Journal
of Econometrics
62 (1994) 129-141.
North-Holland
Moments of the ratio of quadratic
in non-normal
variables with
econometric examples*
forms
Aman Ullah
Unicersiiy
of Cai(forniu,
Virendra
Ricerside,
CA 92521,
USA
K. Srivastava
Lucknow
Uniwrsity,
Lucknow.
226007,
Received
September
1991, final version
India
received January
1993
In this paper we have provided a general result on the higher-order
approxtmate
moments of the
ratio of quadratic forms in the non-normal
vector. The results for the normal case are then presented
as a special case of this result. It is also indicated that the results for a large class of econometric
estimators and test statistics can be obtained by using our general result.
Keq’ words: Moments; Quadratic
JEL classific.ation: Cl; C12; Cl3
forms; Non-normal:
Estimators;
Test statistics
1. Introduction
In analyzing the distributional
properties of estimators and other statistics in
econometric
models, the small disturbance
asymptotic
theory has assumed
a prominent
place for several reasons. For instance, the exact finite-sample
distribution
theory generally provides intricate expressions from which it is hard
to draw any neat inference related to the efficiency properties of estimators,
confidence ellipsoids, and tests of hypotheses. In fact, for the non-normal
cases,
it is even quite difficult to obtain results. On the other hand, the corresponding
Correspondence
to:
Aman
Ullah, Department
of Economics,
University
of California.
Riverside,
CA 92521, USA.
*Some work on this paper was done when the second author visited the University of Western
Ontario. The first author gratefully acknowledges
research support from UCR. The authors are
thankful to a referee and the seminar participants
at UCLA. UCSD, Texas A&M, and SMU for
helpful comments and suggestions.
0304-4076/‘94/$07.00
(0 1994 -~Elsevier
Science B.V. All rights reserved
130
A. Ullah und V.K. Srivasiara.
Moments
of the ratio of’ yuaclraric,forms
approximate
results based on the small-disturbance
asymptotic
theory are
straightforward
to derive and are relatively simple. Moreover, the small-disturbance asymptotic
approximations
have been found to give closer approximations, in comparison
to the large-sample
asymptotic
approximations;
for the
exact results in some specific applications,
see Srivastava
et al. (1980). The
small-disturbance
asymptotic theory is, however, subject to the usual limitations
and qualifications
as any asymptotic theory. It requires the standard deviation
of disturbances
to be small and to approach
zero in the limit. The simple
implication
of this specification is that the validity of the inferences based on the
small-disturbance
asymptotic theory increases as the variance of the disturbance
in the model tends to be small.
In view of the growing interest in the application
of the small-disturbance
asymptotic
theory, particularly
in the evaluation
of moments of econometric
estimators,
we have considered
a general form as a ratio of quadratic
forms
which not only provides a natural extension of the existing work but also gives
a framework from which various other estimators and test statistics come out as
special cases. The evaluation
of the exact moments of the ratio of quadratic
forms in normally distributed
variables has received some attention;
see, for
example, Jones (1987), Magnus (1986) Smith (1989) Ullah (1990), and the
references cited therein. These exact results are quite intricate and do not
generally allow the deduction of any clear inference. Further, when the variables
are not normal, to our knowledge,
hardly any results are available
in the
literature; although see Peters (1989) and Zellner (1976). Our approach
here
permits us to evaluate higher-order
moments in a unified way and to relax the
conventional
assumption
of the normality
of disturbances.
The plan of the paper is as follows. In section 2, we consider a general form of
the ratio of quadratic forms in variables having finite moments of at least order
four and present an expression for the small-disturbance
asymptotic approximation of the mean of the ratio. As a special case, the results for normal variables
are also given. Then in section 3, we present a variety of its applications.
2. Main results
Consider
a general
form of the ratio of quadratic
qV’.” = (y’M1J’)r1(y’M2y)-r*,
where rl 3 0 and r2 > 0 are real numbers,
both MI and M2 are n x n nonstochastic
vector y is such that
y = m +
CM,
m=
Ey,
forms as
(2.1)
y is an n x 1 stochastic vector, and
symmetric matrices. The stochastic
(2.2)
A. U&h and V.K. Srivastnoa.
Moments of the ratio of quadratic.forms
where u is an n x 1 disturbance
vector. We assume that the elements
independently
and identically distributed
such that, for i = 1,
,n,
Eui = 0,
En; = 1,
En; = y,,
Eu; = y2 + 3,
131
of u are
(2.3)
where yi and yZ are Pearson’s measures of skewness and kurtosis of the
distribution.
