In classical econometric models, we typically study linear relationships of the form:
$$ Y = C_1 X_1 + C_2 X_2 + \dots + C_n X_n + \epsilon, $$
where $Y$ is a scalar dependent variable, $X_1, X_2, \dots, X_n$ are scalar independent variables, $C_1, C_2, \dots, C_n$ are scalar coefficients, and $\epsilon$ is the $1$ dimensional error term.
However, in my case:
- $Y$ is a $k$-dimensional vector ($Y \in \mathbb{R}^k$),
- $X_1, X_2, \dots, X_n$ are also $k$-dimensional vectors, and
- $C_1, C_2, \dots, C_n$ are $k \times k$ matrices.
The model takes the form:
$$ Y = C_1 X_1 + C_2 X_2 + \dots + C_n X_n + \epsilon, $$
and $\epsilon$ is the $k$ dimensional error term.
where the addition and multiplication are matrix-vector operations.
My question is: Has such a model been formally described in the literature? If so:
- Who introduced this model, and in what context?
- Are there any specific references or foundational papers/books that discuss this type of model?
Is this a suitable topic for a master’s dissertation? If so, what areas or questions within this framework could be worth exploring?
I would appreciate any pointers to theoretical discussions or applications of this type of vector-valued linear econometric model.
