**3.1 Test for the Nullity of One Coefficient**

The asymptotic normality of the *QML* in Eq. (12) can be exploited to perform tests on the parameters. This problem is very standard, especially when 0 is an interior point of the parameter space and can be done with the trilogy: Wald, LR and LM tests. We treated the former in [25] and in this chapter, we will use the LR test which is based upon the difference between the maximum of the likelihood under the null and under the alternative hypotheses and has the advantage of not estimating information matrix. In this section, we are interested in testing assumptions of the form

$$H\_0: \varphi\_{i,2} = \mathbf{0} \text{ (or} \, H\_0: \varphi\_{i,1} = \mathbf{0} \text{) vs } H\_1: \varphi\_{i,2} \neq \mathbf{0} \text{ (or} \, H\_1: \varphi\_{i,1} \neq \mathbf{0} \text{)},\tag{15}$$

for some given *i:* Under *H*1, we have the *QML* estimator *φ*^*<sup>i</sup>* given by Eq. (11) and mean square error *<sup>Q</sup>*<sup>~</sup> *<sup>i</sup>*,*<sup>m</sup> <sup>φ</sup>*^*<sup>i</sup>* � � given by Eq. (10) and *<sup>φ</sup>*~*<sup>i</sup>* <sup>¼</sup> *<sup>φ</sup>*~*<sup>i</sup>*,1 0 � �, is the *QML* estimator given under *H*<sup>0</sup> where

$$\tilde{\rho}\_{i,1} = \frac{\sum\_{\mathbf{r}=0}^{m-1} Y\_{\mathbf{Sr}+i-1} Y\_{\mathbf{Sr}+i}}{\sum\_{\mathbf{r}=0}^{m-1} Y\_{\mathbf{Sr}+i-1}^2} \tag{16}$$

and the corresponding mean square error under the null

$$\tilde{Q}\_{i,m}\left(\underline{\tilde{\varrho}}\_{i}\right) = \frac{1}{m} \sum\_{\tau=0}^{m-1} \left(Y\_{i+\text{Sr}} - \tilde{\rho}\_{i,1} Y\_{i+\text{Sr}-1}\right)^2. \tag{17}$$

The usual LR statistic is

$$\lambda\_{i,m} = \frac{L\left(\tilde{\underline{\varrho}}\_i, \tilde{\sigma}\_i^2\right)}{L\left(\hat{\underline{\varrho}}\_i, \hat{\sigma}\_i^2\right)} = \left(\frac{\tilde{\mathcal{Q}}\_{i,m}\left(\hat{\underline{\varrho}}\_i\right)}{\tilde{\mathcal{Q}}\_{i,m}\left(\hat{\underline{\varrho}}\_i\right)}\right)^{\frac{\mu}{2}}\tag{18}$$

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

$$\begin{split} LR\_{i,m} &= -2\log \lambda\_{i,m} \\ &= m \log \frac{\tilde{Q}\_{i,m}\left(\tilde{\underline{\rho}}\_{i}\right)}{\tilde{Q}\_{i,m}\left(\dot{\underline{\rho}}\_{i}\right)} > \chi^{2}\_{1}(1-a), \end{split} \tag{19}$$

where *χ*<sup>2</sup> <sup>1</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 1 degree of freedom.

In the same manner we can test the nullity of *<sup>φ</sup>i*,1 by taken *<sup>φ</sup>*~*<sup>i</sup>* <sup>¼</sup> <sup>0</sup> *<sup>φ</sup>*~*i*,2 � � and

$$\check{\mathbf{Q}}\_{i,m}\left(\check{\underline{\boldsymbol{\varrho}}}\_{i}\right) = \frac{\mathbf{1}}{m} \sum\_{\mathbf{r}=\mathbf{0}}^{m-1} \left(\mathbf{Y}\_{i+\mathbf{S}\mathbf{r}} - \,\,\|\,\check{\boldsymbol{\varrho}}\_{i,2}\exp\left(-\gamma \mathbf{Y}\_{i+\mathbf{S}\mathbf{r}-1}^{2}\right) \mathbf{Y}\_{i+\mathbf{S}\mathbf{r}-1}\right)^{2}.\tag{20}$$

Example 1

In the simulation we focused on testing the nullity of *φi*,2 only. We simulated 1000 independent samples of length *n* ¼ 200 and 500 of 3 models.

