Estimates of Regression Coefficients \(\boldsymbol{\hat{\beta}}\) - QR Decomposition

Estimates coefficients of a linear regression model using QR Decomposition. The data matrix \(\mathbf{X}\) is decomposed into $$ \mathbf{X} = \mathbf{Q} \mathbf{R} . $$ Estimates are found by solving \(\boldsymbol{\hat{\beta}}\) in $$ \mathbf{R} \boldsymbol{\hat{\beta}} = \mathbf{Q}^{T} \mathbf{y}. $$

.betahatqr(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: QR decomposition

Wikipedia: Orthogonal decomposition methods

Wikipedia: Design matrix

See also

Other beta-hat functions: .betahatnorm(), .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"]]
.betahatqr(X = X, y = y)
#>       betahat
#> [1,] 4.874251
#> [2,] 0.197486

# Multiple regression----------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
# age is removed
X <- X[, -ncol(X)]
.betahatqr(X = X, y = y)
#>         betahat
#> [1,] -7.1833382
#> [2,] -3.0748755
#> [3,] -1.5653133
#> [4,]  1.0959758
#> [5,]  1.3703010
#> [6,]  0.1666065