A note on calculating autocovariances of periodic ARMA models
aa r X i v : . [ s t a t . M E ] S e p A NOTE ON CALCULATING AUTOCOVARIANCES OFPERIODIC
ARMA
MODELS
Abdelhakim A knouche
Hac`ene B elbachir
Fay¸cal H amdi
ABSTRACT
An analytically simple and tractable formula for the start-up autocovariances of periodic
ARM A ( P ARM A ) models is provided.
Keywords : P ARM A models, autocovariance functions, periodic Yule-Walker equations. . Primary: 62M10; Secondary: 62F15.
1. INTRODUCTION
Autocovariance calculation procedures for
P ARM A models are generally carried out us-ing the periodic Yule Walker equations (e.g. Bentarzi and Aknouche, 2005). This approachhas been considered earlier by Li and Hui (1988) for calculating
P ARM A autocovariances,where the ( p +1)-start-up autocovariances, for all seasons, were given through a matrix equa-tion Aγ = y , which is solved for γ ( γ being the ( p + 1) S × A is not given explicitly but formed through an appropriate algorithm. Adoptingthe same approach, Shao and Lund (2004) showed that the r -start-up ( r = max( p, q ) + 1)autocovariances may be obtained by solving a linear system Γ U γ = κ for γ , where Γ and U are matrices of dimensions rS × ( p + 1) rS and ( p + 1) rS × rS , respectively. While thesematrices are given explicitly, the method remains relatively cumbersome since it requires anincreasing bookkeeping due to the matrix product. This note proposes an improved compu-tation procedure for calculating the P ARM A autocovariances based on the latter approach.1he proposed method computes the ( p + 1)-start-up autocovariances based on a linear sys-tem with a corresponding matrix given explicitly, whose analytical form exhibits a circularstructure, naturally assorted with the model periodicity.
2. THE METHOD
Consider a causal
P ARM A model of orders ( p, q ) and period S p X j =0 φ ( v ) j y v − j + nS = q X j =0 θ ( v ) j ε v − j + nS , 1 ≤ v ≤ S , n ∈ Z , (1)where φ ( v )0 = θ ( v )0 = − { ε t , t ∈ Z } is a periodic white noise process, i.e., a sequence ofuncorrelated random variables with mean zero and variance E ( ε v + nS ) = σ v , for 1 ≤ v ≤ S and n ∈ Z .Let γ ( v ) h = E ( y v + nS y v + nS − h ) be the autocovariance function at season v and lag h ∈ Z .Then, it is well known (Li and Hui, 1988; Shao and Lund, 2004) that multiplying (1) by y v + nS − h and tacking expectation, the (cid:16) γ ( v ) h (cid:17) are completely identified from the differenceequation γ ( v ) h − p X j =1 φ ( v ) j γ ( v − j ) h − j = − q X j = h θ ( v ) j ψ ( v − h ) j − h σ v − j [ h ≤ q ] , h ≥ , (2)where the normalized cross-autocovariances ( ψ ( v ) k ), coefficients of the unique causal represen-tation of the P ARM A process { y t , t ∈ Z } , are given by (see e.g. Lund and Basawa, 2000;Shao and Lund, 2004) ψ ( v ) k = − θ ( v ) k [ k ≤ q ] + min( k,p ) X j =1 φ ( v ) j ψ ( v − j ) k − j , k ≥ , v = 1 , ..., S, (3)with ψ ( v )0 = 1 ( [ . ] stands for the indicator function).Equation (2) needs to be started from the knowledge of γ ( v ) h , 0 ≤ h ≤ p and 1 ≤ v ≤ S .Once these start-up values are given, the γ ( v ) h for h > p may be obtained recursively from(2) while invoking (3). For γ ( v ) h with negative lags, we may use the well-known relation γ ( v ) − h = γ ( v + h ) h . The main result of this note is to formulate a linear system for computing the p + 1 necessary starting autocovariances. Let γ = ( γ (1)0 , ..., γ (1) p , γ (2)0 , ..., γ (2) p , ..., γ ( S )0 , , ..., γ ( S ) p ) ′
2e the S ( p + 1)-vector of such values and ζ be the S ( p + 1)-vector whose entries ζ hS + v = q P j = h θ ( v ) j ψ ( v − h ) j − h σ v − j , for 1 ≤ v ≤ S and 0 ≤ h ≤ p , are the right-hand sides of (2).Define the ( p + 1)-square matrices ϕ ( v ) h ( h ≥
0, 1 ≤ v ≤ S ) and Φ ( v ) k ( k , v ∈ { , ..., S } )as follows ϕ ( v ) h = h × ( p +1 − h ) h × h − φ ( v ) h − φ ( v ) h +1 · · · − φ ( v ) p − φ ( v ) h · · ·
0. . . . . . . . . ...... . . . 00 · · · − φ ( v ) h ( p +1 − h ) × h , for h = 0 , ..., p ( p +1) × ( p +1) for h ≥ p + 1 , (4) Φ ( v ) k = X n ≥ ϕ ( v ) nS + k , v, k ∈ { , ..., S } , (5)and the S ( p + 1)-square matrix Φ = Φ (1)0 Φ (1) S − · · · Φ (1)2 Φ (1)1 Φ (2)1 Φ (2)0 · · · Φ (2)3 Φ (2)2 ... ... . . . ... ... Φ ( S − S − Φ ( S − S − · · · Φ ( S − Φ ( S − S − Φ ( S ) S − Φ ( S ) S − · · · Φ ( S )1 Φ ( S )0 . where m × n denotes the null matrix of dimension m × n . Then, the starting autocovariancevector γ is the unique solution of the following linear system Φ γ = ζ , (6)whenever model (1) is causal. Note that, in view of (4), the infinite sum in (5) contains only p non zero terms. It may be possible to reduce the complexity of forming Φ using its circularproperty. Indeed, equation (5) may be used to only evaluate the first bloc Φ (1) k , k = 1 , ..., S .The blocs Φ ( v ) k ( v = 2 , ..., S ) would be deduced from Φ (1) k by substituting the correspondingparameters φ (1) j by φ ( v ) j for j = 1 , ..., p . 3 . CONCLUDING REMARKS Despite the simplicity of the proposed method, it has the drawbacks that the starting au-tocovariances for all seasons are computed in the same bloc, thereby requiring O (( S ( p + 1)) )operations, which might be very costly for models with a large period. This is the main lim-itation of the periodic Yule Walker approach compared to which the methods that computeautocovariances for distinct seasons separately (e.g. Aknouche, 2007) are more suitable.BIBLIOGRAPHYAknouche, A., (2007). Causality conditions and autocovariance calculations in P V AR mod-els.
Journal of Statistical Computation and Simulation , , 769–780.Bentarzi, M. and Aknouche A. (2005). Calculation of the Fisher information matrix forperiodic ARM A models.
Communications in Statistics-Theory and Methods , , 891-903.Li, W. K and Hui, Y. V. (1988). An Algorithm for the exact likelihood of periodic autore-gressive moving average models. Communication in Statistics- Simulation and Computation , , 1483-1494.Lund, R. and Basawa, I. V. (2000). Recursive prediction and likelihood evaluation forperiodic ARM A models.
Journal of Time Series Analysis , , 75-93.Shao, Q. and Lund, R. (2004). Computation and characterization of autocorrelations andpartial autocorrelations in periodic ARM A models.
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