Testing a null variance ratio in mixed models with zero degrees of freedom for error

Computational Statistics & Data Analysis 46 (2004) 707 – 719 www.elsevier.com/locate/csda

Testing a null variance ratio in mixed models with zero degrees of freedom for error M.Y. El-Bassiouni∗ , H.A. Charif Department of Statistics, UAE University, P.O. Box 17555, Al-Ain, United Arab Emirates Accepted 20 September 2003

Abstract An invariant test that combines the most powerful invariant tests against small and large alternatives is proposed for testing a null variance ratio in mixed models with zero degrees of freedom for error. Such models occur in many applications including plant and animal breeding and time varying regression coe2cients. The proposed test statistic is easily computed and the corresponding test procedure is just as easy to carry out using currently available software. The power of the test is compared with the power of other tests advocated in the literature using two real data sets and is found to maintain high e2ciency all over the parameter space. c 2003 Elsevier B.V. All rights reserved.
Keywords: Animal models; Locally most powerful tests; Most powerful invariant tests; Point optimal tests; Random walk regression coe2cients; Variance components

1. Introduction Mixed models having two sources of random variation are frequently used in biological and economic applications where it is usually of interest to test the null hypothesis that the ratio,
say, of the two variance components is zero. Speci;cally, consider the model y ∼ N (X; (In +
V ));

(1)

where y is an n × 1 observable random vector, X is an n × m known matrix,  is a m × 1 vector of unknown parameters,  ¿ 0 and
¿ 0 are unknown parameters and V is an n × n known nonnegative de;nite matrix. Let U be such that V = UU
. For ∗

Corresponding author. E-mail address: [email protected] (M.Y. El-Bassiouni).

c 2003 Elsevier B.V. All rights reserved. 0167-9473/$ - see front matter  doi:10.1016/j.csda.2003.09.011

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model (1), the error space is de;ned by R(X; U )⊥ , the perpendicular to the range of (X; U ), whose dimension (the error degrees of freedom) is f = n − rank(X; U ). If f ¿ 0, the customary analysis of variance methods such as Wald’s procedures directly apply, see also the alternative tests in Lin and Harville (1991). However, in certain applications f may be zero. For instance, Burch and Iyer (1997) argued that in plant and animal breeding, where the random eFects correspond to genetic sources and the error corresponds to environmental (nongenetic) sources, the observational units are not independent if they possess common genetic material and, in addition, the random eFects may be correlated depending on the genetic relationship among the observational units. In particular, it is shown that f = 0 for full animal models. Another example is provided by LaMotte and McWhorter (1978) who considered a regression model where the coe2cient vector varies over time following a random walk process. They argued that f = 0 whenever the covariance matrix of the random regression coe2cients is positive de;nite. Also, f = 0 for the so-called “intercept variation” models, in which only the intercept varies over time. Assume that f = 0 and consider testing the hypotheses H0 :
= 0

vs:

H1 :
¿ 0:

(2)

LaMotte and McWhorter (1978) suggested a family of exact F-tests, which are based on arbitrary partitions of a set of quadratic forms, and outlined a shortcut procedure for identifying an acceptable partition. Subsequently, LaMotte and McWhorter (1982) compared powers of their test with powers of other tests, including the full and the restricted likelihood ratio tests, and showed that their test compared favorably with these other tests, even after adjusting their critical values to give them the stated size. LaMotte et al. (1988) and Lin and Harville (1991) considered most powerful invariant (MPI) tests against a speci;c alternative. Each such a test has maximum power among translation and scale invariant tests of the same size at the speci;c value in the alternative hypothesis. In addition to providing attainable upper bounds on powers, the MPI tests form a class of exact tests, each of which is admissible among tests of the same size. It should also be noted that LaMotte et al. (1988) and Lin and Harville (1991) considered the case f ¿ 0 as well as the case f = 0. Many authors, e.g., Nyblom and Makelainen (1983), King and Hillier (1985), Nabeya and Tanaka (1988), Lin and Harville (1991) and Jandhyala and MacNeill (1992) discussed locally most powerful invariant (LMPI) tests, which are optimal against small alternatives. On the other hand, Lin and Harville (1991) introduced an MPI test that is optimal against large alternatives. Consider the class of MPI tests and note that it includes the LMPI, as
→ 0, and the MPI against large alternatives, as
→ ∞ (Westfall, 1989). A point optimal invariant test is a member of the class of MPI tests with the speci;c alternative chosen so that the resulting test is more powerful than other members of the class against most alternatives of interest, (King, 1988). Sometimes the speci;c alternative in a point optimal test has been chosen by requiring that the corresponding power envelop equals a pre-assigned value like 0.50 or 0.80 (Shively, 1988). It is interesting to note that such choices lead to the beta-optimal tests of Davies (1969).

