A Difference-in-Differences Approach using Mixed-Integer Programming Matching

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# A Difference-in-Differences Approach<br/>using Mixed-Integer Programming Matching
## Magdalena Bennett
### Brown Bag Seminar, McCombs School of Business - UT Austin<br/>Dec 02, 2020

---

# Diff-in-Diff as an identification strategy

<img src="mbennett_did_files/figure-html/dd-1.svg" style="display: block; margin: auto;" />
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# Diff-in-Diff as an identification strategy

---
# Diff-in-Diff as an identification strategy

---
# Very popular for policy evaluation

<img src="mbennett_did_files/figure-html/gg-1.svg" style="display: block; margin: auto;" />
.source[Source: Google Scholar]
---

# What about parallel trends?

.pull-left[
![:scale 50%](https://raw.githubusercontent.com/maibennett/website_github/master/exampleSite/content/images/data_comic.jpg)]

.pull-right[
- Can matching work to solve this?

- It's complicated (?) .small[(Zeldow & Hatfield, 2019;Lindner & McConnell, 2018; Daw & Hatfield, 2018 (x2); Ryan, 2018; Ryan et al., 2018)]
  
- Most work has focused on **.purple[matching outcomes]**]

---

# This paper

- Identify contexts when matching can recover causal estimates under **violations in the parallel trend assumption**.

- Use **mixed-integer programming matching (MIP)** to balance covariates directly.

- Matching for **panel** and **repeated cross-sectional** data.

--
<br/>
.pull-left[
.box-3.medium.sp-after-half[Simulations:<br/>Different DGP scenarios]
]

.pull-right[
.box-6.medium.sp-after-half[Application:<br/>School segregation & vouchers]
]

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.big[
Let's get started
]
---

# DD Setup

- Let `$Y^z_i(t)$` be the potential outcome for unit `$i$` in period `$t$` under treatment `$z$`.

- Intervention implemented in `$T_0$` `$\rightarrow$` No units are treated in `$t\leq T_0$`

- Difference-in-Differences (DD) focuses on ATT for `$t>T_0$`:
`$$ATT = E[Y_i^1(t) - Y_i^0(t)|Z=1]$$`
- **.purple[Assumptions for DD]**:
  - Parallel-trend assumption (PTA)
  
  - Common shocks
  
  `$$E[Y_i^0(1) - Y_i^0(0) | Z=1] = E[Y_i^0(1) - Y_i^0(0) | Z=0]$$`
---
# DD Setup (cont.)

- Under these assumptions:
$$
`\begin{align}
\hat{\tau}^{DD} = &\color{#900DA4}{\overbrace{\color{black}{E[Y(1)|Z=1] - E[Y(1)|Z=0]}}^{\color{#900DA4}{\Delta_{post}}}} - \\
&\color{#F89441}{\underbrace{\color{black}{(E[Y(0)|Z=1] - E[Y(0)|Z=0])}}_{\color{#F89441}{\Delta_{pre}}}}
\end{align}`
$$
  - Where `$t=0$` and `$t=1$` are the pre- and post-intervention periods, respectively.
  
  - `$Y(t) = Y^1(t)\cdot Z + (1-Z)\cdot Y^0(t)$` is the observed outcome.

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# Violations to the PTA

.pull-left[
- Under PTA, `$g_1(t) = g_0(t) + h(t)$`, where:

- `$g_z(t) = E[Y^0_i(t) | Z=z, T=t]$`
  - `$h(t) = \alpha$`
  
- Bias in a DD setting depends on the structure of `$h(t)$`.

- Confounding in DD affect **.purple[trends]** and not **.purple[levels]**.

- Contextual knowledge is important!
  - Do groups come from different populations? 
]

.pull-right[
![](https://media.giphy.com/media/L8yQ0RQBItqso/giphy.gif)
]
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# Two distinct problems when combining matching + DD

.pull-left[
- **Regression to the mean**:

- Both groups come from different populations
  - Particularly salient when matching on previous outcomes
  
- **Bias when matching on time-varying covariates**:
  
  - Depends on the structure of time variation
]

.pull-right[
<img src="https://raw.githubusercontent.com/maibennett/presentations/main/content/presentations/DD/bb_20201202/images/reg_to_the_mean.svg" alt="diagram" width="500"/>
]
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# How do we match?

- Match covariates or outcomes? Levels or trends?

- Use of **.purple[MIP Matching]** .small[(Zubizarreta, 2015; Bennett, Zubizarreta, & Vielma, 2020)]:
  
  - Balance covariates directly
  
  - Yield largest matched sample under balancing constraints
  
  - Use of template matching to match multiple groups
  
  - Works with large samples
  
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# Panel or repeated cross-sections?

