The Properties of Pure Diagonal Bilinear Models

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1 American Journal of Mathematics and Statistics 016, 6(4): DOI: /j.ajms The roerties of ure Diagonal Bilinear Models C. O. Omekara Deartment of Statistics, Michael Okara University of Agriculture, Umuahia, Nigeria Abstract Stationarity, invertibility and covariance structure of ure diagonal bilinear models have been studied in details in this aer. We transformed the ure diagonal bilinear model into the state sace form and subsequently examined the condition under which it is stationary and invertible. We also derived the covariance structure of the ure diagonal bilinear model and showed that for every ure diagonal bilinear rocess there exists an ARMA rocess with identical covariance structure. Keywords Bilinear models, Stationarity, Invertibility and covariance structure 1. Introduction Let {X t } and {e t } be two stochastic rocesses. We assume that e t, is indeendent, identically, distributed with E(e t ) = 0 and E(e t ) = σσ <. A bilinear model is one which is linear in both {X t } and {e t } but not in those variables jointly. Let a 1,a,,a,b 1,b, b q and b ij, 1 i m, 1 j k, be real constants. The general form of a bilinear model is given by Granger and Andersen(1978) as qq X t = ii=1 aa ii XX tt ii + jj =1 CC jj ee tt jj mm,kk + ii=1,jj =1 bb iiii XX tt ii ee tt jj + e t (1.1) The first art on the right hand side of (1) can be identified as the autoregressive art of the rocess X t, the second art as the moving average art and the third art as the ure bilinear art. Following Subba Rao (1981), we denote this model by BL (,q,m,k), where BL is the abbreviation for bilinear. On the other hand if = q = 0 and b ij = 0, for all i j, the model is called ure Diagonal Bilinear Model of order [DBL()] and we write it as X t = ii=1 bb ii XX tt ii ee tt ii + e t (1.) Subba Rao (1981) obtained second order moments of the bilinear model BL(,0,,1), and Subba Rao and Gabr (1984) obtained third order moments of the BL(1,0,1,1) model. Sessay and SubbaRao (1988, 1991) have also shown that for the BL (,0,,1) model, third order moments satisfy Yule-Walker tye difference equation. It is well known that the linear autoregressive moving average model can be written in the form of first order vector difference equation (See Anderson, 1971; riestly, 1978, * Corresonding author: coomekara@live.com (C. O. Omekara) ublished online at htt://journal.saub.org/ajms Coyright 016 Scientific & Academic ublishing. All Rights Reserved 1980) and this Vector form is known as the State Sace Form. It is convenient to study the roerties of a model when it is in the State Sace Form (Akaike, 1974). we ut the ure diagonal bilinear model in the Vectorial form and subsequently examine the conditions under which it is stationary and invertible.. Vectorial Reresentation of the ure Diagonal Bilinear Model A time series {X t } is said to be a diagonal bilinear rocess if it satisfies the difference equation r q X t = j=1 a j X t j + j=1 C j e t j + j=1 b j X t j e e j + e t (.1) where {e t } is a sequence of indeendent and identically distributed random variables with zero mean and variance σσ <. If r = q = 0, the model (.1) is called ure diagonal bilinear (DBL) model and we write it as X t = j=1 b j X t j e e j + e t (.) where {e t } is as defined reviously. The model (.) is denoted by DBL (). Let A x = (.3) bb jj B jx = (.4) ( j = 1,,,) C T 1x = ( 1, 0, 0,, 0, 0) (.5)

