The joint moment generating function of quadratic forms in multivariate autoregressive series - The case with deterministic components

multivariate autoregressive series - The case with deterministic components

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White Rose Research Online URL for this paper:

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Article:

Larsson, R. and Abadir, K.M. (2001) The joint moment generating function of quadratic

forms in multivariate autoregressive series - The case with deterministic components.

Econometric Theory. pp. 222-246. ISSN 0266-4666

https://doi.org/10.1017/S0266466601171070

[email protected]

https://eprints.whiterose.ac.uk/

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THE JOINT MOMENT GENERATING

FUNCTION OF QUADRATIC

FORMS IN MULTIVARIATE

AUTOREGRESSIVE SERIES

The Case with Deterministic Components

K

AARARRIIIMMM

M. A

BABBAADDDIIIRRR University of York

R

OOOLLLFFF

L

ARAARRSSSSSSOOONNN Stockholm University

Let$Xt%follow a discrete Gaussian vector autoregression with deterministic com-ponents+ We derive the exact finite-sample joint moment generating function ~MGF!of the quadratic forms that form the basis for the sufficient statistic+The formula is then specialized to the limiting MGF of functionals involving multi-variate and unimulti-variate Ornstein–Uhlenbeck processes,drifts,and time trends+Such processes arise asymptotically from more general non-Gaussian processes and also from the Gaussian$Xt% and have also been used in areas other than time series,

such as the “goodness of fit” literature+

1. INTRODUCTION

Let the k3_{1 vector of discrete time series}$Xt%1

T

be generated by the vector autoregression~VAR!

X_t5

(

j5₀ p

m_jtj_1AX

t2₁1«_t[ mt_t1AX_t2₁1«_t, «_t;NID~0,V!, (1) wherem [ @ m₀,m₁, + + + ,m_p#is k3~p11!,t_t[ @1,t, + + + ,tp_#' _is~p11!31,

A is k3k, X₀ is a known constant vector, andV is k3 k positive definite+ The moment generating function~MGF!derivations given subsequently are not affected by the value ofV,which we take for simplicity to be I_k,the identity matrix of order k+Also,because the process includes a drift term,we can take

The first author gratefully acknowledges ESRC~UK!grant R000236627+We thank the co-editor and a referee for helpful comments+Address correspondence to:Karim M+Abadir,Department of Mathematics and Depart-ment of Economics,University of York,Heslington,York Y010 5DD,England;e-mail:kma4@york+ac+uk+

X05 0 without loss of generality+ For example, defining $Yt% [ $Xt 2 X0%

and using~1!,

~Yt1X0!5mtt1A~Yt211X0!1«t,

leading to the VAR

Y_t5~ mt_t1~A2I_k!X₀!1AY_t2₁1«_t[mtI _t1AY_t2₁1«_t,

where Y050,m [ @I mI0,m1, + + + ,mp#,andmI0[ m01~A2Ik!X0+

The likelihood of this model can be written as

L~ m,A;X₀, + + + ,X_T!

5_~2p!2~Tk02!_6V62~T02!

3_etr

F

21 2V

2₁

(

t5₁ T

~X_t2_mt

t2AXt2₁!~X_t2mt_t2AX_t2₁!

'

G

_{, (2)}

where6{6[det~{!and etr~{! [exp@tr~{!#and the corresponding sufficient sta-tistic is extracted from

(

Xt@Xt' tt' Xt'21# and

(

F

t_t

Xt21

G

@tt' Xt'21#,

where henceforth all the summations are from t5_{1, + + + ,T except when stated} explicitly+There are two obvious reductions in our special setting to

S

(

X_tt_t'_,

(

X_t2₁t_t

'_,

(

X_tX_t2₁

' _,

(

X_t2₁X_t2₁

'

D

_,

unlike in the full linear model+First,

₍

tttt'is deterministic and need not

ap-pear+Second,

₍

X_tX_t'and

₍

X_t2₁X_t2₁

' _{differ by X}

TXT

'_{,which is already}

obtain-able from the first element X_T of

₍

X_tt_t'₂

₍

_X

t2₁t_t'+There remains one final and more subtle simplification+To this end,note that

(

X_tt_t'₂

(

X_t2₁t_t

'_5X

TtT1₁

' ₁

(

X_t~t_t'_2t

t1₁

' _!

and

(

X_t~t_t1₁

' _2t

t

'_!5

(

X_t@0, 1, 2t11, 3t2_{13t11, + + +#}

is a function of

₍

Xttt'only+~When p5 0, the term is the null function and

may be omitted altogether+!The reduction is therefore to replace

₍

X_tt_t'_{by X}

T,

and the sufficient statistic is

S

X_T,

(

X_t2₁t_t

'_,

(

X_tX_t2₁

' _,

(

X_t2₁X_t2₁

'

D

₊

The sufficient statistic is minimal if one furthermore excludes terms that are repeated in the symmetric matrix

₍

X_t2₁X_t2₁

' _{+The elimination matrix could be}

follows where the off-diagonal elements of parameter matrices corresponding to symmetric matrices are customarily scaled by 12

_ _anyway+

For the purpose of the asymptotic analysis in Section 3,it is more conve-nient to work with the basis for the sufficient statistic

