Panel Estimation

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.title[
# Panel Estimation
]
.subtitle[
## EC 421, Set 12
]
.author[
### Edward Rubin
]

---

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# Prologue

---
name: schedule

# Schedule

## Last time

Instrumental variables

## Today

**Panel estimation**

- what panel data buy us,
- fixed effects,
- the difference-in-differences estimator,
- *brief extension:* the synthetic-controls estimator.

---
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class: inverse, middle

# Panel estimation

---
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# Panel estimation

---
name: panel_intro

## Introduction

Until now, we mostly used one of two data structures:

1. .hi-orange[Cross section:] many units, one period,
--

2. .hi-purple[Time series:] one unit, many periods.

A .hi[panel] gives us both:

- .orange[units] indexed by `$\color{#FFA500}{i}$`,
- .purple[(time) periods] indexed by `$\color{#6A5ACD}{t}$`,
- outcomes and regressors like `$\color{#e64173}{y_{\color{#FFA500}{i}\color{#6A5ACD}{t}}}$` and `$\color{#e64173}{x_{\color{#FFA500}{i}\color{#6A5ACD}{t}}}$`.

---
## Example

Consider the following panel of six observations:

.pull-left[

```
#> Unit Year y_it x_it
#> <char> <num> <num> <num>
#> 1: A 1 10 2
#> 2: A 2 11 3
#> 3: A 3 13 4
#> 4: B 1 7 1
#> 5: B 2 8 1
#> 6: B 3 9 2
```
]

---
## Example

Consider the following panel of six observations:

.pull-left[

```
#> Unit Year y_it x_it
#> <char> <num> <num> <num>
*#> 1: A 1 10 2
*#> 2: A 2 11 3
*#> 3: A 3 13 4
#> 4: B 1 7 1
#> 5: B 2 8 1
#> 6: B 3 9 2
```
]
.pull-right[
We observe the .orange[same unit] repeatedly.
]

We can compare unit A to itself .hi-purple[over time], not just A to B (potentially helping with *selection bias*).

---
## Example

Consider the following panel of six observations:

.pull-left[

```
#> Unit Year y_it x_it
#> <char> <num> <num> <num>
*#> 1: A 1 10 2
#> 2: A 2 11 3
#> 3: A 3 13 4
*#> 4: B 1 7 1
#> 5: B 2 8 1
#> 6: B 3 9 2
```
]
.pull-right[
Similarly, we observe the .purple[period] across .orange[units].
]

This feature lets us control for shocks shared across .orange[units] at a given .purple[time].

---

## The benefits of panels

Panels help solve (some) econometric problems because they allow us to

- control for .b.orange[time-invariant differences] across units .grey-light.it[removes selection bias from cross-sectional differences across units]
 
--

- implicitly study .b.purple[changes] .grey-light.it[not just levels]

- .note[related:] evaluate policies using their .b.purple[timing] .grey-light.it[policy-timing is often more plausibly exogeneous than policy existence]

Panel methods are workhorses of economics and public policy.

---

## Panels are helpful—not magical

Panel data *can* be helpful for estimating causal effects,
 but they can still cause problems without careful use.

- We need (sufficient) .it.orange[within variation] in `$\color{#FFA500}{x_{it}}$`. .it[I.e.,] `$x_{it}$` must change meaningfully through time for some units.

- .purple[Time-varying] omitted variables can cause bias.

- A .orange[unit]'s disturbances `$u_{\color{#FFA500}{i}\color{#6A5ACD}{t}}$` are often correlated over .purple[time].

- Attrition and missing periods can matter—as can measurement error.

.grey-light.it[(Any tool can be misused.)]

---
name: pooled_problem
## The problem with pooled OLS

Suppose you're interested in a simple linear regression model
`$$y_{it} = \beta_0 + \color{#e64173}{\beta_1} x_{it} + u_{it}$$`
and have .b.slate[panel data] (with individual `$i$` and time `$t$`).

