Estimates of Regression Coefficients \( \boldsymbol{\hat{\beta}} = \left( \mathbf{X}^{T} \mathbf{X} \right)^{-1} \left( \mathbf{X}^{T} \mathbf{y} \right) \)

Estimates coefficients of a linear regression model using $$ \boldsymbol{\hat{\beta}} = \left( \mathbf{X}^{T} \mathbf{X} \right)^{-1} \left( \mathbf{X}^{T} \mathbf{y} \right) . $$ Also know as the normal equation.

.betahatnorm(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 \(\boldsymbol{\hat{\beta}}\), that is, a \(k \times 1\) vector of estimates of \(k\) unknown regression coefficients estimated using ordinary least squares.

References

Wikipedia: Linear regression

Wikipedia: Ordinary least squares

Wikipedia: Inverting the matrix of the normal equations

Wikipedia: Design matrix

See also

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

Author

Ivan Jacob Agaloos Pesigan

Examples

# Simple regression------------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
X <- X[, c(1, ncol(X))]
y <- jeksterslabRdatarepo::wages.matrix[["y"]]
.betahatnorm(X = X, y = y)
#>           betahat
#> constant 4.874251
#> age      0.197486

# Multiple regression----------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
# age is removed
X <- X[, -ncol(X)]
.betahatnorm(X = X, y = y)
#>               betahat
#> constant   -7.1833382
#> gender     -3.0748755
#> race       -1.5653133
#> union       1.0959758
#> education   1.3703010
#> experience  0.1666065