# 9 Mediation modeling and its massive limitations

## 9.1 Mediators (and selection and Roy models): a review, considering two research applications

Originally focused on issues relevant to Parey et al project on ‘returns to HE institution’ using data from the Netherlands (flagged as @NL); also relevant to Reinstein et al work on substitution in charitable giving (flagged as @subst).

## 9.2 DR initial thoughts (for NL education paper)

Suppose we observe treatment \(T\) (e.g., ‘allowed to enter first-choice institution and course’),

intermediate outcome \(M\) (e.g., completion of degree in first-choice course and institution),

and final outcome \(Y\) (e.g., lifetime income.)

*Alternately, in the “substitution between charities” (@subst) context… (unfold)*

The treatment \(T\) is

‘asked to donate in the first round’ (in Reinstein, Riener and Vance-McMullen, henceforth ‘RRV’ experiments, and perhaps in Schmitz 2019)’,

a greater incentive or a nudge to donate in round 1 (Heger and Slonim, 2020; others?),

the inclusion of (an incentive to donate to) an additional charity in that same round (Reinstein 2006; Filiz-Ozbay and Uler; many others),

the intermediate outcome \(M\) is the amount given to that (first-round) charity,

and the final outcome \(Y\) is the amount given to that charity (or other charities) in round 2 (experiments “3”: other charities in that round ).

The treatment \(T\) (may) directly affect the final outcome \(Y\).

Do: show a diagram here

\[T\rightarrow Y\]

\(T\) also may affect an intermediate outcome \(M\).

\[T \rightarrow M\]

The intermediate outcome also may affect the final outcome \(Y\).

\[M \rightarrow Y\]

With exogenous variation in \(T\) *and* \(M\) (or identified instruments for each of these), we should be able to estimate each of these three relationships as functions.

With homogeneous (and in a simplest case linear and separate) effects we can then use these functions to compute the total (direct plus indirect) effect of \(T\) on \(Y\).

We could also compute the share of this effect that occurs *via* the intermediate effect, i.e., \(T \rightarrow M\rightarrow Y\).
This should be merely the composition of these two functions, or, in the linear case, the product of the slope coefficients.

However, there are two major challenges to this estimation.

We (may) have a valid instrument for (exogenous variation in) \(T\) only, and \(M\) may arise through a process involving selection on unobserved variables.

Each of the three above relationships (as well as the selection equation) may involve heterogeneous functions; i.e., differential treatment effects.

Thus we consult the relevant literature, discussed below.

The most influential paper in Economics has been (**Heckman2013?**).

It is cited in more recent applied work such as Fagereng, 2018 (unfold).

… We follow Heckman et al. (2013) and Heckman and Pinto (2015) in using mediation analysis. The goal of this analysis is to disentangle the average causal effect on outcomes that operate through two channels: a) Indirect effects arising from the effect of treatment on measured mediators, and b) direct effects that operate through channels other than changes in the measured mediators (including changes in mediators that are not observed by the analyst and changes in the mapping between mediators and outcomes).

It is therefore necessary to assume that the mediators we do not observe are uncorrelated with both \(\mathbf{X}\) and the measured mediators for all values of \(D\).

Antonakis, coming from the Psychology and Leadership disciplines, considers the mediation question in a much simpler set of models.

## 9.3 Econometric Mediation Analyses (Heckman and Pinto)

**Econometric Mediation Analyses: Identifying the Sources of Treatment Effects from Experimentally Estimated Production Technologies with Unmeasured and Mismeasured Inputs**

### Relevance to Parey et al

We have an instrument for admission to one’s first-choice institution (and course-subject). Our result show an impact of this admission on future income, for at least some groups. However, this effect could come through any of a number of channels. We observe some of these ‘intermediate outcomes,’ including course enrollment, course completion, medical specialization, and location of residence, but we do not have specific instruments for each of these.

a lot of work might yield an instrument for specialization; I hear there is a lottery at that level as well

### 9.3.1 Summary and key modeling

There is a ‘production function’

cf income as a function of human capital, opportunities, etc.

cf donation as a function of income, prices, mood, framing, etc.

