A New Method for Obtaining the Autocovariance of an ARMA Model_ An Exact Form Solution

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A New Method for Obtaining the Autocovariance of an ARMA Model: An Exact Form Solution M. Karanasos Econometric Theory, Vol. 14, No. 5. (Oct., 1998), pp. 622-640.
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Econometric Theory, 14, 1998, 622-640. Printed in the United States of America.

A NEW METHOD FOR OBTAINING
THE AUTOCQVARIANCE OF
AN ARMA MODEL: AN EXACT
FORM SOLUTION

M. KARANASOS
York University

In this article we present a new method for computing the theoretical autocovariance function of an autoregressive moving average model. The importance of our theorem is that it yields two interesting results: First, a closed-form solution is derived in terms of the roots of the autoregressive polynomial and the parameters of the moving average part. Second, a sufficient condition for the lack of model redundancy is obtained.

1, INTRODUCTION

In this paper we present a new method for computing the theoretical autocovariance function of an autoregressive moving average (ARMA) model. In Section 2 of this paper we give a new method for computing the theoretical autocovariance function of an autoregressive (AR) scheme. Exact methods of calculating the autocovariance for autoregressive models are given by Quenouille (1947) and Pagano (1973). We believe that our method is an improvement over those proposed by Quenouille and Pagano. It is exact, easily coded, and can be used for AR models of all orders. In Section 3 of this paper we give a new method for computing the autocovariance function of the following ARMA scheme:

(From now on, we will assume that a. = 0 for ease of calculation.) McLeod (1975) gives an algorithm for the computation of the autocovariances of the pre; , O,, .. .,O,. In ceding process in terms of the parameters of the model +T, ... ,+ our method we express the autocovariance as an explicit function of the roots of
I am grateful to Marika Karanassou, who introduced me to the identification problem that arises when an AR(2) process is expressed as an AR(1) process with an AR(1) error term, which in turn formed the basis for the proof of Theorems 1 and 2. I also thank the co-editor of Econometric Theory, Katsuto Tanaka, and a referee for a careful reading of earlier drafts, which eliminated many errors and generally improved the quality of the paper. Address correspondence to: M. Karanasos, Department of Economics, YorkUniversity, Heslington, York, YO1 5DD, United Kingdom.

622

O 1998 Cambridge University Press

0266-4666/98 $9.50

AUTOCOVARIANCE OF AN ARMA MODEL

623

the AR polynomial (-Z:=o 4; L', 4; = - 1) and the parameters of the moving average (MA) part O,, 02,...,O,. The only restriction that we impose is that all the roots (complex or real) of the AR polynomial are distinct. Nevertheless, the cases of two, three, and four equal roots are presented in the Appendixes. To point out the importance of our theorem we quote Granger and Newbold (1986): "A specific form for the autocovariance sequence covj(y,) of an ARMA(p,q) model is more difficult to find than it was for the AR and MA models. In principle an expression for covj(y,) can be found by solving the difference equation

for the E(y,-j~,-i), substituting into

to find the g's and then solving the difference equation

for the covj(y,). Because in general the results will be rather complicated, the eventual solution is not presented." The reasons we present our proof are that it is simpler than the one based on solving differences equations and that it has two important consequences:
(a) It is convenient for obtaining a solution in closed form. (b) It provides a sufficient condition for the lack of model redundancy for an ARMA process.'

2. AUTOCOVARIANCE OF AN AR( p)

Let y, be an AR(2) process that is given by

(Without loss of generality, we assume that the variance of E , is 1.) PROPOSITION 1. The autocovariancefunction of the preceding AR(2) process is given by

624

M. KARANASOS

Proof. The AR(2) process can be written as an AR(1) process with an A R ( ~ ) error term (2,):

where zr = ( b 2 ~ t - 1

+ E,. The nth (n r 1) autocovariance of y, is given by

Because c ~ v ( z ~ , z , = - ~4; ) var(zt), we get

From (2.4) and (2.6) we get

Using (2.3) and (2.6) we can write the variance of y, as

From equations (2.7) and (2.8) we have

Important Property. Equation (2.3) is symmetric with respect to and $9. Hence the autocovariance function (2.9) is symmetric with respect to 4, and &. We note that the nth autocovariance of an AR(2) process is a function of the and 42of the second-order polyinverse of the two roots (hereafter i-roots)

