# A Statistical Analysis of Industrial Penetration and Internet Intensity in Taiwan +
## 3 [PERSON] Sample Selection Model
\begin{array}{ll}y_{i}=\mathbf{x}_{i}^{\prime}\beta+u_{yi},\text{ observed only if }z_{i}=1\\ corr\big{(}u_{yi},u_{zi}\big{)}=\rho\\ \big{(}u_{zi},u_{yi}\big{)}\sim\text{bivariate normal }\big{[}0,0,1,\sigma_{y}^{2}, \rho\big{]}.\end{array} \]  
In Equation (3), the selection equation is estimated by maximum likelihood (for details, see [30] and [PERSON], 1983) as an independent probit model for determining the decision to join using the available information. However, (25) [PERSON]'s (1979) two-step estimation procedure is typically used for both the selection mechanism and regression model estimations. The first step estimates the selection equation by maximum likelihood to obtain an estimate of \(\gamma\) in Equation (3), and to compute \(\hat{\lambda}_{i}=\varnothing\big{(}\omega_{i}^{\prime}\gamma\big{)}/\Phi \big{(}\omega_{i}^{\prime}\gamma\big{)}\) and \(\hat{\delta}_{i}=\hat{\lambda}_{i}\big{(}\hat{\lambda}_{i}-\omega_{i}^{\prime }\gamma\big{)}\). The second step estimates the regression equation by ordinary least squares to obtain estimates of \(\beta\) and \(\beta_{\lambda}=\rho\sigma_{y}\). (28,31) [PERSON] (1981, 2003) provides the statistical proof for consistency of the estimators of the individual parameters, \(\rho\) and \(\sigma_{y}^{2}\). The mean and variance of the incidentally truncated (or sample selection) bivariate normal distribution are given, respectively, as Equations (4) and (5) (the moments of the incidentally truncated bivariate normal distribution are given in (31) [PERSON] (2003b, p. 781)):  
\[ \begin{array}{ll}\text{E}[y_{i}|z_{i}=1]&=\text{E}\big{[}y_{i}|u_{zi}>-\omega _{i}^{\prime}\gamma\big{]}\\ &=\mathbf{x}_{i}^{\prime}\beta+\text{E}\big{[}u_{yi}|u_{zi}>-\omega_{i}^{ \prime}\gamma\big{]}\\ &=\mathbf{x}_{i}^{\prime}\beta+\rho\sigma_{y}\lambda_{i}(\alpha_{z})\\ &=\mathbf{x}_{i}^{\prime}\beta+\beta_{\lambda}\lambda_{i}(\alpha_{z})\end{array}  \tag{4}\]  
\[\text{Var}[y_{i}|z_{i}=1]=\sigma_{y}^{2}\Big{[}1-\rho^{2}\delta_{i}(\alpha_{z })\Big{]} \tag{5}\]  
where \(\alpha_{z}=-\omega_{i}^{\prime}\gamma/\sigma_{z}\), \(\lambda_{i}(\alpha_{z})=\varnothing(\alpha_{z})/[1-\Phi(\alpha_{z})]\), \(\delta_{i}(\alpha_{z})=\lambda_{i}(\alpha_{z})[\lambda_{i}(\alpha_{z})-\alpha _{z}]\), \(0<\delta_{i}<1\), \(\lambda_{i}(\alpha_{z})\) is called the inverse [PERSON]'s ratio, \(\varnothing(\cdot)\) is the standard normal pdf, and \(\Phi(\cdot)\) is the standard normal cumulative density function (cdf). The regression equation with observed data can be written as Equation (6):  
\[ \begin{array}{ll}y_{i}|(z_{i}=1)&=\text{E}\big{[}y_{i}|z_{i}^{*}>0\big{]}+ \upsilon_{i}\\ &=\mathbf{x}_{i}^{\prime}\beta+\beta_{\lambda}\lambda_{i}(\alpha_{z})+ \upsilon_{i}\end{array}  \tag{6}\]  
where the disturbance \(\upsilon_{i}\) is heteroscedastic. Ordinary least squares regression of \(y_{i}\) on \(\mathbf{x}\) and \(\lambda\) would give a consistent estimator, but if \(\lambda\) is omitted, then the speciation error of an omitted variable is committed ((30) [PERSON], 2003). The marginal effect of the regressors on \(y_{i}\) in Equation (6) is given as Equation (7):  
\[\frac{\partial E\big{[}y_{i}|z_{i}^{*}>0\big{]}}{\partial x_{\text{ik}}}=\beta _{\text{k}}-\gamma_{\text{k}}\Big{(}\frac{\rho\sigma_{y}}{\sigma_{z}}\Big{)} \delta_{i}(\alpha_{z}) \tag{7}\]  
where \(\delta_{i}(\alpha_{z})=\lambda_{i}(\alpha_{z})[\lambda_{i}(\alpha_{z})-\alpha _{z}]\), \(0<\delta_{i}<1\). The full marginal effect of the regressors on \(y_{i}\) in the observed sample consists of two parts: (i) the direct effect, which is \(\beta_{\text{k}}\), and (ii) the indirect effect, which is \(\gamma_{\text{k}}\Big{(}\frac{\rho\sigma_{y}}{\sigma_{\text{u}}}\Big{)}\delta_ {i}(\alpha_{\text{u}})\). Suppose that \(\rho\) is positive and \(\text{E}[y_{i}]\) is greater when \(z_{i}^{*}>0\) than otherwise. As \(0<\delta_{i}<1\), for a particular independent variable, if it appears in the probability as \(z_{i}^{*}>0\), then it will influence \(y_{i}\) through \(\lambda_{i}\), and thereby reduce the marginal effect (see (31) [PERSON] 2003b, p. 783).