dominated convergence theorem, we have that
$\theta(a_{n})=\int_{\mathcal{D}}f_{n}(d)\to\int_{\mathcal{D}}\omega(d)E_{P}[m(aY(d)]=\theta(a)$,
as needed.

To show that $|\theta(a)|\to\infty$ as $a\to\infty$, we will show that

$\frac{\theta(a)}{\log(a)}\to\int_{\mathcal{D}}\omega(d)P[Y(d)>0]$

as $a\to\infty$. Consider $a_{n}\to\infty$, and suppose without loss of
generality that $a_{n}>e$. Observe that

$\frac{\theta(a_{n})}{\log(a_{n})}=\int_{\mathcal{D}}\omega(d)\dfrac{E_{P}[m(a_{n}Y(d))]}{\log(a_{n})}.$

We showed in the proof to 4 that for each $d$,

$\dfrac{E_{P}[m(a_{n}Y(d))]}{\log(a_{n})}\to P(Y(d)>0).$

Letting $f_{n}(d)=\omega(d)\dfrac{E_{P}[m(a_{n}Y(d))]}{\log(a_{n})}$, we thus
have that $f_{n}(d)\to\omega(d)P(Y(d)>0)$ pointwise. Moreover, we showed in
the proof to 1 that

$|m(y)|\leq\kappa+2\cdot\mathbbm{1}[y>0]\cdot|\log(y)|$

where $\kappa$ is a constant not depending on $y$. It follows that

$\frac{|m(a_{n}y)|}{\log(a_{n})}\leq\kappa+2\cdot\mathbbm{1}[y>0]\cdot(1+|\log(y)|)$

and thus that

$|f_{n}(d)|\leq|\omega(d)|\cdot(\kappa+2+2E_{P}[|\log(Y(d)|\mid Y(d)>0])$

where the upper bound is integrable by the fourth part of the proposition. The
result then follows from dominated convergence.

∎

###  B.3 Extension to OLS estimands and standard errors

As noted in 4, our results imply that any consistent estimator of the ATE for
an outcome of the form $m(aY)$ will be (asymptotically) sensitive to scaling
when there is an extensive margin effect. Our results thus cover the OLS
estimator when it is consistent for the ATE for some (sub)-population $P$
(e.g. in an RCT or under unconfoundedness). Given the prominence of OLS in
applied work—and the fact that it is sometimes used for non-causal analyses—we
now provide a direct result on the sensitivity to scaling of the estimand of
an OLS regression of an outcome of the form $m(aY)$ on an arbitrary random
variable $X$.

Specifically, suppose that $(X,Y)\sim Q$, for $Y\in[0,\infty)$ and
$X\in\mathbb{R}^{J}$, where the first element of $X$ is a constant. Consider
the OLS estimand

$\beta(a)=E_{Q}[XX^{\prime}]^{-1}E_{Q}[Xm(aY)],$

i.e. the population coefficient from a regression of $m(aY)$ on $X$. We assume
that $E_{Q}[XX^{\prime}]$ is full-rank so that $\beta(a)$ is well-defined.
Letting $\beta_{j}(a)=e_{j}^{\prime}\beta(a)$ be the $j$th element of
$\beta(a)$, we will show that $\beta_{j}(a)$ can be made to have arbitrary
magnitude via the choice of $a$ if $\gamma_{j}\neq 0$, where

$\gamma=E_{Q}[XX^{\prime}]^{-1}E_{Q}[X\mathbbm{1}[Y>0]]$

is the coefficient from a regression of $\mathbbm{1}[Y>0]$ on $X$.

######  Proposition 6.

Suppose that

  1. 1.

The function $m$ satisfies parts 1 and 2 of 1.

  2. 2.

(Finite expectations) $E_{Q}[\lVert X\rVert]<\infty$ and $E_{Q}[\lVert
X\log(Y)\rVert\mid Y>0]<\infty$ .

Then for every $j\in\\{2,...,J\\}$, $\beta_{j}(a)/\log(a)\to\gamma_{j}$ as
$a\to\infty$. Moreover, if $\gamma_{j}\neq 0$ for some $j\in\\{2,...,J\\}$,
then for every $\beta_{j}^{*}\in(0,\infty)$, there exists $a>0$ such that
$|\beta_{j}(a)|=\beta_{j}^{*}$. In particular $\beta_{j}(a)$ is continuous
with $\beta_{j}(a)\to 0$ as $a\to 0$ and $|\beta_{j}(a)|\to\infty$ as
$a\to\infty$.

We note that 6 implies that the OLS estimator for the $j$th coefficient,
$\hat{\beta}_{j}(a)$, will be arbitrarily sensitive to the choice of $a$ when
the corresponding extensive margin OLS estimator $\hat{\gamma}_{j}$, is non-
zero. This follows immediately from setting $Q$ to be the empirical
distribution of $(Y_{i},X_{i})_{i=1}^{N}$ and applying 6 (note that part 2 of
the Proposition is trivially satisfied for the empirical distribution, since
$X$ and $Y$ are both bounded over the empirical distribution).