S[ ~S1,S2,S3,S4! [

S

XT,

(

Xt21tt',

(

«tXt'21,

(

Xt21Xt'21

D

, (3)

which is a 12_{1 transform of the sufficient statistic,by~1!+}

Distributional results for such models are patchy and are summarized in Tanaka~1996!+See also Nielsen~1997!,Rothenberg~1999!,Gönen,Puri, Ruym-gaart,and van Zuijlen~1999!+In this paper,we present the exact MGF of the general S in Section 2+This extends earlier work by Abadir and Larsson~1996! and opens the way for a systematic study of the effects of including drifts and trends in VAR models+For example,results along the lines of Abadir,Hadri, and Tzavalis~1999! may now be investigated+In Abadir and Larsson~1996!, the marginal MGF for the different basis

S

XTXT',

(

«tXt'21,

(

Xt21Xt'21

D

was derived because there were no deterministic components there,hence the irrelevance of the sign of XT,from a distributional viewpoint,and its inclusion

through X_TX_T'+

In Section 3,we specialize the MGF to the asymptotic case,which happens to allow the process~1! to cover more general error structures$«t%+We do so

while focusing on the casem5_{0+The result is the joint MGF of functionals} involving Ornstein–Uhlenbeck processes,drifts, and time trends,all of which had no known joint MGF ’s except for some~not all!of the univariate special cases+Other potential uses for our results can be found in the literature on good-ness of fit, where these functionals arise~see,e+g+,d’Agostino and Stephens, 1986!+

All the proofs are collected in the Appendix+As for the general notation,we follow the one summarized in the appendix of Abadir and Larsson~1996!+ Ad-ditionally,the change of a variable of integration that maps u°v[ l~u!,for some functionl~{!,will be written in the inverse-mapping form ual21~v!,

whereby u is replaced byl21_{~v!in the integrand+}

2. THE MOMENT GENERATING FUNCTION

Consider the block-tridiagonal nonsingular Tk3Tk symmetric matrix

Ds5

3

P Q' 0 J 0

Q P Q' L I

0 L L L 0

I L Q P Q'

0 J 0 Q M

4

where M,P,Q are k-square full-rank matrices with M and P symmetric and 0 denotes null matrices of appropriate dimensions+The following lemma is ex-tracted from the proofs in Abadir and Larsson~1996!and will be used in our derivations here+

LEMMA 1+ The determinant of Dsis given by

6D_s6562_Q6T21

*

_@I

k 0#

F

2PQ2₁

2I_k

Q'Q21 ₀

G

T2₁

F

M 2_Q'

G

*

+

We need to derive one further lemma about Dsbefore proceeding to obtain the

main result of this section+Following Abadir and Larsson~1996,Theorem 2+1!, define

C0[Ik

C₁[PQ2₁ ,

C_j[C_j2₁PQ 21_2C

j2₂Q'Q

21_{, j[Z,}

whose solution is

@Cj Cj21#5@Ik 0#

F

PQ21 _I

k

2_Q'_Q2₁

0

G

j

5 2_{Q@0 ~Q}'_!2₁

#

F

PQ

21 _I

k

2_Q'_Q2₁

0

G

j11

(5)

for any integer~including negative!j+Then we have the following lemma+

LEMMA 2+ The typical block of the inverse of Dsis given by

D_sT2_t1₁1_j,T2_t1₁

5_~2_1!j_Q2₁

~C_t2_j2₂M2C_t2_j2₃Q

'_!

3

(

n5₀ T2t

~C_t1_n2₁M2C_t1_n2₂Q

'_!2₁

Q'~MC_t1_n2₂

' _2QC

t1_n2₃

' _!2₁

3_~MCt'₂₂2_QCt'_23!~Q'_!2₁ ,

where t5_{1, + + + ,T and j}5_{0, + + + ,t}2_{1 and the superscripts refer to (block-) row} and column numbers,respectively,starting from the top left corner. The terms above the diagonal blocks are obtained by DsT

2_t1₁,T2_t1₁1_j

5_~DsT2_t1₁1_j,T2_t1₁ !'₊

This lemma is general and, as pointed out by the referee, can be used in problems that are not necessarily related to our work~or to statistics!+

We can now derive the main result of this section+

THEOREM 3+ The MGF of S is

w_T,m~u1,U2,U3,U4!5wT,0~0,0,U3,U4!etr

S

22 1

2m'm

(

t

ttt

'

D

_exp~

2 1

2z'D

s 2₁

where~u1,U2,U3,U4!correspond to ~S1,S2,S3,S4!,respectively,U4is

symmet-ric,and

M[I_k

P[I_k22U₄1U₃A1A'U₃'1A'A5P' Q[2_A2_U

3

'

w_T,0~0,0,U3,U4!5det~2Q! ~12_T!02

3det

S

@I_k 0#

F

2_PQ2₁

2_Ik

Q'Q2₁

0

G

T2₁

F

I_k 2Q'

GD

2~102!

+

Finally,Ds 2₁

is defined explicitly in Lemma 2,andz [ ~z1', + + + ,zT'!'with

z_t'_{[ t}

t1₁

' _~U

22m

'_U

3

'_2m'_A!1t

t

'_m'_{, t51, + + + ,T21,}

z_T' _[u

1

'_1t

T

'_m'₊

The theorem can be made more explicit in a variety of directions,depending on the required application+For example,the~p11!3~p11!matrix

(

t_tt_t'

can be written as

(

t_tt_t'₅

(

3

1 t J

t t2 _J

I I L

4

5

3

T T~T

11!

2 J

T~T1_1!

2

T~T1_1!~2T1_1!

6 J

I I L

4

;

3

T

i1j21

i1_j2₁

4

,

where i,j are the row and column numbers,respectively+This theorem can also be simplified to a variety of published univariate asymptotic special cases+ How-ever, more important,we can specialize this theorem to a general asymptotic nearly nonstationary case that arises frequently in connection with limiting dis-tributions in time series+This is the purpose of the next section+

3. THE NEARLY NONSTATIONARY LIMITING CASE

Letm5_{0 and A}5_Ik1_~10T!H,where H is an arbitrary k3k matrix+Then, defining

G [diag~1,T2₁

it can be shown that

S

1

!

