Panel Data

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# Panel Data
### AECN 396/896-002

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# Before we start

## Learning objectives

Understand the new econometric methods that can be used with panel datasets to address endogeneity problems.

---
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# Panel (longitudinal) Data

.content-box-green[**Definition**]

Data follow the .blue[same] individuals, families, firms, cities, states or whatever, across time

<br>

.content-box-green[**Example**]

+ Randomly select people from a population at a given point in time

+ Then the same people are reinterviewed at several subsequent points in time, which would result in data on wages, hours, education, and so on, for the same group of people in different years.

---
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.content-box-green[**Panel Data in data.frame**]

```
##   year  fcode employ    sales
## 1 1987 410032    100 47000000
## 2 1988 410032    131 43000000
## 3 1989 410032    123 49000000
## 4 1987 410440     12  1560000
## 5 1988 410440     13  1970000
## 6 1989 410440     14  2350000
## 7 1987 410495     20   750000
## 8 1988 410495     25   110000
## 9 1989 410495     24   950000
```

+ `year`: year
+ `fcode`: factory id
+ `employ`: the number of employees
+ `sales`: sales in USD

---
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.content-box-red[**Central Question**]

Can we do anything to deal with endogeneity problem taking advantage of the panel data structure?

---
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name: select

# Panel Data Estimation Methods

---
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.content-box-green[**Demand for massage (cross-sectional)**]

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      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Location</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Year</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">P</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Q</th>
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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">75</td>
<td class="gt_row gt_right">2.0</td></tr>
    <tr><td class="gt_row gt_left">Peoria</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">50</td>
<td class="gt_row gt_right">1.0</td></tr>
    <tr><td class="gt_row gt_left">Milwaukee</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">1.5</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">55</td>
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+ `P`: the price of one massage
+ `Q`: the number of massages received per capita

---
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.content-box-green[**Demand for massage (cross-sectional)**]

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<br>

.content-box-green[**Question**]

Across the four cities, how are price and quantity are associated? Positive or negative?

<br>

.content-box-green[**Answer**]

They are positively correlated. So, does that mean people want more massages as their price increases? Probably not.

---
class: middle

.content-box-green[**Demand for massage (cross-sectional)**]

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<td class="gt_row gt_right">2003</td>
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<td class="gt_row gt_right">2003</td>
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<br>

.content-box-green[**Question**]
  
What could be causing the positive correlation?

<br>

.content-box-green[**Answer**]

+ Income (can be observed)
+ Quality of massages (hard to observe)
+ How physically taxing jobs are (?)

---
class: middle

.content-box-green[**Demand for massage (cross-sectional)**]

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      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Location</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Year</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">P</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Q</th>
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    <tr><td class="gt_row gt_left">Chicago</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">75</td>
<td class="gt_row gt_right">2.0</td>
<td class="gt_row gt_right">10</td></tr>
    <tr><td class="gt_row gt_left">Peoria</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">50</td>
<td class="gt_row gt_right">1.0</td>
<td class="gt_row gt_right">5</td></tr>
    <tr><td class="gt_row gt_left">Milwaukee</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">1.5</td>
<td class="gt_row gt_right">7</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">55</td>
<td class="gt_row gt_right">0.8</td>
<td class="gt_row gt_right">6</td></tr>
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</table>
</div>

<br>

.content-box-green[**Key**]

Massage quality was hidden (omitted) affecting .red[both] price and massages per capita.

<br>

.content-box-green[**Problem**]

Massage quality is not observable, and thus cannot be controlled for.

---
class: middle

.content-box-red[**Mathematically**]

`\(Q  = \beta_0 + \beta_1 P + v \;\;( = \beta_2 + Ql + u)\)`

+ `\(P\)`: the price of one massage
+ `\(Q\)`: the number of massages received per capita  
+ `\(Ql\)`: the quality of massages
+ `\(u\)`: everything else that affect `\(P\)`

<br>

.content-box-red[**Endogeneity Problem**]

`\(P\)` is correlated with `\(Ql\)`.

