Consider the seemingly unrelated regression (SUR) model (1): y=XiBi + ei for i = 1, ...,...

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Consider the seemingly unrelated regression (SUR) model (1): y₁=XiBi + ei for i = 1, ..., M where y₁, X, and B, are of sizes T x 1, T x K, and K₁ x 1, respectively. The error term vector e is of size T x 1, with E(e) = 0, for i = 1, ..., M and cov(e; e) = alt for i, j = 1, ..., M. The stacked form of the model above is (2): y = x + e, where y. X are ß are respectively of sizes MT x 1, MT XK and Σ K₁ × 1. Precisely: (3): y = (4): e X = The MT x 1 error term vector e is such that 011/T OIMIT *** = E 0 1 XMJ Page 2 of 4 1₁ IT and B = -w). = W, where Σ = = 011 *** 01M 1 LOMILT UMMIT OMI (a) (6 points) Under the assumption that the individual regressions have the same set of regressors, say X₁ = X (hence K₁= K) for all i = 1, ..., M, show that running GLS or FGLS estimation technique on the whole regression system (2) would be equivalent to running LS on the whole system or the individual regression models (1). MM. (b) (6 points) Show that the result above holds under the assumption that the errors driving the different regression models are contemporaneously uncorrelated regardless of whether the X₂'s are the same. That is, oj = 0 when i # j for i, j = 1, ..., M. Otherwise said, the covariance matrix of e is diagonal, namely, £ = diag{011,...,OMM}. Consider the seemingly unrelated regression (SUR) model (1): y₁=XiBi + ei for i = 1, ..., M where y₁, X, and B, are of sizes T x 1, T x K, and K₁ x 1, respectively. The error term vector e is of size T x 1, with E(e) = 0, for i = 1, ..., M and cov(e; e) = alt for i, j = 1, ..., M. The stacked form of the model above is (2): y = x + e, where y. X are ß are respectively of sizes MT x 1, MT XK and Σ K₁ × 1. Precisely: (3): y = (4): e X = The MT x 1 error term vector e is such that 011/T OIMIT *** = E 0 1 XMJ Page 2 of 4 1₁ IT and B = -w). = W, where Σ = = 011 *** 01M 1 LOMILT UMMIT OMI (a) (6 points) Under the assumption that the individual regressions have the same set of regressors, say X₁ = X (hence K₁= K) for all i = 1, ..., M, show that running GLS or FGLS estimation technique on the whole regression system (2) would be equivalent to running LS on the whole system or the individual regression models (1). MM. (b) (6 points) Show that the result above holds under the assumption that the errors driving the different regression models are contemporaneously uncorrelated regardless of whether the X₂'s are the same. That is, oj = 0 when i # j for i, j = 1, ..., M. Otherwise said, the covariance matrix of e is diagonal, namely, £ = diag{011,...,OMM}.

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a If we assume that the individual regressions have the same set of regressors meaning that each of ... View the full answer