Let the disturbance combine (1) .orange[unit]-specific factor `$\color{#FFA500}{\alpha_i}$` and (2) error `$\color{#2b59c3}{v_{it}}$`
$$
`\begin{align}
  u_{it} = \color{#FFA500}{\alpha_i} + \color{#2b59c3}{v_{it}}
\end{align}`
$$

If `$\mathop{\text{Cov}}\left( x_{it}, \color{#FFA500}{\alpha_i} \right) \neq 0$`, then OLS may be biased. .grey-vlight.it[(Think OVB.)]

.attn[Standard OLS estimates] of `$\color{#e64173}{\beta_1}$` (*i.e.*, without FEs) will reflect both

- .it[between-unit differences] `$\color{#FFA500}{\alpha_i}$`,
- .it[actual changes] in `$x_{it}$`. .note[(We want to isolate this last part.)]
  
---

## Introducing *fixed effects*

.note[The solution:] Let each .orange[unit] have its own intercept `$\color{#FFA500}{\alpha_i}$`, and each .purple[time period] have its own intercept `$\color{#6A5ACD}{\lambda_t}$`:
`$$y_{it} = \beta_1 x_{it} + \color{#FFA500}{\alpha_i} + \color{#6A5ACD}{\lambda_t} + u_{it}$$`

These .orange[unit]- and .purple[period]-specific intercepts are called .attn[fixed effects].

```
#> Unit Year y_it x_it Unit_A Unit_B Year_1 Year_2 Year_3
#> <char> <num> <num> <num> <num> <num> <num> <num> <num>
#> 1: A 1 10 2 1 0 1 0 0
#> 2: A 2 11 3 1 0 0 1 0
#> 3: A 3 13 4 1 0 0 0 1
#> 4: B 1 7 1 0 1 1 0 0
#> 5: B 2 8 1 0 1 0 1 0
#> 6: B 3 9 2 0 1 0 0 1
```

---
## Fixed effects in action

.qa[Q] So what are these fixed effects (FE) doing?

Remember the old interpretation of coefficients on binary variables?

.qa[A] These fixed effects *absorb*

- `$\color{#FFA500}{\alpha_i}$`: all time-invariant .orange[differences across units], .grey-light.it[(including selection bias from cross-sectional differences)]

- `$\color{#6A5ACD}{\lambda_t}$`: all common shocks at a given .purple[time].

The .attn[(two-way) fixed-effects estimator] (OLS that includes unit and time FEs)
 estimates effects using .pink.it[within-unit variation]—net of common time shocks.

---

## The *within* transformation

For intuition, focus on unit-level fixed effects.
`$$y_{it} = \beta_0 + \beta_1 x_{it} + \alpha_i + u_{it}$$`

Take the average of this equation for unit `$i$` across all periods...
`$$\overline{y}_i = \beta_0 + \beta_1 \overline{x}_i + \alpha_i + \overline{u}_i$$`

Finally, subtract the unit mean from both sides of the original equation.
 (This step is what our unit-level fixed effects are doing.)
`$$y_{it} - \overline{y}_i = \beta_1 \left( x_{it} - \overline{x}_i \right) + \left( u_{it} - \overline{u}_i \right)$$`

.hi[So what?] Unit-specific FEs identify `$\beta_1$` using .attn[within-unit variation] in `$x_{it}$`.
 .grey-vlight.it[(The same logic applies to time FEs.)]

---
## Fixed-effects questions

Fixed effects .b[do not] ask questions like
 .white[FE] .note[Why is unit A different from unit B?]
 .grey-light.it.slab[In fact, they try to absorb cross-unit differences.]

Fixed effects also .b[do not] ask questions like
 .white[FE] .note[When period 1 had a shock, how did all units' outcomes change?]
 .grey-light.it.slab[They try to absorb common (shared) time shocks.]

