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The Periodic Restricted EXPAR(1) Model

By Mouna Merzougui

Submitted: July 1st 2020Reviewed: September 17th 2020Published: November 24th 2020

DOI: 10.5772/intechopen.94078

Downloaded: 129

Abstract

In this chapter, we discuss the nonlinear periodic restricted EXPAR(1) model. The parameters are estimated by the quasi maximum likelihood (QML) method and we give their asymptotic properties which lead to the construction of confidence intervals of the parameters. Then we consider the problem of testing the nullity of coefficients by using the standard Likelihood Ratio (LR) test, simulation studies are given to assess the performance of this QML and LR test.

Keywords

  • nonlinear time series
  • periodic restricted exponential autoregressive model
  • quasi maximum likelihood estimation
  • confidence interval
  • LR test

1. Introduction

Since the 1920s, linear models with Gaussian noise have occupied a prominent place, they have played an important role in the specification, prevision and general analysis of time series and many specific problems were solved by them. Nevertheless, many physical and natural processes exhibit nonlinear characteristics that are not taken into account with linear representation and are better explicated and fitted by nonlinear models. For example, ecological and environmental fields present phenomena close to the theory of nonlinear oscillations, such as limit cycle behavior remarked in the famous lynx or sunspot series, leading to the consideration of more complex models from the 1980sonwards. A first nonlinear model possible is the Volterra series which plays the same role as the Wold representation, for linear series. The interest of this representation is rather theoretical than practical, for this reason, specific parametric nonlinear models were presented as the ARCHand Bilinear models suitable for financial and economic data, threshold AutoRegressif TARand exponential AREXPARmodels suitable for ecological and meteorological data. These nonlinear models have been applied with great success in many important real-life problems. Basics of nonlinear time series analysis can be found in [1, 2, 3] and references therein.

Amplitude dependent frequency, jump phenomena and limit cycle behavior are familiar features of nonlinear vibration theory and to reproduce them [4, 5] introduced the exponential autoregressive EXPARmodels. The start was by taking an autoregressive ARmodel Yt,say, and then make the coefficients dependent in an exponential way of Yt12.

Several papers treated the probabilistic and statistic aspects of EXPARmodels. A direct method of estimation is proposed by [5], it consists to fix the nonlinear coefficient in the exponential term at one of a grid of values and then estimate the other parameters by linear least squares and use the AIC criterion to select the final parameters, necessary and sufficient conditions of stationarity and geometric ergodicity for the EXPAR1model are given by [6], the problem of estimation of nonlinear time series in a general framework by conditional least squares CLS and maximum likelihood ML methods is treated by [7] with application in EXPARmodels, a forecasting method is proposed by [8], the LANproperty was shown in [9] and asymptotically efficient estimates was constructed there for the restricted EXPAR1, a genetic algorithm for estimation is used in [10], Bayesian analysis of these models is introduced in [11], a parametric and nonparamtric test for the detection of exponential component in AR1is constructed by [12], sup-tests are constructed by [13] with the trilogy Likelihood Ratio (LR), Wald and Lagrange Multiplier (LM) for linearity in a general nonlinear AR1model with EXPAR1as special cases, the extended Kalman filter EKFis used in [14]. Given that nonlinear estimation is time consuming [15] proposed to estimate heuristically the nonlinear parameter from the data and this is a very interesting remark because when the nonlinear parameter is known we get the Restricted EXPARmodel. The applications of the EXPARmodel are multiple: ecology, hydrology, speech signal, macroeconomic and others see, for example, [16, 17, 18, 19, 20, 21].

On the other hand, fitted seasonal time series exhibiting nonlinear behavior such cited before and having a periodic autocovariance structure by SARIMAmodels will be inadequate. These models are linear and the seasonally adjusted data may still show seasonal variations because the structure of the correlations depends on the season. The solution is the use of a periodic version of EXPARmodels. The notion of periodicity, introduced by [22], was used to fit hydrological and financial series and allowed the emergence of new classes of time series models such as Periodic GARCH, Periodic Bilinear, MPARmodel. Motivated by all this, we introduced recently the Periodic restricted EXPAR1model see [23], which consists of having different restricted EXPAR1for each cycle and we established a most stringent test of periodicity since a periodic model is more complicated than a nonperiodic one and its consideration must be justified. We studied the problem of estimation by the least squares (LS) method in [24] and the test of Student was used for testing the nullity of the coefficients in the application. Traditionally, the step of estimation must be followed by tests of nullity of coefficients and the major tests used are Wald, LR and LM tests. We used a Wald test for testing the nullity of one coefficient and consequently testing linearity in [25].

