The Periodic Restricted EXPAR(1) Model

2.1 Restricted PEXPAR1model

Let Ytt≥1be a seasonal stochastic process with period SS≥2.

Definition 1

The process Ytt≥1is a Periodic Restricted EXPonential AutoRegressive model (restricted PEXPAR1) of order 1 if it is a solution of the nonlinear difference equation given by

where εtt≥1is iid0σt2, φt,1and φt,2are the autoregressive parameters and γ>0is the known nonlinear parameter. A heuristic determination of γfrom data is

where εis a small number and nis the number of observations. (cf. [15]).

The autoregressive parameters and the innovation variance are periodic of period S,that is,

To point out the periodicity, let t=i+Sτ,i=1,…,Sand τ∈N, then Eq. (1) becomes

In Eq. (4), Yi+Sτis the value of Ytduring the i-th season of the cycle τand φi,1,φi,2are the model parameters at the season i.It is clear that the parameters depend on Yi+Sτ−1in the sense that for large Yi+Sτ−1we have φi,1+φi,2exp−γYi+Sτ−12∼φi,1while for small Yi+Sτ−1: φi,1+φi,2exp−γYi+Sτ−12∼φi,1+φi,2of course the change is done smoothly between these regimes. In application, the restricted PEXPAR1model is fitted to seasonal time series displaying nonlinearity features like amplitude dependent frequency.

These forms of models are new in the literature of the time series it is interesting to make several simulations to see their characteristics. An important fact is their property of non normality as is shown by histogram in Figure 1 and confirmed by the test of Shapiro Wilk where the p−value=0.008226is less than 0.05. The realization of the process (A) is given in Figure 1 from it and from the correlogram we can see that the process is stationary in each season due to the fast decay to 0as hincreases. Another interesting fact, that these types of models can exhibit, is the limit cycle behavior which is a well known feature in nonlinear vibrations and is one of possible mode of oscillations. Such phenomena is shown in Figure 2 from model (B).

2.2 QML Estimation

Let φ¯=φ′¯1…φ′¯S′∈R2Sthe parameter vector where φ¯i=φi,1φi,2′, i=1,…,S.We want to estimate the true parameter φ¯0from observations Y1,…,Ynwhere n=mSwhich means that we have mfull period of data. The problem is resolved by the QMLmethod and under the conditions:

A1: The Periodical restricted exponential autoregressive parameters φ¯satisfy the stationary periodically condition of (1). A sufficient condition is given by φi,1<1,φi,2∈R,i=1,…,S.

A2: The periodically ergodic process Ytt∈Nis such that EYt4<∞, for any t∈N.

Periodic stationarity has not been treated for this model so stationarity is required for each season hence A1. We can replace the assumption A2by Eεt4<∞, for any t∈N, since Eεt4<∞⇒EYt4<∞.Under this condition significant outliers are improbable and the existence of the information matrix is guaranteed.

Given initial value Y0,the conditional log likelihood of the observations evaluated at φ¯depends on f. The QMLestimator is obtained by replacing fby the N0σt2:

(assuming) σi≠0.

Let φ¯̂the QMLestimator, one can see that maximizing Lnis equivalent to minimization of the quantity:

The initial value is unknown but its choice is not important for the asymptotic behavior of the QMLestimator so we put Y0=0, which defines the operational criterion

and

The first order condition of the QMLminimization problem is a system of 2Slinear equations with 2Sunknowns. The solution is

We remark that the QMLestimator is the LSestimator and we can proof the next theorem in the same way.

Theorem

The QMLestimator is strongly consistent and we have for i=1,…,S

Furthermore, φ¯̂i,mand φ¯̂j,mare asymptotically independant, i≠j,i,j=1,…,S.

Proof

The proof is very standard in the literature of time series. The consistency is based on an ergodicity argument and for the normality a central limit version for martingale differences is used. The detail is similar to the LSE(see [24]) hence it is omitted. The independence of the εi+Sτimplies that all the terms for i≠jare zero, this implies that mφ¯̂i,m−φ¯iand mφ¯̂j,m−φ¯j,i≠j,are asymptotically uncorrelated.

The QMLestimators (Eq. (4)) yields a point estimator, a confidence interval (CI) gives a region where the parameters fall in with a given probability (usually 95%or 90%). Based on the asymptotic normality of the QMLestimators, with asymptotic probability 1−α,φi,jis in the interval

where

and Φ1−α/2is the 1−α/2quantile of the standard normal distribution. That is, the CIcontains the true parameters in 1001−α%of all repeated samples.

