Ordinary least squares (OLS) is a method of estimating the parameters in a linear regression model.

We have a dataset of n observations (y_i, \mathbf x_i), where \mathbf x_i are the column vectors of length p containing the parameters and y_i are the scalar responses. We set up the linear function for the i'th case as

y_i = \beta_1 x_{i1} + \beta_2 x_{i2} + \dots + \beta_p x_{ip} + \epsilon_i,

y_i = \mathbf x^T_i\beta + \epsilon_i .

Here \beta is an unknown p\times 1 vector and \epsilon_i is the error of the model in the i'th case. So for all i cases, we write the total equation on matrix form as

\mathbf y = \mathbf X\mathbf \beta + \mathbf \epsilon ,

with \mathbf X being an n\times p matrix, often called the design matrix or matrix of regressors. It's worth noting that - contrary to what one might think - the i'th row of \mathbf X equals \mathbf x_i^T. Also, the constant term is included in our equation by setting x_{i1} = 1\ \forall\ i=1,\dots, n.

Now because of the errors, \mathbf y = \mathbf X\mathbf \beta does not have a unique solution. However, we can find the best fit by finding the vector \hat \beta that minimizes some sort of cost function,

\hat \beta = \underset{\beta}{\arg \min} S(\beta) .

\hat \beta will give us an estimate of \mathbf y, namely \hat{\mathbf y} = \mathbf X \hat \beta. In the case of OLS, the cost function we use is the

\text{RSS}(\beta) = \sum_{i=1}^N \left(y_i - \hat y_i\right)^2 .

Changing into matrix notation again, we get

\text{RSS}(\beta) = (\mathbf y - \hat{\mathbf y})^T(\mathbf y - \hat{\mathbf y}) = (\mathbf y - \mathbf X\beta)^T(\mathbf y - \mathbf X\beta) ,

which we can differentiate with respect to \beta to find the minimum.

\frac{\partial \text{RSS}}{\partial \beta} = -2\mathbf X^T (\mathbf y - \mathbf X\beta) .

Assuming full column rank for \mathbf X, (\mathbf X^T \mathbf X) is thus positive definite (and importantly, invertible). Setting the first derivative to $0$, we get1:

\mathbf X^T(\mathbf y - \mathbf X\beta) = 0

\Rightarrow \hat \beta = (\mathbf X^T\mathbf X)^{-1}\mathbf X^T \mathbf y

1

Hastie, T., Tibshirani, R., & Friedman, J. H. (2009). The elements of statistical learning: data mining, inference, and prediction (2nd ed). Springer.

11:42 Wednesday 26 May 2021