Residual Sum of Squares

Calculates the residual sum of squares \(\left( \mathrm{RSS} \right)\) using $$ \mathrm{RSS} = \sum_{i = 1}^{n} \left( Y_i - \hat{Y}_i \right)^2 \\ = \sum_{i = 1}^{n} \left( Y_i - \left[ \hat{\beta}_{1} + \hat{\beta}_{2} X_{2i} + \hat{\beta}_{3} X_{3i} + \dots + \hat{\beta}_{k} X_{ki} \right] \right)^2 \\ = \sum_{i = 1}^{n} \left( Y_i - \hat{\beta}_{1} - \hat{\beta}_{2} X_{2i} - \hat{\beta}_{3} X_{3i} - \dots - \hat{\beta}_{k} X_{ki} \right)^2 . $$ In matrix form $$ \mathrm{RSS} = \sum_{i = 1}^{n} \left( \mathbf{y} - \mathbf{\hat{y}} \right)^{2} \\ = \sum_{i = 1}^{n} \left( \mathbf{y} - \mathbf{X} \boldsymbol{\hat{\beta}} \right)^{2} \\ = \left( \mathbf{y} - \mathbf{X} \boldsymbol{\hat{\beta}} \right)^{\prime} \left( \mathbf{y} - \mathbf{X} \boldsymbol{\hat{\beta}} \right) . $$ Or simply $$ \mathrm{RSS} = \sum_{i = 1}^{n} \boldsymbol{\hat{\varepsilon}}_{i}^{2} = \boldsymbol{\hat{\varepsilon}}^{\prime} \boldsymbol{\hat{\varepsilon}} $$ where \(\boldsymbol{\hat{\varepsilon}}\) is an \(n \times 1\) vector of residuals, that is, the difference between the observed and predicted value of \(\mathbf{y}\) \(\left( \boldsymbol{\hat{\varepsilon}} = \mathbf{y} - \mathbf{\hat{y}} \right)\). Equivalent computational matrix formula $$ \mathrm{RSS} = \mathbf{y}^{\prime} \mathbf{y} - 2 \boldsymbol{\hat{\beta}} \mathbf{X}^{\prime} \mathbf{y} + \boldsymbol{\hat{\beta}}^{\prime} \mathbf{X}^{\prime} \mathbf{X} \boldsymbol{\hat{\beta}}. $$ Note that $$ \mathrm{TSS} = \mathrm{ESS} + \mathrm{RSS}. $$

RSS(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 residual sum of squares \(\left( \mathrm{RSS} \right)\).

Details

If betahat = NULL, betahat computed using betahat().

References

Wikipedia: Residual Sum of Squares

Wikipedia: Explained Sum of Squares

Wikipedia: Total Sum of Squares

Wikipedia: Coefficient of Determination

See also

Other sum of squares functions: .ESS(), .RSS(), ESS(), TSS()

Author

Ivan Jacob Agaloos Pesigan

Examples

# Simple regression------------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
X <- X[, c(1, ncol(X))]
y <- jeksterslabRdatarepo::wages.matrix[["y"]]
RSS(X = X, y = y)
#> [1] 73673.13

# Multiple regression----------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
# age is removed
X <- X[, -ncol(X)]
RSS(X = X, y = y)
#> [1] 54342.54