LARGE SAMPLE PROPERTIES OF PARAMETER ESTIMATES FOR PERIODIC ARMA MODELS

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1 LARGE SAMPLE PROPERIES OF PARAMEER ESIMAES FOR PERIODIC ARMA MODELS BY I. V. BASAWA and ROBER LUND he University of Georgia First Version received November 1999 Abstract. his paper studies the asymptotic properties of parameter estimates for causal and invertible periodic autoregressive moving-average (PARMA) time series models. A general limit result for PARMA parameter estimates with a moving-average component is derived. he paper presents examples that explicitly identify the limiting covariance matrix for parameter estimates from a general periodic autoregression (PAR), a rst-order periodic moving average (PMA(1)), and the mixed PARMA(1,1) model. Some comparisons and contrasts to univariate and vector autoregressive movingaverage sequences are made. Keywords. Least squares; maximum likelihood; periodic time series; PARMA model; ARMA model; VARMA model. 1. INRODUCION ime series with periodically varying parameters are natural modelling vehicles for series with cyclic autocovariances. Such series arise in climatology (Hannan, 1955; Monin, 1963; Jones and Brelsford, 1967; Bloom eld et al., 1994), economics (Parzen and Pagano, 1979), hydrology (Vecchia, 1985a, 1985b), electrical engineering (Gardner and Franks, 1975) and many other disciplines. Analogous to autoregressive moving-average (ARMA) models and short memory stationary series, periodic autoregressive moving-average (PARMA) models are fundamental periodic time series models (Jones and Brelsford, 1967; Vecchia, 1985a, 1985b; Cipra and lusty, 1987; Lund and Basawa, 2000). In this paper, the asymptotic properties of parameter estimates from a general causal and invertible PARMA model are derived. he results extend those for periodic autoregressions rst proven in Pagano (1978) and routman (1979). he rest of this paper proceeds as follows. In Section 2, a brief overview of PARMA models and their properties is presented. Section 3 establishes consistency and asymptotic normality of the PARMA parameter estimates and identi es the asymptotic covariance matrix for these estimates. Section 4 presents three examples where the asymptotic covariance matrix of the PARMA parameter estimates is explicitly computed in terms of the PARMA model parameters /01/06 651±663 JOURNAL OF IME SERIES ANALYSIS Vol. 22, No. 6 # 2001 Blackwell Publishers Ltd., 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.

2 652 I. V. BASAWA AND R. LUND 2. PARMA MODELS Suppose that fx t g is a time series with nite second moments and that E[X t ] ˆ 0 for all t. We call fx t g a PARMA( p, q) series if it satis es the periodic linear difference equation X n í Xp ö k (í)x n í k ˆ E n í Xq è k (í)e n í k (2:1) In (2.1), the notation X n í denotes the series during the íth season, 1 < í <, of period n. he autoregressive and moving-average model orders are p and q, respectively, and ö 1 (í),..., ö p (í) and è 1 (í),..., è q (í) are the autoregressive and moving-average model coef cients, respectively, during season í. here is no mathematical loss of generality in taking p and q to be constant in the season í (Lund and Basawa, 2000). he errors fe t g are mean zero periodic white noise with var(e n í ) ˆ ó 2. 0 for all seasons í. For convenience, the non-periodic notations fx t g, fe t gfö k (t)g, etc. will be used interchangeably with the periodic notations fx n í g, fe n í g, fö k (í)g, etc. Equation (2.1) has the -variate ARMA representation (Vecchia, 1985a) Ö 0 ~X n Xp Ö k ~X n k ˆ È 0 ~E 0 Xq È k ~E n k (2:2) where f ~X n g and f~e n g are the -variate series ~X n ˆ (X n 1,..., X n )9 and ~E n ˆ (E n 1,..., E n )9. he model orders in (2.2) are p ˆdp=e and q ˆdq=e, where dxe denotes the smallest integer greater than or equal to x. he 3 autoregressive and moving-average coef cients are computed as follows. Ö 0 and È 0 have (i, j)th entries 8 8 < 1 i ˆ j < 1 i ˆ j (Ö 0 ) i, j ˆ 0 i, j (È : 0 ) i, j ˆ 0 i, j (2:3) : ö i j (i) i. j è i j (i) i. j (Ö m ) i, j ˆ ö m i j (i) for 1 < m < p, and (È m ) i, j ˆ è m i j (i) for 1 < m < q. Here, the conventions ö k (í) ˆ 0 for k. p and è k (í) ˆ 0 for k. q are made. he PARMA model in (2.1) is assumed to be causal and invertible in the sense that det@ Ö 0 Xp Ö k z k A 6ˆ 0 and È0 Xq È k z k A 6ˆ 0 (2:4) for all complex z satisfying jzj < 1. Bentarzi and Hallin (1993) give alternative causality and invertibility conditions that are equivalent to (2.4). It is tacitly assumed that the coef cients in (2.2) are identi able in the sense of Reinsel (1997), Section or Deistler et al. (1978).