For the normal distribution
yi and y2 are zero, while for the
symmetric distribution
only yr is zero. Thus the nonzero values of yi and yz
indicate a departure from normality.
Our aim is to obtain, under (2.3), the approximation
of the expected value of
the ratio of powers of two quadratic forms,
Eq“J*
= E[(y’M,y)“(y’M&*‘],
(2.4)
for any r1 > 0 and r2 > 0. In the special
moment of the ratio of quadratic forms,
Eq”
=
EqW’
=
case when
ri = r2, (2.4) is the rth
(2.5)
WY'MIY/Y'MZY)~~.
Some econometric
examples of (2.5) are given in section 3. In another special
case when r1 = 0, (2.4) is the rzth inverse moment of y’Mzy. Further, when
rz = 0, (2.4) is the r1 th moment of y’M 1y.
Before presenting Eqr1”2 we introduce the following notations:
0, = m’M,m,
ci = [r,(r,
AZ = c,M,
-
02 = m’Mzm,
1) . ..(rl
-j
+ 1)1/O{.
+ 2c2Mlmm’M1
+ 2cl_clMlmm’Mz,
A3 = 2c2Ml
+ 2c,_c,M,
+ $c,Mlmm’M1
+ 4c,g2M,mm’Mz,
Aa = icZM1
+ ~clclM,
+ 2c3Mlmm’M1
+ 2clg2M2mm’M2
(2.6)
+ 4_c,c,M,mm’M,,
A$ = $cqMlmm’M1
+ 2c2c,M,mm’M2
+ $,c3M,mm’M2;
_Cjis the same as cj with rl replaced by - r2 and 8, by OZ. Note that Cj is defined
for 1 < j < rl. Ifj > ri, then cj = 0. However, no such restriction is needed for Cj.
Further we represent
_Aj below as Aj with “j interchanged
interchanged
with MZ. For our result we need
Cl2= m’M,m
with _Cjand
# 0.
Ml
(2.7)
Using (2.3) and assuming the disturbances
to be small, we can now present the
small-a approximation
for the expectation in (2.4). The result is asymptotic and
requires the disturbance
to be small. For details see Kadane (1971), and also
Srivastava
et al. (1980) where they show a better performance
of small-a
approximations
compared to the large-n approximation,
particularly
when the
sample is small. Further, if the distribution
for the disturbances
is assumed to be
different from normal, analytical
derivations
of the exact expressions
for the
moments may be considerably
intricate and often formidable.
Moreover, the
resulting expressions
will firstly depend upon the specific distribution.
and
secondly would be hard to analyze so as to draw any clear inference regarding
efficiency properties.
Our results are fairly general and require merely the
knowledge of the first four moments which more or less characterize
all the
salient features of any distribution.
Theorem
Jf thr disturbances
1.
,fi,llow
(2.3), NH~ (2.7) holds,
thr expectation
in
(2.4) to 0(a4) is
E(!“M,l.)r’(~‘Mr!‘)~rz
I.2
=
= 0;‘0; ‘,[l
+ 04(;‘2i4 + ;_;)I, (2.8)
+ AJ(I*M,mm’M,)],
(2.9)
tr( A7 + AZ).
2,. = m’[M,(I*A,)
+ M2(I*A,)]i,
i, = tr[A,(I*M,)
+
A4(I*M2)]
+ tr[A~(I*M,mm’M,)
j.x = (trM,)(trAJ
+ (trM2)(trAJ
+ (rn’M:m)(trA,*)
+ 2m’(M,AzM,
A’s and 4’s ure us gioen
Hadumard
+ a2i2 + a”;‘,;.,
product
+ 2tr(M,A,
+ M2AJ
+ (m’M$Tl)(trA%)
+ M2_AzM2)m;
in (2.6), tr represents
of’ two mutrices
A und B.
the truce, and A*B
represents
the
The result in (2.8) is derived
Corollary
1.
