DYNAMIC ECONOMETRIC MODELS vol NICHOLAS COPERNICUS UNIVERSITY - TORUŃ Józef Stawicki and Joanna Górka Nicholas Copernicus University

Transcription

1 DYNAMIC ECONOMETRIC MODELS vol.. - NICHOLAS COPERNICUS UNIVERSITY - TORUŃ 996 Józef Sawicki and Joanna Górka Nicholas Copernicus Universiy ARMA represenaion for a sum of auoregressive processes In he ime analysis essenially hree represenaions are considered: an ARMA represenaion, a represenaion in he sae space and a specral represenaion. The ARMA represenaion is he mos common mus frequenly one and is very ofen applied. The special cases of his represenaion, i. e., AR and MA models are very ofen used in he modeling of economic processes is no. The ARMA models is creaed as fundamenal in building of forecass in so called he Box-Jenkins mehod. In his aricle some problems of he ARMA represenaion will be presened. form Definiion. A saionary mulidimensional process Y of he Y = H ε, Eε = 0, Vε = Ω () j j j= 0 has he ARMA(p,q) represenaion, only if i can be wrien in he form of differenia equaion where Φ( LY ) = Θ( L) ε, ()

2 p Φ( L) = Φ + Φ L Φ L, Φ =, Φ 0 0 p 0 q Θ( L) = Θ + Θ L Θ L, Θ =, Θ 0 0 q 0 p q (3) where Φ( z) = 0 and Θ( z) = 0 have all he roos ouside he uni circle. The ARMA represenaion is no univocal. This problem is illusraed below on an examples. and where Example. Le Φ( LY ) = Θ( L) ε, Φ ( LY ) = Θ ( L) ε, (4) * * * * Φ ( L) = AL ( ) Φ( L), Θ ( L) = AL ( ) Θ( L) (5) be a represenaion of he same sochasic process. Example. The process MA() Y ε ε = Θ (6) can be wrien in he form If Θ ( + Θ LY ) = ( Θ L ) ε (7) = 0 hen he process MA() may be presened in he form of AR(). I is possible, because he marix equaion Θ = 0 has a soluion. Taking ino accoun he above a minimal represenaions of ARMA models should be considered. Definiion. A saionary process Y has a minimal represenaion of ARMA ype, if polynomials Φ( L) and Θ( L ) do no have common roos. Below he imporan heorem of minimal represenaion is quoed according o Beguin, Gourieroux, Monfor (980). This heorem is reaed as a pracical way in verifying he models of sochasic processes and in idenifying he ime series as well.

3 Theorem. A saionary process Y has a minimal represenaion of ARMA ype if and only if where (,) ij= 0 i p j q ( pq, ) 0 (8) ( p, q) 0, (i,j)=dea(i,j) (9) A(i,j) is he marix of auocovariance coefficien γ(h). γ( j+ ) γ( j+ )... γ( j+ i+ ) () j ( j )... ( j i) Aij (,) = γ γ + γ γ( j+ i) γ( j+ i)... γ( j+ ) (0) On he ground of (i,j) values he array (so-called C-array) is buil, which is used o deermine he orders p and q of an ARMA model. If a process has he minimal ARMA represenaion, hen he C-array has he form: Table. i\j 0... q- q q x x x... x x x... x x x... x x x... x x x... x x x p- x x x... x x x x p x x x... x p+ x x x... x x where 0 means zero value of deerminan of he marix A(i,j), where x means non-zero value of deerminan of he marix A(i,j).

4 The mehod presened above allows o idenify a minimal ARMA represenaion for he sum of processes. I refers in paricular o he processes hose sum is general of he ARMA ype of process. This fac was discussed in Granger, Morris (976). Theorem. If he (nx) saionary Y process has he ARMA represenaion Φ( LY ) = Θ( L) ε and Ψ( L) is he (m,n) marix of polynomial, he process Ψ * = Ψ( LY ) has an ARMA represenaion oo. This heorem (see Gourieroux, Monfor (990)) is a simple conclusion derived from he heorem saying ha a process has an ARMA represenaion, when he specral densiy funcion is a funcion wih respec o e iω. In he heorem is no included a way of deermining he represenaion of process Y *. In special cases i is possible, bu in general i is very difficul. Example 3. If Y and Y are of he MA() ype of processes Y = ( Ξ L) ε Vε = σ Y = ( Ξ L) ε Vε = σ () and for all and τ cov( ε, ε τ ) = 0, hen process Y = Y + Y () is also of he ype MA(). In his case he parameers of his process may be deermined (derived), i. e. coefficien of process and a variance of whie noise creae he process Y. If Y and Y are auoregressive processes of he form ( Φ LY ) = ε ( Φ LY ) = ε (3) hen he process Y = Y + Y (4)

