6_xxx_gupta.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 3, 2010, 435-442

ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE ON

GRANGER ECONOMETRICS AND STATISTICAL MODELING

DEDICATED TO THE MEMORY OF PROF. SIR CLIVE W.J. GRANGER

A Generalization of Durbin-Watson Statistic

A. K. Gupta1∗, D. G. Kabe2, and S. NiwitPong3

1 Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, USA
2 5971 Greensboro Dr., Mississauga, Ontario, Canada
3 Department of Applied Statistics, King Mongkut’s University of Technology North Bangkok, Thai-

land

Abstract. Two generalizations of the Durbin-Watson Statistic d, for testing that the serial correlation,

in a given univariate normal regression model, is zero, to its multivariate counter part, are proposed.

In the univariate case the moments of d are obtained in terms of generalized gamma functions. Our

methodology is based on the generalized quadratic form of the central Wishart distribution.

2000 Mathematics Subject Classifications: 62M10,62G10

Key Words and Phrases: Additive outlier; AR(1); Predictor; Prediction interval; Unit toot test

1. Introduction

For the univariate normal linear regression model

Y = Xβ + e, e ∼ N(0,σ2 I) (1)

where Y is an n component (column) vector, β has q components, X is n× q and of rank

q < n, σ2 is unknown, the Durbin-Watson statistic d is defined as follows,

(Y − Xβ)′(Y − Xβ) = (β − β̂)′X ′X (β − β̂) + Y ′(I − X (X ′X )X ′)Y

= (β − β̂)′X ′X (β − β̂) + Y ′QQ′Y,

β̂ = (X ′X )−1X ′Y and Q′Q = I , Q is n×m, m=(n-q) matric of rank m< n. It follows that

Y ′QQ′Y = f ′ f , (2)

∗Corresponding author.

Email address: gupta�bgsu.edu (A. Gupta)

http://www.ejpam.com 435 c© 2010 EJPAM All rights reserved.



A. Gupta, D. Kabe, S Niwitpong / Eur. J. Pure Appl. Math, 3 (2010), 435-442 436

f is m× n, and the density of f is

g( f ) = K exp

¨

−
1

2σ2
f ′ f

«

,−∞ < f <∞, (3)

where K , as a generic letter, denotes the normalizing constants of density functions in this

paper.

Now [6, p. 200] define d to be

d = f ′Af / f ′ f ,A= Q′A1Q, (4)

A1 =















1 −1 0 0 . . . 0

−1 2 1 0 . . . 0

. . . . . .

. . . . . .

0 0 . . . 1 2 1















where n× n A1 is of rank (n− 1).

Next setting f = ht, f ′ f = 1,(4) reduces to

d = h′Ah,h′h= 1, (5)

and hence

E(d g) = K

∫

h′h=1

(h′Ah)gdh=
(1

2
)gC(g)(A)

(m

2
)g

, (6)

where C(θ ) is the zonal polynomial [1, p. 29].

The integrals of the type (6) are known in the literature as the generalized quadratic form

of the central Wishart distribution (GQFCWD) integrals. The power function integrals of the

type (6) may be called the generalized quadraic form of the noncentral Wishart distribution

(GQFNCWD) integrals.

Mathai et al. [6, Chapter 5] list a number of integrals of the type (6) and their gener-

alizations, however, none of them are suitable for the moments problem of d in the present

context. We formulate some suitable integrals in our context for the moments problem.

The model (1) generalizes to the model

Y = Xβ + E, E ∼ N(0, I ⊗Σ), (7)

where Y is p× n, β is p× q, n> (p+ q), X is q× n and of rank q < n, Σ is p× p unknown.

We now write

(Y − βX )(Y − βX )′ = (β − β̂)X X ′(β − β̂)′ + Y (I − X ′(X ′X )−1X )Y ′

= (β̂ − β)X X ′(β̂ −β)′ + YQQ′Y ′,

β̂ = (X ′X )−1X ′Y and the first generalized d to be

d = t r(YAY ′)/t r(Y QQ′Y ′) = t r(FAF ′)/t r(F F ′) = t r(HAH ′) (8)



A. Gupta, D. Kabe, S Niwitpong / Eur. J. Pure Appl. Math, 3 (2010), 435-442 437

where HH ′ = I and H is p× (n− q) or p×m.