Fixed effects .b[do] ask questions like
 .white[FE] .note[When unit A changed its value of] `$x_{it}$`, .note[how did its] `$y_{it}$` .note[change?]
 .grey-light.it.slab[This is the "within" variation.]

---
## Example: State minimum wage laws

Suppose we want to estimate the effect of min. wage laws on employment
`$$\text{Employment}_{it} = \beta_1 \text{(Min. Wage)}_{it} + \alpha_i + \lambda_t + u_{it}$$`
with a .orange[state] (`$\color{#FFA500}{i}$`) by .purple[year] (`$\color{#6A5ACD}{t}$`) panel of employment and min. wage data.

- `$\alpha_i$` absorbs time-invariant, cross-state differences;
- `$\lambda_t$` absorbs national shocks in year `$t$`.

.qa[Q] Should we worry that some states have higher min. wage than others?
--
 .qa[A] No. `$\alpha_i$` absorbs those differences. The FE estimator estimates using within-state variation in min. wage.

---
## Example: State minimum wage laws

- `$\alpha_i$` absorbs time-invariant, cross-state differences;
- `$\lambda_t$` absorbs national shocks in year `$t$`.

.qa[Q] What if some *years* have lower emp. than others (recessions)?
--
 .qa[A] Still not worried! `$\lambda_t$` absorbs *shared* shocks.

---
## Example: State minimum wage laws

- `$\alpha_i$` absorbs time-invariant, cross-state differences;
- `$\lambda_t$` absorbs national shocks in year `$t$`.

.qa[Q] So what should we worry about?
--
 .qa[A] We should worry about .attn[time-varying differences between states].

*E.g.*, if some states have faster-growing economies and also raise their minimum wages.

---

## What FE means in that example

In the minimum-wage panel, fixed effects (FEs) net out factors like

- state geography,
- long-run industry mix,
- baseline labor-market conditions,
- permanent political differences.

`$$\text{Employment}_{it} = \color{#e64173}{\beta_1} \text{(Min. Wage)}_{it} + \alpha_i + \lambda_t + u_{it}$$`

With FEs, `$\color{#e64173}{\hat{\beta}_1}$` comes from .b.pink[states that change their minimum wage], relative to their own histories and common national trends.

---
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Let's see a graphical example. .grey-vlight[(Simulated data but real FE issues.)]

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The dark line is .b.slate[pooled OLS]—a regression without fixed effects.
 
Here, states with higher min. wage laws have higher employment rates.

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However, the .b.purple[within-state relationships] tell another story.
 
The FE estimator focuses on this var.—.it[absorbing] cross-state differences.

---
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Integrating .b.purple[fixed effects], the relationship flips. States that raise their minimum wage see lower teen employment.

---
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This *flipped* relationship is an example of .attn[Simpson's paradox]—when a relationship reverses after controlling for *"group"*.

---
# Panel estimation
## `fixest` syntax

Now you know the intution, logic, and mechanics of fixed effects.

You also now understand the `fix` part of the `fixest` package.

To specify FEs: use `lm` syntax along with `|` followed by the FEs.

``` r
feols(
* teen_emp ~ min_wage | state + year, 
  cluster = ~ state,
  data = mw_df
)
```
Here we have fixed effects for .orange[state] and .purple[year].

---
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class: clear

.b.mono.pink[Output] `feols` output for the *plain* OLS model .it[vs.] the FE model.

```
*#>                         est_ols           est_fe
#> Dependent Var.:        teen_emp         teen_emp
#>                                                 
#> Constant         21.3*** (3.00)                 
#> min_wage        2.93*** (0.345) -1.44*** (0.196)
#> Fixed-Effects:  --------------- ----------------
#> state                        No              Yes
#> year                         No              Yes
#> _______________ _______________ ________________
#> S.E.: Clustered       by: state        by: state
#> Observations                320              320
#> R2                      0.48037          0.96360
#> Within R2                    --          0.13134
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---
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.b.mono.pink[Output] `feols` output for the *plain* OLS model .it[vs.] the FE model.

