lm_per {PerRegMod}R Documentation

Fitting periodic coefficients regression model by using LSE

Description

lm_per() function gives the least squares estimation of parameters, intercept \mu_s, slope \boldsymbol{\beta}_s, and standard deviation \sigma_s, of a periodic coefficients regression model using LSE_Reg_per and sd_estimation_for_each_s functions. \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]\ ,

\mathbf{X}^j_{s}=\left(x^j_{s},...,x^j_{s+(m-1)S}\right)^{'}, Y=(\mathbf{Y}_1^{'},...,\mathbf{Y}_S^{'})^{'}, \mathbf{Y}_{s} =(y_{s},...,y_{(m-1)S+s})^{'}, \mathbf{\epsilon}=(\mathbf{\epsilon}_{1}^{'},...,\mathbf{\epsilon}_{S}^{'})^{'}, \mathbf{\epsilon}_{s} =(\varepsilon_{s},...,\varepsilon_{(m-1)S+s})^{'}, \mathbf{1}_{m} is a vector of ones of dimension m, \textbf{I}_{S} is the identity matrix of dimension S, \otimes denotes the Kronecker product, and \boldsymbol{\vartheta} =\left(\boldsymbol{\mu}^{'} ,{\boldsymbol{\beta}}^{'}\right)^{'} with \boldsymbol{\mu}=(\mu_1,...,\mu_S)^{'} and \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 p.

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)

[Package PerRegMod version 4.4.3 Index]