---
class: middle

.content-box-green[**Demand for massage (two-period panel)**]

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  <tbody class="gt_table_body">
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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">75</td>
<td class="gt_row gt_right">2.0</td>
<td class="gt_row gt_right">10</td></tr>
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<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">85</td>
<td class="gt_row gt_right">1.8</td>
<td class="gt_row gt_right">10</td></tr>
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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">50</td>
<td class="gt_row gt_right">1.0</td>
<td class="gt_row gt_right">5</td></tr>
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<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">48</td>
<td class="gt_row gt_right">1.1</td>
<td class="gt_row gt_right">5</td></tr>
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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">1.5</td>
<td class="gt_row gt_right">7</td></tr>
    <tr><td class="gt_row gt_left">Milwaukee</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">65</td>
<td class="gt_row gt_right">1.4</td>
<td class="gt_row gt_right">7</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">55</td>
<td class="gt_row gt_right">0.8</td>
<td class="gt_row gt_right">6</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">0.7</td>
<td class="gt_row gt_right">6</td></tr>
  </tbody>
  
  
</table>
</div>

.content-box-green[**Key**]

There are two kinds of variations:

+ inte-rcity (across city) variation 
+ intra-city (within city) variation

The cross-sectional data offers only the inte-rcity (across city) variations.

---
class: middle

.content-box-green[**Demand for massage (two-period panel)**]

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    <tr><td class="gt_row gt_left">Chicago</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">75</td>
<td class="gt_row gt_right">2.0</td>
<td class="gt_row gt_right">10</td></tr>
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<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">85</td>
<td class="gt_row gt_right">1.8</td>
<td class="gt_row gt_right">10</td></tr>
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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">50</td>
<td class="gt_row gt_right">1.0</td>
<td class="gt_row gt_right">5</td></tr>
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<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">48</td>
<td class="gt_row gt_right">1.1</td>
<td class="gt_row gt_right">5</td></tr>
    <tr><td class="gt_row gt_left">Milwaukee</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">1.5</td>
<td class="gt_row gt_right">7</td></tr>
    <tr><td class="gt_row gt_left">Milwaukee</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">65</td>
<td class="gt_row gt_right">1.4</td>
<td class="gt_row gt_right">7</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">55</td>
<td class="gt_row gt_right">0.8</td>
<td class="gt_row gt_right">6</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">0.7</td>
<td class="gt_row gt_right">6</td></tr>
  </tbody>
  
  
</table>
</div>

Now, compare the massage price and massages per capita .red[within] each city (over time). What do you see?

<br>

.content-box-green[**Answer**]

Price and quantity are .red[negatively] correlated!

---
class: middle

.content-box-green[**Demand for massage (two-period panel)**]

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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">75</td>
<td class="gt_row gt_right">2.0</td>
<td class="gt_row gt_right">10</td></tr>
    <tr><td class="gt_row gt_left">Chicago</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">85</td>
<td class="gt_row gt_right">1.8</td>
<td class="gt_row gt_right">10</td></tr>
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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">50</td>
<td class="gt_row gt_right">1.0</td>
<td class="gt_row gt_right">5</td></tr>
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<td class="gt_row gt_right">2004</td>
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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">60</td>
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<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">65</td>
<td class="gt_row gt_right">1.4</td>
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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">55</td>
<td class="gt_row gt_right">0.8</td>
<td class="gt_row gt_right">6</td></tr>
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<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">0.7</td>
<td class="gt_row gt_right">6</td></tr>
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</table>
</div>

.content-box-green[**Question**]

Why looking at the .blue[intra-city (within city)] variation seemed to help us estimate the impact of massage price on demand more credibly?

<br>

.content-box-green[**Answer**]

The omitted variable, massage quality, did not change over time within city, which means it is controlled for as long as you look only at the intra-city variations (you do not compare .red[across] cities).

---
class: middle

.content-box-green[**Demand for massage (two-period panel)**]

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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">75</td>
<td class="gt_row gt_right">2.0</td>
<td class="gt_row gt_right">10</td></tr>
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<td class="gt_row gt_right">2004</td>
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<td class="gt_row gt_right">2003</td>
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<td class="gt_row gt_right">2004</td>
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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">1.5</td>
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<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">65</td>
<td class="gt_row gt_right">1.4</td>
<td class="gt_row gt_right">7</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">55</td>
<td class="gt_row gt_right">0.8</td>
<td class="gt_row gt_right">6</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">0.7</td>
<td class="gt_row gt_right">6</td></tr>
  </tbody>
  
  
</table>
</div>

.content-box-green[**Question**]

But, what if massage quality changed from 2003 to 2004?

.content-box-green[**Answer**]

Looking at the intra-city variations is problematic just like looking at the inter-city variations.

---
class: middle

.content-box-green[**Question**]

So, how do we use only the intra-city variations in a regression framework?

.content-box-green[**One way**]

One way to do this is to compute the changes in prices and th changes in quantities in each city `\((\Delta P\)` and `\(\Delta Q)\)` and then regress `\(\Delta Q\)` and `\(\Delta P\)`.

