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\setlength{\parindent}{0pt}\raggedright \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt \textcolor{GJBlue}{\large\bfseries\abstractname\space} }{% \vskip8pt \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt } \usepackage[absolute,overlay]{textpos} \makeatother \usepackage{lineno} \linenumbers \begin{document} \affil[1]{ } \renewcommand\Authands{ and } \date{\small \em Received: 1 January 1970 Accepted: 1 January 1970 Published: 1 January 1970} \maketitle \begin{abstract} \end{abstract} \keywords{} \begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) \textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark} \end{textblock*} \begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) \textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}} \end{textblock*} \begin{textblock*}{10cm}(1.05cm,3cm) {{\textit{CrossRef DOI of original article:}} \underline{}} \end{textblock*}\let\tabcellsep& \section[{INTRODUCTION}]{INTRODUCTION}\par Model selection is a challenging step in statistical modelling. Modelling any data requires characterization of the associated data generating process (DGP). The DGP is unknown and therefore we face several admissible competing models. Model selection consists on selecting a model, which mimics such unknown DGP, from a set of admissible models according to a criterion. One retains the model which makes such criterion optimal, \hyperref[b25]{Tibshirani et al. (2015)}, \hyperref[b6]{Ferraty and Hall (2015)} and \hyperref[b18]{Li et al. (2017)}. There exist various model selection criteria in the literature when admissible models have fully parametric specification, such as the Wald test, the likelihood ratio test, the Lagrange multiplier test, the information criteria and so on, see Hamilton (1994), \hyperref[b11]{Greene (2003)} and \hyperref[b16]{Hooten and Hobbs (2015)}. The other case, when admissible models contain simultaneously parametric and nonparametric specifications, seems underdeveloped, \hyperref[b13]{Hendry et al. (2008)}. Encompassing tests appear to be helpful for the latter situation where an encompassing model is intended to account the salient feature of the encompassed model. Therefore, encompassing test can detect redundant models among the admissible models.\par London Journal of Research in Science: Natural and Formal\par Encompassing tests are based on two points, that the encompassing model ought to be able to explain the predictions and to predict some mis-specifications of the encompassed model, \hyperref[b13]{Hendry et al. (2008)}. We know that there are various considerations and developments of encompassing tests, we refer readers to \hyperref[b19]{Mizon (1984)}, \hyperref[b15]{Hendry and Richard (1989)}, \hyperref[b8]{Gouriéroux and Monfort (1995)} and \hyperref[b7]{Florens et al. (1996)}. For an overview on the concept of encompassing tests, see \hyperref[b2]{Bontemps and Mizon (2008)} and \hyperref[b20]{Mizon (2008)}. Applications of encompassing tests can be found inside the model selection procedure of general to specific (GETS) modelling developed by \hyperref[b12]{Hendry and Doornik (1994)}, \hyperref[b17]{Hoover and Perez (1999)}. For application in real data, see Nazir (2017).\par Recently, {\ref Bontemps et al. (2008)} have developed encompassing tests which cover large set of methods such as parametric and nonparametric methods. Among their results, encompassing tests for kernel nonparametric regression method are established. They provide asymptotic normality of the associated encompassing statistics under the independent and identically distributed hypothesis (i.i.d). We extend their results by relaxing the independent hypothesis. We then focus on processes with some dependence structures. This extension lies on the generalization of encompassing test to dependent processes.\par The paper is organized as follows. In section 2, we provide an overview of encompassing test. In section 3, we study the asymptotic behaviors of the encompassing test associated to the linear parametric modelling and the kernel nonparametric method. In last section, we conclude. The encompassing statistic is given by the difference between ?M 2 and ?