J. Ita!' Statist. Soc. (/997)3, pp. 245-255

IDENTIFIABILITY CONDITIONS FORGENERALISED STARMA MODELS

Giuseppina Guagnano*, Silvia TerziUniversita di Roma «La Sapienza», Italy

Summary

In this paper, starting from some well known results for the structural identifiability ofVARMA models of given maximum time lags p and q, we derive parametric conditionswhich guarantee the identification of Generalized STARMA models.

Keywords: Identification, structural identifiability, STARMA models.

1. Introduction

The problem of identifiability has been extensively studied in literature (see, forexample, Hannan, 1969, 1971; Deistler, 1985) within the framework of VectorAutoregressive Moving-Average (VARMA) models:

p q

LAiYt-i =: LMPt-J;~O J~O

where Yr is a vector of N observations at time t and u, is a vector of white noiseunobserved inputs.

Assume we have a priori information in form of constraints on the parametermatrices Ai and MJ• In such a situation, the question arises whether these a priorirestrictions are sufficient to guarantee identifiability of the free parameters in theAi and Mj matrices.

This problem is often referred to as structural identifiability (Deistler, 1985, p.271); in particular, identifiability conditions have been derived in literature underthe hypothesis that the maximum time lags p and q are given.

Recalling that in STARMA models the parameter matrices Ai and Mj are alinear combination of given weights matrices Wk

), the aim of the present paper is

* Address for correspondence: Dip.to di Studi Geoeconomici, Statistici, Storici perl' Analisi regionale, Facolta di Economia, Universita di Roma «La Sapienza», Via delCastro Laurenziano 9, 00161 Roma. E-mail: guagnano@sec.eco.uniromal.it.

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G. GUAGNANO . S. TERZI

to see how well known identifiability conditions apply in presence of these fur­ther parametric restrictions.

The paper is outlined as follows: in § 2 we introduce the generalised STARMAmodel and the stability and invertibility conditions; in § 3 we define the identifi­cation problem and derive parametric conditions which guarantee structural iden­tifiability; in § 4 we comment the results, pointing out that, even if the conditionswe derive are more restrictive than the usual ones, they result more operative.

2. The Generalised STARMA model

Assume that observations Ym(t)are available at each of N fixed locations in space(sites) m =1, ... , N over T time periods. A Generalised Space-Time Autoregres­sive Moving Average model GSTARMA (Di Giacinto, 1994; Terzi, 1995), oforder p and q in time and 1in space, can be expressed in the following way:

I p I q I

YI =L/PkOW1k)YI+ L L~k;W(k)YI_;+ul+ L Lt7'kjW(kJUt-j (1)k:[ ;:[ k:O je I k:O

where: ".,(k) are NxN contiguity matrices with elements w~~ *' 0 only if sites landN

m are k-th order neighbours, rescaled so that L w~) =1 (and thus WO) =IN)'m:[

Equation (I) generalises the usual STARMA model introducing contempora-neous spatial dependence, namely the parameters ~kO> k =1, ... , l.

By defining the matrices:

i =0, ... , p; ~oo =0;I

A; =L~k;W(k),k:O

I

M; =L t7'kjW(k), j =1,oo.,q; Mo =IN;k:O

model (1) can be rewritten as:

p q

(IN -Ao)YI =LA;YI-; +Ut + LMPt-j;:[ j:o

(2)

p

and thus in the VARMA form A(L)Yr =M(L)u" where A(L) = IN - LA;I! andq ;:0

M(L) =IN+ LM/J.jel

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IDENTIFIABILITY CONDITIONS FOR GSTARMA MODELS

As usual for time series models, the A(L) and M(L) operators can be equiva­lently expressed as polynomial matrices in the complex variable z.

The underlying process is stable if it can be expressed as an infinite MA proc­ess; this is possible if the polynomial matrixA(z) is invertible, that is to say (Lutke­pol, 1991, p. 17) if det A(z) ;t 0, for lzI ::; 1, z E C. Furthermore, the process isinvertible if there is an equivalent AR representation of infinite order; this ispossible if the polynomial matrix M(z) is invertible, that is to say, if det M(z) ;t0,for lzl ~ 1, z E C.

