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.
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.
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.
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.
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.
* 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.
Amemiya, T. (1977). A note on a heteroscedastic model. Journal of Econometrics 6, 365-70.
Anderson, G. J., and R. W. Blundell (1982). Estimation and hypothesis testing in dynamic singular equation systems. Econometrica 50, 1559-72.
Anselin, L. (1988). A test for spatial autocorrelation in seemingly unrelated regressions. Economics Letters 28, 335-41.
Anselin, L. (1990). Spatial dependence and spatial structural instability in applied regression analysis. Journal of Regional Science 30, 185-207.
Atkinson, S. E., and P. W. Wilson (1992). The bias of bootstrapped versus conventional standard errors in the general linear and SUR models. Econometric Theory 8, 258-75.
Attfield, C. L.F. (1995). Bartlett adjustment to the likelihood ratio test for a system of equations. Journal of Econometrics 66, 207-24.
Attfield, C. L.F. (1998). Bartlett adjustments for systems of linear equations with linear restrictions. Economics Letters 60, 277-83.
Avery, R. (1977). Error components and seemingly unrelated regressions. Econometrica 45, 199-209.
Baltagi, B. H. (1980). On seemingly unrelated regressions with error components. Eco - nometrica 48, 1547-51.
Baltagi, B. H. (1989). Applications of a necessary and sufficient condition for OLS to be BLUE. Statistics and Probability Letters 8, 457-61.
Baltagi, B. H., S. Garvin, and S. Kerman (1989). Further evidence on seemingly unrelated regressions with unequal number of observations. Annales D'Economie et de Statistique 14, 103-15.
Bartels, R., and D. G. Fiebig (1991). A simple characterization of seemingly unrelated regressions models in which OLS is BLUE. American Statistician 45, 137-40.
Bartels, R., and D. G. Fiebig (1992). Efficiency of alternative estimators in generalized seemingly unrelated regression models. In R. Bewley, and T. V. Hao (eds.) Contributions to Consumer Demand and Econometrics: Essays in Honour of Henri Theil. London: Macmillan Publishing Company, 125-39.
Bartels, R., D. G. Fiebig, and M. Plumb (1996). Gas or electricity, which is cheaper? An econometric approach with application to Australian expenditure data. Energy Journal 17, 33-58.
Batchelor, R., and D. Gulley (1995). Jewellery demand and the price of gold. Resources Policy 21, 37-42.
Bauwens, L., D. G. Fiebig, and M. F.J. Steel (1994). Estimating end-use demand: A Bayesian approach. Journal of Business and Economic Statistics 12, 221-31.
Bewley, R. A. (1983). Tests of restrictions in large demand systems. European Economic Review 20, 257-69.
Bewley, R. A. (1986). Allocation Models: Specification, Estimation and Applications. Cambridge, MA: Ballinger Publishing Company.
Binkley, J. K. (1982). The effect of variable correlation on the efficiency of seemingly unrelated regression in a two equation model. Journal of the American Statistical Association 77, 890-5.
Binkley, J. K., and C. H. Nelson (1988). A note on the efficiency of seemingly unrelated regression. American Statistician 42, 137-9.
Blattberg, R. C., and E. I. George (1991). Shrinkage estimation of price and promotional elasticities: Seemingly unrelated equations. Journal of the American Statistical Association 86, 304-15.
Bollerslev, T. (1990). Modeling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model. Review of Economics and Statistics 72, 498-505.
Breusch, T. S., and A. R. Pagan (1980). The Lagrange multiplier test and its applications to model specification in econometrics. Review of Economic Studies 47, 239-53.
Brown, B. W., and M. B. Walker (1989). The random utility hypothesis and inference in demand systems. Econometrica 57, 815-29.
Brown, B. W., and M. B. Walker (1995). Stochastic specification in random production models of cost minimizing firms. Journal of Econometrics 66, 175-205.
Buse, A. (1994). Evaluating the linearized almost ideal demand system. American Journal of Agricultural Economics 76, 781-93.
Byron, R. P. (1982). A note on the estimation of symmetric systems. Econometrica 50, 1573-5.
Chavas, J.-P., and K. Segerson (1987). Stochastic specification and estimation of share equation systems. Journal of Econometrics 35, 337-58.
Chesher, A. (1984). Improving the efficiency of probit estimators. Review of Economics and Statistics 66, 523-7.