U&r
the assumption
(2.7), tile r.upectution
and
in
E(.v’M,JV(~‘M,~)
where
i,,
in section
(2.4)
2.1
Corollur~~
disturhorlce
actor
in (2.17)
O(os) is
r* = t);1t)2~r2[1 + 02i2 + o”it],
(2.10)
and j.2 me as in (2.8).
The result in (2.10) follows immediately
ity,
ofthe
of’twrnlulit~*
to
2.
Under
i.e., u _ N(0,
the awimption
o2 Q),
E(fM,j.)“(
of’u nonscmlur
the r.upectation
y’M2?‘)
”
=
by substituting
in (2.4)
o;ltlr”[
1 +
2.2
;‘, = yz = 0 in (2.8).
cowriance
to O(o”)
+
$1.
rnutriv
and normal-
is
(2.1 1)
For r, = r2 = r, the results in (2.8), (2.10), and (2.1 1) provide the rth moment
of the ratio of quadratic forms. Further, when r, = 0, (2.8) and (2.10) provide the
rzth inverse moment of y’Mz~).
We note here that, in general. the behavior of the moments in the non-normal
case can be quite different from those in the normal case. In fact, if the true
disturbances
are not normal. the moments can be under- or overestimated
by
falsely assuming normality, to the order of our approximation.
by the magnitude
determined
by [subtracting
(2.10) from (2.X)]
where i, and & are as in (2.8).
_‘.I.
Drrizatim
of’ tlw
Let us substitute
wsult
(2.2) in (2.1) and write
(2.13)
where, for i = 1, 2, Oi = fn’Minl and
(2.14)
134
Further,
A. Ullah
and Y.K. Srivasrava,
expanding
Moments
(1 + ti)” for small-o
qVI.*>= d;ltp
and retaining
Ojtcj + cji) ,
1+ i
[
of the ratio
i=
1
qf’ quadratic forms
terms to 0(04) we get
(2.15)
1
where
c2
=
u’A,u.
(2.16)
Ed = (u’M,m)(u’A3u),
c4 = (u’Mlu)(u’A4u)
+ (u’M,m)2u’AXu,
the matrices A4 and AZ are as given in (2.6) and ~j is Fj with Ml interchanged
with M2 and Aj in (2.6) replaced by Aj.
Thus we have
O’E(cj + cj)
Eq
To evaluate E(cj + cj) we first note the following
then using (2.3) it can be verified that
1.
(2.17)
results. If A is any n x n matrix,
E(u’Au) = trA,
(2.18)
E(u’Au.u) = g, (I*A)i,
E(u’Au.uu’)
= yz(l*A)
+ (trA)I + A + A’,
where i is an n x 1 vector with all the elements
Hence, using (2.18) in (2.16) we obtain
unity.
E&r = 0,
EC, = tr A,,
Ec3 = ylm’M1(I*A,)i,
(2.19)
A. Ullah and V.K. Srivastavcr, Moments of the ratio qf quadratic ,forms
Ec4 = y2[trA4(1*MI)
135
+ trAX(I*Mrmm’Mr)]
+ (trM,)(trA,)
+ (m’M$n)(trAX)
+ 2trM1A4
+ 2m’M,AXM,m.
Similarly, we can write Egj for j = 1, . . . ,4.
Finally, substituting
(2.19) and Ecj in (2.17) we get the result in (2.8).
3. Special cases
Here we show that the approximate
moments of various econometric
estimators and test statistics, when the disturbances
are not necessarily normally
distributed, can be written by using directly the result in Theorem 1. This is done
by indicating that these estimators and test statistics are essentially the ratio of
quadratic forms. The detailed analysis of each and every one is beyond the scope
of this paper and we therefore present only some examples for illustrative
purposes.
Let us first explain the application of the result (2.8) for the mean of the ratio
of quadratic forms. As rl = r2 = 1, we have
1
cr =-)
0,
(‘j = 0,
1
c, = - -,
0,
j = 2,3,4,
2
-cz = -,
0;
c3=
1
=
2
%“I
-
__
H,O2
6
cd=<.
