Estimated Regression Intercept \(\hat{\beta}_{1}\)

Estimated Regression Intercept \(\hat{\beta}_{1}\)

intercepthat(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 intercept \(\hat{\beta}_1\) of a linear regression model derived from the estimated means and the slopes \(\left( \boldsymbol{\hat{\beta}}_{2, \cdots, k} \right)\) .

Details

The intercept \(\beta_1\) is given by $$ \hat{\beta}_1 = \hat{\mu}_y - \boldsymbol{\hat{\mu}}_{\mathbf{X}} \boldsymbol{\hat{\beta}}_{2, \cdots, k}^{T} . $$

See also

Other beta-hat functions: .betahatnorm(), .betahatqr(), .betahatsvd(), .intercepthat(), .slopeshatprime(), .slopeshat(), betahat(), 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"]]
intercepthat(X = X, y = y)
#>    wages 
#> 4.874251 

# Multiple regression----------------------------------------------
X <- jeksterslabRdatarepo::wages.matrix[["X"]]
# age is removed
X <- X[, -ncol(X)]
intercepthat(X = X, y = y)
#>     wages 
#> -7.183338