We show that the F-test can be both liberal and conservative in the context of a particular type of nonspherical behaviour induced by spatial autocorrelation, and that the conservative variant is more likely to occur for extreme values of the spatial autocorrelation parameter. In particular, it will wipe out the progressive one as the sample size increases. --F-test,spatial autocorrelation

Hanck, Christoph

Krämer, Walter

English

Research Papers in Economics

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Krämer, Walter; Hanck, Christoph
Working Paper
More on the F-test under nonspherical
disturbances
Technical Report // Sonderforschungsbereich 475, Komplexitätsreduktion in Multivariaten
Datenstrukturen, Universität Dortmund, No. 2008,14
Provided in cooperation with:
Technische Universität Dortmund
Suggested citation: Krämer, Walter; Hanck, Christoph (2008) : More on the F-test
under nonspherical disturbances, Technical Report // Sonderforschungsbereich 475,
Komplexitätsreduktion in Multivariaten Datenstrukturen, Universität Dortmund, No. 2008,14,
http://hdl.handle.net/10419/36618More on the F-test under nonspherical disturbances
Walter Kr amery Christoph Hanckz
May 29, 2008
Abstract
We show that the F-test can be both liberal and conservative in the
context of a particular type of nonspherical behaviour induced by spatial
autocorrelation, and that the conservative variant is more likely to occur for
extreme values of the spatial autocorrelation parameter. In particular, it will
wipe out the progressive one as the sample size increases.
Key words: F-test, spatial autocorrelation.
Research supported by Deutsche Forschungsgemeinschaft (DFG) under SFB 475. We are grate-
ful to Anurag Banerjee for providing us the Gauss routines for our calculations.
yFakult at Statistik, TU Dortmund, D-44221 Dortmund, Germany, walterk@statistik.uni-
dortmund.de
zDepartment Quantitative Economics, Universiteit Maastricht, NL-6211 LM Maastricht,
c.hanck@ke.unimaas.nl
11 Introduction and summary
The robustness of the F-test to nonspherical disturbances has concerned applied
statisticians for many decades. The present paper considers the F-test in the context
of the linear regression model
y = X + u = X
(1)
(1) + X
(2)
(2) + u; (1)
where y and u are T 1, X is T K and nonstochastic of rank K < T,  is K 1,
and the disturbance vector u is multivariate normal with mean zero and (possibly)
nonscalar covariance matrix V . The design matrix is partioned into X(1)(T q) and
X(2)(T  (K   q)) and the null hypothesis to be tested is H0 : (1) = b(1).
The standard F-test assumes that V = 2I and rejects for large values of
F =
(~ u0~ u   ^ u0^ u)=q
^ u0^ u=(T   K)
; (2)
where ^ u = y   X ^ , ^  = (X0X) 1X0y, ~ u = y   X(1)b(1)   X(2)~ (2),
~ (2) = (X(2)0X(2)) 1X(2)0(y   X(1)b(1)). Its null distribution is central F with q and
T   K degrees of freedom and the problem to be studied here is the robustness of
this null distribution to deviations from V = 2I.
Vinod (1976) and Kiviet (1980) address this problem for a given disturbance covari-
ance matrix V , and derive bounds for the size of the test when the design matrix
X varies across all T  K matrices of rank K, while Banerjee and Magnus (2000)
and Hillier and King (1982) consider the test statistics themselves. Below we follow
Kr amer (1989), Kr amer et al. (1990) and Kr amer (2003) by xing X and letting
V vary across certain subsets of possible disturbance covariance matrices which are
likely to occur in practice. This seems the more natural approach, as X is always
known in applications, whereas V is an unknown T  T parameter matrix.
The subset of disturbance covariance matrices under study here is implicitly dened
by the spatial autoregressive scheme
u = Wu + "; (3)
where " is a T  1 normal random vector with mean zero and scalar covariance
matrix 2
"I, and W is some known T T-matrix of nonnegative spatial weights with
2wii = 0 (i = 1;:::;T). Although there are many other patterns of spatial dependence
which have been suggested in the literature (see Anselin and Florax (1995) for an
overview), the one dened by (3) is by far the most popular, so it seems worthwhile
to investigate the behaviour of parameter estimates and tests when the regression
disturbances \misbehaves" according to this particular scheme.
Below we build on Kr amer (2003), who shows that the size of the test can tend to
both one and zero as the parameter  varies across its allowable range. While Kr amer
(2003) is silent on the respective empirical relevance of the two extreme cases, we
show here that the conservative variant is far more likely to occur in practice, and
will wipe out the liberal one as sample size increases.
2 The null distribution under spatial
autocorrelation
The coecient  in (3) measures the degree of correlation, which can be both positive
and negative. There is no disturbance autocorrelation where  = 0. Below we focus
on the empirically more relevant case of positive disturbance correlation, where
0   <
1
max
and where max is the Frobenius-root of W (i.e. the unique positive real eigenvalue
such that max  jij for arbitrary eigenvalues i). The disturbances are then given
by
u = (I   W)
 1"; (4)
so V := Cov(u) = 2
" [(I   W)(I   W)0]
 1 and V = 2
"I whenever  = 0.
The behaviour of the test statistic (2) when disturbances are given by (3) is best
seen by rst rewriting it as
F =
u0(M(2)   M)u=q
u0Mu=(T   K)
; (5)
3where M = I   X(X0X) 1X0 and M(2) = I   X(2)(X(2)0X(2)) 1X(2)0. Let F 
q;T K
be the (1   ) quantile of the central F-distribution with q and T   K degrees of
freedom, respectively, where  is the nominal size of the test. Then
P(F  F

