4.1 — Panel Data and Fixed Effects

# 4.1 — Panel Data and Fixed Effects
## ECON 480 • Econometrics • Fall 2021
### Ryan Safner<br> Assistant Professor of Economics <br> <a href="mailto:safner@hood.edu"><i class="fa fa-paper-plane fa-fw"></i>safner@hood.edu</a> <br> <a href="https://github.com/ryansafner/metricsF21"><i class="fa fa-github fa-fw"></i>ryansafner/metricsF21</a><br> <a href="https://metricsF21.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>metricsF21.classes.ryansafner.com</a><br>

---

# Outline

### [Pooled Regression Model](#3)
### [Fixed Effects Model](#)
### [Least Squares Dummy Variable Approach](#)
### [De-Meaned Approach](#)
### [Two-Way Fixed Effects](#)
---

# Pooled Regression Model

---

# Types of Data I

.smallest[
- .red[**Cross-sectional data**]: compare different individual `$i$`’s at same time `$\bar{t}$`

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.smallest[
- .blue[**Time-series data**]: track same individual `$\bar{i}$` over different times `$t$`

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{"columns":[{"label":["state"],"name":[1],"type":["fct"],"align":["left"]},{"label":["year"],"name":[2],"type":["fct"],"align":["left"]},{"label":["deaths"],"name":[3],"type":["dbl"],"align":["right"]},{"label":["cell_plans"],"name":[4],"type":["dbl"],"align":["right"]}],"data":[{"1":"Maryland","2":"2007","3":"10.866679","4":"8942.137"},{"1":"Maryland","2":"2008","3":"10.740963","4":"9290.689"},{"1":"Maryland","2":"2009","3":"9.892754","4":"9339.452"},{"1":"Maryland","2":"2010","3":"8.783883","4":"9630.120"},{"1":"Maryland","2":"2011","3":"8.626745","4":"10335.795"},{"1":"Maryland","2":"2012","3":"8.941916","4":"10393.295"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
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---

# Types of Data I

.smallest[
- .red[**Cross-sectional data**]: compare different individual `$i$`’s at same time `$\bar{t}$`
]

<img src="4.1-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" />
]

.smallest[
- .blue[**Time-series data**]: track same individual `$\bar{i}$` over different times `$t$`
]

<img src="4.1-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" />
]

- .purple[**Panel data**]: combines these dimensions: compare all individual `$i$`’s over all time `$t$`’s

---

# Panel Data I

---

# Panel Data II

.pull-left[
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Island","2":"2007","3":"7.989810","4":"8016.396"},{"1":"Rhode Island","2":"2008","3":"7.939416","4":"8450.801"},{"1":"Rhode Island","2":"2009","3":"10.060606","4":"8478.849"},{"1":"Rhode Island","2":"2010","3":"7.970802","4":"8740.536"},{"1":"Rhode Island","2":"2011","3":"8.352921","4":"9102.998"},{"1":"Rhode Island","2":"2012","3":"8.197905","4":"9825.839"},{"1":"South Carolina","2":"2007","3":"20.857384","4":"7940.633"},{"1":"South Carolina","2":"2008","3":"18.549509","4":"7417.742"},{"1":"South Carolina","2":"2009","3":"18.196621","4":"8541.533"},{"1":"South Carolina","2":"2010","3":"16.488773","4":"8507.438"},{"1":"South Carolina","2":"2011","3":"16.991524","4":"8082.526"},{"1":"South Carolina","2":"2012","3":"17.599247","4":"9155.914"},{"1":"South Dakota","2":"2007","3":"16.213215","4":"7485.425"},{"1":"South Dakota","2":"2008","3":"13.242822","4":"7846.365"},{"1":"South Dakota","2":"2009","3":"14.822422","4":"8382.745"},{"1":"South Dakota","2":"2010","3":"15.790696","4":"8941.512"},{"1":"South Dakota","2":"2011","3":"12.330527","4":"8785.533"},{"1":"South Dakota","2":"2012","3":"14.593781","4":"8999.776"},{"1":"Tennessee","2":"2007","3":"16.999395","4":"8520.772"},{"1":"Tennessee","2":"2008","3":"14.898732","4":"8878.680"},{"1":"Tennessee","2":"2009","3":"14.083103","4":"9392.887"},{"1":"Tennessee","2":"2010","3":"14.636814","4":"9758.742"},{"1":"Tennessee","2":"2011","3":"13.370816","4":"9955.722"},{"1":"Tennessee","2":"2012","3":"14.248202","4":"10042.992"},{"1":"Texas","2":"2007","3":"13.814322","4":"8231.546"},{"1":"Texas","2":"2008","3":"14.368134","4":"8635.682"},{"1":"Texas","2":"2009","3":"13.353091","4":"8816.372"},{"1":"Texas","2":"2010","3":"12.811095","4":"9158.674"},{"1":"Texas","2":"2011","3":"12.702162","4":"9250.748"},{"1":"Texas","2":"2012","3":"14.287167","4":"9422.007"},{"1":"Utah","2":"2007","3":"11.143411","4":"7450.866"},{"1":"Utah","2":"2008","3":"10.587511","4":"7655.977"},{"1":"Utah","2":"2009","3":"9.290283","4":"7778.574"},{"1":"Utah","2":"2010","3":"8.877112","4":"8144.333"},{"1":"Utah","2":"2011","3":"9.152750","4":"8263.460"},{"1":"Utah","2":"2012","3":"8.180087","4":"8436.980"},{"1":"Vermont","2":"2007","3":"8.578113","4":"6470.783"},{"1":"Vermont","2":"2008","3":"9.983588","4":"7001.787"},{"1":"Vermont","2":"2009","3":"9.678263","4":"7446.603"},{"1":"Vermont","2":"2010","3":"9.796406","4":"7750.811"},{"1":"Vermont","2":"2011","3":"7.701961","4":"8093.469"},{"1":"Vermont","2":"2012","3":"10.671238","4":"8274.615"},{"1":"Virginia","2":"2007","3":"12.512641","4":"8319.404"},{"1":"Virginia","2":"2008","3":"10.014828","4":"8824.716"},{"1":"Virginia","2":"2009","3":"9.354109","4":"9197.484"},{"1":"Virginia","2":"2010","3":"9.005652","4":"9492.535"},{"1":"Virginia","2":"2011","3":"9.435128","4":"9605.262"},{"1":"Virginia","2":"2012","3":"9.597490","4":"9650.780"},{"1":"Washington","2":"2007","3":"9.975588","4":"8181.282"},{"1":"Washington","2":"2008","3":"9.377587","4":"8587.277"},{"1":"Washington","2":"2009","3":"8.720776","4":"8727.235"},{"1":"Washington","2":"2010","3":"8.008331","4":"8955.260"},{"1":"Washington","2":"2011","3":"8.023939","4":"9150.754"},{"1":"Washington","2":"2012","3":"7.822134","4":"9314.179"},{"1":"West Virginia","2":"2007","3":"20.958958","4":"6473.385"},{"1":"West Virginia","2":"2008","3":"18.292095","4":"7137.078"},{"1":"West Virginia","2":"2009","3":"18.157707","4":"7698.746"},{"1":"West Virginia","2":"2010","3":"16.403658","4":"8095.007"},{"1":"West Virginia","2":"2011","3":"17.771332","4":"8893.134"},{"1":"West Virginia","2":"2012","3":"17.632570","4":"8612.638"},{"1":"Wisconsin","2":"2007","3":"12.707377","4":"6858.706"},{"1":"Wisconsin","2":"2008","3":"10.528697","4":"7578.225"},{"1":"Wisconsin","2":"2009","3":"9.646302","4":"8039.225"},{"1":"Wisconsin","2":"2010","3":"9.626346","4":"8317.235"},{"1":"Wisconsin","2":"2011","3":"9.939593","4":"8633.056"},{"1":"Wisconsin","2":"2012","3":"10.408328","4":"8796.105"},{"1":"Wyoming","2":"2007","3":"16.015375","4":"8434.864"},{"1":"Wyoming","2":"2008","3":"16.830740","4":"9086.335"},{"1":"Wyoming","2":"2009","3":"14.005016","4":"9498.962"},{"1":"Wyoming","2":"2010","3":"16.554895","4":"9332.430"},{"1":"Wyoming","2":"2011","3":"14.602637","4":"9363.593"},{"1":"Wyoming","2":"2012","3":"13.267186","4":"9455.043"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

