SEEMINGLY UNRELATED REGRESSIONS MODEL
EQUATION SYSTEMS
Often times it makes sense to view two or more equations as a system of equations that are related to one another in some particular way. There are 4 major types of equation systems:
1. Seemingly unrelated equation system
2. Simultaneous equation system
3. Recursive equation system
4. Block recursive system
You can use information about how the equations are related to obtain better estimates of the parameters of the model you are estimating.
INTRODUCTION TO SEEMINGLY UNRELATED EQUATIONS SYSTEMS
In a seemingly unrelated equation system, the equations are related in one or both of the following ways.
1. The error terms in the different equations are related.
The error terms are correlated if there are common unobserved factors that influence the dependent variables in the equations.
2. The parameters in the different equations are related.
This occurs if the same parameter(s) appears in more than one equation, or if a parameter(s) in one equation is a linear or nonlinear function the parameters in the
other equations.
There are many economic processes that are best described by a seemingly unrelated equation system. Some examples are as follows. 1) Investment demand equations for firms in the same industry. 2) Consumer demand equations implied by utility maximizing behavior. 3) Input demand equations implied by cost minimizing and profit maximizing behavior.
EXAMPLE
We want to analyze spending on plant and equipment for two firms: GM and Chrysler. The investment demand equations for the two firms are,
GM:Y1 = β11 + β12X12 + β13X13 + μ1
Chrysler:Y2 = β21 + β22X22 + β23X23 + μ2
Y1 is annual spending on plant and equipment for GM. Y2 is annual spending on plant and equipment for Chrysler. X12 is expected profit for GM. X22 is expected profit for Chrysler. Expected profit is measured by market capitalization. X13 is desired capital stock for GM. X23 is desired capital stock for Chrysler. Desired capital stock is measured by actual capital stock. All variables are measured in millions of dollars. μ1 and μ2 are the error terms for GM and Chrysler. They represent the net effect of all factors other than expected profit and desired capital stock that affect spending on plant and equipment.
Since GM and Chrysler are in the same industry, we would expect the that there are common unobserved factors in the error terms of these two firms,. These factors may include interest rates, capacity utilization, and the general state of the economy. If so, then the error term for GM is correlated with the error term for Chrysler, and therefore these two equations are seemingly unrelated regression equations. It is also possible that the parameters in the equations are related. For example, a one million dollar increase in expected profit may have the same effect on investment spending for both GM and Chrysler. If so β12 = β22. Alternatively, the effect of a one million increase in expected profit for GM may be twice as large as Chrysler, and therefore β12 = 2β22,
If we can use the information about how the equations are related in the estimation procedure, then we should be able to get more precise estimates of the parameters in both equations.
To estimate these two equations, we have annual data for each firm for the years 1935 to 1954 (20 observations for each firm).
SPECIFICATION OF THE SUR MODEL
Assumptions
1. The functional form of each equation is linear in parameters.
2. The error term in each equation has mean zero.
3. The error variance for each equation is constant (no heteroscedasticity).
4. The error variance may differ for different equations.
5. The errors for each equation are uncorrelated (no autocorrelation)
6. The errors for different equations are contemporaneously correlated.
i) For time series data, the errors in different equations in the same time period are
correlated. The errors in different equations for different time periods are not
correlated.
ii) For cross section data, the errors in different equations for the same decision making
unit are correlated. The errors in different equations for different decision making
units are not correlated.
4. The error term in each equation has a normal distribution
5. The error term in each equation is uncorrelated with the explanatory variables.
ESTIMATION
To estimate the parameters of the equations, you use an FGLS estimator to estimate the equations jointly as a system. The two most often used FGLS estimators are Zellners SUR estimator and Zellners ISUR estimator.
Properties of the Zellner’s SUR andZellner’s ISUR Estimators
If the assumptions of the SUR model are a reasonable approximation, then Zellner’s SUR and ISUR estimators are asymptotically unbiased, efficient, and consistent. The small sample properties of are unknown, but Monte Carlo studies suggest it is unbiased and has a smaller variance than the OLS estimator. As a result, you get more precise estimates than if you estimate each equation separately using the OLS estimator.