The second generalized d is, where m is assumed to be n,

d = |HAH ′| = |HΛH ′|, HH ′ = I , (9)

where Λ = diag(λ1,λ2, . . . ,λn) is the diagonal matrix of the roots of A.

From (8), [6, p.270, equation 5.5.6] show that

E(d g) =

∫

HH′=1

(t r(HΛH ′))KdH =

�

p

2

�

g

C(g)(Λ)/

�

pm

2

�

g

, (10)

where Λ = diag(λ1,λ2, . . . ,λm) is the diagonal matrix of the roots of A1.

However, it does not appear that the integral

E(d g) =

∫

HH′=1

|HΛH ′|gdH, (11)

has been suitably evaluated in the literature, and the evaluation of the integral (11) is the

main result of the present paper. The methodology for integrating (11) is based on [2, 3, 4]

and [5, p. 352, equation 5.5.29].

We present our methodology in the next section, and section 3 evaluates the integral (11).

The moments of d can be calculated in terms of gamma functions; however the density of d

is not, as yet, available in the literature, except in trivial cases. Sometimes the same symbol

denotes different quantities; however, its meaning is made explicit in the paper.

2. Methodology

Given the joint density of n gamma variates to be

g(y1, y2, . . . , yn) = K exp{−(λ1 y1 + · · ·+λn yn)}y
g1−1

1 . . . y gn−1
n , (12)

the density of t = (y1 + · · ·+ yn) is desired. The moment generating function ψ(θ) of t is

ψ(θ) = (α1− θ)
−g1 . . . (αn− θ)

−gn

= (α1− θ)
−g1((α1 − θ)− (α1 −α2))

−g2 . . . ((α1− θ)− (α1 −αn))
−gn ,

where α1 is the largest parameter amongst the n positive parameters α1, . . . ,αn.

Now [2] expands ψ(θ) as

ψ(θ) = (α1 − θ)
(−g1+···+gn+r2+···+rn)

∞
∑

r2=0

· · ·
∞
∑

rn=0

�

g2+ r2 − 1

r2

�

. . .

�

gn+ rn − 1

rn

�

(α1 −α2)
r2 . . . (α1 −αn)

rn , (13)



A. Gupta, D. Kabe, S Niwitpong / Eur. J. Pure Appl. Math, 3 (2010), 435-442 438

and inverting (13) finds the density of t to be

g(t) = K exp{−t}t(g1+...+gn−1)φ(g2, . . . , gn; g1+ · · ·+ gn; (α1 −α2)t, . . . , (α1 −αn)t)

where

φ =

∞
∑

r2=0

· · ·
∞
∑

rn=0

Γ(g2+ r2) . . .Γ(gn+ rn)(α1 −α2)
r2 . . . (α1 −α2)

rn

Γ(g1 + · · ·+ gn+ r2 + · · ·+ rn)r2! . . . rn!

= 1F1(g2+ · · ·+ gn; g1 + · · ·+ gn); ((n− 1)α1−α2 − . . .αn)t). (14)

To prove (14), we observe that the sum of two noncentral Wishart p×p matrices A and B, with

noncentrality parameter p× p matrices ∆ and Ω, and n and q degrees of freedom respectively

is again noncentral Wishart with (n + q) degrees of freedom, and noncentrality parameter

matrix (∆+Ω). With 2g = (p+ 1), we write this result as

∫

A+B=D

exp{−t r(A+ B)}|A|n−g |B|q−goF1(n;∆A)oF1(q;ΩB)dAdB

= K exp{−t rD}|D|n+q−g oF1(q; (∆+Ω)D) (15)

or formally that

oF1(n;∆A)oF1(q;ΩB) = oF1(n+ q; (∆+Ω)(A+ B)). (16)

