Glejser
'
s test revisited
Jose
H
A.F. Machado
!
, J.M.C. Santos Silva
",
*
!Faculdade de Economia, Universidade Nova de Lisboa, Portugal "ISEG, Universidade Te&cnica de Lisboa, R. do Quelhas 6, 1200 Lisboa, PortugalAbstract
Godfrey (1996,Journal of Econometrics72, 275}299) has shown that the Glejser test for heteroskedasticity is valid only under conditional symmetry. Here, modi"cations of the Glejser test are suggested. The proposed test statistics are asymptotically valid even when the disturbances are not symmetrically distributed and can be used to test for hetero-skedasticity when conditional location functions other than the conditional mean are estimated. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C52
Keywords: Conditional symmetry; Glejser test; Heteroskedasticity; Regression quantiles
1. Introduction
Godfrey (1996) showed that the well-known Glejser (1969) test for hetero-skedasticity is not valid when the disturbances of the model are not symmetric-ally distributed around zero. In this paper, modi"cations of the Glejser test are suggested. The proposed test statistics do not require assumptions about the symmetry of the distribution of the disturbances and are asymptotically equiva-lent to the Glejser test in certain circumstances.
Given the limitations of the Glejser test and the availability of simple and well-established alternative tests for heteroskedasticity (see Breusch and Pagan,
*Corresponding author. Fax: 351-213922781. E-mail address:[email protected] (J.M.C. Santos Silva).
1979; White, 1980; Koenker, 1981), one might wonder whether there is value in developing new variants of the Glejser test. The main motivation for this development is that the Glejser test is based on considerably weaker assump-tions than its counterparts. The Glejser test places weaker restricassump-tions on the shape of the error distribution under the null hypothesis than the Breusch and Pagan (1979) test based on the normal likelihood function. On the other hand, the Glejser-type tests are less demanding on the existence of moments than the statistics of Koenker (1981) and White (1980) since they require the error distribution to have only "nite second moments rather than "nite fourth moments.
Consequently, the new versions of the Glejser test proposed here are parti-cularly attractive in the presence of errors without third or fourth moments. Moreover, they can be used to test for heteroskedasticity in models for condi-tional location functions other than the condicondi-tional mean (e.g., condicondi-tional quantiles and expectiles). Finally, they are very easy to implement using any standard econometric package.
The remainder of the paper is organized as follows. Section 2 reviews the Glejser test for heteroskedasticity and introduces asymptotically valid Glejser-type tests. The results of a small simulation study on the performance of the suggested statistics are presented in Section 3, and Section 4 o!ers some concluding remarks. All the proofs and the assumptions under which they are valid are presented in a technical appendix.
2. Asymptotically valid Glejser-type tests
2.1. Model and location estimators
Consider the following linear model for a conditional location functionlof the conditional distribution ofygivenx,
y
i"x@ib(l)#u(l)i, i"1,2,n, (1)
wherex
i andb(l) arek-dimensional vectors withk(n,x1i,1 and
u(l)
i"p(z@ic)e(l)i
in whichz
i is a vector ofqfunctions ofxi, p()) is a positive function such that p@(0)"Lp(a)/LaD
a/0O0 and the e(l)i are i.i.d. random variables with
condi-tional location function l equal to zero. This paper is concerned with the detection of departures from the assumption of homoskedasticity, in particular with testing
H
The location parameters,b(l) are estimated by solving
min
b
1 n
n + i/1
ol(y
i!x@ib),
where the choice of the function ol depends on l, the conditional location
function ofybeing modelled. Of course,ol(v)"v2corresponds to the case in
whichlis the conditional mean. Other choices will also play an important role later:ol(v)"DvD, whenlis the conditional median, andol(v)"v[h!I(v)0)], if
lis thehth quantile. The parameters of these location functions will be denoted byb(m), b(12) andb(h), respectively. Under the null of homoskedasticity, these sets of parameters will be identical except for the intercepts, as all the condi-tional location functions are then parallel.
2.2. The Glejser test
Glejser's (1969) test for heteroskedasticity was proposed for the case in which (1) de"nes the conditional mean ofygivenx. Let
q(b(m))" 1
Jn
n + i/1
(z
i!z6)Du(m)iD,
where u(m)
i"yi!x@ib(m) and z6 denotes the sample mean of zi. The test is
essentially a check for the signi"cance of the di!erence from zero of q(bK(m)), wherebK(m) is the least squares estimator of the unknown parametersb(m).
