Ordinary Least Squares

The Ordinary Least Squares (OLS) estimator is \(\hat{\beta} = (X^tX)^{-1}(X^tY)\) for a design matrix \(X\) and an outcome vector \(Y\) (Hansen 2022).

The following code shows how to compute the OLS estimator using Armadillo and sending data from R to C++ and viceversa using cpp11 and cpp11armadillo (Sanderson and Curtin 2016):

#include <cpp11.hpp>
#include <cpp11armadillo.hpp>

using namespace arma;
using namespace cpp11;

[[cpp11::register]] doubles_matrix<> ols_mat_(const doubles_matrix<>& x) {
  // Convert from R to C++
  Mat<double> Y = as_Mat(y);  // Col<double> Y = as_Col(y); also works
  Mat<double> X = as_Mat(x);

  // Armadillo operations
  Mat<double> XtX = X.t() * X;             // X'X
  Mat<double> XtX_inv = inv(XtX);          // (X'X)^(-1)
  Mat<double> beta = XtX_inv * X.t() * Y;  // (X'X)^(-1)(X'Y)

  // Convert from C++ to R
  return as_doubles_matrix(beta);
}

The previous code includes the cpp11 and cpp11armadillo libraries (cpp11armadillo calls Armadillo) to allow interfacing C++ with R. It also loads the corresponding namespaces in order to simplify the notation (i.e., using Mat instead of arma::Mat), and the function as_Mat() and as_doubles_mat() are provided by cpp11armadillo to pass a matrix object from R to C++ and that Armadillo can read and then pass it back to R.

The use of const and & are specific to the C++ language and allow to pass data from R to C++ without copying the data, and therefore saving time and memory.

cpp11armadillo provides flexibility and in the case of the resulting vector of OLS coefficients, it can be returned as a matrix or a vector. The following code shows how to create three functions to compute the OLS estimator and return the result as a matrix or a vector avoiding repeated code:

Mat<double> ols_(const doubles_matrix<>& y, const doubles_matrix<>& x) {
  Mat<double> Y = as_Mat(y);  // Col<double> Y = as_Col(y); also works
  Mat<double> X = as_Mat(x);

  Mat<double> XtX = X.t() * X;             // X'X
  Mat<double> XtX_inv = inv(XtX);          // (X'X)^(-1)
  Mat<double> beta = XtX_inv * X.t() * Y;  // (X'X)^(-1)(X'Y)

  return beta;
}

[[cpp11::register]] doubles_matrix<> ols_mat_(const doubles_matrix<>& y,
                                              const doubles_matrix<>& x) {
  Mat<double> beta = ols_(y, x);
  return as_doubles_matrix(beta);
}

[[cpp11::register]] doubles ols_dbl_(const doubles_matrix<>& y,
                                     const doubles_matrix<>& x) {
  Mat<double> beta = ols_(y, x);
  return as_doubles(beta);
}

In the previous code, the ols_mat_() function receives inputs from R and calls ols_() to do the computation on C++ side, and ols_dbl_() does the same but it returns a vector instead of a matrix.

Additional Examples

The package repository includes the directory cpp11armadillotest, which contains an R package that uses Armadillo, and that provides additional examples for eigenvalues, Cholesky and QR decomposition, and linear models.