- .purple[**Panel**] or .purple[**Longitudinal**] data contains 
  - .blue[repeated observations] `$(t)$`
  - on .red[multiple individuals] `$(i)$`
]

---

# Panel Data II

- .purple[**Panel**] or .purple[**Longitudinal**] data contains 
  - .blue[repeated observations] `$(t)$`
  - on .red[multiple individuals] `$(i)$`

- Thus, our regression equation looks like:

$$
`\begin{align}
  \hat{Y}_{\color{red}{i}\color{blue}{t}}} = \beta_0 + \beta_1 X_{\color{red}{i}\color{blue}{t}} + u_{\color{red}{i}\color{blue}{t}}
\end{align}`
$$

> for .red[individual `\$i\$`] in .blue[time `\$t\$`].

]

---

# Panel Data: Our Motivating Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Do cell phones cause more traffic fatalities?
]
- No measure of cell phones *used* while driving
  - `cell_plans` as a **proxy** for cell phone usage

- State-level data over 6 years

]

---

# The Data I

```r
glimpse(phones)
```

```
## Rows: 306
## Columns: 8
## $ year          <fct> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 20…
## $ state         <fct> Alabama, Alaska, Arizona, Arkansas, California, Colorado…
## $ urban_percent <dbl> 30, 55, 45, 21, 54, 34, 84, 31, 100, 53, 39, 45, 11, 56,…
## $ cell_plans    <dbl> 8135.525, 6730.282, 7572.465, 8071.125, 8821.933, 8162.0…
## $ cell_ban      <fct> 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ text_ban      <fct> 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ deaths        <dbl> 18.075232, 16.301184, 16.930578, 19.595430, 12.104340, 1…
## $ year_num      <dbl> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 20…
```

---

# The Data II

```r
phones %>%
  count(state)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["state"],"name":[1],"type":["fct"],"align":["left"]},{"label":["n"],"name":[2],"type":["int"],"align":["right"]}],"data":[{"1":"Alabama","2":"6"},{"1":"Alaska","2":"6"},{"1":"Arizona","2":"6"},{"1":"Arkansas","2":"6"},{"1":"California","2":"6"},{"1":"Colorado","2":"6"},{"1":"Connecticut","2":"6"},{"1":"Delaware","2":"6"},{"1":"District of Columbia","2":"6"},{"1":"Florida","2":"6"},{"1":"Georgia","2":"6"},{"1":"Hawaii","2":"6"},{"1":"Idaho","2":"6"},{"1":"Illinois","2":"6"},{"1":"Indiana","2":"6"},{"1":"Iowa","2":"6"},{"1":"Kansas","2":"6"},{"1":"Kentucky","2":"6"},{"1":"Louisiana","2":"6"},{"1":"Maine","2":"6"},{"1":"Maryland","2":"6"},{"1":"Massachusetts","2":"6"},{"1":"Michigan","2":"6"},{"1":"Minnesota","2":"6"},{"1":"Mississippi","2":"6"},{"1":"Missouri","2":"6"},{"1":"Montana","2":"6"},{"1":"Nebraska","2":"6"},{"1":"Nevada","2":"6"},{"1":"New Hampshire","2":"6"},{"1":"New Jersey","2":"6"},{"1":"New Mexico","2":"6"},{"1":"New York","2":"6"},{"1":"North Carolina","2":"6"},{"1":"North Dakota","2":"6"},{"1":"Ohio","2":"6"},{"1":"Oklahoma","2":"6"},{"1":"Oregon","2":"6"},{"1":"Pennsylvania","2":"6"},{"1":"Rhode Island","2":"6"},{"1":"South Carolina","2":"6"},{"1":"South Dakota","2":"6"},{"1":"Tennessee","2":"6"},{"1":"Texas","2":"6"},{"1":"Utah","2":"6"},{"1":"Vermont","2":"6"},{"1":"Virginia","2":"6"},{"1":"Washington","2":"6"},{"1":"West Virginia","2":"6"},{"1":"Wisconsin","2":"6"},{"1":"Wyoming","2":"6"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