Godfrey (1996, p. 279) shows thatq(b(m)) is not robust to estimation e!ects, in the sense that, when the errors have an asymmetric distribution, q(b(m)) and q(bK(m)) do not have the same asymptotic null distribution. Speci"cally, because of the additional variability that results from replacingb(m) bybK(m), the variance of the"rst-order asymptotic distributions ofq(bK(m)) andq(b(m)) will not be the same, except under conditional symmetry. Proposition 1, which is due to Godfrey (1996), formalizes this lack of equivalence.
Proposition 1. Under Assumptions 1}3 detailed in the appendix, q(bK(m))"q(b(m))#[1!2P(u(m)
i*0)]Xzx*bK(m)#o1(1),
where *bK(m)"Jn(bK(m)!b(m)) andXzx is the limit of the sample covariance matrix betweenzandx.
This result implies that, when (1) de"nes the conditional mean and the parameters are estimated by least squares, the Glejser test will be valid only under the additional assumption of conditional symmetry, that is, when P(u(m)
1Alternatively, the Glejser test could be corrected by using an appropriate estimate of the variance ofq(bK(m)).
median is the natural choice of conditional location function for the equivalence ofq(b(m)) andq(bK(m)) to hold under general conditions. Then, as Proposition 2 will show,q(bK(1/2))"q(b(1/2))#o
1(1). This result is not surprising as the Glejser test is a natural test for heteroskedasticity in the context of median regression. As pointed out by Godfrey (1988), the Glejser test is the score test for hetero-skedasticity when the least absolute deviation estimator coincides with the maximum likelihood estimator, that is, when the errors have a double exponen-tial distribution.
2.3. New tests
WritingDu(l)
iDas 2u(l)i[I(u(l)i*0)!12], whereI(A) is the indicator function
of the eventA, it is clear that the non-equivalence of q(b(l)) andq(bK(l)) stems from the fact that the expectation of the weights [I(u(l)
i*0)!12] is in general
di!erent from zero. The problem can then be solved by subtracting from these weights their expectation, i.e., by considering u(l)
i[I(u(l)i*0)!g] with g"P[u(l)
i*0].1The modi"ed test can be based on,
qH(b(l), g)" 1
Jn
n + i/1
(z
i!z6)u(l)i[I(u(l)i*0)!g].
For any giveng3(0, 1), the requirement that g"P[u(l)
i*0] implies that
x@b(l) in (1) must be the (1!g)th quantile of the conditional distribution ofy. The sample version ofqH(b(l), g) should then use the residuals from the (1!g)th quantile regression"t,u((1!g)
i,
qH(bK(1!g), g)" 1
Jn
n + i/1
(z
i!z6)u((1!g)i[I(u((1!g)i*0)!g], (2)
wherebK(1!g) are the parameter estimates from the (1!g)th quantile regres-sion.
The next proposition states formally the asymptotic equivalence between qH(b(1!g), g) andqH(bK(1!g), g).
Proposition 2. Under Assumptions 1}3 detailed in the appendix, qH(bK(1!g), g)"qH(b(1!g),g)#o
1(1).
2It has recently to our attention that Im (2000) proposed a similar modi"cation of the Glejser test.
estimators (Koenker and Bassett, 1978) (cf. Section 2.1). Consequently, (2) may be rationalized as suggesting the construction of heteroskedasticity tests based on the correlation between the o-function de"ning the chosen conditional location estimator and functions of the regressors. This is precisely what the tests for heteroskedasticity of Breusch and Pagan, White or Koenker do as there the location function is the conditional mean and o(u)"u2. However, as mentioned before, a test based onqH(b(l),g) is interesting even if the location function being estimated is not a regression quantile.
There is an alternative way of constructing tests based on qH(b(l), g). The approach is arguably less natural than the previous one but yields tests that are applicable to arbitrary location estimators under mild moment restrictions. If b(l) is the vector of parameters of any function of location, not only conditional quantiles,gcan be estimated by
g("(1/n)+ i
I[u((l)
i*0],
withu((l)
i"yi!x@ibK. In this case, the sample version of the statisticqH(b(l), g) is qH(bK(l), g()" 1
Jn
n + i/1
(z
i!z6)u((l)i[I(u((l)i*0)!g(]. (3)
Proposition 3. Under Assumptions 1}3 detailed in the appendix, qH(bK(l), g()"qH(b(l), g)#o
1(1).