M.Y. El-Bassiouni, H.A. Charif / Computational Statistics & Data Analysis 46 (2004) 707 – 719

709

When testing against small or large alternatives the choice of a test is obvious. Otherwise, one has to choose a member of a class of tests such as the class of point optimal tests or the class of F-tests of LaMotte and McWhorter (1978). To avoid the problem of having to make such arbitrary choices, we propose an invariant test combining the optimal tests against small and large alternatives, which is consequently expected to be powerful overall. It should be noticed that all tests under consideration take the form of a ratio of linear combinations of independent chi-squared random variables. Thereby, the distribution function of any such test statistic can be computed using algorithm AS 155 from Davies (1980), which is based on the work of Imhof (1961). The proposed test is developed in Section 2, while Section 3 describes other competitive test procedures. In Section 4, two applications involving real data sets are discussed to illustrate the computations involved. The powers and e2ciencies (relative to the power envelope) of the tests under consideration are assessed in Section 5.

2. The combined invariant test Consider Model (1) and assume that n ¿ p=rank(X ) and that V is positive de;nite, implying f = 0. Let H be an n × (n − p) matrix whose columns form an orthonormal basis for the orthogonal complement of the column space of X . Set z = H
y and W = H
VH . Then, z ∼ N (0; (In−p +
W )). De;ne Qi = z
Ei Ei
z, i = 1; : : : ; k, where the ri columns of Ei are the orthonormal eigenvectors of W corresponding to the eigenvalue i of multiplicity ri , with 1 ¿ 2 ¿ · · · ¿ k ¿ 0. Olsen et al. (1976) showed that the Qi constitute a minimal su2cient statistic for the family of distributions induced by the maximal location invariant statistic z and that the Qi are independently distributed as ([1 + i
])−1 Qi ∼ 2 (ri );

i = 1; : : : ; k:

It can be shown that q=(Q1 =Qk ; : : : ; Qk−1 =Qk ) is a maximal location and scale invariant statistic whose family of distributions can be parameterized in terms of
alone, e.g., see LaMotte et al. (1988). Clearly, any test of H0 that depends on y through q is location and scale invariant. Further, corresponding to any test of H0 that is location and scale invariant, there exists a test based on q that has the same power function, e.g., see Lin and Harville (1991). Consequently, attention is restricted in the sequel to tests based on q. Set P = X (X
X )− X
, where (X
X )− is an arbitrary g-inverse of X
X and let P denote the arithmetic mean of the i . Hence,  k k

P = ri  i ri = tr[(I − P)V ]=(n − p): i=1

i=1

When k = 2, there is only the UMP invariant F-test based on r2 Q1 =r1 Q2 . For k ¿ 2, no such a UMP test exists. The LMPI test, which is optimal against small alternatives

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M.Y. El-Bassiouni, H.A. Charif / Computational Statistics & Data Analysis 46 (2004) 707 – 719