```r
phones %>%
  count(year)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["year"],"name":[1],"type":["fct"],"align":["left"]},{"label":["n"],"name":[2],"type":["int"],"align":["right"]}],"data":[{"1":"2007","2":"51"},{"1":"2008","2":"51"},{"1":"2009","2":"51"},{"1":"2010","2":"51"},{"1":"2011","2":"51"},{"1":"2012","2":"51"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

---

# The Data III

```r
phones %>%
  distinct(state)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["state"],"name":[1],"type":["fct"],"align":["left"]}],"data":[{"1":"Alabama"},{"1":"Alaska"},{"1":"Arizona"},{"1":"Arkansas"},{"1":"California"},{"1":"Colorado"},{"1":"Connecticut"},{"1":"Delaware"},{"1":"District of Columbia"},{"1":"Florida"},{"1":"Georgia"},{"1":"Hawaii"},{"1":"Idaho"},{"1":"Illinois"},{"1":"Indiana"},{"1":"Iowa"},{"1":"Kansas"},{"1":"Kentucky"},{"1":"Louisiana"},{"1":"Maine"},{"1":"Maryland"},{"1":"Massachusetts"},{"1":"Michigan"},{"1":"Minnesota"},{"1":"Mississippi"},{"1":"Missouri"},{"1":"Montana"},{"1":"Nebraska"},{"1":"Nevada"},{"1":"New Hampshire"},{"1":"New Jersey"},{"1":"New Mexico"},{"1":"New York"},{"1":"North Carolina"},{"1":"North Dakota"},{"1":"Ohio"},{"1":"Oklahoma"},{"1":"Oregon"},{"1":"Pennsylvania"},{"1":"Rhode Island"},{"1":"South Carolina"},{"1":"South Dakota"},{"1":"Tennessee"},{"1":"Texas"},{"1":"Utah"},{"1":"Vermont"},{"1":"Virginia"},{"1":"Washington"},{"1":"West Virginia"},{"1":"Wisconsin"},{"1":"Wyoming"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

```r
phones %>%
  distinct(year)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["year"],"name":[1],"type":["fct"],"align":["left"]}],"data":[{"1":"2007"},{"1":"2008"},{"1":"2009"},{"1":"2010"},{"1":"2011"},{"1":"2012"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

---

# The Data IV

```r
phones %>%
  summarize(States = n_distinct(state),
            Years = n_distinct(year))
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["States"],"name":[1],"type":["int"],"align":["right"]},{"label":["Years"],"name":[2],"type":["int"],"align":["right"]}],"data":[{"1":"51","2":"6"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]

---

# Pooled Regression I

- What if we just ran a standard regression:

`$$\hat{Y_{it}}=\beta_0+\beta_1X_{it}+u_{it}$$`

--
  - `$N$` number of `$i$` groups (e.g. U.S. States)
  - `$T$` number of `$t$` periods (e.g. years)

- This is a .hi[pooled regression model]: treats all observations as independent

---

# Pooled Regression II

```r
pooled <- lm(deaths ~ cell_plans, data = phones)
pooled %>% tidy()
```

---

# Pooled Regression III

```r
ggplot(data = phones)+
  aes(x = cell_plans,
      y = deaths)+
  geom_point()+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven")+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=14)
```

]
]

---

# Pooled Regression III

```r
ggplot(data = phones)+
  aes(x = cell_plans,
      y = deaths)+
  geom_point()+
* geom_smooth(method = "lm", color = "red")+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven")+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=14)
```
]
]

---

# Recap: Assumptions about Errors

1. The expected value of the residuals is 0
`$$E[u]=0$$`

2. The variance of the residuals over `$X$` is constant:
`$$var(u|X)=\sigma^2_{u}$$`

3. **Errors are not correlated across observations:**
`$$cor(u_i,u_j)=0 \quad \forall i \neq j$$`

4. There is no correlation between `$X$` and the error term: 
`$$cor(X, u)=0 \text{ or } E[u|X]=0$$`

]
]

---

# Biases of Pooled Regression

`$$\hat{Y_{it}}=\beta_0+\beta_1X_{it}+\epsilon_{it}$$`

- .hi-purple[Assumption 3]: `$cor(u_i,u_j)=0 \quad \forall \, i \neq j$`

- Pooled regression model is **biased** because it ignores:
  - Multiple observations from same group `$i$`
  - Multiple observations from same time `$t$`

- Thus, errors are .hi[serially] or .hi[auto-correlated]; `$cor(u_i, u_j) \neq 0$` within same `$i$` and within same `$t$`

---

# Biases of Pooled Regression: Our Example

`$$\widehat{\text{Deaths}_{it}}=\beta_0+\beta_1 \, \text{Cell Phones}_{it}+u_{it}$$`

- .hi-turquoise[Multiple observations from same state `\$i\$`]
  - Probably similarities among `$u$` for obs in same state
  - Residuals on observations from same state are likely correlated 
  
- .hi-turquoise[Multiple observations from same year `\$t\$`]
  - Probably similarities among `$u$` for obs in same year
  - Residuals on observations from same year are likely correlated
  
---

# Example: Consider Just 5 States

```r
phones %>%
* filter(state %in% c("District of Columbia",
*                     "Maryland", "Texas",
*                     "California", "Kansas")) %>%
ggplot(data = .)+
  aes(x = cell_plans,
      y = deaths,
      color = state)+
  geom_point()+ 
  geom_smooth(method = "lm")+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=14)+
  theme(legend.position = "top")
```
]
]

---

# Example: Consider Just 5 States

```r
phones %>%
  filter(state %in% c("District of Columbia",
                      "Maryland", "Texas",
                      "California", "Kansas")) %>%
ggplot(data = .)+
  aes(x = cell_plans,
      y = deaths,
      color = state)+ 
  geom_point()+ 
  geom_smooth(method = "lm")+ 
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=14)+
* theme(legend.position = "none")+
* facet_wrap(~state, ncol=3)
```
]
]

---

# Look at All States