T XT, 1

!

T3

(

Xt21tt

'_G,1

T

(

«tXt21

' _, 1

T2

(

Xt2₁X_t2₁

'

D

d

&

&

S

J_H~1!,

E

0 1

J_H~r! @1,r, + + + ,rp_#dr,

S

E

0 1

J_H~r!dW~r!'

D

'

,

E

₀1

JH~r!JH~r!'dr

D

[ ~SD1,SD2,SD3,SD4! [S,D (6)

where d&& _{denotes convergence in distribution}, _W~_r! _{is the standard}

k-dimensional Wiener process on r [ @0,1#, and JH~r! is the corresponding

Ornstein–Uhlenbeck process defined by

J_H~r! [

E

0

r

exp@~r2s!H#dW~s!

when X050+These limiting distributions hold under less restrictive

distribu-tional assumptions on$«t%+

In view of~6!,the limiting MGF of interest becomes

f_H~u₁,U₂,U₃,U₄! [E@etr~u₁'SD₁1_U

2SD21U3SD31U4SD4!#

5 _lim

Tr`wT,0

S

1

!

T u1, 1

!

T3 GU2,

1

T U3, 1

T2 U4

D

,

which is the joint MGF ofSD [ ~SD1,SD2,SD3,SD4!+When H50,J0~r!5W~r!and

the resulting MGF is denoted byf~u1,U2,U3,U4! [ f0~u1,U2,U3,U4!,and the

functionals contain no stochastic components other than Wiener processes+

COROLLARY 4+ The joint MGF ofS isD

f_H~u₁,U₂,U₃,U₄!5exp@₂₂1_~,

11,21,3!#etr@2 1 2

__~H'_1U

3!#0

!

det~g~1!!,

where

F[H'H1_H'_U3'1_U3H2_2U4, G[H'_2H1U

32U3',

,_1[u1'

E

0 1

~g~12_z!'_g~12_z!!2₁ dz u1,

,

2[2u1

'

E

0 1

,₃[

E

0 1

f~z!'_~g~12z!'_g~12z!!21_f~z!dz,

g~z! [ @I_k 0#exp

S

z

F

G Ik F 0

G

D

F

Ik

2_H'2_U

3

G

,

f~z!'_[

(

i5₀ p

@u_i1 + + + u_ik, 0'#

(

j5₀ i

~2_i! j~21!j

3

S

_zi2_j

exp

S

~12_z!

F

G I_k

F 0

G

D

2I2k

D

3

F

G I_k

F 0

G

2_j2₁

F

Ik

2_H'2_U

3

G

,

with z being a scalar,uinbeing the typical element ~row i, column n!of U2,

and~2i!_j[

₎

_n5₀ j2₁

~n2i!denoting the Pochhammer symbol+

We have not worked out the integrals in z for the general case,as they are more easily manipulated numerically~the integration is over the interval~0,1! in one dimension!in applications+Note that any function of an n3n matrix can be written as a polynomial of degree n21 in the matrix,by the Cayley– Hamilton theorem,so that infinite series of matrices are not required numeri-cally+ This comment applies to exp~{! and also to matrix functions such as ~g~{!g~{!'_!2₁

+

In f~{!,the finite series in j is Tricomi’s confluent hypergeometric function+ Little additional insight is gained from using the hypergeometric formulation here,so we have refrained from doing so+

We now illustrate our corollary by specializing it to the univariate case with general trend components+In the univariate case,G5_{0 and},furthermore,put F5f,H5h,and U35u3 to stress that these quantities have become scalars+

Because

F

G I₁

F 0

G

5

F

0 1

f 0

G

,

it is now possible to evaluate the exponential of this matrix explicitly+To this end,a series expansion yields

exp

S

z

F

0 1

f 0

G

D

5

3

cosh~z

!

f! 1

!

f sinh~z

!

f!

!

f sinh~z

!

f! cosh~z

!

f!

4

,

implying

g~z!5cosh~z

!

f!2vsinh~z

!

f!, v[h1u3

In connection with ,₁,,₂,,_{3 of Corollary 4}, we have the following further reductions+

COROLLARY 5+ In the univariate case~k51!,we have

,

15u12

sinh

!

f g~1!

!

f,

,

25 2

u₁

g~1!f i

(

5₀ p

u_i1

S

i!~1

1~21!i_!

~

!

f

!

i

2

(

j5₀ i _~2i!

j

~

!

f

!

j

~

e

!f_1~21!j_e2!_f

!

D

,

,₃5 1 2g~1!f i

(

50

p

(

j50

p

u_i1u_j1g_ij,

where ui1is the typical element of the vector u2and

gij5 2

i!~11~21!i_!

~

!

f

!

i11

F

j!

~

~12v!e!f_2~21!j_~11v!e2!f

_!

~

!

f

!

j

1

(

n50 j _~2j!

n~~11v!2~21!n~12v!!

~

!

f

!

n

G

1

(

m50 i _~2i!

m

~

!

f

!

m

F

~12v!e!f_1~21!m_~11v!e2!f j1_i2_m1₁

1~

2_1!m_~j1i2m!!

_~

_~12v!e!f_2~21!i1j_~11v!e2!f

_!

~2

!

f!j1i2m11

1

(

n50

j1i2m_~m2i2j! n

~2

!

f!n11

~

~11v!e

!f_2~21!m1n_~12v!e2!f

_!