##### OLS Standard Errors.

We also show that as $a\to\infty$, the $t$-statistic for the OLS estimate
$\hat{\beta}_{j}$ constructed using heteroskedasticity-robust standard errors
converges to the $t$-statistic for $\hat{\gamma}_{j}$ (again using
heteroskedasticity-robust standard errors). Formally, let

$\hat{\Omega}_{\beta}(a)=\left(\frac{1}{N}\sum_{i}X_{i}X_{i}^{\prime}\right)^{-1}\left(\frac{1}{N}\sum_{i}X_{i}X_{i}^{\prime}\hat{\epsilon}_{i}(a)^{2}\right)\left(\frac{1}{N}\sum_{i}X_{i}X_{i}^{\prime}\right)^{-1}$

denote the estimator of the heteroskedasticity-robust variance matrix for
$\hat{\beta}(a)$, where
$\hat{\epsilon}_{i}(a)=m(aY_{i})-X_{i}^{\prime}\hat{\beta}(a),$ and
$\hat{\beta}(a)$ is the OLS estimate of $\beta(a)$. The $t$-statistic for
$\hat{\beta}_{j}(a)$ is then
$\hat{t}_{\beta_{j}}(a)=\hat{\beta}_{j}(a)/\hat{\sigma}_{\beta_{j}}(a)$, where
$\hat{\sigma}_{\beta_{j}}(a)=\sqrt{e_{j}^{\prime}\hat{\Omega}_{\beta}(a)e_{j}}/\sqrt{N}$.
Analogously, let

$\hat{\Omega}_{\gamma}=\left(\frac{1}{N}\sum_{i}X_{i}X_{i}^{\prime}\right)^{-1}\left(\frac{1}{N}\sum_{i}X_{i}X_{i}^{\prime}\hat{u}_{i}^{2}\right)\left(\frac{1}{N}\sum_{i}X_{i}X_{i}^{\prime}\right)^{-1}$

be the heteroskedasticity-robust variance estimator for $\hat{\gamma}$, the
OLS estimate of $\gamma$, where
$u_{i}=\mathbbm{1}[Y_{i}>0]-X_{i}^{\prime}\hat{\gamma}$. The $t$-statistic for
$\hat{\gamma}_{j}$ is then
$\hat{t}_{\gamma_{j}}=\hat{\gamma}_{j}/\hat{\sigma}_{\gamma_{j}}$, where
$\hat{\sigma}_{\gamma_{j}}=\sqrt{e_{j}^{\prime}\hat{\Omega}_{\gamma}e_{j}}/\sqrt{N}$.

######  Proposition 7.

Suppose that $\left(\frac{1}{N}\sum_{i}X_{i}X_{i}^{\prime}\right)$ is full-
rank and that $\hat{\sigma}_{\gamma_{j}}>0$. If the function $m$ satisfies
parts 1 and 2 of 1 and $\hat{\gamma}_{j}>0$, then
$\hat{t}_{\beta_{j}}(a)\to\hat{t}_{\gamma_{j}}$ as $a\to\infty$.

It follows that when the units of $Y$ are made large, the $t$-statistic for a
treatment effect estimate for $m(Y)$ estimated using OLS will converge to the
$t$-statistic for the OLS estimate of the extensive margin. LABEL:fig:tstats
shows that, indeed, the $t$-statistics for estimates using
$\operatorname{arcsinh}(Y)$ in the _AER_ tend to be close to the
$t$-statistics for the extensive margin, and tend to become even closer after
rescaling the units by a factor of 100.

###### Proof of 6.

Fix $j\in\\{2,...,J\\}$. Note that $\beta(0)=E_{Q}[XX^{\prime}]^{-1}E[Xm(0)]$,
is the coefficient from a regression of a constant outcome $m(0)$ on $X$, and
thus $\beta_{1}(0)=m(0)$ while $\beta_{k}(0)=0$ for $k\geq 2$. Thus
$\beta_{j}(0)=0$. To complete the proof, we will first show that
$\beta_{j}(a_{n})=\gamma_{j}\log(a_{n})+o(\log(a_{n}))$. Hence, if
$\gamma_{j}>0$, then $|\beta_{j}(a)|\to\infty$ as $a\to\infty$. We will then
establish that $\beta_{j}(a)$ is continuous for $a\in[0,\infty)$. The fact
that one can obtain any positive value for $|\beta_{j}(a)|$ then follows from
the intermediate value theorem.

For ease of notation, let
$\nu^{\prime}=e_{j}^{\prime}E_{Q}[XX^{\prime}]^{-1}$, so that
$\beta_{j}(a)=E_{Q}[\nu^{\prime}Xm(aY)]$.