```r
*ggplot(data = phones)+
  aes(x = cell_plans,
      y = deaths,
      color = state)+ 
  geom_point()+ 
  geom_smooth(method = "lm")+ 
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed")+
  theme(legend.position = "none")+
* facet_wrap(~state, ncol=7)
```
]
]

---

# The Bias in our Pooled Regression

.smallest[
`$$\widehat{\text{Deaths}_{it}}=\beta_0+\beta_1 \, \text{Cell Phones}_{it}+\text{u}_{it}$$`

- `$\text{Cell Phones}_{it}$` is .hi-purple[endogenous]:
]

`$$cor(\text{u}_{it}, \text{Cell Phones}_{it}) \neq 0 \quad \quad E[\text{u}_{it}|\text{Cell Phones}_{it}] \neq 0$$`
]

.smallest[
- Things in `$u_{it}$` correlated with `$\text{Cell phones}_{it}$`:
  - infrastructure spending, population, urban vs. rural, more/less cautious citizens, cultural attitudes towards driving, texting, etc
]

.smallest[
- A lot of these things vary systematically **by State**!
  - `$cor(\text{u}_{it_1}, \text{u}_{it_2})\neq 0$`
      - Error in State `$i$` during `$t_1$` correlates with error in State `$i$` during `$t_2$`
      - things in State `$i$` that don’t change over time
]

---

# Fixed Effects Model

---

# Fixed Effects: DAG

- Many (often unobservable) factors that determine both Phones & Deaths
  - Culture, infrastructure, population, geography, institutions, etc
]
]

---

# Fixed Effects: DAG

- Many (often unobservable) factors that determine both Phones & Deaths
  - Culture, infrastructure, population, geography, institutions, etc

- But the beauty of this is that .hi-turquoise[most of these factors systematically vary by U.S. State and are stable over time!]

- We can simply .hi-purple[“control for State”] to safely remove the influence of all of these factors!
]
]

]

---

# Fixed Effects: Decomposing `$\text{u}_{it}$`

- Much of the endogeneity in `$X_{it}$` can be explained by systematic differences across `$i$` (groups)

- Exploit the systematic variation across groups with a .hi[fixed effects model]

- *Decompose* the model error term into two parts:

`$$\text{u}_{it} = \alpha_i + \epsilon_{it}$$`

---

# Fixed Effects: `$\alpha_i$`

- *Decompose* the model error term into two parts:

`$$\text{u}_{it} = \color{#6A5ACD}{\alpha_i} + \epsilon_{it}$$`

- `$\color{#6A5ACD}{\alpha_i}$` are .hi-purple[group-specific fixed effects]
  - group `$i$` tends to have higher or lower `$\hat{Y}$` than other groups given regressor(s) `$X_{it}$`
  - estimate a separate `$\alpha_i$` for each group `$i$`
  - essentially, estimate a separate constant (intercept) _for each group_
  - notice this is stable over time within each group (subscript only `$i$`, no `$t)$`

- .hi-purple[This includes **all** factors that do not change _within_ group *i* over time]

---

# Fixed Effects: `$\epsilon_{it}$`

- *Decompose* the model error term into two parts:

`$$\text{u}_{it} = \color{#6A5ACD}{\alpha_i} + \color{#D7250E}{\epsilon_{it}}$$`

- `$\color{#D7250E}{\epsilon_{it}}$` is the .hi-red[remaining random error]
  - As usual in OLS, assume the 4 typical assumptions about this error:
    - `$E[\epsilon_{it}]=0$`, `$var[\epsilon_{it}]=\sigma^2_{\epsilon}$`, `$cor(\epsilon_{it}, \epsilon_{jt})=0$`, `$cor(\epsilon_{it}, X_{it})=0$`

- `$\color{#D7250E}{\epsilon_{it}}$` includes all other factors affecting `$Y_{it}$` *not* contained in group effect `$\alpha_i$`
  - i.e. differences *within* each group that *change* over time
  - Be careful: `$X_{it}$` **can still be endogenous due to other factors!**
      - `$cor(X_{it}, \epsilon_{it}) \neq 0$`

---

# Fixed Effects: New Regression Equation

`$$\widehat{Y}_{it} = \beta_0+\beta_1 X_{it} +\color{#6A5ACD}{\alpha_i} + \color{#D7250E}{\epsilon_{it}}$$`

- We've pulled `$\color{#6A5ACD}{\alpha_i}$` out of the original error term into the regression

- Essentially we’ll estimate an intercept **for each .purple[group]** (minus one, which is `$\beta_0)$`
  - avoiding the dummy variable trap

- Must have multiple observations (over time) for each group (i.e. panel data)

---

# Fixed Effects: Our Example

`$$\widehat{\text{Deaths}}_{it} = \beta_0+\beta_1 \text{Cell phones}_{it} +\color{#6A5ACD}{\alpha_i}+ \color{#D7250E}{\epsilon_{it}}$$`

- `$\color{#6A5ACD}{\alpha_i}$` is the .hi-purple[State fixed effect]
  - Captures everything unique about each state `$i$` that *does not change over time*
    - culture, institutions, history, geography, climate, etc!

- There could *still* be factors in `$\color{#D7250E}{\epsilon_{it}}$` that are correlated with `$\text{Cell phones}_{it}$`!
  - things that do change over time within States
  - perhaps individual States have cell phone bans for *some* years in our data

---

# Estimating Fixed Effects Models

`$$\widehat{Y}_{it} = \beta_0+\beta_1 X_{it} +\alpha_i+\epsilon_{it}$$`

- Two methods to estimate fixed effects models:

1. Least Squares Dummy Variable (LSDV) approach

2. De-meaned data approach

---

# Least Squares Dummy Variable Approach

---

# Least Squares Dummy Variable Approach

.smallest[
`$$\widehat{Y_{it}}=\beta_0+\beta_1X_{it}+\beta_2 D_{1i}+ \beta_3 D_{2i} + \cdots +\beta_N D_{(N-1)i}+\epsilon_{it}$$`

- A dummy variable `$D_{i} = \{0,1\}$` for each possible group
  - `$=1$` if observation `$it$` is from group `$i$`, otherwise `$=0$`
]

.smallest[
- If there are `$N$` groups:
  - Include `$N-1$` dummies (to avoid **dummy variable trap**) and `$\beta_0$` is the reference category<sup>.magenta[†]</sup>
  - So we are estimating a different intercept for each group
]

.footnote[<sup>.magenta[†]</sup> If we do not estimate `\$\beta_0\$`, we could include all N dummies. In either case, `\$\beta_0\$` takes the place of one category-dummy.]

---

# Least Squares Dummy Variable Approach: Our Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: 
`$$\widehat{\text{Deaths}_{it}}=\beta_0+\beta_1\text{Cell Phones}_{it}+\text{Alaska}_i+ \cdots +\text{Wyoming}_i$$`
]

- Let Alabama be the reference category `$(\beta_0)$`, include all other States

---

# Our Example in R I

`$$\widehat{\text{Deaths}_{it}}=\beta_0+\beta_1\text{Cell Phones}_{it}+\text{Alaska}_i+ \cdots +\text{Wyoming}_i$$`

- If `state` is a `factor` variable, just include it in the regression

- `R` automatically creates `$N-1$` dummy variables and includes them in the regression
  - Keeps intercept and leaves out first group dummy

---

# Our Example in R II