In this chapter, we will present the quasi maximum likelihood (QML) estimation of the parameters, which are the LSestimators in [24] under the assumption that the density is Gaussian, these estimators are asymptotically normal under quite general conditions. This will play a role in the construction of the confidence interval for the parameters and then we treat the problem of testing the nullity of parameters which lead us to a linearity test using the standard and well known LR test. This test is based on the comparison between the maximum of the constrained and unconstrained quasi log likelihood, see for example [26] or [27], the null hypothesis is accepted, if the difference is small enough or equivalently H0ought to be rejected for large values of the difference. The problem is standard because the periodic model is restricted, i.e. the nonlinear parameter is known and for the other parameters 0is an interior point of the parameter space, then the LR statistic asymptotically follows the χ2distribution under H0just like the Wald test, but we chose the former because it does not require estimation of the information matrix. It is known that the two tests are asymptotically equivalent and may be identical see [26] for more details.

The chapter is organized as follows. In Section 2, we introduce the Restricted PEXPARmodel and we present the asymptotic normality of the QML estimators and we construct confidence intervals of the parameters. Section 3 provides the LR test for nullity of one coefficient and a test for linearity, a small simulation shows the efficiency of these tests.

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2. The Periodic Restricted EXPAR1model and QML estimation

2.1 Restricted PEXPAR1model

Let Ytt1be a seasonal stochastic process with period SS2.

Definition 1

The process Ytt1is a Periodic Restricted EXPonential AutoRegressive model (restricted PEXPAR1) of order 1 if it is a solution of the nonlinear difference equation given by

Yt=φt,1+φt,2expγYt12Yt1+εt,tN,E1

where εtt1is iid0σt2, φt,1and φt,2are the autoregressive parameters and γ>0is the known nonlinear parameter. A heuristic determination of γfrom data is

γ̂=logεmax1tnYt2,E2

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,

φt+kS,1=φt,1,φt+kS,2=φt,2andσt+kS2=σt2,k,tN.E3

To point out the periodicity, let t=i+,i=1,,Sand τN, then Eq. (1) becomes

Yi+=φi,1+φi,2expγYi+12Yi+1+εi+,i=1,,S,τNE4

In Eq. (4), Yi+is 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+1in the sense that for large Yi+1we have φi,1+φi,2expγYi+12φi,1while for small Yi+1: φi,1+φi,2expγYi+12φi,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.

Figure 1.

Realization of (A) with corresponding histogram and correlogram.

Figure 2.

Limit cycle fromPEXPAR21model.

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 pvalue=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).

Model (A):Y1+2τ=0.3+2expY2τ2Y2τ+ε1+2τY2+2τ=0.8+expY1+2τ2Y1+2τ+ε2+2τ.E5
Model (B):Y1+2τ=0.21.5expY2τ2Y2τ+ε1+2τY2+2τ=0.8+0.3expY1+2τ2Y1+2τ+ε2+2τ.E6

2.2 QML Estimation

Let φ¯=φ¯1φ¯SR2Sthe 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,2R,i=1,,S.

A2: The periodically ergodic process YttNis such that EYt4<, for any tN.

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 tN, 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:

Lnφ¯Y1Yn=mS2log2πm2i=1Slogσi2i=1Sτ=0m1Yi+φi,1+φi,2expγYi+12Yi+12σi22,E7

(assuming) σi0.

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

Qnφ¯=1nt=1nYtφt,1+φt,2expγYt12Yt12.E8

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

Q˜nφ¯=1Si=1SQ˜i,mφ¯iE9

and

Q˜i,mφ¯i=1mτ=0m1Yi+φi,1+φi,2expγYi+12Yi+12.E10

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

φ̂i,1φ̂i,2=τ=0m1Y+i12τ=0m1Y+i12expγY+i12τ=0m1Y+i12expγY+i12τ=0m1Y+i12exp2γY+i121×τ=0m1Y+i1Y+iτ=0m1Y+i1Y+iexpγY+i12σ̂i2=1mτ=0m1Y+iφ̂i,1+φ̂i,2expγY+i12Y+i12.E11

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

mφ̂i,1φi,1φ̂i,2φi,2mLN0¯2σi2EYi12EXi12expγYi12EYi12expγYi12EYi12exp2γYi121.E12

Furthermore, φ¯̂i,mand φ¯̂j,mare asymptotically independant, ij,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+implies that all the terms for ijare zero, this implies that mφ¯̂i,mφ¯iand mφ¯̂j,mφ¯j,ij,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 1α,φi,jis in the interval

