RD-A12@ 662 MODEL IDENTIFICATION AND ESTIMATION OF NONGAUSSIRN ARMA i/iPROCESSES(U) CALIFORNIA UNIV SRN DIEGO LA JOLLA DEPT OFMATHEMATICS K LII 82 N88@14-81-K-8883

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0Model Identification and Estimation of NonGaussian ARMA Processes

- By

Keh-Shin Li

University of California, Riverside

Research supported In part by Office of Naval Research Contract100014-81-K-0003

Key words: ARMA model, Identification, Estimation, C-table, Pade

table, Asymptotic8, Bispectrum.

AMS 1970 subject classifications: Primary 62415; Secondary 62G05,62E70.

,TIG'K OCT 2 5 1982

jhdcr.tbgufOs82 10 22 004 AA!A

for publ.ic teloae cnd salet to

ditrlibutlom Is uzzlld.

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Summary

Finite parameter model$of ARMA type have been used extensively in

many applications. Under the usual Gaussian assumption, the second

order analysis will not be able to discriminate among competing models

which give the same correlation structure. In many applications the

innovation process is non-Gaussian. In this case, analysis using

higher order moments will identify- the model uniquely without the

usual invertibility assumption. This in turn will affect the fore-

casting based on the non-Gaussian. model. We present a method which

uses bispectral analysis and the Pade approximation. We show that the

method will consistently identify the order of the ARMA model and

estimate the parameters of the model. One could also deconvolve the

process to estimate the innovation process ,,iich will provide infor-

mation for possible more efficient aximn likelihood estimation of

the parameters. Asymptotic distributions are given, and a few ex-

amples are presented to illustrate the effectiveness of the method.

Je.son For

VT q GRAkI

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iAvvil azid/or4, :6p c1l3

1. Introduction

Finite parameter autoregressive moving average models have been

used extensively in time series modeling, forecasting and control.

Host of the literature is concerned with Gaussian processes. Let ran-

don variables et, t - ...,-1,0,1,... be independent and identically

distributed with mean zero, tat M 0, and variance one Ee 2 - 1. Let

{aj } be a sequence of real constants with

Ea2<rEs <..

Consider the linear process generated by fa and let)

Xt Z a . (.I)lt j.-o3 et-i

The frequency response function is given by

A(e - m jeJ)= . (1.2)

If the process X is normally distributed then its full probabilityt

structure Is completely determined by its spectral density function

fiA) = I y _A 13

Hence the phase information In A(e - ) Is not Identifiable In the

Gaussian case. If A(z) Is a rational function

A(z) - Q (z)/P (s) (1.4)q p

with

q I

Qqlz) - I qz q 0 00

' (1.5)

.Pp(Z =rp~z PO0 -1

1-o

I

-2-

Nthen we say that the process (Xt satisfies a finite parameter auto-

regressive moving average model or simply ARhA(p,q). Usually we writeP (B)X t Q (B)e

(1.6)p t q t

where B is the backshift operator. There are two related problems to

be considered here. The first is to determine the orders of the poly-

nomials P (z) and Q (W). This Is the model identification problem.p q

The second is the problem of estimating the coefficients in P (z) andp

Q (z) after the model is identified. Given a model of the form (1.6),q

most of the literature assumes that P (z) and Q (W) have no root onp q

the unit disk, <~ <1. The condition P (z) * 0 for all I~ 1 is

called the realizability condition so that at has a one-sided infinite

*i moving average representation

X - A(B)e t - Ea e (1.7)tJ O i t-J

with

A(B) - E aJ-0

This is the same as saying a - 0 for all J < 0 in (1.1). The condi-

ti q Q 2<, z 1. called the Invertibility condition, is not

needed for stationarity. If Ix d satisfies (1.6) and Is a Gaussian

process then it Is well known that any real root rj * 0 of P () orp

Q" (z) can be replaced by its inverse r 1 and paired conjugate complex

roots can be replaced by their conjugated inverses rj without' chang-

Ing the correlation structure of (XtJ. This mans that if all the

roots are real and distinct then there are 2p + q different ways to

specify the roots and they are indistinguishable by examining the

autocorrelation function. Since different set of roots correspond to

q " ,' ", ,, .- , - ". '..- ": .. ' ." " ". '" '. . " "".. ." " ... .. ".. . ..