```r
fe_reg_1 <- lm(deaths ~ cell_plans + state, data = phones)
fe_reg_1 %>% tidy()
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
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Hampshire","2":"-6.127382094","3":"0.6660295641","4":"-9.19986503","5":"1.357559e-17"},{"1":"stateNew Jersey","2":"-5.707365784","3":"0.6688922440","4":"-8.53256386","5":"1.314338e-15"},{"1":"stateNew Mexico","2":"-1.719158568","3":"0.6716721629","4":"-2.55952035","5":"1.106160e-02"},{"1":"stateNew York","2":"-4.754827932","3":"0.6694121494","4":"-7.10299019","5":"1.225067e-11"},{"1":"stateNorth Carolina","2":"-1.529215200","3":"0.6654724423","4":"-2.29793918","5":"2.237838e-02"},{"1":"stateNorth Dakota","2":"0.595880660","3":"0.6659608543","4":"0.89476830","5":"3.717581e-01"},{"1":"stateOhio","2":"-4.683202635","3":"0.6654474519","4":"-7.03767461","5":"1.815281e-11"},{"1":"stateOklahoma","2":"-0.065432340","3":"0.6660893891","4":"-0.09823357","5":"9.218243e-01"},{"1":"stateOregon","2":"-4.271677763","3":"0.6667266398","4":"-6.40694028","5":"7.158251e-10"},{"1":"statePennsylvania","2":"-1.922414465","3":"0.6658670533","4":"-2.88708453","5":"4.223056e-03"},{"1":"stateRhode Island","2":"-6.533207516","3":"0.6655747188","4":"-9.81588893","5":"1.725455e-19"},{"1":"stateSouth Carolina","2":"2.566280693","3":"0.6685038446","4":"3.83884209","5":"1.561121e-04"},{"1":"stateSouth Dakota","2":"-0.889041844","3":"0.6673510107","4":"-1.33219525","5":"1.839901e-01"},{"1":"stateTennessee","2":"0.543722021","3":"0.6675006813","4":"0.81456399","5":"4.160850e-01"},{"1":"stateTexas","2":"-1.215305938","3":"0.6654309611","4":"-1.82634414","5":"6.897223e-02"},{"1":"stateUtah","2":"-6.393350380","3":"0.6723735941","4":"-9.50862800","5":"1.547018e-18"},{"1":"stateVermont","2":"-7.070384958","3":"0.6803787706","4":"-10.39183652","5":"2.613517e-21"},{"1":"stateVirginia","2":"-4.468646712","3":"0.6660140850","4":"-6.70953785","5":"1.263262e-10"},{"1":"stateWashington","2":"-6.236757171","3":"0.6654881222","4":"-9.37170321","5":"4.069919e-18"},{"1":"stateWest Virginia","2":"2.106291665","3":"0.6744981466","4":"3.12275382","5":"1.999170e-03"},{"1":"stateWisconsin","2":"-5.356989528","3":"0.6712322806","4":"-7.98082822","5":"5.015723e-14"},{"1":"stateWyoming","2":"0.773613122","3":"0.6660728030","4":"1.16145430","5":"2.465475e-01"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

]

---

# De-meaned Approach

---

# De-meaned Approach I

- Alternatively, we can control our regression for group fixed effects without directly estimating them

- We simply .hi-purple[de-mean the data for each group] to remove the group fixed-effect

- For each group `$i$`, find the means (over time, `$t)$`:
`$$\bar{Y}_i=\beta_0+\beta_1 \bar{X}_i+\bar{\alpha}_i+\bar{\epsilon}_{it}$$`
--

- Where:
  - `$\bar{Y}_i$`: average value of `$Y_{it}$` for group `$i$`
  - `$\bar{X}_i$`: average value of `$X_{it}$` for group `$i$`
  - `$\bar{\alpha}_i$`: average value of `$\alpha_{i}$` for group `$i$` `$(=\alpha_i)$`
  - `$\bar{\epsilon}_{it}=0$`, by assumption 1 about errors

---

# De-meaned Approach II

.smallest[
`$$\begin{align*}
\widehat{Y_{it}}&=\beta_0+\beta_1X_{it}+u_{it}\\
\bar{Y}_i&=\beta_0+\beta_1 \bar{X}_i+\bar{\alpha}_i+\bar{\epsilon}_i\\
\end{align*}$$`
]

`$$\begin{align*}
Y_{it}-\bar{Y}_i&=\beta_1(X_{it}-\bar{X}_i)+\tilde{\epsilon}_{it}\\
\tilde{Y}_{it}&=\beta_1 \tilde{X}_{it}+\tilde{\epsilon}_{it}\\
\end{align*}$$`
]

.smallest[
- Within each group `$i$`, the de-meaned variables `$\tilde{Y}_{it}$` and `$\tilde{X}_{it}$`'s all have a mean of 0<sup>.magenta[†]</sup>

