Nonlinear AR(1) and MA(1) time series models

Transcription

1 Nonlinear AR(1) and MA(1) time series models Vesna M. Ćojbašić Abstract The simple nonlinear models of autoregressive and moving average structure are analyzed in the paper 1. In this paper we obtain some information on distributions of random parameters of two models. 1 An autoregressive process In this section we shall deal with the following model for a stationary sequence {X t } t {0 ±1 ±2...} : X t = AX t 1 + Bξ t (1) where A and B are random coefficients with distributions ( ) β γ A : p r q r ( ) 0 1 B :. q p p q r are probabilities with p+q = 1 0 < < β < γ < 1 and ξ t are independent identically distributed random variables. We assume A is independent of B A B are independent of X t and A B are independent of ξ t all t. We also assume X t is independent of ξ s for all s > t. To derive moments of X t from model (1): X t w.p.pq βx t w.p.rq γx X t+1 = t w.p.(q r)q X t + ξ t+1 w.p.p 2 (2) βx t + ξ t+1 w.p.rp γx t + ξ t+1 w.p.(q r)p we use the Laplace transforms 1 Presented at the IMC Filomat 2001 Niš August Mathematics Subject Classification: 62M10 Keywords: Time series model autoregressive process 227

2 228 Vesna M. Ćojbašić L x (s) = E (exp( sx)) L ξ (s) = E (exp( sξ)). Transforming both sides of (2) gives: L x (s) = pql x (s) + rql x (βs) + (q r)ql x (γs) + p 2 L x (s)l ξ (s) + (3) and from (3) follows + rpl x (βs)l ξ (s) + (q r)pl x (γs)l ξ (s) EX = L X(0) = EX 2 = L X(0) = peξ 1 p rβ (q r)γ 2pEξEX(p + rβ + (q r)γ) + peξ2 1 p 2 rβ 2 (q r)γ 2. The method of moments is quite complicated for estimating unknown parameters p q r and we shall estimate p q and r using the method of maximum likelihood. For model (2) we can consider the conditional probability: ψ(s s t ) = P (X t+1 < s s t X t < s t + h) and the conditional density given by: g(s s t ) = d ds P (X t+1 < s X t = s t ) = = pqδ(s s t ) + rqδ(s βs t ) + (q r)qδ(s γs t ) + + p 2 g ξ (s s t )H(s s t ) + rpg ξ (s βs t )H(s βs t ) + + (q r)pg ξ (s γs t )H(s γs t ) when h 0. δ(.) is the Dirac delta function H(.) is the Heaviside function defined by: { 1 s = st H (0) (s s t ) = 0 s s t H(s s t ) = H (0 ) (s s t ) = { 1 s > st 0 s s t and g ξ (.) is the density of random variables ξ t for all t. An alternative form for the conditional density is: g(s s t ) = [pqδ(s s t )] H (0)(s s t ) [rqδ(s βs t )] H (0)(s βs t) [(q r)qδ(s γs t )] H (0)(s γs t ) [p 2 g ξ (s s t )] H(s s t) [rpg ξ (s βs t )] H(s βst) [(q r)pg ξ (s γs t )] H(s γst).

3 Nonlinear AR(1) and MA(1) time series models 229 Having observed (X 2... X n+1 ) and fixed X 1 = s 1 from the model (2) we can estimate the parameters p q r of the model (2) using conditional likelihood function: L(p r) = n g(s t+1 s t ). For fixed β γ the maximum likelihood estimators of p q and r are: ˆp = A 1 + 2A 4 + A 5 + A 6 2n ˆr = (A 2 + A 5 )(A 1 + 2A 2 + 2A 3 + A 5 + A 6 ) (A 2 + A 3 + A 5 + A 6 )2n where ˆq = 1 ˆp = A 1 + 2A 2 + 2A 3 + A 5 + A 6 2n A 1 = A 3 = A 5 = H (0) (s t+1 s t ); A 2 = H (0) (s t+1 γs t ); A 4 = H (0 ) (s t+1 βs t ); A 6 = H (0) (s t+1 βs t ); H (0 ) (s t+1 s t ); H (0 ) (s t+1 γs t ). If the exponential distribution with mean λ 1 is used for the sequence {ξ t } then the maximum likelihood estimator of λ is: ˆλ = n [H (0 )(s t+1 s t ) + H (0 ) (s t+1 βs t ) + H (0 ) (s t+1 γs t )] S where S = [ (s t+1 s t )H (0 ) (s t+1 s t ) + + (s t+1 βs t )H (0 ) (s t+1 βs t ) + ] + (s t+1 γs t )H (0 ) (s t+1 γs t ).

4 230 Vesna M. Ćojbašić 2 A moving-average process Let us consider a first order moving-average process given by ξ t+1 w.p. p 1 q 1 βξ X t+1 = t+1 w.p. q1 2 ξ t+1 + ξ t w.p. p 2 (4) 1 βξ t+1 + ξ t w.p. p 1 q 1 where ξ t are i.i.d. random variables with exponential ε(λ) distribution (with mean λ 1 ) and p 1 q 1 are probabilities with p 1 + q 1 = 1 and 0 < < β < 1. We assume X t is independent of ξ s for all s > t. Using Laplace transform of (4) we can derive: EX = p 1 + q 1 β + p 1 λ Let s denote ˆp 1 and ˆq 1 the estimators of p 1 and q 1 produced by the method of moments (for fixed β). Then we have and ˆp 1 = λ X n β β + 1 ˆq 1 = 1 ˆp 1 = λ X n β + 1 where X n = 1 X i for given sample (X 1... X n ). It can be easily shown n i=1 that ˆp 1 and ˆq 1 are consistent estimators for n. For model (4) we can also consider the conditional probability: ψ(s s t ) = P (X t+1 < s s t X t < s t + h). We can not give the maximum likelihood estimators of p 1 and q 1 here. In one of 16 (4 4 combinations in(4)) different cases (where for example X t+1 = ξ t+1 + ξ t and for example X t = ξ t + ξ t 1 ) we have very complicated form of the conditional density:. g(s s t ) = λ( 1) exp( λ s st st )(exp( λ ) exp(λ ) exp( λs 2 t )) ( 2 + 1)(exp( λ st ) exp( λs t)) for s > s t+h and h 0. Then it is difficult to derive the likelihood and estimate unknown parameters. Some other aspects of such a process are under investigation.

5 Nonlinear AR(1) and MA(1) time series models 231 Acknowledgement I would like to thank Prof. J. Mališić for drawing my attention to this problem. The author wishes also to express his thanks and appreciations to the referee for several comments and suggestions. References [1] V. Jevremović Two examples of nonlinear processes with a mixed exponential marginal distribution Statistics & Probability Letters 10 (1990) [2] J.L. Hutton Non-negative time series models for dry river flow J. Appl. Prob. 27 (1990) Faculty of Mechanical Engineering 27. marta Beograd vesnac@alfa.mas.bg.ac.yu