G

₊

It is worth observing that,in the asymptotics for the univariate case,matters simplify further+This is because,from~6!,

D

S₃5

E

0 1

J_h~r!dW~r!5Jh~1!

2₂₁

2 2h

E

0 1

J_h~r!2_dr

5 SD1

2₂₁

2 2hSD4,

so that we may set u350+This, however,induces no simplifications for the

preceding derivations,other than settingv[h

YY

!

f ~i+e+,u350!in~7!+For the

vector case,inequalities such as

E

₀1

J_h~r!dW~r!'_Þ

S

E

0 1

J_h~r!dW~r!'

D

'

Some special cases follow directly from these corollaries by setting some components of u•to zero,hence obtaining marginal MGF ’s+For example,when

only a constant is fitted to the model,we get the MGF of sufficient statistics associated with de-meaned Brownian motions which are particularly useful in connection with the goodness-of-fit literature~see,e+g+,d’Agostino and Stephens, 1986!+In this case,the joint MGF of the sufficient statistic

S

W~1!,

E

0 1

W~r!dr,

E

0 1

W~r!2_dr

D

is

f₀~u₁,u₂,0,u₄!5 exp@

1 2 __~,

11,21,3!#

!

cosh

~

!

22u₄

!

with the ordering of u₁,u₂,u₄ corresponding to the respective variates given earlier,and

,₁5 u1

2

!

2_2u

4

tanh

~

!

22u₄

!

,

,

25

u₁u₂

u4

S

1

cosh

~

!

22u₄

!

2₁

D

_,

,₃5 u2

2

2u₄

S

tanh

~

!

2_2u

4

!

!

2_2u

4

2₁

D

+

The marginal MGF of a famous related stochastic integral has been obtained by Anderson and Darling~1952!and used further by Abadir and Paruolo~1997!; see also Rothenberg~1999!+

REFERENCES

Abadir,K+M+,K+Hadri,& E+Tzavalis~1999!The influence of VAR dimensions on estimator bi-ases+Econometrica 67,163–181+

Abadir,K+M+& R+Larsson~1996!The joint moment generating function of quadratic forms in multivariate autoregressive series+Econometric Theory 12,682–704+

Abadir,K+M+& P+Paruolo~1997!Two mixed normal densities from cointegration analysis+ Econ-ometrica 65,671– 680+

d’Agostino,R+B+& M+A+Stephens~eds+! ~1986!Goodness-of-fit Techniques+New York:Marcel Dekker+

Anderson,T+W+& D+A+Darling ~1952!Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes+Annals of Mathematical Statistics 23,193–212+

Gönen,M+,M+L+Puri,F+H+Ruymgaart,& M+C+A+van Zuijlen~1999!The limiting density of unit root test statistics:A unifying technique+Journal of Time Series Analysis,forthcoming+

Nielsen,B+~1997!On the Distribution of Cointegration Tests+Ph+D+Thesis,University of Copen-hagen,Institute of Mathematical Statistics+

Rothenberg,T+J+~1999!Some Elementary Distribution Theory for an Autoregression Fitted to a Random Walk+Mimeo,University of California,Berkeley+

Tanaka,K+~1996!Time Series Analysis: Nonstationary and Noninvertible Distribution Theory+New York:Wiley+

APPENDIX

:

PROOFS

Proof of Lemma 1. See Theorem 2+3 of Abadir and Larsson~1996!+

Proof of Lemma 2. Let

Ds

21₅

3

cT CT '

C_T c_T21 CT21 '

C_T21 L L

L c2 C2'

C2 c1

4

,

wherectis k3_{k andCtis the~t}2_1!k3_{k matrix of all the blocks belowct,therefore}

expanding with t+Abadir and Larsson~1996,BTof Theorems 2+2 and 2+3 there!derive cTby recursive partitioned inverse,but here we need to derive the whole matrix+

Define Ds~t!as the matrix Ds~T! [Dsbut of dimensions tk3tk instead of Tk3Tk and denote the leftmost upper block of Ds

21_{~t! by B}

t+Notice that BtÞct except for t5T+First,partition Dsinto

Ds5

3

P Q' ₀

Q

0 Ds~T21!

4

+

Then,using the partitioned inverse formula,the first component of the second diagonal block is

cT215@Ik 0#

S

Ds

21_~T21!1D s

21_~T21!

F

Ik

0

G

QcTQ

'_@

Ik 0#Ds

21_~T21!

D

F

Ik

0

G

5_B

T211BT21QcTQ '

BT21+ Second,partition Dsinto

Ds5

3

P Q'

Q P

0 0 J

Q' 0 J

0 Q

0 0

I I

and repeat a similar operation to get

cT225@Ik 0#

1

Ds

21_~T22!1D s

21_~T22!

3

0 Q

0 0

I I

4

3

F

+ +

+ cT21

GF

0 0 J

Q' 0 J

G

Ds

21_~T22!

2

F

Ik

0

G

5_B

T221BT22QcT21Q '

BT22

as before+By induction,this relation

ct215Bt211Bt21QctQ'Bt21

holds for all partitions of Dsbecause B15M

21_{+As a result,}

ct5Bt1BtQBt11Q '

Bt1{{{1BtQBt11+ + +QBTQ '_{+ + +}

Bt11Q '

Bt+ (A.1)

For the explicit solution of this formula,we need to work out BtQ+ + +Bt1jQ using

BtQ5@0 ~Q'!