( ?M 1 ) scaled by a coefficient a n . Specification of the encompassing test will depend on the estimation of the regression method: parametric or nonparametric methods. \section[{II. ENCOMPASSING TEST}]{II. ENCOMPASSING TEST} \section[{London Journal of Research in Science: Natural and Formal}]{London Journal of Research in Science: Natural and Formal}\par Let S = (Y, X, Z) be a zero mean random process with valued in RxR d xR q where d, q ? N * . For\par x ? R d and z ? R q , consider the two models M 1 and M 2 as the conditional expectations m(x) and g(z), respectively. These can be defined as follows:M 1 : m(x) = E[Y |X = x] and M 2 : g(z) = E[Y |Z = z] (2.1)\par Moreover, the general unrestricted model is given by r(x, z) = E[Y |X = x, Z = z].\par Following the encompassing test in {\ref Bontemps et al. (2008)}, we are interested in testing the hypothesis that M 1 encompasses M 2 , and then introducing the null hypothesis:H : E[Y |X = x, Z = z] = E[Y |X = x].\textbf{(2.2)}\par This null states that M 1 is the owner model, and M 2 will be served on validating this statement and is called the rival model. We test this hypothesis H through the following implicit encompassing hypothesis:H * : E[E[Y |X = x]/Z = z] = E[Y |Z = z].\textbf{(2.3)}\par 1 Kullback-Leiber Information Criterion\par The following homoskedasticity condition will be assumed all along this work:V ar[Y |X = x, Z = z] = ? 2 .\textbf{(2.4)}\par Moreover, a necessary condition for the encompassing test relies on the errors of both models where the intended encompassing model M 1 should have smaller standard error than the encompassed model M 2 .\par In general, M 1 or M 1 can be estimated using nonparametric or parametric regression method.\par We will consider these different situations when the processes (S n ) n are dependent. We begin by constructing the encompassing statistic associated to each of these four situations and then discuss their asymptotic behaviors. \section[{III. ASYMPTOTIC BEHAVIOR OF THE ENCOMPASSING STATISTIC}]{III. ASYMPTOTIC BEHAVIOR OF THE ENCOMPASSING STATISTIC}\par We are interested on the asymptotic behavior of the encompassing statistic associated to the null hypothesis M 1 EM 2 . We can encounter the following four situations: M 1 and M 2 are both estimated parametrically, M 1 and M 2 are both estimated nonparametrically, M 1 is estimated nonparametrically and M 2 parametrically and M 1 is estimated parametrically and M 2 nonparametrically. We will consider the kernel regression estimate for nonparametric methods and the linear regression for parametric methods. For both dependent processes, we will study and establish the asymptotic normality of the corresponding four encompassing tests. \section[{London Journal of Research in Science: Natural and Formal}]{London Journal of Research in Science: Natural and Formal}\par Consider a sample S i = (Y i , X i , Z i ), i = 1, ..., n, which can be viewed as realization of the random process S = (Y, X, Z) with valued in RxR d xR q where d, q ? N * . We suppose that S i , i = 1, ..., n has a joint density f . Moreover, ?(., .), ?(. | .) and ?(.) will denote the joint, the conditional and the marginal densities of the process (Y, Z), respectively. That is, for y ? R and z ? R q , ?(y, z), ?(y | z) and ?(z) correspond to the density of the following processes (Y, Z) at point (y, z), (Y | Z = z) at point y and Z at point z, respectively. Similarly, h will denote the joint, the conditional and the marginal densities of the process (Y, X), according to the argument that it takes.\par To get the asymptotic normality of the associated encompassing statistic, we need the following assumptions from \hyperref[b4]{Bosq (1998)}. The first assumption characterizes the dependence structure.Assumption 3.1. (S t ) is ?-mixing with ?(n) = O(n -? ) where ? > ? 2 +4\par 2? for some positive ?.\par The next assumption collects regularity conditions on the continuity and on the differentiability of the density functions. with ? (2) denotes any partial derivative of order 2 for ?. Next,Sup t?k ||?(Z 1 , Z t )|| ? < ? and last, ?(.)E[Y 2 1 |Z 1 = .\par ] is continuous at z.\par The last assumption concerns finiteness of the moments of (Y n , Z n ) n .Assumption 3.3. ||E[|Y 1 | 4+? |Z 1 = .]|| ? < ?; E[|Z 1 | 4+? ] < ? for some positive ?; Sup t?N ||E[Y i 1 Y j t |Z t = ., Z 1 = .]|| ? < ? where i ? 0, j ? 0, i + j = 2.\par Throughout this section, we assume the existence of continuous version of the various joint and marginal density functions and of the three conditional means m, g and r. In addition, the square integrability will be assumed.