-3-

different set of coefficients, it is customary to assume that all

roots of Pp(z) and Qq( ) are outside the unit circle and to estimate

p q2.the coefficients of Pp(Z) and Q (s) under this condition. We will

present a method that can be applied without imposing the Invertibil-

ity assumption.

There are various procedures In the literature concerning the

identification of the orders p of P (z) and q of Q (z). Most of these

procedures involve the examination of the residuals or estimates of

6 tIs. In doing so, invertibility Is assumed. The distribution of t

is assumed to be Gaussian or a known one so that maximum likelihood

estimation of the coefficients can be carried out. Box and Jenkins

[1976] considered an Iterative procedure by examining the autocorrela-

tion function and partial autocorrelation function. In a series of

papers, Akaike [1969, 1971, 1978 proposed a final prediction error

criteria (FPF), an information criteria (AIC) and a Bayesian version

of it (BIC). These methods are based on multiple decision procedure

and were studied by others. (See Priestley [19811). Hypothesis

testing methods were considered by Godfrey [19791 and Poskitt and

Tremayne [19801. Gray, Kelly and Woodward 119781 considered the S

Array method using a pattern recognition technique. Nore recently

Woodward and Gray 119811 proposed a generalized partial autocorrela-

tion method. Tiso and Tasy [19811 proposed an Iterative regression

approach based on extended sutocorrelation function. Hannan and

Rlissanen [19821 considered a recursive method to identify an ARNAi model.

-4-

Parameter estimation methods have been developed by Hannan

[1969], Box and Jenkins [ 1976], Anderson 11977] and Ansley [1979).

If the process x tI is nonGaussian, LIi and Rosenblatt [19821

proved that, under broad conditions, (1.2) is Identifiable up to a

sign change and/or index shift of the aj'a requiring only that Pp (z)

and Q (z) have no root of absolute value one.q

It has been observed that in many geophysical and economic con-

text that data Is often nonGauselan. In this paper we propose a

method to Identify the orders p and q of the model (1.6) and estimate

the corresponding coefficients without the usual invertibillty assump-

tion. In section 2 we adopt the higher order spectrum method proposed

in Rosenblatt [19801 and Lii and Rosenblatt [19821 to estimate the

aj's In (1.7) and obtain their asymptotic distributions. In section

3 we Introduce C-table and the Pade table and give a method to

Identify the model and to estimate the underlying parameters. Asymp-

totic results are given. Section 4 consists of a few examples and a

discussion.

2. Asymptotics of the higher order spectral method

Let the frequency response function from (1.2) be

1

.e - 2f() explih()) (2.1)

There are many references concerning the estimation of the spectral

*. density function f(A). (See Anderson [19711 or Jenkins and Watts

119681). Lii and Rosenblatt 119821 proposed a method to estimate the

phase information h() when the process Xt (and hence the Innovation

process •t) is nonGaussian. Some basic results from this paper are

summarized in the following lemmas.

' " ".................... "" ......... . -" "'-,, .v, ,.. *. . ,-,

.1* -5-

Lama 2.1. Let {X I be a nonGausslan linear process given In (1.1)t

with the independent random variable {et ) having all moments finite.

Assuming that

IE IJa fand

,'. -i AA(e ) 0 for all I

andh(O) - 0 (2.2)

,! Then the phase h()) in (2.1) is given by

ho)) - hi(A) - Xhi(w)/, + aX (2.3)

where a Is an integer and

S1(1) - (b'u) - h'(0))du (2.4)

with

h'(0) - h'(A) - la 1 (2.5)

A.O

where aO2 is an integer such that C3, the ath order cumulant of (Xtj,

is nonzero and

h(o) +...+ h(_) - h( +...+ A 1)

A C -1 b(A1 ,...,9 _)1 (2.6)ag I AO)I a A-

! th

where b(.) Is the a order cumulant spectral density of the process

{Xtj discussed in BrIllinger and Rosenblatt (1967).

Remark 1. From (1.2) and (2.1) we have

A(.) - Ia. - ,2f(O)I exp.i,(o) •

Since a's are real, we have either Ia > 0 or Ea < 0. The assump-

tion h(O) - 0 in Lemms 2.1 represents an arbitrary choice of the signs

o 4 4 *~4* ~ . . . . . . . . .