```
#>                         est_ols           est_fe
#> Dependent Var.:        teen_emp         teen_emp
#>                                                 
#> Constant         21.3*** (3.00)                 
*#> min_wage        2.93*** (0.345) -1.44*** (0.196)
#> Fixed-Effects:  --------------- ----------------
#> state                        No              Yes
#> year                         No              Yes
#> _______________ _______________ ________________
#> S.E.: Clustered       by: state        by: state
#> Observations                320              320
#> R2                      0.48037          0.96360
#> Within R2                    --          0.13134
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

Both `$\color{#e64173}{\hat{\beta}_1}$` are stat. significant, but they tell .pink[very different stories].

---
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.b.mono.pink[Output] `feols` output for the *plain* OLS model .it[vs.] the FE model.

```
#>                         est_ols           est_fe
#> Dependent Var.:        teen_emp         teen_emp
#>                                                 
*#> Constant         21.3*** (3.00)                 
#> min_wage        2.93*** (0.345) -1.44*** (0.196)
#> Fixed-Effects:  --------------- ----------------
#> state                        No              Yes
#> year                         No              Yes
#> _______________ _______________ ________________
#> S.E.: Clustered       by: state        by: state
#> Observations                320              320
#> R2                      0.48037          0.96360
#> Within R2                    --          0.13134
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

.qa[Q] Why is there .pink[no intercept] `$(\beta_0)$` in the FE model?
--
 
.qa[A] The .pink[fixed effects are collinear] with the intercept—they each provide a separate intercept for each unit and time period.

---
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.b.mono.pink[Output] `feols` output for the *plain* OLS model .it[vs.] the FE model.

```
#>                         est_ols           est_fe
#> Dependent Var.:        teen_emp         teen_emp
#>                                                 
#> Constant         21.3*** (3.00)                 
#> min_wage        2.93*** (0.345) -1.44*** (0.196)
#> Fixed-Effects:  --------------- ----------------
#> state                        No              Yes
#> year                         No              Yes
#> _______________ _______________ ________________
#> S.E.: Clustered       by: state        by: state
#> Observations                320              320
#> R2                      0.48037          0.96360
*#> Within R2                    --          0.13134
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

.qa[Q] What is .pink.b[*within* R.super[2]]?
--
 
.qa[A] It is the `$R^2$` of the regression .pink[after accouting for the FEs]: how much of the outcome's variation regressors explain, *after controlling for FEs*.