Treatments (e.g., RCTs) may affect outcomes through the following channels:

- observable or proxied inputs

Cf degree obtained, specialization entered, years of study, moving away from parents, location of residence as proxy for job opportunities

Cf donation in first stage (to targeted charity), measured/self reported attitudes towards charities, self-reported mood

- unobservable/unmeasured inputs

- cf human capital, social connections
- cf unobservable generosity, wealth, or temporary mood

- the production function itself, the ‘map between inputs and outputs for treatment group members’

Cf does the institution itself directly shift the income?, does it change the impact of entering a specialization, does human capital ‘matter more’ at some institutions?

Cf Does he treatment affect the impact of having made the first donation on later donations , or the effect of mood on donating.. ; what else?

If treatments affect unmeasured inputs in a way not statistically independent of measured inputs, this biases estimates of the effect of measured inputs.

RCTs unaided by additional assumptions do not allow the analyst to identify the causal effect of increases in measured inputs on outputs ... [nor distinguish effects from changes in production functions].

Here “we can test some of the strong assumptions implicitly invoked.”

“Direct effects” as commonly stated refer to the impact of both channels 2 and 3 above.

DR: Channel 2 isn’t really a direct effect imho (what was this?)

**Standard potential outcomes framework:**

\[Y=DY_{1}+(1-D)Y_{0}\]

\[ATE=E(Y_{1}-Y_{0})\]

**Production function**

\[Y_{d}=f_{d}(\mathbf{\mathbf{{\theta}}}_{d}^{p},\mathbf{\mathbf{{\theta}}}_{d}^{u},\mathbf{{X}}),d\in\left\{ 0,1\right\}\]

... the function under treatment \(d\); of proxied and unobserved inputs that occur under state \(d\), and baseline variables.

The production function implies:

\[ATE=E\Big(f_{1}(\mathbf{\mathbf{{\theta}}}_{1}^{p},\mathbf{\mathbf{{\theta}}}_{1}^{u},\mathbf{{X}})-f_{0}(\mathbf{\mathbf{{\theta}}}_{0}^{p},\mathbf{\mathbf{{\theta}}}_{0}^{u},\mathbf{{X}})\Big)\]

We also consider counterfactual outputs, fixing treatment status and proxied inputs:

\[Y_{d,\bar{\theta_{d}}^{p}}=f_{d}(\mathbf{\mathbf{{\bar{{\theta}}}}}_{d}^{p},\mathbf{\mathbf{{\theta}}}_{d}^{u},\mathbf{{X}}),d\in\left\{ 0,1\right\}\]

This allows us to decompose (‘as in the mediation literature’):

\[ATE(d)=IE(d)+DE(d)\]

*IE, Indirect effect*: allows only the proxied inputs to vary with the treatment (holds the rest fixed at one of the two treatment statuses)*DE, Direct effect*: allows technology and the distribution of unobservables to vary with the treatment (holds proxied inputs fixed at one of the two treatment statuses)

HP further decompose the direct effect into:

\(DE'(d,d')\): The impact of letting the treatment vary the map only (fixing the rest at one of the two appropriate values)

\(DE''(d,d')\): The impact of letting the treatment vary the unmeasured inputs only (fixing the rest at one of the two appropriate values)

They use this to give two further ways of decomposing the ATE.

### 9.3.2 Common assumptions and their implications

“The __standard literature__ on mediation analysis in psychology regresses outputs on mediator inputs” ... often adopts the strong assumptions of:

- no variation in unmeasured inputs conditional on the treatment (implying the effects of these are summarized by a treatment dummy) and

Cf ‘winning institution’ impacts human capital, social networks, etc identically for everyone; e.g., not a greater effect for men then for women, nor a greater effect for those entering particular specializations.

- full invariance of the production function: \(f_{1}=f_{0}\).

... which implies \(Y_{d}=f(\mathbf{\theta}_{d}^{p},d,\mathbf{X})\).

__Sequential ignorability (Imai et al, 10, ’11)__:
Essentially, independent randomization of both treatment status and measured inputs.

Cf ‘winning institution’ does not effect the specialization entered nor the location of residence, nor are both determined by a third factor.

*This sentence is hard to follow:*

In other words, input \(\theta_{d'}^{p}\) is statistically independent of potential outputs when treatment is fixed at \(D=d\) and measured inputs are fixed at \(\bar{\theta_{d'}^{p}}\) conditional on treatment assignment \(D\) and same preprogram characteristics \(X\).

