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 |
y |
A response variable. |
s |
A period of the regression model. |
Value
check_periodicity() |
returns the value of observed statistic, |
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)