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

The most important case to test is when *φ<sup>i</sup>*,2 ¼ 0, ∀*i*, which correspond to the linear periodic autoregressive model ð Þ *PARS*ð Þ1 of period *S*. The null hypothesis is then

$$H\_0: \rho\_{i,2} = 0, \forall i \text{ vs } H\_1: \exists i / \rho\_{i,2} \neq \mathbf{0}. \tag{21}$$

*H*<sup>1</sup> correspond to the restricted *PEXPARS*ð Þ1 model, that is, the linear *PARS*ð Þ1 model is nested within the nonlinear restricted model and it can be obtained by limiting the parameters *φ<sup>i</sup>*,2 to be zero ∀*i*, hence we have a problem of testing the linearity hypothesis.

The standard LR test statistic is

$$\lambda\_m = \left(\sum\_{i=1}^S \frac{\tilde{Q}\_{i,m}\left(\underline{\hat{\varrho}}\_i\right)}{\tilde{Q}\_{i,m}\left(\underline{\hat{\varrho}}\_i\right)}\right)^{\frac{m}{2}}.\tag{22}$$

The test rejects *H*<sup>0</sup> at the asymptotic level *α* when

$$\begin{split} LR\_m &= -2\log \lambda\_m \\ &= m \sum\_{i=1}^{S} \log \frac{\tilde{Q}\_{i,m}\left(\tilde{\underline{\rho}}\_i\right)}{\tilde{Q}\_{i,m}\left(\dot{\underline{\rho}}\_i\right)} > \chi\_S^2(1-a), \end{split} \tag{23}$$

where *χ*<sup>2</sup> *<sup>S</sup>*ð Þ <sup>1</sup> � *<sup>α</sup>* is the 1ð Þ� � *<sup>α</sup>* quantile of the *<sup>χ</sup>*<sup>2</sup> distribution with *<sup>S</sup>* degrees of freedom which is simply the number of supplementary parameters in *H*1.

#### *Recent Advances in Numerical Simulations*


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 *EXPAR* to the

periodic case.

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

**Author details**

Mouna Merzougui

**205**

LaPS Laboratory, University Badji Mokhtar Annaba, Algeria

\*Address all correspondence to: merzouguimouna@yahoo.fr

provided the original work is properly cited.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Table 6.**

*The rejection frequency computed on 10000 replications.*

#### **Figure 4.** *Asymptotic distribution of LRm.*

Example 2

**Table 6** shows the rejection frequency computed on 10000 replications of simulations of length *n* ¼ 200 and 500 from the 2 models.

Model I: *PAR*4ð Þ1 with the parameters *φ* ¼ �ð Þ 0*:*8, 0*:*5, 0*:*9, �0*:*4 <sup>0</sup> .

Model II: Restricted *PEXPAR*4ð Þ1 with *φ* ¼ �ð Þ 0*:*8, 2, 0*:*5, �1*:*5, 0*:*9, 1*:*1, �0*:*4, 0*:*6 <sup>0</sup> and *γ* ¼ 1*:*, **Figure 4** shows the asymptotic distribution of *LRm* under the null hypothesis. The results show that the empirical levels are acceptable, for *n* ¼ 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 *n* which means that the test is consistent. The rejection region is *LRm* >*χ*<sup>2</sup> <sup>4</sup>ð Þ <sup>1</sup> � *<sup>α</sup>* , where *<sup>χ</sup>*<sup>2</sup> <sup>4</sup>ð Þ 1 � *α* is the ð Þ� <sup>1</sup> � *<sup>α</sup>* quantile of the *<sup>χ</sup>*<sup>2</sup> distribution with 4 degrees of freedom. From **Figure 4**, we see that the asymptotic distribution of *LRm* (in full line) is close to the *χ*<sup>2</sup> <sup>4</sup> (in dashed lines), this confirm the above theoretical result.
