The purpose of this paper is to clarify certain issues related to the incidence
of heteroskadisticity in the General Linear Model (GLM); to provide simpler
and more accessible proofs for a number of propositions, and to allow
the results to stand under conditions considerably less stringent that
those hitherto available in the literature

Dhrymes, Phoebus J.

Columbia University Academic Commons

A Note on Heteroskedasticity IssuesbyPhoebus J. Dhrymes, Columbia UniversityOctober 1992Discussion Paper Series No. 629A Note on Heteroskedasticity IssuesPhoebus J. DhrymesOctober 14, 19921 IntroductionThe purpose of this paper is to clarify certain issues related to the incidenceof heteroskadisticity in the General Linear Model (GLM); to provide sim-pler and more accessible proofs for a number of propositions, and to allowthe results to stand under conditions considerably less stringent thatthose hitherto available in the literature. A systematic treatment ofthe heteroskedasticity problem in the context of the GLM is given in thevery interesting paper by White (1980) and the follow-up paepr by Nichollsand Pagan (1983). White's context consists of stochastic regressors that areuncorrelated with the error process, but no lagged dependent variables;the context in Nicholl and Pagan consists only of lagged dependent vari-ables, and the error process is assumed to be a martingale difference. Ifthe underlying difference equation is stable, their dependent variable is alsoa martingale difference. Although this may appear to be a generalizationof White's result, in fact it is not, since the admission of "exogenous" (con-temporaneous) regressors will destroy the martingale difference property ofthe ?/-process.In our paper the context is one of stochastic exogenous and lagged de-pendent explanatory variables, and an error process as in White. Thus, ourcontext is a generalization of the White context, but it is at once more* This is a preliminary draft. Not to be quoted without the author's permission. Com-ments are solicited.I would like to thank H. White for an interesting discussion on the subject of this paper.general and more restrictive than the one in Nicholls and Pagan, in that ourscan accommodate lagged dependent and current exogenous variables, butthe error process is one of independent non-identically distributed randomvariables, while in Nicholls and Pagan the error process is one of martingaledifferences.Although a number of the results presented therein had been availablein the statistics literature for some time past, e.g., Eicker (1963), (1967),Rao (1970), (1973), White's paper is quite important in having introducedthem in the econometrics literature and having employed them in addressingcertain important problems arising in empirical applications. On the otherhand, the presentation is overly complex and constitutes a barrier to thewider understanding and appreciation of the material contained therein.2 Formulation of the ProblemConsider the GLMyt = zt-Po + uu * = 1,2, . . . , T , (1)or more compactly, y = X(30-\-u , where (30 is the true parameter vector, xt.is an (n -f 1) -element row vector containing the explanatory variables, y , uand X are the corresponding observation vectors and matrix, respectively,etc. The problem is: (a) to establish the strong consistency of the OLSestimator of (3 and (b) to establish the limiting distribution of the OLSestimator of /?, when the error process is one of zero mean independentrandom variables withCov(u) = E = diag(<jj, <r\,..., <rT), (2)and (c) to produce a (strongly) consistent estimator of the covariance matrixof the limiting distribution noted above.Following White (1980), we assume(A.I) The sequence {zt. = (xt.,nt) : t > 1} is one of independent (but notnecessarily identically distributed) random variables, such thatMxx = lim ]=E(X'X), E(x'tut) = 0,X—K3O 1for all t,* and Mxx is a nonsingular matrix;(A.2). the sequence {zt. = (xt.,ut) : t > 1} possesses fourth order mo-ments obeyingE(\ztiztjztkztr\) <cta a£ [0,1).3 Consistency and Asymptotic Normality ofOLS Estimators3.1 Strong ConsistencyProposition 1. The OLS estimator of p0 of the GLM of the previoussection, under (A.I) and (A.2), obeys^(X'xy'X'y, p^p0. (3)Proof: By Corollary 3, Ch. 3 in Dhrymes (1989), it will suffice to show that(X X/T) and (X u/T), converge to their respective limits a.c. (almostcertainly). This is so since (3 is a continuous functions of the entities above,and