Tilburg University

Identifiability in multiple time series

Tigelaar, H.H.

Publication date:1977

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Citation for published version (APA):Tigelaar, H. H. (1977). Identifiability in multiple time series. (Ter discussie FEW; Vol. 77.066). UnknownPublisher.

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CBMR

7627197766

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CATHOLIEKE HOGESCHOOL TILBURGTITDSCHRIFTENBUREAU I Nr.Bestemming, giBLI :?THI:LKKr~Tii::i.:I-~Y.EHOG1rSCEiJOL

TiLBUE~G

REEKS "TER DISCUSSIE"

FACULTEIT DER ECONOMISCHE WETENSCHAPPEN

KATHOLIEKE HOGESCHOOL TILBURG

"REEKS TER DISCUSSIE"

No 77.066 Oktober 1977

Identifiability in Multiple Time Series

Drs. H.H. Tigelaar. d`"3 ~~

T 7~~ ~~~~.~

SUBFACULTEIT ECONOMETRIE

SLJNLNiARY .

This paper deals with identifiability of matrixcoëfficiënts inmultivariate stochastic difference-equation-models, where identifiabilityshould be interpreted as in [5], that is, we are only concerned withidentifiability with respect to the classf(n), the class of possibledistributions of a sample taken from the observable process at n consecutivetime points. (For a discussion on this: see [ 2] and [ 51 ).

CONTENTS.

1. Introduction -2. Notations and preliminary results3. The basic lemma - - - - -4. The AR~case - . - - -5. The ARMA-case - - . - - .6. References - - - - - -

- - - - - - 1

- - - - - - 2

- - - - 5- - - - 10- - - - 12

~8

1

1. Introduction.

In this paper we study m-variate stochastic processes with complex-valuedcomponents and finite second moments, which are assumed to be the unique(a.s.} weakly stationairy solution of a difference-equation of the form:

P q(1.1) E Ak Xt-k - E Bj et-j t- O,f1,t2,...

k-0 j-0

where {~} is (unobservable) m-variate white noise, with unknown covariance-matrix EE. Furthermore, the mxm matrices Ak, k- 1,...,p and Bj, j- 1,...,qare supposed to be unknown. As in [5) we introduce a parameter ~, characteri-zing the distribution of the process {et}, and we put:

0 - (A1,...,Ap ; B1,...~Bq~~)

In section 3 identifiability will be proved for

~Y(0) - (A1,...,Ap , B1,...,Bq, Ee(~))

for the special case p - 0(Moving Average).In section 5 the general case will be treated.

2. Notations and preliminary results.

Let {xt} be a m-variate weakly stationary process. Then its matrix-valuedspectral measure (see: [3]) will be denoted by FX and its spectral represen-tation by:

I eitaz {d~}-x

where ~(a) is the associated m-variate process with orthogonal incrementswith.

E{x {dJ~} z~{da}} - F {dl}.X -X X

(The ~ indicates the complex conjugate transpose). Furthermore we introducethe real-valued measure:

~x{da} - tr Fx{da}

It is easily seen that ~x{D} - 0 i.f.f. (if and only if) FX{D} - 0.Thus we can write:

FX{d1} - fx(a) ~X{da}.

When the measure ~X is absolutely continuous with respect to Lebesgue measure,(with density cpX(~)), FX is said to be so and we can write:

FX{a} - fX(a)da

where fX(a) - fX(a) cpx(a) is cailed the spectral density matrix.

When we are dealing with processes, satisfying the differer:ce eo,uation (1.1),it is usefull to define the matrix-generating functions:

PA(z) - E Ak zk~

k-0

~B(z) - E Bk zk.

k-0

To achieve maximum analogy between the univariate and the multivariatecase, we introduce the concept of a singularity-point:

Definition 2.1.Let Q(z) be a square matrix-valued function of the complex variable z. Thenthe point z~ is zalled a singularity point of Q(z) if det Q(z~) - 0.

Definition 2.2.P

A matrixvalued function of the form Ek-0

zk is called a matrixpolynomial;

if Ap ~ 0 p is the degree of the matrixpolynomial.It will be clear now, that singularitypoints of matrixpolynomials

will take over the role of the zeros of polynomials in the univariate case.

Definition 2.3.The matrixfunction Q(z) is called analytic (or holomorphic), if all itseléments are analytic functions of z.