02
Mlmm’M2,
+ 2clclM2mm’M,
-~M2+$M2mm’M,-&
2
2
24
that
A2 =-cl Mz + 2c,M2mm’M2
=
--,
Q;
Using these in (2.6) we observe
A2
... .
M2mm’M,,
2
I
8
__
A3 = - eru2 M2 + 0,e;
M,mm’M,,
2
A3 = 2czM,
+ 2r,c,M,
+ ic3Mzmm’M,
4
__
20102 M2 + o,e:
A4z
-___
1
A4 = i c2M2
+ i~,c,
+ 4r,czM,mm’MI
M,mm’M,,
M, + 2c3 M,mm’M,
+ 2~,c2M,mm’M1
+ 4rIcZM2mm’M,
12
MI - zM2mm’M2
8
+ ~
o,8i
+ 2_c,c2 MImm’M,
+ ~~,csM2mm’MI
M2mm’MI,
A$ = 0,
&
= f e4M2mm’M2
= !$ M,mm’M,
?
Substituting
- 3
16
M2mm’M,.
(9102
the above expressions
in A,, i3, I.,, and $
as defined by (2.9), we
get
(3.1)
to the order of our approximation.
In order to demonstrate
the application
of this result for analyzing the bias of
R2 in regression analysis, let us consider the following linear regression model:
y =
xg +
ml,
(3.2)
where y is an n x 1 vector, X = [i X*] is an tz x k nonstochastic
matrix with
a constant vector and matrix X* of (k - 1) regressors, p is a k x 1 parameter
vector, and u is as in (2.2). Note that the model (3.2) is the same as (2.2) with
m = Xg.
A. Uilah rend V.K. Sricastaaa,
The RZ for (3.2) can be written
e’e
Moments
of the ratio
qf quadratic
jbrms
137
as
Y’MIY
R2=1-(y-j)‘(y_j)=L.‘M,L.J
(3.3)
where e is the least squares residual vector, M, = I - if/n, and Ml = M2 - p,,
with p; = (I - X(X’X)- IX’). Thus R2 is a ratio of quadratic forms in the vector
y, and its rth moment can be written immediately by substituting
rl = r2 = r in
(2.8). For example, consider the mean of R2. For this purpose, we observe that
M,m = M2m - &Xb = M,m,
0, = e2 = m’M,m = p’X’M2Xp
= 0,
whence it is easy to verify that
- M, + 5 M,XPP’X’M,
A2 = f
A3 = ;
- M, + ;M2Xfi/WM2
2
1
1
1
1,
43 = $
M2 + PI - ; M,XfiB’X’M,
A4 = $
- ; M, + f M2Xjl,6’X’M2
A$ = 0,
A4* = 0.
,
,
,
138
Utilizing
A. Ullah and V.K. Srioastaca,
Moments
c~f the ratio
c~f quadratic.,form
these in (2.9) we find
2, =
i(k- n),
;Lj =
$ P’X’M,(Z*P,)i,
3.2 = $ [(n - k)(n - 3)1,
which when substituted
of R2 as
E(R2) = 1 +
+
in (3.1) yields the approximate
expression
;(k- n) + ?!+M,(Z*P,)i
$
y2
trM,(Z*P,)
- i P’X’M,(I*P,)M,XP
[i
I
+ (n - k)(n - 3)
to the order 0(a4).
When disturbances
following:
E(R*) = 1 +
1
are normally
(3.4)
distributed,
f(k - !‘I) + $[(n- k)(n -
Defining the population
up to order O(o’) is
Bias(R*)=
for the mean
counterpart
1 -<2+$(h-~)=$,
this expression
3)l.
reduces
to the
(3.5)
of R2 as iz = o/(0 + na2), the bias of R*
(3.6)
A. (Illah
and V.K. Srivastaca.
Moments
c~f the ratio
oj’quadratic
forms
139
where we use 1 - i2 = na2/(8 + na2) = (no2/8)(l + no2/O)-’ rr. no210 to the
order of our approximation.
It follows from (3.6) that the bias of the adjusted R2 defined as R2 =
(1 + l)R2 - 1 with I = (k - l)(n - k))*, up to order O(d), is
Bias (R2) = (1 + I) Bias R2 - I(1 - [‘) = s.