3 Under causality and invertibility, it is possible to relate fx t g and fe t g through the in nite order moving-average and autoregressive expansions X n í ˆ X1 ø j (í)e n í j (2:5) and E n í ˆ X1 jˆ0 jˆ0 ð j (í)x n í j (2:6) In (2.5) and (2.6), the `seasonal weights' ø k (í) and ð k (í) satisfy X 1 X 1 max jø k (í)j, 1 and max jð k (í)j, 1 (2:7) 1<í< 1<í< kˆ0 kˆ0 he weight sequences fø k (í)g and fð k (í)g can be computed by setting ø 0 (í) ˆ ð 0 (í) ˆ 1 for all seasons í, and calculating values recursively via and PARAMEER ESIMAES FOR PERIODIC ARMA MODELS 653 ø k (í) ˆ è k (í)1 [k<q] ð k (í) ˆ ö k (í)1 [k,q] min(k, X p) jˆ1 min(k, X p) jˆ1 ö j (í)ø k j (í j) k > 1; 1 < í < (2:8) è j (í)ð k j (í j) k > 1; 1 < í < (2:9) he notation used in (2.8), (2.9) and elsewhere interprets ø k ( j) and ð k ( j) for each k > 0, ö k ( j) for 1 < k < p, ó 2 ( j), etc. periodically in j with period. When causal, (2.1) has a unique (in mean square) solution fx t g with a periodic autocovariance structure in the sense that cov(x t, X s ) ˆ cov(x t, X s ) for all integers t and s (Lund and Basawa, 2000). Such series are also called periodically correlated (Gladyshev, 1961; Hurd, 1989), cyclostationary (Gardner and Franks, 1975), and periodically stationary (Monin, 1963). he autocovariance structure of the PARMA model is easily computed from its causal representation. Let ã í (h) ˆ cov(x n í, X n í h ) be the season í autocovariance at lag h > 0. Manipulations with (2.5) give (Lund and Basawa 2000) ã í (h) ˆ X1 ø k h (í)ø k (í h)ó 2 (í k h) (2:10) kˆ0 3. ASYMPOIC PROPERIES OF PARMA PARAMEER ESIMAES his setion studies the asymptotic properties of parameter estimates from a causal and invertible PARMA model. As seen in Section 2, any PARMA model

