Estimates of Regression Slopes \(\boldsymbol{\hat{\beta}}_{2, \cdots, k}\)

Source: R/betahat_matrix.R

Estimates of Regression Slopes \(\boldsymbol{\hat{\beta}}_{2, \cdots, k}\)

slopeshat(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 slopes \(\boldsymbol{\hat{\beta}}_{2, \cdots, k}\) of a linear regression model derived from the estimated variance-covariance matrix.

Details

Estimates of the linear regression slopes are calculated using $$ \boldsymbol{\hat{\beta}}_{2, \cdots, k} = \boldsymbol{\hat{\Sigma}}_{\mathbf{X}}^{T} \boldsymbol{\hat{\sigma}}_{\mathbf{y}, \mathbf{X}} $$

where

  • \(\boldsymbol{\hat{\Sigma}}_{\mathbf{X}}\) is the \(p \times p\) covariance matrix of the regressor variables \(X_2, X_3, \cdots, X_k\) and

  • \(\boldsymbol{\hat{\sigma}}_{\mathbf{y}, \mathbf{X}}\) is the \(p \times 1\) column vector of the covariances between the regressand \(y\) variable and regressor variables \(X_2, X_3, \cdots, X_k\)

See also

Other beta-hat functions: .betahatnorm(), .betahatqr(), .betahatsvd(), .intercepthat(), .slopeshatprime(), .slopeshat(), betahat(), intercepthat(), slopeshatprime()

Author

Ivan Jacob Agaloos Pesigan

Examples

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

#>       slopes
#> age 0.197486

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

#>                slopes
#> gender     -3.0748755
#> race       -1.5653133
#> union       1.0959758
#> education   1.3703010
#> experience  0.1666065