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We apply a Box-Cox transformation as a generalization to our fixed-effects model (where is a site-specific fixed effect). The model can be specified as
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With the results from the Box-Cox regression we can also peform two likelihood-ratio tests, 2[L(X) — Z(0)] ~ x2(l) and 2[L(X) — L(l)] ~ X2(l), that allow us to test the Box-Cox transformation against the log-linear (our baseline) model and the simple linear model.
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We find that the signs of our estimates remain stable and significant. The optimal Box-Cox transformation parameter is approximately 0.2. When we test this specification against either the log-linear or pure-linear case, the log-likelihood test statistics reject both the log-linear and pure-linear specifications in favour of the Box-Cox transformation. Observe, though, that the pure-linear model is rejected by a much larger margin than the log-linear model. Also note that the interpretation of the parameters changes and cannot be compared across the three models.

C.3 Simultaneity

Yet another concern in our work has been the possibility of a simultaneous determination of pollution and (current-period) per-capita income. We did not pursue a simultaneous-equations approach for our main analysis because it is our belief that the likely effect of pollution on per-capita income is rather small. This belief appears to be validated by Dean (1998), who finds no significant relationship in her 2SLS procedures. Contemporaneous per-capita income only enters through our scale variable but not through our technique variable; recall that we use lagged per-capita income to determine the technique effect because income increases will typically take a number of years to translate into policy changes.