To examine the performance of the QMLestimators, we construct CIof the parameters from the simulation of restricted PEXPAR21model with parameters: φ¯1=−0.8,1.2'and φ¯2=0.4−0.9'with sizes n=200,500and 1000and for the significance levels: α=10%and 5%and 1000replications. From the Tables 1–3 we deduce that the parameters are well estimated and when nincreases the length of CIdecreases showing that the estimates are consistent. Obviously, a higher confidence level produces wider CI.

3.1 Test for the Nullity of One Coefficient

The asymptotic normality of the QMLin Eq. (12) can be exploited to perform tests on the parameters. This problem is very standard, especially when 0is an interior point of the parameter space and can be done with the trilogy: Wald, LR and LM tests. We treated the former in [25] and in this chapter, we will use the LR test which is based upon the difference between the maximum of the likelihood under the null and under the alternative hypotheses and has the advantage of not estimating information matrix. In this section, we are interested in testing assumptions of the form

for some given i.Under H1,we have the QMLestimator φ̂¯igiven by Eq. (11) and mean square error Q˜i,mφ̂¯igiven by Eq. (10) and φ˜¯i=φ˜i,10,is the QMLestimator given under H0where

and the corresponding mean square error under the null

The usual LR statistic is

then the test rejects H0at the asymptotic level αwhen

where χ121−αis the 1−α−quantile of the χ2distribution with 1 degree of freedom.

In the same manner we can test the nullity of φi,1by taken φ˜¯i=0φ˜i,2and

Example 1

In the simulation we focused on testing the nullity of φi,2only. We simulated 1000independent samples of length n=200and 500of 3 models.

Model I: Periodic autoregressive (PAR21)with the parameters φ¯=−0.7,0.4'.

Model II: Restricted PEXPAR21with the parameters φ¯=−0.7,0,0.4−2'and γ=1

Model III: Restricted PEXPAR21with the parameters φ¯=−0.7,1.5,0.4−2'and γ=1.

The model I is chosen to calculate the level, the model III is chosen to calculate the power, the choice of model II is to show that the test is efficient since in the first cycle we have an AR1and in the second cycle a restricted EXPAR1. On each realisation we fitted a restricted PEXPAR21model by QMLand carried out tests of H0:φi,2=0against H1:φi,2≠0.The rejection frequencies at significance level 5%and 10%are reported in Tables 4 and 5. Figure 3 shows the asymptotic distribution of LRi,munder the null hypothesis. From the tables we see that the levels of the LR test are pretty well controlled since for n=500,we note a relative rejection frequency of 5.5%for φ1,2and 5.1%for φ2,2, which are not meaningfully different from the nominal 5%, the same remark is made for α=10%where the relative rejection frequency is of 9.5%and 10.3%. From model III, the rejection frequencies which represent the empirical power increase with the length nindicating the good performance and the consistency of the test. To illustrate that the asymptotic distribution of LRi,munder the null hypothesis is the standard χ12we have the histograms in Figure 3 where we see that the distribution of LRi,mhas the well known shape of χ12.

3.2 Test for linearity in Restricted PEXPAR(1) model

The most important case to test is when φi,2=0,∀i,which correspond to the linear periodic autoregressive model PARS1of period S. The null hypothesis is then

H1correspond to the restricted PEXPARS1model, that is, the linear PARS1model is nested within the nonlinear restricted model and it can be obtained by limiting the parameters φi,2to be zero ∀i, hence we have a problem of testing the linearity hypothesis.

The standard LR test statistic is

The test rejects H0at the asymptotic level αwhen

where χS21−αis the 1−α−quantile of the χ2distribution with Sdegrees of freedom which is simply the number of supplementary parameters in H1.

Example 2

Table 6 shows the rejection frequency computed on 10000replications of simulations of length n=200and 500from the 2 models.

Model I: PAR41with the parameters φ¯=−0.8,0.5,0.9−0.4'.

Model II: Restricted PEXPAR41with φ¯=−0.8,2,0.5−1.5,0.9,1.1−0.4,0.6'and γ=1., Figure 4 shows the asymptotic distribution of LRmunder the null hypothesis. The results show that the empirical levels are acceptable, for n=500, we have a relative rejection frequency of 5.81%(resp. 10.75%) which is very close to 5%(resp. 10%), the empirical power increases with the size nwhich means that the test is consistent. The rejection region is LRm>χ421−α,where χ421−αis the 1−α−quantile of the χ2distribution with 4degrees of freedom. From Figure 4, we see that the asymptotic distribution of LRm(in full line) is close to the χ42(in dashed lines), this confirm the above theoretical result.