the limit of the expectation of the matrix is nonsingular by assumption(A.I). By the same result it will suffice to consider, for arbitrary conformable7,t=l1 The assumption that E(xt.ut) = 0 , for all t, is not quite in character with theGLM where, almost invariably, we are interested in E(yt\xt.) , as e.g., in forecasting.The assumption above does not rule out the case E(ut \xt.) = h(xt.) ; when this is so,we have E(yt \xt.) = Xt.flo + h(xt) . But this is not what we are interested in, nor isit what is commonly forecast. Thus it would be more in character, if the assumption of"uncorrelatedness" were framed asE(ut\xt.) = 0,Cov(utut' \xt.,xt>.) = of, if t — t= 0, otherwise.From a technical point of view, White's assumption makes the derivation of the limitingdistribution of the OLS estimator considerably simplerBy Proposition 22, Ch. 3 in Dhrymes (1989), (especially Remark 11) wehave that the Kolmogorov criterion reduces tosince 2 — a > 1. Hence, by Proposition 22, (X'X/T) ^ Mxx . An entirelysimilar argument will show that (X u/T) ^ 0 .q.e.d.Corollary 1. If we strengthen the assumption (A.I) by requiringE(\ztlztJztkztr\) <ctQ ae [0,-),the result of the Proposition holds for sequences {zt. = (xt.,ut) : t > 1} ,which are merely uncorrelated nonidentically distributed.Proof: See Proposition 26, Ch. 3 in Dhrymes (1989).3.2 Asymptotic NormalityTo show the asymptotic normality of the estimator (3 we assume, in additionto (A.I) and (A.2),(A.3). For every real conformable vector, 7 , the sequence{0 — Xt.^fUt '. t > 1} obeyslim -J2 f _^ (2dFt(() = 0,for abritrary integer r , where Ft is the distribution function of (t.(A.4). The matrix $ is positive definite (nonsingular), where1 T$ = lim — y^ E(u2fxi xt.).~"°° ± t = lRemark 1. Our assumptions (A.I), (A.2) are considerably weaker thanthe assumptions A.2 through A.4 in White, which involve, inter alia, theuniform boundedness of E\e^xtiXtj\1+6, i.e., the uniform boundedness of(4 + 4:S)th moments.Moreover, this (White's) assumption ensures that the sufficient condi-tions for the Liapounov CLT hold, i.e., they imply the Liapounov conditions.Our assumption (A.3) is also weaker than White's assumptions, since theLiapounov conditions imply our assumption (A.3), but our assumption doesnot imply the Liapounov conditions. Finally, our assumption A.4 is preciselyassumption A.3 (b) in White.In view of the preceding we obtainPropos i t ion 2. Consider the model in Eq. (1.1), together with assumptionsA.I through A.4. Then- (30) - i iV(0,Proof: We note thatBy Proposition 26, Corollary 6 in Ch. 4, Dhrymes (1989)VT0 - ft) ~ M~Xwhere (X'u/y/f) - i £. By Proposition 34 in Ch. 4, Dhrymes (1989)X'u d ^ ., , j X u d /£, 11 and only it —j= > 7 <f,y/Twhere 7 is an arbitrary (real) conformable vector. By assumption (A.3)the sequence {(t = xt.^ut : t > 1} obeys the Lindeberg condition; conse-quently, by Proposition 45 in Ch. 4, Dhrymes (1989)Thus, we conclude y/T0 - f3o) ~ i\T(O, M~^M^).q.e.d.The next problem is to produce (at least) a consistent estimator of for $ .4 Convergence of the CovarianceEstimatorThe proposed estimator for $ is1 Twhere ut = yt - xt.fi = ut - xt.(fi - fio). Thus,w^  = ut — 2xt\fi — fio)ut + (p ~ Po) \xt-xt-)\fi ~ fio) = ut ~~ 2</> i^ -f (j>tT2-> (6)and, in the obvious notation, $ = $! — 2$2 + $3 . We have,Proposition 3. Under assumptions A.I through A.4, and in addition,2(A.5) Var[(zt.7)2ut2] < cf\ t > 1, for a G [0,1),1 T\xt. *A lim - J] £(u2 .^a;t.).l i 1 T$ = >Proof: We note that< T[(fi-fi0) (fi-fio)]and by Propositions 1, 2, we conclude that the right member above convergesa.c. to the zero matrix. We further note that' \ (1/2)The proof of the proposition will be concluded if we show that <!>! convergesa.c. to its limit, $ . But this is ensured, in view of assumption (A.5), byKolmogorov's criterion as cited above.q.e.d.This concludes the derivation of results in White (1980), under assumptionsequal to or less stringent than those employed by that author.2 The corresponding assumption in White is that E\(xt.j)2u^\1+i is uniformlybounded. In our case we require the existence of the second moment but allow thelatter to grow at the rate ta , so that we have basically similar kinds of restrictive as-sumptions. Which set one chooses to work with is a matter of taste. In this paper one cantake White's assumptions and prove the same result by relying on a certain SLLN, see forexample Chung (1968), Corollary (ii.) p. 119, as also cited