We first show that $\beta_{j}(a_{n})=\gamma_{j}\log(a_{n})+o(\log(a_{n}))$.
Consider a sequence $a_{n}\to\infty$, and assume without loss of generality
that $a_{n}>e$. Let $f_{n}(x,y)=\nu^{\prime}x\cdot m(a_{n}y)/\log(a_{n})$.
Observe that $f_{n}(x,y)\to\nu^{\prime}x\cdot\mathbbm{1}[y>0]$ pointwise,
since $f_{n}(x,0)=\nu^{\prime}x\cdot m(0)/\log(a_{n})\to 0$, while for $y>0$,

$f_{n}(x,y)=\nu^{\prime}x\cdot\frac{m(a_{n}y)}{\log(a_{n})}=\nu^{\prime}x\cdot\frac{m(a_{n}y)}{\log(a_{n}y)}\frac{\log(a_{n})+\log(y)}{\log(a_{n})}\to\nu^{\prime}x,$

where we use the fact that $m(y)/\log(y)\to 1$ as $y\to\infty$. We showed in
the proof to 4 that

$\frac{|m(a_{n}y)|}{\log(a_{n})}\leq\kappa+2\cdot\mathbbm{1}[y>0]\cdot(1+|\log(y)|),$

which implies that

$|f_{n}(x,y)|\leq|\nu^{\prime}x\cdot(\kappa+2\cdot\mathbbm{1}[y>0]\cdot(1+|\log(y)|))|=:\overline{f}(x,y).$

Moreover, part 2 of the proposition implies that $\overline{f}(X,Y)$ is
integrable. From the dominated convergence theorem, it follows that

$\frac{\beta_{j}(a_{n})}{\log(a_{n})}=E_{Q}[f_{n}(X,Y)]\to
E_{Q}[\nu^{\prime}X\mathbbm{1}[Y>0]]=\gamma_{j}.$

Hence, we see that $\beta_{j}(a_{n})=\gamma_{j}\log(a_{n})+o(\log(a_{n}))$. It
follows that $|\beta_{j}(a_{n})|\to\infty$ when $\gamma_{j}\neq 0$.

To complete the proof, we show continuity of $\beta_{j}(a)$. Fix
$a\in[0,\infty)$, and consider a sequence $a_{n}\to a$. Assume without loss of
generality that $a_{n}<a+1$ for all $n$. Let $f_{n}(x,y)=\nu^{\prime}x\cdot
m(a_{n}y)$. From the continuity of $m$, we have that
$f_{n}(x,y)\to\nu^{\prime}x\cdot m(ay)$ pointwise. We showed in the proof to 1
that there exists some $\kappa$ (not depending on $n$) such that

$|m(a_{n}y)|\leq\kappa+2\mathbbm{1}[y>0]\cdot|\log(y)|.$

Hence,

$|f_{n}(x,y)|\leq|\nu^{\prime}x\cdot(\kappa+2\mathbbm{1}[y>0]|\log(y)|)|.$

Moreover, the bounding function is integrable over the distribution of $(X,Y)$
by part 2 of the proposition. Applying the dominated convergence theorem
again, we obtain that

$\beta_{j}(a_{n})=E_{Q}[f_{n}(X,Y)]\to E_{Q}[\nu^{\prime}X\cdot
m(aY)]=\beta_{j}(a),$

as needed. ∎

###### Proof of 7.

Consider $a_{n}\to\infty$. Applying 6 to the empirical distribution, we have
that $\hat{\beta}(a_{n})/\log(a_{n})=\hat{\gamma}+o(1)$. It follows that

$\frac{1}{\log(a_{n})}\hat{\epsilon}_{i}(a_{n})=\frac{m(a_{n}Y_{i})}{\log(a_{n})}-\frac{\hat{\beta}(a_{n})^{\prime}X_{i}}{\log(a_{n})}=\mathbbm{1}[Y_{i}>0]-\hat{\gamma}^{\prime}X_{i}+o(1)=\hat{u}_{i}+o(1).$

Since $\hat{\Omega}_{n}(a_{n})$ is a continuous function of the
$\hat{\epsilon}_{i}(a_{n})^{2}$, we obtain that
$\log(a_{n})^{-2}\hat{\Omega}_{\beta}(a_{n})\to\hat{\Omega}_{\gamma}$, and
thus that
$\log(a_{n})^{-1}\hat{\sigma}_{\beta_{j}}(a_{n})=\hat{\sigma}_{\gamma_{j}}+o(1)$.
It follows that

$\hat{t}_{\beta_{j}}(a_{n})=\frac{\hat{\beta}_{j}(a_{n})/\log(a_{n})}{\hat{\sigma}_{\beta_{j}}(a_{n})/\log(a_{n})}=\frac{\hat{\gamma}_{j}+o(1)}{\hat{\sigma}_{\gamma_{j}}+o(1)}\to\frac{\hat{\gamma}_{j}}{\hat{\sigma}_{\gamma_{j}}}=\hat{t}_{\gamma_{j}},$

as needed. ∎

##  Appendix C Connection to structural equations models

Previous work has considered a variety of estimators for settings with zero-
valued outcomes beginning with structural equations models rather than the
potential outcomes model that we consider. These papers have reached different
results, with some concluding that regressions with
$\operatorname{arcsinh}(Y)$ have the interpretation of an elasticity, and
others showing that they are inconsistent and advocating for other methods
(e.g. Poisson regression) instead. In this section, we interpret the results
in those papers from the perspective of the potential outcomes model, and show
that these diverging conclusions stem from different implicit assumptions
about the potential outcomes, as well as a focus on different causal
parameters.