Since singularity points of an analytic matrix Q(z) are in factzeros of the analytic function det Q(z), they are either isolated points,and hence the set of singularity points has lebesgue-measure 0, or det Q(z)is identically zero.We can now state:

Theorem 2.1.The homogeneous difference-equation

P(2.1.) k~0 ~ Xt-k - 0 t - O,tl,f2,...

has a non-trivial weakly stationary solution i.f.f. A(e 1~ has as a functionof a at least one singularitypoint. Furthermore, if S is the set of sinrity points and S' its complement, we have for all solutions {~};

F {S'} - 0x

Proof. Suppose:

xt - ! eita zX{da}

is a weakly stationary solution of (2.1), thus:

! A(e-ia) eita zX{da} a-s p p

! A(e-ia) FX{d~} A~(e-ia) - 0 p

I A(e-ia) fX(~) A3(e-la) ~X{da} - 0 p

A(e-la) fX(~) A~(e-ia) a.e 0, w.r.t. ~X (or FX).

Hence, for a E S' we must have:

fX(~) a.e 0 w.r.t. ~X ~

FX{S'} - ! FX{d~} - ! fX(a) ~X{da} - 0S' S'

Remark: In contrast to the univariate case, the m-variate homogeneousdifference equation may have a non trivial solution with absolutelycontinuous spectrum, due to the fact that det A(e-l~) can be identi-

cally zero.

- 5

,3. The basic lemma.

Lemma 3.1.

Let Q(z) be a mxm matrixpolynomial of degree p~ 0, which is non-singular for at least one point zl, and let E be an arbitrary hermitianpositive definite mxm matrix.Then there exists a matrixpolynomial Q~(z) with:

a) QD(z) nonsingular for Izl ~ 1.

b)Q(e-ia) ~ Qi~(e-ia)

- QD(e-ia) ~ QO(e-i~) ~~E[-~r,~r]

Furthermore Q~(z) is of degree p and is uniquely determined except for aconstant mxm matrix H, with H E H~ - E.

Proof. Since E can be decomposed into TT~, and the matrix T can be absorbedinto Q(z), it is no restriction to take E- Im, the mxm unit-matrix.

~ Existance.

Let zC be a singularitypoint of Q(z) with IzCI ~ 1, and let cl,...,ck be anorthonormal basis for ker Q(z~).The matrix with columns cl,...,ck is denotedby C.Let dl,...,dm-k be an orthonormal basis for [ker Q(z~)]1, the orthogonalcomplement of ker Q(z~) and denote the matrix with columns dl,...,dm-k by D.Put:

U-[C;D]

then we have:

U~U-W~-I .m

Since:

Q(z~)C - 0

there exists an integer a~ ~ 0, such that:

qQ,(z) C-(z-z~) ~ Q1(z) tlz,

where Q1(z) is a mxm matrix-polynomial of degree p-a~ and rank(Q1(z~)) - k.Thus :

Q(z)U - [ (z-zQ)a~ Q1(z), Q(Z)D] .

Post-multiplying by iJx and using (2.2) yields:

Q(z) - [ (z-zD)a~ Q1(z); Q(z)D] Ux

Put :

r

Q(z) - [ (zzD-1).a0 Q1(z)~ Q(z)D ]L~

Then we have:

„á~kdet Q(z) -(zz~-1) det [Q1(z); Q(z)D]' det iJ~

zz~-1 a~k- (z-z ) det Q(z).

0

Thus, ignoring multilicities, Q(z) has at least one singularity point lessthan Q(z) in Izl ~ 1. Furthermore:

-ia t -ia -ia- ~2.~u -ia x -ia -ia x x -ia)Q(e }Q (e ) - ~e z~-1~ Q1(e )Q1(e ) } Q(e )DD Q (e

- le-ia-z0l`a0 Q1{e-ia)Qx(e-ia)}Q(e-ia)DDxQx(e-ia)

- Q(e-ia) Qx{e-í`)~

so that Q(z) satisfies conditior. b). Since this procedure can be repeated forevery singularity point, we can find in finitely many steps a matrix Q~(ziof degree p satisfying a) and b).

r Uniqueness.

Suppose we have two matrixpolynomials Q~ and P~, both of degree p, satisfyinga) and D). If z~ is a singularitypoint of QS and PS with Iz~l - 1, thenb) implies:

ker Q~(z~) - ker P~(z~).