This assumption yields the ‘mediation formulas’:

\[\begin{aligned} E(IE(d)|X)= & \int E(Y|\theta^{p}=t,D=d,X)\underbrace{\Big(dF_{(\theta^{p}|D=1,X)}(t)-dF_{(\theta^{p}|D=1,X)}(t)\Big)}_{{\text{Difference in distribution of proxy inputs}}} & (9)\\ E(DE(d)|X)= & \int\underbrace{\Big(E(Y|\theta^{p}=t,D=1,X)-E(Y|\theta^{p}=t,D=0,X)\Big)}_{\text{Dfc in expectations (unobservables, function) between treatments given proxy inputs }}expe\underbrace{{dF_{(\theta^{p}|D=1,X)}(t)}}_{\text{Distn proxy inputs for D=1}} & (10) \end{aligned}\]*(??F is presumably the distribution over the observables; where did the
unobservables go? They are in the expectations, I guess.)*

__Difference from RCT__

*What RCT doesn’t do:*

[sequential ignorability] translates into ...

no confounding effectson both treatments and measured inputs ... does not follow from a randomized assignment of treatment ...[which] ensures independence between treatment status and counterfactual inputs/outputs ... [butnot] between proxied inputs \(\theta_{d}^{p}\) and unmeasured inputs \(\theta_{d}^{u}\). [Thusnotbetween counterfactual outputs and measured inputs is assumed in condition (ii).]

Cf, randomizing ‘win first-choice institution’ does not guarantee that the choice (potential choice under winning/losing institution) to enter a particular specialty is independent of (potential after winning/losing institution) unobserved human capital gains at an institution. The (potential) choiceof specialty is alsonot guaranteed choice independent of potential incomes (holding proxy inputs like specialty constant) if winning/losing institution.

*What RCT* __does__*do:*

RCT ensures “independence between treatment status and counterfactual inputs/outputs,” thus identifying ’treatment effects for proxied inputs and for outputs.

CF, we can identify the impact of the treatment ‘win first chosen institution’ on proxied input like ‘enters a specialization’ and on outputs like ‘income in observed years.’

## 9.4 Pinto (2015), Selection Bias in a Controlled Experiment: The Case of Moving to Opportunity

### Summary

... 4000+ families targeted, incentive to relocate from projects to better neighbourhoods.

Easy to identify impact of vouchers

Challenge (here) is to assess impact of

*neighborhoods*on outcomes.Method here to decompose the TEOT into unambiguously interpreted effects. Method applicable to ‘unordered choice models with categorical instrumental variables and multiple treatments’

Finds significant causal effect on labour market outcomes

### Relevance to Parey et al

We also have an instrument (DUO lottery numbers) cleanly identifying the effect of the ‘opportunity to do something’ (in our case, to enter the course at your preferred institution). However, we also want to measure the impact of choices ‘encouraged’ by the instrument, such as (i) attending the first choice course and institution and (ii) completing this course. We also deal with unordered choices (i. enter course and institution, enter course at other institution, enter other course at institution, enter neither) (ii. choice of medical specialisation)

The geographic outcome is relevant to our second paper (impact on ‘lives close to home’)

### Introduction

The causal link between neighborhood characteristics and resident’s outcomes has seldom been assessed.

**Treatments:**

Control (no voucher)

Experimental: could use voucher to lease in low-poverty neighborhood

Section 8: Could use voucher in any () neighborhood

*Many papers evaluate the ITT or TOT effects of MTO.*

ITT: effect of being

*offered*voucher- estimated as difference in average outcome of experimental vs control families

TOT: effect for ‘voucher compliers’ (assuming no effect of simply being

*offered*voucher on those who don’t use it)- estimated as ITT/compliance rate

[ITT and TOT] are the most useful parameters to investigate the effects of

offering[EA] rent subsidising vouchers to families.

### Identification strategy brief

- Vouchers as IVs for choice among 3 neighborhood alternatives (no relocation, relocate bad, relocate good)

Cf @NL: enter course and fp-institution, enter course at other institution, do not enter course

Neighborhood causal effects as difference in counterfactual outcomes among 3 categories

Challenge: “MTO vouchers are insufficient to identify the expected outcomes for all possible counterfactual relocation decisions”

- ... “compliance with the terms of the program was highly selective [Clampet-Lundquist and M, 08]”

Solution: Uses theory and ‘tools of causal inference. Invokes SARP to identify ’set of counterfactual relocation choices that are economically justifiable’

__Identifying assumption__: “the overall quality of the neighborhood is not directly caused by the unobserved family variables even though neighborhood quality correlates with these unobserved family variables due to network sorting”‘Partition sample ... into unobserved subsets associated with economically justified counterfactual relocation choices and estimate the causal effect of neighborhood relocation conditioned on these partition sets.’ [

*what does this mean?*]

### Results in brief

“Relocating from housing projects to low poverty neighborhoods generates statistically significant results on labor market outcomes ... 65% higher than the TOT effect for adult earnings.”

