Seemingly unrelated regressions

Seemingly Unrelated Regressions (SUR or SURE) is a statistical technique used to estimate multiple regression models that are potentially correlated. Introduced by Arnold Zellner in 1962, the method provides a way to improve the efficiency of the estimators by taking into account the correlation between the error terms of the different equations. This approach is particularly useful in econometrics and other fields where simultaneous equations with correlated error terms are common.

Overview

Seemingly Unrelated Regressions involve a system of equations where each equation has its own dependent variable and set of explanatory variables. Despite the name, the regressions are "seemingly unrelated" because the error terms across these equations can be correlated. The SUR method estimates the parameters of these equations jointly rather than separately, which is the traditional approach in ordinary least squares (OLS) regression.

Mathematical Formulation

Consider a system of \\(n\\) regression equations:

\\[ y_i = X_i\beta_i + \epsilon_i, \quad i = 1, \ldots, n \\]

where \\(y_i\\) is a \\(T \times 1\\) vector of observations on the dependent variable for the \\(i\\)-th equation, \\(X_i\\) is a \\(T \times k_i\\) matrix of observations on \\(k_i\\) explanatory variables, \\(\beta_i\\) is a \\(k_i \times 1\\) vector of unknown parameters, and \\(\epsilon_i\\) is a \\(T \times 1\\) vector of error terms.

The key assumption in SUR is that while the error terms \\(\epsilon_i\\) are uncorrelated across \\(T\\) for a given equation, there may be correlation across equations, i.e., \\(Cov(\epsilon_i, \epsilon_j) \neq 0\) for \\(i \neq j\\).

Estimation

The estimation of the SUR model involves constructing a system-wide covariance matrix of the error terms and using generalized least squares (GLS) to estimate the parameters. This approach takes into account the cross-equation error correlations, leading to more efficient estimators compared to estimating each equation separately via OLS.

Applications

SUR models are widely used in econometrics, particularly in the analysis of panel data, time series, and cross-sectional data where simultaneous relationships between variables are of interest. Applications include studies on consumer demand, investment behavior, and the interdependence of financial markets.

Advantages and Limitations

The main advantage of SUR is its efficiency gain over separate OLS estimations when the error terms are correlated across equations. However, the method requires knowledge of the covariance structure of the error terms, which may not always be available or accurately estimable, especially with small sample sizes.

See Also

• Econometrics
• Generalized least squares
• Ordinary least squares
• Panel data
• Time series analysis

References

• Zellner, A. (1962). An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias. Journal of the American Statistical Association, 57(298), 348-368.