- Variables that don't change over time will drop out of analysis altogether

- .hi-turquoise[Removes any source of variation .ul[across] groups (all now have mean of 0) to only work with variation .ul[within] each group]
]

.footnote[<sup>.magenta[†]</sup> Recall **Rule 4** from the [2.3 class notes](class/2.1-class/#the-summation-operator) on the Summation Operator: `\$\sum(X_i-\bar{X})=0\$`]

---

# De-meaned Approach III

`$$\tilde{Y}_{it}=\beta_1 \tilde{X}_{it}+\tilde{\epsilon}_{it}$$`

- Yields identical results to dummy variable approach

- More useful when we have many groups (would be many dummies)

- Demonstrates **intuition** behind fixed effects:
  - Converts all data to deviations from the mean of each group
    - All groups are “centered” at 0, no variation across groups
  - Fixed effects are often called the .hi-purple[“within” estimators], they exploit variation *within* groups, not *across* groups

---

# De-meaned Approach IV

- We are basically comparing groups *to themselves* over time
  - apples to apples comparison
  - e.g. Maryland in 2000 vs. Maryland in 2005

- Ignore all differences *between* groups, only look at differences *within* groups over time

---

# De-Meaning the Data in R I

```r
# get means of Y and X by state
means_state <- phones %>%
  group_by(state) %>%
  summarize(avg_deaths = mean(deaths),
            avg_phones = mean(cell_plans))

# look at it
means_state
```
]

.pull-right[
.tiny[
<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
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  </script>
</div>
]
]
---

# De-Meaning the Data in R II

```r
ggplot(data = means_state)+
  aes(x = fct_reorder(state, avg_deaths),
      y = avg_deaths,
      color = state)+
  geom_point()+
  geom_segment(aes(y = 0,
                   yend = avg_deaths,
                   x = state,
                   xend = state))+
  coord_flip()+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=10)+
  theme(legend.position = "none")
```
]
]

]

---

# Visualizing “Within Group” Estimates for the 5 States

# Visualizing “Within Group” Estimates for All 51 States

---

# De-meaned Approach in R I

- The `fixest` package is designed for running regressions with fixed effects

- `feols()` function is just like `lm()`, with some additional arguments:

```r
#install.packages("fixest")
library(fixest)
fe_reg_1_alt <- feols(deaths ~ cell_plans | state,
                      data = phones)
```

---

# De-meaned Approach in R II

```r
fe_reg_1_alt %>% summary()
```

```
## OLS estimation, Dep. Var.: deaths
## Observations: 306 
## Fixed-effects: state: 51
## Standard-errors: Clustered (state) 
##             Estimate Std. Error  t value  Pr(>|t|)    
## cell_plans -0.001204   0.000143 -8.41708 3.792e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 1.05007     Adj. R2: 0.886524
##                 Within R2: 0.357238
```

```r
# or using broom's tidy()
fe_reg_1_alt %>% tidy()
```

---

# Two-Way Fixed Effects

---

# Two-Way Fixed Effects

.pull-left[
.smallest[
- State fixed effect controls for all factors that vary by state but are stable over time

- But there are still other (often unobservable) factors that affect both Phones and Deaths, that *don’t* vary by State
  - The country’s macroeconomic performance, federal laws, etc
]
]

---

# Two-Way Fixed Effects

- But there are still other (often unobservable) factors that affect both Phones and Deaths, that *don’t* vary by State
  - The country’s macroeconomic performance, federal laws, etc

- If these factors systematically vary over time, but are the same by State, then we can .hi-purple[“control for Year”] to safely remove the influence of all of these factors!
]
]
.pull-right[
<img src="4.1-slides_files/figure-html/unnamed-chunk-33-1.png" width="504" />

]

---

# Two-Way Fixed Effects

- A .hi[one-way fixed effects model] estimates a fixed effect for **groups**

- .hi[Two-way fixed effects model] estimates fixed effects for *both* **groups** *and* **time periods**
`$$\hat{Y_{it}}=\beta_0+\beta_1 X_{it}+ \color{#6A5ACD}{\alpha_{i}} + \color{#44C1C4}{\theta_{t}} + \color{#e64173}{\nu_{it}}$$`

- `$\color{#6A5ACD}{\alpha_{i}}$`: group fixed effects
  - accounts for **time-invariant differences across groups**

- `$\color{#44C1C4}{\theta_{t}}$`: time fixed effects
  - accounts for **group-invariant differences over time**

- `$\color{#e64173}{\nu_{it}}$` remaining random error
  - all remaining factors that affect `$Y_{it}$` that vary by state *and* change over time
  
---

# Two-Way Fixed Effects: Our Example

`$$\widehat{\text{Deaths}}_{it} = \beta_0+\beta_1 \text{Cell phones}_{it} +\color{#6A5ACD}{\alpha_{i}} + \color{#44C1C4}{\theta_{t}} +  \color{#e64173}{\nu_{it}}$$`

- `$\color{#6A5ACD}{\alpha_{i}}$`: .purple[State fixed effects]
    - differences **across states** that are **stable over time** (note subscript `$i$` only)
    - e.g. geography, culture, (unchanging) state laws

- `$\color{#44C1C4}{\theta_{t}}$`: .turquoise[Year fixed effects]
    - differences **over time** that are **stable across states** (note subscript `$t$` only)
    - e.g. economy-wide macroeconomic changes, *federal* laws passed

---

# Visualizing Year Effects I

```r
# find averages for years
means_year <- phones %>%
  group_by(year) %>%
  summarize(avg_deaths = mean(deaths),
            avg_phones = mean(cell_plans))
means_year
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["year"],"name":[1],"type":["fct"],"align":["left"]},{"label":["avg_deaths"],"name":[2],"type":["dbl"],"align":["right"]},{"label":["avg_phones"],"name":[3],"type":["dbl"],"align":["right"]}],"data":[{"1":"2007","2":"14.00751","3":"8064.531"},{"1":"2008","2":"12.87156","3":"8482.903"},{"1":"2009","2":"12.08632","3":"8859.706"},{"1":"2010","2":"11.61487","3":"9134.592"},{"1":"2011","2":"11.36431","3":"9485.238"},{"1":"2012","2":"11.65666","3":"9660.474"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]

---

# Visualizing Year Effects II