---
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.b.mono.pink[Output] `feols` output for the *plain* OLS model .it[vs.] the FE model.

```
#>                         est_ols           est_fe
#> Dependent Var.:        teen_emp         teen_emp
#>                                                 
#> Constant         21.3*** (3.00)                 
#> min_wage        2.93*** (0.345) -1.44*** (0.196)
#> Fixed-Effects:  --------------- ----------------
#> state                        No              Yes
#> year                         No              Yes
#> _______________ _______________ ________________
#> S.E.: Clustered       by: state        by: state
#> Observations                320              320
*#> R2                      0.48037          0.96360
#> Within R2                    --          0.13134
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

The FE model has as high `$\color{#e64173}{R^2}$` because fixed effects explain a lot of variation (in essence, means of states and years).

---
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# Panel estimation
## Summary

Fixed effects help one key problem: .hi[time-invariant omitted variables].

But FEs .b.slate[do not] solve *all* causal problems, *e.g.*,

- time-varying confounders,
- reverse causality
- measurement error,
- poor measurement of treatment timing,
- weak within variation.
  
As with *all* econometric tools, careful use is required.

---
layout: true
# Difference-in-differences

---
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---
name: did_intuition

## Core idea

.attn[Difference-in-differences] (DiD).super.pink[†] is a popular panel-data approach for estimating causal effects of policies/events.

.footnote[.pink[†] Also called .it[double-differences] or .it[diff-in-diff]. Increasingly combined with .it[two-way fixed effects (TWFE)].]

DiD compares observations across two dimensions:

1. .b.purple[Group:] comparing
  - .purple.it[treated] individuals (potentially affected by the policy)
  - .purple.it[control] individuals (unaffected by the policy)
2. .b.orange[Time:] comparing
  - .orange.it[before] the policy takes effect .grey-vlight.it[(pre-treatment)]
  - .orange.it[after] the policy takes effect .grey-vlight.it[(post-treatment)]

---
## Intuition

DiD functions by taking the difference between two differences:

1. Δ.b[Y̅.sub.orange.tran50[Pre]] `$=$` .b[Y̅](.purple[Trt]; .orange.tran50[Pre]) `$-$` .b[Y̅](.purple.tran50[Ctrl]; .orange.tran50[Pre]) the average difference between .purple[trt.] and .purple.tran50[control] .b.orange.tran50[before] the policy, accounting for pre-existing differences between the groups;

2. Δ.b[Y̅.sub.orange[Post]] `$=$` .b[Y̅](.purple[Trt]; .orange[Post]) `$-$` .b[Y̅](.purple.tran50[Ctrl]; .orange[Post]) the average difference between .purple[trt.] and .purple.tran50[control] .b.orange[after] the policy, which includes pre-existing differences *and* the .pink[treatment effect].

The .attn[DiD estimate] is the difference between these differences
 
   .pink[τ̂] `$=$` Δ.b[Y̅.sub.orange[Post]] `$-$` Δ.b[Y̅.sub.orange.tran50[Pre]]
 
.note[I.e.,] how much did the .purple[trt]-.purple.tran50[control] difference change from .orange.tran50[pre] to .orange[post]?

---
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class: clear

.mono.b.slate[Ex.] Two .purple[groups (trt/ctrl)] and two .orange[time periods (pre/post policy)].

```
#> Group Period Mean
#> <char> <char> <num>
#> 1: Treated Pre 1.965
#> 2: Control Pre 0.323
#> 3: Treated Post 6.172
#> 4: Control Post 1.036
```

---
layout: false
class: clear

.mono.b.slate[Ex.] Two .purple[groups (trt/ctrl)] and two .orange[time periods (pre/post policy)].

```
#> Group Period Mean
#> <char> <char> <num>
*#> 1: Treated Pre 1.965
*#> 2: Control Pre 0.323
#> 3: Treated Post 6.172
#> 4: Control Post 1.036
```

1. Δ.b[Y̅.sub.orange.tran50[Pre]] `$=$` .b[Y̅](.purple[Trt]; .orange.tran50[Pre]) `$-$` .b[Y̅](.purple.tran50[Ctrl]; .orange.tran50[Pre]) `$=$` 1.642.

---
layout: false
class: clear

.mono.b.slate[Ex.] Two .purple[groups (trt/ctrl)] and two .orange[time periods (pre/post policy)].

```
#> Group Period Mean
#> <char> <char> <num>
#> 1: Treated Pre 1.965
#> 2: Control Pre 0.323
*#> 3: Treated Post 6.172
*#> 4: Control Post 1.036
```

1. Δ.b[Y̅.sub.orange.tran50[Pre]] `$=$` .b[Y̅](.purple[Trt]; .orange.tran50[Pre]) `$-$` .b[Y̅](.purple.tran50[Ctrl]; .orange.tran50[Pre]) `$=$` 1.642.

2. Δ.b[Y̅.sub.orange[Post]] `$=$` .b[Y̅](.purple[Trt]; .orange[Post]) `$-$` .b[Y̅](.purple.tran50[Ctrl]; .orange[Post]) `$=$` 5.136

---
layout: false
class: clear

.mono.b.slate[Ex.] Two .purple[groups (trt/ctrl)] and two .orange[time periods (pre/post policy)].

