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 $$ \boldsymbol{\hat{\varepsilon}} = \mathbf{y} - \mathbf{\hat{y}} \\ = \mathbf{y} - \mathbf{X} \boldsymbol{\hat{\beta}} . $$

.yminusyhat(y, yhat = NULL, X = NULL, betahat = NULL)

Arguments

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.
yhat	Numeric vector of length `n` or `n` by `1` numeric matrix. $n \times 1$ vector of predicted values of $\mathbf{y}$ $\left( \mathbf{\hat{y}} \right)$.
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.
betahat	Numeric vector of length `k` or `k` by `1` matrix. The vector $\boldsymbol{\hat{\beta}}$ is a $k \times 1$ vector of estimates of $k$ unknown regression coefficients.

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

Details

If yhat = NULL, the yhat vector is computed using Xbetahat() with X as a required argument and betahat as an optional argument. If yhat is provided, X and betahat are not needed.

References

Wikipedia: Errors and Residuals

Author

Ivan Jacob Agaloos Pesigan

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.
yhat	Numeric vector of length `n` or `n` by `1` numeric matrix. \(n \times 1\) vector of predicted values of \(\mathbf{y}\) \(\left( \mathbf{\hat{y}} \right)\).
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.
betahat	Numeric vector of length `k` or `k` by `1` matrix. The vector \(\boldsymbol{\hat{\beta}}\) is a \(k \times 1\) vector of estimates of \(k\) unknown regression coefficients.

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

Arguments

Value

Details

References

See also

Author