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Class Notes in Statistics and Econometrics Part 33

CHAPTER 65 Disturbance Related (Seemingly Unrelated) Regressions. One has m timeseries regression equations y i = X i β i + εi . Everything is different: the dependent variables, the explanatory variables, the coefficient vectors. Even the numbers of the observations may be different, | CHAPTER 65 Disturbance Related Seemingly Unrelated Regressions One has m timeseries regression equations yi Xi i ei. Everything is different the dependent variables the explanatory variables the coefficient vectors. Even the numbers of the observations may be different The ith regression has ki explanatory variables and ti observations. They may be time series covering different but partly overlapping time periods. This is why they are called seemingly unrelated regressions. The only connection between the regressions is that for those observations which overlap in time the disturbances for different regressions are con-temperaneously correlated and these correlations are assumed to be constant over 1375 1376 65. SEEMINGLY UNRELATED time. In tiles this model is 65.0.18 m m 65.1. The Supermatrix Representation One can combine all these regressions into one big supermatrix as follows 65.1.1 yi y2 . . 1 o H- O X 2 . . O O . . . w 2 . . ei E2 . . . . . . V . . 65.1. THE SUPERMATRIX REPRESENTATION 1377 The covariance matrix of the disturbance term in 65.1.1 has the following striped form 65.1.2 1 U11111 U12112 2 U21I21 U22I22 V . . . . . . . . . _ m_ _Um 1I m 1 Um2I m2 U1mI1m im12m . . . UmmImm Here I j is the ti x tj matrix which has zeros everywhere except at the intersections of rows and columns denoting the same time period. In the special case that all time periods are identical i.e. all ti t one can define the matrices Y y1 ym and E e1 em and write the equations in matrix form as follows 65.1.3 Y X1P1 . X m m E H B E The vector of dependent variables and the vector of disturbances in the supermatrix representation 65.1.1 can in this special case be written in terms of the vectorization operator as vec Y and vec E. And the covariance matrix can be written as a Kronecker product V vec E S I since all Iij in 65.1.2 are t x t .