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
\( \widehat{\phi}(x)=\frac{1}{b^2_n}\frac{\sum\limits_{\underset{ }{s=1}}^{S}\sum\limits_{\underset{}{r=0}}^{m-1}\left(x-Z_{s+Sr}\right)\exp\left(-\frac{\left(x-Z_{s+Sr} \right)^2}{2b_n^2}\right) }{\sum\limits_{\underset{}{s=1}}^{S}\sum\limits_{\underset{}{r=0}}^{m-1}\exp\left(-\frac{\left(x-Z_{s+Sr} \right)^2}{2b_n^2}\right) } \text{ with }b_n\rightarrow 0\).
GAMMA(x,phi,s,z,sigma)returns the matrix \(\mathbf{\Gamma}\). See Regui et al. (2024) for simple periodic coefficients regression model.
A list of independent variables with dimension \(p\).
phi_n.
A period of the regression model.
The residuals vector.
sd_estimation_for_each_s.
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. tools:::Rd_expr_doi("https://doi.org/10.1080/03610918.2024.2314662")