In the terminology of Godfrey, qH(b(l),g) is robust to estimation e!ects. Moreover, choosing b(l) to be the coe$cients of the conditional mean, the statistic qH(bK(m), g() is directly comparable and as simple to compute as Glejser's.2
The next result provides a convenient way of testing for the null hypothesis of homoskedasticity based onqH(b(l),g).
Proposition 4. Let R2 denote the coezcient of determination of a least squares regression of thevariableu((l)
i[I(u((l)i*0)!g(]on a constant andzi.Then,under
Assumptions 1}3 detailed in the appendix and ifH
0:c"0is true,
nR2PD s2
Remark 1. When (1) de"nes the (1!g)th quantile regression, an identical result holds if for the left-hand-side variable in the arti"cial least squares regression de"ning theR2, g( is replaced byg.
Remark 2. It is not hard to show that under the sequence of local alternatives u(l)
i"p(z@ic/Jn)e(l)i,
nR2PD s2q(d)
with non-centrality parameterd"(p@(0)/p(0))2[k(w)2/u(w)]c@Xzzc, whereX
zz is
the covariance matrix of the q-vectorz, and k(w) and u(w) are the "rst and second moments about zero ofw,e(l)[I(e(l)*0)!g].
When (1) de"nes a mean regression and its errors are symmetrically distrib-uted around zero, the test based onqH(b(l), g) is asymptotically equivalent to the Glejser (1969) test. In that case, the Glejser test is a valid test and the test now proposed should share its good properties, (see Newey and Powell, 1987 for a study of the performance of the Glejser test). For asymmetric distributions, the tests based onqH(b(l), g) are still valid, while the Glejser test overrejects the null.
3. Monte-Carlo evidence
In this section, the results of a small simulation study are presented to illustrate the"nite sample behaviour of the Glejser type tests. The design of the Monte-Carlo experiments is similar to the one used in Section 6 of Godfrey's (1996) paper. In particular, the data are generated by
y
i"b(m)1# k + j/2
x
jib(m)j#p(z@ic)e(m)i, i"1,2,n.
Given the way the test statistics are computed, the results of the experiments are invariant to the value of the regression coe$cients. Therefore, as in Godfrey (1996), the parametersb(m)
j, j3M1,2,kN, are set to zero in every experiment.
Following Godfrey (1996), k3M3, 4N and two sets of slope regressors are considered. In the"rst set (Set 1) the slope regressors are generated by autoreg-ressive schemes of the form
x
ji"jjxji~1#lji,
in whichjj"0.3(j!1),j"2, 3, 4. The observationsx
j1are taken from a
stan-dard normal andlji is drawn from a normal distribution with mean zero and variance 1!j2
j. In the second set of regressors (Set 2),x4is obtained as in Set 1
but x
3Simulations results for the rejection frequencies under a di!erent alternative are available from the authors on request.
4Simulation results using as test variables the two"rst principal components of the variables in set one are available from the authors on request.
N(3, 1), andx
3from an uniform distribution in the range (1, 31). In both designs
the regressors are newly drawn in each replication. As in Godfrey (1996), the disturbances e(m)
i are obtained as independent
draws from a given distribution. Five di!erent distributions fore(m)
i are
con-sidered: normal, Student'st with"ve degrees of freedom; uniform; chi-square with two degrees of freedom; and log normal with log(e(m)
i) distributed as
N(0, 1), which is denotedK(0, 1). In each casee(m)
iis standardized to have mean
zero and unit variance. The functionp(z@
ic) controls the presence of heteroskedasticity. Under the null p(z@
ic) is set to 1. In the experiments that study the behaviour of the tests under
the alternative, the following speci"cation is adopted:3
p(z@
j) denotes the second moment about 0 of xj. This speci"cation of p(z@
ic) corresponds to the hypothesis that b(m)3 is a random variable with
variance that depends onu(x
3), and weights are used to ensure that the expected
value of p2(z@
ic) is one. This kind of alternative is chosen because random
variation of the regression parameters is a major source of heteroskedasticity in empirical applications.
