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1.2 The Problem of Boundary Violations

An alternative approach is using the LSDV estimator instead. While LSDV does not suffer from the above problem, the boundary restrictions of the dependent variable still pose a challenge to the feasibility of the parameter estimates for the panel data regression. In fact, neither the OLS (equivalent to LSDV when dummy variables are specified) nor the truncated regression model (such as truncreg) (Cong, 2000) can solve the boundary violations problem. If we seek nonlinear programming techniques, such as constrained optimization (Bertsekas, 1996), to resolve this issue, the dimensionality of the parameter space then becomes a critical concern, given that $k$ regressors will engender $(2k+6)$ boundary constraints in a constrained optimization problem (hereafter COP).3 If $k$ is in the thousands, which is usually the case, solving a COP becomes nearly impossible given the limited capacity of any personal computer (Greene, 2008: 195).

The essential problem of the panel regression for a truncated dependent variable is an out-of-bounds violation of the predicted value$-$ a common scenario in political science research that is seldom reported. For instance, in Benjamin Fordham and Thomas Walker's 2005 article published in International Studies Quarterly, "Kantian Liberalism, Regime Type, and Military Resource Allocation: Do Democracies SpendLess?", the authors analyze three dependent variables, "Military spending as a percentage of GDP," "Military personnel as a percentage of population," and "Regression-based index of military allocation" using 14 panel regressions in total. All three dependent variables have a lower bound value of 0. However, nine of the 14 models have negative predicted values and apparently suffer from boundary violations. While Fordham and Walker do identify this problem in a footnote (n.4, p.147), they do not evaluate the admissibility of the parameter estimates, nor do they discuss how we can meaningfully interpret those out-of-bounds results.4

The above discussion pinpoints the protracted problem regarding analysis of the TSCS data of a truncated dependent variable via panel regression. First, the most significant and pressing issue is boundary violations. Though the problem is widespread, the political science community pays it little attention.5 Second, if we intend to solve boundary violations with the within- and between-groups estimators by applying the constrained optimization technique, the demeaning operation generates an invalid estimate of the within-groups variations. Third, if we use the LSDV estimator, the large number of spatial units makes constrained optimization implementation impractical. Consequently, despite the fact that the panel regression model has the same problem as the linear probability model in terms of out-of-bounds violations (Aldrich and Nelson 1984:24), political scientists do not question the validity of the panel regression as they would to the linear probability model. 6

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Footnote

3 For a truncated regression model with $k$ covariates, we need to specify two constraints for the maximal and minimal predicted values, $(2k+2)$ constraints for the lower and upper limits for all beta coefficients, and two for boundary constraints of the scale parameter. See supplementary document A for detail (p.9).

4 For similar cases, see the supplementary material A for detail (p.3).

5 While scholars have been aware of this issue and some efforts were made in the field of econometrics, for example Honorè (1992), we have not seen the emergence of a standardized approach that is widely accepted and available in statistical packages such as SPSS, SAS, or Stata.

6 The usage of the logit or probit has already become the norm in social science when a binary dependent variable is analyzed. However, almost no political science literature contains a truncated regression model. Since binary and truncated dependent variables share the same nature of the problem, truncated regression certainly receives much less attention than it should.

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