L - . 4-

I,'. ':, .- ., " " ". .', ",' '.- .-. '..' -' --. -. ..- .-. . ..- --. -. . - .... -. . ..-. -.° • ,,;. .. . -- - - . .-.-... .. ..•.--. -.. -,- --

K -6-

of a 's. Observing Xt I only, the signs of the aj ' are-intrinsically

undecidable since we can mltiply all a 's and at's by minus one with-

out changing (1.1).

The integer a in (2.3) is Intrinsically undecidable also since it

corresponds to reindexing the Xf'a.Xt

Remark 2. However, in the usual normalization of model (1.6) or (1.7)

we assume so > 0. Under this assumption we can use Theorem 2.1 to be

proved later to ascertain the first nonzero ea and shift the Index and

adjust the sign accordingly. Without loss of generality, in what fol-

lows, we will assume, that u - 3 in order to illustrate the techniques

of the method.

Lema8 2.2. Under the assumptions of Lesa 2.1. An estimate of h1(X)

is, from (2.4 - 2.6),

k-IH(X)-- E arg b (jA,6) (2.7)n J-i-I

where kA - A and it is understood that the bispectral estiuates b (.)n

based on a sample of size n are weighted averages of third order

periodogram values. If b(X,u) e C2 and the weight function W Is

symetric and band limited with band width A, then

Hn(X) - hI(X) - %(X) + o p(H(X) -with

ERn( A) - AG( A) + o(A)eand Cov(R ()),R (U)) - f(O) Ortn( A, v) f2(u) du

W2(u,v)dudv (2.8)

2w 2 mn(AU) W2(u,v)dudv

a 3nC2

for A(n) 0 0, A2n * . as n + -

. - .. -.. - . . . . . . . .

*.... -- - - -. . . . . . ---.. . . ......-. ........... .. .. ,... .-- .. "°.. . .. ". o.. . -

-7-

where G ) is a function involving b(,p). Further ER (A) + h I(I)

and H (A )'s are asymptotically jointly normally distributed with co-n j

variances given by (2.8).

Since

1 2w A~e ix)eJ: ai f- Me )e dX

an estimate as of a Is given byA, ^1 2 w MA

aj 2 I A~-l)et~ dl

1H an(T)-- E (2wfn(Xk)k7 exp{i(H n(X) -' + J") (2.9)

k-i I l

which by symetry can be written as

I2 E (2fnk-i 1n~ -

where 2L - M - 2w/A and "k represent a discretization and f is anIt n

estimate of f(.) similar to that of b(.). For a given sample of size

n, let the bandwidth of the weight function WI in fn(X) be A, and the

bandwidth of the weight function W2 in b(Xp) be A2, we will derive

the asymptotic joint distribution of the aj 's given in (2.9) to the

first order.

It is proved in Brillinger and Rosenblatt (1967) that if for

1-1,2 A + 0 and nA1 . " as n + W, then asymptotically as n + -

fn (X) and ba(X jPi) are independent normally distributed with

var(fnAk)) . f 21) W2 u)du O .

Since Hn is a function of bn, hence Hn and f are asymptotically inde-e nde n

pendent. Let

;,dA f2 (A)Cos(Zn(A )+kX)

withH (W)

Zn(A) - Hn(l) - 2 A

and observe that the asymptotic distribution of the vector (f n(AI)

H l(A), f (X ), H ( ) (w)) is normal with mean (f( h(X).aI n j n j nf(X ), h (A j) h I(w)) and covariance matrix

a A 0 a 0 0

' 0 t 0 t I t T

5 0 a 0 0

0 ta ,l 0 tj ,j tj ,W

0 t 0 t to t 0,t t~

L, i. jW .V

A where

8 s£j Cov(fn(A 1) If(j))

t Cov(H(),Hn( )) with A w

we note that the magnitude of st,1 Is smaller than that of t, . An

application of a multivariate 6-method (see Bishop, Fienberg and

Holland [1975] p. 493) we can show that the asymptotic distribution of

(d Ik dj,m ) is bivarlate normal with mean

1 1

(f 2( AX,)CoS(Z( 1A)+klX) ,f 2 ( A )Cos(Z( A1 )4UA ))

* whereZ(Ax) = hi(x) - Ath (T)/W

and coveriance matrix'.

A1

-9-

C(l,k;L ,k) C(tk;jm)

with

C(l,k;j,) f 2 (A)f 2 (A) Sin(Z(Xt)+kXt)

ndSin(Z(XA )ftm A) (min(A., A~ - Xtjw (.0and

22 2X W 3 2w 2 W(u,v)dudv

Using this and a straightforward calculation, we have the following

theorem.