Thus we can write as before:

Q~(z) - [((z-z0)aSQ~(z))~~ Q~(z)D ]U~

bPó(Z) - [((Z-Zo) oPi(Z))~, Pó(Z)D ]ti~

where det [Q~(z0), Q~(zQ)D] ~ 0 and

det [P~(z~), P~(z~)D] ~ ~

Since b) implies:

(det Q~(e-1~)I - Idet Pp(e-1~)I , ~~

it is easily seen that we must have a~ - b~.Therefore it suffices to prove the uniqueness of;

W(Z) -

Put:

T(z) -

Then we have:

Q(z) - ~(z) W(z), PS(z) - ~(z) T(z)

where:

l~(~(z} -

a(z-zp) pIk

0

Thus for all z we have:

0

Im-k

W(z) - lim ~r~n) Qp(z), T(z) - lim1f~ ~~) Pp(z).n-'z IJ ~.~z ({

Hence:

W(e-ia)Wt(e-ia) - lim ~ (~1)Qp(e-ia)Qó(e-ia) ~

-(n) --ian~e

~- lim ~ (

~)PO(e-ia)PO(e-ia)V (~)

~-~e -i a

- T(e-ia}T~(e-ia).

Thus W(z) and T(z} also satisfy b), but have at least one singularity pointless than Pp(z) and Qp(z) on Iz~ - 1.In this way we can remove all singularitypoints from the unitcircle, andtherefore it is no restriction to assume Qp(z) and Pp(z) non-singular onIzl - 1.Consider:

V(z) - Pp~(z) Qp(z)

Using b) we have:

V(e-ia)V~(e-ia) - P~~(e-i~)Qp(e-ia)Q~(e-ia)P~-1(e-ia) -

- P~~(e-ia)Pp(e-ia)PÓ(e-ia)PÓ-1(e-ia) - Im

or equivalently: V(z) is unitary on ~zl - 1. Furthermore V(z) is analyticinside the unitcircle and has r.o singularitypoints on Iz~ ~ 1.Therefore both V(z) and V~(z) can óe expanded ír~to a powerseries:

V(z) - E Mk zkk-0

--1, , ~ ,. k

k-0k z~ ~zl ~ 1.

Since the elements of V(z) and V-1(z) are rational functions with no poleson Izl ~ 1, both series converge on the unitcircle.Hence:

~ ~V 1(e-ia) - E N e-ika - V~(e-ia) - E~ eika ~

k-0 k k-0

NO - Ó

Nk-Mk-O, k ~ 1

by a simple equation of coëfficiënts.It follows that V(z) - N0, or equivalently:

Q~(z) - P~(z) N0, with N~ unitary.

This proves the lemma.

As an immediate consequence of the lemma we have:

Theorem 3.1.

When in the m-variate Moving Average cas~~ the parameterspace A is such thatB(z) has no singularitypoints in ~zl ~ 1, and B~ - Im, then Y~(0) -(B1,...,Bq, Ec) is identifiable w.r.t.~(q}1).

The proof is exactly that of the univariate case (see [5]) and will be omitted.

~ (qf1 )Remark: The distributions of the class f are in fact m(qf1)-dimensionaldistributions, and there are m2q t 2 m(mf1) unknown parameters. Thus form~ 1 one has identification with less observations than unknown parameters.

- 10 -

4. The AR-case.

Theorem ~.1.

When in the AR-case 6 is such that AO - Im, A(z) has no singularitypointson IzI ~ 1, and Es is non-singular, then:

4'(0) - (A1 ... Ap, EE)

~pt 1~is identifiable w.r.t.~

Proof .

As in the univariate case, we may write down the Y"JLE-WALKER-equations:~)

(~t. 1 ) ~Y(0

where

and

R - (I'0,...,I'p) , I's - E{xt~-s}

~I' 1 I' 0 ..... I'p-1

~ ~I' 2 I' 1 ..... 1'p-2

~ ~rr rr-1..... r0

~-1 0 ..... om

Thus we have identifiability when there is no non-trivial linear combination

PE c~ xt-j , c j E Cm

j-1

with variance zero.

x) The condition that A(z) has no singularity-points on ~zl ~ 1 impiies theexistance of a non-sided MA-representation which in turn implies (4.1)

- 11 -

Suppose

p a-sí4.2) E c! xJ ~-Jj-1

Since the process {xt} has a spectral density given by:

~fX(~) - Á 1(e-ia) ~e Á1(e-ia)

we must have:

P ~E I c! Á1(e-ia) ~e A- 1 (e-ia) c~~ da - 0

Jj-1

But then it follows easily that cj - 0 j- 1,...,p since all terms are real

and nonnegative.