5 is of he ARMA(,) ype. I can be wrien in he form where Φ( LY ) = ( θl) ε (5) Φ( L) = ( Φ + Φ ) L+ ΦΦL (6) and Vε may be derived from a soluion of he equa- he parameer θ ions σ ( + θ ) = σ ( + θ ) + σ ( + θ ) θσ = θσ + θσ (7) The process Φ( LY ) = ( θl) ε can be idenified as he ARMA(,0) process, if he condiion min { θ Φ, θ Φ } < K is saisfied (value K is close o zero). The process Φ( LY ) = ( θl) ε can be idenified as he ARMA(,0) process, if θ<k (value K is close o zero). The able presens he orders p and q of he ARMA(p,q) process for a sum of AR() processes wih indicaed parameers Φ and Φ. The assumpion was arbirary made ha K = K =.. Table. 0 Φ /Φ (,0) (,0) (,0) (,0) (,) (,) (,) (,) (,) (,) -0.8 (,0) (,0) (,0) (,0) (,) (,) (,) (,) (,) (,) -0.7 (,0) (,0) (,0) (,0) (,0) (,) (,) (,) (,) (,) -0.6 (,0) (,0) (,0) (,0) (,0) (,0) (,) (,) (,) (,) -0.5 (,) (,) (,0) (,0) (,0) (,0) (,0) (,) (,) (,) -0.4 (,) (,) (,) (,0) (,0) (,0) (,0) (,0) (,) (,) -0.3 (,) (,) (,) (,) (,0) (,0) (,0) (,0) (,0) (,) -0. (,) (,) (,) (,) (,) (,0) (,0) (,0) (,0) (,0) -0. (,) (,) (,) (,) (,) (,) (,0) (,0) (,0) (,0) 0 (,) (,) (,) (,) (,) (,) (,) (,0) (,0) (,0) 0. (,) (,) (,) (,) (,) (,) (,) (,0) (,0) (,0) 0. (,) (,) (,) (,) (,) (,) (,0) (,0) (,0) (,0) 0.3 (,) (,) (,) (,) (,) (,0) (,0) (,0) (,) (,)

6 0.4 (,) (,) (,) (,) (,0) (,0) (,0) (,) (,) (,) 0.5 (,) (,) (,) (,0) (,0) (,0) (,) (,) (,) (,) 0.6 (,) (,) (,0) (,0) (,0) (,) (,) (,) (,) (,) 0.7 (,) (,0) (,0) (,0) (,) (,) (,) (,) (,) (,) 0.8 (,0) (,0) (,0) (,) (,) (,) (,) (,) (,) (,) 0.9 (,0) (,0) (,) (,) (,) (,) (,) (,) (,) (,) To evaluae he ARMA (p,q) represenaion for sum of auoregressive of he AR() ype he heorem was used. For he process Y he auocovariance funcion can be found. For he sake of simpliciy i was assumed ha σ = σ =. The auocovariance funcion for considered processes may expressed as follows: γ γ i ( 0) = ( i =,) Φ i ( h) = Φ γ ( h ) ( i =,) i i i (8) Assuming, ha cov( Y, Yτ) = 0 for all and τ, he auocovariances funcions componen-processes. Example 4. Le Φ = 07. and Φ = 03.. Then he C-array has he form Table 3. p/q The analysis of above array leads o he saemen, ha he sum of AR() processes wih indicaed parameers Φ and Φ is he ARMA(,) ype of process.