Chib, S., and E. Greenberg (1995). Hierarchical analysis of SUR models with extensions to correlated serial errors and time-varying parameter models. Journal of Econometrics 68, 339-60.
Conniffe, D. (1985). Estimating regression equations with common explanatory variables but unequal numbers of observations. Journal of Econometrics 27, 179-96.
Conniffe, D. (1997). Improving a linear regression through joint estimation with a probit model. The Statistician 46, 487-93.
Creel, M., and M. Farell (1996). SUR estimation of multiple time-series models with heteroscedasticity and serial correlation of unknown form. Economic Letters 53, 23945.
de Jong, P., and R. Thompson (1990). Testing linear hypothesis in the SUR framework with identical explanatory variables. Research in Finance 8, 59-76.
Deschamps, P. J. (1998). Full maximum likelihood estimation of dynamic demand models. Journal of Econometrics 82, 335-59.
Dhrymes, P. J. (1994). Autoregressive errors in singular systems of equations. Econometric Theory 10, 254-85.
Dufour, J. M., and O. Torres (1998). Union-intersection and sample-split methods in econometrics with applications to MA and SURE models. In A. Ullah, and D. E.A. Giles (eds.) Handbook of Applied Economic Statistics. New York: Marcel Dekker, 465-505.
Eakin, B. K., D. P. McMillen, and M. J. Buono (1990). Constructing confidence intervals using the bootstrap: An application to a multi-product cost function. Review of Economics and Statistics 72, 339-44.
Efron, B. (1992). Jackknife-after-bootstrap standard errors and influence functions. Journal of the Royal Statistical Society, B 54, 83-127.
Fernandez, F. J., and A. C. Harvey (1990). Seemingly unrelated time series equations and a test for homogeneity. Journal of Business and Economic Statistics 8, 71-82.
Fiebig, D. G., and J. H. Kim (2000). Estimation and inference in SUR models when the number of equations is large. Econometric Reviews 19, 105-130.
Fiebig, D. G., and Theil, H. (1983). The two perils of symmetry-constrained estimation of demand systems. Economics Letters 13, 105-11.
Fiebig, D. G., R. Bartels, and D. J. Aigner (1991). A random coefficient approach to the estimation of residential end-use load profiles. Journal of Econometrics 50, 297-327.
Freedman, D. A., and S. C. Peters (1984). Bootstrapping a regression equation: Some empirical results. Journal of the American Statistical Association 79, 97-106.
Frees, E. W. (1995). Assessing cross-sectional correlation in panel data. Journal of Econometrics 69, 393-414.
Fry, J. M., T. R.L. Fry, and K. R. McLaren (1996). The stochastic specification of demand share equations: Restricting budget shares to the unit simplex. Journal of Econometrics 73, 377-86.
Goldberger, A. S. (1991). A Course in Econometrics. Cambridge, MA: Harvard University Press.
Green, R., Z. A. Hassan, and S. R. Johnson (1992). The bias due to omitting income when estimating demand functions. Canadian Journal of Agricultural Economics 40, 475-84.
Griffiths, W. E., and D. Chotikapanich (1997). Bayesian methodology for imposing inequality constraints on a linear expenditure system with demographic factors. Australian Economic Papers 36, 321-41.
Griffiths, W. E., and R. Valenzuela (1998). Missing data from infrequency of purchase: Bayesian estimation of a linear expenditure system. In T. B. Fomby, and R. C. Hill (eds.) Advances in Econometrics, 13: Messy Data - Missing Observations, Outliers and Mixed - Frequency Data. Greenwich CT: JAI Press, 47-74.
Griliches, Z., and M. D. Intriligator (1983). Preface. In Z. Griliches, and M. D. Intriligator (eds.) Handbook of Econometrics. Amsterdam: Elsevier Science Publishers B. V., xi-xvii.
Hasegawa, H. (1995). On small sample properties of Zellner's estimator for the case of two SUR equations with compound normal disturbances. Communications in Statistics, Simulation and Computation 24, 45-59.
Hashimoto, N., and K. Ohtani (1990). An exact test for linear restrictions in seemingly unrelated regressions with the same regressors. Economics Letters 32, 243-6.
Henley, A., and J. Peirson (1994). Time-of-use electricity pricing: Evidence from a British experiment. Economics Letters 45, 421-6.
Hill, R. C., P. A. Cartwright, and J. F. Arbaugh (1990). Using aggregate data to estimate micro-level parameters with shrinkage rules. American Statistical Association: Proceedings of the Business and Economic Statistics Section 339-44.
