Time Series Analysis. Asymptotic Results for Spatial ARMA Models

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1 Communications in Statistics Theory Methods, 35: , 2006 Copyright Taylor & Francis Group, LLC ISSN: print/532-45x online DOI: 0.080/ Time Series Analysis Asymptotic Results for Spatial ARMA Models AUDE ILLIG AND BENOÎT TRUONG-VAN Laboratoire de Statistique et Probabilités, Université Paul Sabatier, Toulouse, France Causal quadrantal-type spatial ARMAp q models with independent identically distributed innovations are considered. In order to select the orders p q of these models estimate their autoregressive parameters, estimators of the autoregressive coefficients, derived from the extended Yule Walker equations are defined. Consistency asymptotic normality are obtained for these estimators. Then, spatial ARMA model identification is considered simulation study is given. Keywords Asymptotic properties; Causality; Estimation; Order selection; Spatial autoregressive moving-average models. Mathematics Subject Classification Primary 62M30, 62M0; Secondary 60F05.. Introduction Unlike the time series autoregressive moving-average ARMA case, several kind of spatial ARMA models may be defined cf. Guyon, 995, Tjøstheim, 978, 983. Most of the different ARMA representations appear to depend on the order chosen on the lattice d For spatial autoregressive AR model having quadrantal representation, Tjøstheim 983 considered Yule Walker least squares LS estimators for the parameters proved the strong consistency of both these estimators the asymptotic normality for the LS estimator, this even when the innovations of these AR models are strong martingale-differences. Ha Newton 993 proved that the Yule Walker estimator considered in Tjøstheim 983 is in fact biased for the asymptotic normality. They gave the explicit expression of this bias for a causal AR rom field indexed by a two-dimensional lattice proposed an new estimator called unbiased Yule Walker estimator that has the asymptotically normal property. Received August 6, 2004; Accepted September 23, 2005 Address correspondence to Aude Illig, Laboratoire de Statistique et Probabilités, UMR 5583, Université Paul Sabatier, 8, Route de Narbonne 3062, Toulouse Cedex 4, France; Aude.Illig@math.ups-tlse.fr 67

2 672 Illig Truong-Van Tjøstheim 983 used a central limit theorem for strong martingale-differences, relative to the partial ordering on d, to obtain the asymptotic normality of the LS estimators. For weaker martingale-differences, defined with the lexicographic order, Huang 992 also obtained a central limit theorem. While using this total ordering, non symmetric half-space ARMA models can be defined. Huang Anh 992 considered such models on a two-dimensional lattice. They gave a method based on an inverse model associated with the original one to estimate the orders the parameters of the model avoiding in this way the estimation of the innovations. Other kind of spatial ARMA process, not depending on the order on d, can also be defined. Etchison et al. 994 introduced two-dimensional lattice separable ARMA models, which are in fact products of two ARMA processes indexed over one-dimensional lattice set. They studied the properties of the sample partial autocorrelation function in order to identify orders in AR models. Furthermore, Shitan Brockwell 995 proposed an asymptotic test of separability using the properties of the autocovariance function of a spatial separable AR model. In this article, we are interested in model order selection in AR parameters estimation of a causal quadrantal ARMA rom fields as defined in Tjøstheim 978 with independent identically distributed innovations. In this purpose, we introduce estimators of the AR coefficients based on a derivation of extended Yule Walker equations. These estimators correspond in fact to an extension to spatial case of the sample generalized partial autocorrelation GPAC function known in time series Woodward Gray, 98. Unlike the one-dimensional lattice case, several estimators of the GPAC function, that are unbiased for asymptotic normality, could be defined. Our estimators differ from those retained in Ha Newton 993. We establish that they are consistent asymptotically normal. For this, we prove in the first sections some asymptotic properties of sample autocovariances sample crosscovariances of two linear rom fields. These results extend those in Choi 2000 for moving-average MA rom fields also generalize the known corresponding results for time series Brockwell Davis, 987. In the last section, we apply some obtained asymptotic results when considering the order selection of a spatial ARMA model the estimation of its AR parameters. A simulation study is also given. 2. Preliminaries Assumptions In this section, we recall some main notations on rom fields indexed over d.in all the following, all the rom fields are indexed over d, with d 2, d is endowed with the usual partial order that is for s = s s d, t = t t d in d, we write s t if for all i = d, s i t i. For a b d, such that a b a b, the following indexing subsets in d, will be considered: Sa b = x d a x b Sa = x d a x Sa b = Sa b\a Sa = Sa \a Let X t t d be a real valued square integrable rom field. Its autocovariance function is defined by u v = X u X u X v X v for u v in d.