7 In he same way, as in he example 4, he orders p and q mere deermined for he ARMA represenaion of a sum of auoregressive processes assuming differen welficiens. The resuls are abulaed below. Table 4. Φ /Φ (,0) (,) (,) (,) (,) (,) (,) (,) (,) -0.8 (,) (,0) (,) (,) (,) (,) (,) (,) (,) -0.7 (,) (,) (,0) (,) (,) (,) (,) (,) (,) -0.6 (,) (,) (,) (,0) (,) (,) (,) (,) (,) -0.5 (,) (,) (,) (,) (,0) (,) (,) (,) (,) -0.4 (,) (,) (,) (,) (,) (,0) (,) (,) (,) -0.3 (,) (,) (,) (,) (,) (,) (,0) (,) (,) -0. (,) (,) (,) (,) (,) (,) (,) (,0) (,0) -0. (,) (,) (,) (,) (,) (,) (,) (,0) (0,0) 0 (,) (,) (,) (,) (,) (,) (,0) (,0) (,0) 0. (,) (,) (,) (,) (,) (,) (,0) (,0) (,0) 0. (,) (,) (,) (,) (,) (,0) (,0) (,0) (,0) 0.3 (,) (,) (,) (,) (,0) (,0) (,0) (,0) (,0) 0.4 (,) (,) (,) (,0) (,0) (,0) (,0) (,0) (,) 0.5 (,) (,) (,) (,0) (,0) (,0) (,0) (,) (,) 0.6 (,) (,) (,0) (,0) (,0) (,0) (,) (,) (,) 0.7 (,) (,0) (,0) (,0) (,) (,) (,) (,) (,) 0.8 (,) (,0) (,0) (,) (,) (,) (,) (,) (,) 0.9 (,0) (,) (,) (,) (,) (,) (,) (,) (,) I seen, ha in special cases a sum of auoregressive processes is a auoregressive process. However, his process has in general be idenified as he ARMA(,) process. The following ables were consruced nucler he assumpion abou he correlaion of componen-processes. The orders p and q of he ARMA(p,q) represenaion for differen value of covariance funcion and coefficiens Φ and Φ are presened below.

8 I is no possible o creae he process Y = Y + Y for all values of Φ and Φ covariance funcion cov( Y, Y ). These parameers and covariance funcion should saisfy he following equaion cov( ε, ε) cov( Y, Y ) = Φ Φ. To deermine cov( Y, Y ) he covariance cov( ε, ε ) should be known. For example covariance cov( Y, Y ) =0.8 may be deermined for saisfying he condiion Φ Φ 05.. Table 5. The orders p and q of he ARMA process, when cov( Y, Y ) =. for = τ τ 08 Φ /Φ (,) (,) -0.8 (,) (,) (,) -0.7 (,) (,) (,) -0.6 (,) (,) (,) (,) -0.5 (,) (,) (,) (,) (,) -0.4 (,) (,) (,) (,) (,) (,) -0.3 (,) (,) (,) (,) (,) (,) (,) (,) -0. (,) (,) (,) (,) (,) (,) (,) (0,) (0,) -0. (,) (,) (,) (,) (,) (,) (,) (0,) (0,0) 0. (,) (,) (,) (,) (,) (,) (,) (0,) (,0) 0. (,) (,) (,) (,) (,) (,) (,) (,0) (0,) 0.3 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.4 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.5 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.6 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.7 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.8 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.9 (,) (,) (,) (,) (,) (,) (,) (,) (,)

9 Table 6. The orders p and q of he ARMA process, when cov( Y, Y ) =. for = τ τ 04 Φ /Φ (,) (,) (,) (,) (,) (,) (,) (,) (,) -0.8 (,) (,) (,) (,) (,) (,) (,) (,) (,) -0.7 (,) (,) (,) (,) (,) (,) (,) (,) (,) -0.6 (,) (,) (,) (,) (,) (,) (,) (,) (,) -0.5 (,) (,) (,) (,) (,) (,) (,) (,) (,) -0.4 (,) (,) (,) (,) (,) (,) (,) (,) (,) -0.3 (,) (,) (,) (,) (,) (,) (,) (,) (,) -0. (,) (,) (,) (,) (,) (,) (,) (0,) (0,) -0. (,) (,) (,) (,) (,) (,) (,) (0,) (0,0) 0. (,) (,) (,) (,) (,) (,) (,0) (,0) (,0) 0. (,) (,) (,) (,) (,) (,) (,0) (,0) (,0) 0.3 (,) (,) (,) (,) (,) (,) (,0) (,0) (,0) 0.4 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.5 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.6 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.7 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.8 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.9 (,) (,) (,) (,) (,) (,) (,) (,) (,) Table 7. The orders p and q of he ARMA process, when cov( Y, Yτ) = 04. for = τ Φ /Φ (,3) (,3) (,3) (,3) (,3) (,3) (,) (,) (,) -0.8 (,3) (,3) (,3) (,3) (,3) (,) (,) (,) (,) -0.7 (,3) (,3) (,3) (,3) (,) (,) (,) (,) (,) -0.6 (,3) (,3) (,3) (,) (,) (,) (,) (,) (,) -0.5 (,3) (,3) (,) (,) (,) (,) (,) (,) (,) -0.4 (,3) (,) (,) (,) (,) (0,) (0,) (,) (,) -0.3 (,) (,) (,) (,) (,) (0,) (0,) (,) (,) -0. (,) (,) (,) (,) (,) (,) (,0) (,0) (,0) -0. (,) (,) (,) (,) (,) (,) (,) (0,) (0,) 0. (,) (,) (,) (,) (,) (,) (,) (0,) (0,) 0. (,) (,) (,) (,) (,) (,) (,) (0,) (0,) 0.3 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.4 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.5 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.6 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.7 (,) (,) (,) (,) (,) (,) (,) (,) (,)