21_#

F

PQ

21 _I k

2Q'_Q21 ₀

G

t21

F

M

2_Q'

G

3

S

_{@0 ~Q}'_!21_#

F

PQ

21 _I k

2_Q'_Q21 ₀

G

t

F

M

2_Q'

G

D

21

, (A.2)

which we deduce from Abadir and Larsson~1996,Theorem 2+3!+We have reformulated the power of the first~2k!3~2k! matrix in their formula for Bt, so that the required typical product simplifies sequentially to

BtQ+ + +Bt1nQ5@0 ~Q

'_!21_#

F

PQ

21 _I k

2_Q'_Q21 ₀

G

t21

F

M

2_Q'

G

3

S

@0 ~Q'_!21_#

F

PQ

21 _I k

2_Q'_Q21 ₀

G

t1n

F

M

2Q'

G

D

21

5_Q21_~C

t22M2Ct23Q '_!~

Ct1n21M2Ct1n22Q

'_!21_{Q (A.3)}

by~5!+We also need to work out Q'_B

t1n21+ + +Q'Btto solve~A+1!explicitly+Because the B+matrices are symmetric,Q'Bt1n21+ + +Q'Bt5~BtQ+ + +Bt1n21Q!',and using the trans-pose of~A+3!with n2_{1 in place of n,}

Q'_B

t1n21+ + +Q'Bt5Q'~MCt'1n222QCt'1n23!

21_~MC

t'222QCt'23!~Q'!

Combining it with~A+3!,

BtQBt11+ + +QBt1nQ'+ + +Bt11Q'Bt

5Q21_~C

t22M2Ct23Q'!~Ct1n21M2Ct1n22Q'!

21

3Q'_~MC

t'1n222QCt'1n23!

21_~MC

t'222QCt'23!~Q'!

21_, which upon substitution in~A+1!gives

ct5

(

n50 T2t

Q21_~C

t22M2Ct23Q '_!~

Ct1n21M2Ct1n22Q '_!21

3_Q'_~MC

t'1n222QCt'1n23!

21_~MC

t'222QCt'23!~Q'!

21_{, (A.4)}

as the required diagonal blocks+

Now we turn to the off-diagonal blocks of Ds

21_{~T!+ By symmetry of D} s

21_{~T!, we} only need the blocksCtthat are below any typical diagonal blockct+For this,we need to calculate the complete first block-column of Ds

21_{~t!,which we denote by}

Ds

21_~t!

F

Ik

0

G

[

F

Bt

Bt

G

+

By the recursive partitioned inversion of D_s21_{~t! for successive t,}

Bt5 2Ds

21_~t21!

F

Q

0

G

Bt5 2

F

Bt21

Bt21

G

QBt5

3

2B_t21QBt

1_B

t22QBt21QBt

2Bt23QBt22QBt21QBt

I

4

+

Similarly,

CT5 2Ds

21_~T21!

F

Q

0

G

cT5 2

F

BT21

BT21

G

QcT,

and by repeated use of the partitioned inverse formula as we did earlier forct,

CT215 2Ds

21_~T22!

3

0 Q

0 0

I I

4

F

+ +

+ c_T21

GF

0

I_k

G

5 2Ds

21_~T22!

F

QcT21

0

G

5 2

F

BT22

BT22

G

QcT21+

By induction,

Ct5 2

F

Bt21

Bt21

G

Qct5

3

2Bt21Qct

1_B

t22QBt21Qct

2Bt23QBt22QBt21Qct

I

4

Changing ta_t2_{j and n}a_j2_{1 in~A+3!,we have}

Bt2jQ+ + +Bt21Q5Q

21_~C

t2j22M2Ct2j23Q '_!~

Ct22M2Ct23Q

'_!21_{Q, (A.6)}

which together with~A+4!and~A+5!gives the stated result+

_n

Proof of Theorem 3. The MGF of S [ ~S1,S2,S3,S4! [ ~XT,

(

Xt21tt '_,

(

«tXt'21,

(

Xt21Xt'21!is wT,m~u1,U2,U3,U4!

[E

F

exp

S

u1 '

XT1

(

tt '

U2Xt211

(

Xt21 '

U3«t1

(

Xt21 '

U4Xt21

D

G

5_~2p!2~Tk02!

E

exp

S

u1 '

xT1

(

xt21 '

U2 '_t

t1

(

xt21 '

U3«t

1

(

x_t21 ' _U

4xt212 1 2

(

«t

'_«

t

D

~dx! (A.7)

by using~2!withV5_{Ikand where the integral is over}RTk_{with~dx!being the exterior} product dx11dx12+ + +dxkT+ Using the VAR formulation~1! to substitute for«t, we can decompose these sums into deterministic and stochastic components

(

xt21 '

U3«t5

(

xt21 '

U3~xt2mtt2Axt21!,

2 1

2

(

«t

'_« t5 2

1

2

(

~xt

2_mt

t2Axt21!'~xt2mtt2Axt21!

5 21

2

(

tt

'_m'_mt t

1

S

(

_x

t '_mt

t2 1 2

(

xt

'

xt1

(

xt21 '

A'

S

xt2mtt2 1

2Axt21

DD

Substituting back into~A+7!,rearranging,then using x050,these give

wT,m~u1,U2,U3,U4!

5~2p!2~Tk02!_exp

S

₂1

2

(

tt

'_m'_mt t

D

3

E

exp

F

u₁'_x T1

(

xt

'_mt t2

1 2

(

xt

'_x t

1

(

_x

t21 '

S

_~

U2 '₂

U3m2A '_m!t

t1~U31A '_!

xt

1

S

U42U3A2 1 2A

'_A

D

_x

Define x[vec~x1, + + + ,xT!,and Dsas in~4!with M,P,Q stated in the theorem+Then,we can rewrite~A+8!as

wT,m~u1,U2,U3,U4!