### Framework: first for binary/binary (simplification)

**First, for binary outcomes (simplified)**

\(Z_{\omega}\): whether family \(\omega\) receives a voucher *(cf
institution-winning lottery number)*

\(T_{\omega}\): whether family \(\omega\) relocates (*cf enters first choice
institution and course)*

**Counterfactuals**

\(T_{\omega}(z)\): relocation decision \(\omega\) would choose if it had been assigned voucher \(z\in{0,1}\)’: vector of potential relocation decisions (

*cf education choices)*for each voucher assignment (*cf lottery number)*- Can partition into never-takers, compliers, always takers, and defiers

\((Y_{\omega}(0);Y_{\omega}(1\))): (Potential counterfactual) outcomes (

*cf income, residence, etc*) when relocation decision is fixed at 0 and 1, respectively

**Key ( standard) identification assumption: instrument independent of
counterfactual variables**

\[(Y_{\omega}(0),Y_{\omega}(1),T_{\omega}(0),T_{\omega}(1))\perp Z_{\omega}\]

**Standard result 1: ITT**

\[\begin{aligned} ITT=E(Y_{\omega}|Z_{\omega}=1)-(Y_{\omega}|Z_{\omega}=0)\\ =E(Y_{\omega}(1)-Y_{\omega}(0)|S_{\omega}=[0,1]')P(S_{\omega}=[0,1])+E(Y_{\omega}(1)-Y_{\omega}(0)|S_{\omega}=[1,0]')P(S_{\omega}=[0,1])\end{aligned}\]

i.e., ITT computation yields the sum of the ‘causal effect for compliers’ and the ’causal effect for defiers, weighted by the probability of each.

**Standard result 2: LATE**

\[\begin{aligned} LATE=\frac{{ITT}}{P(T_{\omega}=1|Z_{\omega}=1)-P(T_{\omega}=1|Z_{\omega}=0)}= & & E(Y_{\omega}(1)-Y_{\omega}(0)|S_{\omega}=[0,1]')\\ if\:P(S_{\omega}=[0,1])=0\end{aligned}\]

i.e., the LATE, computed as the ITT divided by the ‘first stage’ impact of the instrument, is the causal effect for compliers if there are no defiers.

### Framework for MTO multiple treatment groups, multiple choices

\(Z_{\omega}\in\{z_{1,}z_{2,}z_{3}\}\) for no voucher, experimental voucher, and section 8 voucher, respectively

\(T_{\omega}\in\{1,2,3\}\) ... no relocation, low poverty neighborhood relocation, high poverty relocation

\(T_{\omega}(z)\): relocation decision for family \(\omega\) if assigned voucher \(z\)

\(\rightarrow\) Response type for each family \(\omega\) is a three-dimensional vector:

\[S_{\omega}=[T_{\omega}(z_{1}),T_{\omega}(z_{2}),T_{\omega}(z_{3})]\].

\(\rightarrow\)

**ITT** computation now measures a weighted sum of effects across a subset of those response types whose responses vary between the assignments being compared.

*Cf:*

Considering the ‘treatments’: ‘1: enter other course at fp-inst, ’2: enter course at fp-inst,’ ‘3: enter course at non-fp inst’

- (I ignore other course at other institution for now)

Looking among those who won the course lottery (so we have a binary instrument: wininst \(Z_{\omega}\in{0,1\}}\)

Our reduced-form estimates (regressions on the ‘lottery number wins institution’ dummy) measures the probablility-weighted sum of:

impact of institution within course ($T_{}=$2 versus 3); for those who would ‘fully comply’ (enter course at institution if \(Z_{\omega}=1\), enter course at other institution if 0)

impact of the course at fp-institution versus second-best course at fp-institution for ‘institution-loving’ noncompliers; those who would enter the course

*only*if they get the fp-institution and otherwise another course at the same institutioneffects for perverse defiers

## 9.5 Antonakis approaches

Insert notes here