4 654 I. V. BASAWA AND R. LUND has a -variate ARMA representation. Hence, in principle, the asymptotic properties of the PARMA parameter estimates can be deduced from the wellestablished multivariate ARMA asymptotic results in Dunsmuir and Hannan (1976), Deistler et al. (1978), Hannan and Deistler (1988), LuÈtkepohl (1991) and Reinsel (1997) (among others). Because of this, we will omit lengthy technical arguments in favour of an estimating equation outline. We will work in the univariate PARMA setting rather than transform to a - variate ARMA model via (2.2). here are two primary reasons for this. First, one would have to invert the -variate ARMA transformation to learn about the individual PARMA model coef cients. Results developed directly in terms of the univariate PARMA model would be more readily usable. Second, (2.2) is not in standard multivariate ARMA form as Ö 0 and È 0 are not necessarily the identity matrix. Rescaling the vector noises in (2.2) via ~ç n ˆ È 0 ~E n and then multiplying both sides by Ö 1 0 yields a standard vector ARMA model; however, the covariance matrix of f~ç n g and the MA parameters would then depend on both the PARMA autoregressive parameters and ~ó 2. For these reasons, we will work directly in the PARMA setting. Suppose that X 1,..., X N is a data sample from a causal and invertible PARMA model. he sample contains N full periods of data which are indexed from 0 to N 1. Note that 1 < n í < N when 0 < n < N 1 and 1 < í <. In this section, the large sample properties of the least squares estimates of the PARMA parameters are studied. We assume that fe t g is periodic i.i.d. mean zero noise with a nite fourth moment. For notation, let ~ö(í) ˆ (ö 1 (í),..., ö p (í))9 and è(í) ~ ˆ (è 1 (í),..., è q (í))9 denote the autoregressive and moving-average parameters during season í, respectively. he collection of all PARMA parameters will be denoted by ~á ˆ (~ö(1)9, è(1)9, ~ ~ö(2)9, è(2)9, ~..., ~ö()9, è()9)9. ~ he dimension of ~á is ( p q) 3 1. he white noise variances ~ó 2 ˆ (ó 2 (1),..., ó 2 ())9 will be treated as nuisance parameters. he weighted least squares estimate ^~á LS of ~á is obtained by minimizing the weighted sum of squares S(~á) ˆ XN 1 X íˆ1 ó 2 (í)e n í (~á)2 (3:1) where E t (~á) is determined recursively in t via a truncated version of (2.1): E n í (~á) ˆ X n í Xp ö k (í)x n í k Xq è k (í)e n í k (~á) (3:2) In (3.2), it is understood that E t (~á) ˆ X t ˆ 0 for t < 0. As a matter of notation, we will write E t (~á) to emphasize explicit dependence of E t on ~á (see (2.6) and (2.9)). We will freely interchange E t (~á) and E t (~á) in the asymptotic arguments that follow. hat this interchange does not alter any of the derived asymptotic distributions follows from straightforward modi cations of the proofs of

5 PARAMEER ESIMAES FOR PERIODIC ARMA MODELS 655 heorem in Fuller (1996), or Sections 8.11 and 10.8 of Brockwell and Davis (1991). he estimate ^~á LS is a solution to the ( p q)-dimensional estimating equation X N 1 X ó 2 (í)e n í n í (~á) ˆ ~0 íˆ1 (3:3) In t (~á)= is numerically evaluated by taking partial derivatives in (3.2); see also (3.10) below. Equation (3.3) assumes that ó 2 (í) is knownp for each season í. When ó 2 (í) is unknown it can be replaced by any N - consistent estimate without altering the plimit distribution of ^~áls. Using E[E n í (~á) 4 ], 1, it can be shown that one N -consistent estimate of ó 2 (í) is ^ó 2 (í) ˆ N 1 XN 1 E n í ( ^~á 0 ) 2 (3:4) where ^~á 0 is the ordinary least squares estimate of ~á obtained as a solution to the estimating equation (3.3) that is not weighted for ó 2 (í): X N 1 X E n í n í (~á) ˆ ~0 (3:5) íˆ1 Note that solution of (3.5) does not require a value of ~ó 2. Now set ~S N (~á) ˆ XN 1 ~Z n (~á) (3:6) where ~Z n (~á) ˆ X ó n í (~á) (í)e n í (~á) (3:7) íˆ1 he estimating equation in (3.3) is asymptotically equivalent to ~S N (~á) ˆ ~0. he asymptotic distribution of ^~á LS can be obtained from that of N 1=2 ~S N (~á). o quantify this, we will use the following ergodic result. LEMMA 3.1. Consider a causal and invertible PARMA model with the above assumptions on fe t g. hen the following convergence holds as N!1. (i) N 1 ~S N (~á)! P ~0; (ii) N 1P N 1 ~ Z n (~á)~z n (~á)9! P A(~á, ~ó 2 ) where A(~á, ~ó 2 ) ˆ X íˆ1 ó 2 (í)ã í (~á, ~ó 2 ) (3:8)

