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
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.
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.