Before discussing specific papers, we first note that, broadly speaking,
structural equation models can be viewed as constraining the joint
distribution of potential outcomes. Observe that, for any pair of potential
outcomes $(Y(1),Y(0))$, we can represent them as $(Y(1,U),Y(0,U))$ for some
function $Y(d,u)$ and individual-level unobservable (or “structural error”)
$U$. The potential outcomes framework we work with in this paper does not
impose any functional form assumptions on $Y(d,u)$. Structural equation
models, on the other hand, tend to specify explicit functional forms for
$Y(d,u)$. In what follows, we consider the implicit restrictions placed on the
potential outcomes as well as the target estimand in work related work that
starts with a structural equations model.

###  C.1 Bellemare and Wichman (2020) and Thakral and Tô (2023)

Bellemare and Wichman (2020) consider OLS regressions of the form494949They
also consider specifications with additional covariates on the right-hand
side, although we abstract away from this for expositional simplicity.

$\operatorname{arcsinh}(Y)=\beta_{0}+D\beta_{1}+U.$ (C.1)

Note that when $D$ is binary and randomly assigned,
$D\perp\\!\\!\\!\perp(Y(1),Y(0))$, then from the perspective of the potential
outcomes model, the population coefficient $\beta_{1}$ is the ATE for
$\operatorname{arcsinh}(Y)$. Bellemare and Wichman (2020) instead consider the
interpretation of $\beta_{1}$ when (C.1) is treated as structural, i.e. if
there are constant treatment effects of $D$ on $\operatorname{arcsinh}(Y)$.
From the perspective of the potential outcomes model, this amounts to imposing
that the potential outcomes $Y(d):=Y(d,U)$ take the form

$\operatorname{arcsinh}(Y(d,U))=\beta_{0}+d\beta_{1}+U,$ (C.2)

where the individual-level random variable $U$ takes the same value for all
values of $d$. Under (C.2), we have that

$\beta_{1}=\operatorname{arcsinh}(Y(1,U))-\operatorname{arcsinh}(Y(0,U)).$

Since $\operatorname{arcsinh}(y)\approx\log(2y)$ for $y$ large, it follows
that $\beta_{1}\approx\log(Y(1,U)/Y(0,U))$ when $Y(1,U)$ and $Y(0,U)$ are
large. Thus, Bellemare and Wichman (2020) argue that $\beta_{1}$ approximates
the semi-elasticity of the outcome with respect to $d$ when the outcome is
large. They likewise provide similar results for the elasticity of $Y(d,U)$
with respect to treatment when treatment is continuous. Their results thus
imply that the ATE for $\operatorname{arcsinh}(Y)$ has a sensible
interpretation as a (semi-)elasticity when the structural equation for the
potential outcomes given in (C.2) holds.

It is worth emphasizing, however, that (C.2) will generally be incompatible
with the data when both $Y(1)$ and $Y(0)$ have point-mass at zero, and
$\beta_{1}\neq 0$. Specifically, note that (C.2) implies that for all values
of $U$,

$\operatorname{arcsinh}(Y(1,U))-\operatorname{arcsinh}(Y(0,U))=\beta_{1}.$

If $\beta_{1}>0$, for example, this implies that
$\operatorname{arcsinh}(Y(1,U))>\operatorname{arcsinh}(Y(0,U))$, and hence
$Y(1,U)>Y(0,U)$, since the $\operatorname{arcsinh}(y)$ function is strictly
increasing for $y\geq 0$. However, since $Y(0,U)\geq 0$ by assumption, this
implies that $Y(1,U)>0$ with probability 1. Thus, the model in (C.2) is
incompatible with $P(Y(1)=0)>0$ if $\beta_{1}>0$. By similar logic, the model
is also incompatible with $P(Y(0)=0)>0$ if $\beta_{1}<0$. In settings where
there is point-mass at zero, the model that Bellemare and Wichman (2020) show
gives $\beta_{1}$ an interpretation as a semi-elasticity will therefore
typically be rejected by the data. It is also worth noting that even if there
are no zeros in the data, the model in (C.2) will generally be sensitive to
units, in the sense that if (C.2) holds for $Y$ measured in dollars, it will
generally not hold when $Y$ is measured in cents. The validity of the
interpretation of $\beta_{1}$ as an elasticity thus depends on having chosen
the “correct” scaling of the outcome such that (C.2) holds.

Similar issues apply if we consider alternative transformations on the left-
hand side of (C.1). For example, Thakral and Tô (2023) consider versions of
(C.1) that replaces $\operatorname{arcsinh}(Y)$ with the power function
$Y^{k}$. They then consider the implied “semi-elasticities” of the form
$\eta(y_{0})=\beta_{1}/(ky_{0}^{k})$. The parameter $\eta(y_{0})$ has the
interpretation as a structural semi-elasticity when $d$ has a contant effect
on $Y^{k}$. Specifically, if $D$ is continuous and the structural equation

$Y(d,U)^{k}=\beta_{0}+d\beta_{1}+U,$ (C.3)

holds, then $\eta(y_{0})=\left(\frac{\partial}{\partial
d}Y(d,U)\right)/Y(d,U)$ evaluated at $Y(d,U)=y_{0}$, so $\eta(y_{0})$
corresponds to the semi-elasticity of $Y(d,U)$ with respect to $d$. However,
as with (C.2), (C.3) is generally incompatible with settings in which
$P(Y(d,U)=0)$ for multiple values of $d$. For example, if $\beta_{1}>0$, then
$Y(0,U)\geq 0$ implies that $Y(1,U)>0$. Equation (C.3), which gives a causal
interpretation to $\eta(\beta_{0})$ as a semi-elasticity, will thus generally
be incompatible with settings in which some units have $Y=0$ under multiple
treatment statuses.