3 Asymptotics for Spatial ARMA Models 673 The rom field X t is said to be stationary if X t = m t d u v = u + h v + h u v h d strictly stationary if for all h a b d, X j j Sab X j+h j Sab have the same joint distributions. As u v = u v 0 for all u v d when X t t d is a stationary rom field, it is convenient to redefine the autocovariance function as a function of one argument as follows: h = h 0 Throughout this work, t t d denotes a family of independent identically distributed i.i.d centered rom variables with variance 2 > 0. We say that a rom field X t t d is linear if for any t, X t = j t j j d with j <. Remark 2.. Linear rom fields as defined above are strictly stationary. They are called MA rom fields in Choi For two linear rom fields X t t d Y t t d, we define their crosscovariance function as xy h = X t Y t+h for every h in d. Given two samples X t t S N Y t t S N, we define the sample crosscovariance function as follows: ˆ xy h = N h t+h SN X t Y t+h 2 with N h = i=dn h i where for N \0, we denote by N the element of d whose all components equal N. In the particular case when X t = Y t, ˆ are, respectively denoted for xy ˆ xy. Following Tjøstheim 978 Guyon 995, we say that a rom field X t t d is a spatial ARMAp q with parameters p q d if it is stationary satisfies the following equation X t j S0p j X t j = t + k S0q k t k 3 where j j S0p k k S0q denotes, respectively the autoregressive the moving-average parameters with the convention that 0 = 0 =. If p respectively, q equals 0, the sum over S0p respectively, S0q is supposed to be 0 the process is called an ARp respectively, MAq rom field.

4 674 Illig Truong-Van The ARMA rom field is called causal if it has the following unilateral expansion with j <. X t = j S0 j t j 4 Remark 2.2. Let z = j S0p j z j z = + j S0q j z j where z = z z d. Then, a sufficient condition c.f. Tjøstheim, 978 for the rom field to be causal is that the autoregressive polynomial z has no zeroes in the closure of the open disc D d in d. For the identifiability of model 3, we assume in addition that these two polynomials have no common irreductible factors in the factorial ring z z d. The ARMA order selection procedure that we consider herein is based on extended Yule Walker equations. More precisely, for spatial ARMAp q rom field X t t d with p 0, we define for S0 S0, the coefficients = j j S0 as the solution, when it exists, of the following extended Yule Walker equations, where Y t = X t j S0 j X t j. Y t X t j = 0 j S0 5 If we arrange vectors matrix in the lexicographic order, these equations can be rewritten under the form of the following single matricial equation with for all i j S0 = 6 j i = + j i j = + j Given a sample X t t S N, we could have considered the extended Yule Walker estimator of defined as the solution of the following matricial equation where for i j S0, = j i = R + j i j = R + j

5 Asymptotics for Spatial ARMA Models 675 with for h S0, Rh = t h SN X t X t h As mentioned above, Ha Newton 993 have proved for an ARp rom field, indexed over 2, that the estimator is biased for the asymptotic normality. The bias appears because, in the terms Rh, the summation set t S N t h S epends on h simultaneously, the normalization coefficient does not match to its cardinal. By modifying the normalization coefficient, they proposed then an unbiased estimator called unbiased Yule Walker estimator for which they proved the asymptotic normality. Here we take another approach to obtain unbiased estimators from the extended Yule Walker equations 5. More precisely, we define the estimators ˆ = ˆ j j S0 of the coefficients j j S0 by the following equations t SN Ŷ t X t j = 0 j S0 7 where Ŷ t = X t j S0 ˆ j X t j = +. These equations can also be rewritten under the following matricial form with for all i j S0, j i = ˆ j = ˆ = ˆ 8 t SN t SN X t i X t j X t X t j For the convenience of the reader, we list below assumptions that will be considered. Assumption A. X t t d is a spatial ARMAp q with p 0. Assumption A2. X t d is causal. Assumption A3. t t d is a family of i.i.d centered rom variables with variance 2 > 0. Assumption A4. t 4 = 4 with being some positive constant.

