Fixing Boundary Violations: Applying Constrained
Optimization to the Truncated Regression Model
5.3 Replication Results
Table 3 shows replication results of Model I by three methods. TRMCO is significantly different from TRM and OLS. For the TRMCO model, Economy, Trade Openness, and Income are no more significant, while they are positively significant in the other two models. None of the regional dummies is significant in the TRMCO model, but the OLS and TRM models have one and two significant results, respectively. Besides, the negative relationship of the interaction term Economy$\times$Trade Openness does not hold in the TRMCO model either. Among the three methods, the OLS model has the greatest number of significant results (9), larger than TRM (8) and TRMCO (4). This result indicates a great variability of causal analysis when different methods are applied.
Table 3 Replication of Model I, Hellwig and Samuels (2007: 292)
In terms of model performance, the TRM suffers the problem of inadmissible predicted values ${{\hat{y}}^{\min }}=-7.938$ and 13 boundary violations. The OLS model also has an inadmissible predicted value ${{\hat{y}}^{\min }}=-4.797$ and 13 boundary violations. The only eligible solution is generated from the TRMCO model, which has a slightly lower log psedolikelihood value, but the solution is admissible and no boundary violation occurs. Apparently, the TRMCO model has the best performance among the three.
Similar findings are concluded in the replication results of Model II in Table 4. Again, Economy is not significant in the TRMCO model, and the adjusted constant is significantly larger than zero but not in the OLS or TRM model. For Presidential Election and Economy$\times$Presidential Election, the TRMCO model shows a significant negative and positive relationship, but the two findings do not appear in the OLS and TRM models. For the rest of the parameter estimates findings, while the significance tests show the same result, the beta estimates are somewhat different. In sum, the TRMCO model has the greatest number of significant results (8), larger than OLS (6) and TRM (6), and apparently, the regression results also show a great variability when different methods are applied.
Table 4 Replication of Model II, Hellwig and Samuels (2007: 292)
The OLS and TRM models still suffer the out-of-bounds predicted values in Model II, and they have 12 and 13 boundary violations, respectively. In contrast, the TRMCO model does not have the above problems and performs better. Based on the above results, we can conclude that the TRMCO model is a superior method to the current models in use. This conclusion casts doubt on the inferential validity of the current methods, such as the OLS or TRM model when the dependent variable fits the truncated normal assumption better.
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