Theorem 2.1. Under the assumption of Lemma 2.1 w have ( - ak) for

k-I,...,K are asymptotically jointly Gaussian with mans zero and co-

variances given by

? Cv( k~a ) -2(2w)3 fW2(u~vdd L LCov(a, ) - vdu X E C(l,k;j,m) (2.11)

where C(lk;j,m) is given in (2.10)

We will now assume that the stationary process Ix d satisfies

(1.6) with a representation given in (1.7) such that a0 > 0. As

usual, we assume P p(z) and Qq (z) given In (1.5) have no common factor

with P0 - I and q0 > 0. Under these assumptions, equation (2.11) canbe used to estimate the variance of a with f(A) and h (A) estimated

by fa(X) and Hn(X I ) respectively. These results can be used to deter-

mine the smallest integer k such that k * 0. We then reindex the

aj's and change their signs if necessary. This gives a complete pro-

cedure to estimate a 's in equation (1.7) consistently. We use these

estimated A ts to identify and estimate the polynomials P (z) anda pQ (z) by the C-table and the Pads tc .I- as discussed in Lii [19821q

-10-

dealing with a distributed lag model.

3. Asymptotics of the C-table and the Pade approximant

Given a pair of nonnegative integers q and p we denote Pade

rational approximants to a formal power series A(z) - £ a zj by1-0 J

[q/pj - Qq (z)/Pp (z) where Qq (z) and Pp(z) are polynomials of degrees

at most q and p respectively. We assume P (0) - 1 and Q (z) and P (z)P q p

have no common factors. The coefficients of Q (z) and P (z) are

determined by A(z) - (Q )(z)/F ()) - o(z P+q+). The following three

lemmas can be found in Baker [19751.

Lemma 3.1. When it exists, the (q/p] Pade approximant for A(z) is

uniquely determined. Further

a a too aq-p+l q-p+2 q 1

aq-p+2 aq-p+3 a q+2

Qq(z) det a a ... a (3.1)q q q+1 q+p(31

q q qEa z Z E a z ... q a.

jMp i-P J-p-I -ar J=O

and

aq-~l aq-p42 too aq+1

aq-p+2 aq-p+3 q+2

Pp(z) det (3.2)

aq aq+1 aq+p

zp z p - I

where a = 0 if j < 0 and the summation is set to zero if the lower

index on a sum exceeds the upper index.

i

i? -11-

Given nonnegative integers r and s, we define

ar-s+1 ar-s+2 •. ar

C det a a (3.3)• :r,s ar-s+2 ar-s+3 "" r+1

r r+1 r+8-i

5(5-1)

- (-1) = det[(ari.j)i,j a 11

The C-table,which is a doubly infinite array, is defined by

C - (C ) . We further define C -1.rs rs rO

Lemma 3.2. (i) C * 0 implies that [q/p] exists. (ii) Every zeroqP

a

entry in the C-table for a formal power series A(z) - I + E ajzJ-1

occurs in a square block of zero entries and Is completely boarded by

nonzero entries.

Lemma 3.3. Given a formal power series A(z) the following three

conditions are equivalent

£ m

(1) A(z). E c zJ/(1 + E d zj)

(2) [q/pJ A(z) for all q > I and p m

(3) C 0 and C - 0 for all r > I and s > m.

qp r,s

If condition (3) in Lemna 3.3 is satisfied, we call the entry

(1+1, m+1) in the C-table the "breaking point".

Lemmas 3.1, 3.2 and 3.3 lead to the following.

Theorem 3.1. The process JXt given in (1.7) is ARMA (p,q) given in

(1.6) if and only if the C-table associated with A(z) has the breakingI

point (p+l,q+l). Further C * 0 and the coefficients of P (z) and•q ,p p

Q (z) in (1.6) are obtained from (3.1) and (3.2). To normalize theseq

.

-12-

coefficients we divided both P (z) and Qq (z) by C qp so that P- 1.

Whether the roots of Q (z) are inside or outside of the unit circle isq

Immaterial here.