###  C.2 Cohn et al. (2022)

Cohn et al. (2022) consider structural equations of the form

$Y=\exp(\alpha+D\beta)U.$ (C.4)

When $E[U\mid D]=1$, they show that Poisson regression is consistent for
$\beta$, whereas regressions of $\log(1+Y)$ or $\log(Y)$ on $D$ may be
inconsistent for $\beta$.505050We thank Kirill Borusyak for an insightful
discussion on this topic. Relatedly, in an influential paper, Santos Silva and
Tenreyro (2006) consider the structural equations model
$Y_{i}=\exp(X_{i}^{\prime}\beta)U_{i}$ where $E[U_{i}\mid X_{i}]=1$, and show
that Poisson regression consistently estimates $\beta$ while a regression
using $\log$ on the left-hand side does not, although they do not provide any
formal results on log-like transformations. Although Cohn et al. (2022) do not
consider a potential outcomes interpretation of $\beta$, we can give $\beta$ a
causal interpreation if we impose that the potential outcomes take the form

$Y(d,U)=\exp(\alpha+d\beta)U(d),$ (C.5)

where $E[U(d)]=1$. Under (C.5), it follows that $\exp(\beta)=E[Y(1)]/E[Y(0)]$,
i.e. the parameter $\theta_{\mathrm{ATE}\%}$ considered in
LABEL:subsec:_interpretableunits.515151Bellégo et al. (2022) also consider
(C.4), but consider the more general class of identifying restrictions of the
form $E[D\log(U+\delta)]=0$, where $\delta$ is a tuning parameter.

We note, however, that if one were instead to impose (C.4) with the assumption
that $E[\log(U(d))\mid D]=0$, then the regression of $\log(Y)$ on $D$ would be
consistent for $\beta$, whereas Poisson regression would generally be
inconsistent for $\beta$. Indeed, under the potential outcomes model in (C.5)
with the assumption that $E[\log(U(d))]=0$, we have that
$\beta=E[\log(Y(1))-\log(Y(0))]$, the ATE in logs.525252Note that the
assumption that $E[\log(U)]=0$ implicitly implies that $U>0$, and thus $Y>0$.

This discussion highlights that whether or not an estimator is consistent
depends on the specification of the _target parameter_. Our results help to
illuminate what parameters can be consistently estimated by enumerating the
properties that identified causal parameters can (or cannot) have.

###  C.3 Tobit models

An alternative structural approach is to explicitly model the extensive
margin, a classic example of which is the Tobit model (Tobin, 1958). Following
the discussion of Tobit models in Angrist and Pischke (2009), suppose there
exist latent potential outcomes $Y^{*}(d)=\mu_{d}+U$, where
$U\sim\mathcal{N}(0,\sigma^{2})$ and $D\perp\\!\\!\\!\perp U$. The observed
potential outcome $Y(d)$ is then the latent potential outcome truncated at
zero, $Y(d)=\max(Y^{*}(d),0)$. We note that in this model, the treatment has a
constant additive effect of $\mu_{1}-\mu_{0}$ on the latent outcome, and the
latent potential outcomes are assumed to be normally distributed.

Thanks to the parametric assumptions, the unknown parameters
$\mu_{1},\mu_{0},\sigma^{2}$ are identified and estimable via, e.g., maximum
likelihood. As a result, the entire joint distribution of potential outcomes
is identified, since this depends only on $(\mu_{1},\mu_{0},\sigma)$. This
implies, in turn, that all of the possible target parameters considered in
Section 4 are point-identified. For example, under this model

$E[\log Y(d)\mid Y(1)>0,Y(0)>0]=E\left[{\log\left({\mu_{d}+U}\right)\mid
U>-\mu_{1},U>-\mu_{0}}\right],$

where the right-hand side can be computed numerically since
$U\sim\mathcal{N}(0,\sigma^{2})$. Thus, the intensive margin treatment effect
in logs, $\theta_{\text{Intensive}}$, is actually point-identified under the
Tobit model.535353Likewise, the intensive margin treatment effect in levels,
$E[Y(1)-Y(0)\mid Y(1)>0,Y(0)>0]$ is simply $\mu_{1}-\mu_{0}$.

It is worth nothing that unlike some of the models considered above, the Tobit
model is consistent with a nonzero extensive margin. However, the assumptions
of normal errors and constant treatment effects on the latent index are
restrictive. As discussed in Section 4, imposing these assumptions is not
necessary for identification if one is ultimately interested in, say,
$E[Y(1)-Y(0)]/E[Y(0)]$, and one can obtain bounds on the intensive margin
effect without imposing these assumptions.545454We note that the assumptions
of the Tobit model imply (but are strictly stronger than) the assumption of
rank preservation of the potential outcomes. However, rank preservation alone
suffices to point identify $E[\log Y(1)-\log Y(0)\mid Y(1)>0,Y(0)>0]$.
Moreover, as Angrist and Pischke (2009) and Angrist (2001) point out, it is
often not clear what the economic meaning of the latent potential outcome
$Y^{*}(d)$ is—if $Y(d)$ is earnings, for example, what is the meaning of
having negative latent earnings ($Y^{*}(d)<0$)?