5~2p!2~Tk02!_etr

S

₂1

2m

'_m

(

t_tt_t'

D

E

_exp

S

_z'_x21

2x

'_D sx

D

~dx!

56Ds6

2~102!_etr

S

₂1

2m

'_m

(

tttt'

D

exp

S

1 2z

'_D s

21_z

D

5_w

T,0~0,0,U3,U4!etr

S

2 1

2m

'_m

(

tttt

'

D

exp

S

1 2z

'

Ds

21_z

D

_{, (A.9)} where we have completed the square and integrated x out and wherewT,0~0,0,U3,U4!is

obtained by Lemma 1+

_n

Proof of Corollary 4. From Theorem 3,

fH~u1,U2,U3,U4!5 lim Tr`wT,0

S

1

!

T u1, 1

!

T3GU2, 1

T U3, 1

T2U4

D

5 lim

Tr`wT,0

S

0,0,

1 T U3,

1

T2U4

D

exp

S

1 2z

'_D s

21_z

D

₊

The limit of the first component is available from Corollary 3+2 of Abadir and Larsson ~1996!as

lim

Tr`wT,0

S

0,0,

1 T U3,

1

T2U4

D

5etr

F

2 1 2~H

'_1U 3!

G

3

*

_@I

k 0#exp

S

F

G Ik F 0

G

D

F

I_k

2H'_2U 3

G

*

2102

+

Define E [ G 2 _~10T!G~H 1 _U

3 '_!+

Then for the second component, limTr`

exp~₂₂1_z'_D s

21_{z!,we note the reductions} M5_I

k P52Ik1

1 T ~H1H

'_1U

31U3'!1 1 T2F

2_Q5_I

k1 1

T ~H

1_U

3'!

2_Q21_5I k2

1

T ~H1U3

'_!1 1

T2~H1U3

'_!2_1O

S

1

T3

D

(A.10)

2PQ21_52I k1

1 T E1

1

T2F1O

S

1 T3

D

Q'_Q21_5I

k1 1 T E1O

S

and dropping lower-order terms henceforth for clarity of exposition,~5!gives

~21!q_@C

q Cq21#

F

M

2Q'

G

5@Ik 0#

F

2PQ21 _2I k Q'_Q21 ₀

G

q

F

M

2Q'

G

; @Ik 0#

3

2Ik1

1 T E1

1

T2F 2Ik Ik1

1

T E 0

4

q

3

F

Ik

Ik1 1 T ~H

'₁

U3!

G

; @Ik 0#exp

S

q

T

F

G Ik F 0

G

D

F

I_k

2H'_2U 3

G

[g

S

q T

D

+

(A.11)

The latter step follows because, for any fixed ~2k!3_{~2k! matrixJ, and q} [N an increasing function of T of maximal order O~T!,

lim Tr`

S

I2k

1 1

T J

D

q

5 _lim

Tr`exp

F

q log

S

I2k

1 1

T J

DG

5 _lim

Tr`exp

F

q

S

1

T J1O

S

1

T2

DDG

5_Tlim_r`exp

F

q T J

G

,

so that we could use the same steps of Corollary 3+2 of Abadir and Larsson~1996!+

Then, ~A+11! allows us to write the required blocks of Ds

21_{of Lemma 2 for large T,} with t51, + + + ,T and j50, + + + ,t21,as

DsT

2t111j,T2t11_;g

S

t2j T

D

n

(

50

T2t

S

g

S

t

1n

T

D

'

g

S

t

1n

T

DD

21 g

S

t

T

D

'

+

We need the limit

lim Tr`z

'

Ds

21_{z [ lim} Tr`_t

(

51

T

(

j50 t21

~11_sgn~j!!z

T2t111j '

DsT

2t111j,T2t11_z T2t11

5 _lim

Tr`_t

(

51 T

(

j50 t21

~11_sgn~j!!z

T'2t111jg

S

t2j

T

D

3

(

n50 T2t

S

g

S

t

1_n

T

D

'

g

S

t

1_n

T

DD

21 g

S

t

T

D

' z_T2t11

5 _lim

Tr`_t

(

51 T

(

j50 t21

~11_sgn~j!!z

T2t111j '

g

S

t

2_j

T

D

3

(

n50 T2t

S

g

S

12 n

T

D

'

g

S

12 n

T

DD

21 g

S

t

T

D

'

where sgn~{!is the signum~sign!function and we have reversed the sum in n by replac-ing na_T2t2n+In the context of this corollary,

zt'5 1

!

T3tt11 ' _GU

2, t51, + + + ,T21,

zT ' ₅ 1

!

