check_periodicity {PerRegMod}R Documentation

Checking the periodicity of parameters in the regression model

Description

check_periodicity() function allows to detect the periodicity of parameters in the regression model using pseudo_gaussian_test. See Regui et al. (2024) for periodic simple regression model. T^{(n)}=\left(\mathbf{\Delta}_{1}^{\circ(n)'},\mathbf{\Delta}_{2}^{\circ(n)'},\mathbf{\Delta}_{3}^{\circ(n)'} \right) \left(\begin{array}{ccc} \mathbf{\Gamma}^{\circ} _{1} & \mathbf{\Gamma}^{\circ}_{12} & \mathbf{0} \\ \mathbf{\Gamma}^{\circ}_{12} &\mathbf{\Gamma}^{\circ}_{22} & \mathbf{0} \\ \mathbf{0} &\mathbf{0} & \mathbf{\Gamma}^{\circ}_{33} \end{array} \right)^{-1} \left(\begin{array}{c} \mathbf{\Delta}_{1}^{\circ(n)} \\ \mathbf{\Delta}_{2}^{\circ(n)}\\ \mathbf{\Delta}_{3}^{\circ(n)} \end{array} \right), where \boldsymbol{\Delta}_{1}^{\circ(n)}= n^{\frac{-1}{2}} \sum\limits_{\underset{ }{r=0}}^{m-1} \left(\begin{array}{c} \widehat{\phi}(Z_{1+Sr})-\widehat{\phi}(Z_{S+Sr}) \\ \vdots\\ \widehat{\phi}(Z_{S-1+Sr})-\widehat{\phi}(Z_{S+Sr}) \end{array} \right),

\mathbf{\Delta}_{2}^{\circ(n)}= \frac{n^{\frac{-1}{2}}}{2\widehat{\sigma} }\sum\limits_{\underset{ }{r=0}}^{m-1} \left(\begin{array}{c} \widehat{\psi}(Z_{1+Sr})- \widehat{\psi}(Z_{S+Sr}) \\ \vdots\\ \widehat{\psi}(Z_{S-1+Sr})- \widehat{\psi}(Z_{S+Sr}) \\ \end{array}\right),

\mathbf{\Delta}_{3}^{\circ(n)}=n^{\frac{-1}{2}} \sum\limits_{\underset{ }{r=0}}^{m-1} \left( \begin{array}{c} \widehat{\phi}(Z_{1+Sr}) \mathbf{K}_1^{(n)}\mathbf{X}_{1+Sr}- \widehat{\phi}(Z_{S+Sr}) \mathbf{K}_S^{(n)}\mathbf{X}_{S+Sr}\\ \vdots\\ \widehat{\phi}(Z_{S-1+Sr})\mathbf{K}_{S-1}^{(n)}\mathbf{X}_{S-1+Sr}- \widehat{\phi}(Z_{S+Sr})\mathbf{K}_S^{(n)}\mathbf{X}_{S+Sr} \end{array} \right), \mathbf{\Gamma}^{\circ} _{11}=\frac{\widehat{I}_n }{S} \Sigma , \mathbf{\Gamma}^{\circ} _{22}=\dfrac{\widehat{I}_n}{4S\widehat{\sigma}^2} \Sigma, \mathbf{\Gamma}^{\circ} _{12}=\frac{ \widehat{N}_n }{2S\widehat{\sigma}} \Sigma, and \mathbf{\Gamma}^{\circ} _{33}=\frac{\widehat{I}_n }{S} \Sigma \otimes \mathbf{I}_{p\times p} with \widehat{I}_n=\frac{1}{nT}\sum\limits_{\underset{ }{s=1}}^{S}\sum\limits_{\underset{}{r=0}}^{m-1}{\widehat{\phi}^{2}\left(\frac{\widehat{Z}_{s+Sr}}{\widehat{ \sigma}_s} \right)}, \widehat{N}_n=\frac{1}{nT}\sum\limits_{\underset{ }{s=1}}^{S}\sum\limits_{\underset{ }{r=0}}^{m-1}{\widehat{\phi}}^{2}\left( \frac{\widehat{Z}_{s+Sr}}{\widehat{ \sigma}_s}\right)\frac{\widehat{Z}_{s+Sr}}{\widehat{ \sigma}_s},

\Sigma=\left[\begin{array}{cccc} 2 & 1& \ldots&1 \\ 1&\ddots & \ddots& \vdots\\ \vdots& \ddots &\ddots & 1 \\ 1&\ldots &1 & 2 \end{array}\right]\ , Z_{s+Sr}=\frac{y_{s+Sr}-\widehat{\mu}_s-\sum\limits_{\underset{}{j=1}}^{p}\widehat{\beta}^j_{s}x^j_{s+Sr}}{\widehat{\sigma}_s}, \mathbf{ X}_{s+Sr}=\left(x^1_{s+Sr},...,x^p_{s+Sr} \right)^{'}, \mathbf{K}^{(n)}_{s}=\left[\begin{array}{ccc} \overline{(x^1_{s})^2 } & &\overline{x^i_{s}x^j_{s} }\\ &\ddots & \\ \overline{x^j_{s}x^i_{s} } & &\overline{(x^p_{s})^2 } \end{array}\right]^{\frac{-1}{2} } ,

\overline{x^i_{s}x^j_{s} } =\frac{1}{m}\sum\limits_{\underset{ }{r=0}}^{m-1}{x^i_{s+Sr}x^j_{s+Sr}}, \overline{(x^i_{s})^2 } =\frac{1}{m}\sum\limits_{\underset{ }{r=0}}^{m-1}{(x^i_{s+Sr})^2 }, \widehat{\psi}(x)=x\widehat{\phi}(x)-1, and

\widehat{\phi}(x)=\frac{1}{b^2_n}\frac{\sum\limits_{\underset{ }{s=1}}^{S}\sum\limits_{\underset{}{r=0}}^{m-1}\left(x-Z_{s+Sr}\right)\exp\left(-\frac{\left(x-Z_{s+Sr} \right)^2}{2b_n^2}\right) }{\sum\limits_{\underset{}{s=1}}^{S}\sum\limits_{\underset{}{r=0}}^{m-1}\exp\left(-\frac{\left(x-Z_{s+Sr} \right)^2}{2b_n^2}\right) } with b_n\rightarrow 0.

Usage

check_periodicity(x,y,s)

Arguments

x

A list of independent variables with dimension p.

y

A response variable.

s

A period of the regression model.

Value

check_periodicity()

returns the value of observed statistic, T^{(n)}, degrees of freedom, (S-1)\times(p+2), and p-value

References

Regui, S., Akharif, A., & Mellouk, A. (2024). "Locally optimal tests against periodic linear regression in short panels." Communications in Statistics-Simulation and Computation, 1–15. doi:10.1080/03610918.2024.2314662

Examples

library(expm)
set.seed(6)
n=400
s=4
x1=rnorm(n,0,1.5)
x2=rnorm(n,0,0.9)
x3=rnorm(n,0,2)
x4=rnorm(n,0,1.9)
y=rnorm(n,0,2.5)
x=list(x1,x2,x3,x4)
check_periodicity(x,y,s)

[Package PerRegMod version 4.4.3 Index]