##  Appendix D Connection to two-part models

One approach recommended for settings with weakly-positive outcomes is to
estimate a two-part model (Mullahy and Norton, 2023). In this section, we
briefly review two-part models, and show that the marginal effects implied by
these models do not correspond with ATEs for the intensive margin without
further restrictions on the potential outcomes. Thus, while two-part models
strike us as a reasonable approach if the goal is to model the conditional
expectation function of observed outcomes $Y$ given treatment $D$ (as in
Mullahy and Norton (2023)), they will often not be appropriate if instead the
goal is to learn about a causal effect along the intensive margin.555555We are
particularly grateful to John Mullahy for an enlightening discussion of this
topic.

The idea of a two-part model is to separately model the conditional
distribution $Y\mid D$ using (a) a first model for the probability that $Y$ is
positive given $D$, $P(Y>0\mid D)$ (b) a second model for the conditional
expectation of $Y$ given that it is positive, $E[Y\mid D,Y>0]$. Common
specifications include logit or probit for part (a), and a linear regression
of the positive values of $Y$ on $D$ for part b); see, e.g., Belotti et al.
(2015). After obtaining estimates of the two-part model, it is common to
evaluate the marginal effects of $D$ on both parts, i.e. the implied values of

$\displaystyle\tau_{a}=P(Y>0\mid D=1)-P(Y>0\mid D=0)$
$\displaystyle\tau_{b}=E[Y\mid Y>0,D=1]-E[Y\mid Y>0,D=0].$

We now consider how the parameters of the two-part model relate to causal
effects in the potential outcomes model. Suppose, for simplicity, that the
two-part model is well-specified, so that it correctly models $P(Y>0\mid D)$
and $E[Y\mid Y>0,D]$. Suppose further that $D$ is randomly assigned,
$D\perp\\!\\!\\!\perp Y(1),Y(0)$. In this case, we have that

$\displaystyle\tau_{a}=P(Y(1)>0)-P(Y(0)>0)$ $\displaystyle\tau_{b}=E[Y(1)\mid
Y(1)>0]-E[Y(0)\mid Y(0)>0].$

From the previous display, we see that the marginal effect on the first
margin, $\tau_{a}$, has a causal interpretation: it is the treatment’s effect
on the probability that the outcome is positive.

The interpretation of the marginal effect on the second margin, $\tau_{b}$, is
more complicated, however. For simplicity, suppose are willing to impose the
“monotonicity” assumption discussed in Section 4, $P(Y(1)=0,Y(0)>0)=0$, so
that anyone with a zero outcome under treatment also has a zero outcome under
control. Then, letting $\alpha=P(Y(0)=0\mid Y(1)>0)$, we can write $\tau_{b}$
as

$\displaystyle\tau_{b}$ $\displaystyle=(1-\alpha)E[Y(1)\mid Y(1)>0,Y(0)>0]$
$\displaystyle\quad\quad+\alpha E[Y(1)\mid Y(1)>0,Y(0)=0]-E[Y(0)\mid
Y(1)>0,Y(0)>0]$ $\displaystyle=\underbrace{E[Y(1)-Y(0)\mid
Y(1)>0,Y(0)>0]}_{\text{Intensive margin effect}}$
$\displaystyle\quad\quad+\alpha\underbrace{\left(E[Y(1)\mid
Y(1)>0,Y(0)=0]-E[Y(1)\mid Y(1)>0,Y(0)>0]\right)}_{\text{Selection term}},$

where the first equality uses iterated expectations, and the second re-
arranges terms.

The previous display shows that $\tau_{b}$ is the sum of two terms. The first
is the ATE for individuals who would have a positive outcome regardless of
treatment status (similar to $\theta_{\mathrm{Intensive}}$ in Section 4,
except using $Y$ instead of $\log(Y)$). The second term is not a causal
effect, but rather represents a selection term: it is proportional to the
difference in the average value of $Y(1)$ for “compliers” who would have
positive outcomes only under treatment versus “always-takers” who would have
positive outcomes regardless of treatment status. In many economic contexts,
we may expect this selection effect to be negative. For example, we may
suspect that individuals who would only get a job if they receive a particular
training have lower ability, and hence lower values of $Y(1)$, than
individuals who would have a job regardless of training status. The marginal
effect $\tau_{b}$ thus only has an interpretation as an ATE along the
intensive margin if either (a) there is no extensive margin effect
($\alpha=0$) or (b) we are willing to assume that the selection term is zero.
Angrist (2001) provided a similar decomposition (without imposing
monotonicity), concluding that the two-part model “seems ill-suited for causal
inference,” at least without further restrictions on the potential outcomes.
See, also, Mullahy (2001) for additional discussion.

##  Appendix E Details on Lee bounds using IV in Berkouwer and Dean (2022)

We now describe in detail our approach for constructing Lee (2009)-type bounds
in the IV setting of Berkouwer and Dean (2022).

