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.

Highlights

  • 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

  • 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 ðTARÞ and exponential AR ðEXPARÞ models suitable for ecological and meteorological data

  • We introduced recently the Periodic restricted EXPARð1Þ model see [23], which consists of having different restricted EXPARð1Þ 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

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Summary

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. 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 ðTARÞ and exponential AR ðEXPARÞ models suitable for ecological and meteorological data. 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 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.

Restricted PEXPARð1Þ model
QML Estimation
A2: The periodically ergodic process f
SτþiÀ1 exp ÀÀγY
Test for the Nullity of One Coefficient
Test for linearity in Restricted PEXPAR(1) model
Conclusion
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