Job Amenities & Earnings Inequality

A Non-Parametric Proof of Identification

The proof below shows how compensating differentials can be identified from a
large amount of data. i.e., setting aside finite-sample issues
Underlined words have mouse-overs.

Object of Interest

$\psi(w,z|\eta)$ The frontier of wage $w$ and amenity $z$ values associated with total compensation $ \eta$

Assumptions

  1. $(w,z)\perp h|\eta$ conditional independence means $h$ is irrelevant to the wage-amenity split
  2. $E[h|\eta]$ is everywhere strictly monotone in $\eta$ monotonicity is helpful but not necessary
  3. $\eta$ is degenerate given values of $w$ and $z$ there is only 1 omitted variable for which to proxy
    ; i.e., there exists a non-stochastic function $g$ such that $\eta\equiv g(w,z)$

Proposition

Under these assumptions, $\psi(w,z|\eta)$ is identified for all levels of
$\eta$. units of $\eta$ are not important: we want to learn the values of $w$ and $z$ associated with the same $\eta$, so we will remain agnostic about the cardinal units of $\eta$

Proposed Estimator: $\hat{h}$ stands-in for $\eta$

Define the random variable $\hat{h}$ as an approximation to
$E[h|w,z]$. e.g., predicted values from a (linear) regression of $h$ on $w$ and $z$
Denote quantiles by subscripts: e.g.,
$\hat{h}_{j}$quantile $j$ of the constructed variable $\hat{h}$
and
$\eta_{k}$quantile $k$ of the unobserved variable $\eta$
. For all observed $w$ and $z$,
$\psi(w,z|\hat{h}_{j})$the pairs of $w$ and $z$ with a particular constructed control value
identifies
$\psi(w,z|\eta_{k})$the pairs of $w$ and $z$ associated with a particular level of compensation
(for
unknownwe don't need to know these, as we are not interested in the units of $\eta$
$j$ and $k$).

Proof of Identification

With lots of data, the observed $\hat{h}=\hat{E}[h|w,z]$
approximates $\eta$ is degenerate given $w$ and $z$ (Assumption 3), so adding it to the conditioning is redundant
$E[h|w,z,\eta]$, which is also
equivalent the wage-amenity split is not related to $h$ (Assumption 1)
to $E[h|\eta]$. And every quantile of $E[h|\eta]$, and thus $\hat{h}$,
corresponds to a different quantile of because $E[h|\eta]$ is strictly monotone in $\eta$ (Assumption 2)
$\eta$. Thus, the $w$ and $z$ values belonging to arbitrarily small ranges of $\hat{h}$ are exactly the $w$ and $z$ combinations associated with different values of $\eta$.

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