Package 'PerRegMod'

Title: Fitting Periodic Coefficients Linear Regression Models
Description: Provides tools for fitting periodic coefficients regression models to data where periodicity plays a crucial role. It allows users to model and analyze relationships between variables that exhibit cyclical or seasonal patterns, offering functions for estimating parameters and testing the periodicity of coefficients in linear regression models. For simple periodic coefficient regression model see Regui et al. (2024) <doi:10.1080/03610918.2024.2314662>.
Authors: Slimane Regui [aut, cre] , Abdelhadi Akharif [aut], Amal Mellouk [aut]
Maintainer: Slimane Regui <[email protected]>
License: GPL
Version: 4.4.2
Built: 2024-11-26 20:47:09 UTC
Source: CRAN

Help Index

A Kronecker product B

Description

A_x_B() function gives A Kronecker product B

Usage

A_x_B(A,B)

Arguments

A

A matrix.
B

A matrix.

Value

A_x_B(A, B)

returns the matrix A Kronecker product B, A⊗BA\otimes B

Examples

A=matrix(rep(1,6),3,2)
B=matrix(seq(1,8),2,4 )
A_x_B(A,B)

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)=(Δ1∘(n)′,Δ2∘(n)′,Δ3∘(n)′)(Γ1∘Γ12∘0Γ12∘Γ22∘000Γ33∘)−1(Δ1∘(n)Δ2∘(n)Δ3∘(n))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 Δ1∘(n)=n−12∑r=0m−1(ϕ^(Z1+Sr)−ϕ^(ZS+Sr)⋮ϕ^(ZS−1+Sr)−ϕ^(ZS+Sr))\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),

Δ2∘(n)=n−122σ^∑r=0m−1(ψ^(Z1+Sr)−ψ^(ZS+Sr)⋮ψ^(ZS−1+Sr)−ψ^(ZS+Sr))\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),

Δ3∘(n)=n−12∑r=0m−1(ϕ^(Z1+Sr)K1(n)X1+Sr−ϕ^(ZS+Sr)KS(n)XS+Sr⋮ϕ^(ZS−1+Sr)KS−1(n)XS−1+Sr−ϕ^(ZS+Sr)KS(n)XS+Sr)\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), Γ11∘=I^nSΣ\mathbf{\Gamma}^{\circ} _{11}=\frac{\widehat{I}_n }{S} \Sigma, Γ22∘=I^n4Sσ^2Σ\mathbf{\Gamma}^{\circ} _{22}=\dfrac{\widehat{I}_n}{4S\widehat{\sigma}^2} \Sigma, Γ12∘=N^n2Sσ^Σ\mathbf{\Gamma}^{\circ} _{12}=\frac{ \widehat{N}_n }{2S\widehat{\sigma}} \Sigma, and Γ33∘=I^nSΣ⊗Ip×p\mathbf{\Gamma}^{\circ} _{33}=\frac{\widehat{I}_n }{S} \Sigma \otimes \mathbf{I}_{p\times p} with I^n=1nT∑s=1S∑r=0m−1ϕ^2(Z^s+Srσ^s)\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)}, N^n=1nT∑s=1S∑r=0m−1ϕ^2(Z^s+Srσ^s)Z^s+Srσ^s\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]\, Zs+Sr=ys+Sr−μ^s−∑j=1pβ^sjxs+Srjσ^sZ_{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}, Xs+Sr=(xs+Sr1,...,xs+Srp)′\mathbf{ X}_{s+Sr}=\left(x^1_{s+Sr},...,x^p_{s+Sr} \right)^{'}, Ks(n)=[(xs1)2‾xsixsj‾⋱xsjxsi‾(xsp)2‾]−12\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} },

xsixsj‾=1m∑r=0m−1xs+Srixs+Srj\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}}, (xsi)2‾=1m∑r=0m−1(xs+Sri)2\overline{(x^i_{s})^2 } =\frac{1}{m}\sum\limits_{\underset{ }{r=0}}^{m-1}{(x^i_{s+Sr})^2 }, ψ^(x)=xϕ^(x)−1\widehat{\psi}(x)=x\widehat{\phi}(x)-1, and

ϕ^(x)=1bn2∑s=1S∑r=0m−1(x−Zs+Sr)exp⁡(−(x−Zs+Sr)22bn2)∑s=1S∑r=0m−1exp⁡(−(x−Zs+Sr)22bn2)\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 bn→0b_n\rightarrow 0.