The choice of test variables is critical for the performance of any hetero-skedasticity test. In these experiments, two sets of test variables are considered. In the"rst set, the test variables are the regressors and their squares and cross products, which corresponds to the test variables of White's (1980) test. Because the number of variables used in White's test can be relatively large when compared to the sample size, in the second setx23is the only test variable used. This is the optimal set of test variables under the alternative considered here.4
The test statistics considered in this study are: the Breusch and Pagan (1979) score test for heteroskedasticity as studentized by Koenker (1981), denotedK; the adjusted form of this test proposed by Godfrey (1996), denotedK
A; the test
based onqH(bK(m),g(), denotedG
W; and the Glejser test based on least absolute
deviations residuals, denotedG
LAD. The original Glejser test is not studied since
it is not valid under several of the designs considered here. Godfrey and Orme (1999) study the relative performance of the original Glejser test and many other heteroskedasticity tests, includingK, K
A andGW.
Table 1
Percentage of rejections at the 5% level, under the null, using the"rst set of test variables
Regressor set 1 Regressor set 2
Error distribution Error distribution
k Test N (0, 1) t(5) Unif. K(0, 1) s2(2) N (0, 1) t(5) Unif. K(0, 1) s2(2) 4 K 5.16 7.88 4.22 11.80 9.66 4.54 6.26 3.38 9.88 8.66 4 K
A 4.40 5.86 3.96 7.14 5.48 3.64 4.52 3.60 5.60 5.10
4 G
W 3.84 4.94 3.72 9.88 6.88 3.50 4.70 3.36 8.68 6.84
4 G
LAD 3.88 5.08 4.38 7.92 6.14 4.66 5.66 5.74 8.06 6.82
3 K 5.10 6.96 4.64 9.82 8.40 4.22 5.16 4.38 7.56 6.06 3 K
A 4.78 6.10 4.50 7.92 6.38 3.74 4.18 4.28 5.84 4.82
3 G
W 4.04 5.70 4.86 8.50 6.28 3.24 4.54 4.50 6.94 5.30
3 G
LAD 4.52 5.36 5.66 7.20 6.12 4.96 5.46 7.04 6.80 5.48
(1996) found some evidence that in small samples this test tends to overreject the null when the disturbances are asymmetric. Therefore,K
A, the skewness
ad-justed version of Koenker's test proposed by Godfrey (1996), is also investigated. Note that ad hoc adjustments for skewness of the type used in K
A could be
introduced in the Glejser tests, but that was not done, mainly because this type of adjustment is di$cult to justify.
Under the null hypothesis, all statistics are asymptotically distributed as s2variates with the appropriate numbers of degrees of freedom. All the results presented here are for the case in whichn"60 and are based on 5000 replica-tions.
Tables 1 and 2 present the rejection frequencies under the null for the various tests, at the 5% level, for the di!erent combinations of designs and test variables. Following Godfrey (1996), it is considered that rejections frequencies in the range 3.54}6.55% are evidence of good agreement between the empirical and nominal size of the tests, while rejections frequencies in the range 2.60}7.59% are evidence of adequate agreement.
Starting with the results obtained with the"rst set of test variables, it is clear that, when the errors follow an asymmetric distribution, even tests that are asymptotically valid tend to overreject the null in small samples. This tendency is specially noticeable for theKtest, which clearly overrejects the null both for thes2and log normal errors. As for the other tests, this tendency only becomes apparent in the extreme case of log normal errors. Not surprisingly, in this case the best results are obtained with theK
Atest followed by theGLADandGWtests.