6 656 I. V. BASAWA AND R. LUND and " # Ã í (~á, ~ó n í í (~á) 9 ) ˆ E (iii) N 1P N ~Z n (~á)=! P A(~á, ~ó 2 ). (3:9) PROOF. aking partial derivatives in (2.1) n í (~á) ˆ Xp Xq ˆ Xp k (í) X n í k k (í) E n í k(~á) ~e ( p q)(í 1) k X n í k Xq è k n í k(~á) è k n í k(~á) ~e ( p q)(í 1) p k E n í k (~á) (3:10) where ~e j denotes a ( p q) 3 1 unit vector whose entries are all zero except for a one in the jth row. Stability properties of (3.10) can be used to show that f@e t (~á)=g satis es a differentiated version of n í (~á) ˆ k (í, ~á) X n í k (3:11) where max 1<í< P 1 j@ð k(í, ~á)=j, 1. It now follows that f@e t (~á)=g is a strictly stationary (in a periodic sense) mean zero ( p q)-variate series with nite second moments. Causality implies t (~á)= and E t (~á) are independent for each xed t; note that the k ˆ 0 term is absent in (3.11). From (3.11), heorems 6.21 and Proposition 6.32 in Breiman (1968), and the niteness of second moments of f@e t (~á)=g, the law of large numbers relations and XN 1 1 N N 1 XN n í í n í (~á)! P ~0 (3:12) 9! P Ãí (~á, ~ó 2 ) (3:13) as N!1 now follow. Using (3.13) and the independence of E t (~á) t (~á)= for each xed t establishes the result in part (i).

7 From (3.7), (3.11), and causality, one sees that f~z n (~á)g is mean zero strictly stationary white noise with E[~Z n (~á)~z n (~á)9] ˆ A(~á, ~ó 2 ). Hence, part (ii) is the law of large numbers for f~z n (~á)~z n (~á)9g and follows in a similar manner to part (i) using (3.13). o prove part (iii), take a partial derivative in (3.6) and use (3.13) to reduce the task to showing that XN 1 2 E n í (~á) N E n í (~á)! P ~0 (3:14) 9 as N!1. Using independence of E t (~á) 2 E t (~á)=9 for each xed t, we need only show that XN 1 2 E n í (~á) N! P 0 (3:15) 9 as N!1. aking a second derivative in (3.10) and arguing as with the rst derivative shows that f@ 2 E t (~á)=9g is a mean zero ( p q) 3 ( p q) dimensional series satisfying the law of large numbers in (3.15). QED HEOREM 3.1. For a causal and invertible Gaussian PARMA model with the above assumptions on fe t g, min( p, q) > 1, and ó 2 (í). 0 for each season í, as N!1. N 1=2 ( ^~á LS ~á)! D N(~0, A 1 (~á, ~ó 2 )) (3:16) PROOF. Let F n 1 ˆ ó (X n, X n 1,...) and use (3.7) to get " # Ef~Z n (~á)jf n 1 ] ˆ X ó n í (~á) (í)e E n í (~á) F n 1 íˆ1 ˆ X íˆ1 " # ó n í (~á) (í)e[e n í (~á)]e F n 1 ˆ ~0 (3:17) since E n í (~á) is independent of X n, X n 1,... for each season í and E[E n í (~á)] 0. Hence, f~s n (~á)g is a mean zero martingale with respect to ff n g. Using part (ii) of Lemma 3.1, one can verify that the central limit theorem for martingales (Hall and Heyde, 1980, ch. 3) applies to f~s n (~á)g: as N!1. PARAMEER ESIMAES FOR PERIODIC ARMA MODELS 657 N 1=2 ~S N (~á)! D N(~0, A(~á, ~ó 2 )) (3:18)