by White.65 Extension to Dynamic ModelsIn this section we reexamine the model in Eq. (1.1) with two importantchanges: first, among the explanatory variables we take yt-\ , and second,we adopt the framework of footnote 1. Thus, the model isVt = Zt-Po + 2/t-iAo + «t, t = 1,2. . . ,T, (7)so that it is the same model as in Eq. (1.1), except that it is dynamic.Since here we need greater detail in our exposition we adopt a more formalframework. Let ( fi , A, V ) be the probability space and let zt. = (#*., yt-i)-Assumptions made in previous sections on the vector xt. , remain in effect,except as modified. Moreover, putCt = cr(xs. :s<t), Vt = a(us :s<t), t > 1, Co = ( 0 , ft), Vo = ( 0 , 0 ) .defineAt = <T(Ct+iUVt), t>l, Ao = a[C1UVo], (8)and note that AQ C A\ C A^ • • • •In this segment of the paper we have the following framework:(B.I) The sequence {(xt.,Ui) : t > 1} is one of independent (but not neces-sarily identically distributed) random variables, such thatMxx = lim ^E(X'X), E{ut\Ct) = 0, E{u]\Ct) = CJ\,for all t, where Mxx is a well defined nonsingular (positive definite)matrix;(B.2) the sequence {xt. : t > 1} possesses fourth order moments obeyingE(\xtixtjxtkxtr\) <cta a e [0,1);(B.3) the sequence {u] : t > 1} is uniformly integrable;(B.4) E[{v?t - a]fv\\ <cf, a e [0, | ) , v, = Ej=S Ifa-i-i ;(B.5) the matrix $ is positive definite (nonsingular), where1 T2tE(izt-)' a n d A O G ( - 1To investigate the problem and at the same time maintain maximal corre-spondence between the argument here and that in the earlier part of the pa-per, we adopt the notation yt = zt.90-\-ut , where, evidently, zt. = (xt-iVt-i) •>5.1 ConsistencyProposition 4. The OLS estimator of 0o of the GLM of the previoussection, under (B.I) and (B.2), obeys= (ZZ)~1Zy, 0 (9)where Z = (X, y-\) •Proof: By the reasons given in the proof of Proposition 1, it will suffice toshow that (Z Z/T) and (Z u/T), converge to their respective limits a.c. .Since1 1-Z Z = -T TXXwe need only be concerned about X y_\ and y_1y_i , in view of Proposition1. Define the matrix1AA201A001. . . 0. . . 0. . . 0DXT-1 AT-2 AT-3and note that, solving the difference equation of the model we havey = (10)Since1 , ^Ty y ~ - A 0 ) 2 T ^etc., the existence of the least squares estimator is proved, i.e.4^ l im^f3)>0;T T—KX> ±as for its convergence a.c. we note that proving<^ rprp rp < rpis exactly the same problem dealt with in Proposition 1, except for the termv u/T, which has mean zero. Sincev uis a sequence of zero mean uncorrelated random variables whose variance iscertainly bounded by (B.4), it follows that ip? converges a.c. to zero, andthe strong consistency of the least squares estimator is proved.q.e.d.5.2 Asymptotic NormalityWe note that1 Ty/T t=1and it will suffice to show that, for arbitrary (confomrable) vector 7 thesequence%T = ZX*T, (tT = —7=Zt."1Ut, (12)converges in distribution. We havePropostion 5. Under assumptions (B.I) through (B.4),Vf(0 - eo).Proof: By the discussion just above, it will suffice to show that the sequenceZf converges in distribution. We note that At, or modified to take accountof division by \JT as Atr , ^ < T, T > 1 is a stochastic basis, i.e. Atr C>At+i,T C At+i,r+i •> etc. Moreover, ((t,At) is a martingale differencestochastic sequence, sinceE{\Ct\)< 00, and E((t\At-i) = 0.Moreover, the sequence Zy satisfies a Lindeberg condition, as follows.Let, for arbitrary integer r ,AT = {u • |Ctrl > - } , BtT = {OJ : \ut\ > },r ar/ / / v l / 2Jand note that Atx C Btx , CXT ~^ 0 by Propositions 1 and 4. Letting I (A)denote the indicator function of the set A, we havet=i2 2 I ] (13)rp / j \ C'The convergence a.c. to zero is valid by (B.3) and Proposition 4. Next weobserve thatconsequently, by Proposition 21 (i.) in Ch. 5, Dhrymes (1989) we concludeZT ^ N(0,7'$7), and Vf(0 - 6o) ~ 7V(0, M~^M~^).q.e.d.5.3 Convergence of the CovarianceEstimator in the Dynamic ModelThis is by far the most complex issue in the extension. Since we have alreadyshown that (Z u/T) -1> 0, the arguments given in Proposition 3 apply tothe three components of the estimator1 Trr~\ ^ _j t Z' ^' -*- Z • O5 \ /1 t-\10the three components above corresponding to the three components of u2 inthe equation below,u] = u\ - 2zt.(9 - eo)ut + (§- 0o)'{zt.zt.)