Consistent estimates of the public/private wage gap

Abstract

Existing estimates of the public/private wage gap allow for possible sorting of individuals into one sector, but they rely on parametric assumptions that may introduce substantial bias in the parameter of interest. Solutions are semi- and nonparametric approaches. For Italy, the latter methods yield a gap of approximately 20%, whereas the bias from parametric assumptions is as large as 10%.

Appendices

Appendix A: Asymptotic bias from parametric assumptions

Let the true model be given by Eqs. 1–3 and the true distribution of $\varepsilon _{ji}$ be unknown for $j=\left\{ 0,1\right\}$. Define $\theta _{j}\equiv \left( \beta _{j},\xi _{j}\right)$ and $\omega _{ji}\equiv \left( x_{i}',\lambda _{j}\left( z_{i}\right) \right) '$, where $\lambda _{j}\left( z_{i}\right)$ is the correction term ($\lambda _{0}\left( z_{i}\right) =\frac{\phi \left( z_{i}'\gamma \right) }{1-\Phi \left( z_{i}'\gamma \right) }$ and $\lambda _{1}\left( z_{i}\right) =\frac{\phi \left( z_{i}'\gamma \right) }{\Phi \left( z_{i}'\gamma \right) }$). It is possible to rewrite the outcome equation as $y_{ji}=\omega _{ji}'\theta +u_{ji}$, where $u_{ji}=\varepsilon _{ji}-\xi _{j}\lambda _{j}\left( z_{i}\right)$, which is a zero-mean error term. Hence, it can be interpreted in terms of the Heckman two-step estimator. The probability limit of this estimator is given by:

$$\begin{aligned} \hat{\theta }_{j}^{H2S}\equiv \left( \frac{1}{N}\sum _{i=1}^{N}\omega _{ji}\omega _{ji}'\right) ^{-1}\frac{1}{N}\sum _{i=1}^{N}\omega _{ji}y_{ji}\overset{P}{\rightarrow }\theta _{j}+{\mathbb {E}}\left[ \omega _{ji}\omega _{ji}'\right] ^{-1}{\mathbb {E}}\left[ \omega _{ji}u_{ji}\right] \end{aligned}$$

Even though the mean regression model is correctly specified, $\beta _{j}$ may suffer from an asymptotic bias equal to:

$$\begin{aligned} ABias\left( \hat{\theta }_{j}^{H2S}\right) ={\mathbb {E}}\left[ \omega _{ji}\omega _{ji}'\right] ^{-1}{\mathbb {E}}\left[ \omega _{ji}{\mathbb {E}}\left( \varepsilon _{ji}|d_{i}=j,z_{i}\right) -\omega _{ji}\xi _{j}\lambda _{j}\left( z_{i}\right) \right] \end{aligned}$$

Hence, the severity of the bias is proportional to ${\mathbb {E}}\left[ \omega _{ji}{\mathbb {E}}\left( \varepsilon _{ji}|d_{i}=j,z_{i}\right) \right. \left. -\omega _{ji}\xi _{j}\lambda _{j}\left( z_{i}\right) \right]$, which equals 0 only under normality (if an exogenous and relevant instrument exists).

Appendix B: Additional table (available on the website)

See Table 6.

Table 6 Stability of the estimates to the inclusion of covariates