8 658 I. V. BASAWA AND R. LUND Now consider the rst-order aylor expansion ~S N ( ^~á LS ) ˆ ~S N ~S N (~á) ( ^~á LS ~á) ~R N (3:19) where ~R N denotes the remainder. A straightforward modi cation of the proof of heorem in Fuller (1996) with Lemma 3.1 shows that ^~áls is a consistent estimator of ~á. It can also be shown that ~R N ˆ O P (1) as N!1. Now use ~S N ( ^~á LS ) ˆ ~0 in (3.19) to get N ~S N (~á) N 1=2 ( ^~á LS ~á) ˆ N 1=2 ~S N (~á) o P (1) (3:20) Using part (iii) of Lemma 3.1, equation (3.18), and Slutzky's heorem in equation (3.20) completes our work. QED REMARK 3.1. If fe t g is Gaussian, then the maximum likelihood estimate ^~á ML of ~á has the same asymptotic distribution as the weighted least squares estimate ^~á LS. Lund and Basawa (2000) discuss ef cient computation of ^~á ML. he maximum likelihood estimates of ó 2 (í), denoted by ^ó 2 ML (í) for 1 < í <, have the large sample form ^ó 2 ML XN 1 1 (í) ˆ N E n í ( ^á ML ) 2 (3:21) heorem 3.1 establishes asymptotic normality of PARMA maximum likelihood estimates from Gaussian models. Lund and Basawa (2000) discuss ef cient computation of ^~á ML. REMARK 3.2. In the stationary ARMA setting (ö k (í) ö k, è k (í) è k ó 2 (í) ó 2 ), heorem 3.1 shows that and (N) 1 ( ^~á LS ~á)! D N(~0, ó 2 Ã 1 (~á, ó 2 )) (3:22) as N!1where Ã(~á, ó t (~á) 9 ) ˆ E (3:23) and ~á ˆ (ö 1,..., ö p, è 1,..., è q ). In the ARMA setting, the limiting covariance matrix ó 2 Ã 1 (~á, ó 2 ) does not depend on ó 2 as Ã(~á, ó 2 ) is ó 2 times a function of the AR and MA parameters only. As the examples in the next section show, this factorization will not carry over the PARMA setting.

9 PARAMEER ESIMAES FOR PERIODIC ARMA MODELS EXAMPLES Application of heorem 3.1 requires computation of A(~á, ~ó 2 ) (or à í (~á, ~ó 2 ) for each season í). he rest of this paper addresses this matter with speci c PARMA examples. EXAMPLE 4.1. Consider the causal pth order periodic autoregression (PAR( p)) satisfying X n í Xp ö k (í)x n í k ˆ E n í (4:1) Equation (3.10) reduces n í (~á) ˆ Xp ~e p(í 1) k X n í k (4:2) Multiplying (4.2) by the transpose of itself and taking expectations gives à í (~á, ~ó 2 ) ˆ Xp X p E p(í 1) k, p(í 1) j cov(x n í k, X n í j ) (4:3) jˆ1 where E i, j ˆ ~e i ~e9 j denotes a p 3 p matrix whose entries are all zero except for a one in the ith row and jth column. Using (4.3) in (3.8) reveals the block diagonal form A(~á, ~ó 2 ) ˆ block diag(q 1 (~á, ~ó 2 ),..., Q (~á, ~ó 2 )) (4:4) where Q í (~á, ~ó 2 )isa p 3 p matrix for each season í with (i, j)th entry Q í (~á, ~ó 2 ) i, j ˆ ó 2 (í)cov(x n í i, X n í j ) 1 < i, j < p (4:5) his classical result was rst proven in heorem 3 of Pagano (1978) using spectral-based methods (Hannan, 1970). REMARK 4.1. In the general PARMA setting, A(~á, ~ó 2 ) can depend on the white noise variances. o see this more clearly, consider Example 4.1 with p ˆ 1. Equations (4.4) and (4.5) show that A(~á, ~ó 2 ) is a 3 diagonal matrix with (í, í)th entry A(~á, ~ó 2 ) í,í ˆ ó 2 (í) var(x n í 1 ) 1 < í < (4:6) heorem 1 of Bloom eld, et al. (1994) explicitly computes the PAR(1) periodic variances as! X í var(x n í ) ˆ r 2 ó 2 (k) r 2 X í r 2 k 1 r 2 r 2 ó 2 (k) í r 2 1 < í < (4:7) k where r í ˆ Ð í lˆ1 ö 1(l) for each season í. Causality implies that jr j, 1

