Improved estimation

In a series of papers Hill, Cartwright, and Arbaugh (1990, 1991, 1992) consider the performance of conventional FGLS compared to various improved estimators when applied to a basic SUR model. The improved estimators include several variants of the Stein-rule family and the hierarchical Bayes estimator of Blattberg and George (1991). The primary example is the estimation of a price-promotion model, which captures the impact of price reductions and promotional activities on sales.

In Hill, Cartwright, and Arbaugh (1996) they extend their previous work on estimator performance by investigating the possibility of estimating the finite sample variability of these alternative estimators using bootstrap standard errors. Conclusions based on their Monte Carlo results obtained for conventional FGLS are found to be somewhat contrary to those of Atkinson and Wilson (1992) that we discussed previously. Hill et al. (1996, p. 195) conclude, "the bootstrap stan­dard errors are generally less downward biased than the nominal standard errors" concluding that the former are more reliable than the latter. For the Stein-rule estimators they find that the bootstrap may either overestimate or underestimate the estimator variability depending on whether the specification errors in the restrictions are small or not.

An SUR pre-test estimator can be readily defined based on an initial test of the null hypothesis that Ъ is diagonal. Ozcam, Judge, Bera, and Yancey (1993) define such an estimator for a two-equation system using the Lagrange multiplier test of Breusch and Pagan (1980) and Shiba and Tsurumi (1988) and evaluate its risk properties under squared error loss.

5.2 Misspecification

Green, Hassan, and Johnson (1992) and Buse (1994) investigate the impact of model misspecification on SUR estimation. In the case of Green et al. (1992) it is the omission of income when estimating demand functions, the motivating example being the use of scanner data where demographic information such as income is not typically available. They conclude that anomalous estimates of own-price elasticities are likely due to this misspecifiaction. Buse (1994) notes that the popular linearization of the almost ideal demand system introduces an errors-in-variables problem, which renders the usual SUR estimator inconsistent.

5.3 Robust estimation

In the context of single-equation modeling there has been considerable work devoted to the provision of estimators that are robust to small perturbations in the data. Koenker and Portnoy (1990) and Peracchi (1991) extend this work by proposing robust alternatives to standard FGLS estimators of the basic SUR model. Both papers illustrate how their estimators can guard against the potential sens­itivity due to data contamination. Neither paper addresses the equally important issue of drawing inferences from their robust estimators.

5.4 Model extensions

A natural extension to the basic SUR model is to consider systems that involve equations which are not standard regression models. In the context of time series modeling Fernandez and Harvey (1990) consider a multivariate structural time series model comprised of unobserved components that are allowed to be con­temporaneously correlated. King (1989) and Ozuna and Gomez (1994) develop a seemingly unrelated Poisson regression model, which somehow is given the acronym SUPREME. Both applications were to two-equation systems, with extensions to larger models not developed. King (1989) analyses the number of presidential vetoes per year for the period 1946-84 allowing for different explana­tions to be relevant for social welfare and defense policy vetoes. Ozuna and Gomez (1994) apply the approach to estimate the parameters of a two-equation system of recreation demand functions representing the number of visits to one of two sites.

6 Conclusion

The SUR model has been the source of much interest from a theoretical stand­point and has been an extremely useful part of the toolkit of applied econo­metricians and applied statisticians in general. According to Goldberger (1991, p. 323), the SUR model "plays a central role in contemporary econometrics." This is evidenced in our chapter by the breadth of the theoretical and applied work that has appeared since the major surveys of Srivastava and Dwivedi (1979) and Srivastava and Giles (1987). Hopefully this new summary of recent research will provide a useful resource for further developments in the area.

Note

* I gratefully acknowledge the excellent research assistance of Hong Li and Kerri Hoffman. Badi Baltagi, Bob Bartels, Mike Smith, and three anonymous referees also provided helpful comments.

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