The Periodic Restricted EXPAR(1) Model

*Mouna Merzougui*

## **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 1920*s*, 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 1980*s* onwards. 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 *ARCH* and Bilinear models suitable for financial and economic data, threshold AutoRegressif ð Þ *TAR* and exponential *AR EXPAR* ð Þ models 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 [1–3] 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 ð Þ *EXPAR* models. The start was by taking an autoregressive ð Þ *AR* model *Yt*, say, and then make the coefficients dependent in an exponential way of *Y*<sup>2</sup> *t*�1.

Several papers treated the probabilistic and statistic aspects of *EXPAR* models. 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 *EXPAR*ð Þ1 model 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 *EXPAR* models, a forecasting method is proposed by [8], the *LAN* property was shown in [9] and asymptotically efficient estimates was constructed there for the restricted *EXPAR*ð Þ1 , 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 *AR*ð Þ1 is 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 *AR*ð Þ1 model with *EXPAR*ð Þ1 as special cases, the extended Kalman filter ð Þ *EKF* is 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 *EXPAR* model. The applications of the *EXPAR* model are multiple: ecology, hydrology, speech signal, macroeconomic and others see, for example, [16–21].

test for nullity of one coefficient and a test for linearity, a small simulation shows

The process f g *Yt <sup>t</sup>*≥<sup>1</sup> is a Periodic Restricted EXPonential AutoRegressive model

*γ* >0 is the known nonlinear parameter. A heuristic determination of *γ* from data is

^*<sup>γ</sup>* ¼ � log *<sup>ε</sup>* max 1≤*t*≤*n Y*2 *t*

where *ε* is a small number and *n* is the number of observations. (cf. [15]). The autoregressive parameters and the innovation variance are periodic of

To point out the periodicity, let *t* ¼ *i* þ *Sτ*, *i* ¼ 1, … , *S* and *τ* ∈ , then Eq. (1)

In Eq. (4), *Yi*þ*S<sup>τ</sup>* is the value of *Yt* during the *i*-th season of the cycle *τ* and *φi*,1, *φ<sup>i</sup>*,2 are the model parameters at the season *i:* It is clear that the parameters depend

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* ¼ 0*:*008226 is 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 0 as *h* increases. 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

� � � � *Yi*þ*Sτ*�<sup>1</sup> <sup>þ</sup> *<sup>ε</sup><sup>i</sup>*þ*S<sup>τ</sup>*, *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>S</sup>*, *<sup>τ</sup>* <sup>∈</sup> (4)

*i*þ*Sτ*�1

2*τ* � � � � *<sup>Y</sup>*2*<sup>τ</sup>* <sup>þ</sup> *<sup>ε</sup>*<sup>1</sup>þ2*<sup>τ</sup>*

1þ2*τ* � � � � *<sup>Y</sup>*<sup>1</sup>þ2*<sup>τ</sup>* <sup>þ</sup> *<sup>ε</sup>*<sup>2</sup>þ2*<sup>τ</sup>*

� � � *<sup>φ</sup><sup>i</sup>*,1<sup>þ</sup> *<sup>φ</sup><sup>i</sup>*,2 of course the

*t*�1

� � � � *Yt*�<sup>1</sup> <sup>þ</sup> *<sup>ε</sup>t*, *<sup>t</sup>*<sup>∈</sup> , (1)

� �, *φ<sup>t</sup>*,1 and *φ<sup>t</sup>*,2 are the autoregressive parameters and

*<sup>t</sup>*þ*kS* <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>

, (2)

*<sup>t</sup>* , ∀*k*, *t*∈ *:* (3)

*i*þ*Sτ*�1 � � �

*:* (5)

(restricted *PEXPAR*ð Þ1 ) of order 1 if it is a solution of the nonlinear difference

**2. The Periodic Restricted** *EXPAR*ð Þ**1 model and QML estimation**

Let f g *Yt <sup>t</sup>*≥<sup>1</sup> be a seasonal stochastic process with period *S S*ð Þ ≥2 .

*Yt* <sup>¼</sup> *<sup>φ</sup><sup>t</sup>*,1 <sup>þ</sup> *<sup>φ</sup><sup>t</sup>*,2 exp �*γY*<sup>2</sup>

*<sup>φ</sup><sup>t</sup>*þ*kS*,1 <sup>¼</sup> *<sup>φ</sup>t*,1, *<sup>φ</sup><sup>t</sup>*þ*kS*,2 <sup>¼</sup> *<sup>φ</sup><sup>t</sup>*,2 and *<sup>σ</sup>*<sup>2</sup>

*i*þ*Sτ*�1

on *Yi*þ*Sτ*�<sup>1</sup> in the sense that for large j j *Yi*þ*Sτ*�<sup>1</sup> we have *<sup>φ</sup><sup>i</sup>*,1<sup>þ</sup> *<sup>φ</sup><sup>i</sup>*,2 exp �*γY*<sup>2</sup>

change is done smoothly between these regimes. In application, the restricted *PEXPAR*ð Þ1 model is fitted to seasonal time series displaying nonlinearity features

mode of oscillations. Such phenomena is shown in **Figure 2** from model (B).

*<sup>Y</sup>*<sup>1</sup>þ2*<sup>τ</sup>* ¼ �0*:*<sup>3</sup> <sup>þ</sup> 2 exp �*Y*<sup>2</sup>

*<sup>Y</sup>*<sup>2</sup>þ2*<sup>τ</sup>* ¼ �0*:*<sup>8</sup> <sup>þ</sup> exp �*Y*<sup>2</sup>

*t*

the efficiency of these tests.

**Definition 1**

equation given by

period *S*, that is,

becomes

**197**

**2.1 Restricted** *PEXPAR*ð Þ**1 model**

*The Periodic Restricted EXPAR(1) Model DOI: http://dx.doi.org/10.5772/intechopen.94078*

where f g *<sup>ε</sup><sup>t</sup> <sup>t</sup>*≥<sup>1</sup> is *iid* 0, *<sup>σ</sup>*<sup>2</sup>

*Yi*þ*S<sup>τ</sup>* <sup>¼</sup> *<sup>φ</sup><sup>i</sup>*,1 <sup>þ</sup> *<sup>φ</sup><sup>i</sup>*,2 exp �*γY*<sup>2</sup>

like amplitude dependent frequency.

Model ðAÞ :

(

*<sup>φ</sup><sup>i</sup>*,1 while for small j j *Yi*þ*Sτ*�<sup>1</sup> : *<sup>φ</sup><sup>i</sup>*,1<sup>þ</sup> *<sup>φ</sup><sup>i</sup>*,2 exp �*γY*<sup>2</sup>

On the other hand, fitted seasonal time series exhibiting nonlinear behavior such cited before and having a periodic autocovariance structure by *SARIMA* models 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 *EXPAR* models. 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, *MPAR* model. Motivated by all this, we introduced recently the Periodic restricted *EXPAR*ð Þ1 model see [23], which consists of having different restricted *EXPAR*ð Þ1 for 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 *LS* estimators 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 *H*<sup>0</sup> ought 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 0 is an interior point of the parameter space, then the LR statistic asymptotically follows the *χ*<sup>2</sup> distribution under *H*<sup>0</sup> just 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 *PEXPAR* model 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.