q;T K) = P(u
0(M
(2)   M)u  
q
T   K
F

q;T Ku
0Mu  0)
= P(u
0(M
(2)   dM)u  0)
(where d = 1 +
q
T   K
F

q;T K)
= P(
0(I   W)
0(M
(2)   dM)(I   W)  0)
(where  =
1
"
"  N(0;I))
= P(
T X
i=1
i
2
i  0)
= P((1   max)
2
T X
i=1
i
2
i  0); (6)
where the 2
i are iid 2
(1) and the i are the eigenvalues of
(I   W)
0(M(2)   dM)(I   W), and therefore also of V (M(2)   dM).
The limiting rejection probability as  ! 1=max depends upon the limiting be-
haviour of (1   max)2V . We conne ourselves to the case where W is symmetric,
which appears to be the more important one in practice. This will for instance occur
if spatial dependence follows the j-ahead-and-j-behind or the equal-weight criteria
(see section 3 below). Then W admits a spectral decomposition
W =
T X
i=1
i!i!
0
i; (7)
where we have without loss of generality arranged the eigenvalues i in increasing
order, and
V =
T X
i=1
2
"
(1   i)2!i!
0
i (8)
is the resulting spectral decomposition of V , which always exists as V is symmetric.
The point of our argument now is that
lim
!1=max
(1   max)
2V = 
2
"!T!
0
T; (9)
4a matrix of rank 1. Therefore, all limiting eigenvalues of
(1   max)
2V (M
(2)   dM) (10)
are zero except one, which is given by
cT = tr(!T!
0
T(M
(2)   dM)) = !
0
T(M
(2)   dM)!T: (11)
This constant cT is crucial for our analysis. It determines whether the F-test will
eventually be conservative or liberal. If cT is positive, the rejection probability of the
F-test will tend to 1 as  approaches 1=max. The test is then liberal in the extreme,
at least for values of  close to the edge of the parameter space.
If cT is negative, the rejection probability will tend to zero, and the test will
eventually be extremely conservative. And if cT = 0, the limiting behaviour of
the test cannot be determined from the limiting behaviour of the eigenvalues of
(1   max)2V (M(2)   dM) (which are all zero). Section 3 now sheds some light on
which of these cases is more likely to occur in empirical applications.
3 Exact rejection probabilities in nite samples
The rst important point to make is that the crucial constant cT depends only on
X and W and the nominal size of the test, all of which are known. Therefore, cT is
known as well and can guide the user in interpreting a test: If cT < 0, one has to
beware of a loss in power, and if cT > 0, one has to beware of spurious rejections.
The following argument shows that the rst problem is far more likely to occur in
practice: Rewrite the critical constant as
cT = !
0
TM
(2)!T   !
0
T
T   K + q
T   K
F

q;T KM!T: (12)
Then it is easily seen that in general cT < 0 (i.e. except in very contrived cases).
This follows from the fact that
T   K + q
T   K
F

