Wald Tests in the Restricted Periodic EXPAR (1) Model
Merzougui M*
LaPS Laboratory, University Badji Mokhtar Annaba, Algeria
Submission: May 18, 2020;Published: July 09, 2020
*Corresponding author: Merzougui M, LaPS Laboratory, University Badji Mokhtar Annaba, Algeria
How to cite this article: Merzougui M. Wald Tests in the Restricted Periodic EXPAR (1) Model Biostat Biom Open Access J. 2020; 10(1): 555780.DOI:10.19080/BBOAJ.2020.10.555780
Abstract
In this paper, we study the problem of testing the nullity of coefficients in restricted periodic exponential autoregressive model of order 1. We consider two cases: tests for the nullity of one coefficient and test of linearity this will be achieved by using the standard Wald test.
Keywords: Nonlinear time series, Restricted Periodic exponential autoregressive model, Least squares estimation, Wald test
Abbreviations:EXPAR: Exponential Autoregressive Models; PEXPAR: Periodic Exponential Autoregressive Models; LS: Least Squares; LSE: Least Squares Estimator
Mathematics Subject Classification: 62F12; 62M10
Introduction
Since the introduction of the exponential autoregressive models (EXPAR) by Ozaki [1] to capture some features of the non-linear vibration theory, several papers discussed their theoretical and practical aspects as Chan and Tong [2], Al-Kassam and Lane[3]; Allal and El Melhaoui [4]; Ghosh, et al. [5]; Azouagh and El Melhaoui [6] and many others and when we have data exhibiting nonlinear behaviour such amplitude dependent frequency and periodic autocovariance structure, it will be suitable to use a periodic version of the EXPAR model. The notion of periodicity due to Gladyshev [7], was exploited to introduce the restricted periodic EXPAR(1) model (PEXPAR) in Merzougui et al. [8] and an optimal test of periodicity is given there, the parameters were estimated by the least squares (LS) method in Merzougui [9] and the test of Student was used for testing the nullity of the coefficients in the application but the problem of testing linearity has not been treated before.
Nonlinear time series models are generally more complex than linear ones so it is important first to do the linearity test before considering them. In our testing problems the parameter is not at the boundaries of stationarity so the solution is very standard, we choose the Wald test and we begin by testing the coefficients of the restricted PEXPAR(1) model as it is a novel model and then we approach the linearity test. The Wald statistic is defined in the usual way see for example Bierens [10]. It is given by a quadratic form based on the difference between the unrestricted estimated value of the restrictions and their value under the null hypothe sis. The paper is organized as follows. In section 2, we discuss the model and we remind the asymptotic normality of Least Squares Estimator (LSE) of the parameters and define the Wald test for nullity of one coefficient which is based on the LSE. Section 3 provides a test for linearity after rearranging the parameters.
Test for the nullity of one coefficient
Suppose that a time series Y1,......Yn is generated by the restricted PEXPAR(1), with period S(S≥2):
Where is a Gaussian white noise process with mean 0 and finite variance σ2t. The autoregressive parameters and the innovation variance are periodic, in time, with period S and the nonlinear parameter, γ>0, is known. Putting ,t=i+Sr, i =1,...,S and ,τ ∈ Z, one can rewrite equation (1) in a form that emphasize the periodicity:
which means that Yi+Sτis the value of Yt during the i−th season of the cycle τ. Figure 1 shows a simulated series and a month plot of the restricted PEXPAR4 (1) model with ϕ= (-0.7,1.2;0.6,-1.5;0.9,2;-0.7,0.5)′ γ=1 and n=500.The Figure 2 gives the scatterplot which clearly indicates nonlinear behaviour and Figure 3 gives the histogram where we see the non-Gaussian characteristic of the time series, this is confirmed by the Shapiro Wilk test which rejects the normality with a p-value = 2.186e-07.Let the parameter vector where ( i=1,...,S
The problem of estimation is resolved by the LSE method because γ is known so it is a linear optimisation. Under the conditions:
A1 : The Periodical restricted exponential autoregressive parameters ϕ satisfy the stationary periodically condition of (1): A sufficient condition is given by
A2: The periodically ergodic process {Yt;t∈R} is such that E(Yt4)<α, for any t∈R. It can be shown (see Merzougui, [9]) that the LSE are strongly consistent and we have for i=1,...,S
The asymptotic normality of the LSE in (3) can be exploited to perform tests on the parameters. It is clear from A1 that 0 is an interior point of the parameter space. Let
Consider the null hypothesis where R= [0 1] and for some given i=1,...,S. The usual Wald test rejects H0 at the asymptotic level α when the test statistic is
The Wald test looks whether ˆϕi,2, is close to 0 and we reject the null hypothesis for large values of Wi,m. In application we must replace Γi by a consistent estimator. Of course, for the nullity of one coefficient, the test of Student can be used. In the same manner we can test the nullity of ϕi,1 by taken R=[1 0].
Test for linearity in Restricted PEXPAR(1) model
When ϕi,2=0,∀i the restricted PEXPAR(1) model reduces to the periodic autoregressive model (PARS (1)) of period S. This case corresponds to testing the linearity hypothesis: To reorder the parameters, we introduce the 2S×2S matrix K where
where the S×2 matrices K1,k and k2,K, k=1,2,....s are given, their general elements, as follows:
Then We consider testing the nullity of the last S parameters of β, which is split into two components β(β1,β2)′= where βi∈RS. The null hypothesis is then
Where R=(0SIS) From (3) we have
Where
and since hence,
where Σ=KΓK′. Then the testing problem can be easily solved by a standard Wald test.
Where ˆΣis a consistent estimator of Σ..
References
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