T u1

'_,

wherett11 ' _G

is a normalized bounded matrix for any t,T+For large T,the sums become integrals,and we convert successively

t

T ax[~0,1!, j

T ay[~0,x!, n

T az[~0,12x!, and

tt'11G ;

F

1, t

T, + + + ,

S

t

T

D

p

G

a@₁,_x, + + + ,_xp# [_h~_x!'+ _(A.13)

With these normalizations,the quadratic terms inztfor t5₁, + + + ,T2_{1~which are}

ob-tained when j5_{0!vanish at a rate of T}21_{+The remaining quadratic term is the one in} zT,which is obtained when j50 and t51 in~A+12!and which we denote by,1in the limit+The cross-product terms are split in two,namely,when j5_t2₁._{0 in~A+12!}

and we have products involvingz_T' _andz

T2t11, whose limit we denote by,2;and the remaining terms whose limit we denote by,_{3+This gives}

lim Tr`

z'_D s

21_z5,

11,21,3, where

,

1[u1 '

E

0 1

~g~12_z!'_g~12_z!!21_{dz u} 1,

,₂[2u1 '

E

0 1

E

₀12x

~g~12_z!'_g~12_z!!21_{dz g~x!}'

U2 '

h~12_x!dx,

,₃[2

E

0 1

E

₀x

h~12_x1_y!'_U

2g~x2y!dy

3

E

0 12x

~g~12z!'_g~12z!!21_{dz g~x!}'_U

The integral in z is the least tractable of the three,because of the inversion of g~{!+It is therefore beneficial to reverse the order of integration in,₂to

,₂5_2u

1 '

E

0 1

~g~12_z!'_g~12z!!21

E

0

12z g~x!'_U

2'h~12x!dxdz+ (A.15)

For,₃,a change of variable of yax2y,followed by a change of order of integration ~x and z!,gives

,

352

E

0 1

E

₀x

h~12_y!'_U

2g~y!dy

E

0

12x

~g~12_z!'_g~12_z!!21_{dz g~x!}'

U2 '

h~12_x!dx

52

E

0 1

E

₀12z

E

₀x

h~12y!'_U

2g~y!dy~g~12z!

'_g~12z!!21_g~x!'_U 2

'_h~12x!dxdz+

(A.16)

By symmetry of this particular integrand in x and y,we have here

E

₀12z

E

₀x

+ + +dydx5

E

0 12z

E

₀y

+ + +dxdy,

whereas we know that,from standard changing of the order of double integrals,

E

₀12z

E

₀x

+ + +dydx[

E

0

12z

E

_y12z

+ + +dxdy

for any integrand;so that

2

E

0

12z

E

₀x

+ + +dydx5

E

0 12z

E

₀12z

+ + +dxdy

and

,₃5

E

0 1

S

E

₀12z g~x!'_U

2'h~12x!dx

D

'

~g~12_z!'_g~12z!!21

E

0

12z g~x!'_U

2'h~12x!dxdz+

(A.17)

We now work out the row vector

f~z!'_[

E

0

12z

h~12_x!'_U

2g~x!dx

From the definitions of g~{!and h~{!in~A+11!and~A+13!,

f~z!'₅

E

0

12z

@

@1,~12_{x!, + + + ,~1}2_x!p_#U 2 0

'

_#

exp

S

x

F

G Ik F 0

G

D

dx

F

Ik

2_H'_2U 3

G

5

E

21

2z

@

@1,~2_{x!, + + + ,~}2_x!p_#U 2 0

'

_#

exp

S

x

F

G Ik F 0

G

D

dx

3_exp

S

F

G Ik F 0

G

D

F

Ik

2_H'_2U 3

G

,

where the latter equality follows from xax11 and from the~2k!3~2k!matrix com-muting with itself+Notice that the first matrix is now a row vector and we can write

f~z!'₅

(

i50 p

@ui1 J uik, 0'#

E

21

2z

~2x!i_exp

S

_x

F

G Ik F 0

G

D

dx

3exp

S

F

G Ik F 0

G

D

F

Ik

2H'_2U 3

G

5

(

i50 p

~21!i_@u

i1 + + + uik, 0'#exp

S

~x11!

F

G I_k F 0

GD

j

(

50

i ~2i!jxi

2j

*

21

2z

3

F

G I_k F 0

G

2j21

F

Ik

2_H'2_U

3

G

5

(

i50 p

@ui1 + + + uik, 0'#

(

j50 i

~2i!j~21!j

S

zi

2j_exp

S

_~12z!

F

G Ik

F 0

G

D

2I2k

D

3

F

G I_k F 0

G

2j21

F

Ik

2_H'2_U

3

G

,

which is the required result+

_n

Proof 1 of Corollary 5. To evaluate,_{1,via~7!and because}

d

dz

sinh~~12z!

!

f! g~12_z!

5 2

!

f

g~12_z!2, (A.18)

we simply have

,

15u12

E

0

1 _dz

g~12z!25u12 sinh

!

f

g~1!

!

f+ (A.19)

As for,₂,we retrace the steps of the proof of the previous corollary and observe that U2

'

h~12_x!5

(

i50 p

to find

,

252u1

(

i50 p

u_i1

E

0 1

E

₀12x _dz

g~12z!2g~x!~12x!

i_{dx+ (A.21)}

Here,via~A+18!and using the identity

sinh a cosh b2_{cosh a sinh b}5_sinh~a2_{b!, (A.22)}

we find

,₂5 2u1

!

f i

(

50 p

ui1

E

0 1

S

sinh

!

f g~1! 2

sinh~x

!

f!

g~x!

D

g~x!~12x!

i_{dx (A.23)}

5 2u1

g~1!

!

f i

(

50 p

ui1

E

0 1

sinh~~12_x!

!

_f!~12_x!i_dx

5 2u1

g~1!

!

f i

(

50 p

ui1

E

0 1

xi_sinh~x

!

_f!dx+

Further,successive integration by parts yields

E

₀y

xi_sinh~x

!

