Residuals \( \left( \boldsymbol{\hat{\varepsilon}} = \mathbf{y} - \mathbf{\hat{y}} \right) \)

Calculates residuals using $$ \hat{\varepsilon}_{i} = Y_{i} - \hat{Y}_{i} \\ = Y_{i} - \left( \hat{\beta}_{1} + \hat{\beta}_{2} X_{2i} + \hat{\beta}_{3} X_{3i} + \dots + \hat{\beta}_{k} X_{ki} \right) \\ = Y_{i} - \hat{\beta}_{1} - \hat{\beta}_{2} X_{2i} - \hat{\beta}_{3} X_{3i} - \dots - \hat{\beta}_{k} X_{ki} . $$ In matrix form $$ \boldsymbol{\hat{\varepsilon}} = \mathbf{y} - \mathbf{\hat{y}} \\ = \mathbf{y} - \mathbf{X} \boldsymbol{\hat{\beta}} . $$

epsilonhat(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 an \(n \times 1\) matrix of residuals \(\left( \boldsymbol{\hat{\varepsilon}} \right)\), that is, the difference between the observed \(\left( \mathbf{y} \right)\) and predicted \(\left( \mathbf{\hat{y}} \right)\) values of the regressand variable \(\left( \boldsymbol{\hat{\varepsilon}} = \mathbf{y} - \mathbf{\hat{y}} \right)\).

References

Wikipedia: Errors and Residuals

See also

Other residuals functions: .My(), .tepsilonhat(), .yminusyhat(), My(), tepsilonhat(), yminusyhat()

Author

Ivan Jacob Agaloos Pesigan

Examples

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

# Multiple regression----------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
# age is removed
X <- X[, -ncol(X)]
epsilonhat <- epsilonhat(X = X, y = y)
hist(epsilonhat)