Note that (1~.2) does not imply a singular spectral measure (see ~ 2).

- 12 -

5. The ARMA-case.

The multivariate ARMA-case needs special attention.'The generalization fromthe univariate case is far from obvious: matrixpolynomials don't factorize assimple as polynomials do, and there is no 1-1-correspondence between singula-ritypoints and factorizations~besides singularitypoints, also the correspon-ding nullspaces play an important role.The following terminology will be usefull:

Definition 5.1.

The square matrices A and B are called comparable when they have the samenullspaces. They are called completely incomparable when the nullspaces havenull-ir.tersection.

(~) .HANNAN, who proved the identifiability with respect to the class ~ in [4],

Pshowed that a necessary condition for the matrixpolynomials A(z) - E Ak zk

k-0q

and B(z) - E Bk zk to have no common left factor~), is that AP and Bp arek-0

completely incomparable, and he achieves identifiability by the condition thatA(z) and B(z) have the unitmatrix as a greatest common left-divisor (g.~.l.d)( see mc . Duffee, ( 1] p 35 ).The last condition is difficult to verify and a condition in terms of singu-laritypoints and nullspaces looks more appropriate.The following lemma shows what kind of condition will be needed.

Lemma 5.1.

ti:1~Then the mxm-matrix polynomials A(z) and B(z) have a common singularitypointz0, and A~(z~) and B~`(z~) are not completely imcomparable, then A(z) and B(z)have a common left factor.

~e) Obviously this is necessary for identifiability when only second-orderproperties are considered.

- 13 -

Proof:Let C be the mxc matrix with columns an orthogonal basis for the intersectionof the nullspaces of A~(zC) and B~(zC), and let D be the matríx with columnsan orthonormal basis for the orthogonal complement.In exactly the same way as in the proof of the basic lemma 3.1. we have:

U~A(z) -(z-z~)A1(z)~D A(z)

(z-zp)B1(z)tD B(z)

where U-[ C ; D] , UU~ - U~CJ - Im.

Hence:

(z-z~)Ici 0 A1(z)A(z) - U - - - - ~ - - -

O ~ Im-c D~A(z)

(z-z~)Ic~ 0B(z) - U - - - - ~ - - -

which proves the lemma.

D~ B(z)

We now have:

Theorem 5.2.

When in the ARMA-case A is such that:

a ) A~ - BD - Im,

b) A(z) has no singularitypoints on ~zl ~ 1B(z) has no singularitypoints on Izl ~ 1

c) A~(z) and B~(z) are completely imcomparable ~zAP and Bq are completely incomparable.

then:

Y'(0) - (A1~...,Ap,B1,...,Bq,EE)

is identifiable w.r.t.~~p}q}1~

Proof.Put:

~r.~0) - (A1,...,Ap), ~2(0) - (B1,...,Bq,EE).~Postmultiplying both sides of the difference-equation (1.1) with x~-s,

s- qf1, qt2,... and taking expectations yiel3s:

P(5.1) ~ ~ rs-k - 0 , s - qfl,qt2,...

k-0

where

rs - E{Xt Xt-s}.

Since AQ - Im this can for s- qf1,...,qtp be written:

where

R -

I rq}1 I

~ rq~p ~

q -rq-1; - , rqt1

rq-pf1 rq-1 " ' rq

( rq rq}1 ... rq}p-1 1

Suppose ~ is singular; then there exist vectors c,...,c , not all zero,q 0 p-1such that:

ie) The condition that A(z) has no singularitypoints on ~z~ ~ 1 implies thatthere exists a one-sided moving average representation, which in turn im-plies (5.1).

- 15 -

p-1(5.2) E I'jts cs - 0 j- q-~t1, q-pt2,...~q

s-0

From (5.1) we have:

Prs -- E~ rs-k s- qtl,qt2,...

k-1

or equivalently:

P

rqtlfs --kE1 --k

I'qtlts-k s- 0,1,...,p-1

Postmultiplying by c and summing over s yields:s

p-1 p p-1E rqtl-s cs -- E--k E rqtlts-k cs - 0

s-0 k-1 s-0

using (5.2). Continuing this way (5.2) is easily seen to be valid for j~ q-pt1.Let

p-1~t -~ ck xt-k t- O,t 1, t 2,...

k-0

and p-1c(z) - E ck zk.

k-0

Then {y~} is a weakly stationary process with spectral density matrix:

2~rc(e-ia) A-1(e-ia) B(e-ia) ~e B~(Xia) A-1~(e-ia) C~(e-ia)

(see: [ ~ ] ~ II.4).