##### Estimating the instrument-complier distributions.

The first step is to estimate the distribution of $Y(0)$ and $Y(1)$ for
instrument-compliers. As shown in Abadie (2002), the CDF for $Y(1)$ for
instrument-compliers at a point $y$ can be consistently estimated by using
two-stage least squares to estimate the effect of treatment on the outcome
$D_{i}1[Y_{i}\leq y]$. The CDF for $Y(0)$ for instrument-compliers can
analogously be obtained using the outcome $(D_{i}-1)1[Y_{i}\leq y]$. We
estimate these TSLS regressions using analogues to (11) (except replacing
$\operatorname{arcsinh}(Y_{i})$ with the outcomes just described) for all
values of $y$ contained in the data. We thus obtain empirical estimates of the
CDFs for instrument-compliers, $\hat{F}_{Y(d)}(y)$ for $d=0,1$.

##### Constructing bounds.

Note that if $U\sim U[0,1]$, then $Y(d)\sim F_{Y(d)}^{-1}(U)$, where
$F_{Y(d)}^{-1}(u):=\inf\\{y\mid F_{Y(d)}(y)\geq u\\}$. With this formulation
in mind, Lee (2009)’s bounds for $E[\log(Y(1))\mid Y(1)>0,Y(0)>0]$ can be
written as

$\displaystyle E[\log(F_{Y(1)}^{-1}(U))\mid
U\in[\theta_{NT},\theta_{NT}+\theta_{AT}]]$ $\displaystyle\leq
E[\log(Y(1))\mid Y(1)>0,Y(0)>0]$ $\displaystyle\leq
E[\log(F_{Y(1)}^{-1}(U))\mid U\in[1-\theta_{AT},1]],$ (E.1)

where $\theta_{AT}=P(Y(1)>0,Y(0)>0)$, $\theta_{NT}=P(Y(1)=0,Y(0)=0)$, and
$\theta_{C}=P(Y(1)>0,Y(0)=0)$. We estimate the bounds in (E.1) by plugging in
the estimated CDFs for instrument-compliers described above, as well as the
values of $\theta_{AT},\theta_{NT},\theta_{C}$ implied by the estimated CDFs.
We approximate the expectation over $U$ by taking the average over 100,000
uniform draws.565656We note that in finite samples, the estimated CDF
$\hat{F}_{Y(d)}(y)$ may be non-monotonic. Nevertheless, the inverse
$\hat{F}^{-1}_{Y(d)}(u):=\inf\\{y\mid\hat{F}_{Y(d)}(y)\geq u\\}$ remains well
defined. Finally, to compute the bounds on the treatment effect, we must
estimate $E[Y(0)\mid Y(0)>0]$. To do this, we use the fact that

$E[Y(0)\mid Y(0)>0]=E[F^{-1}_{Y(0)}(U)\mid U\in[\theta_{NT}+\theta_{C},1]].$

As before, we then estimate the right-hand-side in the previous display by
plugging-in the estimated CDF for instrument-compliers, and simulating over
100,000 uniform draws. The Lee bounds for $\theta_{\text{Intensive}}$ are then
obtained by subtracting the estimate of $E[Y(0)\mid Y(0)>0]$ from the
estimates of the lower and upper bounds in (E.1). We estimate standard errors
for the bounds using 1,000 draws from a non-parametric clustered
bootstrap.575757One complication that arises is that for some draws from the
bootstrap distribution, the sign of the extensive margin can be the opposite
of that in the original data. In our bootstrap procedure, we construct Lee-
type bounds assuming monotonicity in whichever direction matches the
bootstrapped data. The resulting bootstrap estimates of the bounds appear to
be approximately normally distributed, but we think a formal theoretical
evaluation of the bootstrap in this setting is an interesting topic for future
work.

##  Appendix F Appendix Tables and Figures

  * –

LABEL:tbl:selected-quotes-aer contains information on the _AER_ papers
discussed.

  * –

LABEL:fig:tstats shows how $t$-statistics change in the replication exercise.

  * –

LABEL:tbl:recale-by-100-aer-log1p shows the analogue of Table 1 for
$\log(1+Y)$.