AUTOCOVARIANCE OF AN ARMA MODEL

625

nomial(1- 4: L - 4; L2) and can be expressed as the sum of two terms. The first term is the product of the first i-root raised to the power of n (+?) and a coefficient that depends on the two i-roots {4,/[(1- 4 1 4 2 ) ( 4 1- d2)(1 - 4;)I). The second term is the product of the second i-root raised to the power of n (4,") and a coefficient that depends on the two i-roots {qb2/[(1 - 4 2 ) ( 4 2- &)(I - 42)]). Given the symmetry of the autocovariance function in the i-roots of the AR polynomial the coefficients of the two terms are such that if in the first term we interchange and we will get the second one. The preceding form of the autocovariance of an AR(2) process leads us to the first principal result of this paper. Let y, be an AR(p) process that is given by

THEOREM 1. The autocovariancefinction of the preceding AR(p) process is given by

Proof (1st Method). We prove the theorem by induction: In Proposition 1 we proved that it holds for an AR(2) process; if we assume that it holds for an A R ( p - 1) process, then it will be sufficient to prove that it holds for an AR(p). Let y, be an AR(p) process. Then it can be written as anAR(1) process with an AR(p - 1) error term [y, = 4, yr_l + z , where z, is an AR(p - 1) process given by1

Notice that the AR(p) process is symmetric with respect to thep i-roots of the AR polynomial . .,&). Hence, the autocovariance function will also be symmetric with respect to the p i-roots of the AR polynomial. For easy reference we rewrite (2.4) and (2.5):

In equation(2.13), when n = 0, the lower limit exceeds the upper limit of the summation. When this is the case we will say that the summation vanishes. In other words for n = 0 equation (2.13) becomes a tautology.

626

M. KARANASOS

Because z, is an AR(p - 1) process, its autocovariance is given by

Substituting (2.15) into (2.14) we get

Substituting (2.16) into (2.13) we get

where

The autocovariance function is the sum ofp terms. The first term is the product of c$r and a coefficient that depends on $q, 4 2 , ...,q5p,(A).Each of the rest of the p - 1terms is the product of 4; where j = 2,3,. . ., p and a coefficient that depends on #q,. ..,4P,(ejo).Given the symmetry of the autocovariance function in the i-roots of thepth-order AR polynomial, if we interchange the inverse of thej-root 4j with the inverse of the first root 4, in the coefficient ejo we will get the coefficient of 47, that is, coefficient A. Hence, A is given by

Finally, substituting the preceding equation into (2.17) we get equation (2.11).

AUTOCOVARIANCE OF AN ARMA MODEL

627

Proof (2nd Method). The preceding proof does not hold for the variance of y,, because, as we mentioned earlier on, when n = 0, equation (2.13) becomes a tautology. However, there is an alternative proof that holds for the variance as well. The covariance between yt and z t - , is given by

Substituting (2.15) into (2.19) we get

Note that, when n = 0, the first term in the right-hand side of the preceding equation becomes zero, and this is in accordance with the fact that, when n = 0, the lower limit exceeds the upper limit of the summation of the first term in the right-hand side of equation (2.19). The nth (n 2 0) autocovariance of y, is given by

Substituting (2.20) into (2.21), and after some algebra, we get

where

We employ the same reasoning with the one used in the first method (p. 626) to get

Finally, substituting the preceding equation into (2.22) we get equation (2.1 1).

s


628

M. KARANASOS

3. AUTOCOVARIANCE OF AN ARMA(p, q )

Let z, be an ARMA(1, q) process given by

where Oo = - 1 and E,

- i.i.d. (0,a2).2

PROPOSITION 2. The autocovariancefunction of z , is given by

where

Proof. It is not difficult to show that

where

a i = -2

q-i

q-1-i ei+m6?