For the GSTARMA model class, stationarity and invertibility parameter con­ditions are given by the following theorem.

Theorem 2.1. Sufficient conditions for the stability and invertibility of the proc­ess underlying model (3) are:

, p

I, I,lqJk,!<1k=O .=0

, q

I, I, 16k)I <1.kd.O )=1

(4)

Proof For a given matrix G, it is known that (Rizzi, 1988), if its absolute norm(N(G») is less than 1, then G' r->~)0 and thus (l + G + c' + ...) exists and isequal to (l - os'.

pp'

Let us define a polynomial matrix G == [g"m] as G == I, A,z' == I, I, qJki W(k) z·,Z E C. For lzI ::; 1 we have: '=0 '=0 k=O

="'f't, ~{I~ (0.'; I}wl:!

="'f'~{I~(O,/I}t, wl:! ="'f'~I~ (0·';1'..,f, 'p , p

::;I, LlqJk.Zil==I, I,lqJkillzil::;I, I,lqJkJk=O i=O k=O i=O k=O '=0

247

since wi~ 2:0,

N

since L wi~ == 1m=1

G. GUAGNANO' S. TERZI

Thus, the first of conditions (4) ensures the invertibility of A(z) for Izl ~ 1.q N

In similar manner, by defining G =-L Mjz j, we have N(G) =max L

j;J h m;J

q J

-L L tJkjwi~zj and the second of (4) ensures the invertibility of M(z).j;J k;O

The theorem is thus proved.

3. Identifiability conditions for GSTARMA models

Given a stable and invertible process, the problem is that in general the VARMArepresentation A(L)YI =M(L)u, is not unique. In fact if the polynomial matricesA(z) and M(z) admit a common left factor D(z) such that A*(z) =D(z) A(z) andM*(z) =D(z) M(z) and thus A(z) =A*(zrJ M*(z) =A(zr' M(z), we can have anequivalent VARMA representation in terms of polynomial matrices A*(z) and M*(z).

To ensure the uniqueness of the VARMArepresentation, we must impose (iden­tification) conditions on the operators A(L) and M(L), in order for them to beuniquely determined by the corresponding A(L) operator.

To avoid redundancy of description, attention is restricted to irreducible A(z)and M(z) matrices, that is to say, to left prime matrices. Two matrices are leftprime if and only if the degree of det A(z) is minimal among the class of equiva­lent representations.

If the orders of the maximum time lags p and q are given, the only possiblecommon left factor has to be unimodular (i.e. a matrix with constant non zerodeterminant), since any other would increase the lag lengths. (This can easily beseen bearing in mind that the degrees of det A(z) and of det M(z) are not greaterthan Np and Nq respectively).

By imposing further conditions for det A(z) (or det M(z)) to be of prescribeddegree, the matrices A(z) and M(z) will be identified.

Assuming that the matrices A(z) and M(z) are invertible and that the maximumtime lags p and q are given, we thus have that necessary and sufficient conditionsfor the identification are (Hannan, 1969, 1971):

a) A(z) and M(z) are left prime matrices, or equivalently (Deistler, 1985):a-bis) rank[A(z), M(z)] =N, \;;fz E C;

b) rank[Ap• Mq ] =N;c) IN - Aa =IN or analogous norrning condition I •

1. A complete definition of condition c) is the following:Let F = [M{)o ... , Mq, IN - A{)o ... , Ap] , then there must be N - I prescribed zeros in each

row of F and one unit element in each row of IN - Aa; furthermore, the rank of each

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IDENTIFIABILITY CONDITIONS FOR GSTARMA MODELS

For VARMA models identifiable parameterizations are obtained by means ofthe final equations form or the echelon form. However, since STARMA modelscan be seen as VARMA models satisfying particular restrictions (due to theknown contiguity matrices Wk»), a natural question which arises is whetherthese particular restrictions provide useful conditions for the identification. Ouraim is thus to find under which conditions a GSTARMA models satisfies re­quirement a-bis) and b). The following theorem provides, besides a mere ex­tension of existing results, sufficient conditions derived exclusively forGSTARMA models.