(
→ 0), rejects H0 for large values of  k k

P i Qi Qi ; T0 = i=1

i=1

e.g., see Nyblom and Makelainen (1983), King and Hillier (1985), Nabeya and Tanaka (1988), Lin and Harville (1991) and Jandhyala and MacNeill (1992). On the other hand, the MPI test that is optimal against large alternatives (
→ ∞) rejects H0 for large values of  k k

P Qi −1 Qi ; T∞ = i

i=1

i=1

see Lin and Harville (1991). We propose to combine T0 and T∞ , which are optimal against small and large alternatives, respectively, through their geometric mean to obtain a test that is expected to have good power properties overall. This leads to the equivalent test which rejects H0 for large values of  k k

2 P  i Qi −1 Qi : (3) TC = i

i=1

i=1

The eigenvalues and eigenvectors of W are not needed for the computation of TC . For instance, using Lemma 1 of King (1980), ˆ P2 "˜
V −1 ";˜ TC = z
Wz= P2 z
W −1 z = "ˆ
V "=

(4)

where "ˆ = (I − P)y denotes the least-squares residuals for the model y = X + ", var(") = I , and "˜ = (I − X (X
V −1 X )−1 X
V −1 )y denotes the generalized least-squares residuals for the model y = X + ", var(") = V . The power function of TC , with size #, is given by   k
[a(i ) − b(i )c# ] (1 + i
)Ui ¿ 0 ; (5) P(
) = Pr i=1

P where the Ui are independent random variables such that Ui ∼ 2 (ri ), a(i ) = i = , P b(i ) = =i , and c# is the solution to the equation P(0) = #. To show that the power function (5) is monotonic, with a(i ) and b(i ) as given above, let j be de;ned such that a(i ) − b(i )c# ¿ 0; ¡ 0;

i = 1; : : : ; j; i = j + 1; : : : ; k:

For

¿
, note that (1 + j

)=(1 + j
) ¿ 1 and that (1 + i
) (1 + j

)=(1 + j
) ¡ (1 + i

); ¿ (1 + i

);

i = 1; : : : ; j; i = j + 1; : : : ; k:

M.Y. El-Bassiouni, H.A. Charif / Computational Statistics & Data Analysis 46 (2004) 707 – 719

Hence,



P(
) ¡ Pr  ¡ Pr

k





[a(i ) − b(i )c# ](1 + i
)(1 + j
)=(1 + j
)Ui ¿ 0

i=1 k


711




[a(i ) − b(i )c# ](1 + i
)Ui ¿ 0

= P(

):

i=1

Thus, the combined test TC is not only location and scale invariant, but also unbiased. P P so that
= )= (1 P − )). For 0 6
¡ ∞, note that 0 6 ) ¡ 1. Let ) = 
=(1 + 
) The power function (5) can be rewritten as  k 
P − ))]Ui ¿ 0 ; P()) = Pr [a(i ) − b(i )c# ][1 + i )= (1 (6) i=1

P of the eigenvalues which indicates that only relative magnitudes (with respect to ) aFect power. To gain an insight into how the eigenvalues aFect the power function, consider the case k = 2 where (6) reduces to P P P()) = Pr{r2 U1 =r1 U2 ¿ F(1 − #; r1 ; r2 ) [1 − ) + )(2 = )]=[1 − ) + )(1 = )]} with F(1 − #; r1 ; r2 ) denoting the 1 − # percentile of the F-distribution with r1 and r2 degrees-of-freedom. The best situation occurs when 2 is near zero, while the worst occurs when 1 and 2 are nearly equal. For k ¿ 2, one might expect in a similar fashion that the power is going to be low when the eigenvalues are close together and higher when they fall into two groups, one of large positive values and one of small values near zero. The test statistic TC can be computed using (3), which requires the eigenvalues and eigenvectors of W , and, consequently, the critical value c# (or the p-value) and the power function (4) can be computed using the method of Imhof (1961) as outlined in Lin and Harville (1991). Otherwise, TC can be computed using (4), which requires the inversion of the n × n matrix V , and, by virtue of location and scale invariance, the critical and/or p-values can be easily simulated with su2cient accuracy assuming that the null distribution of the observation vector y is N (0; In ). 3. Other test procedures g k Let g be an index such that 0 ¡ g ¡ k, and de;ne ng = i=1 ri , mg = i=g+1 ri . LaMotte and McWhorter (1978) suggested test statistics which reject H0 for large values of  k g