+

x

q-i-n

C

eken+i+k&.

It should be noted that, for i = q, the preceding double summation vanishes because the lower limit exceeds the upper limit of the summation operator. It can be seen that the variance of z,' is given by

We substitute equations (3.3a) and (3.4) into (3.3) and (after some algebra) we get a equation (3.2). Theorem 1 and Proposition 2 lead us to the second principal result of this paper. Let y, be an ARMA(p,q) process given by

where O0 = - 1 and E, i.i.d.(O, 1).

-

AUTOCOVARIANCE OF AN ARMA MODEL

629

THEOREM 2. The autocovariance function o f the preceding ARMA(p,q) process (y,) is given by

where

and

Note that, for j = q, the third term of the right-hand side of the preceding equation disappears because the lower limit exceeds the upper limit of the first summation operator. Proof (Case i, j r q). We prove the theorem by induction: In Proposition 2 we proved that the theorem holds for an ARMA(1,q) process; if we assume that it holds for an ARMA(p - l,q) process, then it will be sufficient to prove that it holds for an ARMA(p,q) process. Let y, be an ARMA(p,q) process. Then it can be written as an AR(1) process with anARMA(p - 1,q) error term (y, = y,-l + z,), where z, is an ARMA ( p - 1,q) process given by

Notice that the ARMA(p,q) process is symmetric with respect to the i-roots of the pth-order AR polynomial. Hence, the autocovariance will also be symmetric with respect to the i-roots of the AR polynomial. Because z, is anARMA(p - 1,q) process, its autocovariance is given by

630

M. KARANASOS

where

and hi, is given by (3.6b). Note that, when j = q - 1, the second term in the preceding equation vanishes because the lower limit exceeds the upper limit of the summation operator. Moreover, the relation between the 2's is gij = +igi, j - =

... = + ! - l e ^ i l .

In equation (3.8) we expressed the hii coefficients as functions of the hi, coefficients. The covariance of z,and yi-j (for 1 5 j 5 q - 1) is given by

Substituting (3.8) into (3.9) we get

where

and r = 0, for j r q, r = 1, for 1 5 j 5 q - 1. The jth autocovariance of y, is given by

Substituting (3.10) into the preceding equation, and after some algebra, we get

AUTOCOVARIANCE OF AN ARMA MODEL

631

where

The autocovariance of the ARMA(p, q) process is the sum of p terms. The last term is the product of 4: and a coefficient that depends on ..,&,and el,. .. ,Oq. Each of the firstp - 1terms is the product of 4; where i = 2,. .. , p and a coefficient that depends on ..,q$,,el,. ..,Oq(eio hi,). Given the symmetry of the autocovariance function in the i-roots of the AR polynomial, if we interchange the i-root c$i with the first i-root in the coefficient eioh,, we will get the coefficient of 4[, that is, coefficient [. Hence, i is given by

Finally, substituting (3.13) into (3.12) we get (3.6). Proof (Case ii, 1 5 j 5 q - 1, 1st Method). Because z, is an ARMA(p - 1,q) process its autocovariance function is given by

where dinis given by (3.8a), v j jis given by (3.8b), and by
q-j-

and hij,b-are given

=

u=o

c (-eq-.
1

+ I=

,,,-,,) (mi(q-d-2-ij+H
1

- 4;-2j-u

1,

(3.14a)

w h e r e . r r 1 = 1 f o r 0 5 ~ I q - j - l , . r r l = -1foru?q-jand.rr2=1for0ccvs q-j-1,.rr2=Oforv~q-jand25jIq-1,lIbIj-1. In equation (3.14) we express the hiq,hi,j-b, and Ai,j+, as functions of the hi, coefficients. Note that in equations (3.14a) and (3.14b), when v = 0, the second

632

M. KARANASOS

summation term vanishes because the lower limit exceeds the upper limit of the summation operator. The covariance between z, and yt-, is given by