6 676 Illig Truong-Van 3. Some Asymptotic Results for Linear Rom Fields We consider now two linear rom fields X t t d, Y t t d under the form X t = j t j Y t = j t j 9 j d j d with the crosscovariance function xy. We first establish below the consistency of their sample crosscovariance function ˆ xy as defined in 2. Proposition 3.. Let X t t d Y t t d be two linear rom fields as defined in 9. Under Assumptions A3 A4, for all h d, where ˆ xy h xy h N means the convergence in probability when N tends to infinity. N Proof. We first note that xy h = 2 i d i i+h ˆ xy h = i d i j N j d h t+h SN For i = j h, from the weak law of large numbers N h Now, let us prove that for i j h, We have N h t+h SN N h t+h SN t+h SN t i t+h j 2 = N 2 h 2 t i t i t+h j t SN t +h SN t i t+h j N 2 0 N 0 t 2 SN t 2 +h SN t i t +h j t2 i t2 +h j If t t 2 t i t 2 + h j, t i t +h j t2 i t2 +h j = 0 by the independence property. If t t 2 t i = t 2 + h j, t i t +h j t2 i t2 +h j = 2 t i t 2 i t +h j = 0 again by the independence property.

7 Asymptotics for Spatial ARMA Models 677 Thus, N h t+h SN t i t+h j 2 = N 2 h t+h SN 2 t i 2 t+h j Since i j h, we have N h t+h SN t i t+h j 2 = 4 N h The right-h side of the above equality converges to 0 as N tends to so that follows from Chebyshev s inequality. Let us denote for K, Z NK = i S KK j S KK i j N h t+h SN t i t+h j Z N = i d i j N j d h t+h SN t i t+h j From 0, we deduce that Z NK N Z K = i S KK i+h S KK i i+h 2 Moreover, Z K converges to xy h as K tends to. Following Brockwell Davis 987, Proposition 6.3.9, it suffices then to show that lim K lim sup Z N Z NK =0 N But, this is immediately deduced from the absolute summability of j j. Remark 3.. Tjøstheim 983 has shown the consistency of ˆ under martingaledifference assumptions for linear rom fields having unilateral expansions like 4. It could be shown that Proposition 3. also holds under martingale-difference assumptions as in Tjøstheim 983 by using the same proof with only some slight modifications. Choi 2000 proved consistency for various estimators of the autocovariance function of a spatial linear rom field. This result is a special case of Proposition 3. when X t = Y t. We consider again the two linear rom fields X t t d Y t t d denote by x h y h the covariance functions of Y t X t, respectively.

8 678 Illig Truong-Van Proposition 3.2. Let X t t d Y t t d be two linear rom fields as in 9. If Assumptions A3 A4 hold, then for p S0 q S0, with lim Cov N Y t X t q u Y t X t q v = Vu v u v S0p Vu v = 3 yx q u yx q v + y h x h u + v + yx h q v yx h q u h d Proof. Observe first that for i j k l d 4 if i = j = k = l i j k l = 4 4 if i = j k = l if i = k j = l 4 if i = l j = k 0 otherwise. Now for h d, From above, Y t X t q u Y t h q u X t h 2q u v = i d j d k d Y t X t q u Y t h q u X t h 2q u v = 3 4 It follows that Cov l d i j q u k q u h l 2q u v h t i t j t k t l i d i i q u i q u h i 2q u v h + yx q u yx q v + y q + u + h x q + v + h + yx 2q u v h yx h = N 2d = N 2d s SN Y t X t q u Y t X t q v Y t X t q u Y s X s q v yx q u yx q v [ 3 4 i i q u i+s t i q+s t v s SN i d ] + y t s x t s u + v + yx s t q v yx t s q u