Tg = (mg =ng ) Qi Qi : i=1

i=g+1

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M.Y. El-Bassiouni, H.A. Charif / Computational Statistics & Data Analysis 46 (2004) 707 – 719

Since Tg ∼ F(ng ; mg ), under H0 , an exact test is available for any choice of the index g. Hence, they proposed to select g so as to minimize    g k

s = ng mg r i i ri i  F(1 − #; ng ; mg ); (7) i=g+1

i=1

where F(1 − #; ng ; mg ) is the 1 − # percentile of the F-distribution with ng and mg degrees of freedom. The MPI test against an alternative
1 ¿ 0, rejects H0 for large values of  k k

T
1 = Qi (1 + i
1 )−1 Qi ; i=1

i=1

e.g., see LaMotte et al. (1988) and Lin and Harville (1991). To use a most powerful test, when k ¿ 2, an alternative must be speci;ed. If this is not feasible or is undesirable, the LMPI test T0 is optimal against small alternatives, while T∞ is optimal against large alternatives. Davies (1969) provided another approach to the choice of an alternative. To this end, note that the power function of the level # MPI test is given by  k 
−1 P(
|
1 ) = Pr (1 − (1 + i
1 ) c# ) (1 + i
)Ui ¿ 0 ; i=1

P where c# is the solution to the equation P(0 |
1 )=#. Choosing
P such that P(
P |
)=, 0 ¡ # ¡  ¡ 1, leads to the beta-optimal test statistic which rejects H0 for large values of  k k

T
P = Qi (1 + i
) P −1 Qi : i=1

i=1

However, one is still left with the problem of choosing a value for . Shively (1988) advocated the choice P(
P |
) P = 0:50:

(8)

Actually, he found T
P to perform well versus T0 and Tg when testing for a stochastic coe2cient in a time series regression model. The eigenvalues and eigenvectors of W are only required for the computation of Tg , but are not needed for the computation of the other tests. For instance, P
z = "ˆ
V "= T0 = z
Wz= z ˆ P"ˆ
"ˆ and P
W −1 z = "ˆ
"= T∞ = z
z= z ˆ P"˜
V −1 ":˜ In a similar fashion, let V
= I +
V and "˜
= (I − X (X
V
−1 X )−1 X
V
−1 )y denote the generalized least-squares residuals for the model y = X + ", var(") = V
. Then, both T
1 and T
P can be computed upon replacing
by
1 and
, P respectively, in the

M.Y. El-Bassiouni, H.A. Charif / Computational Statistics & Data Analysis 46 (2004) 707 – 719

713

Table 1 Coe2cients in the power function (5)

Test

a(i )

b(i )

