A Note on Two-Way ECM Estimation of SUR Systems on Unbalanced Panel Data

Abstract

This article considers the two-way error components model (ECM) estimation of seemingly unrelated regressions (SUR) on unbalanced panel by generalized least squares (GLS). As suggested by Biørn (2004) for the one-way case, in order to use the standard results for the balanced case the individuals are arranged in groups according to the number of times they are observed. Thus, the GLS estimator can be interpreted as a matrix weighted average of the group specific GLS estimators with weights equal to the inverse of their respective covariance matrices.

Heteroscedastic stratified two-way EC models of single equations and SUR systems

“…Assuming homoscedasticity and no serial correlation (i.e., the assumption RE.3 in Appendix B of Platoni et al, 2012), the variance-covariance matrix Ω has the following form:…”

“…The homoscedastic time variance is σ 2 ν = 6.271, while the heteroscedastic variances have been generated with ϕ 2 a = σ 2 µ (1 + λx 2 a ) 2 , where σ 2 µ = 6.488, and ψ 2 a = σ 2 u (1 + λx 2 a ) 2 , where σ 2 u = 6.039; λ is assigned values 0, 1, and 2, where λ = 0 denotes the homoscedastic case and the degree of heteroscedasticity increases as the value of λ becomes larger. 12 Finally, the independent variables' values x kit (k = 1, 2, 3) are generated according to a modified version of the scheme introduced by Nerlove (1971) and used, among others, by Baltagi (1981), Wansbeek and Kapteyn (1989), and Platoni et al (2012):…”

“…where ι p is a p × 1 vector of ones (see Platoni et al, 2012). The pM × pM heteroscedastic variance-covariance matrix of the pM × 1 composite error terms ε i(a,p) for the individual i ∈ I a,p , with I a,p = I a ∩ I p the set of individuals belonging to stratum a and observed in p periods, is given by:…”

“…The estimated variance-covarianceσ u,m j is obtained by equating q n,m j to its expected value (see Platoni et al, 2012). Furthermore, the estimator of ϕ a,m j is: ϕ a,m j = q N a ,m j − N a − 2 n a n ψ a,m j − k N a ,m j − k 0 a ,m j + n a n k 0,m j + n a n σ u,m j n a − 2λ µ a + − n a n λ µσµ,m j − N a − 2λ ν a + n a n λ ν σ ν,m j n a − 2λ µ a ,…”

“…Primal and dual production models are a common case, when systems of input demands and/or output supply equations have to be estimated; the same is true for systems of demand equations in consumer analysis. Baltagi (1980) and Magnus (1982) extended the estimation procedure of the single-equation model to the case of seemingly unrelated regressions (SURs) for balanced panels; Biørn (2004) proposed a parsimonious technique to estimate one-way SUR systems on unbalanced panel data; Platoni et al (2012) extended the procedure suggested by Biørn (2004) to the two-way case. Although heteroscedasticity is a frequent and relevant issue also in the multi-equation models applied to (unbalanced) panel data, to our knowledge very few papers concerning heteroscedastic SUR systems have been published.…”