\par For more precision, N (µ, v) will denote the Gaussian distribution with mean µ and variance v.\par We now consider the first case that is the encompassing test when the two models M 1 and M 2 have parametric specification. London Journal of Research in Science: Natural and Formal \section[{Parametric modelling for M1 and M2}]{Parametric modelling for M1 and M2}\par Encompassing test for parametric modelling has been developed a lot in the literature. We discuss briefly one parametric encompassing test where models M 1 and M 2 have linear parametric specification. In that case, the two models M 1 and M 2 are given in relation (3.1) with the nesting model r:m(x) = ? x with ? = (E[XX ]) -1 E[XY ] g(z) = ? z with ? = (E[ZZ ]) -1 E[ZY ] r(x,z) = ? w with ? = E[W W ] -1 E[W Y ] and W = (X, Z). (3.1)\par We can get the estimates ?, ? and ? of the paramaters ?, ? and ?, respectively, using the sampleS i = (Y i , X i , Z i ), i = 1, ..., n. Now, testing M 1 EM 2 corresponds \section[{to the test of the null hypothesis}]{to the test of the null hypothesis}\par H where the conditional mean is just the linear projection. Therefore, the encompassing statistic of the null M 1 EM 2 can be written as follows.??,? = ? -?L ( ?),\textbf{(3.2)}\par where ?L ( ?) is an estimate of the pseudo-true value ? L (?) associated with ? on H 1 . Remarking that the pseudo-true value is defined by ? L (?) = (E[ZZ ]) -1 E[ZX ]?,? n ??,? ? N (0, ? 2 ?) in distribution as n ? ?. (3.3)\par where? = V ar(Z) -1 E[V ar(Z | X)]V ar(Z) -1 .\par For development on this asymptotic behavior of the encompassing statistic, we refer to Gouriéroux Next, we will study the completely nonparametric case.\par London Journal of Research in Science: Natural and Formal \section[{Nonparametric modelling for M1 and M2}]{Nonparametric modelling for M1 and M2}\par We now consider the case where the two models M 1 and M 2 defined in (2.1) are estimated using nonparametric techniques. To test the hypothesis " M 1 encompasses M 2 ", we build the corresponding encompassing statistic and establish asymptotic property of such statistic.\par Considering the functional estimates m n and g n of the unknown functions m and g in relation (2.1) respectively, we define the encompassing statistic as follows:?m,g (z) = g n (z) -?(m n )(z),\textbf{(3.4)}\par where ?(m n ) is an estimate of the pseudo true value G(m) associated with g n on H, which isdefined by G(m) = E[m | Z = z].\par Using the sample S i = (Y i , X i , Z i ), i = 1, ..., n, the kernel regression estimates m n of the function m, and g n of the function g have the following expressions:m n (x) = 1 nh d 1n n i=1 K 1 ( x-X i h 1n )Y i 1 nh d 1n n i=1 K 1 ( x-X i h 1n ) g n (z) = 1 nh q 2n n i=1 K 2 ( z-Z i h 2n )Y i 1 nh q 2n n i=1 K 2 ( z-Z i h 2n )\textbf{(3.5)}\par where h jn and K j , j = 1, 2 are window widths and kernel densities, respectively. The kernel densities satisfyK j (u) ? 0 and K j (u)du = 1 j = 1, 2.\textbf{(3.6)}\par We provide in the following, a theorem establishing the asymptotic convergence of the encompassing statistic.\par Theorem 3.2. Suppose that assumptions 3.1-3.3 hold. Moreover, suppose that relation (3.6) is satisfied. Then, under H, we get:nh q 2n ?m,g (z) ? N (0, ? 2 K 2 2 (u)du ?(z) ) in distribution as n ? ?.\textbf{(3.7)}\par ?(z) is the marginal density of the Z at z and? 2 = V ar[Y |X = x, Z = z].\par Proof of theorem 3.2 The proof of this theorem will be based on the decomposition of the expression of the encompassing statistic into two parts as follows: London Journal of Research in Science: Natural and Formalnh q 2n ?m,g (z) = nh q 2n (g n (z) -?(m n )(z)) = nh q 2n ( n t=1 K 2 ( z-Zt h 2n ) n t=1 K 2 ( z-Zt h 2n ) Y t - n t=1 K( z-Zt hn ) n t=1 K( z-Zt hn ) m n (x t )) = nh q 2n n t=1 K 2 ( z-Zt h 2n ) n t=1 K 2 ( z-Zt h 2n ) (Y t -m(x t )) + nh q 2n n t=1 K( z-Zt hn ) n t=1 K( z-Zt hn ) (m(x t ) -m n (x t )) = C 1 + C 2 .\par (3.8)\par The first part C 1 coincides to the kernel regression of the residuals t = Y t -m(x t ) onto Z t .\par When assumptions 3.1-3.3 hold, then under H, we achieved the convergence in distribution of the first part to a normal distribution using Rhomari's result in \hyperref[b4]{Bosq (1998)}. The second part C 2 reflects the limit in probability of the supremum of the difference m n (x t ) -m(x t ) at x t ? R d scaled by nh q n and its convergence can be derived from the rate of covergence of the uniform convergence of the estimate m n (x t ) which has been provided by Bosq (1998). \section[{Parametric modelling M1 vs nonparametric modelling M2}]{Parametric modelling M1 vs nonparametric modelling M2}\par We consider the case that model M 1 is a linear parametric model and M 2 is estimated by kernel nonparametric technique. Therefore, the hypothesis H will have linear parametric specification.