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

Source: R/betahat_matrix.R

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

.slopeshat(SigmaXhat = NULL, sigmayXhat = NULL, X, y)

Arguments

SigmaXhat

p by p numeric matrix. \(p \times p\) matrix of estimated variances and covariances between regressor variables \(X_2, X_3, \cdots, X_k\) \(\left( \boldsymbol{\hat{\Sigma}}_{\mathbf{X}} \right)\).
sigmayXhat

Numeric vector of length p or p by 1 matrix. \(p \times 1\) vector of estimated covariances between the regressand \(y\) variable and regressor variables \(X_2, X_3, \cdots, X_k\) \(\left( \boldsymbol{\hat{\sigma}}_{\mathbf{y}, \mathbf{X}} = \left\{ \hat{\sigma}_{y, X_2}, \hat{\sigma}_{y, X_3}, \cdots, \hat{\sigma}_{y, X_k} \right\}^{T} \right)\).
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 estimated slopes \(\boldsymbol{\hat{\beta}}_{2, \cdots, k}\) of a linear regression model derived from the estimated variance-covariance matrix.

Details

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

where

  • \(\boldsymbol{\hat{\Sigma}}_{\mathbf{X}}\) is the \(p \times p\) covariance matrix of the regressor variables \(X_2, X_3, \cdots, X_k\) and

  • \(\boldsymbol{\hat{\sigma}}_{\mathbf{y}, \mathbf{X}}\) is the \(p \times 1\) column vector of the covariances between the regressand \(y\) variable and regressor variables \(X_2, X_3, \cdots, X_k\)

See also

Other beta-hat functions: .betahatnorm(), .betahatqr(), .betahatsvd(), .intercepthat(), .slopeshatprime(), betahat(), intercepthat(), slopeshatprime(), slopeshat()

Author

Ivan Jacob Agaloos Pesigan