Table 2
Percentage of rejections at the 5% level, under the null, using the second set of test variables
Regressor set 1 Regressor set 2
Error distribution Error distribution
k Test N (0, 1) t(5) Unif. K(0, 1) s2(2) N (0, 1) t(5) Unif. K(0, 1) s2(2) 4 K 3.88 4.64 4.90 4.76 4.22 5.02 4.66 5.42 5.82 5.78 4 K
A 3.76 4.52 4.84 4.34 3.98 4.90 4.42 5.36 4.62 5.08
4 G
W 5.18 4.92 5.48 4.86 4.24 5.20 5.28 5.08 6.30 6.04
4 G
LAD 5.30 4.68 5.92 4.40 4.60 4.92 4.84 4.56 5.10 4.74
3 K 3.76 4.50 5.12 4.62 4.42 5.10 4.70 5.40 5.92 5.76 3 K
A 3.62 4.42 5.06 4.24 4.12 4.92 4.50 5.38 5.02 5.06
3 G
W 4.96 4.84 5.44 4.72 4.18 4.98 5.40 5.20 6.50 5.78
3 G
LAD 5.38 4.48 6.06 4.24 4.64 5.08 5.20 4.48 5.22 4.88
Table 3
Percentage of rejections at the 5% level, under the alternative, using the"rst set of test variables
Regressor set 1 Regressor set 2
Error distribution Error distribution
k Test N (0, 1) t(5) Unif. K(0, 1) s2(2) N (0, 1) t(5) Unif. K(0, 1) s2(2) 4 K 67.62 53.40 87.80 32.72 46.08 36.62 23.22 74.68 14.62 19.78 4 K
A 65.04 48.12 87.24 22.74 35.34 32.76 19.80 72.88 9.26 12.74
4 G
W 67.94 56.68 82.90 33.06 44.28 44.56 31.53 71.28 16.96 23.54
4 G
LAD 66.50 55.64 77.30 34.74 47.28 45.56 32.08 66.02 15.80 22.86
3 K 77.22 61.04 93.86 35.40 52.22 53.66 30.80 91.08 13.66 22.28 3 K
A 76.02 58.62 93.62 30.60 46.66 51.72 28.06 90.74 10.90 18.00
3 G
W 77.66 66.82 90.14 37.08 52.52 64.90 46.34 88.36 17.68 29.94
3 G
LAD 75.92 65.38 86.08 41.74 56.82 65.32 47.52 85.88 18.46 30.74
Using the second set of test variables, there is evidence of good agreement between the empirical and nominal size of the tests in all the cases considered. Tables 3 and 4 contain the rejection frequencies, again at the 5% level, under the alternative described above. Generally speaking, in these experiments, the Glejser-type tests tend to have a power advantage over the studentized score tests, except for the case of uniform errors. As expected, the advantage of the tests proposed in Section 2 is clearer when the kurtosis of the errors is high. However, even for the case of normal errors, there are several cases in which the Glejser-type tests outperform their competitors. Finally, note that the power of all these tests tends to drop as the kurtosis of the error increases.
Table 4
Percentage of rejections at the 5% level, under the alternative, using the second set of test variables
Regressor set 1 Regressor set 2
Error distribution Error distribution
k Test N (0, 1) t(5) Unif. K(0, 1) s2(2) N (0, 1) t(5) Unif. K(0, 1) s2(2) 4 K 87.32 72.06 97.38 38.60 61.98 86.82 65.00 98.48 28.30 50.98 4 K
A 87.06 71.10 97.32 36.50 60.28 86.34 63.54 98.44 25.72 48.14
4 G
W 88.02 80.96 96.00 45.20 67.16 87.60 76.04 96.14 35.96 58.88
4 G
LAD 86.08 79.60 92.56 54.26 72.92 85.70 75.92 93.72 38.04 61.16
3 K 87.74 72.44 97.70 38.84 62.68 87.80 65.98 98.58 28.48 51.40 3 K
A 87.48 71.86 97.66 37.10 60.96 87.48 65.06 98.56 26.24 48.92
3 G
W 88.70 81.90 96.26 45.10 67.08 88.58 77.26 96.50 36.34 59.24
3 G
LAD 86.70 80.26 93.78 54.80 72.96 87.32 76.96 94.98 38.80 62.18
samples, when the errors are asymmetric and large sets of test variables are used. Although less sensitive to this problem, the Glejser-type tests also tend to overreject the null in these circumstances. The skewness adjusted version of K, K
A, has the best performance in these cases. However, using shorter lists
of test variables, this problem is much attenuated and, under the null, there is little to choose between the four tests compared here (see also Godfrey and Orme, 1999).
Under the alternative considered in this study, the Glejser-type tests led to very positive results, outperforming their competitors even is situations where that was not expected. Although these simulation results should be taken only as indicative, they are very encouraging as they suggest that the tests proposed here can outperform their competitors in many situations that are likely to occur in practice.
4. Conclusions
interesting alternatives to the more cumbersome tests of Koenker and Bassett (1982) and Newey and Powell (1987).