Usage

check_periodicity(x,y,s)

Arguments

x

A list of independent variables with dimension pp.
y

A response variable.
s

A period of the regression model.

Value

check_periodicity()

returns the value of observed statistic, T(n)T^{(n)}, degrees of freedom, (S−1)×(p+2)(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)

Calculating the component of vector DELTA

Description

DELTA() function gives the value of the component of vector DELTA Δ\boldsymbol{\Delta}. See Regui et al. (2024) for periodic simple regression model. \mathbf{\Delta}= \left[\begin{array}{c} \mathbf{\Delta}_1 \\ \mathbf{\Delta}_2\\ \mathbf{\Delta}_3 \end{array}\right]\, where Δ1\mathbf{\Delta}_1 is a vector of dimension SS with component n−12σ^s∑r=0m−1ϕ^(Zs+Sr,t)\frac{n^{\frac{-1}{2} } }{\widehat{ \sigma}_s}\sum\limits_{\underset{ }{r=0}}^{m-1}\widehat{\phi}(Z_{s+Sr,t}), Δ2\mathbf{\Delta}_2 is a vector of dimension pSpS with component n−12σ^s∑r=0m−1ϕ^(Zs+Sr)Ks(n)Xs+Sr\frac{ n^{\frac{-1}{2} } }{\widehat{\sigma}_{s}}\sum\limits_{\underset{ }{r=0}}^{m-1} \widehat{\phi}(Z_{s+Sr})K_{s}^{(n)} \mathbf{X}_{s+Sr}, Δ3\mathbf{\Delta}_3 is a vector of dimension SS with component n−122σ^s2∑r=0m−1Zs+Srϕ^(Zs+Sr)−1\frac{n^{\frac{-1}{2} } }{2\widehat{\sigma}_{s}^{2}}\sum\limits_{\underset{ }{r=0}}^{m-1}{Z_{s+Sr} \widehat{\phi}(Z_{s+Sr})-1 }.

Usage

DELTA(x,phi,s,e,sigma)

Arguments

x

A list of independent variables with dimension pp.
phi

phi_n.
s

A period of the regression model.
e

The residuals vector.
sigma

sd_estimation_for_each_s.

Value

DELTA()

returns the values of Δ\mathbf{\Delta}. See Regui et al. (2024) for simple periodic coefficients regression model.

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

Adaptive estimator for periodic coefficients regression model

Description

estimate_para_adaptive_method() function gives the adaptive estimation of parameters of a periodic coefficients regression model.

Usage

estimate_para_adaptive_method(n,s,y,x)

Arguments

n

The length of vector yy.
s

A period of the regression model.
y

A response variable.
x

A list of independent variables with dimension pp.

Value

beta_ad

Parameters to be estimated.

Examples

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)
model=lm(y~x1+x2+x3+x4)
z=model$residuals
estimate_para_adaptive_method(n,s,y,x)