~Here we use the fact that Ee is nonsingular; otherwise ~ could be identicallyzero.~

~ p-1E{xt-j ~t-s}- k~1 rs-jtk ck - 0 s . ~ q-pt 1

J -

Premultiplying with c~, and summing over j yields:

E{~ ,y~ts }- 0 ~ ~ s ~' q

- 16 -

This can equivalently be written as:

Ieisac(e-ia)A-1(e-i~)B(e-i~)~EB:t(e-ia)A-1~(e-ia)cx(e-ia)da - 0~ ~s~ ' q

But then it follows that c(z)A 1(z)B(z) must be a matrixpolynomial whose degree

is at most q-1.Let

K(z) - (det A(z)) A-1(z)

then also K(z) is a matrixpolynomial in z with:

det K(z) - (det A(z))m-1

Denote the degree of c(z)K(z) by s and the degree of det A(z) by a.Since c(z)Á 1(z)B(z) is a polynomial of degree at most q-1, say q-r, and

det A(z) is a scalar polynomial, we must have:

sfq-a ~ q-r

thus:

a ~ str , r~` 1

1) Suppose: a ~ s.Put :

ss(z) - c(z) K(z) - E sj z~

j -0

Since s(z)B(z) is a matrixpolynomial of degree at most q-1}a we must have:

sE

sk Bl-k -0 , 1~ qfa

k-0

In the same way for s(z) A(z) - c(z) det A(z) we must have:

sE sk

An-k - 0, n~ p~-ak-0

- 17

Choosing 1- qts and n- pts yields:

s B - s A - 0s q s p

which contradicts the complete incomparability of A~ and B~.P q

2~ When a~sf1, it is easily seen that there exists at least one zero, say zCof det A(z) with:

s(zC) ~ 0

s(z) B(z) - (z-z0) V(z),

for some polynomial V(z).Thus B(zC) is singular and B~(zC) s~(zC) - 0.

~n the other hand we have:

s(z~) A(z~) - c(z~) det A(z~) - 0,

thus also A~(zC)s~(zC) - 0, which contradicts the complete incomparability ofA(z) and B(z) for all z, and completes the proof of the identifiability of`~1(0).

We now fix Y'1(0) and put:

P qzt - kEC --k

xt-k - ~C Bj ~t-jj-

Thus {zt} is a MA-process, with distribution ir.dependent of ~Y1(0). If P~,n(0)2

denotes the joint distribution of z1,...,~, ar,d

~(n) - {P~2(p) ~ C E e})

Then we have the implication:

P(n) ~ P(n) ~ P(ntP) ~ P(nfP)`~2(C1 ) 4'2(02) C1 02

Thus since `Y2(0) is identifiable w.r.t.~(q}1) (according to theorem 3.1), wealso have identifiability w.r.t f(p}q}1). This proves the theorem.

- 18

REFERENCES.

1 MAC DUFFEE, C.C., (1956). The Theory of matrices. New York Chelsea.

2 GENUGTEN, B.v.d., Identifiability in Statistical inference.Statistica Neerlandica.

3 HANNAN, E.J., (1970) Multiple Time Series. Wiley, New York.

1~ HANNAN, E.J., (1969) The identification of vector mixed autoregressive-moving average systems.Biometrika 56, 223-225.

5 TIGELI~R, H.H., (1976) Identifiability in models with lagged variablesReeks "Ter Diskussie" 76.020Catholic University, Tilburg.

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2.J.P.C.Kleijnen

3.J.J. Kriens

4.L.R.J. Westermann

5 . W. van HtilstJ.Th. ~~ran Lieshout

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B.J. Kriens

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J.RitzenJ.Roemen

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januari '76februari '76

februari '76

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juni ''(6

ju].i ' 76

augustus '76

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september ' 7E;

augus t us ' "(t,

auguatus ''íf~.

26.F. Cole }.~orerr~sting uy c,xpc~~eictit~l september '76:;moot,liin~., Ltie Itc~x an~j Jenkinsproceciurt~ u.nci cpe.cLrul iuialy-sis. A sitnulr~tiun :;Lud,y.