Theorem 3.1. Given the GSTARMA process (1), assuming that it is stable andinvertible and that the maximum time lag orders p and q are given, a necessaryand sufficient condition for its identification is that the following conditions hold:

a) rank[A(zy), M(zy)] = N, 'ilz y: det M(zy} = 0

b) rank[Ap, M q] =N

Furthermore, condition a) is implied by

at) {z: det A(z) = O} n {z: det M(z) = O} = 0;

and condition b) is implied by:

[qJOP) {(qJ/p) (qJ/p)}bl) ~ span ,... , .1JOq 1J1q 1J1q

The proof comprises several propositions.

Proposition 1. Let W(O), W(/), ... , WI), be real matrices NxN; let rank[WO)] =N;

let B(z), NxN polynomial matrix in z, be defined as:

r /

B(z)=LBizi; Bj=LvrkiW(k). i=O•... .r. vrkj E 9\

;=0 k=O

A sufficient condition for B(z) to have full rank in the space of polynomial matri­ces (which from now on will be denoted by (.)) is that:

submatrix of F, obtained by taking the columns of F with prescribed zeros in a certainrow, must be N -1. I

For a GSTARMA model we have: Mo = IN and Aa = L'PkOW(kj, thus condition c) isalways satisfied. k=1

249

G. GUAGNANO . S. TERZI

'Po espan{'PI ., ••• , 'P;J,

where 'Pk. = {'l'ktJ' 'Pk/, ..., 'l'kr}T, k = 0,1• ..., I.

(5)

Proof Let the columns of WO) = [w l • • • •, wN] be a base in 9tN; let wjkl, Biz) and

Bi! }) denote thej-th columns of, respectively, ",,(k),B(z) and B; We have:

N

wjk)=L,am(k;j)Wm, k=O, .... I; j=l•...• Nm=1

The polynomial matrix B(·) will have full rank if and only if:

N

L,f3jBj{e)=0 ~ f3j =0 \:fj =1...., N.j=1

However, since:

(6)

(7)

N N r IN r N

L,f3jBj{Z)=L,f3jL,ZiL, L,lJfkiam(k;j)Wm = L,Zi L,Cm(i)Wm (8)j=1 j=1 ;=0 k=O m=1 ;=0 m=1

where:N I

Cm(i) = L,f3jL,lJfkiam(k;j)' i = O•...• r; m = 1, ... ,Nj=1 k=O

for the principle of identity among polynomials, a linear combination of the col­umns B/z) will give rise to the null vector if and only if:

N

L,Cm(i)Wm = 0, \:fi = O,...,r.m=1

(9)

It remains to prove that, under condition (5), the system of equations (9) has aunique solution /31 =... = /3N =O.

To prove this, consider that W m are linearly independent vectors; thus (9) holdsif and only if em(i) =0, Vm, Vi. Equivalently, (9) holds if and only if:

I N

L, L,f3j a m(k;j)lJfki = 0, m = 1,... ,N; i = O•.•.• r.k=O j=1

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IDENTIFIABILITY CONDITIONS FOR GSTARMA MODELS

This system of N x (r + 1) equations can be rewritten in matrix form as:

±{ff3Pm(k,jl}'l'k. = 0, m =1,... , Nk=O j=1

but also, bearing in mind that according to (6) am(o:j) =om:j (where om:j is Kroneck­er symbol), as:

f3m'PO. +±{±f3P m(k;j)}'Pk. =0, m =1,... N.k=1 j=1

It can immediately be seen that under condition (5) this system has the uniquesolution f31 = ... = f3N = 0 and this proves proposition 1.

Proposition 2. The polynomial matrix M(e) always has rank N.

q

Proof. Referring to proposition 1, let B(z)=M(z)=1N + IMjZj.

Let alsoj=1

I

tJoo = 1, tJkO = 0, k = 1,... ,1, so as to write Mo = IN =I tJkO W(k) e M(z) =k=O

q /

I I tJkjW(k) zj .The sufficient condition for M(e) to have full rank now becomes:j=O k=O

(10)

where: eo ={l, tJOb ... , tJoqf, e k. ={O, tJkl, ... , tJkq(However, since each vector ek. , for k :;C 0 has zero as the first element, it can be

immediately verified that this condition is always met.