Singular Seemingly Unrelated Regressions Models
For some types of seemingly unrelated regressions models (e.g., consumer demand equations implied by utility maximizing behavior; input demand equations implied by cost minimizing and profit maximizing behavior) the variance-covariance matrix of errors for the big equation is singular, and therefore the entire system of equations cannot be estimated jointly. These are called singular SUR models. To solve the singularity problem, you drop one of the equations and estimate the remaining equations jointly. The ISUR parameter estimates are invariant to the equation dropped; that is, you will always get the same parameter estimates regardless of the equation you eliminate. This is not true for the SUR parameter estimates. Thus, when estimating the parameters of a singular SUR model you should use the ISUR estimator.
Common Properties of the SUR AND, ITSUR Estimators
- If the error terms across equations are not contemporaneously correlated, then the SUR and ISUR estimators produce the same estimates as the OLS estimator and there are no efficiency gains.
- If each of the equations have the same data for the explanatory variables, then the SUR and ISUR estimators produce the same estimates as the OLS estimator. This occurs if each of the M-equations have identical explanatory variables with identical observations. In this case, there are no efficiency gains from using the SUR or ISUR estimator.
- If there are cross equation restrictions, then there are efficiency gains from using the SUR or ISUR estimator, even if the error terms across equations are not correlated or the data for the explanatory variables is the same.
SPECIFICATION TESTING
A specification test, tests an assumption that defines the specification of a statistical model. An often used specification test for the SUR model is the Breusch-Pagan Test of Independent Errors.
Breusch-Pagan Test of Independent Errors
The Breusch Pagan Test is used to test the assumption that the errors across equations are contemporaneously correlated. The null hypothesis is no contemporaneous correlation. The alternative hypothesis is contemporaneous correlation. For a two equation SUR model, the test statistic is the following Lagrange multiplier statistic that has a chi-square distribution with M(M-1)/M degrees of freedom, where M is the number of equations estimated jointly.
LM = Tr212 ~ χ2(M(M-1)/M), where r212 = (12^)2/(211^222^)
Where T is the sample size, (12^)2 is the square of the sample covariance of the errors for the two equations, and 211^ and 222^ are the sample error variances for the two equations. This test statistic can be generalized for more than two equations.
To perform the test, use the 5-step hypothesis test procedure. For the auto company example, Stata reports a test statistic of LM = 1.49 with a p-value of 0.22. Thus, we accept the null hypothesis of no contemporaneous correlation of errors for the GM and Chrysler investment demand equations. This suggests the classical linear regression model may be the appropriate model, if there are no cross-equation restrictions.
HYPOTHESIS TESTING
The following statistical tests can be used to test hypotheses in the SUR model. 1) Asymptotic t-test. 2) Approximate F-test. 3) Wald test. 4) Likelihood ratio test. 5) Lagrange multiplier test. You must choose the appropriate test to test the hypothesis in which you are interested. Note the following. 1) The small sample t-test and F-test cannot be used. This is because if the sample data are generated by the SUR model we don’t know the sampling distribution of the t-statistic or the F-statistic. 2) Each of these tests is applied to the big equation.
Cross-Equation Restrictions
Economic theory and other sources of prior information often times imply that the values of two or more parameters in two or more equations are identical, or a parameter in one equation is a linear or nonlinear function of one or more parameters in one or more other equations. These are called cross equation restrictions. For example, in a system of M consumer demand equations implied by utility maximizing behavior the same parameters appear in different demand equations. These cross-equation restrictions can be easily tested and/or imposed in the context of the SUR model.
GOODNESS-OF-FIT
The R2 statistic that is used to measure the goodness-of-fit of a classical linear regression model is not appropriate for the SUR regression model. Many statistical programs will report an R2 statistic for each individual equation for the SUR model, but these R2 statistics have little if any meaning. They do not measure the proportion of the variation in the dependent variable which is explained by variation in the explanatory variables for the individual equation, and they can take values of less than zero and greater than one.