```r
ggplot(data = phones)+
  aes(x = year,
      y = deaths)+
  geom_point(aes(color = year))+
  # Add the yearly means as black points
  geom_point(data = means_year,
             aes(x = year,
                 y = avg_deaths),
             size = 3,
             color = "black")+
  # connect the means with a line
  geom_line(data = means_year,
            aes(x = as.numeric(year),
                y = avg_deaths),
            color = "black",
            size = 1)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size = 14)+
  theme(legend.position = "none")
```
]
]

---

# Estimating Two-Way Fixed Effects

`$$\widehat{Y}_{it} = \beta_0+\beta_1 X_{it} +\alpha_i+\theta_t+\nu_{it}$$`

- As before, several equivalent ways to estimate two-way fixed effects models:

1) **Least Squares Dummy Variable (LSDV) Approach**: add dummies for both groups and time periods (separate intercepts for groups and times)

2) **Fully De-meaned data**: 
`$$\tilde{Y}_{it}=\beta_1\tilde{X}_{it}+\tilde{\nu}_{it}$$`

where for each variable: `$\tilde{var}_{it}=var_{it}-\overline{var}_{t}-\overline{var}_{i}$`

3) **Hybrid**: de-mean for one effect (groups or years) and add dummies for the other effect (years or groups)

---

# LSDV Method

```r
fe2_reg_1 <- lm(deaths ~ cell_plans + state + year,
                data = phones)

fe2_reg_1 %>% tidy()
```

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of Columbia","2":"-3.5044963616","3":"1.9710939218","4":"-1.7779449","5":"7.663326e-02"},{"1":"stateFlorida","2":"-1.1904908389","3":"0.5959900409","4":"-1.9975012","5":"4.685883e-02"},{"1":"stateGeorgia","2":"-2.4422504400","3":"0.5975174655","4":"-4.0873290","5":"5.890829e-05"},{"1":"stateHawaii","2":"-3.2811475461","3":"0.5978042894","4":"-5.4886651","5":"9.961835e-08"},{"1":"stateIdaho","2":"-1.6145029533","3":"0.6167129515","4":"-2.6179164","5":"9.388795e-03"},{"1":"stateIllinois","2":"-5.2488047162","3":"0.5963135278","4":"-8.8020890","5":"2.304408e-16"},{"1":"stateIndiana","2":"-4.5580058469","3":"0.6088089548","4":"-7.4867589","5":"1.219751e-12"},{"1":"stateIowa","2":"-2.5117857625","3":"0.6116310108","4":"-4.1067011","5":"5.446936e-05"},{"1":"stateKansas","2":"-1.4042963587","3":"0.5953392440","4":"-2.3588170","5":"1.910646e-02"},{"1":"stateKentucky","2":"1.5143772249","3":"0.6006791561","4":"2.5211083","5":"1.232396e-02"},{"1":"stateLouisiana","2":"3.1093174334","3":"0.6148710122","4":"5.0568613","5":"8.270383e-07"},{"1":"stateMaine","2":"-4.0022269640","3":"0.6070911650","4":"-6.5924645","5":"2.566949e-10"},{"1":"stateMaryland","2":"-4.9202430607","3":"0.6090775854","4":"-8.0781877","5":"2.839121e-14"},{"1":"stateMassachusetts","2":"-8.2276855095","3":"0.6009756368","4":"-13.6905475","5":"3.483280e-32"},{"1":"stateMichigan","2":"-5.1840811542","3":"0.5962924572","4":"-8.6938567","5":"4.798059e-16"},{"1":"stateMinnesota","2":"-7.3888657967","3":"0.5962743623","4":"-12.3917214","5":"8.630449e-28"},{"1":"stateMississippi","2":"2.1741873311","3":"0.6065902849","4":"3.5842765","5":"4.066096e-04"},{"1":"stateMissouri","2":"-2.1422872290","3":"0.5958403283","4":"-3.5954049","5":"3.904420e-04"},{"1":"stateMontana","2":"4.6383935108","3":"0.6074334287","4":"7.6360524","5":"4.795866e-13"},{"1":"stateNebraska","2":"-3.9461735450","3":"0.5995547993","4":"-6.5818396","5":"2.728158e-10"},{"1":"stateNevada","2":"-1.9595530016","3":"0.5956187600","4":"-3.2899451","5":"1.146946e-03"},{"1":"stateNew Hampshire","2":"-5.8751508656","3":"0.5971883463","4":"-9.8380200","5":"1.662472e-19"},{"1":"stateNew Jersey","2":"-6.3140121777","3":"0.6063733658","4":"-10.4127466","5":"2.592960e-21"},{"1":"stateNew Mexico","2":"-0.9037640047","3":"0.6151984859","4":"-1.4690608","5":"1.430785e-01"},{"1":"stateNew York","2":"-5.4055532053","3":"0.6080307982","4":"-8.8902622","5":"1.263646e-16"},{"1":"stateNorth Carolina","2":"-1.4618984034","3":"0.5953889564","4":"-2.4553670","5":"1.475848e-02"},{"1":"stateNorth Dakota","2":"0.3585402358","3":"0.5969666397","4":"0.6006035","5":"5.486504e-01"},{"1":"stateOhio","2":"-4.7265870183","3":"0.5953081507","4":"-7.9397317","5":"6.948185e-14"},{"1":"stateOklahoma","2":"0.1990835341","3":"0.5973813359","4":"0.3332604","5":"7.392182e-01"},{"1":"stateOregon","2":"-3.9007020747","3":"0.5994342611","4":"-6.5073059","5":"4.174801e-10"},{"1":"statePennsylvania","2":"-1.7070557163","3":"0.5966638749","4":"-2.8610006","5":"4.581739e-03"},{"1":"stateRhode Island","2":"-6.4092098245","3":"0.5957195832","4":"-10.7587697","5":"2.028760e-22"},{"1":"stateSouth Carolina","2":"3.1378079173","3":"0.6051330489","4":"5.1853190","5":"4.465807e-07"},{"1":"stateSouth Dakota","2":"-0.4374085492","3":"0.6014407918","4":"-0.7272678","5":"4.677451e-01"},{"1":"stateTennessee","2":"0.0748011108","3":"0.6019210702","4":"0.1242706","5":"9.012013e-01"},{"1":"stateTexas","2":"-1.2268812972","3":"0.5952548236","4":"-2.0611027","5":"4.033173e-02"},{"1":"stateUtah","2":"-5.5331388814","3":"0.6174110376","4":"-8.9618399","5":"7.742095e-17"},{"1":"stateVermont","2":"-5.8044649960","3":"0.6422792890","4":"-9.0372912","5":"4.609738e-17"},{"1":"stateVirginia","2":"-4.7176009071","3":"0.5971384048","4":"-7.9003475","5":"8.948304e-14"},{"1":"stateWashington","2":"-6.1580557474","3":"0.5954396533","4":"-10.3420317","5":"4.347999e-21"},{"1":"stateWest Virginia","2":"3.0901095302","3":"0.6240786511","4":"4.9514745","5":"1.359339e-06"},{"1":"stateWisconsin","2":"-4.5709778908","3":"0.6138080620","4":"-7.4469173","5":"1.561890e-12"},{"1":"stateWyoming","2":"0.5124452910","3":"0.5973278354","4":"0.8578962","5":"3.917748e-01"},{"1":"year2008","2":"-1.0106324960","3":"0.2165232747","4":"-4.6675467","5":"4.983806e-06"},{"1":"year2009","2":"-1.6830081686","3":"0.2458855624","4":"-6.8446807","5":"5.933357e-11"},{"1":"year2010","2":"-2.0721270275","3":"0.2751069693","4":"-7.5320776","5":"9.198413e-13"},{"1":"year2011","2":"-2.2176558795","3":"0.3187735015","4":"-6.9568389","5":"3.058267e-11"},{"1":"year2012","2":"-1.8728161563","3":"0.3425094663","4":"-5.4679252","5":"1.105984e-07"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

---

# With fixest