q;T K ! 
2;
q =q (13)
as T ! 1, which is larger than 2 for moderate values of  and q. (It takes the
values 3.84, 2.99 and 2.60 for  = 0:05 and q =1, 2 und 3, respectively). This will in
general be more than enough to counterbalance the fact that !0
TM(2)!T > !0
TM!T.
5Figure 1: An example of the queen matrix
W =
0
B B B B B B
B B B B B B
B B B B B
B B B B
@
0 1 0 1 1 0 0 0 0
1 0 1 1 1 1 0 0 0
0 1 0 0 1 1 0 0 0
1 1 0 0 1 0 1 1 0
1 1 1 1 0 1 1 1 1
0 1 1 0 1 0 0 1 1
0 0 0 1 1 0 0 1 0
0 0 0 1 1 1 1 0 1
0 0 0 0 1 1 0 1 0
1
C C C C C C
C C C C C C
C C C C C
C C C C
A
Of course one can always construct a weighting matrix and regressor matrices W,
X and X(2) such that !0
TM(2)!T = 1 and !0
TM!T = 0 and therefore cT > 0. For
instance, let  = (1;:::;1)0 be a (T  1)-vector and choose X(2) orthogonal to .
E.g., for T even, pick X(2) = (1; 1;1; 1;:::;1; 1)0. Let
X =


. . . X
(2)
(14)
(that is, test H0 : (1) = 0) and let
W = W
EW = (w
EW
ij ) =
8
<
:
1 for i 6= j
0 for i = j
; (15)
the equal weight matrix. This and similar cases will however rarely happen in the
natural course of events, and will become ever more unlikely as sample size increases.
Figure 2 gives an example where W is derived from the queen-criterion (see Figure 1
for an illustration of the criterion with N = 9): There is a square of cells, and all cells
around a given cell obtain a weight of 1. The sample size is then a square number.
We choose X to have K = 2 and T = 16 or 25 such that, for T = 16, the second
column is the (normalized) eigenvector corresponding to the largest eigenvalue of
W, (which happens to be max = 5:85), and the rst column is any (16  1)-vector
6Figure 2: Rejection probabilities for the queen matrix
orthogonal to !16. For t > 16, xt1 = 1 and xt2 = t 16. Then we have c16 = 0:316 and
c25 =  0:850, and so our theoretical result predicts that the rejection probabilities
will tend to one as  ! 1=max for T = 16 and will tend to zero as  ! 1=max for
T = 25. Figure 1 shows that this is indeed the case.
The case cT = 0, where our analysis does not apply, will occur for instance whenever
!T is in the column space of X(2). The most important special case is when W is row-
normalized and therefore !T = 1 p
T (1;:::;1)
0 and where in addition X(2) contains
an intercept. However, row-normalization will often destroy the symmetry of W, so
this case is not covered by our discussion above.
7References
Anselin, L. and Florax, R. (eds.) (1995): New directions in spatial economet-
rics. Berlin (Springer).
Banerjee, A. N. and Magnus, J. (2000): \On the sensitivity of the usual t-
and F-tests to covariance misspecications." Journal of Econometrics 95, 157
{ 176.
Hillier, O.H. and King, M.L (1982): \Linear regression with correlated errors:
bounds on coecient estimates and t-values" In: King, M. L. and Giles, D. E.
A. (editors): Specication Analysis in the Linear Model. London (Routledge
and Kegan-Paul), 74 { 80.
Kiviet, J.F. (1980): \Eects of ARMA errors on tests for regression coecients:
comments on Vinod's article, improved and additional results." Journal of the
American Statistical Association 75, 333 { 358.
Kr amer, W. (1989): \On the robustness of the F-test to autocorrelation among
disturbances." Economics Letters 30, 37 { 40.
Kr amer, W. (2003): \The robustness of the F-test to spatial autocorrelation
among regression disturbances" Statistica 63, 435-440.
Kr amer, W.; Kiviet, J. and Breitung, J. (1990): \The null distribution of
the F-test in the linear regression model with autocorrelated disturbances."
Statistica 50, 503 { 509.
Vinod, H.D. (1976): \Eects of ARMA errors on the signicance tests for re-
gression coecients." Journal of the American Statistical Association 71, 929
{ 933.
8

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