2 140 C. O. Omekara: The roerties of ure Diagonal Bilinear Models H T 1x = ( 1, 0, 0,, 0, 0) (.6) X t T = ( X t, X t 1,, X t +1 ) (.7) where T stands for oeration transose of a matrix. We now reresent the model (.) in vectorial form. THEOREM.1 If {X t } satisfies (.) then X t = A X t 1 + j=1 B j X t j e t j + Ce t (.8) is the vectorial form of (.) roof. By direct Verification..1. Stationarity We want to examine the conditions under which a rocess {X t } satisfying (.8) exists. This tye of roblem has been tackled by Bhaskara Rao et al (1983) for the secial class of models satisfying q X t = j=1 a j X t j + j=1 b j X t j e t j + e t (.9) After utting this model in vectorial form, they gave a sufficient condition for the existence of a strictly stationary rocess {X t } satisfying (.9). Earlier, Subba Rao and Gabr (1981) gave a sufficient condition for the existence of a second order stationary rocess {X t } satisfying (.9) with =q. The sufficient conditions in both cases were the same. Subba Rao and Gabr (1981) also obtained the same sufficient conditions for the existence of a second order stationary rocess {X t } satisfying, X t = e t + j=1 a j X t j + b ij X t j e t j i=1,j=1 (.10) Under some conditions involving a,b, and σσ. Akamanam et al (1986) have shown that under some conditions on the sectral radius of a matrix, the rocess {X t } satisfying (.1) do exist, are stationary, ergodic and unique Vectorial Reresentaion Method We now give a set of sufficient conditions under which there is a strictly stationary and ergodic rocess X t, t ɛ Z, satisfying (.8). We use the methods of Akamanam et al (1986). THEOREM. (AKAMANAM, BHASKARA RAO, SUBRAMANYAM, 1986) Let {e t } be sequence of indeendent and identically distributed random variables with zero mean and variance σ <. Let A x, B j x, j = 1,,,, C T 1x, T H 1x and X T be the matrices given by (.3) to (.7) resectively. Γ 1 = A A + (B 1 B 1 )σ Γ j = σ [ B j (A j 1 B AB j 1 + B j ) + (A j 1 B AB j 1 ) + B j ] (for j =,, ) where is the symbol for Kronecker roduct of matrices. Suose all the eigenvalues of the block comanion matrix Γ 1 Γ Γ 1 Γ I Γ 3 3=.. (.11) I 0 have moduli less than unity, where I n stands for the identity matrix of order n x n. Then there exists a strictly stationary and ergodic rocess {X t } conforming to the model (.8). Further, if a rocess {U t } conforms to the above bilinear model (.8), then U t = XX tt roof. See Akamanam et al (1986)..1.. Characteristic Equation Method We use the above theorem to show that a sufficient condition for the existence of the strictly stationary rocess {X t } satisfying (.) is that the roots (in modulus) of the characteristic equation Γ + βγ 1 + β Γ + + β 1 Γ 1 β I X = 0(.1) are in absolute value less than unity. We roceed by considering the following cases. CASE 1 = Γ = Γ 1 Γ I x 0 The eigenvalues of Γ are obtained as follows: Γ 1 ββii Γ Γ βii = I x β I x = 0 Alying the rocedure for obtaining the determinant of a artitioned matrix, we can show that Γ βi x = β x ( Γ 1 βi x ) (Γ (1 β)i x. I x = 0 (See Morrison 1976). Simlifying the left-hand side, we have ββ 1 Γ + ββγ 1 - ββ II = 0 This imlies Γ + ββγ 1 - ββ II = 0, since ββ 1 0 CASE =3 Γ 1 Γ Γ Γ = II II 0 roceeding as in =, we have Γ 1 βι x Γ Γ 3 Γ - β I x = Ι x βι x 0 0 Ι x βι x

3 American Journal of Mathematics and Statistics 016, 6(4): Simlifying, we have This imlies that, Γ - β I x = β x Γ 1 β I x + (1 β)γ + (1 β )Γ 3 = 0 Γ - β I x = β x Γ 3 + β Γ + β Γ 1 - β 3 Γ x = 0 Γ 3 - β Γ + β Γ 1 - β 3 Γ x = 0 since β x 0 Thus, based on the behavior of the above two cases considered, it can be shown in general that Γ - β Γ 1 + β Γ + + β 1 Γ 1 β I x = 0 a sufficient condition for the existence of a strictly stationary rocess {X t } satisfying (.8) is that the roots (in modulus) of the characteristic equation (.1) are in absolute value less than unity. The following examle will illustrate further the work of this section. Let β j = b j σ From case one, Imlies Imlies This imlies that We have shown that It is also easy to show that β β 1 β β 1 β b 1 σ Γ 4xx4 = b 1 b σ b 1 b σ β Γ 1 = β Γ = β 1 β β 1 β Γ + YΓ 1 - Y I = 0 Yβ Y Y Y = 0 0 Y 0 0 Y β + Yβ 1 Y β 1 β Y 0 0 = 0 β 1 β 0 Y 0 Y 0 0 Y Y 4 β + Yβ 1 Y 0 β 1 β Y 0 = 0 Y 4 (-Y ) (β + Yβ 1 Y ) - 0 = 0 Y 6 Y β 1 Y β = 0 Y - Yβ 1 β = 0 since Y 6 0 Γ + YΓ 1 - Y I = 0 imlies Y Yβ 1 β = 0