Acknowledgements
We are indebted to Andrew Chesher, Les Godfrey and the anonymous referees for helpful comments on previous versions of this paper. The usual disclaimer applies. JoseH Machado's research was partially supported by FEDER, Praxis XXI program grant 2/21/ECO/94, which is gratefully acknow-ledged. Joa8o Santos Silva is thankful to Fundaia8o para a Cie(ncia e a Tecnologia, programme Praxis XXI, for"nancial support.
Appendix A.
A.1. Assumptions
The results in the paper will be derived under the following assumptions.
Assumption 1. My
i,x@iNni/1 is a random sample from an absolutely continuous
distribution with respect to the Lebesgue measure onR]Rk. The conditional
density ofygivenxis continuous inyfor almost allx. Moreover, the conditional distribution ofygivenx has"nite moments of order two.
Assumption 2. There exists a solutionbK(l) to the program min
b
1 n
n + i/1
ol(y i!x@ib)
such that (bK(l)!b(l))"O
1(n~1@2), withb(l) de"ned by (1).
Assumption 3. Almost all sequencesMx
iNandMziNsatisfy:
(a) (i) +n
ix2ij"O(n), j"1,2,k
(ii) +n
iz2ij"O(n), j"1,2,q
(iii) +n
ix2ijzil"O(n), j"1,2,k, l"1,2,q
(b) 1n+n
i/1(zi!z6)x@iPXzx
and
1n+n
i/1(zi!z6)(zi!z6)@PXzz, positive de"nite
(c) sup
A.2. A useful decomposition
LetqH(bK(l), g) be the sample version ofqH(b(l), g), evaluated at any estimated location function de"ned by Assumption 3. That is
qH(bK(l), g)" 1
It is very useful to derive for this statistic a decomposition analogous to that in Godfrey (1996) and presented in our Proposition 1. The results will be explicitly derived under the sequence of local alternatives u(l)
i"p(z@ic/Jn)e(l)i, with
Using Assumption 3(a)(iii) and a Law of Large Numbers,
S(0)PP [P(e*0)!g]X
zx.
A Taylor series expansion of the expectation of the second term on the right-hand side ofS(D) shows that its expectation converges to
!fe(0)1
e denotes the density ofe. Therefore, by JureckovaH 's (1977) Theorem 4.1
(see also Ruppert and Carroll's (1980) Lemma A.3), the second term on the right-hand side ofS(D) is o
1(1).
Turning now to <(D), by a Taylor expansion it is not di$cult to show that
E[<(D)]!E[<(0)]"o(1). Therefore, using the aforementioned Theorem of JureckovaH,<(D)"<(0)#o
1(1). Combining all these results one has,
qH(bK(l), g)"<(0)!S(0)D#o
or, in a less compact notation
qH(bK(l), g)" 1
Jn+i (zi!z6)p(z@ic/Jn)ei[I(ei*0)!g]
#MP[u*0]!gNXzxD#o 1(1)
"qH(b(l), g)#MP[u*0]!gNXzxD#o 1(1).
A.3. Proofs
Proof (Proposition 1). Proposition 1 is proved in Godfrey (1996). It can also be obtained as a special case of the decomposition in (A.2) whenc"0,b(l)"b(m) andgis taken to be12regardless ofP[u(l)*0].
Proof (Proposition 2). Proposition 2 follows easily from the decomposition in (A.2), since P[u(1!g)*0]"g.
Proof (Proposition 3). To simplify notation, let qH"qH(b(l), g), q(H"
qH(bK(l), g(), I
i"I[u(l)i*0] andIKi"I[u((l)i*0].
First note that using an argument identical to the one in (A.2),
g(!g"!fe(0)1 n+
i
h
in p
in
#o
1(1)"O1(n~1@2).
Note also that, de"ningS(D) as in (A.2),
1 n+
i
(z
i!z6)x@iD(IKi!g()"S(D)!
1 n+
i
(z
i!z6)x@iD(g(!g)"o1(1),
since now P(e*0)!g"0, by de"nition.
Therefore, appealing again to the results in (A.2),
q(H"qH# 1
Jn+i (zi!z6)ui[(IKi!Ii)#(g(!g)]#o1(1)
"qH#[<(D)!<(0)]#o
1(1)
"qH#o 1(1).
Proof (Proposition 4). Proposition 3 and the Central Limit Theorem imply that, under the null, qH(bK(l),g() has an asymptotic normal distribution with mean 0 and covariance matrixp(0)2u(w)X
zz, whereu(w) is the second moment about
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