Calculating the component of matrix GAMMA

Description

GAMMA() function gives the value of the component of matrix GAMMA Γ\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 Γ11=I^ndiag(1σ^12,...,1σ^S2)\mathbf{\Gamma}_{11}=\widehat{I}_{n}\text{diag}(\frac{1}{\widehat{\sigma}_{1}^{2}},...,\frac{1}{\widehat{\sigma}_{S}^{2}} ), Γ13=N^n2diag(1σ^13,...,1σ^S3)\mathbf{\Gamma}_{13}=\frac{\widehat{N}_{n}}{2}\text{diag}(\frac{1}{\widehat{\sigma}_{1}^{3}},...,\frac{1}{\widehat{\sigma}_{S}^{3}} ), Γ22=I^ndiag(1σ^12,...,1σ^S2)⊗Ip\mathbf{\Gamma}_{22}=\widehat{I}_{n}\text{diag}(\frac{1}{\widehat{\sigma}_{1}^{2}},...,\frac{1}{\widehat{\sigma}_{S}^{2}} ) \otimes \mathbf{I}_{p}, Γ33=J^n4diag(1σ^14,...,1σ^S4)\mathbf{\Gamma}_{33}=\frac{\widehat{J}_{n}}{4}\text{diag}(\frac{1}{\widehat{\sigma}_{1}^{4}},...,\frac{1}{\widehat{\sigma}_{S}^{4}} ), I^n=1nT∑s=1S∑r=0m−1ϕ^2(Z^s+Srσ^s)\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)}, N^n=1nT∑s=1S∑r=0m−1ϕ^2(Z^s+Srσ^s)Z^s+Srσ^s\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}, J^n=1nT∑s=1S∑r=0m−1ϕ^2(Z^s+Srσ^s)(Z^s+Srσ^s)2−1\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

ϕ^(x)=1bn2∑s=1S∑r=0m−1(x−Zs+Sr)exp⁡(−(x−Zs+Sr)22bn2)∑s=1S∑r=0m−1exp⁡(−(x−Zs+Sr)22bn2) with bn→0\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 pp.
phi

phi_n.
s

A period of the regression model.
z

The residuals vector.
sigma

sd_estimation_for_each_s.

Value

GAMMA()

returns the matrix Γ\mathbf{\Gamma}. See Regui et al. (2024) for simple periodic coefficients regression model.

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

Fitting periodic coefficients regression model by using LSE

Description

lm_per() function gives the least squares estimation of parameters, intercept μs\mu_s, slope βs\boldsymbol{\beta}_s, and standard deviation σs\sigma_s, of a periodic coefficients regression model using LSE_Reg_per and sd_estimation_for_each_s functions. ϑ^=(X′X)−1X′Y\widehat{\boldsymbol{\vartheta}}=\left(X^{'}X\right)^{-1}X^{'} Y where X= \left[\begin{array}{ccccccccccc} &\mathbf{X}^1_{1}&0&\ldots & 0& &\mathbf{X}^p_{1}&0&\ldots & 0 \\ & 0&\mathbf{X}^1_{2} &\ldots &0 & &0&\mathbf{X}^p_{2} &\ldots &0\\ \textbf{I}_{S}\otimes \mathbf{1}_{m} &0&0& \ddots&\vdots&\ldots&0& 0&\ddots&\vdots \\ & 0 &0&0 &\mathbf{X}^1_{S}& &0 &0&0 &\mathbf{X}^p_{S} \end{array}\right]\,

Xsj=(xsj,...,xs+(m−1)Sj)′\mathbf{X}^j_{s}=\left(x^j_{s},...,x^j_{s+(m-1)S}\right)^{'}, Y=(Y1′,...,YS′)′Y=(\mathbf{Y}_1^{'},...,\mathbf{Y}_S^{'})^{'}, Ys=(ys,...,y(m−1)S+s)′\mathbf{Y}_{s} =(y_{s},...,y_{(m-1)S+s})^{'}, ϵ=(ϵ1′,...,ϵS′)′\mathbf{\epsilon}=(\mathbf{\epsilon}_{1}^{'},...,\mathbf{\epsilon}_{S}^{'})^{'}, ϵs=(εs,...,ε(m−1)S+s)′\mathbf{\epsilon}_{s} =(\varepsilon_{s},...,\varepsilon_{(m-1)S+s})^{'}, 1m\mathbf{1}_{m} is a vector of ones of dimension mm, IS\textbf{I}_{S} is the identity matrix of dimension SS, ⊗\otimes denotes the Kronecker product, and ϑ=(μ′,β′)′\boldsymbol{\vartheta} =\left(\boldsymbol{\mu}^{'} ,{\boldsymbol{\beta}}^{'}\right)^{'} with μ=(μ1,...,μS)′\boldsymbol{\mu}=(\mu_1,...,\mu_S)^{'} and β=(β11,...,βS1;...;β1p,...,βSp)′\boldsymbol{\beta}=(\beta^1_{1},...,\beta^1_{S};...;\beta^p_{1},...,\beta^p_{S})^{'}.

