Regression Slopes $\boldsymbol{\beta}_{2, \cdots, k}$

Derives the slopes $\boldsymbol{\beta}_{2, \cdots, k}$ of a linear regression model ($\boldsymbol{\beta}$ minus the intercept) as a function of covariances.

.slopes(SigmaX = NULL, sigmayX = NULL, X, y)

Arguments

SigmaX	`p` by `p` numeric matrix. $p \times p$ matrix of variances and covariances between regressor variables ${X}_{2}, {X}_{3}, \cdots, {X}_{k}$ $\left( \boldsymbol{\Sigma}_{\mathbf{X}} \right)$.
sigmayX	Numeric vector of length `p` or `p` by `1` matrix. $p \times 1$ vector of covariances between the regressand $y$ variable and regressor variables $X_2, X_3, \cdots, X_k$ $\left( \boldsymbol{\sigma}_{\mathbf{y}, \mathbf{X}} = \left\{ \sigma_{y, X_2}, \sigma_{y, X_3}, \cdots, \sigma_{y, X_k} \right\}^{T} \right)$.
X	`n` by `k` numeric matrix. The data matrix $\mathbf{X}$ (also known as design matrix, model matrix or regressor matrix) is an $n \times k$ matrix of $n$ observations of $k$ regressors, which includes a regressor whose value is 1 for each observation on the first column.
y	Numeric vector of length `n` or `n` by `1` matrix. The vector $\mathbf{y}$ is an $n \times 1$ vector of observations on the regressand variable.

Value

Returns the slopes $\boldsymbol{\beta}_{2, \cdots, k}$ of a linear regression model derived from the variance-covariance matrix.

Details

The linear regression slopes are calculated using $$ \boldsymbol{\beta}_{2, \cdots, k} = \boldsymbol{\Sigma}_{\mathbf{X}}^{T} \boldsymbol{\sigma}_{\mathbf{y}, \mathbf{X}} $$

where

$\boldsymbol{\Sigma}_{\mathbf{X}}$ is the $p \times p$ covariance matrix of the regressor variables $X_2, X_3, \cdots, X_k$ and
$\boldsymbol{\sigma}_{\mathbf{y}, \mathbf{X}}$ is the $p \times 1$ column vector of the covariances between the regressand $y$ variable and regressor variables $X_2, X_3, \cdots, X_k$

Author

Ivan Jacob Agaloos Pesigan

SigmaX	`p` by `p` numeric matrix. \(p \times p\) matrix of variances and covariances between regressor variables \({X}_{2}, {X}_{3}, \cdots, {X}_{k}\) \(\left( \boldsymbol{\Sigma}_{\mathbf{X}} \right)\).
sigmayX	Numeric vector of length `p` or `p` by `1` matrix. \(p \times 1\) vector of covariances between the regressand \(y\) variable and regressor variables \(X_2, X_3, \cdots, X_k\) \(\left( \boldsymbol{\sigma}_{\mathbf{y}, \mathbf{X}} = \left\{ \sigma_{y, X_2}, \sigma_{y, X_3}, \cdots, \sigma_{y, X_k} \right\}^{T} \right)\).
X	`n` by `k` numeric matrix. The data matrix \(\mathbf{X}\) (also known as design matrix, model matrix or regressor matrix) is an \(n \times k\) matrix of \(n\) observations of \(k\) regressors, which includes a regressor whose value is 1 for each observation on the first column.
y	Numeric vector of length `n` or `n` by `1` matrix. The vector \(\mathbf{y}\) is an \(n \times 1\) vector of observations on the regressand variable.

Regression Slopes \(\boldsymbol{\beta}_{2, \cdots, k}\)

Arguments

Value

Details

See also

Author