This theorem provides a consistent procedure to identify the

model by determining the orders p and q and to estimate the parameters

of the identified model. We use the a 's obtained from (2.9) to con-

struct estimates Cr,s of C r,s, ascertain the breaking point in

C-table to identify the model and finally by substituting the a 's

for ai's in (3.1) and (3.2) to obtain estimates of the coefficients of

F P(z) and Qq (z). The following lemmas can now be proved with simple

modifications of the proofs given in section 4 of Lit [1982].Lemma 3.4. If Ii.aJKj are asymptotically jointly Gaussian for a fixed

integer K with mean {aj jK.! and covariance matrix X(Cn) where2Cn n 0 6 > 0 as the sample size n then the asymptotic dis-

tribution of R, the determinant of an M*1 matrix,

R - det (aJ-)IjJ with a a ia~1, Is Gaussian with mean

R - det [(a

I4

and variance

O2 . )Gt where Ct is the transpose of C - ( ...,gK) with

g R.4' i j a.

We note that computationally g1 is the sm of the cofactors of ea in

the matrix .[(a1 ,1 ,j..

[HF .

-13-

Theorem 3.2. If the estimates of faJ in (1.7) are given by fajI

obtained from (2.9), then for fixed r and a,

r,s r+i-j ij-1

is asymptotically normally distributed with mean C given In (3.3)

and variance GEG' where C - (g , ) with L - r-s+1,

U - r+s-l, gj = C a nd I is the covariance matrix of ( .Let'a'U)

from (2.11).

This theorem gives a method to construct the C-table and to find

the breaking point (q+l, p+1). If the breaking point can not be

- uniquely determined, the C-table will reduce the number of possible

competing models to only a few for further testing. If the process

IXtI does not have a rational frequency response function [q/p], the

C-table will still suggest a possible ARMA (pq) approximation to A(z)

using the principle of parsimony. Once we have, identified the model

to be ARMA (pq), replacing the a 's by their estimates a 'a in equa-

tions (3.2), we obtain estimates P^ and q of Pa and q respectively

by

Qq(z) q!4+1 lz + ... + 1-A q

Pp(-) j 14z + ... + At P

-" O z + ... + q q (3.4)

1+ ^ Ap p

with* 0

A(3.5)qi I ; I OVl,...,q *

To ottalsn the asymptotic distributions of p 's and ^q's we evaluate

P •

-14-

the determinant In (3.2) by cofactor expansion of the last row. (zP sP-1

( , • , 1) and w obtain

i *z _ 1 "~z 2 (-1)p A z (3.6)p

where I is the cofactor of ZI in (3.2).

Similarly, w have from (3.1)

-q a._pJ),. J.0 i - - pi_

Sa 0Ao + (a1 -aOA) +...+ +...+ (-) PA )qq-p

"BO + it3 + + B3z (3.7)

0 1 q

Using em 3.4 and equations (3.4) - (3.7) we can prove

Theorem 3.3. For fixed p and q, let L - maxfO,q-p+l and U - q+p-1.

Then the asymptotic distributions of ( j-Pj. -P ) - and

(~qj q-ql) are, each bivariate normal with mean (0.0) and covariance

matrices

':, E~PP P ,,ii .j~~ "(Ci.L.U) Z.LU(P.,-U. L~U).. ,LU

d Oi: . LU) L,U( L t t

zij~~. PLU 1,L,Uj q

where''~41**'U~i0,.,

OJ ..U (hL AhL+ 1 $**h U ) J-Op99,q

with oo , o

9 A -, qht B I -L,...,U

and

ZL,U - CoV( L ,P.", ) from (2.11).

A I and Bt are the theoretical values of A and BI respectively in

(3.6) and (3.7). Furthermore, the asymptotic distribution of pI

.and -q are normal with mean zero and variancesi J

1

P" P6 - )0 o ? )

4. Snple andDiscpslo2 q q fo

j 0 3he p o qj 2)P6J

-6

1.

h4. Examples and Discussion

Examples In this section are simulated according to the model of

the form P p(e)X - %q'B*e with

andP (S) * 0 when 1:< I

* The Innovation process are obtained from

* t a t

-16-

where e' are independent, identical, exponentially distributed witht

Sa' - 1 and 2 Var(e') - 1. Hence Et - 0, Var(e t ) - 1. The

sample size for X is 640. Some computational details are discussedt

in Lii and Helland 119811 and Lii and Rosenblatt [19821.

Example 1.