T0 T∞ TC T
1 T
P Tg : i = 1; : : : ; g : i = g + 1; : : : ; k

i = P 1 i = P 1 1 mg 0

1 P i = P i = 1=(1 + i
1 ) 1=(1 + i
) P 0 ng

following formula ˆ "˜

V
−1 "˜
: T
= z
z=z
(I +
W )−1 z = "ˆ
"= Thus, while T0 is relatively easy to compute, both T∞ and TC require the inverse of V . In contrast, T
P involves the additional prerequisite of solving the nonlinear equation (8) for
P besides inverting V
P . The power function of any size # test under consideration is given by (5), where the coe2cients a(i ), b(i ) are given in Table 1 and c# is the solution to the equation P(0) = #. Using the same argument outlined in Section 2, it can be shown that the power function (5), with a(i ) and b(i ) as given in Table 1, is monotonic. Thus, the various location and scale invariant tests under consideration are also unbiased. Except for Tg , whose critical values can be obtained from the readily available tables of the F-distribution, the critical values c# (or the p-values) for the other tests, and the power function P(
), can be either computed using the method of Imhof (1961) or simulated as outlined in Section 2. 4. Applications 4.1. Full animal models Let y, X and  be as de;ned in (1) and consider the model y = X + A# + "; where A is an n×a known matrix, and # and " are unobservable random vectors of size a × 1 and n × 1, respectively. Further, assume that " ∼ N (0; In ),  ¿ 0, independently of # ∼ N (0; 1 Ba×a ), 1 ¿ 0, where B is a known matrix. This leads to (1) with
= 1 = and V = A
BA. An e2cient computing strategy for calculating predictions and their standard errors in such mixed linear models are given by Gilmour et al. (2004). In animal breeding contexts the known matrix B is referred to as the relationship matrix since it describes the degree to which the elements of # are related. For instance,

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Table 2 p-Values and minimum e2ciencies of tests for the lion-eye data

Test

T0

T∞

TC

T
P

Tg

Statistic p-Value Min. e2ciency

1.069 0.227 0.939

0.825 0.117 0.860

0.882 0.196 0.966

1.299 0.180 0.980

0.970 0.426 0.846

if the elements #1 and #2 of # are the additive genetic eFects corresponding to a parent and oFspring, then cov(#1 ; #2 ) = 0:51 , see Falconer (1989, p. 150). It is interesting to note that if A = I , then every unobservable random variable #i has an associated observable random variable Yi . This particular model is referred to as a full animal model. 4.1.1. Example: lion-eye data Burch and Iyer (1997) considered the data of Evans et al. (1995) on 171 yearling bulls from a Red Angus seed stock herd in Montana. The dependent variable y was the lion-eye, i.e., rib-eye, muscle area measured in square inches. The ;xed eFect was age of dam with ;ve categories. The random eFects are the animal’s additive genetic eFect and error. They assumed that A = I and determined the relationship matrix B via a recursive method given in Henderson (1976) using knowledge of the animal’s sire, dam and grandparents. Thus, X is a 171 × 5 incidence matrix, n = a = 171, m = p = 5, f = 0 and k = 165. The eigenvalues range in magnitude from 1 = 8:56925 to 165 = 0:56569 with an average of P = 0:92894. Further, except for 105 = 0:67188 whose multiplicity is two, all eigenvalues have a multiplicity of one. It is found in this example that g = 4 minimizes (7) while
P = 0:347 satis;es (8). Since the p-values for the various tests, which appear in Table 2, are greater than 0.10, the evidence provided by this set of data is not enough to reject the null hypothesis of no random genetic eFects. 4.2. Random walk regression coe8cients Consider the model yt = Xt
t + "t ; t = t−1 + ut ;

(9) t = 1; : : : ; T;

(10)

where yt and "t are scalars and Xt , t and ut are p × 1 vectors. It is assumed that "t ∼ N (0; ) independently of ut ∼ N (0; 1 Dp×p ), where D is a known diagonal matrix. Thus, the model is characterized by the parameters  ¿ 0, 1 ¿ 0 and 0 ∈ Rp , the starting vector needed in (10). The above model belongs to the class of state-space models developed in control engineering for representing the stochastic behavior of a dynamic system. It is also regarded as representing coe2cient instability in time series regression and is often referred to as a varying coe2cient regression model, e.g., LaMotte and McWhorter

M.Y. El-Bassiouni, H.A. Charif / Computational Statistics & Data Analysis 46 (2004) 707 – 719