.content-box-green[**First-differenced Data**]

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    <tr><td class="gt_row gt_left">Chicago</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">75</td>
<td class="gt_row gt_right">2.0</td>
<td class="gt_row gt_right">10</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
    <tr><td class="gt_row gt_left">Chicago</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">85</td>
<td class="gt_row gt_right">1.8</td>
<td class="gt_row gt_right">10</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">10</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-0.2</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">50</td>
<td class="gt_row gt_right">1.0</td>
<td class="gt_row gt_right">5</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
    <tr><td class="gt_row gt_left">Peoria</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">48</td>
<td class="gt_row gt_right">1.1</td>
<td class="gt_row gt_right">5</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-2</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0.1</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
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<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">1.5</td>
<td class="gt_row gt_right">7</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
    <tr><td class="gt_row gt_left">Milwaukee</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">65</td>
<td class="gt_row gt_right">1.4</td>
<td class="gt_row gt_right">7</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">5</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-0.1</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">55</td>
<td class="gt_row gt_right">0.8</td>
<td class="gt_row gt_right">6</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">0.7</td>
<td class="gt_row gt_right">6</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">5</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-0.1</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
  </tbody>
  
  
</table>
</div>

.content-box-green[**Key**]

Variations in quality is eliminated after first differentiation!! (quality is controlled for)

---
class: middle

.content-box-red[**A new way of writing a model**]

`\(Q_{i,t}  = \beta_0 + \beta_1 P_{i,t} + v_{i,t} \;\; ( = \beta_2 Ql_{i,t} + u_{i,t})\)`

+ `i`: indicates city
+ `t`: indicates time

<br>

.content-box-red[**First differencing**]

`\(Q_{i,1}  = \beta_0 + \beta_1 P_{i,1} + v_{i,1} \;\; ( = \beta_2 Ql_{i,1} + u_{i,1})\)`

`\(Q_{i,2}  = \beta_0 + \beta_1 P_{i,2} + v_{i,2} \;\; ( = \beta_2 Ql_{i,2} + u_{i,2})\)`

`\(\Rightarrow\)`

`\(\Delta Q  = \beta_1 \Delta P + \Delta v ( = \beta_2 \Delta Ql + \Delta u)\)`

<br>

.content-box-red[**Endogeneity Problem?**]

Since `\(Ql_{i,1} = Ql_{i,2}\)`, `\(\Delta Ql = 0\)`.

`\(\Rightarrow\)`

`\(\Delta Q  = \beta_0 + \beta_1 \Delta P + \Delta u\)`

No endogeneity problem after first differentiation!

---
class: middle

.content-box-green[**Data**]

```r
massage_data_fd %>% head(5)
```

```
## # A tibble: 5 × 8
##   Location   Year     P     Q    Ql P_dif  Q_dif Ql_dif
##   <chr>     <dbl> <dbl> <dbl> <dbl> <dbl>  <dbl>  <dbl>
## 1 Chicago    2003    75   2      10    NA NA         NA
## 2 Chicago    2004    85   1.8    10    10 -0.2        0
## 3 Peoria     2003    50   1       5    NA NA         NA
## 4 Peoria     2004    48   1.1     5    -2  0.100      0
## 5 Milwaukee  2003    60   1.5     7    NA NA         NA
```

.left5[

.content-box-green[**OLS on the original data:**]

```r
feols(Q ~ P, data = massage_data_fd)
```

```
## OLS estimation, Dep. Var.: Q
## Observations: 8 
## Standard-errors: IID 
##              Estimate Std. Error   t value Pr(>|t|)    
## (Intercept) -0.496511   0.614204 -0.808381  0.44973    
*## P            0.028659   0.009696  2.955833  0.02542 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.27893   Adj. R2: 0.525004
```

]

.right5[

.content-box-green[**OLS on the first-differenced data:**]

```r
feols(Q_dif ~ P_dif, data = massage_data_fd)
```

```
## OLS estimation, Dep. Var.: Q_dif
## Observations: 4 
## Standard-errors: IID 
##              Estimate Std. Error   t value Pr(>|t|)    
## (Intercept)  0.039041   0.012750   3.06213 0.092145 .  
*## P_dif       -0.025342   0.002055 -12.33333 0.006510 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.012414   Adj. R2: 0.980534
```

]

---
class: middle

.content-box-red[**Summary**]
 
+ As long as the omitted variable that affects both the dependent and independent variables are constant over time (time-invariant), then using only the variations over time (ignoring variations across cross-sectional units) can eliminate the omitted variable bias

+ First-differencing the data and then regressing changes on changes does the trick of ignoring variations across cross-sectional units

+ Of course, first-differencing is possible only because the same cross-sectional units are observed multiple times over time.