φ̂i,j±Φ1α/2σ̂miΓijj,j=1,2,i=1,,S,E13

where

Γi=EYi12EYi12expγYi12EYi12expγYi12EYi12exp2γYi121,E14

and Φ1α/2is the 1α/2quantile of the standard normal distribution. That is, the CIcontains the true parameters in 1001α%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=0.40.9'with sizes n=200,500and 1000and for the significance levels: α=10%and 5%and 1000replications. From the Tables 13 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.

n=200CIφ1,1CIφ1,2CIφ2,1CIφ2,2
α=10%0.82060.77961.1136,1.25760.3801,0.41190.95590.8156
α=5%0.81260.76611.0893,1.26180.3808,0.41940.99670.8260

Table 1.

CI of parameters for n = 200.

n=500CIφ1,1CIφ1,2CIφ2,1CIφ2,2
α=10%0.80380.78791.1551,1.21130.3874,0.40010.90150.8470
α=5%0.80970.79121.1783,1.24530.3873,0.40200.91040.8448

Table 2.

CI of parameters for n = 500.

n=1000CIφ1,1CIφ1,2CIφ2,1CIφ2,2
α=10%0.80270.79521.1909,1.21910.3978,0.40400.90720.8793
α=5%0.80300.79381.1797,1.21390.3958,0.40340.91190.8791

Table 3.

CI of parameters for n = 1000.

3. Likelihood Ratio tests

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

H0:φi,2=0orH0:φi,1=0vsH1:φi,20orH1:φi,10,E15

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

φ˜i,1=τ=0m1Y+i1Y+iτ=0m1Y+i12E16

and the corresponding mean square error under the null

Q˜i,mφ˜¯i=1mτ=0m1Yi+φ˜i,1Yi+12.E17

The usual LR statistic is

λi,m=Lφ˜¯iσ˜i2Lφ̂¯iσ̂i2=Q˜i,mφ̂¯iQ˜i,mφ˜¯im2E18

then the test rejects H0at the asymptotic level αwhen

LRi,m=2logλi,m=mlogQ˜i,mφ˜¯iQ˜i,mφ̂¯i>χ121α,E19

where χ121αis the 1αquantile 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

Q˜i,mφ˜¯i=1mτ=0m1Yi+φ˜i,2expγYi+12Yi+12.E20

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¯=0.7,0,0.42'and γ=1

Model III: Restricted PEXPAR21with the parameters φ¯=0.7,1.5,0.42'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,20.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.

Modelαφ1,2φ2,2
I5%10%0.0520.1050.0660.103
II5%10%0.0540.1170.9981
III5%10%0.96710.9300.997

Table 4.

The rejection frequency computed on 1000 replications of simulations of length n=200.

Modelαφ1,2φ2,2
I5%10%0.0550.0950.0510.103
II5%10%0.0530.09811
III5%10%0.991111

Table 5.

The rejection frequency computed on 1000 replications of simulations of length n=500.

Figure 3.

Asymptotic distribution of LR.

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

H0:φi,2=0,ivsH1:i/φi,20.E21

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

λm=i=1SQ˜i,mφ̂¯iQ˜i,mφ˜¯im2.E22

The test rejects H0at the asymptotic level αwhen

LRm=2logλm=mi=1SlogQ˜i,mφ˜¯iQ˜i,mφ̂¯i>χS21α,E23

where χS21αis the 1αquantile 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.

ModelαLRtest n=200LRtest n=500
I5%10%0.06150.12250.05810.1075
II5%10%0.99990.999911

Table 6.

The rejection frequency computed on 10000 replications.

Model I: PAR41with the parameters φ¯=0.8,0.5,0.90.4'.

Model II: Restricted PEXPAR41with φ¯=0.8,2,0.51.5,0.9,1.10.4,0.6'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>χ421α,where χ421αis the 1αquantile 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.

Figure 4.

Asymptotic distribution ofLRm.

4. Conclusion

The periodic restricted EXPARmodel is added to the family of nonlinear and periodic models. Interest is focused on estimation and testing problems. The periodic stationarity allows to calculate the QML estimators and derived tests of coefficients, cycle by cycle, and therefore use standard techniques. From this point of view, we can extend several results concerning the classical EXPARto the periodic case.

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Mouna Merzougui (November 24th 2020). The Periodic Restricted EXPAR(1) Model [Online First], IntechOpen, DOI: 10.5772/intechopen.94078. Available from:

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