9 Asymptotics for Spatial ARMA Models 679 Setting h = t s permutating the two summations yield Cov = N 2d Y t X t q u h i N Y t X t q v [ 3 4 i i q u i h i q h v h S N N i=d i d ] + y h x h u + v + yx h q v yx h q u As a consequence, lim Cov N Y t X t q u Y t X t q v = 3 yx q u yx q v + y h x h u + v h d + yx h q v yx h q u which is exactly Vu v. Remark 3.2. Proposition 3.2 extends to rom fields the well-known results for two time series. More precisely, for the special case d = X t = Y t, Proposition 3.2 corresponds to Proposition 7.3. in Brockwell Davis 987. Furthermore, for rom fields, Lemma 8 in Choi 2000 is a particular case of Proposition 3.2 when Y t = t. 4. Estimation Identification for a Spatial ARMA We are now interested in asymptotic behavior of estimators for the autoregressive parameters of a spatial ARMAp q rom field X t t d with p 0. First the consistency of the autoregressive coefficients estimators is stated in the following proposition. Proposition 4.. If Assumptions A, A2, A3, A4 hold, then for all i j S0, j i + j i N ˆ j + j N Proof. For i S0, j S0, we have t SN X t i X t j = t i SN t j SN X t i X t j R N

10 680 Illig Truong-Van with R N = t I N X t i X t j I N = t S N t i S N t j S N t S N But, #I N #t S N t S N i i where #E denotes the cardinal of a finite subset E of d. Therefore, R N i i x0 N which implies that R N = o. Furthermore, t i SN t j SN X t i X t j = t+i j SN X t X t+i j R 2 N with 0 if i = 0 R 2 N = X t X t+i j if i 0 t I 2 N where IN 2 = t S N t + i j S N t + i S N As for R N, we show that R2 N = o. Consequently, we have t SN X t i X t j = N i j N i j t+i j SN X t X t+i j + o which converges in probability to i j by Proposition 3.. From Eqs. 6 8, the following theorem is established. Theorem 4.. If Assumptions A, A2, A3, A4 hold if for some S0 S0, is invertible, then ˆ N

11 Asymptotics for Spatial ARMA Models 68 Remark 4.. For causal ARp models, Tjøstheim 983 has proved the strong consistency of the LS estimator of the vector of autoregressive coefficients under the assumption that the t are strong martingale-differences strictly stationary. Since, for an ARp model, ˆ coincides with the LS estimator when = p = 0, in the case of i.i.d. innovations, Theorem 4. extends in fact Tjøstheim s result to ARMAp q rom fields. In order to study the asymptotic normality of the estimator ˆ of the vector of coefficients, we consider the S0-indexed vector ˆ whose jth component equals t SN Y t X t j First, we consider truncated rom fields sums over S N. More precisely, for some K, we study the S0-indexed vector with components Y K t X K t j 2 where X K t = j S0K Y K t = X K t j S0 j t j j XK t j which can be rewritten as follows Y K t = j S0K+ K j t j while using the expansion of X K t. According to the Cramér Wold device, the asymptotic normality of the vector whose jth component is defined in 2 amounts to that of the term W NK = Y K t j S0 r j X K t j where r j j S0 are arbitrary vectors of reals. As in Choi 2000, the tool we use to establish the asymptotic normality of the quantities W NK is based on m-dependent rom fields whose definition is recalled below. Definition 4.. For m S0, a rom field Z t t d is m-dependent if it is stationary if for any s t d, Z s Z t are independent when t i s i >m i for at least one i = d.

12 682 Illig Truong-Van For such rom fields, we quote from Choi 2000 the following central limit theorem. Theorem 4.2. Let Z t t be a stationary m-dependent rom field with mean zero autocovariance function. Ifv m = h S mm h <, then lim Var N Z t = v m Z /2 t L 0v m N where L means the convergence in distribution as N tends to infinity. N The following lemma shows that W NK is a sum of N + -dependent rom variables. Lemma 4.. If t t satisfies Assumption A3, then Y K t j S0 r j Xt j K t is a d K + -dependent rom field. Proof. Let Z t = Y K t Xt j K j S0. Then Y K t j S0 r j Xt j K is K + dependent as soon as Z t t d is. t d Observe that Z t is strictly stationary because the t are i.i.d. For h d such that h l l + K, for example h l > l + K, we show that Z t Z t+h are independent. For this, observe that Z t = K i t i k j t k i S0K+ k Sj+K+j+ so that Z t is a function of the t i for i l 0. In the same way, Z t+h = K i+h t i k j+h t k i S hk+ h k S+j hk++j h only depends on the i for i l < 0. The independence of Z t Z t+h follows from that of the t s. Hence Z t is K + -dependent. Let us denote by K xy h the crosscovariance function Yt K Xt+h K by K x h, K y h the covariances of XK t Y K t, respectively. The following lemma gives the asymptotic normality of the truncated term W NK as N tends to infinity. Lemma 4.2. Assume that t t satisfies Assumptions A3 A4. Then /2 W NK W NK L 0v K N