```
#> Group Period Mean
#> <char> <char> <num>
#> 1: Treated Pre 1.965
#> 2: Control Pre 0.323
#> 3: Treated Post 6.172
#> 4: Control Post 1.036
```

1. Δ.b[Y̅.sub.orange.tran50[Pre]] `$=$` .b[Y̅](.purple[Trt]; .orange.tran50[Pre]) `$-$` .b[Y̅](.purple.tran50[Ctrl]; .orange.tran50[Pre]) `$=$` 1.642.

2. Δ.b[Y̅.sub.orange[Post]] `$=$` .b[Y̅](.purple[Trt]; .orange[Post]) `$-$` .b[Y̅](.purple.tran50[Ctrl]; .orange[Post]) `$=$` 5.136

The .attn[DiD estimate]—the difference between these differences
 
   .pink[τ̂] `$=$` Δ.b[Y̅.sub.orange[Post]] `$-$` Δ.b[Y̅.sub.orange.tran50[Pre]] `$=$` 3.494

---
layout: false
class: clear

.note[Note] You can also write the DiD estimator as two other differences,

.pink[τ̂] `$=$` Δ.b[Y̅.sub.orange[Post]] `$-$` Δ.b[Y̅.sub.orange.tran50[Pre]]

.white[τ̂] `$=$` `$\big[$` .b[Y̅](.purple[Trt]; .orange[Post]) `$-$` .b[Y̅](.purple.tran50[Ctrl]; .orange[Post]) `$\big]$`
`$-$`
`$\big[$` .b[Y̅](.purple[Trt]; .orange.tran50[Pre]) `$-$` .b[Y̅](.purple.tran50[Ctrl]; .orange.tran50[Pre]) `$\big]$`

.white[τ̂] `$=$`.b[Y̅](.purple[Trt]; .orange[Post]) `$-$` .b[Y̅](.purple.tran50[Ctrl]; .orange[Post]) `$-$` .b[Y̅](.purple[Trt]; .orange.tran50[Pre]) `$+$` .b[Y̅](.purple.tran50[Ctrl]; .orange.tran50[Pre])

.white[τ̂] `$=$` `$\big[$` .b[Y̅](.purple[Trt]; .orange[Post]) `$-$` .b[Y̅](.purple[Trt]; .orange.tran50[Pre]) `$\big]$` `$-$` `$\big[$` .b[Y̅](.purple.tran50[Ctrl]; .orange[Post]) `$-$` .b[Y̅](.purple.tran50[Ctrl]; .orange.tran50[Pre]) `$\big]$`

.white[τ̂] `$=$` Δ.b[Y̅].sub.purple[Trt] `$-$` Δ.b[Y̅].sub.purple.tran50[Ctrl]

.note[I.e.,] the change in the .purple[treated group] (.orange[post] `$-$` .orange.tran50[pre]) relative to the change in the .purple.tran50[control group].

We're using the .purple.tran50[control group's .b[change]] as a .b[counterfactual] for the .purple[treated group's .b[change]] had it not received treatment.

---
layout: true
# Difference-in-differences

---
## Classic example: Card and Krueger (1994)

One famous DiD [application](https://davidcard.berkeley.edu/papers/njmin-aer.pdf) estimates the .b.slate[impact of increasing the minimum wage on fast-food employment].

- .note[Who:] .purple[New Jersey (trt)] *vs.* .purple.tran50[Pennsylvania (ctrl)]
- .note[When:] .orange.tran50[before 01 April 1992 (pre)] *vs.* .orange[after 01 April 1992 (post)]

DiD estimates .grey-vlight.it[(both questions = same answer)]

1. How did NJ-PA empl. diff. change after the policy, relative to before?
2. How did NJ's empl. change after the policy relative to PA's change?

.note[Key assumption:] PA needs to be a good counterfactual for NJ. In the absence of the policy, NJ and PA would have followed the same empl. trends.

---

## DiD as a regression

With two groups and two periods
`$$y_{it} = \beta_0 + \beta_1 \color{#6A5ACD}{\text{Trt}_i} + \beta_2 \color{#FFA500}{\text{Post}_t} + \delta \left( \color{#6A5ACD}{\text{Trt}_i} \times \color{#FFA500}{\text{Post}_t} \right) + u_{it}$$`
where `$\color{#6A5ACD}{\text{Trt}_i}$` and `$\color{#FFA500}{\text{Post}_t}$` are binary indicators for the .purple[treated group] and .orange[post-treatment period].

The interaction asks if if the .orange[post] change in the .purple[treated group] differs from the .orange[post] change in the .purple.tran50[control group]: the .attn[treatment effect].

With panel data, we often write

`$$y_{it} = \alpha_i + \lambda_t + \delta D_{it} + u_{it}$$`

where `$D_{it}=1$` only for treated units after treatment starts.

---
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.b.slate[DiD in pictures] Starting with time series for each unit.
 
.grey-vlight.it[(Treatment starts between periods 5 and 6.)]

---
class: clear

Here we have .b.purple[group]-by-.b.orange[time] means.

---
class: clear

The .attn[DiD estimate] compares the .b.orange.tran50[pre-trt] difference to the .b.orange[post-trt] diff.

---
class: clear

The difference in the .b.orange.tran50[pre-trt] and .b.orange[post-trt] diffs is the .attn[DiD estimate].

---
class: clear

.b[DiD in `fixest`]

