Seemingly Unrelated Regression: Theory And Its Application

Seemingly unrelated regression comes from the fact that each equation appears to be unrelated in a system. Nevertheless, correlation across the errors in different equations can provide links that can be exploited in estimation. It is well known that we can employ the sample information and error term's variance-covariance matrix to improve the precision of estimation of parameters. That is the new, operational GLS best linear unbiased estimators of parameters of a set of regression equations. However, the efficient GLS estimators depend on some assumptions seriously, such as stationary regressors, and independent and identical errors. If the system of regression equations consists of nonstationary time series regression models that allow for endogenous regressors, the previous conclusion will be not true in general, and the GLS estimators will be biased and have nonstandard limit distributions. Consequently, we have to test whether the variables are nonstationary and then cointegration in a system. However, most of existing panel unit root tests and panel cointegration tests hinge critically on the assumption of cross-sectional independence. What about the results if we allow for the dependence across equations? So, it is an active field to extend such SUR methodology to panel unit root tests and panel cointegration tests that coincide with cross-sectional dependence. Another one is to change the stationary variables into nonstationary variables, and extend the SUR model to seemingly unrelated dynamic cointegrating regressions. For example, Mark et al(2005) develops this model and its DSUR estimators, to pure the effect of endogeneity. They also show that the DSUR estimators have asymptotically mixed normal distribution and the tests of parametric restrictions can be constructed by the Wald statistic which is asymptotically distributed as a chi-square variate.However, we find using Monte Carlo methods that the tests may suffer from serious size distortion in finite samples, and suggest bootstrapping the tests to correct this inference problem. Furthermore, we extend the SUR method to panel unit root test and panel cointegration test to accommodate dependence across the cross-section. In most of recent papers, where unit root behavior is rejected, the conclusion reached is that all members of the panel are either stationary or nonstationary. Although the SUR-ADF test proposed by Breuer et al(2001) are informative about the behavior of each individual time series, it is highly sensitive to the selection of panel members. As such, this paper suggests fast double bootstrap to improve the reliability of SUR-ADF test by computing double p-values. As to the panel cointegration test under the cross-sectional dependence, we, according to the classical EG two steps, estimate the panel model by SUR at first, then conduct a panel unit root test to the residuals. If the null hypothesis of panel stationary can not be rejected, it is implied that the original models are panel cointegrated. Otherwise, it is much more favorable to spurious regression equations.In empirical analysis, we, adopting the previous models, focus our attentions on the stock market, regional capital mobility and the effects of environmental quality on health expenditures. Firstly, we investigate the relationship between stock returns and return volatility in China, and find that the stock price decline raises the firm's financial leverage, resulting in an increase in the volatility of equity. There is a negative statistical relation between current stock returns and changes in future stock return volatility, which is documented by leverage effect. Secondly, when we use the newly developed seemingly unrelated dynamic cointegrating regressions to examine the regional capital mobility, the results prove that China's east section represents a net capital inflow, however, the sustaining inflow has reduced the investment return, which induces the long-run solvency declining. Meanwhile, the west section marks a net capital outflow, which should account for the creeping economic growth in this area, and the central section stays in a basically balanced position. Finally, this paper estimates the role of environmental quality in determining per capita health expenditures, which is taken as an example for panel cointegration application. Our empirical analysis reveals that the environmental quality exerts a statistically significant positive effect on health expenditures in the long run. The seriousness of environmental pollution results to the deterioration of people's health status, which will induce an increase in health expenditures. While the fact of insignificant short-run impacts of environmental quality just provides some evidences for authorities pursuing economic growth, regardless of environmental conservation. This issue has become much more important from a policy point of view. Therefore, it is arduous, but also urgent to protect our environment. Starting from the classical SUR model, this paper introduces its applications and developments both in theoretical and empirical analysis. As we know, the SUR model does not only allow for correlation across the errors in different equations, but have a simple and general exposition. It has become a basis of the econometric theory, which also underlines the great significance of this paper.