Say you want to compute the least squares estimate of $w$ from a data-set: $$ \begin{bmatrix}d_1 \\d_2 \\\vdots\\d_N \end{bmatrix} =\begin{bmatrix} x_1 \\x_2 \\ \vdots \\x_N\end{bmatrix}w + \begin{bmatrix} q_1 \\q_2 \\ \vdots \\q_N\end{bmatrix} $$ Note: $q$ is white noise. This can be compactly written as $$ \mathbf{d}= \mathbf{H}w + \mathbf{q} $$ The least squares estimate of $w$ which I shall call $w^*$ is given by \begin{align} w^* =& (\mathbf{H}^T\mathbf{H})^{-1}\mathbf{H}\mathbf{d}\\ = & \Big(\sum_{i =1} ^{N}x_i^2 \Big) ^{-1} \Big(\sum_{i =1} ^{N}x_id_i \Big) \end{align}
The Problem:
I have $N/2$ of the sample and my friend has the other half of the data set. We compute the least squares estimate separately. How do we now combine our estimates of $w^*$ besides 'averaging' it?