\par The encompassing statistic associated to the null M 1 EM 2 can be written as follows:??,g (z) = g n (z) -?L ( ?)(z),\textbf{(3.9)}\par where ?L ( ?) is an estimate of the pseudo-true value G L (?)(z) associated with g n on H, which is defined by GL (?)(z) = ? E[X | Z = z].\par For the nonparametric specification of M 2 , we consider the estimate g n as the kernel regression estimate of g given in (2.1). Since the rival model g is estimated using kernel method, the various assumptions on kernel density and window width will be maintained.\par Even the process exhibits some dependences, we can still establish the asymptotic normality of the encompassing statistic defined in relation (3.9). London Journal of Research in Science: Natural and Formal with linear specification and when the bandwidth h 2n satisfy kernel regularity condition, we get:nh q 2n ??,g (z) ? N (0, ? 2 K 2 2 (u)du ?(z)\par ) in distribution as n ? ?.\par (3.10) ?(z) is the marginal density of the Z at z.\par Proof of theorem 3.3 Using similar techniques as previously, we can write the encompassing statistic as follows: converges in distribution to zero. This completes the proof.nh q 2n ??,g (z) = nh q 2n (g n (z) -?L ( ?)(z)) = nh q 2n ( n t=1 K 2 ( z-Zt hn ) n t=1 K 2 ( z-Zt h 2n ) Y t - n t=1 K( z-Zt hn ) n t=1 K( z-Zt hn ) ? X t ) = nh q 2n n t=1 K 2 ( z-Zt hn ) t=1 K 2 ( z-Zt h 2n ) (Y t -? X t ) + nh q \section[{Nonparametric modelling M1 vs parametric modelling M2}]{Nonparametric modelling M1 vs parametric modelling M2}\par We consider the owner model M 1 to be estimated using a nonparametric method and the rival model M 2 to be a linear parametric method. Therefore, the encompassing statistic associated to the null M 1 EM 2 is given by:?m,? = ? -?(m n ),\textbf{(3.12)}\par where ?(m n ) is an estimate of the pseudo-true value ?(m) associated with ? on H, which isdefined by ?(m) = (E[ZZ ]) -1 E[Zm].\par When the estimate of model M 1 is obtained from the kernel regression and the model M 2 is from linear parametric modelling, we summarize the asymptotic results in the following theorem. London Journal of Research in Science: Natural and Formal\begin{figure}[htbp] \noindent\textbf{32}\includegraphics[]{image-2.png} \caption{\label{fig_0}Assumption 3 . 2 .}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-3.png} \caption{\label{fig_1}}\end{figure} \begin{figure}[htbp] \noindent\textbf{33}\includegraphics[]{image-4.png} \caption{\label{fig_2}Theorem 3 . 3 .}\end{figure} \begin{figure}[htbp] \noindent\textbf{22}\includegraphics[]{image-5.png} \caption{\label{fig_3}2 ?(z) K 2 2}\end{figure} \label{foot_0}\footnote{\label{foot_0} © 2023 Great ] Britain Journals Press} \label{foot_1}\footnote{\label{foot_1} © 2023 Great ] Britain Journals Press} \label{foot_2}\footnote{\label{foot_2} ©} \label{foot_3}\footnote{\label{foot_3} Great ] Britain Journals Press} \label{foot_4}\footnote{\label{foot_4} London Journal of Research in Science: Natural and Formal} \label{foot_5}\footnote{\label{foot_5} © 2023 Great ] Britain Journals Press} \backmatter \par Funding statement : this work was performed as part of my employement at the Ministry of Higher Education and Scientific Research of Madagascar. London Journal of Research in Science: Natural and Formal \par Theorem 3.4. Assume that relation {\ref (2.4}) is satisfied. When the kernel K 1 and the bandwidth h 1n satisfy the usual regularity condition and when we have one of the following points: Assumption 3.1 holds and the kernel regression estimate m n and the process (Y n , X n ) n satisfy assumptions 3.2 and 3.3.\par Then, under H, we get:\par where ? = plim n?? V ar( ? n ?m,? ).\par Proof of theorem 3.4 We split the encompassing statistic ? n ?m,? into two parts. The first part yields\par ) which gives the asymptotic normality of the theorem, Peligrad and Utev (1997).\par The second part is\par . Again, we bound this by taking the supremum with respect to x i . Thus, F 2 vanishes to zero from the uniform convergence of m n (x i ), \hyperref[b4]{Bosq (1998)}. 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