I have the following panel data problem:
Let $x_{it}^{\prime}$ represent a vector of dimensions $1\times k$, $u_{it}$ a scalar vector with dimensions $1\times 1$, and $f_{t}$ a vector with dimensions $1\times r$. It's worth noting that when expressed in matrix form, $X$ becomes an $(N\times T)\times k$ matrix, $u_{it}$ transforms into $U$, an $(N\times T)\times 1$ matrix, and $f_{t}$ into F, an $T\times r$ matrix. The objective is to compute $z_{it}$ as defined by:
$z_{it}=x_{it} -p_{i}(r_{i})^{-1}f_{t}$
Here, the intermediary quantities are defined as:
$p_{i}=\frac{1}{T}\sum_{t=1}^{T}u_{it}x_{it}f_{t}^{\prime}$
$r_{i}=\frac{1}{T}\sum_{t=1}^{T}u_{it}f_{t}f_{t}^{\prime}$
Compute $z_{it}$.
My solution: Start by constructing $F_{B}$ as a block diagonal matrix with dimensions $(N\times T)\times (N\times r)$. This matrix can be generated using $F_{B}=I_N \otimes F$, where $\otimes$ denotes the Kronecker product, and $I_N$ is an $N\times N$ identity matrix. Additionally, I introduced $U_{d}$, a diagonal matrix with dimensions $(N\times T)\times (N\times T)$, containing the elements of $U$ along its diagonal. Now i did the following matrix operations:
$P=\dfrac{1}{T}F_{B}^{\prime}U_{d}X$
$R=\dfrac{1}{T}F_{B}^{\prime}U_{d}F_{B}$
Subsequently, I arrived at the matrix $Z^{\prime}$ by applying the equation:
$Z^{\prime}= X-P^{\prime}(R)^{-1}F_{B}^{\prime}$
I have two questions:
Is this approach correct?
How does the formula for $p_{i}$ and $r_{i}$ translated into the matrix multiplications i provided (to get to this answer i employed my understanding of the final dimensions of $Z^{\prime}$ ( that i estimated as $p\times (N\times T)$) because i couldn't understand the formulas for $p_{i}$, $r_{i}$ and $p_{i}(r_{i})^{-1}f_{t}$)?
Thank you :)