_{f!dx5 2}i!~11~21! i_! 2

~

!

f

!

i11 1

yi 2

!

f j

(

50

i _~2i!_j ~y

!

f!j~e

y!f_1~21!j_e2y!f_!,

(A.24)

giving the desired expression for,_{2by letting y}5₁₊

Looking at,₃,we use the proof of the preceding corollary to obtain

,₃5 1

2g~1!fi

(

50 p

(

j50 p

ui1uj1gij, (A.25)

where,by~A+20!and~A+16!,

gij[4g~1!f

E

0 1

E

₀x

~12y!j_g~y!dy

E

0

12x _dz

g~12_z!2g~x!~12x! i_dx+

Now,as in~A+21!–~A+23!again,

E

₀12x _dz

g~12_z!2g~x!5 1

g~1!

!

f sinh~~12x!

!

f!,

and

gij54

!

f

E

0 1

E

₀x

~12y!j_{g~y!dy sinh~~12x!}

!

_f!~12x!i_dx

5₄

!

_f

E

0 1

E

₀12x

~12_y!j_{g~y!dy sinh~x}

!

_f!xi_dx

5₄

!

_f

E

0 1

~12_y!j_g~y!

E

0

12y

sinh~x

!

f!xi_dxdy

54

!

f

E

0 1

yj_g~12y!

E

0 y

by xa ₁2_{x, changing the integration order and y}a ₁2_{y,respectively+ The inner}

integral is given by~A+24!,so that

gij5 22

E

0 1

yj_g~12y!

S

i!~11~21! i_!

~

!

f

!

i 2m

(

50

i _~2i! myi

2m

~

!

f

!

m ~e

y!f_1~21!m_e2y!f_!

D

_dy

5 2

E

0 1

yj_~~12v!e~12y!!f_1~11v!e2~12y!!f_!

3

F

i!~1

1_~2_1!i_!

~

!

f

!

i 2m

(

50

i _~2i! m

~

!

f

!

my

i2m_~ey!f_1~21!m_e2y!f_!

G

_dy

5 2i!~1

1~21!i_!

~

!

f

!

i

F

~12v!e

!f

E

0 1

yj_e2y!f_dy1~11v!e2!f

E

0 1

yj_ey!f_dy

G

1

(

m50 i _~2i!

m

~

!

f

!

m

F

~~1

2_v!e!f_1~21!m_~11v!e2!f_!

E

0 1

yj1i2m_dy

1~21!m_~12v!e!f

E

0 1

yj1i2m_e22y!f_dy

1~11v!e2!f

E

0 1

yj1i2m_e2y!f_dy

G

by the definition of g~{!in~7!+Using

E

₀1

yq_eay_dy5 q! ~2a!q111e

a

₍

n50 q _~2q!

n an11

where q[Nø$0% gives the stated result+

_n

Proof 2 of Corollary 5. A more conventional proof may be obtained from exploiting Girsanov’s theorem as in Perron~1991,Theorem 2!or Tanaka~1996,Ch+4!+

We want to calculate the MGF of

~S1,S2 '_,

S3,S4! [

S

Jh~1!,

E

0 1

~1,r, + + + ,rp_!J

h~r!dr,

E

0 1

Jh~r!dW~r!,

E

0 1

Jh~r!2dr

D

,

which we define as

fh~u1,u2,u3,u4! [E$exp~u1S11u2 '

S21u3S31u4S4!%, (A.27)

where u1,u3,u4are scalars and u2 '_{[ ~}

u20, + + + ,u2p! is a~p11!-dimensional vector+At first,apply Itô’s formula to write

d~Jh~t!2!52Jh~t!dJh~t!1dt52hJh~t!2dt12Jh~t!dW~t!1dt, so that,by integrating and solving,

E

₀1

Jh~t!dW~t!5 1 2$Jh~1!

2_21%2h

E

0 1

and~A+27!may be reformulated as

f_h~u₁,u₂,u₃,u₄!5exp

S

2u3

2

D

E

H

exp

S

u1Jh~1!1u2

'

E

0 1

~1,t, + + + ,tp_!'_J h~t!dt

1 u3

2 Jh~1! 2_1~u

42u3h!

E

0 1

Jh~t!2dt

D

J

+

Now,following Tanaka~1996,p+110!,we define the auxiliary process Y~t!through

dY~t!5 2_bY~t!dt1_{dW~t!, Y~0!}5₀₊

Putting X~t!5_{Jh~t! ~witha}5 2_{h in Tanaka’s notation!and lettingmJ andmY be the}

probability measures governing Jhand Y,respectively,we get

dmJ dmY

~Y!5exp

F

~h1b!

E

0 1

Y~t!dY~t!2h

2_2b2

2

E

0 1

Y~t!2_dt

G

5_exp

F

2h 1_b

2 1 h1_b

2 Y~1! 2₂h

2_2b2

2

E

0 1

Y~t!2_dt

G

_,

the second line following from Itô’s formula+Now,as in Tanaka~1996,p+111!,

fh~u1,u2,u3,u4!

5_exp

S

2u3

2

D

E

H

exp

S

u1Y~1!

1_u

2 '

E

0 1

~1,t, + + + ,tp_!'

Y~t!dt

1 u3

2 Y~1! 2_1~u

42u3h!

E

0 1

Y~t!2_dt

D

dmJ dmY

~Y!

J

5_exp~2_g!E

H

_exp

S

_u

1Y~1!1u2 '

E

0 1

~1,t, + + + ,tp_!'

Y~t!dt1_gY~1!2

D

J

_(A.28)

by choosing

b [

!

h2_12hu 322u4

and defining

g [ u31h1b 2 +

To go further,the idea is to define a process Z~t!5_Y~t!2_{utY~1!,whereutis a}

con-stant, chosen in such a way that Z~t! becomes independent of Y~1!+ But as is easily shown,we have that for 0,_s#t#1,

E$Y~s!Y~t!%5e2btsinh~ bs!