Q (o) - 1 - 0.6B + 0

PI(3) -.1 + 0.6B

AU the roots are outside of unit circle. Table I gives the C-table

associated with this model. Each entry has two numbers, the upper one

is C and the lower one is the estimated standard deviation of Crs rts

computed from Theorem 3.2. We also exhibit table 2 which gives the

* ratio of the C and its estimated standard deviation of each entryrps

in table 1. We call table 2 the "resolution table" of table 1. It is

such easier to recognize the pattern in a resolution table when there

is a sudden drop of resolution at entry (tm) and thereafter (I,=) is

;. likely to be the breaking point. From table 2, it is clear that (3,2)

- is the breaking point and the model is correctly identified as ARMA

(2,1). The Pads approximant 12/11 gives, from (3.4) and (3.5),

Q2 (B) - 1.068 - 0.585 1 + 0.763 3(0.446) (0.212) (0.097)

- andPI(B) - I + 0.594 B

(0.092)

where the numbers In the parentheses are estimated standard deviations

from theorem 103.

Example 2. In this example both roots, -0.5 and -0.75, of

02(B) - I + 3.53 + 332

",~~~~~~~~~~~~~~~~~~~~~. --. , -.. . . ....... . ..', ' . . .- -. .-'" -... _- _ . -. .-. ".•...-i.

-17-

* are inside of the unit circle while the roots of P2 (B) - 1 + 0.35B +

0.5B are outside of unit circle. The associated a-table Is table 3

and Its resolution table Is table 4. It seems reasonable to identify

the model to be ARIA(2,2) with breaking point at (3,3). The Pade

approximant [2/21 gives

Q2(B) = 0.907 + 3.64 B + 2.86 B2(0.94) (13.7) (10.5)

and P2(B) I 1 + 0.55 B + 0.53 B2 (4.1)

(0.60) (0.62)

The large estimated standard deviations in Example 2 may be due

to the complicated formula In Theorem 3.3 and the number of parameters

are large relative to the sample size. Nevertheless, the estimates of

the parameters provide good starting values for possible more effl-

cient Iterative methods. We note that the usual Iterative type of

fitting procedure can be used here. We can deconvolve the process X

ttSand estimate the Innovation process e t by s^t". Diagnostic checking can

be performed on et to discriminate among possible competing models.

The probability distribution or density function of e can be esti-

mated to facilitate a non-Gaussian maxima likelihood estimation. It

seems that in building a finite parameter ARMA model of a stationary

tim series {X t, one should use the procedure suggested In Lii and

Rosenblatt [19821 to deconvolve Xt and see if jetj is near Gaussian or

not. If not, one should use the procedure suggested in this paper to

build the AIMA model without Imposing the invertibility condition.

Alternatively, one may want to use any one of those methods mentioned

in the introduction section, using mainly the second order structure,

to identify the orders of the model; however one should still use the

Pade approximant to estimate necessary coefficients and to identify

p.-.v................................................ ,

77-7-18-

whether the roots lie inside or outside the unit circle. Even in the

Gaussian case, one may want to first fit an HA(K) for a moderate inte-

ger K (say 15). Then following the procedure in section 3, one can

identify the equivalent parsimonious ARMA(p,q) model and obtain estl-

mates of parameters. As a comparison, we employed the usual Box-

Jenkins type estimation procedure as it is implemented in the sub-

routine FTHL of the International Mathematical and Statistical LAbrary

(ISL). Given the right orders in the model, we obtain estimates

Q2(B) - 1.0 + 1.1611 + 0.302752

and (4.2)

?2 (B) - 1.0 + 0.3307B + 0.478932

. with estimated white noise variance o2 - 8.765.

Using Q2(3) in (4.2) to Interprete the model may be quite

different from that of using Q2(3) in (4.1).

Example 2 show that e can discriminate models which are Indis-

tinguishable using only second order properties. The method proposal

in this paper produce estimates that are consistent. For moderate

sample size this method can be a valuable tool for AlMA model identi-

fication and estimation.

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-23-

References

1. Akalke, H. (1969). "Fitting autoregressive model for predic-tion." Annals of the Institute of Statistical Mathematics21, 243-247.

2. Akaike, H. (1973). "Information theory and an extension of themaximum likelihood principle". In "2nd InternationalSymposium on Information Theory", Ed. by B. N. Petrov andF. Caski, 267-281, Akademia Kiado, Budapest.

3. Akalke, H. (1978). "A Bayesian analysis of the minimum AICprocedure." Annals of the Institute of Statistical Mathe-matics 30, 9-14.