10 0.8 (,) (,) (,) (,) (,) (,) (,) (,) (,) 0.9 (,) (,) (,) (,) (,) (,) (,) (,) (,) Table 8. The orders p and q of he ARMA process, when cov( Y, Yτ) = 04. for = τ Φ /Φ (,4) (,4) (,4) (,4) (,4) (,3) (,3) (,3) (,3) -0.8 (,4) (,4) (,4) (,4) (,3) (,3) (,3) (,3) (,3) -0.7 (,4) (,4) (,4) (,3) (,3) (,3) (,3) (,3) (,3) -0.6 (,4) (,4) (,3) (,) (,) (,3) (,3) (,) (,) -0.5 (,4) (,3) (,3) (,) (,) (,) (,) (,) (,) -0.4 (,3) (,3) (,3) (,3) (,) (0,) (0,) (0,) (0,) -0.3 (,3) (,3) (,3) (,3) (,) (0,) (0,) (0,) (0,) -0. (,3) (,3) (,3) (,) (,) (0,) (0,) (0,) (0,) -0. (,3) (,3) (,3) (,) (,) (0,) (0,) (0,) (0,) 0. (,3) (,3) (,3) (,) (,) (0,) (0,) (0,) (0,) 0. (,3) (,3) (,3) (,) (,) (0,) (0,) (0,) (0,) 0.3 (,3) (,3) (,3) (,) (,) (,) (0,) (0,) (0,) 0.4 (,3) (,3) (,3) (,) (,) (,) (,) (0,) (0,) 0.5 (,3) (,3) (,3) (,3) (,) (,) (,) (,) (,) 0.6 (,3) (,3) (,3) (,3) (,) (,) (,) (,) (,) 0.7 (,3) (,3) (,3) (,3) (,3) (,3) (,3) (,3) (,3) 0.8 (,3) (,3) (,3) (,3) (,3) (,3) (,3) (,3) (,3) 0.9 (,3) (,3) (,3) (,3) (,3) (,3) (,3) (,3) (,3) The following conclusions may be derived:. The sum of AR() processes is in general ARMA ype of process and as a such ype of process should be idenificaion. The order of auoregressive is no larger ha, and he order q of MA depends on he parameers of componen process and on correlaion of hose processes.. The order of he obained ARMA process increases wih an increasmen of parameers and a disance τ, for which he auocovariance funcion akes he non-zero value. 3. In special cases he sum of AR() processes is a auoregressive process of AR(p) ype. The order p may be also equal o one. 4. There are same cases when a sum of auoregressive processes of order one gives he minimal represenaion ARMA(0,q), i.e., he process is

11 idenified as MA(q). I happens for low values of parameers Φ i and auocovariance funcion aking non-zero values for... -τ=. REFERENCES Beguin J.M., Gourieroux C., Monfor A. (980), Idenificaion of an ARIMA process: he corner mehod, Time Series, ed. T. Anderson, Norh- Holland. Gourieroux C., Monfor A. (990), Séries emporelles e modéles dynamiques, Economica, Paris. Granger C.W.J., Morris M.J. (976), Time Series Modeling and Inerpreaion, J.R. Sais. Soc. A(976), 39, Par.