GAMMA {PerRegMod} | R Documentation |
Calculating the component of matrix GAMMA
Description
GAMMA() function gives the value of the component of matrix GAMMA \boldsymbol{\Gamma}
. See Regui et al. (2024) for periodic simple regression model.
\mathbf{\Gamma}=\frac{1}{S}
\left[\begin{array}{ccc}
\left(\mathbf{\Gamma}_{11}\right)_{S \times S }&\mathbf{0} & \mathbf{\Gamma}_{13} \\
\mathbf{0} &\left(\mathbf{\Gamma}_{22} \right)_{pS\times pS } &\mathbf{0} \\
\mathbf{\Gamma}_{13} & \mathbf{0}& \left(\mathbf{\Gamma}_{33} \right)_{S\times S}
\end{array}\right]\
, where \mathbf{\Gamma}_{11}=\widehat{I}_{n}\text{diag}(\frac{1}{\widehat{\sigma}_{1}^{2}},...,\frac{1}{\widehat{\sigma}_{S}^{2}} )
, \mathbf{\Gamma}_{13}=\frac{\widehat{N}_{n}}{2}\text{diag}(\frac{1}{\widehat{\sigma}_{1}^{3}},...,\frac{1}{\widehat{\sigma}_{S}^{3}} )
,
\mathbf{\Gamma}_{22}=\widehat{I}_{n}\text{diag}(\frac{1}{\widehat{\sigma}_{1}^{2}},...,\frac{1}{\widehat{\sigma}_{S}^{2}} ) \otimes \mathbf{I}_{p}
,
\mathbf{\Gamma}_{33}=\frac{\widehat{J}_{n}}{4}\text{diag}(\frac{1}{\widehat{\sigma}_{1}^{4}},...,\frac{1}{\widehat{\sigma}_{S}^{4}} )
, \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}
, \widehat{J}_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)\left( \frac{\widehat{Z}_{s+Sr}}{\widehat{
\sigma}_s}\right)^{2}-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) } \text{ with }b_n\rightarrow 0
.
Usage
GAMMA(x,phi,s,z,sigma)
Arguments
x |
A list of independent variables with dimension |
phi |
|
s |
A period of the regression model. |
z |
The residuals vector. |
sigma |
Value
GAMMA() |
returns the matrix |
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