Mathai [5, p. 339, Theorem 5.5] defines

φ(b1, b2; c; X1, X2) =

∫

|U1|
d1−g |U2|

d2−g |I − U1 − U2|
c−d1−d2−g

1F1(b1; d1; X1U1)1F1(b2; d2; X2U2)dU1dU2

=

∫

|U1|
d1−g |U2|

d2−g |I − U1 − U2|
c−d1−d2−g

exp{−t r(Z1 + Z2)}|Z1|
b1−g |Z2|

b2−goF1(d1; X1U1Z1)

oF1(d2; X2U2Z2)dZ1dZ2dU1dU2

=

∫

|U1|
d1−g |U2|

d2−g |I − U1 − U2|
c−d1−d2−g

exp{−t r(Z1 + Z2)}|Z1|
b1−g |Z2|

b2−goF1(d1+ d2;

(X1+ X2)(U1+ U2)(Z1+ Z2)dZ1dZ2dU1dU2

=

∫

|U1|
d1−g |U2|

d2−g |I − U1 − U2|
c−d1−d2−g

exp{−t r(Z1 + Z2)}|Z1|
b1−g |Z2|

b2−g

1F1(b1+ b2; d1+ d2; (X1+ X2)(U1+ U2))dU1dU2

= (K)2F2(d1+ d2; b1 + b2; d1 + d2; c; X1 + X2)



A. Gupta, D. Kabe, S Niwitpong / Eur. J. Pure Appl. Math, 3 (2010), 435-442 439

= (K)1F1(b1+ b2; c; X1 + X2), (17)

and hence (16) follows. In (17) all matrices are p× p positive symmetric matrices.

Obviously now we have the integral

g(T ) = K

∫

A1+···+An=T

exp{t r(Σ1A1+ · · ·+ΣnAn)}|A1|
g1−g . . . |An|

gn−g

dA1 . . . dAn

= K exp{−t rT}|T |g1+···+gn−g
1F1(g2 + · · ·+ gn; g1 + · · ·+ gn; ((n− 1)

Σ1 −Σ2− · · · −Σn)T ) (18)

where all matrices in (18) are p× p positive definite symmetric matrices.

The moment generating function φ(θ) of T is

φ(θ) = |Σ1− θ |
−g1 . . . |Σn− θ |

−gn , (19)

and (18) is obtained by inverting (19), see e.g., [4], [5, p. 352, equation 5.5.29].

If now X p× p has the density

g(X ) = K exp{−
1

2
t r(XΛX ′)},−∞ < X <∞, (20)

then the moment generating function M(θ) of T = X X ′ is

M(θ) = |λ1I − θ |−1/2 . . . |λnI − θ |−1/2, (21)

and hence from (18), the density function of the GQFCWD of T is

g(T ) = K exp{−
1

2
t r(T )}|T |

1

2
(n−p−1)

1F1(
1

2
(n− 1);

1

2
n; ((n− 1)λ1−λ2 − · · · −λn)T ). (22)

Now [3] proves the following results. Let p × n Y , −∞ < X < ∞ and q× n D of rank d be

given, then we have that

∫

Y Y ′=T,DY ′=V ′
f (Y Y ′, DY ′)dY

= K |DD′|
−1

2
p f (T, V )|T − V (DD′)−1V ′|

1

2
(n−q−p−1). (23)

Next from (23) it follows that

∫

X X ′=T=F F ′+V (µΛ2µ′)−1V ′,µΛX ′=V ′
exp{−

1

2
t r(XΛX ′) + t r(µΛX ′)}dX

= K

∫

exp{−
1

2
t r(T ) + t r(V )}|T − V (µΛ2µ′)−1V ′|

1

2
(n−2p−1)dV



A. Gupta, D. Kabe, S Niwitpong / Eur. J. Pure Appl. Math, 3 (2010), 435-442 440

1F1(
1

2
(n− 1);

1

2
n;

1

2
((n− 1)λ1−λ2− · · · −λn)T ))

= K exp{−
1

2
t r(T )}|T |

1

2
(n−p−1)

1F1(
1

2
(n− 1);

1

2
n;

1

2
((n− 1)λ1−λ2− · · · −λn)T ))

oF1(
1

2
n;

1

4
µΛ2µ′T ), (24)

which is known as (GQNCWD). The integration with respect to V is a known integral in the

theory of noncentral Wishart distribution. Here F is p× (n− p) matrix of rank (n-p), and the

integral is first evaluated with respect to F and then with respect to V .