- **Panel data:** Straightforward

- **Repeated cross-section data:** Representative template matching

.center[
<img src="https://raw.githubusercontent.com/maibennett/presentations/main/content/presentations/DD/bb_20201202/images/diagram3v2.svg" alt="diagram" width="600"/>]

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.big[
Simulations
]

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# Different scenarios

.box-1.medium.sp-after-half[S1: Time-invariant covariate effect]

.box-2.medium.sp-after-half[S2: Time-varying covariate effect]

.box-3.medium.sp-after-half[S3: Treatment-independent covariate]

.box-4.medium.sp-after-half[S4: Parallel evolution]

.box-6.medium.sp-after-half[S5: Evolution differs by group]

.box-7.medium.sp-after-half[S6: Evolution diverges in post]
  
.source[Following Zeldow & Hatfield (2019)]
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# Covariate evolution: Time-invariant

.center[
<img src="https://raw.githubusercontent.com/maibennett/presentations/main/content/presentations/DD/bb_20201202/images/cov_1.svg" alt="diagram" width="900"/>]

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# Covariate evolution: Time-varying

.center[
<img src="https://raw.githubusercontent.com/maibennett/presentations/main/content/presentations/DD/bb_20201202/images/cov_2.svg" alt="diagram" width="900"/>]

---

# Different ways to control

<div class="center"><table>
<thead>
<tr>
<th>Model</th>
<th>Pseudo <code class="remark-inline-code">R</code> code</th>
</tr>
</thead>
<tbody>
<tr>
<td>Simple</td>
<td><code class="remark-inline-code">lm(y ~ a*p + t)</code> </td>
</tr>
<tr>
<td>Covariate Adjusted (CA)</td>
<td><code class="remark-inline-code">lm(y ~ a*p + t + x)</code> </td>
</tr>
<tr>
<td>Time-Varying Adjusted (TVA)</td>
<td><code class="remark-inline-code">lm(y ~ a*p + t*x)</code> </td>
</tr>
<tr>
<td>Match on pre-treat outcomes</td>
<td><code class="remark-inline-code">lm(y ~ a*p + t, data=out.match)</code> </td>
</tr>
<tr>
<td>Match on pre-treat 1st diff</td>
<td><code class="remark-inline-code">lm(y ~ a*p + t, data=out.lag.match)</code> </td>
</tr>
<tr>
<td>Match on pre-treat cov (PS)</td>
<td><code class="remark-inline-code">lm(y ~ a*p + t, data=cov.match)</code> </td>
</tr>
<tr>
<td id="highlight">Match on pre-treat cov (MIP)</td>
<td id="highlight"><code class="remark-inline-code">Event study (data=cov.match.mip)</code></td>
</tr>
<tr>
<td id="highlight">Match on all cov (MIP)</td>
<td id="highlight"><code class="remark-inline-code">Event study (data=cov.match.mip.all)</code></td>
</tr>
</tbody>
</table>
</div>

.bottom[
.source[Following Zeldow & Hatfield (2019)]]

---
#Parameters:

.center[
Parameter                            | Value
-------------------------------------|----------------------------------------------
Number of obs (N)                               | 1,000 
`Pr(Z=1)`              | 0.5 
Time periods (T)          | 10 
Last pre-intervention period (T_0)          | 5 
Matching PS          | Nearest neighbor
MIP Matching tolerance          | .05 SD
Number of simulations | 1,000
]

- Estimate compared to sample ATT (_different for matching_)
- When matching with post-treat covariates `$\rightarrow$` compared with direct effect `$\tau$`
---
#Results: Time-constant covariates

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# Results: Time-varying covariates

<img src="mbennett_did_files/figure-html/res2-1.svg" style="display: block; margin: auto;" />
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# Other simulations

- Test **.purple[regression to the mean]** under no effect:

- Vary autocorrelation of `$X_i(t)$` (low vs. high)
  - `$X_0(t)$` and `$X_1(t)$` come from the same or different distribution.

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.big[
Application
]
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#Preferential Voucher Scheme in Chile

- Universal **flat voucher** scheme `$\stackrel{\mathbf{2008}}{\mathbf{\longrightarrow}}$` Universal + **preferential voucher** scheme

- Preferential voucher scheme:
  - Targeted to bottom 40% of vulnerable students

- Additional 50% of voucher per student

- Additional money for concentration of SEP students.