Substituting (3.14) into (3.15), and after some algebra, we get

where

Note that inf,:, when v = j, the first term becomes zero, and when j = q - 1, the second term becomes zero because the lower limit exceeds the upper limit of the corresponding summation operator. The jth autocovariance of y, is given by

Substituting (3.16) into the preceding equation, and after some algebra, we get

where

We employ the same reasoning with the one used for the j r q case (p. 63 1) to get

Finally, substituting the preceding equation into (3.18) we get (3.6).

rn

AUTOCOVARIANCE OF AN ARMA MODEL

633

Proof (Case ii, 0 5 j 5 q - 1, 2nd Method). The covariance between yt and z,-(~+,, is given by

Substituting (3.14) into the preceding equation, and after some algebra, we get

where rrl = 0 for v 5 q -j - 1, T ,= 1otherwise. Note that, forj = v = 0, the first term in equation (3.20) becomes zero. In accordance with the latter, the first four terms in equation (3.21) become zero. After some algebra equation (3.21) gives

where T ,= 1 for u > q - j - 1, T ,= 0, otherwise, and rr2 = 0 fork =j, rr2 = 1, otherwise.

634

M. KARANASOS

The jth autocovariance of y, (0 5 j 5 q - 1) is given by

Substituting (3.22) into the preceding equation, and after some algebra, we get

where

where n-, = 1 fork =j, n-I = 0 otherwise. We employ the same reasoning with the one used in the first method (p. 631) to get

Finally, substituting the preceding equation into (3.24) we get (3.6).
4. ARMA MODEL REDUNDANCY

COROLLARY 1. A sufficient conditionfor the lack of the ARMA(p, q ) model redundancy is given by

where A,, is given b y (3.6b).

AUTOCOVARIANCE OF AN ARMA MODEL

635

Proof. The proof follows immediately from Theorem 2. A Note on ARMA Model Parameter Redundancy. When the first inverse root of the AR polynomial is equal to the first inverse root of the MA polynomial (0;) the order of the autoregressive part reduces from p t o p - 1. In accordance with the latter, the coefficient of the +I in thejth autocovariance, for j r q, (A,,) becomes zero. Moreover, the order of the MA part reduces from q to q - 1. In accordance with the latter, the autocovariance of order q becomes

because

where

and

and the autocovariance of order q - 1 becomes

because

636

M. KARANASOS

Example.
Let y, be an ARMA(2,2) process given by 2 where and 42are the two i-roots where z, = 42zt-1+ E, - el E , - ~ - 0 2 ~ r -and . preceding equation says of the second-order polynomial 1 - q5TL - 4; L ~ The that an ARMA(2,2) process can be expressed as an AR(1) process with an ARMA(1,2) error term (z,). The nth autocovariance of y, (for n 2 2) is given by

The nth autocovariance of y, has two terms. The first is the product of the first i-root raised to the power of n(4Y) and a coefficient that depends on 4 1 , 42, el, and 02. The second term is the product of the second i-root raised to the power of n(@) and a coefficient that also depends on gS1, $3, O,, and 02. These two terms are such that if in the one we interchange the two roots we will get the other. And that is in accordance with the fact that the ARMA(2,2) process (4.3) is symmetric with respect to the two i-roots of the second-order AR polynomial. Note also that, if one of the two i-roots &, k = 1,2 is equal to one of the two i-roots of the MA polynomial [(I - O1 L - 02L2)= (1 - O,* L)(1 - 0; L), -8; 0; = 82, 0; + 0; = O,] (say c$l = Of), then the process reduces to an ARMA(1,l) process and the coefficient of 4: in the autocovariance function becomes zero.

5. CONCLUSIONS

In this paper we showed that an ARMA(p,q) model can be expressed as a sequence of p AR(1) processes with ARMA(p - i,q) (where i = 1,2,. .. , p ) error terms. The first AR(1) process has an ARMA(p - l,q) error term, the second has an ARMA(p - 2,q) error term, . . . , and the last one has a MA(q) error term.