Proposition 3. The polynomial matrix [A(e), M(e)] has always rank N.

Proof. It follows immediately from proposition 2.

Proposition 4. A sufficient condition for

rank[A(zv), M(zv)] =N, VZv: det M(zv) =0

is that: {z: det A(z) = O}n {z: det M(z) = O} = 0.

251

(11)

G. GUAGNANO . S. TERZI

Proof Let z. be a root of det M(z) =O. Under condition (11), det A(z) "#0 and thusrank[A(zy), M(zy)] =N. Obviously if A(e) is not full rank, det A(z) =0, \/z E C;thus condition (11) will never be verified.

Proposition 5. A sufficient condition for rank[Ap, Mq] =N is:

(12)

Proof Referring to proposition 1, let W?l and Bj. denote the j-th rows of the ma-I I

trices W(k) and B = [Ap, Mq]. Recalling that Ap =L ({Jkp W(k) and Mq =L 1'JkqW(k)

h~ ~

we ave:

(13)

N I N I

where: cm =L!Jj L({Jkpam(k;j)' dm=L!Jj LtJqam(k;j)' m =1,... ,N.j=1 k=O j=J k=O

Since W m are linearly independent vectors, the linear combination (13) willgive rise to the null vector if and only if Cm =d.; =0, "fm.

However, defining Pk = «({JkP' tJkq)T, we have Cm = d.;= 0 "fm if and only if:

(14)

N

It is easy to see that, under condition (12), the system L !JjB! =0 has the uniquej=1

solution !Jj =0 "fj =1, ..., N. Notice however that for I> 1 the condition is far toorestrictive since it will be met only if the vectors 'l'", k =1, ..., I, are all propor­tional to each other.

ProofofTheorem 3.1. The sufficiency of condition at) has been proved in prop­osition 4. The sufficiency ofbl) has been proved in proposition 5.

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IDENTIFIABILITY CONDITIONS FOR GSTARMA MODELS

Recalling that a necessary and sufficient condition (Hannan, 1969) for theidentification of a VARMA model is that the polynomial matrices A(z) and M(z)are left prime and that rank[Ap, M q] =N, it remains to be proved that

rank[A(zv), M(zv)] = N, "fzv: det M(zv) = 0 <=> rank[A(z), M(z)] = N "fz e C.

Let R = {z: det M(z) = O} (obviously C = R u R). For z eR, these two conditionsare equivalent. For z eR, being M(e) of full rank, it immediately descends thatrank[A(z), M(z)] =N.

4. Concluding remarks

The aim of the paper is to derive conditions for the identification of GSTARMAmodels; these conditions are presented in theorem 3.1 and in proposition 1-5.

In particular, having proved that M(e) is of full rank, a GSTARMA model withgiven orders p and q is identifiable if and only if:

a) rank[A(zv), M(zv)] = N, "fzv: det M(zv) = 0

b) rank[Ap, M q] =N.

Condition a) is necessary and sufficient for A(z) and M(z) to be left prime. How­ever, to check whether this condition is met, or to impose restriction on the pa­rameters CPki and 1'Jkj in order for it to be verified, requires much computational

effort. In fact, it requires the computation of (2;1 minors, for each of the S S Nq

roots {z~, ... , z~} of det M(z) = 0; and this IS pdssible only through numericalcalculus but not analytically.

It could be easier to impose further conditions, no longer necessary, but suffi­cient tqguarantee identifiability. In particular, a sufficient condition for a) is al);thus, imposing rank[A(e)] =N, and seeking for the parametric restrictions whichsatisfy al), namely detA(zJ::;: 0, "fzv; det M(zv) =0, would require the computa­tion of one determinant for each root of det M(z). Proposition I provides as asufficient condition for rank[A(e)] =N the condition:

tPo. ~ span { tPJ , ••• , tPd

where we have defined tPo. ={I, -CPOlt ..•, -CPopf and 4>k. = {CPkOt CPkJ, ••., CPkp}T. Itcan be noted that if in the STARMA model there is no simultaneous spatial ef­fect, that is to say if CPw =0 "fk =1, ... , l, A(e) will always be of full rank.

