 25.2 30.6 34.1 100 3.8 4.1 3.1 47.2 59.2 69.7 66.9 92.7 98.5 63.6 85.9 94.6 200 3.9 3.2 4.0 84.1 96.3 98.9 97.2 100 100 97.1 100 100 ^ n 50 4.4 4.6 5.2 21.3 17.7 16.4 22.9 23.4 22.9 18.8 18.4 16.1 100 5.0 5.4 4.3 41.6 40.5 39.6 55.4 61.6 56.7 45.8 42.3 35.8 200 4.7 4.1 5.9 76.3 83.2 81.4 93.8 96.2 94.7 86.2 84.2 76.7 ^ Tn 50 4.5 4.8 5.7 18.2 20.2 22.7 22.0 48.4 27.2 15.8 42.7 19.8 100 4.2 4.9 4.7 44.8 55.3 36.5 67.0 61.5 42.8 48.8 54.8 39.6 200 4.9 4.8 4.5 71.1 94.0 53.5 97.2 97.7 53.6 89.0 89.8 52.2 Table 3. Percentage of times H0 was rejected, 5% of signi…cance ( k1 = 1, = 0:25 and

= 1) 0.25 0.5 1 2 3 4 5 15 1 2 3 4 5 15 ^ '_n0:02 50 3.8 4.4 5.3 6.6 7.5 11.8 4.1 5.4 8.3 11.6 16.6 35.5 100 4.0 4.7 6.1 7.8 10.6 25.2 3.9 6.2 11.7 21.5 33.6 76.2 200 3.9 4.3 6.4 11.2 18.3 56.3 4.1 6.5 19.0 39.7 61.5 98.7 ^ n 50 5.6 5.4 5.8 6.0 6.5 7.7 5.6 5.7 6.6 8.6 10.9 19.2 100 4.9 5.6 6.7 7.7 9.0 13.7 5.2 7.2 10.0 14.4 20.8 49.7 200 4.3 4.7 6.7 8.6 12.0 24.8 4.6 6.8 13.9 25.4 40.0 87.1

Table 4. Percentage of times H0 was rejected, 5% of signi…cance (k1 = 0, k2 = 1 and

= 1)

0.25 0.5 L 1 2 3 4 1 2 3 4 ^ '_n0:02 50 11.8 7.2 4.4 2.9 35.5 16.4 6.1 2.9 100 25.2 11.8 6.3 4.7 76.2 38.4 11.7 4.9 200 56.3 24.9 7.3 3.9 98.7 77.9 19.6 6.2 ^ n 50 7.7 5.3 6.2 6.1 19.2 8.2 6.0 5.9 100 13.7 6.0 5.6 6.2 49.7 10.5 5.7 6.1 200 24.8 6.3 4.3 4.2 87.1 18.7 5.4 4.6

Table 5. Percentage of times H0 was rejected, 5% of signi…cance (k1 = 0, k2 = 1, = 15

and = 1) 1 2 3 15 k1 1 2 3 1 2 3 1 2 3 1 2 3 ^ '_n0:02 50 3.6 3.3 2.8 4.2 3.7 3.8 4.8 4.4 5.4 8.5 13.8 16.7 100 3.7 5.4 3.0 4.8 5.8 5.4 6.1 8.8 11.6 19.8 38.8 51.6 200 3.8 3.8 4.9 4.9 6.7 10.4 8.1 16.6 26.7 48.3 84.1 94.0 ^ n 50 4.5 6.1 7.0 4.6 6.5 6.6 4.9 7.0 7.8 7.8 9.4 10.3 100 5.3 7.1 4.8 6.4 6.9 5.6 7.1 8.7 7.3 12.1 15.5 11.2 200 4.5 5.9 5.1 4.9 7.1 6.8 8.6 9.0 10.6 21.2 34.1 27.5

Table 6. Percentage of times H0 was rejected, 5% of signi…cance ( k2 = 0, = 0:5 and

= 1)

L 1 2 3 ^ '_n0:02 50 13.8 7.0 4.6 100 38.8 17.3 6.9 200 84.1 47.4 10.4 ^ n 50 9.4 7.7 8.7 100 15.5 8.5 6.3 200 34.1 14.5 7.5

Table 7.Percentage of times H0 was rejected, 5% of signi…cance (k1 = 2, k2 = 0, = 0:5

= 15 and = 1)

H1 :V ar( 00(IQ))>0 H2 :V ar( 00(IQ)) = 0 H2 :V ar( 00(IQ))>0

Test or and and

V ar( ₀₁(IQ))>0 V ar( ₀₁(IQ))>0 V ar( ₀₁(IQ)) = 0 ^ '_n0:003 0.012 0.017 0.08 ^ n 0.734 ^ Tn 0.041 0.009 0.009

Table 8. p-value of testingH0 versus H1 and H2

H1 :V ar( 00(IQ))>0 H2 :V ar( 00(IQ)) = 0 H2 :V ar( 00(IQ))>0

Test or and and

V ar( ₀₁(IQ))>0 V ar( ₀₁(IQ))>0 V ar( ₀₁(IQ)) = 0 ^

'_n0:003 0.6489 0.405 0.484

^

n 0.491 0.653 0.543

Table 9. p-value of testingH0 versus H1 and H2