AUTOCOVARIANCE OF AN ARMA MODEL

637

Moreover, we showed that the ARMA(p,q) process is symmetric with respect to the i-roots of the AR polynomial (+,, ...,&). Hence, the autocovariance of the process is symmetric with respect to the i-roots of the AR polynomial. This recursive and symmetric nature of the ARMA(p,q) model led us to find an exact-form solution for the jth autocovariance. We expressed it as an explicit function of the roots of the AR polynomial and the parameters of the MA part. It is the sum of p terms. The ith term is the product of the inverse of the ith root of the AR polynomial, raised to the power of j, times a coefficient that depends on . .,+p,el,. .. ,By.Because of the aforementioned symmetry, if in the coefficient of 4: we interchange +i with cPk we will get the coefficient of 4 : . Furthermore, in our proof the i-roots of the AR polynomial can be either real or complex. The only restriction that we imposed is that all the roots are distinct.
NOTES
1. McLeod (1993) gives a simple condition, expressed in terms of the ARMA model parameters, for determining ARMA model redundancy. He derived the algebraic condition by setting the determinant of the auxiliary matrix of the ARMA model ( J ) equal to zero. 2. Without loss of generality we assume that the variance of 6, is 1. 3. I am grateful to an anonymous referee from Econometric T h e o v for suggesting this method to me.

REFERENCES
Granger, C.W.J. & P. Newbold (1986) Forecasting Economic Time Series, 2nd ed. Academic. McLeod, I. (1975) Derivation of the theoretical autocovariance function of autoregressive-moving average time series. Applied Statistics 24, 255-257. McLeod, I. (1993) A note on ARMA model parameter redundancy. Journal of Time Series Analysis 14,207-208. Pagano, M. (1973) When is an autoregressive scheme stationary? Communications in Statistics 1, 533-544. Quenouille, M.H. (1947) Notes on the calculation of the autocorrelations of linear autoregressive schemes. Biometrlca 34, 365-367.

APPENDIX A: CASE OF TWO EQUAL ROOTS
Proof (1st Method). Let y, be an AR(2) process that is given by

The covariance between zt and y,-, is given by

638

M. KARANASOS

Because zt is an AR(1) process, its autocovariance is given by

Substituting (A.3) into (A.2) we get

From (A.l) we get the variance of y,,

Substituting (A.3) and (A.4) into (AS) we get

The covariance of y, is given by

Substituting (A.4) and (A.6) into the preceding equation we get

Proof (2nd M e t h ~ d ) .Taking ~ the limit of (2.2) as ( 4 2 -+4 1 )we get
COV,(Y,) = lim
d2++1

1

1

4 2 n 4 2
lim
$2-41

(1 - 4 1 42)(41 - 42)
1

= lim
&-+I

(1 - 4 1 & ) ( 1 - 4?)(1 - 4 3

4i'4,(1 - 42) - 42n42(1 - 43

4 1 - 42

This method will be applicable to all multiple roots cases.

AUTOCOVARIANCE OF AN ARMA MODEL

639

APPENDIX B: CASE OF THREE EQUAL ROOTS (AR(3))
Proof. Let y, be anAR(3) process that we express as a sequence of threeAR(1) processes:
YZ=

41~~ +-~I t

,

ZI =

4[Zr-1 +xt,

xt = 4 1 x t - 1

+~

t

.

(B.1)

Substituting (A.8) into (A.2) we get

Substituting (A.8) and (B.2) into (AS) we get the variance of y,,

Substituting equations (B.3) and (B.2) into (A.7) we get the nth autocovariance of y,,

APPENDIX C: CASE OF FOUR EQUAL ROOTS (AR (4))
Proof. Let y, be anAR(4) process that we express as a sequence of f o u r ~ R ( 1processes: )

Substituting equation (B.4) into (A.2) we get the covariance between z,and yr-,,

Substituting equations (C.2) and (B.3) into (AS) we get the variance of y,,

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M. KARANASOS

Substituting equations (C.2) and (C.3) into (A.7) we get the nth autocovariance of y,,


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