We now proceed with d statistic generalizations. All other results given by [6, Chapter 5]

relating to the GQFNCWD can be simply and elegantly rewritten by our methodology.

3. d Statistic Generalizations

We observe from (10) that

E(d g) = K
� d

dθ

�g

θ=0

∫

HH′=I

exp{t r(θHΛH ′)}dH

= K
� d

dθ

�g

θ=0
1F1

1

2
(n− 1);

1

2
n; ((n− 1)λ1− · · · −λn)θ)). (25)

Mathai et al. [6, p. 302] show that

� d

dθ

�g

1F1(α;β ;θ) =
Γp(α+ g)

Γp(β + g)
1F1(α+ g;β + g;θ), (26)

and hence (25) yields

E(d)g = K((n− 1)λ1−λ2 − · · · −λn)
pg
Γp(

1

2
(n− 1) + g)

Γp(
1

2
n+ g)

. (27)

It follows from (27) that (5) may be written as

E(d)g =

�

1

2

�

g

C(g)(Λ)

�

m

2

�

g

=
Γ(1

2
(m− 1)g)((m− 1)λ1− · · · −λm)

g

Γ(1

2
m+ g)

(28)

Further it follows from (9) that

E(d)g =

�

p

2

�

g

C(g)(Λ)

�

pm

2

�

g



REFERENCES 441

=
Γ(1

2
(m− 1)p+ g)((m− 1)λ1− · · · −λm)

g

Γ(1

2
mp+ g)

(29)

Once again we write Kabe’s [2] result as

g(t) =

∫

y1+···+yn=t

exp{−(Λ1 y1 + · · ·+Λn yn)}y
g1−1

1
. . . y gn−1

n d y1 . . . d yn

= K exp{−t}t g1+···+gn−1
1F1(g2+ · · ·+ gn; g1 + · · ·+ gn;

((n− 1)λ1− · · · −λn)t) (30)

and hence
∫

y1+···+yn=1

(λ1 y1 + · · ·+λn yn)
g y

g1−1

1 . . . y gn−1
n d y1 . . . d yn

= K
Γ(g2 + · · ·+ gn+ g

Γ(g1+ · · ·+ gn+ g)
((n− 1)λ1− · · · −λn))

g .

Similar to (16) it holds that

1F1(a; b;∆A)1F1(c; d;ΩB) = 1F1(a+ c; b+ d; (∆+Ω)(A+ B))

and hence
∫

A+B=D

exp{−
1

2
t r(A+ B)}|A|

1

2
(n−p−1)|B|

1

2
(q−b−1)

1 F1(
1

2
(n− 1);

1

2
n;ΛA)

1F1(
1

2
(q− 1);

1

2
q;θB)oF1(

1

2
n;

1

4
∆A)oF1(

1

2
q;ΩB)dAdB

= K exp{
1

2
t r(D)}|D|

1

2
(n−q−b−1)

1F1(
1

2
(n+ q− 2);

1

2
(n+ q); (∆+ θ)D)

oF1(
1

2
(n+ q);

1

4
(∆+Ω)D).

References

[1] A Gupta and D Nagar. Matrix Variate Distributions. Chapman & Hall/CRC, Boca Raton,

2000.

[2] D Kabe. On the exact distribution of a class of multivariate test criteria. Annals of Mathe-

matical Statistics, 61:1197–1200, 1962.

[3] D Kabe. Generalization of suerdrup’s lamma and its applications to multivariate distribu-

tion theory. Annals of Mathematical Statistics, 36:671–676, 1965.

[4] D Kabe. Hypergeometric functions of marix argument. Industrial Mathematics, 41:125–

136, 1991.



REFERENCES 442

[5] A Mathai. Jacobians of Matrix Transformations and Functions of Matrix Argument. World

Scientific, London, England, 1997.

[6] A Mathai, S Provost, and T Hayakawa. Bilinear Forms and Zonal Polynomials. Springer-

Verlag, New York, 1995.