715

Table 3 p-Values and minimum e2ciencies of tests for the investment demand data

Test

T0

T∞

TC

T
P

Tg

Statistic p-Value Min. e2ciency

3.917 0.0020 0.877

0.572 0.0011 0.742

2.241 0.0008 0.960

2.280 0.0022 0.965

3.963 0.0257 0.790

(1978), Nicholls and Pagan (1985). For such models, it is usually of interest to test if the coe2cient vector varies over time following a random walk process as described by (10). However, testing for the constancy of t is equivalent to testing hypotheses (2), where
= 1 =. The problem of testing for the constancy of regression coe2cients has been studied also under the change point alternative, e.g., see Jandhyala and MacNeill (1992) who demonstrated a duality between the random walk and change point formulations. Hall et al. (2003) discussed various approaches to the estimation of change points in addition to the problem of testing for a common change point. Successive substitution of (10) in (9) leads to yt = Xt
0 + Xt
(u1 + · · · + ut ) + "t . Let y = (Y1 ; Y2 ; : : : ; YT )
, X = (X1 ; X2 ; : : : ; XT )
, and assume that rank(X ) = p ¡ T . It follows that y ∼ N (X0 ; (IT +
V )), where the elements of V are given by vij = min(i; j)Xi
DXj . It can be shown that V is positive de;nite if D is positive de;nite. Moreover, if D = diag(1; 0; : : : ; 0), the elements of V reduce to vij = min(i; j), so that V is also positive de;nite for the so-called “intercept variation” models, in which only the intercept varies over time, LaMotte and McWhorter (1978). The parameters of model (9)–(10) can be estimated by maximum likelihood implemented by the Kalman ;lter. However, Stock and Watson (1998) noted that if
is small, the maximum likelihood estimator has point mass at 0. When Xt = 1 this is related to the so-called “pile-up problem” in the ;rst-order moving average model with a unit root. In general, the pile-up probability depends on the properties of Xt and can be large. The reader is referred to Stock and Watson (1998) and the references therein for more details. 4.2.1. Example: Investment demand Table 16.3 in Greene (1993, p. 446) contains a time series data set for the US Steel ;rm. The time series consists of 20 yearly observations on gross investment (y), market value of ;rm at the end of the previous year (x1 ) and value of the stock of plant and equipment at the end of the previous year (x2 ). Thus, T = 20 and m = p = 3 (counting the intercept). Under the assumption D = I , it follows that f = 0 and k = 17 (all eigenvalues have a multiplicity of one). The eigenvalues range in magnitude from 1 = 67; 206; 472 to 17 = 794; 017:9 with an average of P = 8; 890; 809. In this example, it is found that g = 4 minimizes (7) while Eq. (8) yields
P = 9:65 × 10−8 . Since the p-values that appear in Table 3 are fairly small, the ;xed regression coe2cients hypothesis, H0 :
= 0, is rejected in favor of the random walk alternative, H1 :
¿ 0, for all tests.

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M.Y. El-Bassiouni, H.A. Charif / Computational Statistics & Data Analysis 46 (2004) 707 – 719

In fact, for the ;xed regression coe2cients model Y = X0 + ", the assumption var(") = I is doubtful. The autocorrelation test statistic is 0.91, which is smaller than the Durbin–Watson lower limit 1.100 at the 5% level of signi;cance and falls in the inconclusive interval (0.863, 1.271) at the 1% level. However, the corresponding test statistic based on the generalized least squares residuals, using the covariance matrix V with D = I , exceeds the Durbin–Watson upper limits at the 5% and 1% levels of signi;cance. Thus, the random walk assumption (8) apportions some of the variation to the regression coe2cients instead of putting all of it in the disturbances.