4. Anderson, T. W. (1971). "The spectral analysis of time series."John Wiley, New York.

5. Anderson, T. W. (1977). "Estimation for autoregressive movingaverage models in the time and frequency domains." TheAnnals of Statistics 6, 842-865.

6. Ansley, G. F. (1979). "An algorithm for the exact likelihood ofa mixed autoregressive moving average process." Biometrica66, 59-65.

7. Baker, G. A., Jr. (1975). "Essentials of Pade approximations."Academic Press, New York.

8. Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W. (1975)."Discrete miltivariate analysis - theory and practice." TheMIT Press, Cambridge, Massachusetts.

9. Box, G. E. P., and Jenkins, G. H. (1976). "Time series analysis:forecasting and control." Holden-Day, San Francisco.

10. Brillinger, D. R. and Rosenblatt, M. (1967). "Asymptotic theoryof estimates of kth order spectra." In "Spectral Analysisof Time Series", Ed. by B. Harris, 153-188. John Wiley, NewYork.

11. Chatfield, C. (1979). "Inverse autocorrelation." Journal of the

Royal Statistical Society A 142, 363-377.

Fl 12. Godfrey, L. G. (1979). "Testing the adequacy of a time seriesmodel." Biometrica 66, 67-72.K 13. Gray, H. L., Kelley, G. D., and Mclntire, D. D. (1978). "A newapproach to ARMA modeling." Communications in StatisticsB7, 1-77.

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14. Hannan, E. J. (1969). "The estimation of mixed moving averageautoregressive system." Biometrica 56, 579-594.

15. Hannan, E. J. and Rissanen, J. (1982). "Recursive estimation ofmixed autoregressive-moving average order." Biometrica 1,81-94.

16. Jenkins, G. M. and Watts, D. G. (1968). "Spectral analysis andits application." Holden-Day, San Francisco.

17. Lii, K. S. and Helland, H. N. (1981). "Cross-bispectrum computa-tation and variance estimation." ACM Transaction of Mathe-matical Software 7, 284-294.

18. Lii, K. S. and Rosenblatt, H. (1982). "Deconvolution and estima-tion of transfer function phase and coefficients for non-Gaussian linear processes." To appear in The Annals ofStatistics, December.

19. Lii, K. S. (1982). "Model identification in a transfer functionmodel." Technical report No. 95, Statistics Department,University of California, Riverside.

20. Poskitt, D. S. and Tremayne, A. R. (1980). "Testing the specifi-cation of a fitted autoregressive-moving average model."Biometrica 67, 359-363.

21. Priestley, M. B. (1981). "Spectral analysis and time series,Vol. 1." Academic Press, New York.

22. Rosenblatt, M. (1980). "Linear processes and Bispectra."Journal of Applied Probability 17, 265-270.

23. Tiao, G. C., and Tsay, R. S. (1981). "Identification of non-stationary and stationary ARMA models." In Proceedings ofthe Business and Economic Statistics Section. American Sta-tistical Association, Washington, D.C.

24. Woodward, W. A., and Gray, H. L. (1981). "On the relationshipbetween the S array and the Box-Jenkins method of ARMA modelidentification." Journal of the American Statistical Assoc-iation 76, 579-587.

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MODEL IDENTIFICATION AND ESTIMATION OF Research

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19. KEY WORDS (Continue on reverse side If neccesary and Identify by block number)

ARMA model, identification, estimation, C-table, Pade table, Asymptotics,

bispectrum

20. ABSTRACT (Continue on reverse aide If necesary and Identify by block number)

Finite parameter models of ARMA type have been used extensively in many applica-tions. Under the usual Gaussian assumption, the second order analysis will notbe able to discriminate among competing models which give the same correlationstructure. In many applications the innovation process is nonGaussian. In thiscase, analysis using higher order moments will identify the models uniquelywithout the usual invertibility assumption. This in turn will affect the fore-casting based on the nonGaussian model. We present a method which uses bi-spectral analysis and the Pade avvroximi.tion. We show that the (continued)

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20. ABSTRACT (Continued)

method will consistently identify the order of the ARMA model and estimate theparamet ers of the model. One could also deconvolve the process to estimatethe innovation process which will provide information for possible more effi-cient maximum likelihood estimation of the parameters. Asymptotic distributionsare given, and a few examples are presented to illustrate the effectiveness ofthe method.

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