y-hat \(\left( \mathbf{\hat{y}} = \mathbf{P} \mathbf{y} \right)\)

Calculates y-hat \(\left( \mathbf{\hat{y}} \right)\), that is, the predicted value of \(\mathbf{y}\) given \(\mathbf{X}\) using $$ \mathbf{\hat{y}} = \mathbf{P} \mathbf{y} $$ where $$ \mathbf{P} = \mathbf{X} \left( \mathbf{X}^{T} \mathbf{X} \right)^{-1} \mathbf{X}^{T} . $$

Py(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 y-hat \(\left( \mathbf{\hat{y}} \right)\).

References

Wikipedia: Linear Regression

Wikipedia: Ordinary Least Squares

See also

Other y-hat functions: .Py(), .Xbetahat(), Xbetahat(), yhat()

Author

Ivan Jacob Agaloos Pesigan

Examples

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

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