4 14 C. O. Omekara: The roerties of ure Diagonal Bilinear Models In general Imlies Γ 3 + YΓ - Y Γ 3 Y 3 I = 0 imlies Y 3 Y β 1 Yβ β 3 = 0 Γ + βγ 1 + β Γ + + β 1 Γ 1 β I = 0 Y - Y 1 β 1 Y β Y 3 β 3 β 1 Y - β = 0 An alternative aroach for obtaining the characteristic equation is given below THEOREM.3 Let {e t } be sequence of indeendent and identically distributed random variable with zero mean and E ee tt = σσ <. Suose there exist a stationary and ergodic rocess {X t } Satisfying Let β 1 β 1 β 1 β H x = : Where β j = b j σσ Then e (H) < 1 Where e(h) is the sectral radius of the matrix H. roof: Squaring both sides of (.13) and taking exectations, we obtain Where Let X t = e t + JJ =1 bb jj XX tt jj ee tt jj (.13) (.14) EX t = e t + J=1 β 1 EX t j + C (.15) C = σ (1 + J=1 β 1 + i<jj β i β j ) W T = (EX t, EX t 1, EX t +1 ) R T = (C, 0, 0,, 0, 0)and H is defined in (.14). With this notation, we can write (.13) as the first order difference equation. W t = HW t 1 + R, t = 1,,. Because of stationarity of {X t } we have W t = W t 1 for all t. consequently e(h) < 1. This therefore imlies that the root (in modulus) of the equation Lies inside the unit circle... Invertibility Y - β 1 Y 1 β Y β 1 Y - β = 0 (.16) For a time series to be useful for forecasting uroses, it is necessary that it should be invertible. We do not know of any nice conditions under which the general bilinear autoregressive moving average model is invertible. The invertibility of secial cases of (.1) have been studied by Granger and Anderson (1978), Subba Rao (1981), ham and Tran (1981), Quinn (198) and Iwueze (1988). THEOREM.4 (IWUEZE 1988) Let {e t } be sequence of indeendent and identically distributed random variables with E(e t ) = 0 and EE(ee 4 tt ) <. Then the second order strictly stationary and ergodic rocess {X t } satisfying rr rr XX tt = jj =1 aa jj XX tt jj + (b + jj =1 XX tt qq jj )ee tt qq (.17) for every t is invertible if mm E log b + jj =0 bb jj XX tt jj < 0 Iwueze (1988) established that the resence of autoregressive art makes no imact on the invertibility of his secial case (.17).

5 American Journal of Mathematics and Statistics 016, 6(4): A sufficient condition for invertibility of diagonal bilinear models (.9) have been derived by Guegan and ham (1987). It follows that our ure diagonal bilinear model (.) is invertible..3. Covariance Structure The second order roerties of various forms of the bilinear model have been shown in the literature to be similar to those of some linear time series models. In articular, the second order covariance structure of the bilinear model BL(,0,,1) studied by Subba Rao (1981) is similar to that of an ARMA(,1) model. ham (1985) also arrived at the same conclusion after obtaining a Markovian reresentation of bilinear models. Akamanam (1983) showed that for a secial case of the bilinear rocess, there exists an ARMA rocess with identical covariance structures. In this section, we show that for every ure diagonal bilinear rocess (.), there exists an ARMA rocess with identical covariance structure. We give the covariance function in section Autocovariances of DBL() Model It can be shown that for the model (.), the following are true E(XX tt ) = σσ jj =1 bb jj = μμ (.18) E(XX tt ee tt ) = σσ (.19) E (XX tt ee tt ) = jj =1 bb jj σσ 4 = μμ σσ (.0) E (XX tt ee tt ) = σσ jj =1 bb jj EE XX tt jj ee tt jj + σσ ii<jj bb ii bb jj + 3 σσ 4 = 3 σσ 4 + σσ 6 ii<jj bb ii bb jj E(XX tt ) = jj =1 bb jj EE XX tt jj ee tt jj + σσ 4 ii<jj bb ii bb jj + σσ = σσ + σσ 4 bb ii bb jj We have that the autocovariance function of a stationary rocess {X t } is given by Substituting (.18) and (.) into (.3), we obtain Simlifying, we can show that Where σσ jj =1 bb jj < 1 Now Now, when k But Hence, Now, for k >, we have 1 σσ bb ii<jj + σσ 4 jj =1 bb jj 1 σσ bb, σσ jj =1 bb jj < 1 (.1), σσ jj =1 bb jj < 1 (.) R(K) = E(X t - µ) (X t+k - µ) = E X t X t+k - µ (.3) R (0) = σσ + σσ 4 R (0) = σσ +(1+ σσ 4 ii<jj bb ii bb jj + σσ 4 jj =1 bb jj 1 σσ bb jj =1 bb jj + σσ ii<jj bb ii bb jj )+ jj =1 bb jj σσ 1 σσ bb - bb jj σσ 4 (.4) (.5) X t+k = jj =1 bb jj X t+k-j e t+k-j + e t+k (.6) E(X t X t+k ) = jj =1 bb jj E(X t+k-j e t+k-j X t ) + E(X t e t+k ) (.7) E(X t X t+k ) = b k E(XX tt ee tt ) + jj =1,jj kk bb jj E(X t+k-j e t+k-j X t ) (.8) (X t+k-j e t+k-j X t ) = jj =1 bb jj σσ 4 (.9) E(X t X t+k )= jj =1 bb jj bb kk σσ 4 + jj =1,jj kk bb jj jj =1 bb jj σσ 4 (.30) R(k) = jj =1 bb jj bb kk σσ 4 + jj =1,jj kk bb jj jj =1 bb jj σσ 4 - jj =1 bb jj σσ 4 - ii<jj bb ii bb jj σσ 4, k (.31) E(X t X t+k )= jj =1 bb jj EE(Xt + k j et + k jxt) = jj =1 bb jj σσ 4 - ii<jj bb ii bb jj σσ 4 (.3)

6 144 C. O. Omekara: The roerties of ure Diagonal Bilinear Models Thus, R(K) = jj =1 bb jj σσ 4 - ii<jj bb ii bb jj σσ 4 - ( jj =1 bb jj σσ 4 ii<jj bb ii bb jj σσ 4 ) = 0, K> (.33) R(k)= σ +(1+σ j=1 b j + σ b i b j ) i<jj j=1 b j σ 4 1 σ b j=1 j k = 0 R(K)= j=1 b j b k σ 4 + j=1,j k b j j=1 b j σ 4 j=1 b j σ 4 i<jj b i b j σ 4, k 0 K > (.34) This is similar to the autocovariance function of an ARMA(0,) = MA() model. We see that the autocovariance function of a DBL() rocess like that of a MA() rocess is zero beyond the order of the rocess. In other words, the autocovariance function of a DBL() rocess has a cut off at lag. 4. Conclusions In this aer we reviewed the roerties of ure diagonal bilinear model. We looked at the conditions under which the ure diagonal bilinear model will be stationary and invertible. We derived the autocovariance function of the ure diagonal bilinear model and observed that it was similar to that of a moving model of order. This imlies that to distinguish between a bilinear model and an ARMA model we have to calculate the autocovariance of the squares of the data. REFERENCES [1] AKAIKE H. (1974). Markovian reresentation of stochastic rocess and its alication to the analysis of autoregressive moving average rocesses. Ann. Inst. Stat. Math. 6, [] AKAMANAM, S.I. (1983). Some contributions to the study of Bilinear time series models. Unublished h.d. thesis submitted to the University of Sheffield. [3] AKAMANAM, S.I. BHASKARA RAO, M. AND SUBAMANYAM, K. (1986). On the ergodicity of bilinear time series models. J. time series analysis, 7(3), [4] ANDERSON, T.W. (1971). The statistical analysis of time series. New York and London, Wiley. [5] BHASKARA RAO, M., SUBBA RAO, T., and WALKER, A.M. (1983). On the existence of some bilinear time series models. J. Time series analysis, 4(), [6] GRANGER, C. W. J. and ANDERSON, A.. (1978). An introduction to bilinear time series models. Gottingen: Vandenhock and Rerecht. [7] GUEGAN, D. and HAM, D. T. (1987). Minimality and inversibility of discrete time bilinear models c.r.a.s., series 1, t. 448, [8] IWUEZE, I.S. (1988). On the invertibility of bilinear time series models. To aear is STATISTICA. [9] MORRISSON, D. F. (1976). Multivariate statistical methods, nd ed. Mc Graw Hill, New York. [10] HAM D. T. (1985). Bilinear markovian reresentation and bilinear models. Stochastic rocess Al. 0, [11] HAM D. T. and TRAN, T.L. (1981). On the first Order Bilinear model. J. Al. rob. 18, [1] RIESTLEY, M.B. (1978). Non linearity in time series analysis. The statistician, 7, [13] RIESTLEY, M.B. (1980). System identification, Kalman filtering and Stochastic Control. In: Directions in Time Series. Institute of Mathematics Statistics, U.S.A. [14] QUINN, B. G. (198). Stationarity and invertibility of simle bilinear models. Stochastic rocess Alications, 1, [15] SUBBA RAO, T. (1981). On the theory of bilinear time series. J.R. Stat. Soc., B, 43, [16] SUBBA RAO, T. and GABR, M.M. (1981). The roerties of bilinear time series models and their usefulness in forecasting. University of Manchester Institute of Science and Technology.