13 Asymptotics for Spatial ARMA Models 683 where v K = r V K r with A denoting the transpose of a matrix A for u v S0, V K u v = 3 K yx uk yx v + K y hk x h u + v + K yx h vk yxh u h d Proof. From Theorem 4.2, it follows that /2 W NK W NK L 0v K N where From Proposition 3.2, v K = lim VarW NK N [ = lim r Var N = r V K r ] Y K t X K t j r j S0 V K u v = 3 K xy uk xy v + K y hk x h u + vk xy h vk xyh u h d for u v S0. The following lemma establishes the asymptotic normality of the quantities W N = Y t j 0 which correspond to the original untruncated terms. r j X t j Lemma 4.3. If Assumptions A, A2, A3, A4 hold, then where for u v S0 Vu v = Proof. From Lemma 4.2, /2 W N L 0r Vr N h d y h x h u + v + yx h v yx h u /2 W NK W NK L W K = 0v K N

14 684 Illig Truong-Van with v K = r V K r. But for u v S0, lim K V Ku v = 3 yx u yx v + h d y h x h u + v + yx h v yx h u Moreover, from Eqs. 5, the first term in the asymptotic variance equals 0. As a L consequence, W K 0r Vr K Now to prove the lemma, we use again Proposition in Brockwell Davis 987 show that for all >0, lim K lim sup /2 W N W NK W NK >= 0 N But according to Chebyshev s inequality, it amounts to prove that for every j S0, lim K lim sup Var N Y t X t j Y K t X K t j = 0 For this, observe that Var = Var 2Cov By Proposition 3.2, lim Var N Y t X t j Y K t X K t j Y t X t j + Var Y t X t j Y t X t j = 3 yx j 2 + Since yx j = 0, the last term equals Vj j. Again by Proposition 3.2, lim Var N Y K t X K t j Y K t X K t j h d y h x h + yx h j yx h j Y K t X K t j = 3 K yx j2 + K y hk x h + K yx h jk yxh j h d

15 Asymptotics for Spatial ARMA Models 685 But, K x h = j j+h 2 j S0K h which converges to x h when K tends to infinity. Furthermore, K y h = j S0 k S0 j k K x h k + j Thus, K y h converges to yh when K tends to infinity. In the same way, it can be shown that K yx h converges to yxh so that finally, lim lim Var K N Using the same arguments as in Proposition 3.2, lim /2 Cov N Y K t X K t j = Vj j Y t X t j K Y t X K t j = 3 yx j yx K j + yy Kh xx Kh h d + yx K h j y xh j K Observing that yx j = 0 taking the limit when K tends to infinity yields lim lim /2 Cov K N Now, from Lemma 4.3, r /2 Y t X t j Y t X L t j 0r Vr N j S0 Y K t X K t j = Vj j for all arbitrary vector of reals r = r j j S0. Hence, from the Cramér Wold device, Proposition 4.2 follows. Proposition 4.2. If Assumptions A, A2, A3, A4 hold, then /2 Y t X L t j 0 V N j S0 The next proposition states the asymptotic normality of the statistics t SN Y t X t j

