CI of parameters for n = 200.

## 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

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

Several papers treated the probabilistic and statistic aspects of

On the other hand, fitted seasonal time series exhibiting nonlinear behavior such cited before and having a periodic autocovariance structure by

In this chapter, we will present the quasi maximum likelihood (

The chapter is organized as follows. In Section 2, we introduce the Restricted

## 2. The Periodic Restricted EXPAR 1 model and QML estimation

### 2.1 Restricted PEXPAR 1 model

Let

**Definition 1**

The process

where

where

The autoregressive parameters and the innovation variance are periodic of period

To point out the periodicity, let

In Eq. (4),

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

### 2.2 QML Estimation

Let

Periodic stationarity has not been treated for this model so stationarity is required for each season hence

Given initial value

(assuming)

Let

The initial value is unknown but its choice is not important for the asymptotic behavior of the

and

The first order condition of the

We remark that the

**Theorem**

The

Furthermore,

**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

The

where

and

To examine the performance of the

## 3. Likelihood Ratio tests

### 3.1 Test for the Nullity of One Coefficient

The asymptotic normality of the

for some given

and the corresponding mean square error under the null

The usual LR statistic is

then the test rejects

where

In the same manner we can test the nullity of

Example 1

In the simulation we focused on testing the nullity of

Model I: Periodic autoregressive (

Model II: Restricted

Model III: Restricted

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

Model | |||
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I | |||

II | |||

III |

Model | |||
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I | |||

II | |||

III |

### 3.2 Test for linearity in Restricted PEXPAR(1) model

The most important case to test is when

The standard LR test statistic is

The test rejects

where

Example 2

Table 6 shows the rejection frequency computed on

Model | |||
---|---|---|---|

I | |||

II |

Model I:

Model II: Restricted

## 4. Conclusion

The periodic restricted