Standard Errors of Standardized Estimates of Regression Coefficients (Yuan and Chan (2011))

sehatslopeshatprimedelta(X, y, adjust = FALSE)

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.
adjust	Logical. Use $n - 3$ adjustment for small samples.

Details

The $p$th estimated standard error is calculated using $$ \mathbf{\widehat{se}}_{\boldsymbol{\hat{\beta}}_{\text{p}}^{\prime}} = \sqrt{ \frac{\hat{\sigma}_{X_{p}}^{2} \hat{c}_{p} \hat{\sigma}_{\hat{\varepsilon}}^{2}}{n \hat{\sigma}_{y}^{2}} + \frac{\hat{\beta}_{p}^{2} \left[ \hat{\sigma}_{X_{p}}^{2} \left( \boldsymbol{\hat{\beta}}^{T} \boldsymbol{\hat{\Sigma}}_{X} \boldsymbol{\hat{\beta}} \right) - \hat{\sigma}_{X_{p}}^{2} \hat{\sigma}_{\hat{\varepsilon}}^{2} - \hat{\sigma}_{y, X_{p}}^{2} \right]}{n \hat{\sigma}_{y}^{4}} } $$ where

$p = \left\{2, 3, \cdots, k \right\}$
$\hat{\sigma}_{\hat{\varepsilon}}^{2}$ is the estimated residual variance
$\boldsymbol{\hat{\beta}}_{2, 3, \cdots, k} = \left\{ \hat{\beta}_{2}, \hat{\beta}_{3}, \cdots, \hat{\beta}_{k}\right\}^{T}$ is the $p \times 1$ column vector of estimated regression slopes
$\hat{\sigma}_{y}^{2}$ is the variance of the regressand variable $y$
$\boldsymbol{\hat{\Sigma}}_{\mathbf{X}}$ is the $p \times p$ estimated covariance matrix of the regressor variables $X_2, X_3, \cdots, X_k$
$\hat{\sigma}_{X_p}^{2}$ is the variance of the corresponding $p$th regressor variable.
$\hat{\sigma}_{y, X_{p}}^{2}$ is the covariance of the regressand variable $y$ and the regressor variables $X_2, X_3, \cdots, X_k$
$c_p$ is the diagonal element that corresponds to the regressor variable in $\boldsymbol{\Sigma}_{\mathbf{X}}^{-1}$
$n$ is the sample size

References

Yuan, K., Chan, W. (2011). Biases and Standard Errors of Standardized Regression Coefficients. Psychometrika 76, 670-690. doi:10.1007/s11336-011-9224-6.

Author

Ivan Jacob Agaloos Pesigan

Examples

# Simple regression------------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
X <- X[, c(1, ncol(X))]
y <- jeksterslabRdatarepo::wages.matrix[["y"]]
sehatslopeshatprimedelta(X = X, y = y)
#>      sehatslopeshatprime
#> [1,]          0.02556128

# Multiple regression----------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
# age is removed
X <- X[, -ncol(X)]
sehatslopeshatprimedelta(X = X, y = y)
#>      sehatslopeshatprime
#> [1,]          0.02282716
#> [2,]          0.02317122
#> [3,]          0.02342286
#> [4,]          0.02113537
#> [5,]          0.02330714