Residual Variance \(\hat{\sigma}_{\hat{\varepsilon} \ \textrm{biased}}^{2}\)

Calculates an estimate of the error variance $$ \mathbf{E} \left( \sigma^2 \right) = \hat{\sigma}_{\hat{\varepsilon}}^{2} $$ $$ \hat{\sigma}_{\hat{\varepsilon} \ \textrm{biased}}^{2} = \frac{1}{n} \sum_{i = 1}^{n} \left( \mathbf{y} - \mathbf{X} \boldsymbol{\hat{\beta}} \right)^2 \\ = \frac{\boldsymbol{\hat{\varepsilon}}^{\prime} \boldsymbol{\hat{\varepsilon}}}{n} \\ = \frac{\mathrm{RSS}}{n} $$ where \(\boldsymbol{\hat{\varepsilon}}\) is the vector of residuals, \(\mathrm{RSS}\) is the residual sum of squares, and \(n\) is the sample size.

sigma2hatepsilonhatbiased(X, y)

Arguments

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 residual variance \(\hat{\sigma}_{\hat{\varepsilon} \ \textrm{biased}}^{2}\) .

References

Wikipedia: Linear Regression

Wikipedia: Ordinary Least Squares

See also

Other residual variance functions: .sigma2hatepsilonhatbiased(), .sigma2hatepsilonhat(), sigma2hatepsilonhat()

Author

Ivan Jacob Agaloos Pesigan

Examples

# Simple regression------------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
X <- X[, c(1, ncol(X))]
y <- jeksterslabRdatarepo::wages.matrix[["y"]]
sigma2hatepsilonhatbiased(X = X, y = y)
#> [1] 57.15526

# Multiple regression----------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
# age is removed
X <- X[, -ncol(X)]
sigma2hatepsilonhatbiased(X = X, y = y)
#> [1] 42.15868