(3.7)
From the above results we observe that the bias of R2 depends on the number
of regressors k and it is a decreasing function of 0. Since 0J02 can be written as
nc2/(1 - [‘), the bias of R2 is also a decreasing function of both l2 and n. In
contrast the bias of R2, although a decreasing function of n, c2, and 0, is free from
k. Also the bias of l?’ is considerably
smaller than the bias of R2, especially for
large k. It is interesting to note that these are straightforward
findings based on
the approximate
analytical results and they compare with those obtained by the
numerical evaluation
of the exact moments in Cramer (1987). Results for the
variance can be similarly analyzed. Finally, we observe from (3.4) that these and
Cramer’s findings may not go through when the disturbances
are not normal.
Next, consider a dynamic model
.!Jo=~y-,
where
+xp+&l-au-,),
INI< 1,
(3.8)
y. = [y2, . . . . y,J,
y-r = [yr, . . . ,y_r]‘,
X is (n - 1)x k, u. =
and u_ 1 = [ul, . , u, _ 1]‘. The least squares estimator of c( is
[u2, . . . ,u,J’,
&=
Y’- I
MY, _ Y’MlY
Y’-IMY-l
(3.9)
Y’M~Y’
where M = I - X(X’X)-‘X’,
y = [yl, . . ..y.]‘,
Ml = (D;MD,
+ D;MD,)/2,
M2 = D;MD1, with Dr = [Zn_ r, 0] and D2 = [0, I,_ r]. The second equality on
the right-hand
side of (3.9) follows by noting that y’_ 1My, = y’D;MD2y
= y’(D’,MD2 + D$MD,)y/2
= y’M,y and y’_ 1My_ 1 = y’D; MD,y = y’Mly.
The mode1 in (3.8) can be written in the form y = m + GU,where y is as defined
above, u = [ur, . . . ,zr,,]‘, and m = am_, + X/l. Substituting
the m, Ml, and M,
given above and rl = r2 = r in (2.8), we get the rth moment of &.This generalizes
the results of Ullah and Maasoumi (1986), where normality is assumed and the
small-a approximations
of only the first two moments are obtained from the
exact moments.
The results for the dynamic regression models with other error structures
(including white noise) can be obtained by using Corollary 2.
Similarly,
I’,
consider
=
“yz
+
a single structural
x,/j, + ou,
equation
of the form
(3.10)
where JJ, and y2 are n x 1 vectors of observations
on the endogenous
variables,
X, is an Mx LJ, nonstochastic
matrix of exogenous
variables, Y is a scalar
parameter, /I, is a y, x 1 parameter vector, and u is an II x 1 disturbance
vector.
We denote X = [X,
X,], where Xz is an n x gz matrix of exogenous variables
appearing in the remaining, say G - 1, equations of the model.
The double k-class estimators of (Y,proposed by Nagar (1962), can now be
written as
6 = y;N,y,/y;Nzy,,
(3.11)
where N, = k,(P, - P,,) + (1 - k,)P,, and Nz = k,(P, - P,,) + (1 - k,)P,,;
and k2 are arbitrary
nonstochastic
constants,
P, = X(X’X)m’X’,
and
4, = I - I’,. The advantage
of considering
the estimator L? is that the least
squares (k, = k2 = 0), the two-stage
least squares (k, = k2 = 1), and the
k - class (k, = k2 = k), among others, are special cases of this estimator.
To obtain the moments of j, we rewrite it as
k,
#ii= y'M,y/yM2y,
(3.12)
where
(3.13)
Note
that
E_Y= [I?,‘, m;]’ = RI and
V(J)) = C, where
ml = X/l,
and
m2 = Xx, (;?, and ;i, are y x 1 reduced-form
parameter vectors; g = g, + g2)
and
(3.14)
Thus, substituting
the above values of m, M,, M2 and C in Corollary 2 we get the
moments of the double k-class estimators
in the normal case. Note that the
result for the non-normal
case (Theorem 1 with y, # 0 and y2 # 0) cannot be
used for g because y is not identically distributed.
However, the modification
indicated in Corollary 2 may serve the purpose.
Numerous
other examples of the application
of Theorem
1 exist. Some of
these include: the F-test under misspecification,
estimators
in macro models
with generated regressors, estimators in cointegrated
models.
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