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35. W. Derks Structuuranalyse van EconometrischeModellen met behulp van Grafentheorie.Deel IV. Formulé van Mason en dyna-mische modellen met éé,n vertraging. oktober '76

36. J. Roemen De ontwikkeling van de omvangsverde-ling in de levensmiddelenindustriein de D.D.R. oktober ''(f,

37. W. Derks Structuuranalyse van Econometrischemodellen met behizlp van grafentheo-rie.Peel V. De gre.af' van dynamische mo-dellen met meerdere vertragingen. ~kt~~hPr ''(f~

3t3. A. vt~n Schaik l~;en direkt verband tussen economi-sche verouderirrg en be2ettings-graadverliezen.Deel II: gevoeligheidsanalyse. decemLer '7c,

3'). W. Derks Structuuranalyse van Econometri-sChe modellen met behulp vanGrafentheorie.Deel VI. Model I van Klein, sta-tisch, dec~~mber '7{-

40. J. Kleijnen Information Economics: Inleidingen kritiek novembF~r ''!~.

41. M. v.d. Tillaart. De spectrale representatie varimutivaxiate zwak-stationairestochastische processen met c31s-crete tijdparameter. n~,v~~mt,~ r ' rc

~t2. W. Groenendaal Een econometrisch model vanTh. Dunnewijk Engeland december '(~.

43. R. Heuts Capital market models for portfo-lio selection septemLer '"(~

1~4. J. Kleijnen enP. Rens

45. J. Kleijnen eriP. Rens

46. A. Willemsteín

47. W. Derks

1~8. L. Westermann

1~9. W. Derks

50. W.v. Groenendaal enTh. Dunnewijk

51. J. Kleijnen enP. Rens

52. J.J.A. Moors

53. R.M.J. Iieuts

5~. B.B. v.d. Gentzgten

55. P.A. Verheyen

56. W.v.den Bogaard enJ.Kleijnen

57. W. Derks

58. R. Heuts

59. A.P. Willemstein

60. Th. DunnewijkW. van Groener.daal

b1. A. PlaisierA. Hempenius

A critical analysis of IBM's inventorypackage impact. december '7EComputerized ínver.tory management:A critical analysia of' IBM's ímpactsystem.Evaluatie en foutenanalyse van eco-nometrische modelïen.Deel I.

december '7E

Een identificatie methode voor een li-neair discreet systeem met storingenop input, output en structuur. januari '77Structuuranalyse van econometrischemodellen met behulp var. grafer~theorie.Deel VII. Model I van Klein, dyriamisch.februari '7(On sysnems of linear inequalitiesover IR .Structuuranslyse vati econometriscriemodellen met behulp van Grafentheorie.Deel VIII.Klein-Goldberger model.Een ecor,ometrisch m~del van hetVerenigd KoninkrijkA critical analysis ~f IBN1'sinventory package "IMPACT"Estimation ir~ truncated parameter-spacesDynamic transfer function-noisemodelling (Some thecretical con-siderations)Limit theorems for LS-estimatorsin linear regression models withindependent errors.Economische interpretatie in model-len betreffende levensduur vankapitaalgoederenMinimizing wasting times usingpriority classes

Februari '7(

februari '77

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juniStructuuranalyse van EconometrischeModellen met behulp van Grafen-theorie. Deel IX. Dlodeï van lar,-den van de E.E.G. juni ';"~Capital market models for pcrtfolioselection (a revised version) j~:ni '7i'Evaluatie en foute:zanalyse van eco-nometrische modellen. Deel II.Het Modeï I van L.R, Klein.An econometric Model of the Federal

aug.

Republic of Germany 1953-1y73 aug. '"''iSlagen of zakken. Eer. intern rapportover de studieresaltaten propedeuse-economie 197~~19f5 aug .

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62. A. Hempenius Over een maat voor de juistheidvan voorspellingen aug. '77

63. R.M.J. Heuts

64. R.M.J. Heuts

Some reformulations and extensionsin the univariate Box-Jenkins timeseries analysis approach(a revised version) sept. '77Applications of univariate timeseries modelling of U.S, monetaryand business indicator data sept. '77

65. A. Hempenius en J. Frijns Soorten van prijsheteroskedasti-citeit in marktvraagfunkties. okt. '77

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