---
class: middle

# Multi-year panel datasets

+ If we have lots of years of data, we could, in principle, compute all of the first differences (i.e., 2004 versus 2003, 2005 versus 2004, etc.) and then run a single regression. But there is an easier way.

+ Instead of thinking of each year’s observation in terms of how much it differs from the prior year for the same city, let's think about how much each observation differs from the average for that city.

---
class: middle

How much each observation differs from the average for that city?

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<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Location</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Year</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">P</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">P_mean</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">P_dev</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Q</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Q_mean</th>
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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">Ql_dev</th>
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    <tr><td class="gt_row gt_left">Chicago</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">75</td>
<td class="gt_row gt_right">80.0</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-5.0</td>
<td class="gt_row gt_right">2.0</td>
<td class="gt_row gt_right">1.90</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0.10</td>
<td class="gt_row gt_right">10</td>
<td class="gt_row gt_right">10</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
    <tr><td class="gt_row gt_left">Chicago</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">85</td>
<td class="gt_row gt_right">80.0</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">5.0</td>
<td class="gt_row gt_right">1.8</td>
<td class="gt_row gt_right">1.90</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-0.10</td>
<td class="gt_row gt_right">10</td>
<td class="gt_row gt_right">10</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
    <tr><td class="gt_row gt_left">Peoria</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">50</td>
<td class="gt_row gt_right">49.0</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">1.0</td>
<td class="gt_row gt_right">1.0</td>
<td class="gt_row gt_right">1.05</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-0.05</td>
<td class="gt_row gt_right">5</td>
<td class="gt_row gt_right">5</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
    <tr><td class="gt_row gt_left">Peoria</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">48</td>
<td class="gt_row gt_right">49.0</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-1.0</td>
<td class="gt_row gt_right">1.1</td>
<td class="gt_row gt_right">1.05</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0.05</td>
<td class="gt_row gt_right">5</td>
<td class="gt_row gt_right">5</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
    <tr><td class="gt_row gt_left">Milwaukee</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">62.5</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-2.5</td>
<td class="gt_row gt_right">1.5</td>
<td class="gt_row gt_right">1.45</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0.05</td>
<td class="gt_row gt_right">7</td>
<td class="gt_row gt_right">7</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
    <tr><td class="gt_row gt_left">Milwaukee</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">65</td>
<td class="gt_row gt_right">62.5</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">2.5</td>
<td class="gt_row gt_right">1.4</td>
<td class="gt_row gt_right">1.45</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-0.05</td>
<td class="gt_row gt_right">7</td>
<td class="gt_row gt_right">7</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2003</td>
<td class="gt_row gt_right">55</td>
<td class="gt_row gt_right">57.5</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-2.5</td>
<td class="gt_row gt_right">0.8</td>
<td class="gt_row gt_right">0.75</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0.05</td>
<td class="gt_row gt_right">6</td>
<td class="gt_row gt_right">6</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
    <tr><td class="gt_row gt_left">Madison</td>
<td class="gt_row gt_right">2004</td>
<td class="gt_row gt_right">60</td>
<td class="gt_row gt_right">57.5</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">2.5</td>
<td class="gt_row gt_right">0.7</td>
<td class="gt_row gt_right">0.75</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">-0.05</td>
<td class="gt_row gt_right">6</td>
<td class="gt_row gt_right">6</td>
<td class="gt_row gt_right" style="background-color: #FFA500;">0</td></tr>
  </tbody>
  
  
</table>
</div>
<br>

.content-box-green[**Note**]

We call this data transformation .red[within-transformation] or .red[demeaning].

.content-box-green[**Fixed Effects Regression**]

+ Dependent variable: `Q_dev`
+ Independent variable: `P_dev`

.content-box-green[**Key**]

In calculating `P_dev` (deviation from the mean by city), `Ql_dev` is eliminated.