--
<br/>
.pull-left[
.box-3b.medium.sp-after-half[Students:<br/>- Verify SEP status<br/>- Attend a SEP school]
]

.pull-right[
.box-6b.medium.sp-after-half[Schools:<br/>- Opt-into the policy<br/>- No selection, no fees<br/>- Resources ~ performance]
]

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#Impact of the SEP policy

- **Positive impact on test scores** for lower-income students (Aguirre, 2019; Nielson, 2016)

- Design could have **increased** socioeconomic segregation
  - Incentives for concentration of SEP students
  
- Key decision variables: Performance, current SEP students, competition, add-on fees.

- **Diff-in-diff (w.r.t. 2007) for SEP and non-SEP schools**:
  - Only for **.purple[private-subsidized schools]**
  
  - Matching between 2005-2007 --> Effect estimated for 2008-2011
  
  - Outcome: Average students' household income
  
---
#Before Matching
  
.pull-left[
<img src="https://raw.githubusercontent.com/maibennett/presentations/main/content/presentations/DD/bb_20201202/images/dd_all.svg" alt="diagram" width="800"/>
]
.pull-right[
- No (pre) parallel trend

- Covariates evolve differently in the pre-intervention period
]

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# [Pre] parallel trends

.pull-left[
<img src="https://raw.githubusercontent.com/maibennett/presentations/main/content/presentations/DD/bb_20201202/images/pta_all.svg" alt="diagram" width="800"/>
]
.pull-right[
<img src="https://raw.githubusercontent.com/maibennett/presentations/main/content/presentations/DD/bb_20201202/images/pta_match.svg" alt="diagram" width="800"/>
]

---
#After Matching
  
.pull-left[
<img src="https://raw.githubusercontent.com/maibennett/presentations/main/content/presentations/DD/bb_20201202/images/dd_match.svg" alt="diagram" width="800"/>
]
.pull-right[
- **MIP Matching**: 
  - Mean balance (0.05 SD): Rural, enrollment, number of schools in county, charges add-on fees

- Fine balance: Test scores, monthly average voucher.

- **6% increase in the income gap** between SEP and non-SEP schools
]
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.big[
Let's wrap it up
]
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#Conclusions

.pull-left[
- **Matching can be an important tool to address violations in PTA**.

- Relevant to think whether groups come from the **.purple[same]** or **.purple[different]** populations.

- **Serial correlation** also plays an important role: Don't match on random noise.

- Adopt **flexibility** when estimating effects (event study)
  
.box-7.medium.sp-after-half[Match well and match smart!
]]
.pull-right[
![](https://media.giphy.com/media/drwxYI2fxqQGqRZ9Pe/giphy.gif)
]

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#A Difference-in-Differences Approach<br/>using Mixed-Integer Programming Matching

##Magdalena Bennett

---
# Time-invariant Covariates

.box-1a.medium.sp-after-half[S1: Time-invariant covariate effect]
.small[
`$$X_i \stackrel{ind}{\sim} N(m(z_i),v(z_i))$$`
`$$Y_i(t) \stackrel{ind}{\sim} N(1+z_i+treat_{it}+u_i+x_i+f(t),1)$$`]

--
.box-2a.medium.sp-after-half[S2: Time-varying covariate effect]
.small[
`$$X_i \stackrel{ind}{\sim} N(m(z_i),v(z_i))$$`
`$$Y_i(t) \stackrel{ind}{\sim} N(1+z_i+treat_{it}+u_i+x_i+f(t)+g(x_i,t),1)$$`]

--
.box-3a.medium.sp-after-half[S3: Treatment-independent covariate]
.small[
`$$X_i \stackrel{ind}{\sim} N(1,1)$$`
`$$Y_i(t) \stackrel{ind}{\sim} N(1+z_i+treat_{it}+u_i+x_i+f(t)+g(x_i,t),1)$$`]

---
# Time-varying Covariates

.box-4a.medium.sp-after-half[S4: Parallel evolution]
.small[
`$$X_{it} = x_{(t-1)i} + m_1(t)\cdot z$$`
`$$Y_i(t) \stackrel{ind}{\sim} N(1+z_i+treat_{it}+u_i+x_i+f(t)+g(x_i,t),1)$$`]

--
.box-6a.medium.sp-after-half[S5: Evolution differs by group]
.small[
`$$X_{it} = x_{(t-1)i} + m_2(z_i,t)\cdot z$$`
`$$Y_i(t) \stackrel{ind}{\sim} N(1+z_i+treat_{it}+u_i+x_i+f(t)+g(x_i,t),1)$$`]

--
.box-7a.medium.sp-after-half[S6: Evolution diverges in post]
.small[
`$$X_{it} = x_{(t-1)i} + m_1(t)\cdot z - m_3(z_i,t)$$`
`$$Y_i(t) \stackrel{ind}{\sim} N(1+z_i+treat_{it}+u_i+x_i+f(t)+g(x_i,t),1)$$`]