10 660 I. V. BASAWA AND R. LUND (Vecchia, 1985a). Combining (4.6) and (4.7) gives an example where A(~á, ~ó 2 ) depends on ó 2 (í) for all seasons í. Speci cally, one obtains 2! 3 s(í 1) X A(~á, ~ó 2 ) í,í ˆ ó 2 (í) r 2 ó 2 (k) r 2 X 4 s(í 1) r 2 k 1 r 2 r 2 ó 2 (k) 5 s(í 1) r 2 k 1 < í < (4:8) where s(í 1) is the season `corresponding' to index í 1: í 1 if 2 < í < s(í 1) ˆ (4:9) if í ˆ 1 EXAMPLE 4.2. Consider an invertible rst-order periodic moving average (PMA(1)) satisfying X n í ˆ E n í è 1 (í)e n í 1 (4:10) Equation (3.10) n í (~á) ˆ è 1 n í 1(~á) ~e í E n í 1 (~á) (4:11) Multiplying both sides of (4.11) by its own transpose, taking expectations, and using independence of E t (~á) t (~á)= for each t gives à í (~á, ~ó 2 ) è 2 1 (í)ã í 1(~á, ~ó 2 ) E í,í ó 2 (í 1) 1 < í < (4:12) with the boundary condition à 0 (~á, ~ó 2 ) ˆ à (~á, ~ó 2 ). It is a tedious but straightforward algebraic matter to use (4.12) times and apply the boundary condition to get! X í à í (~á, ~ó 2 ) ˆ r 2 ó 2 (l 1) r 2 X í r 2 E l,l lˆ1 l 1 r 2 r 2 ó 2 (l 1) í r 2 E l,l 1 < í < lˆ1 l (4:13) where r í ˆ Ð í iˆ1 è 1(i) for each season í. Invertibility implies that jr j, 1. Equation (4.13) identi es à í (~á, ~ó 2 ) as a diagonal matrix with (k, k)th entry " # à í (~á, ~ó 2 ) k,k ˆ r 2 í r 2 ó 2 r 2 (k 1) k 1 r 2 1 [k<í] 1 < k < (4:14) Using (4.14) in (3.8) shows that A(~á, ~ó 2 ) is a diagonal matrix with (k, k)th entry A(~á, ~ó 2 ) k,k ˆ ó 2 (k 1)r 2 X r 2 k (1 r 2 ) lˆ1 r 2 l ó 2 (l) ó 2 (k 1) X r 2 l r 2 k ó lˆk 2 (l) 1 < í < j (4:15)

11 Equation (4.15) explicitly identi es the asymptotic covariance matrix for the PMA(1) parameter estimates, complementing the PAR(1) result in (4.8). REMARK 4.2. In the stationary ARMA setting, the asymptotic covariance matrix for AR and MA parameter estimates has an interchangeable structure (Brockwell and Davis, 1991, Section 8.7). For example, the asymptotic variance of the AR(1) estimate of ö 1 is n 1 (1 ö 2 1 ) and the asymptotic variance of the MA(1) estimate of è 1 is n 1 (1 è 2 1 ) ± merely interchange ö 1 and è 1. Examples 4.1 and 4.2 show that this interchangeability does not hold in the PARMA setting. his can be explicitly seen in the rst-order case by comparing the expressions in (4.8) and (4.15): for a PAR(1), à í (~á, ~ó 2 ) ˆ E í,í Var(X n í 1 ) in (4.3) has one nonzero entry whereas à í (~á, ~ó 2 ) for a PMA(1) in (4.14) has nonzero entries. Our last example considers the mixed PARMA(1,1) model. EXAMPLE 4.3. Consider a causal and invertible rst-order PARMA(1,1) series satisfying X n í ˆ ö 1 (í)x n í 1 E n í è 1 (í)e n í 1 (4:16) hen (3.10) n í (~á) PARAMEER ESIMAES FOR PERIODIC ARMA MODELS 661 ˆ ~e 2í 1 X n í 1 è 1 n í 1(~á) ~e 2í E n í 1 (~á) (4:17) Multiplying both sides of (4.17) by its own transpose, taking an expectation, and simplifying gives the matrix-valued periodic difference equation à í (~á, ~ó 2 ) ˆ è 1 (í) 2 à í 1 (~á, ~ó 2 ) M í (4:18) with the boundary condition à 0 (~á, ~ó 2 ) ˆ à (~á, ~ó 2 ). In (4.18), M í ˆ ó 2 (í 1)[E 2í,2í 1 E 2í 1,2í E 2í,2í ] var(x n í 1 )E 2í 1,2í 1 (4:19) In obtaining (4.18), we have used n í 1(~á) he solution to (4.18) is à í (~á, ~ó 2 ) ˆ r 2 è,í X n í 1 X í M k r 2 è,k E[E n í 1(~á)X n í 1 ] [ó 2 (í 1)] ˆ ~0 (4:20) r 2 è, 1 r 2 è,! X r 2 M k è,í r 2 è,k (4:21)