``` r
did_mod =
  feols(
    y ~ i(post, treat, ref = 0) | id + period,
    cluster = 'id',
    data = base_did
  )
```

```
#> OLS estimation, Dep. Var.: y
#> Observations: 1,080
#> Fixed-effects: id: 108,  period: 10
#> Standard-errors: Clustered (id) 
#>               Estimate Std. Error t value   Pr(>|t|)    
*#> post::1:treat  3.49339   0.521434 6.69959 1.0069e-09 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> RMSE: 4.74465     Adj. R2: 0.202735
#>                 Within R2: 0.032761
```

The interaction btn .orange[post] and .purple[treat] is the .attn[DiD estimate] of the trt effect.

---
layout: true
# Difference-in-differences

---
## The key DiD assumption

DiD relies upon a critical assumption often called .attn[parallel trends]:

> In the absence of treatment, treated and control units would have followed the .pink[same average trend].

Again, we're assuming that the trend of the control units provide a .b.slate[good counterfactual] for the treated group.

---
## What threatens DiD?

So what violates this .note[parallel trends] assumption?

- treated units were already on a different trend,
- anticipation changes behavior before treatment,
- other shocks hit treated units at the same time as treatment.

What else can *mess up* DiD estimates?

- standard errors ignore within-unit dependence,
- measurement error,
- spillovers from trt to control,
- dynamic treatment effects,
- het trt effects *and* het trt timing...

---

## DiD's place in the toolbox

DiD is especially useful when

- treatment .orange[timing] is clear,
- .purple[treatment definition] is clear,
- you have good .orange[pre-treatment] data
- .purple.tran50[ctrl group] provide a plausible counterfactual for .purple[trt group],
- the .pink[treatment effect] is tied to a discrete policy or event.
  
--

.qa[Q] What are the other options?
--
 
.qa[A] Synthetic control and friends—build a weighted average of control units to better match the treated unit's pre-treatment path.

---
## Takeaways

Key concepts from panel estimation, FEs, and DiD:

- .b.slate[Panel data] let us compare a unit to itself .b.purple[over time].

- .b.slate[Fixed effects] remove .b.orange[time-invariant omitted variables].

- `$\implies$` .b.slate[Identifying variation] is .b.purple[within variation].

- .b.slate[DiD] focuses on how trt-ctrl .b.slate[differences] change over time.

Panels can certainly help, but they still depend on strong assumptions.