Supplement to the paper "Asymptotic expansions in multi-group analysis of moment structures with an application to linearized estimators" and

Supplement to the paper "Asymptotic expansions in multi-group analysis of moment structures with an application to linearized estimators" and

errata for the paper on the ADF chi-square statistic.

Haruhiko Ogasawara

This note is to supplement Ogasawara (in press), and gives errata for Ogasawara (2009).

10 Supplement to Ogasawara (in press) Let

n- I - (mAB) - = ^{(nCok) n('2k») =}

(

n(k) n(k,2k) J ^(n(k

J

- - nC2k,k) n(2k) n(2k ^{o )}

(A, B = 1, ... , G * ^{p **),}

n- ^{I - n(2k) d E (0)} = ^{E E} (0) L

(2klk) - an ul- u(k)lr U(2k) 1!l(2klk),U(k)

et

(·)(A-D) = ()(ABCD) and (\A-F) = (\ABCDEF) . Then, the expectation of b(2)

with the redefinition of U(2k) - N(n(2k,k)n~IU(k)' n(2klk») IS

E _ol _ (h(2) )(AB) = {n- ¹ E _{ul_} (uu ')n- ^{1 } AB - mAE}

(135J

136

r = ^gll2 (I _ 2itgIl2 J gIl2 )-1 gIl2 = ^{gll2 (,hp} P '+ P P ^')g1l2

7l G* p* 7l 0 7l 7l 7l tp 1 1 2 2 7l

= gl/2pWp'gll2 = gIl2(,hQ +Q )g1/2

7l 7l 7l tp 1 2 7 l '

(

¢I ⁺ 0 J

P = (PI P _{2 ),} PP' = IG*p*' W = ^P ,QI = PIP _{1 ',Q2} = P ₂ P _{2 '.}

o ^Iq

Then,

E (h) - (g(ok)r ^{g(k o ) )} + ^(g(02k)g g-lr ^{g(k o ) )}

ujo (2) (AB) - AB (2k,k) J[ AB

+(g(ok)rg-l g ^{g(2k o))} + ^(g(02k)g g-lrg-I g ^{g(2k o))}

J[ (k,2k) AB (2k,k) J[ J[ (k,2k) AB + (g(02k) g ^{g(2k o ) ) _ mAB}

. (2klk) AB

= da{ g(ok)g1l2Q g1l2 ^{g(k o )} + ^g(02k)g g-II2Q g1l2 g(ko)

tp J[ 1 J[ (2k,k) J[ 1 J[

+g(ok)gIl2Q g-I12g g(02k)

J[ 1 J[ (k,2k)

+ g(02k)g g-1I2Q g-1I2g ^{g(02k) }}

(2k,k) J[ 1 ^J[ (k,2k) AB

+ { g(ok)gIl2Q g1l2 g(k ^o ) + ^g(02k)g g-I12Q gII2 ^{g(k o )}

J[ 2 ^J[ (2k,k) J[ 2 . ^J[

+ g(ok)gII2Q g-ll2g ^g(02k)

J[ 2 J[ (k,2k)

+ g(02k)g g-1/2Q g-ll2g ^g(02k)

(2k,k) ^J[ 1 J[ (k,2k) + g(02k)g g(02k)} _ mAB

(2klk) AB

=¢(Q;)AB + ^(Q;)AB + ^{(g*)AB _m AB}

= ¢(Q;)AB + ^{(R*)AB _m AB •}

~45 * * AB * CD * AB CD * +¢ ~ {(R )AB(R )CD -OJ (R )CD -OJ (R)AB +OJ OJ }(Ql)EF

+ 2: ¹⁵ (R*)ABCR*)CDCR*)EF _2: ⁴⁵ OJABCR*)CDCR*)EF

+ 2: ⁴⁵ OJABOJCDCR*)EF _ 2: ¹⁵ OJABOJCDOJEF CA,B,C,D,E,F = 1, ... ,G* p**).

Define

A~~F ⁼ 2: ¹⁵ (Q;)AB(Q;)CD(Q;)EF' A~~D = 2: ³ (Q;)AB(Q;)CD'

A A-F - ^{(5) - ~45} L...J {(R*) (R*) AB CD ^{- ( j )} AB(R*) CD ^-(j) CD(R*) AB ^+(j) AB. CD}(Q*) ^{(j) 1} EF' A~~ = ^CR*) AB - OJAB,

A~~D ⁼ 2: ³ (R*)AB(R*)CD - 2: ⁶ OJAB(R*)CD + 2: ^{3 OJABOJCD,}

A~~F ⁼ LIS (R*)AB(R*)CDCR*)EF - 2: ⁴⁵ WABCR*)CDCR*)EF

+ 2: ⁴⁵ wABwCD(R*)EF - LIS wABWCDOJEF,

B~~D ⁼ L3 CUQ;)ADCUQ;U)BC'

B~2}D ⁼ 2: ³ {CUQ;)AD(UR*U)BC +CUR*)AD CUQ;U)BC} '

B~~D ⁼ 2: ³ CUR*)ADC!lR*U)BC'

B~~F ⁼ 2: ⁶ (UQ;)ADcnQ;)BEcnQ;)CF + 2: ⁹ cnQ;n)AB(Q;)DE(nQ;)CF'

138

C~2}F = L: ⁴⁵ (!lQ;!l)AB (!lQ;!l)CD (!lR*!l)EF'

c~3lF = L: ⁴⁵ (!lQ;!l)AB(!lR*!l)CD(!lR*!l)EF'

C(4) A-F = ,,15 L.J (!lR*!l) (!lR*!l) (!lR*!l) AB CD EF (A, ... ,F = 1, ... ,G* p**).

Then, after some algebra (see Ogasawara, 2009),

C r (t*) = tjJP+ 12 {I + N:I (a(3) tjJ3 + a(2) tjJ2 + a(1) tjJ + a(O)

+ it* bel) +(it*)2 b(2»)} + O(N.-2),

where

(3) _ ,,[_1 ^{{(R<2» A(I) +} 8 ^{3 Po (R) B(4) }}

a - L.J 7· 2 ^P(3) (A-F) A-F 8 8 8 ^P(3) (DEF) A-F

A-F . TA TB Tc

1 8 3 _{Po 8} 3 _Po ",(1)]

+-- ₂₈₈ C AF ,

8TA8TB8Tc 8TD8TE8TF -

140

2. Errata for Ogasawara (2009)

Ogasawara's (2009, Equations (4.4) and (4.5» result with k=2 and without mean structures corresponding to the result in the previous section is a special case of this note, and should be corrected by using the definitions of

* * n* *

QJ' Q2'~" and R given earlier. See also Ogasawara (2010) for an additional minor erratum in Ogasawara (2009, Equations (4.6) and (4.7».

References

Ogasawara, H. (2009). Asymptotic expansions of the distributions of the chi-square statistic based on the asymptotically distribution-free theory in covariance structures. Journal of Statistical Planning and Inference, 139, 3246-3261.

Ogasawara, H. (2010). Supplement to the paper "Asymptotic expansions of the distributions of the chi-square statistic based on the asymptotically distribution-free theory in covariance structures". Economic Review (Otaru University of Commerce), 60 (4), 187-200.

http://www.res.otaru-uc.ac.jp/~hogasa/

Ogasawara, H. (in press). Asymptotic expansions in multi-group analysis of moment structures with an application to linearized estimators.

Communications in Statistics - Theory and Methods.