I work on implementing a GREG estimator, using these materials. I’m styck at the expression of the $\beta$-coefficient and can't really make heads or tails out of it.
$$\hat \beta = \left( \sum_{i \in s} d_i q_i x_i x’_i \right)^{-1} \sum_{i \in s} d_i q_i x_i y_i,$$
So $\beta$ represents a vector of $\beta$-coefficients. In order for this to make sense, the rest of these variables would have to be vectors as well. This is also implied by the apostrophe on $x_i$, indicating a transpose. But if the variables represent vectors, then how exacly are they multiplied? There are several ways of taking products of vectors. Are we taking the cross product, since we don’t want to end up with a scalar? Also, why would we be taking the cross product of $d_iq_ix_i$ with the transpose of $x_i$?
Would the cross product with a transpose of a vector somehow differ from the cross product of the original vector?
Consider a 'standard' linear regression in the form $\mathbf{y}=\mathbf{Xb+e}$. The estimator is known to be $\hat{\mathbf{b}}=\left( \mathbf{X}^T\mathbf{X} \right)^{-1} \mathbf{X}^T\mathbf{y}$.
Now if the $i$th row of $\mathbf{X}$ is denoted $\mathbf{x}_i^T$. It follows that $\mathbf{X}^T\mathbf{y}=\sum y_i \mathbf{x}_i$ and $\mathbf{X}^T\mathbf{X}=\sum \mathbf{x}_i \mathbf{x}_i^T$. So in the formula $y_i$ are scalars and simply denote the components of the response variable. The rest should follow easily.