Model I: Periodic autoregressive (*PAR*2ð ÞÞ 1 with the parameters *φ* ¼ �ð Þ 0*:*7, 0*:*4 <sup>0</sup> . Model II: Restricted *PEXPAR*2ð Þ1 with the parameters *φ* ¼ �ð Þ 0*:*7, 0, 0*:*4, �2 <sup>0</sup> and *γ* ¼ 1

Model III: Restricted *PEXPAR*2ð Þ1 with the parameters *φ* ¼ �ð Þ 0*:*7, 1*:*5, 0*:*4, �2 <sup>0</sup> and *γ* ¼ 1.

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 *AR*ð Þ1 and in the second cycle a restricted *EXPAR*ð Þ1 . On each realisation we fitted a restricted *PEXPAR*2ð Þ1 model by *QML* and carried out tests of *H*<sup>0</sup> : *φ<sup>i</sup>*,2 ¼ 0 against *H*<sup>1</sup> : *φ<sup>i</sup>*,2 6¼ 0*:* The rejection frequencies at significance level 5% and 10% are reported in **Tables 4** and **5**. **Figure 3** shows the asymptotic distribution of *LRi*,*<sup>m</sup>* under the null hypothesis. From the tables we see that the levels of the LR test are pretty well controlled since for *n* ¼ 500, we note a relative rejection frequency of 5*:*5% for *φ*1,2 and 5*:*1% for *φ*2,2, which are not meaningfully different from the nominal 5%, the same remark is made for *α* ¼ 10% where the relative rejection frequency is of 9*:*5% and 10*:*3%. From model III, the rejection frequencies which represent the empirical power increase with the length *n* indicating the good performance and the consistency of the test. To illustrate that the asymptotic distribution of *LRi*,*<sup>m</sup>* under the null hypothesis is the standard *χ*<sup>2</sup> <sup>1</sup> we have the

histograms in **Figure 3** where we see that the distribution of *LRi*,*<sup>m</sup>* has the well

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*<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

> *<sup>Q</sup>*<sup>~</sup> *<sup>i</sup>*,*<sup>m</sup> <sup>φ</sup>*^*<sup>i</sup>* � �

> *<sup>Q</sup>*<sup>~</sup> *<sup>i</sup>*,*<sup>m</sup> <sup>φ</sup>*~*<sup>i</sup>* � �

*<sup>Q</sup>*<sup>~</sup> *<sup>i</sup>*,*<sup>m</sup> <sup>φ</sup>*~*<sup>i</sup>* � �

*<sup>Q</sup>*<sup>~</sup> *<sup>i</sup>*,*<sup>m</sup> <sup>φ</sup>*^*<sup>i</sup>*

� � <sup>&</sup>gt; *<sup>χ</sup>*<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

*<sup>λ</sup><sup>m</sup>* <sup>¼</sup> <sup>X</sup> *S*

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

*LRm* ¼ �2 log *λ<sup>m</sup>*

<sup>¼</sup> *<sup>m</sup>*<sup>X</sup> *S*

*i*¼1

log

freedom which is simply the number of supplementary parameters in *H*1.

0 @

*i*¼1

*H*<sup>0</sup> : *φ<sup>i</sup>*,2 ¼ 0, ∀*i* vs *H*<sup>1</sup> : ∃*i=φ<sup>i</sup>*,2 6¼ 0*:* (21)

1 A

*m* 2

*:* (22)

*<sup>S</sup>*ð Þ <sup>1</sup> � *<sup>α</sup>* , (23)

known shape of *χ*<sup>2</sup>

*Asymptotic distribution of LR.*

**Figure 3.**

linearity hypothesis.

where *χ*<sup>2</sup>

**203**

1.

