Seemingly Unrelated Regressions - The Model

The Model

Suppose there are m regression equations

y_{it} = x_{it}'\;\!\beta_i + \varepsilon_{it}, \quad i=1,\ldots,m.

Here i represents the equation number, and t = 1, …, T is the observation index. The number of observations is assumed to be large, so that in the analysis we take T → ∞, whereas the number of equations m remains fixed.

Each equation i has a single response variable yit, and a ki-dimensional vector of regressors xit. If we stack observations corresponding to the i-th equation into T-dimensional vectors and matrices, then the model can be written in vector form as

y_i = X_i\beta_i + \varepsilon_i, \quad i=1,\ldots,m,

where yi and εi are T×1 vectors, Xi is a T×ki matrix, and βi is a ki×1 vector.

Finally, if we stack these m vector equations on top of each other, the system will take form

\begin{pmatrix}y_1 \\ y_2 \\ \vdots \\ y_m \end{pmatrix} = \begin{pmatrix}X_1&0&\ldots&0 \\ 0&X_2&\ldots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\ldots&X_m \end{pmatrix} \begin{pmatrix}\beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end{pmatrix} + \begin{pmatrix}\varepsilon_1 \\ \varepsilon_2 \\ \vdots \\ \varepsilon_m \end{pmatrix} = X\beta + \varepsilon\,.

(1)

The assumption of the model is that error terms εit are independent across time, but may have cross-equation contemporaneous correlations. Thus we assume that E = 0 whenever t ≠ s, whereas E = σij. Denoting Σ = ] the m×m skedasticity matrix of each observation, the covariance matrix of the stacked error terms ε will be equal to

\Omega \equiv \operatorname{E} = \Sigma \otimes I_T,

where IT is the T-dimensional identity matrix and ⊗ denotes the matrix Kronecker product.

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