Instrumental Variables for Taxable-Income Elasticities
A tax reform changes the schedule, but a person’s realized marginal rate also changes when earnings respond. A simulated instrument holds baseline earnings fixed and applies the new schedule, removing that mechanical feedback. The remaining variation can still be related to mean reversion or differential income trends, as emphasized by (Gruber and Saez 2002; Weber 2014). The calculations below separate those identification issues from heterogeneous-effect weights.
1. Start with the population IV estimand
Write \(X=\Delta\log(1-\tau)\) and \(Y=\Delta\log z\). In a heterogeneous-elasticity model,
For a scalar instrument \(Z\) with nonzero first-stage covariance,
There are two departures from the unweighted mean elasticity: the instrument may place unequal weight on responses, and it may be correlated with the structural disturbance. A strong first stage alone eliminates neither term.
2. What a valid binary instrument averages
Proposition 1. Suppose a binary \(Z\) is independent of potential treatment changes, elasticities, and the structural disturbance, with exclusion and \(X(1)\ge X(0)\) almost surely. If \(\mathbb{E}[X(1)-X(0)]>0\), the Wald ratio is
Subtracting expected outcomes across instrument values leaves each person’s elasticity multiplied by the first-stage change. Dividing by the mean change normalizes these nonnegative weights. This illustrates the taxable-income weighting problem studied in (Kumar and Liang 2020); exogeneity and exclusion are still required to remove disturbance terms.
3. Why baseline-income instruments remain delicate
Suppose baseline log income is permanent income plus a transitory shock, \(y_0=a+e_0\). Even if future shocks are independent, income growth contains \(-e_0\). A simulated reform instrument that changes with \(y_0\) can therefore correlate with untreated income growth. Flexible baseline-income controls attempt to account for this dependence, but their adequacy is a substantive identifying assumption.
Changing the lag changes both the relation to transitory shocks and the instrument’s relevance. Exogeneity must be justified for that choice; it does not follow from using an earlier observation.
4. A covariance simplification worth keeping
Let \(\overline Z(X)=\mathbb{E}[Z\mid X]\). Then
The first-stage identity remains \(\operatorname{Cov}(Z,X)=\operatorname{Cov}(\overline Z(X),X)\), since \(\operatorname{Cov}(Z,X\mid X)=0\). Thus the residual conditional covariance in the numerator is an additive term, not the numerator of a second IV ratio: its proposed denominator would be identically zero.
The bunching approach uses local density information and faces a different counterfactual problem. Comparing its elasticity with an IV estimate requires comparing their effect weights as well as their identifying assumptions.
5. Deriving the weights rather than naming an average
For a binary instrument with \(\pi=\mathbb P(Z=1)\in(0,1)\),
The same factor multiplies the treatment difference, so it cancels in the IV ratio. Under the assumptions of the proposition,
Writing \(\Delta X_i=X_i(1)-X_i(0)\), the implicit weight is \(w_i=\Delta X_i/\mathbb{E}\Delta X_i\). Monotonicity ensures \(w_i\geq0\) and \(\mathbb{E}w_i=1\). Individuals whose net-of-tax rates respond more to the instrument contribute more to the estimand. Even a perfectly valid instrument therefore need not recover the unweighted population elasticity.
If individual responses are nonlinear, replace \(Y_i(x)=\varepsilon_ix+U_i\) by a differentiable response function. The fundamental theorem of calculus gives
For a monotone first stage, the Wald ratio averages these marginal responses over the treatment intervals moved by the instrument, divided by their expected length. This is the continuous-treatment analogue of the simple weighting result. Calling it an elasticity also requires that both variables have the stated logarithmic interpretation and that other channels of the reform are excluded.
6. A calculation of mean-reversion bias
Take \(y_0=a+e_0\) and \(y_1=a+e_1\) in the absence of a behavioral response. Assume \(a,e_0,e_1\) are independent and centered, with variances \(\sigma_a^2,\sigma_e^2,\sigma_e^2\). Untreated growth is \(U=e_1-e_0\). For the illustrative instrument \(Z=y_0\),
If the population first stage is \(X=\pi_ZZ+v\) with \(\operatorname{Cov}(Z,v)=0\), a homogeneous-elasticity IV regression yields
Baseline income contains a transitory shock that enters subsequent growth with the opposite sign. Independence of future shocks does not remove this correlation. Its effect on IV bias depends on the reform’s first-stage direction.
Under joint normality, the conditional untreated trend is explicit:
Baseline-income controls address the conditional trend in this example, but real trends may depend on permanent income, shock persistence, and the nonlinear schedule. There is also a relevance constraint: if the instrument is determined entirely by baseline income, unrestricted conditioning on that income removes all instrument variation. Identification then needs remaining variation, such as reform differences across groups or periods, and a restriction on their untreated trends.
7. What residualization does to the identifying moment
Let \(W\) denote included controls and, for a population calculation, define \(\widetilde Z=Z-\mathbb{E}[Z\mid W]\) and \(\widetilde X=X-\mathbb{E}[X\mid W]\). In the homogeneous model \(Y=\varepsilon X+g(W)+U\),
The control function \(g(W)\) disappears because \(\mathbb{E}[\widetilde Z\mid W]=0\). But residualization does not force \(\mathbb{E}[\widetilde ZU]=0\). That is the conditional identifying restriction. In a finite-dimensional regression, the corresponding calculation uses linear projections onto the specified controls; it only removes components in that span.
Simulation removes post-reform earnings from the calculation of the instrument. It leaves a separate requirement: that the simulated variation remaining after controls be unrelated to \(U\). Baseline controls, lag choices, and pre-reform trends belong to that identifying argument.
For a binary instrument, residualization also changes the effect weights. Let \(p(W)=\Pr(Z=1\mid W)\), and suppose instrument independence and exclusion hold conditional on \(W\). Define conditional first-stage and reduced-form changes \(\Delta_X(W)\) and \(\Delta_Y(W)\). Iterating over \(W\) gives
and the analogous denominator is \(\mathbb{E}[p(W)(1-p(W))\Delta_X(W)]\). Where \(\Delta_X(W)>0\), write \(\tau_{IV}(W)=\Delta_Y(W)/\Delta_X(W)\). Then
Thus covariate groups receive more weight when the instrument varies within the group and when it moves the net-of-tax rate more strongly. Conditional monotonicity makes the weights nonnegative; groups with zero first stage contribute no denominator weight. A conditional exclusion restriction remains essential for interpreting each \(\tau_{IV}(W)\) causally.
Valid IV specifications can estimate different weighted elasticities. Controls change the residual instrument and the groups supplying first-stage variation; reform episodes change the income ranges moved. A comparison with a local bunching elasticity must account for these weights before interpreting differences as evidence against either design.
8. Keeping the covariance decomposition well defined
For the final identity above, decompose \(Z=\overline Z(X)+(Z-\overline Z(X))\). The residual is orthogonal to every integrable function of \(X\), so
and
The split preserves one first-stage denominator. Conditioning on pre-treatment controls may express an identifying restriction; conditioning on the endogenous change \(X\) in this calculation is only algebra. It does not produce an additional instrument.