Section 5 Predictability

**Chapter 12**

Model

**Abstract**

**1. Introduction**

exponential way of *Y*<sup>2</sup>

**195**

*Mouna Merzougui*

The Periodic Restricted EXPAR(1)

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

**Keywords:** nonlinear time series, periodic restricted exponential autoregressive model, quasi maximum likelihood estimation, confidence interval, LR test

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 analy-

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

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

given to assess the performance of this QML and LR test.

sis can be found in [1–3] and references therein.

*t*�1.