Usage

lm_per(x,y,s)

Arguments

x

A list of independent variables with dimension pp.
y

A response variable.
s

A period of the regression model.

Value

Residuals

the residuals, that is response minus fitted values
Coefficients

a named vector of coefficients
Root mean square error

The root mean square error

Examples

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)
lm_per(x,y,s)

Fitting periodic coefficients regression model by using Adaptive estimation method

Description

lm_per_AE() function gives the adaptive estimation of parameters, intercept μs\mu_s, slope βs\boldsymbol{\beta}_s, and standard deviation σs\sigma_s, of a periodic coefficients regression model. θ^AE=ϑ^LSE+1nΓ−1Δ\widehat{\boldsymbol{\theta}}_{AE} ={\widehat{\boldsymbol{\vartheta} }_{LSE} }+\frac{1}{\sqrt{n}}{\mathbf{\Gamma}}^{-1}\mathbf{\Delta}.

Usage

lm_per_AE(x,y,s)

Arguments

x

A list of independent variables with dimension pp.
y

A response variable.
s

A period of the regression model.

Value

Residuals

the residuals, that is response minus fitted values
Coefficients

a named vector of coefficients
Root mean square error

The root mean square error

Examples

set.seed(6)
n=200
s=2
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)
lm_per_AE(x,y,s)

Least squares estimator for periodic coefficients regression model

Description

LSE_Reg_per() function gives the least squares estimation of parameters of a periodic coefficients regression model.

Usage

LSE_Reg_per(x,y,s)

Arguments

x

A list of independent variables with dimension pp.
y

A response variable.
s

A period of the regression model.

Value

beta

Parameters to be estimated.
X

Matrix of predictors.
Y

The response vector.

Examples

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)
LSE_Reg_per(x,y,s)

Calculating the value of ϕ\phi function

Description

phi_n() function gives the value of ϕ^(x)=1bn2∑s=1S∑r=0m−1(x−Zs+Sr)exp⁡(−(x−Zs+Sr)22bn2)∑s=1S∑r=0m−1exp⁡(−(x−Zs+Sr)22bn2)\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 bn=0.002b_n=0.002.

Usage

phi_n(x)

Arguments

x

A numeric value.

Value

returns the value of ϕ^(x)\widehat{\phi}(x)

Detecting periodicity of parameters in the regression model

Description

pseudo_gaussian_test() function gives the value of the statistic test, T(n)T^{(n)}, for detecting periodicity of parameters in the regression model. See check_periodicity function.

Usage

pseudo_gaussian_test(x,z,s)

Arguments

x

A list of independent variables with dimension pp.
z

The residuals vector.
s

A period of the regression model.

Value

returns the value of the statistic test, T(n)T^{(n)}.

Estimating periodic variances in a periodic coefficients regression model

Description

sd_estimation_for_each_s() function gives the estimation of variances, σ^s2=1m−p−1∑r=0m−1ε^s+Sr2\widehat{\sigma}_s^2=\frac{1}{m-p-1}\sum\limits_{\underset{ }{r=0}}^{m-1}\widehat{\varepsilon}^2_{s+Sr} for all s=1,...,Ss=1,...,S,in a periodic coefficients regression model.

Usage

sd_estimation_for_each_s(x,y,s,beta_hat)

Arguments

x

A list of independent variables with dimension pp.
y

A response variable.
s

A period of the regression model.
beta_hat

The least squares estimation using LSE_Reg_per.

Value

returns the value of σ^s2\widehat{\sigma}_s^2.

Examples

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)
beta_hat=LSE_Reg_per(x,y,s)$beta
sd_estimation_for_each_s(x,y,s,beta_hat)