## Appendix

This appendix explains that structural factors with positive coefficient $ (\beta_{k}) $ do not necessarily increase the probability of y=0, 1, or 2. The marginal effect of independent variables in the OLM depends on which values of those variables we evaluate. Let $ Pr(.) $ , $ F(.) $ and $ f(.) $ denote probability, probability distribution function, and probability density function, y denote the dependent variable, x denote a vector of independent variables, $ \beta $ denote a vector of coefficients, and $ x_{k} $ and $ \beta_{k} $ represent the k-th element of those vectors, respectively. Furthermore, $ \tau_{i}(i=0,1,2) $ denotes the threshold values in the OLM. Without loss of generality, $ \tau_{i} $ 's is ordered in $ \tau_{2} > \tau_{1} > \tau_{0} = 0 $ .

By formulation of the OLM, the probabilities when y is 0, 1, or 2 are given by:

$$ P r(y=0|\mathbf{x})=F(\tau_{0}-\mathbf{x}\pmb{\beta}), $$

$$ P r(y=1|\mathbf{x})=F(\tau_{1}-\mathbf{x}\pmb{\beta})-F(\tau_{0}-\mathbf{x}\mathbf{B}),\mathrm{a n d} $$

$$ P r(y=2|\mathbf{x})=1-F(\tau_{2}-\mathbf{x}\pmb{\beta}). $$

Next, taking the partial derivative of these probabilities with respect to $ x_{k} $ will give marginal effects of $ x_{k} $ on those probabilities:

$$ \frac{\partial P r(y=0|\mathbf{x})}{\partial x_{k}}=\frac{\partial F(\tau_{0}-\mathbf{x}\boldsymbol{\beta})}{\partial x_{k}}=-\beta_{k}f(\tau_{0}-\mathbf{x}\boldsymbol{\beta}), $$

$$ \frac{\partial P r(y=1|\mathbf{x})}{\partial x_{k}}=\frac{\partial F(\tau_{1}-\mathbf{x}\boldsymbol{\beta})}{\partial x_{k}}-\frac{\partial F(\tau_{0}-\mathbf{x}\boldsymbol{\beta})}{\partial x_{k}} $$

$$ =\beta_{k}f(\tau_{0}-\mathbf{x}\boldsymbol{\beta})-\beta_{k}f(\tau_{1}-\mathbf{x}\boldsymbol{\beta}) $$

$$ =\beta_{k}[f(\tau_{0}-\mathbf{x}\pmb{\beta})-f(\tau_{1}-\mathbf{x}\pmb{\beta})],\mathrm{a n d} $$

$$ \frac{\partial P r(y=2|\mathbf{x})}{\partial x_{k}}=-\frac{\partial F(\tau_{2}-\mathbf{x}\boldsymbol{\beta})}{\partial x_{k}}=\beta_{k}f(\tau_{2}-\mathbf{x}\boldsymbol{\beta}). $$

Note that the probability density function $ f(\cdot) $ is always positive by its definition.

Now, because the last term of (A.4) has a negative, the marginal effect with y = 0 is the opposite to the sign of the coefficient $ \beta_{k} $ , which is to be estimated. That is, if our estimate for $ \beta_{k} $ is positive, its structural factor has a negative marginal impact on the probability with y = 0, and thus reduces the probability, and vice versa. For y = 1, with equation (A.5), even when $ \beta_{k} $ is a positive, if $ f(\tau_{0} - x\beta) - f(\tau_{1} - x\beta) $ is negative, then the marginal effect of $ x_{k} $ is negative, and thus it reduces the probability with y = 0, and vice versa. For y = 2, contrary to cases of y = 0, the equation (A.6) indicates that the structural factors with positive $ \beta_{k} $ increase the probability of y = 2, and vice versa.

The above discussion can be summarized as follows. The structural factors in industries with the estimated coefficient with positive sign increase the probability of cartels with two or more cases, but decrease the probability of non-cartels. Conversely, the structural factors with the estimated coefficient with negative sign increase the probability of non-cartels, but decrease the probability of cartels with two or more cases.