Understanding seemingly unrelated regression

SUR regression

• A generalization of a linear model that consists of several regression equations, each having its own dependent variable and potentially differnts sets of exogenous explanatory variables.

• Error terms are assumed to be correlated across the equations

• Consider a basic function:
  • Urban function: $Y_{ij} = X_{ij}^{\beta_{ij}} + \mu_{ij}$
  • Rural function: $Y_j = X_{j}^{\beta_{j}} + \mu_j$
• If $Corr(\mu_{ij}, \mu_j) = 0$, we can employ OLS regression directly
• While $Corr(\mu_{ij}, \mu_j) \neq 0$, the separate OLS estimates are no longer valid (the standard errors of the coefficients is too high). A joint estimation (GLS) of the two equations to take into account the correlation between the interference terms of the two can produce a more confidence estimate
  • Simultaneous equation models: right side includes dependent variable
  • Seemingly unrelated models: right side doesn’t include dependent variable
• Model assumption:
  • $E(\mu_{ij} \mu_{ik} | X_i) = \sigma_{jk} \neq 0$
• According to the variance-covariance matrix $E(\mu_i \mu_i)$, for a equation $j$ and $k$:
  • $E(\mu_j | X) = 0$
  • $E(\mu_j \mu_j | X) = \sigma_j I_n$
  • $E(\mu_j \mu_k | X) = \sigma_{jk} I_n$
• Variance-covariance matrix of the system of equations:
  • $\Omega = E(\mu \mu’ | X) = \sum \bigotimes I_n$
• Estimation:
  • $\widehat{\sum} = = \frac{1}{n} \widehat{\mu}_{ols} \widehat{\mu’} _{ols}$
  • $\widehat{\Omega} = \widehat{\sum} \bigotimes I_n$
  • $\widehat{\beta} _{sur} = \widehat{\beta} _{gls} \widehat{\Omega} = (X’ \widehat{\Omega}^{-1} X) ^{-1} X’ \widehat{\Omega}^{-1} y$

Test

• In general, we may carry group differences if the sample includes groups in advance:
  • t-test
  • ANOVA
  • …
• Based on significant difference between groups, we further apply coefficient differences in SUR estimation:
  • $\chi$: if $p \geq 0.05$, null hypothesis cannot be rejected.

Scenario

• Compare various groups, with correlation