```r
fe2_reg_2 <- feols(deaths ~ cell_plans | state + year,
                 data = phones)

fe2_reg_2 %>% summary()
```

```
## OLS estimation, Dep. Var.: deaths
## Observations: 306 
## Fixed-effects: state: 51,  year: 6
## Standard-errors: Clustered (state) 
##            Estimate Std. Error   t value Pr(>|t|) 
## cell_plans   -3e-04   0.000305 -0.980739  0.33144 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.930036     Adj. R2: 0.909197
##                  Within R2: 0.011989
```

```r
fe2_reg_2 %>% tidy()
```

---

# Adding Covariates

.pull-left[
.quitesmall[
- State fixed effect absorbs all unobserved factors that vary by state, but are constant over time

- Year fixed effect absorbs all unobserved factors that vary by year, but are constant over States

- But there are still other (often unobservable) factors that affect both Phones and Deaths, that *vary* by State *and* change over time!
  - *Some* States *change* their laws during the time period
  - State *urbanization* rates *change* over the time period

- We will also need to .hi-purple[control for these variables] (*not* picked up by fixed effects!)
  - Add them to the regression
]
]
.pull-right[
<img src="4.1-slides_files/figure-html/unnamed-chunk-36-1.png" width="504" />
]

---

# Adding Covariates I

`$$\widehat{\text{Deaths}_{it}}=\beta_1\text{Cell Phones}_{it}+\alpha_i+\theta_t+\text{urban pct}_{it}+\text{cell ban}_{it}+\text{text ban}_{it}$$`

- Can still add covariates to remove endogeneity not soaked up by fixed effects
  - factors that change within groups over time 
  - e.g. some states pass bans over the time period in data (some years before, some years after)

---

# Adding Covariates II

```r
fe2_controls_reg <- feols(deaths ~ cell_plans + text_ban + urban_percent + cell_ban | state + year,
                          data = phones)

fe2_controls_reg %>% summary()
```

```
## OLS estimation, Dep. Var.: deaths
## Observations: 306 
## Fixed-effects: state: 51,  year: 6
## Standard-errors: Clustered (state) 
##                Estimate Std. Error  t value Pr(>|t|)    
## cell_plans    -0.000340   0.000277 -1.22780 0.225269    
## text_ban1      0.255926   0.243444  1.05127 0.298188    
## urban_percent  0.013135   0.009815  1.33822 0.186878    
## cell_ban1     -0.679796   0.335655 -2.02528 0.048194 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.920123     Adj. R2: 0.910039
##                  Within R2: 0.032939
```

```r
fe2_controls_reg %>% tidy()
```

---

# Comparing Models

```r
library(huxtable)
huxreg("Pooled" = pooled,
       "State Effects" = fe_reg_1,
       "State & Year Effects" = fe2_reg_1,
       "With Controls" = fe2_controls_reg,
       coefs = c("Intercept" = "(Intercept)",
                 "Cell phones" = "cell_plans",
                 "Cell Ban" = "cell_ban1",
                 "Texting Ban" = "text_ban1",
                 "Urbanization Rate" = "urban_percent"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 4)
```
]
]

.tiny[
<table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto;  " id="tab:unnamed-chunk-38">
<col><col><col><col><col><tr>
<th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Pooled</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">State Effects</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">State & Year Effects</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">With Controls</th></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Intercept</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">17.3371 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">25.5077 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">18.9305 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.9754)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(1.0176)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(1.4511)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Cell phones</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.0006 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.0012 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.0003    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.0003  </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.0001)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.0001)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.0002)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.0003) </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Cell Ban</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.6798 *</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.3357) </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Texting Ban</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.2559  </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.2434) </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Urbanization Rate</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.0131  </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">         </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.0098) </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">N</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">306         </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">306         </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">306         </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">306       </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">R-Squared</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.0845    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.9055    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.9259    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.9274  </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">SER</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">3.2791    </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1.1526    </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1.0310    </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1.0262  </td></tr>
<tr>
<th colspan="5" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> *** p < 0.001;  ** p < 0.01;  * p < 0.05.</th></tr>
</table>

]