12 662 I. V. BASAWA AND R. LUND where r è,í ˆ Ðíˆ1 è 1(í). Invertibility of the model implies that jr è, j, 1. he information matrix A(~á, ~ó 2 ) is easily obtained by using (4.21) in (3.8); var(x n í ) is computed explicitly in Lund and Basawa (2000) as! X í var(x n í ) ˆ r 2 ä(k) r 2 X ö, ö,í r 2 ö,k 1 r 2 r 2 ä(k) ö,í ö, r 2 (4:22) ö,k where ä(í) ˆ ó 2 (í) è 1 (í)ó 2 (í 1) 2ö 1 (í)è 1 (í)ó 2 (í 1) (4:23) and r ö, í ˆ Ð9 íˆ1 ö,(í). Causality of the model implies that jr ö, j, 1. ACKNOWLEDGEMENS Robert Lund's research was supported by NSF Grant DMS he comments of two referees greatly improved this paper. REFERENCES BENARZI, M. and HALLIN, M. (1993) On the invertibility of periodic moving-average models. Journal of ime Series Analysis 15, 263±8. BLOOMFIELD, P., HURD, H. L. and LUND, R. B. (1994) Periodic correlation in stratospheric ozone data. Journal of ime Series Analysis 15, 127±50. BREIMAN, L. (1968) Probability. Reading, MA: Addison-Wesley. BROCKWELL, P. J. and DAVIS, R. A. (1991) ime Series: heory and Methods (2nd edn). New York: Springer Verlag. CIPRA,. and LUSY, P. (1987) Estimation in multiple autoregressive-moving average models using periodicity. Journal of ime Series Analysis 8, 293±300. DEISLER, M., DUNSMUIR, W. and HANNAN, E. J. (1978) Vector linear time series models: corrections and extensions. Advances in Applied Probability 10, 360±72. DUNSMUIR, W. and HANNAN, E. J. (1976) Vector linear time series models. Advances in Applied Probability 8, 339±64. FULLER, W. A. (1996) Introduction to Statistical ime Series (2nd edn). New York: John Wiley and Sons. GARDNER, W. and FRANKS, L. E. (1975). Characterization of cyclostationary random signal processes. IEEE ransactions on Information heory 21, 4±14. GLADYSHEV, E. G. (1961) Periodically correlated random sequences. Soviet Math 2, 385±8. HALL, P. and HEYDE, C. C. (1980) Martingale Limit heory and its Applications. New York: Academic Press. HANNAN, E. J. (1955) A test for singularities in Sydney rainfall. Australian Journal of Physics 8, 289±97. б (1970) Multiple ime Series. New York: John Wiley and Sons. б and DEISLER, M. (1988) he Statistical heory of Linear Systems. New York: Wiley. HURD, H. L. (1989) Representation of strongly harmonizable periodically correlated processes and their covariances. Journal of Multivariate Analysis 29, 53±67. JONES, R. H. and BRELSFORD, W. M. (1967) imes series with periodic structure. Biometrika 54, 403±8. LUND, R. B. and BASAWA, I. V. (2000) Recursive prediction and likelihood evaluation for periodic ARMA models. Journal of ime Series Analysis 20, 75±93. LUÈ KEPOHL, H. (1991) Introduction to Multiple ime Series Analysis. Berlin: Springer-Verlag.

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