Model-Implied Variance-Covariance Matrix \(\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)\)

Model-implied variance-covariance matrix \(\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)\) from parameters of a \(k\)-variable linear regression model.

Sigmatheta(slopes, sigma2epsilon, SigmaX)

Arguments

slopes	Numeric vector of length p or p by 1 matrix. \(p \times 1\) column vector of regression slopes \(\left( \boldsymbol{\beta}_{2, 3, \cdots, k} = \left\{ \beta_2, \beta_3, \cdots, \beta_k \right\} \right)\) .
sigma2epsilon	Numeric. Variance of the error term \(\varepsilon\) \(\left( \sigma_{\varepsilon}^{2} \right)\).
SigmaX	p by p numeric matrix. \(p \times p\) matrix of variances and covariances between regressor variables \({X}_{2}, {X}_{3}, \cdots, {X}_{k}\) \(\left( \boldsymbol{\Sigma}_{\mathbf{X}} \right)\).

Value

Returns the model-implied variance-covariance matrix \(\boldsymbol{\Sigma} \left( \boldsymbol{\theta} \right)\). Note that the first item corresponds to y. The rest of the items correspond to how SigmaX is arranged.

Details

The following are the parameters of a linear regression model for the covariance structure

• \(\boldsymbol{\beta}_{2, \cdots, k}\) is the \(p \times 1\) column vector of regression slopes,

• \(\sigma_{\varepsilon}^{2}\) is the variance of the error term \(\varepsilon\), and

• \(\boldsymbol{\Sigma}_{\mathbf{X}}\) is the \(p \times p\) matrix of variances and covariances of \({X}_{2}, {X}_{3}, \cdots, {X}_{k}\).

See also

Other model-implied functions: mutheta()

Author

Ivan Jacob Agaloos Pesigan

Examples

slopes <- c(0.207648, 0.451039)
sigma2epsilon <- 0.9310598
SigmaX <- matrix(
  data = c(1.2934694, 0.4379592, 0.4379592, 1.0779592),
  ncol = 2
)
Sigmatheta(slopes = slopes, sigma2epsilon = sigma2epsilon, SigmaX = SigmaX)
#>           [,1]      [,2]      [,3]
#> [1,] 1.2471452 0.4661230 0.4862016
#> [2,] 0.2685863 1.2934694 0.0000000
#> [3,] 0.5771430 0.4379592 1.0779592