Human capital versus signaling is empirically unresolvable

Abstract

Economists offer two main explanations for the causal labor market returns to education. The first is human capital accumulation: Education improves ability. The second is signaling: Education allows high-ability students to distinguish themselves. A major point of interest is the relative contributions of these effects. I demonstrate the theoretical and empirical conditions necessary to identify the relative contribution of the two models. Then, I review the existing literature to evaluate whether the feasible set of empirical estimates is capable of meeting those conditions and so informing theory. Empirical evidence is capable of rejecting pure human capital and signaling models and usually does so. I argue that, for the general question of relative contribution, necessary identification conditions are not met, and partial identification bounds are wide. Two models with different nonzero contributions of human capital and signaling cannot be empirically distinguished, limiting the usefulness of human capital versus signaling as a framing for understanding the return to education and for policy.

The effect of education on the labor market in potential outcomes notation

This appendix draws from Imai et al. (2010), albeit with the use of superscript counterfactual notation rather than function notation.

We are interested in the causal effect of education $Ed_i$ on some outcome $Y_i$ for individual i. For simplicity, assume that the margin of interest compares $Ed_i = 1$ against $Ed_i = 0$. Let $Y_i^{C}$ be the possibly counterfactual outcome for individual i under the condition C.

The average causal effect of education is

$$\begin{aligned} E(Y_i^{Ed_i=1} - Y_i^{Ed_i=0}) \equiv B \equiv \varSigma + K + \varOmega \end{aligned}$$

(3)

where $\varSigma$, K, and $\varOmega$ are the parts of the education return attributable to signaling, human capital, and other explanations, respectively.

In observed data, either $Y_i^1$ or $Y_i^0$ is missing, such that the raw relationship between $Y_i$ and $Ed_i$ does not identify the effect of education:

$$\begin{aligned} E(Y_i|Ed_i =1) - E(Y_i|Ed_i = 0) \ne E(Y_i^{Ed_i=1} - Y_i^{Ed_i=0}) \end{aligned}$$

(4)

However, for the purposes of this paper, we will assume that the causal effect of education can be identified. For simplicity, assume that we do this by conditioning on a set of controls $W_i$ such that

$$\begin{aligned} E(Y_i|Ed_i =1, W_i) - E(Y_i|Ed_i = 0, W_i) = E(Y_i^{Ed_i=1} - Y_i^{Ed_i=0}) \end{aligned}$$

(5)

The effect of $Ed_i$ on $Y_i$ is fully mediated by a set of mediating variables ${x_1,x_2,\ldots ,x_J}$. For simplicity of notation, assume that each of these mediators is binary. We have

$$\begin{aligned}&\displaystyle E(Y_i^{Ed_i=1} - Y_i^{Ed_i=0} | x_1,x_2,\ldots ,x_J) = 0 \end{aligned}$$

(6)

$$\begin{aligned}&\displaystyle E(Y_i^{x_{ij}=1}-Y_i^{x_{ij}=0}) \equiv \sigma _j + \kappa _j + \omega _j + \epsilon _j \ \forall \ j \in \{1, \ldots , J\} \end{aligned}$$

(7)

where $\sigma _j$, $\kappa _j$, $\omega _j$, and $\epsilon _j$ are the parts of the effect of $x_j$ on Y that are attributable to signaling, human capital, other educational explanations, and other non-educational explanations. $\varSigma = \sum _j \sigma _j, K = \sum _j \kappa _j$, and $\varOmega = \sum _j \omega _j$. Isolating the part of each mediator that is driven by education excludes non-educational explanations such that

$$\begin{aligned} E\left( Y_i^{Ed_i = 1, x_{ij}^{Ed_i = 1}}-Y_i^{Ed_i = 0, x_{ij}^{Ed_i = 0}}\right) = \sigma _j + \kappa _j + \omega _j \ \forall \ j \in \{1, \ldots , J\} \end{aligned}$$

(8)

and for most points of discussion assume also that $\omega _j = 0 \ \forall \ j$.

Divide the set ${x_1,x_2,\ldots ,x_J}$ into the subset $\chi _\sigma$ for which $\kappa _j = 0 \ \forall \ x_j \in \chi _\sigma$, $\chi _\kappa$ for which $\sigma _j = 0 \ \forall \ x_j \in \chi _\kappa$, and $\chi ^C$ for which $\kappa _j \ne 0$ and $\sigma _j \ne 0 \ \forall \ x_j \in \chi ^C$. For an element of $\chi ^C$, we could identify $\sigma _j$ or $\kappa _j$ separately if there were a way to vary the mediator while holding the other explanation constant, or by varying only the part of the mediator associated with one explanation, expressed as

$$\begin{aligned} E\left( Y_i^{Ed_i = 1, x_{ij}^{Ed_i = 1}|\kappa _j}-Y_i^{Ed_i = 0, x_{ij}^{Ed_i = 0}|\kappa _j}\right) = \sigma _j \end{aligned}$$

(9)

The share of the educational return that is due to, for example, signaling, is defined as $\varSigma /B = (B-K)/B = \varSigma /(\varSigma +K)$ under the assumption that $\varOmega =0$. Identifying this share requires following the steps given in Sect. 2, which includes the tasks of identifying at least some of:

• The total effect of education B, which is $E\left( Y_i^{Ed_i=1} - Y_i^{Ed_i=0}\right)$

• The part of the effect of education that goes through $\chi _\sigma$, which is

  $\sum _{j|x_j\in \chi _\sigma }E\left( Y_i^{Ed_i = 1, x_{ij}^{Ed_i = 1}}-Y_i^{Ed_i = 0, x_{ij}^{Ed_i = 0}}\right)$

• The part of the effect of education that goes through $\chi _\kappa$, which is

  $\sum _{j|x_j\in \chi _\kappa }E\left( Y_i^{Ed_i = 1, x_{ij}^{Ed_i = 1}}-Y_i^{Ed_i = 0, x_{ij}^{Ed_i = 0}}\right)$

• The parts of the effect of education that go through $\chi ^C$ that are signaling-related, which is

  $\sum _{j|x_j\in \chi ^C}E\left( Y_i^{Ed_i = 1, x_{ij}^{Ed_i = 1}|\kappa _j}-Y_i^{Ed_i = 0, x_{ij}^{Ed_i = 0}|\kappa _j}\right)$

• The parts of the effect of education that go through $\chi ^C$ that are human capital-related, which is

  $\sum _{j|x_j\in \chi ^C}E\left( Y_i^{Ed_i = 1, x_{ij}^{Ed_i = 1}|\sigma _j}-Y_i^{Ed_i = 0, x_{ij}^{Ed_i = 0}|\sigma _j}\right)$

with the particular elements that must be identified varying depending on which explanation is being identified and what strategy is being taken for following the steps in Sect. 2.