A MODEL FOR NONLINEAR INNOVATION IN TIME SERIES D

This paper introduces a class of nonlinear innovation process that has similar properties as the white noise process. Consequently the process can be a replacement of the white noise process in cases where the latter is inadequate as residual process.


INTRODUCTION
There has been growing interest in non-linear time series of recent mainly as a result of the observation that many practical time series possess nonlinear components [Hinich & Patterson (1985), Granger & Andersen (1978), Baillie & Bollerslev (1989), Ding, Granger & Engle (1993), Tong (1990)].Nonlinear AR models with conditionally heteroscedastic errors are common in financial and econometric time series.The conditional variance may be specified as nonlinear autoregressive conditional heteroscedasticity (ARCH) model [see e.g.Chen & Chen (2001)], or as linear generalized ARCH (GARCH) model of Bollerslev (1986), or even a GARCH model with complicated nonlinear structure.Models with threshold type nonlinearities characterized by discontinuous functions are also in use [see Tong (1990) Chen &Tsay (1993) for models for conditional mean; Glosten, Jaganathan & Runkle (1993), Rabemananjara & Zakoïan (1993) for GARCH models].The functional AR model of Chen & Tsay(1993) encompasses various well-known nonlinear autoregressive models such as the smooth transition AR models [see Teräsvirta (1994), van Dijk, Terasvirta & Fances (2002)].For nonlinear model involving moving average see, for example, Brännes et al (1998).For tests for nonlinearity under various assumptions on the type of nonlinearity see, for example, Tsay(1986), Lee(1991), Pera & Rodriguez (2005), Hinich, Mendes & Stone (2005), Subba Rao & Gabr (1980), Aparicio (1998), Barnett & Wolf (2005).The current paper wishes to introduce a class of nonlinear processes that can be used to model certain nonlinearities in residual process.The nonlinear process will be introduced in the next section.Thereafter we will explore some properties of the process that are similar to those of the white noise process.The final section will give a generalization of the nonlinear process.

THE PROCESS
Let { } t ε be zero mean white noise (independent and identically distributed) process with finite fourth moment.In particular let the white noise (WN) process{ } t ε .However, unlike the latter, { } t W is not independent sequence.Nonetheless, it can serve as a non-linear innovation process in many modeling situations where the WN process is currently used, for example in a linear process, say ARMA process.To see this we first consider MA(q) process: For this process ( ) 0, ( ) 0 See for example Fuller (1976) p. 20.Now consider the processes (3) and It is easily seen (see e.g.Fuller(1976) p. 20) that ( ) 0, ( ) 0 ( ) Consequently we can consider MA process in the non-linear process { } t W as an alternative to the one in WN process and still preserve the autocorrelation function structure.The process (3) seems to be of more practical importance in view of Volterra expansion (see Volterra, 1959) for nonlinear stationary series.However, notice that process (4) is which is an extension of process (3) as MA(q) plus nonlinear terms.
. This is again similar to what is obtained when WN process is used instead of{ } t W .A similar result holds for ARMA process generally.Terms similar to the right hand side of (1) can be included in ARCH and GARCH models to take care of certain nonlinearities.
Consequently the { } t W process (1) can be used as innovation in situations where the WN process is inappropriately used to account for non-linear innovation.This leads to a new class of, for example, ARMA processes in non-linear innovation process { } t W .The rest of the paper will be devoted to more properties of the process { } t W that are similar to those of the WN process.

SOME PROPERTIES OF W T PROCESS
We have seen that { } t W is uncorrelated ( ) 0 0, ( ) W γ non-linear process.We will prove the following: This theorem shows that, just like WN process, the sample mean W process based on sample size n is asymptotically zero mean normal with variance is the sample autocovariance function at lag k based on sample size n.We also have the following results.
Theorem 2: Theorem 2 indicates that for all lags k greater than s+1, the usual ±2/√n bounds for the (approximate) 95% confidence interval of WN ( ) k ρ ε (k ≠ 0) also holds for { } t W process. Theorem 3 indicates that for positive lags not greater than s + 1, the above ±2/√n bounds are too conservative to be bounds for . These results incidentally also provide information on the parameter s of the process { } t W when it has to be estimated from data.It particular, if theorem 2 holds for all k > 0 we would be sure that the WN process should be used instead of the process { } t W .

Proofs
The proofs will make use of central limit theorem for m-dependent random sequences given below in lemma 1.The result is due to Hoeffding and Robbins (1948) and is proved in Fuller (1976) The next lemma is proved, for example, in Rao (1973) is a differentiable function in the neighbourhood of µ and J is q x p Jacobian matrix at µ, then n Proof of Theorem 1 W t is a function of ε t-j for j = 0, 1, , s+1.Consequently { } t W is (s + 1) -dependent sequence.Since

{ } ( )
, 1 ( ) ( ) and 1 where A is as defined in lemma 1 with 2 2 ( ) Here we have used the fact that W t and Therefore, for k> s+1, ( ) Next we compute the second term in (5).

( ) (
) The first term on the right vanishes since − .If k > 1 and u > 1, the second term also vanishes since then + − − are uncorrelated.For u = 1 the second term vanishes whenever k > s+1 since then . This also equals A since A t is a constant.Hence 2 , 0 ( ) (0, ( )) as whenever 1 We now compute the asymptotic distribution of 0, ( ).
where A is as defined in lemma 1 with . To compute the latter it suffices to compute 2 2 ( , ) . Straight forward computation using the properties of WN process gives Since A t is independent of t we see that the asymptotic variance A ≡ 0 ( ) A c is given by To round up the proof we use lemma 2 with the function g defined by ( , ) , 0, , ( )) ', 0, ( ) ', 0 The Jacobian of the transformation at µ is By lemma 2 we have ( ) as n → ∞ for all k > s + 1.The proof of theorem 2 is now complete.

Proof of Theorem 3
Using the same transformation g and same variable V n,k as in the proof of theorem 2, we see that if where JVJ = To compute A t we proceed as in the proof of theorem 2 and use equations ( 5) to (8).Thus for These values are evidently greater than By allowing the parameters i β in (1) to be functions of t, heteroscedasticity in { } t W can be introduced.In that case var (Y t ) in the MA model given by equation (3) will be heteroscedastic but the autocovariances at non-zero lags will not be affected.In comparison, in the MA model given by (4) both var (Z t ) and the autocovariances would be affected. c) The non-linear process (1) is a member of a class of uncorrelated non-linear processes that have properties similar to the WN process.indeed for any integer a ≥ 0, the process U is (s + a)-dependent sequence and theorems 1-3 given above (with s + 1 replaced by s + a) also hold for each member of this class of This provides some flexibility in the choice of non-linear innovation process.The problem of identifying suitable values of the parameters a and s for a given problem will be taken up elsewhere.
k ≤ s Consequently, to prove the theorem it suffices to compute ( ) k A c and show that it is greater than ) may be quite close to 1 especially if s is large.b) p. 246.For the avoidance of doubt, a