Paper |  Interprets Units as % |  Original Units |  Quote About Percents / Notes
---|---|---|---
Azoulay et al (2019) |  Yes |  Publications (yearly) |  “In this case, coefficient estimates can be interpreted as elasticities, as an approximation.”
Beerli et al (2021) |  Yes |  Patent applications (yearly) |  “The estimates thus reflect an approximate percentage increase.”
Berkouwer and Dean (2022) |  Yes |  Weekly expenditure (dollars) |  “A 0.50 IHS reduction corresponds to a 39 percent reduction relative to the control group.”
Cabral et al (2022) |  Yes |  Costs (dollar) per $10K risk-adjusted covered payroll |  Refers to estimates as “the elasticities reported in panel A”
Carranza et al (2022) |  Yes |  Hours worked (weekly) |  “Weekly earnings increase by 34% (Table 1, column 3)”
Faber & Gauber (2019) |  Yes |  Municipality GDP (1000s of Pesos) |  “A one standard deviation increase in tourism attractiveness increases local manufacturing GDP by about 40 percent.”
Hjort and Poulsen (2019) |  Yes |  KB per second |  “We find that cable arrival increases measured speed in connected locations, relative to unconnected locations, by around 35 percent”
Johnson (2020) |  Yes |  Violations (monthly) |  “[T]he regression coefficient estimates the ITT effect of a press release on the percent change in the number of violations. The point estimate (-0.18) is identical to the baseline estimate in percent terms -0.40/2.29 = 17.5%).”
Mirenda et al (2022) |  Yes |  Contract size (euros) |  “The amount of public funds awarded raises by 3.4 percent.”
Norris et al (2021) |  Yes |  Criminal charges |  “We measure both the extensive margin (using a binary indicator for the outcome ever occurring) and the intensive margin (taking the inverse hyperbolic sine, IHS, of the number of times the outcome occurred, so the coefficient is interpreted as a percent change)”
Ager et al (2021) |  No interpretation |  Wealth (1870 dollars) |
Arora et al (2021) |  No interpretation |  Publications (yearly) |
Bastos et al (2018) |  No interpretation |  Sales (yearly, euros) |
Fetzer et al (2021) |  No interpretation |  Incidents (quarterly) |
Moretti (2021) |  No interpretation |  Patents (yearly) |
Rogall (2021) |  No interpretation |  Perpetrators |
Cao and Chen (2022) |  No |  Rebellions per million population in 1600 |  They compute $\exp(\hat{\beta})-1$ and multiply by the baseline mean, then interpret this as the effect in levels

Appendix Table 1: Papers in the _AER_ estimating effects for
$\operatorname{arcsinh}(Y)$ with selected quotes

appendixtable]tbl:selected-quotes-aer Note: this table lists papers in the
_AER_ estimating treatment effects for $\operatorname{arcsinh}(Y)$. The second
column classifies papers by whether they interpret the units of the treatment
effect as a percent/elasticity, with categories “yes”, “no”, or “no
interpretation given.” The third column describes the units of the outcome
before applying the $\operatorname{arcsinh}$ transformation, and the final
column provides selected quotes and notes about the interpretation of the
estimates. See Section 2.3 for details.

Appendix Figure 1: $t$-statistics for effect on $\operatorname{arcsinh}(Y)$,
versus extensive margin $t$-statistic

appendixfigure]fig:tstats Note: this table shows the $t$-statistic for the
extensive margin effect on the $x$-axis, and the $t$-statistic for the
treatment effect using $\operatorname{arcsinh}(Y)$ on the $y$-axis. The circle
shows the $t$-statistic using the original units, whereas the arrow shows the
change if we first multiply the units by 100 before applying the
$\operatorname{arcsinh}$ transformation. We omit two papers where there is no
extensive margin. The plot shows that the $t$-statistics are close to the 45
degree line when the extensive margin is not close to zero, and tend to become
closer when the units are made larger.

| Treatment Effect Using: |  |  Change from rescaling units:
---|---|---|---
Paper | $\log(1+Y)$ | $\log(1+100\cdot Y)$ | Ext. Margin |  Raw |  %
Azoulay et al (2019) | 0.002 | 0.015 | 0.003 |  0.012 |  529
Fetzer et al (2021) | -0.138 | -0.410 | -0.059 |  -0.272 |  197
Johnson (2020) | -0.139 | -0.408 | -0.057 |  -0.269 |  194
Carranza et al (2022) | 0.166 | 0.415 | 0.055 |  0.249 |  149
Cao and Chen (2022) | 0.032 | 0.076 | 0.010 |  0.044 |  136
Rogall (2021) | 1.109 | 2.015 | 0.195 |  0.906 |  82
Moretti (2021) | 0.041 | 0.067 | 0.000 |  0.026 |  64
Berkouwer and Dean (2022) | -0.412 | -0.484 | 0.010 |  -0.072 |  17
Arora et al (2021) | 0.110 | 0.111 | -0.001 |  0.001 |  1
Hjort and Poulsen (2019) | 0.354 | 0.354 | 0.000 |  0.001 |  0

Appendix Table 2: Change in estimated treatment effects using $\log(1+Y)$ from
re-scaling the outcome by a factor of 100 in papers published in the _AER_

appendixtable]tbl:recale-by-100-aer-log1p Note: this table repeats the
exercise in Table 1 but replacing $\operatorname{arcsinh}(Y)$ with $\log(1+Y)$
as the outcome in the second column, and $\operatorname{arcsinh}(100Y)$ with
$\log(1+100Y)$ in the third column. The fourth column shows the estimated
extensive margin effect, which is identical to the fourth column of Table 1.
The final two columns show the raw difference and percentage difference
between the second and third columns. The rows are sorted based on the
percentage differences. Among the papers surveyed, which by construction
report at least one specification using $\operatorname{arcsinh}(Y)$, Arora et
al. (2021); Fetzer et al. (2021); Moretti (2021); Rogall (2021) also report
specifications that contain $\log(1+Y)$ on the left-hand side, and Johnson
(2020) reports a specification with $\log(c+Y)$ on the left-hand side, where
$c$ is the first nonzero percentile of the distribution of the observed
outcome variable.

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