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

The standard LR test statistic is

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


**Table 4.**

*The rejection frequency computed on 1000 replications of simulations of length n* ¼ 200*.*


**Table 5.**

*The rejection frequency computed on 1000 replications of simulations of length n* ¼ 500*.*

In the same manner we can test the nullity of *<sup>φ</sup>i*,1 by taken *<sup>φ</sup>*~*<sup>i</sup>* <sup>¼</sup> <sup>0</sup>

1000 independent samples of length *n* ¼ 200 and 500 of 3 models.

distribution of *LRi*,*<sup>m</sup>* under the null hypothesis is the standard *χ*<sup>2</sup>

10%

10%

10%

10%

10%

10%

*The rejection frequency computed on 1000 replications of simulations of length n* ¼ 500*.*

*The rejection frequency computed on 1000 replications of simulations of length n* ¼ 200*.*

**Model** *α φ***1,2** *φ***2,2**

I 5%

II 5%

III 5%

I 5%

II 5%

III 5%

**Model** *α φ***1,2** *φ***2,2**

0*:*052 0*:*105

0*:*054 0*:*117

0*:*967 1

0*:*055 0*:*095

0*:*053 0*:*098

0*:*991 1

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

In the simulation we focused on testing the nullity of *φi*,2 only. We simulated

Model I: Periodic autoregressive (*PAR*2ð ÞÞ 1 with the parameters *φ* ¼ �ð Þ 0*:*7, 0*:*4 <sup>0</sup>

Model III: Restricted *PEXPAR*2ð Þ1 with the parameters *φ* ¼ �ð Þ 0*:*7, 1*:*5, 0*:*4, �2 <sup>0</sup>

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 *AR*ð Þ1 and in the second cycle a restricted *EXPAR*ð Þ1 . On each realisation we fitted a restricted *PEXPAR*2ð Þ1 model by *QML* and carried out tests of *H*<sup>0</sup> : *φ<sup>i</sup>*,2 ¼ 0 against *H*<sup>1</sup> : *φ<sup>i</sup>*,2 6¼ 0*:* The rejection frequencies at significance level 5% and 10% are reported in **Tables 4** and **5**. **Figure 3** shows the asymptotic distribution of *LRi*,*<sup>m</sup>* under the null hypothesis. From the tables we see that the levels of the LR test are pretty well controlled since for *n* ¼ 500, we note a relative rejection frequency of 5*:*5% for *φ*1,2 and 5*:*1% for *φ*2,2, which are not meaningfully different from the nominal 5%, the same remark is made for *α* ¼ 10% where the relative rejection frequency is of 9*:*5% and 10*:*3%. From model III, the rejection frequencies which represent the empirical power increase with the length *n* indicating the good performance and the consistency of the test. To illustrate that the asymptotic

Model II: Restricted *PEXPAR*2ð Þ1 with the parameters *φ* ¼ �ð Þ 0*:*7, 0, 0*:*4, �2 <sup>0</sup>

� �<sup>2</sup>

*i*þ*Sτ*�1 � �*Yi*þ*Sτ*�<sup>1</sup>

*<sup>Q</sup>*<sup>~</sup> *<sup>i</sup>*,*<sup>m</sup> <sup>φ</sup>*~*<sup>i</sup>* � �

Example 1

and *γ* ¼ 1

and *γ* ¼ 1.

**Table 4.**

**Table 5.**

**202**

¼ 1 *m* X*m*�1 *τ*¼0

*Recent Advances in Numerical Simulations*

*φ*~*i*,2 � �

<sup>1</sup> we have the

0*:*066 0*:*103

0*:*998 1

0*:*930 0*:*997

0*:*051 0*:*103

> 1 1

> 1 1

*:* (20)

and

.

**Figure 3.** *Asymptotic distribution of LR.*

histograms in **Figure 3** where we see that the distribution of *LRi*,*<sup>m</sup>* has the well known shape of *χ*<sup>2</sup> 1.