16 686 Illig Truong-Van Proposition 4.3. Under Assumptions A, A2, A3, A4, here for u v S0, Vu v = N d/2 ˆ L N 0 V h d y h x h u + v + yx h v yx h u Proof. Let us prove that for all j S0, /2 Y t X t j t SN Y t X t j = o By the same calculations as in Proposition 3.2, /2 = \S+N 2 Y t X t j [ 3 4 i i j i+s t i +s t j \S+N s SN\S+N i d ] + yx s t j yx t s j + y t s x t s Now, set h = t s T h = 3 4 i d i i j i h i h j + yx h j yx h j + y h x h Since j < j <, it follows that T h <. Hence, /2 \S+N 2 Y t X t j \S+N i=d i N h d T h h d T h Since the last term converges to 0 when N tends to infinity, the proof then ends. The following theorem is immediately deduced from Proposition 4.3. Theorem 4.3. If Assumptions A, A2, A3, A4 hold if for some S0 S0, is invertible, then with /2 ˆ L N 0 = V as defined in Proposition 4.3. V

17 Asymptotics for Spatial ARMA Models 687 Remark 4.2. The asymptotic normality for the LS estimator of the vector of autoregressive coefficients in the case of a spatial causal ARp model was obtained by Tjøstheim 983. If the innovations are i.i.d., this result is a special case of Theorem 4.3, for = 0 = p. 5. Application Simulation Results We illustrate now an application of some asymptotic results obtained above to identify the following two-dimensional lattice spatial ARMA : X t 08X t t X t t 2 07X t t 2 = t + 05 t t t t t t 2 where t t 2 is a family of i.i.d. normal rom variables with mean 0 variance. 500 replications from the above ARMA model over the rectangle S 500 are simulated the sample mean the sample stard deviation of the ˆ are computed for several values of. These results are summarized in Table below where the values into brackets are the stard deviations the symbol denotes a value greater than 0. The criterion we choose here for identifying spatial ARMA model is based on the estimators ˆ that correspond to a spatial version of the sample GPAC. Indeed, as for the one-dimensional lattice case see Choi, 99, 992; Woodward Gray, 98, we have the following properties that could be deduced from Remark 2.2: pp = p for q q = 0 for p p Table Sample mean sample stard deviation of ˆ over 500 replications \

18 688 Illig Truong-Van Furthermore, from Theorems , we have ˆ N ˆ L n V Applying the properties of the ˆ to the ARMA model considered here, we have i ˆ = ˆ for q =, ii ˆ 0 for p = p. Table in conjunction with both the above properties clearly allows identifying the ARMA model. Indeed, when inspecting the values of the sample stard deviation of the estimators in Table, first, property i is well revealed at the intersection of the 3rd row with the 4th column bold characters, secondly, property ii is well detected at the intersection of the 4th column with the 6th, 7th, 8th rows bold characters. For one replication X t t S 500, the obtained estimates of the AR coefficients are References Brockwell, P. J., Davis, R. A Time Series: Theory Methods. 2nd ed. Springer Verlag. Choi, B. 99. On the asymptotic distribution of the generalized partial autocorrelation function in autoregressive moving-average processes. J. Time Ser. Anal. 2: Choi, B ARMA Model Identification. Springer Verlag. Choi, B On the asymptotic distributions of mean, autocovariance, autocorrelation, crosscovariance impulse response estimators of a stationary multidimensional rom field. Commun. Statist. Theor. Meth. 298: Etchison, T., Pantula, S. G., Brownie, C Partial autocorrelation function for spatial processes. Statist. Probab. Lett. 2:9 9. Guyon, X Rom Fields on a Network. New York: Springer. Ha, E., Newton, H. J The bias of the estimators of causal spatial autoregressive processes. Biometrika 80: Huang, D. W Central limit theorem for two-parameter martingale differences with application to stationary rom fields. Sci. China Ser. A 35: Huang, D. W., Anh, V. V Estimation of spatial ARMA models. Austral. J. Statist. 343: Shitan, M., Brockwell, P. J An asymptotic test for separability of a spatial autoregressive model. Commun. Statist. Theor. Meth. 248: Tjøstheim, D Statistical spatial series modelling. Adv. Appl. Prob. 0: Tjøstheim, D Statistical spatial series modelling II: some further results on unilateral lattice processes. Adv. Appl. Prob. 5: Woodward, W. A., Gray, H. L. 98. On the relationship between the S array the Box-Jenkins method of ARMA model identification. J. Amer. Statist. Assoc. 76: