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The Mystery of Multiple Regression: Why Estimating Different Equations Simultaneously Improves Efficiency?
In the field of econometrics, the seemingly unrelated regression model (SUR) was proposed by Arnold Zellner in 1962, which is an extension of the linear regression model. This model contains multiple regression equations, each with its own independent dependent variable and possibly different exogenous explanatory variables. Although the design of these equations seems to be independent of each other, in fact their error terms are related to each other. This situation has aroused strong interest among econometricians.
According to the assumptions of the SUR model, error terms are independent between observations, but error terms within the same observation may be correlated across equations.
According to Zellner's theory, each equation in the SUR model can be estimated independently, usually using the ordinary least squares method (OLS). However, this method is generally not as efficient as the SUR method, which estimates using the feasible generalized least squares (FGLS) method through a specific variant-covariance matrix.
In most cases, the SUR method can effectively improve the accuracy of estimation, especially when there is correlation between error terms. This allows the SUR model to better reflect real-world situations, because in many economic problems, variables influence each other, and this influence relationship tends to emerge over time.
When the covariance matrix of the error term is a known diagonal matrix, the results of SUR estimation will be the same as the results of equation-wise OLS estimation.
This means that in some specific cases, using OLS for separate regression can also give the same results as SUR. For example, when the explanatory variables of each equation are exactly the same, the estimates of the SUR model and the results of OLS will be highly consistent.
In addition, the application of SUR models is not limited to only a few equations, but also extends to more complex systems, such as simultaneous equation models. In these cases, the explanatory variables on the right-hand side of the equation may also be endogenous, which has motivated further developments in econometric techniques.
Effective estimation techniques
SUR models are usually estimated using the feasible generalized least squares method (FGLS), which is a two-step method. First, we perform a regression using the ordinary least squares method, from which the residuals are used to estimate the elements of the covariance matrix. In the second step, we use the variation matrix for generalized least squares estimation, which can effectively improve the accuracy of estimation.
In addition to the FGLS method, there are several other estimation techniques to choose from, including maximum likelihood estimation (ML), as well as iterative generalized least squares (IGLS) and iterative ordinary least squares (IOLS). Each of these methods has advantages and disadvantages, but research shows that they tend to produce numerically the same results, which allows researchers to choose the appropriate technique based on actual needs.
Applications of econometrics
With the development of econometrics, SUR models are being used in more and more statistical software. For example, the "systemfit" package can be used in R language to estimate the SUR model; in Stata, the "sureg" and "suest" instructions can be used to complete the corresponding estimation.
The development of this series of technologies has greatly enriched the toolbox of econometrics, allowing researchers to provide more accurate analysis and predictions when facing complex economic problems.
In summary, the power of the SUR model is that it can fully take into account possible interactions between different regression equations, which gives us more advantages when dealing with multivariate problems. However, does this mean that using SUR is the best option in all situations?