Alternatively, the same goal could be achieved imposing det A p :;I; O. In fact, ifrank[Ap] =N, the polynomial matrix A(z) is regular, thus its determinant cannot

253

G. GUAGNANO . S. TERZI

be identically zero (Barnett, 1971). Furthermore, since the condition det A,,:F- 0implies rank[A", M q ] =N, no further restrictions are needed for condition b) to beverified.

In both cases, in order for A(z) and M(z) to be left prime, according to condi­tion al), parameteric restrictions can be obtained by solving, with respect to ({Jki

and tJkj, the following system of equations:

z: ;cz~, v« = 1, ..., r(r~Np), \::Iv= 1, ..., S(S ~Nq),

z:: detA(z) = 0; z~: detM(z) = O.

For example, let us consider a simple model where p =q = I = 1, N = 3 and

[0 1 O}

W(J) = 0.5 0 0.5

010

In order to verify condition al) we have to compute det A(zv) for the roots z.:det M(zv) = O. The expression for det M(z) is:

while the expression for det A(z) is:

1- ({J;o - (3({Jo/ - ({Jo/({J;o + 2({J1O({JII )z +

(3({Jg/ + 2({J0l({J1O({J// - ({J;/ )Z2 - (({J~/ - ({Jo/({J;/ )Z3.

First of all, it can be checked that, in this particular example, det A(z) is neveridentically zero; consequently A(e) is always of full rank. However, if we ignoredthis fact, to guarantee the condition we should impose cI>o. ~ span{(/>,.}, that is tosay ({JII ;c O.

In order for condition b) to be satisfied, it is sufficient to impose det A/ ;c 0,which means: ({JOI ;c ± ({JII and ({JOI ;c O. Since A, and M , have a similar structure,analogous conditions for tJOI and tJlI would ensure det M/ ;c O. Alternatively, inview of Proposition 5, this condition is ensured by tJO/({J1I :F- ({JOI tJlI•

Furthermore, the three roots of det M(z) =0 are non linear functions of 1JOI andtJlI and it is very heavy to work leaving all five parameters free. To simplify theexample we can assign numerical values to the moving-average parameters. Forinstance, assigning tJo/ =0.1 and tJlI =0.3, so that det M , :F- 0 and condition b) issatisfied, we find z, =-10, Zz =-2.5 and Z3 =5.

Assuming no contemporaneous spatial effect, so that ({JIO = 0, and solving detA(zv) :F- 0 for ({JII, we find:

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IDENTIFIABILITY CONDITIONS FOR GSTARMA MODELS

q>J/(Z,) =I: :to.1 (1 + IOq>o,) and

q>J/(zz) =I: :to.2 (2 + 5q>o,) and

q>J/(zJ) =I: :to.2 (1 + 5q>o,) and

q>o, =I:-f).I,

q>o, =I: -o.«,q>o, =I: 0.2.

It should be underlined that det A(zv) =0, v = 1, 2, 3 for q>o, = -Oil, -f).4, 0.2,respectively, so that, besides restrictions on q>J/' we need to impose q>o, =I: {-f). 1; ­0.4; 0.2}.

Alternatively, assuming q>1O =I: 0 and solving det A(zv) =I: 0 for q>J/' we find:

q>J/(Z,) =I: ±O.I (1 + lOq>o,) + 0.1q>1O

q>J/(zz) =I:±0.2(2 + 5q>o,) + O.4q>,o·

q>J/(Z3) =I: ±0.2 (1 + 5q>o,) - 0.2q>,o

and

and

and

q>o, =I: -f). 1,

q>o, =I: -o.«,q>o, =I: 0.2.

so that, besides restrictions on q>J/, we need again to impose q>o, =I: {-f). 1; -0.4;0.2}.

Acknowledgements

The authors would like to thank Prof. B.. Bassan who provided helpful discussionand suggestions while preparing the manuscript.

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DI GIACINTO V. (1994), «Su una generalizzazione dei modelIi spazio-temporali autore­gressivi media mobile (STARMAG»>, Atti della XXXVII Riunione Scientifica SIS,Sanremo, aprile 1994, vol. H.

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