5. Power comparisons To investigate the power properties of TC opposite those of the other tests, the power functions were computed from (6) using the eigenvalues of the lion-eye and investment demand data sets for ) = 0:05(0:05)0:90 (values of ) ranging from 0.05 to 0.90 with increments of 0.05) and the power curves are depicted in Figs. 1 and 2, respectively. 1.0 0.9 0.8 T_rho-bar

Power

0.7

T_g

0.6

T_c T_0

0.5 0.4 0.3

T_inf

0.2 0.1 0.0

0.1

0.2

0.3

0.4

∆

0.5

0.6

0.7

0.8

0.9

Fig. 1. Power functions for the lion-eye data.

0.9 0.8 0.7 T_rho-bar

Power

0.6

T_g

0.5

T_0

0.4

T_c

0.3 T_inf

0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

∆

Fig. 2. Power functions for the investment demand data.

M.Y. El-Bassiouni, H.A. Charif / Computational Statistics & Data Analysis 46 (2004) 707 – 719 0.45

T_rho-bar

0.35 Power

717

T_c

T_0 T_g

0.25

0.15

T_inf

0.05 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

∆

Fig. 3. Power functions for random harmonic regression with 10 observations.

Both ;gures show that T0 and T∞ were only dominant at small and large values of ), respectively. Also, Tg was relatively ine2cient since it was dominated by TC and T
P in both cases. In contrast, TC and T
P performed well overall, with TC doing slightly better (worse) than T
P against small (large) alternatives. Further, since the power of T
1 is an attainable upper bound on the power of location and scale invariant tests at
=
1 , it was used as a benchmark (power envelope) against which the e2ciency of the competitive tests were assessed. The minimum e2ciencies (relative powers with respect to the power envelope) of T0 , T∞ , TC , T
P and Tg over the range ) = 0:05(0:05)0:90 are given in Tables 2 and 3 for the lion-eye and investment demand data sets, respectively. Such results con;rm the overall superiority of TC and T
P whose e2ciencies exceeded 0.960 for both data sets. To gain more insight into how the eigenvalues aFect the powers of TC and T
P , various patterns of eigenvalues were explored including those of random polynomial and harmonic regression models, e.g., see Jandhyala and MacNeill (1992). To save space, we consider only the case of a random ;rst order harmonic regression model with 10 observations, where all eigenvalues have a multiplicity of one and range in magnitude from 1 = 8:96098 to 7 = 0:54293 with an average of P = 2:26524. The resultant power curves, which are depicted in Fig. 3, show that TC was superior (inferior) to T
P for ) 6 0:75 () ¿ 0:75). Furthermore, the e2ciencies of T0 , T∞ , TC , T
P and Tg surpassed 0.909, 0.916, 0.974, 0.872 and 0.920, respectively, over the range ) = 0:05(0:05)0:90. Table 4 gives some indicators of the eigenvalues’ structure for the lion-eye data, the investment demand data and the random harmonic regression model. The measures of skewness and kurtosis indicate that the eigenvalues of the random harmonic regression model, though highly unbalanced, are less extreme than those of the lion-eye data. On the other hand, the ratio min =max is larger for harmonic regression than for investment demand. It seems that the power gain realized by TC over T
P , against small-to-intermediate alternatives (Fig. 3), fades away as the ratio min =max approaches zero (Fig. 2) or as the eigenvalues become extremely unbalanced (Fig. 1). This reasoning agrees with the remarks of LaMotte et al. (1988, p. 1162) that the poor performance of the LM tests

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Table 4 Some indicators of the eigenvalues’ structure

Indicator

Lion-eye

Investment demand

Harmonic regression

min =max Measure of skewness Measure of kurtosis

0.066 5.775 36.896

0.012 2.870 8.111

0.061 2.333 5.581

(Tg ) relative to the NP tests (T
1 ) may be linked to having min = 0 as well as to extreme unbalance. In summary, the LMPI test T0 should be used against small alternatives, while T∞ should be used to test against large alternatives. Otherwise, the above results provide support for recommending the proposed combined invariant test TC . Nonetheless, if the eigenvalues are extremely unbalanced and the smallest eigenvalue is nearly zero, then T
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