(§ - 0o) = u2 - 2<f>tT1 + <j>tT2. ( 15 )By an argument identical to the one we made in the previous section, $ 3A, A.converges a.c. to zero and $2 converges a.c. to zero if $1 converges. Thus,the (strong) consistency of the estimator of the covariance matrix in Eq. (14)will be proved, if we prove that $1 converges a.c. to its limit.Proposition 5. Under assumptions (B.I) through (B.5),1 T 1 T<P = "— > u+z+ zf. —> l i m —Proof: We first recall that zt. = zt. + (0,^) •> a n d consequently,u2tz'tzt- = u2tzt.zt. + u?4(°> vt) + «?(0, vt)'zt. + f 2 j u\.The first term evidently converges to its limit, by the arguments of Proposi-tion 3; since the mean of the second and third terms is zero these terms willconverge to zero a.c, if the last term is shown to converge a.c. To do so, notethat we are dealing only with the scalar sequenceT T t-2t=2 t-2 j=0and that ST — E{ST) is a sequence of zero mean dependent variables.Unfortunately, there is no general theorem to which we can appeal for itsa.c. convergence; thus, we resort to first principles. WriteST - y \v2(u2t - a2) + a2Av2 - (3(t - 2)1 + a2(3(t - 2)1 = ST1] + 4 2 ) + 4 3 ) .t=2Since 5^ is a sum of zero mean uncorrelated random variables, by (B.3)and Proposition 26, in Chapter 3 Dhrymes (1989), (S^/T) ^ 0. Next notethat, it being understood that ii-s = 0, for 3 > 0,oo / T \ , / Tj=0 \t=j+2 ) 3'<3 \t=j+211In both terms we are dealing with sums of uncorrelated random variables;thus, the sums over the index t, when divided by T, converge a.c. tozero. Since the sums over j and/or j converge absolutely by (B.5), itfollows again by (B.4) and Proposition 26, in Chapter 3 Dhrymes (1989)that (SW/T)^0. Hence,which shows that <t ^ $ .q.e.d.Remark 2. The results herein may be easily extended to the case where themodel errors are a stable autoregressive process, in the following sense. If themodel is transformed so that it contains exogenous, lagged exogenous, andlagged dependent explanatory variables, the error process would be one of in-dependent nonidentically distributed random variables, and hence amenableto the analysis of this section. If the model initially contains lagged depen-dent variables, the same remark would apply if one is not interested inidentifying the autoregressive parameters in the error process. Ifone is, nonlinear methods would have to be employed and, while one mightconjecture that the same results would hold, the problem is not so transpar-ent ab initio.If all one is willing to specify is that the error process is a martin-gale difference, i.e., if all one is willing to specify is that E\ut\ < oo andE(ut\At-i) = 0, one is forced to rely only on martingale convergencetheorems, as in Nicholls and Pagan. The results one obtains, however, limitthe nature of the explanatory variables one may employ. In Nicholls andPagan, one has a stable AR model with martingale difference errors. Thisimmediately implies that the dependent variable process is also a martingaledifference. This simplifies the problem a great deal, but would not admitexogenous variables unless they were also specified to be a zero meanmartingale difference process. If one did that one would have no diffi-culty in extending the Nicholls and Pagan results to the case where there areexogenous variables, lagged as well as contemporaneous. It would appear,12however, that such a model would not be very useful in empirical applica-tions. At least the Nicholls and Pagan result provides an interesting twist tothe VAR model, although it is quite contrary to the stochastic orientationembodied in the latter.BIBLIOGRPHYDhrymes, P. J. (1989) Topics In Advanced Econometrics: Probability Foun-dations, New York, Springer-VerlangEicker, F. (1967) "Limit Theorems for Regressions with Unequal and De-pendent Errors", in Proceedings of the Fifth Berkeley Symposium on Math-ematical Statistics and Probability, vol. 1, Berkeley, University of CaliforniaPress.(1963) "Asymptotic Normality and Consistencyof the Least Squares Estimators for Families of Linear Regressions",Annalsof Mathematic Statistics, 34, pp. 447-56.Nicholls, D.F. and A.R. Pagan (1983) "Heteroskedasticity in Models withLagged Dependent Variables", Econometrica, 51, pp.1233-42.Rao, C. R. (1973) Linear Statistical Inference and its Applications, NewwYork, Wiley.(1970) "Estimation of Heteroskedastic Variancesin Linear Models", Journal of the American Statistical Association, 65, pp.161-72.White, H. (1980) "A Heteroskedasticity-Consistent Covariance Matrix Esti-mator and a Direct Test for Heteroskedasticity", Econometrica, 48, pp.817-38.13

A Note on Heteroskedasticity Issues

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