Hill, R. C., P. A. Cartwright, and J. F. Arbaugh (1991). Using aggregate data to estimate micro-level parameters with shrinkage rules: More results. American Statistical Association: Proceedings of the Business and Economic Statistics Section 155-60.
Hill, R. C., P. A. Cartwright, and J. F. Arbaugh (1992). The finite sample properties of shrinkage estimators applied to seemingly unrelated regressions. American Statistical Association: Proceedings of the Business and Economic Statistics Section 17-21.
Hill, R. C., P. A. Cartwright, and J. F. Arbaugh (1996). Bootstrapping estimators for the seemingly unrelated regressions model. Journal of Statistical Computation and Simulation 54, 177-96.
Hill, R. C., P. A. Cartwright, and J. F. Arbaugh (1997). Jackknifing the bootstrap: Some Monte Carlo evidence. Communications in Statistics, Simulation and Computation 26, 125-239.
Hirschberg, J. G. (1992). A computationally efficient method for bootstrapping systems of demand equations: A comparison to traditional techniques. Statistics and Computing 2, 19-24.
Holt, M. T. (1998). Autocorrelation specification in singular equation systems: A further look. Economics Letters 58, 135-41.
Hwang, H. S. (1990). Estimation of a linear SUR model with unequal numbers of observations. Review of Economics and Statistics 72, 510-15.
Hwang, C. J., F. A. Sloan, and K. W. Adamache (1987). Estimation of seemingly unrelated Tobit regressions via the EM algorithm. Journal of Business and Economic Statistics 5, 425-30.
Im, E. I. (1994). Unequal numbers of observations and partial efficiency gain. Economics Letters 46, 291-4.
Jensen, M. J. (1995). A Monte Carlo study on two methods of calculating the MLE's covariance matrix in a seemingly unrelated nonlinear regression. Econometric Reviews 14, 315-30.
Jeong, J., and G. S. Maddala (1993). A perspective on application of bootstrap methods in econometrics. In G. S. Maddala, C. R. Rao, and H. D. Vinod (eds.) Handbook of Statistics, Volume 11. Amsterdam: Elsevier Science Publishers B. V., 573-610.
Judge G. G., W. E. Griffiths, R. C. Hill, H. Lutkepohl, and T.-C. Lee (1985). The Theory and Practice of Econometrics, 2nd edn. New York: John Wiley and Sons.
King, G. (1989). A seemingly unrelated Poisson regression model. Sociological Methods and Research 17, 235-55.
Kiviet, J. F., G. D.A. Phillips, and B. Schipp (1995). The bias of OLS, GLS and ZEF estimators in dynamic SUR models. Journal of Econometrics 69, 241-66.
Koenker, R., and S. Portnoy (1990). M estimation of multivariate regressions. Journal of the American Statistical Association 85, 1060-8.
Kontoghiorghes, E. J. and M. R.B. Clarke (1995). An alternative approach to the numerical solution of seemingly unrelated regression equation models. Computational Statistics and Data Analysis 19, 369-77.
Kumbhakar, S. C., and A. Heshmati (1996). Technical change and total factor productivity growth in Swedish manufacturing industries. Econometric Reviews 15, 275-98.
Laitinen, K. (1978). Why is demand homogeneity so often rejected? Economics Letters 1, 187-91.
Lee, B.-J. (1995). Seemingly unrelated regression on the autoregressive (AR(p)) singular equation system. Econometric Reviews 14, 65-74.
MacKinley, A. C. (1987). On multivariate tests of the CAPM. Journal of Financial Economics 18, 341-71.
Mandy, D. M., and C. Martins-Filho (1993). Seemingly unrelated regressions under additive heteroscedasticity. Journal of Econometrics 58, 315-46.
McLaren, K. R. (1990). A variant on the arguments for the invariance of estimators in a singular system of equations. Econometric Reviews 9, 91-102.
McLaren, K. R. (1996). Parsimonious autocorrelation corrections for singular demand systems. Economics Letters 53, 115-21.
Meisner, J. F. (1979). The sad fate of the asymptotic Slutsky symmetry test for large systems. Economics Letters 2, 231-3.
Meng, X. L., and D. B. Rubin (1996). Efficient methods for estimation and testing with seemingly unrelated regressions in the presence of latent variables and missing observations. In D. A. Berry, K. M. Chaloner, and J. K. Geweke (eds.) Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner. New York: John Wiley and Sons, 215-27.
Mizon, G. E. (1995). A simple message for autocorrelation correctors: Don't. Journal of Econometrics 69, 267-88.
Moschini, G., and D. Moro (1994). Autocorrelation specification in singular equation systems. Economics Letters 46, 303-9.
Ozcam, A., G. Judge, A. Bera, and T. Yancey (1993). The risk properties of a pre-test estimator for Zellner's seemingly unrelated regression model. Journal of Quantitative Economics 9, 41-52.
Ozuna, T., and I. A. Gomez (1994). Estimating a system of recreation demand functions using a seemingly unrelated Poisson regression approach. Review of Economics and Statistics 76, 356-60.
Peracchi, F. (1991). Bounded-influence estimators for the SURE model. Journal of Econometrics 48, 119-34.
Percy, D. F. (1992). Prediction for seemingly unrelated regressions. Journal of the Royal Statistical Society, B 54, 243-52.
Percy, D. F. (1996). Zellner's influence on multivariate linear models. In D. A. Berry, K. M. Chaloner, and J. K. Geweke (eds.) Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner. New York: John Wiley and Sons, 203-13.
Richard, J. F., and M. F.J. Steel (1988). Bayesian analysis of systems of seemingly unrelated regression equations under a recursive extended natural conjugate prior density. Journal of Econometrics 38, 7-37.
Rilstone, P., and M. Veall (1996). Using bootstrapped confidence intervals for improved inferences with seemingly unrelated regression equations. Econometric Theory 12, 569-80.
Rocke, D. M. (1989). Bootstrap Bartlett adjustment in seemingly unrelated regression. Journal of the American Statistical Association 84, 598-601.
Rosalsky, M. C., R. Finke, and H. Theil (1984). The downward bias of asymptotic standard errors of maximum likelihood estimates of non-linear systems. Economics Letters 14, 207-11.
Schmidt, P. (1977). Estimation of seemingly unrelated regressions with unequal numbers of observations. Journal of Econometrics 5, 365-77.
Seaks, T. G. (1990). The computation of test statistics for multivariate regression models in event studies. Economics Letters 33, 141-5.
Shiba, T., and H. Tsurumi (1988). Bayesian and non-Bayesian tests of independence in seemingly unrelated regressions. International Economic Review 29, 377-95.
Silk, J. (1996). Systems estimation: A comparison of SAS, SHAZAM and TSP. Journal of Applied Econometrics 11, 437-50.
Silver, J. L., and M. M. Ali (1989). Testing Slutsky symmetry in systems of linear demand equations. Journal of Econometrics 41, 251-66.
Srivastava, V. K., and T. D. Dwivedi (1979). Estimation of seemingly unrelated regression equations: a brief survey. Journal of Econometrics 10, 15-32.
Srivastava, V. K., and D. E.A. Giles (1987). Seemingly Unrelated Regression Models: Estimation and Inference. New York: Marcel Dekker.
Srivastava, V. K., and K. Maekawa (1995). Efficiency properties of feasible generalized least squares estimators in SURE models under non-normal disturbances. Journal ofEcono - metrics 66, 99-121.
Steel, M. F. (1992). Posterior analysis of restricted seemingly unrelated regression equation models: A recursive analytical approach. Econometric Reviews 11, 129-42.
Stewart, K. G. (1997). Exact testing in multivariate regression. Econometric Reviews 16, 32152.
Telser, L. G. (1964). Iterative estimation of a set of linear regression equations. Journal of the American Statistical Association 59, 845-62.
Ullah, A., and J. Racine (1992). Smooth improved estimators of econometric parameters. In W. E. Griffiths, H. Lutkepohl, and M. E. Bock (eds.) Readings in Econometric Theory and Practice. Amsterdam: Elsevier Science Publishers B. V., 198-213.
Wan, G. H., W. E. Griffiths, and J. R. Anderson (1992). Using panel data to estimate risk effects in seemingly unrelated production functions. Empirical Economics 17, 35-49. Williams, M. A. (1986). An economic application of bootstrap statistical methods: Addyston Pipe revisited. American Economist 30, 52-8.
Woodland, A. D. (1979). Stochastic specification and the estimation of share equations. Journal of Econometrics 10, 361-83.
Zellner, A. (1962). An efficient method of estimating seemingly unrelated regressions and tests of aggregation bias. Journal of the American Statistical Association 57, 348-68.