---
class: middle

.content-box-red[**Within-transformation**]

`\(Q_{i,1}  = \beta_0 + \beta_1 P_{i,1} + v_{i,1} \;\; ( = \beta_2 Ql_{i,1} + u_{i,1})\)`

`\(Q_{i,2}  = \beta_0 + \beta_1 P_{i,2} + v_{i,2} \;\; ( = \beta_2 Ql_{i,2} + u_{i,2})\)`

`\(\vdots\)`

`\(Q_{i,T}  = \beta_0 + \beta_1 P_{i,T} + v_{i,T} \;\; ( = \beta_2 Ql_{i,T} + u_{i,T})\)`

`\(\Rightarrow\)`

`\(Q_{i,t} - \bar{Q_{i}}   = \beta_1 [P_{i,t} - \bar{P_{i}}] + [v_{i,t} - \bar{v_{i}}] ( = \beta_2 [Ql_{i,t} - \bar{Ql_{i}}] + [u_{i,t} - \bar{u_{i}}])\)`

.content-box-red[**Endogeneity Problem?**]

`\(Ql_{i,1} = Ql_{i,2} = \dots = Ql_{i,T} = \bar{Ql_i}\)`

`\(\Rightarrow\)`

`\(Q_{i,t} - \bar{Q_{i}}   = \beta_1 [P_{i,t} - \bar{P_{i}}] + [u_{i,t} - \bar{u_{i}}]\)`

No endogeneity problem after the within-transformation!

---
class: middle

# Fixed Effects (FE) Estimation (in general)

Consider the following general model

`\(y_{i,t}=\beta_1 x_{i,t} + \alpha_i + u_{i,t}\)`

+ `\(\alpha_i\)`: the impact of time-invariant <span style = "color: blue;"> unobserved </span> factor that is specific to `\(i\)` (also termed .blue[individual fixed effect]) 
+ `\(\alpha_i\)` is thought to be correlated with `\(x_{i,t}\)`

For each `\(i\)`, average this equation over time, we get

`\(\frac{\sum_{t=1}^T y_{i,t}}{T} = \frac{\sum_{t=1}^T x_{i,t}}{T} + \alpha_i  + \frac{\sum_{t=1}^T u_{i,t}}{T}\)`

Note, `\(\frac{\sum_{t=1}^T \alpha_{i}}{T} = \alpha_i\)`

Subtracting the second equation from the first one,

`\((y_{i,t}-\frac{\sum_{t=1}^T y_{i,t}}{T})=\beta_1 (x_{i,t} -\frac{\sum_{t=1}^T x_{i,t}}{T}) + (u_{i,t} -\frac{\sum_{t=1}^T u_{i,t}}{T})\)`

.content-box-red[**Important**]

`\(\alpha_i\)` is gone!

We then regress `\((y_{i,t}-\frac{\sum_{t=1}^T y_{i,t}}{T})\)` on `\((x_{i,t}-\frac{\sum_{t=1}^T x_{i,t}}{T})\)` to estimate `\(\beta_1\)`.

---
class: middle

# When is FE estimation unbiased?

Here is the data after within-transformation:

`\((y_{i,t}-\frac{\sum_{t=1}^T y_{i,t}}{T})=\beta_1 (x_{i,t} -\frac{\sum_{t=1}^T x_{i,t}}{T}) + (u_{i,t} -\frac{\sum_{t=1}^T u_{i,t}}{T})\)`

So,

`\((x_{i,t} -\frac{\sum_{t=1}^T x_{i,t}}{T})\)` needs to be uncorrelated with `\((u_{i,t} -\frac{\sum_{t=1}^T u_{i,t}}{T})\)`.

The above condition is satisfied if

`\(E[u_{i,s}|x_{i,t}] = 0 \;\; ^\forall s, \;\; t, \;\;\mbox{and} \;\;j\)`

e.g., `\(E[u_{i,1}|x_{i,4}]=0\)`

---
class: middle

.content-box-green[**Fixed effects estimation**]

Regress within-transformed `Q` on within-transformed `P`:

```r
feols(Q_dev ~ P_dev, data = massage_data_wth)
```

```
## OLS estimation, Dep. Var.: Q_dev
## Observations: 8 
## Standard-errors: IID 
##                  Estimate Std. Error       t value   Pr(>|t|)    
## (Intercept)  2.780000e-17   0.006044  4.590000e-15 1.0000e+00    
*## P_dev       -2.077922e-02   0.001948 -1.066667e+01 4.0041e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.014804   Adj. R2: 0.941558
```

---
class: middle

# An alternative way to view the Fixed Effects estimation methods

.content-box-green[**Important**]

The two approached below will result in the same coefficient estimates (mathematically identical).

+ Running OLS on the within-tranformed (demeaned) data
+ Running OLS on the untransformed data but including the dummy variables for the individuals

---
class: middle

You can use the original data (no within-transformation) and include dummy variables for all the cities except one.

```r
(
  massage_data_d <- massage_data_2p %>%
    mutate(
      Peoria_D = ifelse(Location == "Peoria", 1, 0),
      Milwaukee_D = ifelse(Location == "Milwaukee", 1, 0),
      Madison_D = ifelse(Location == "Madison", 1, 0)
    )
)
```

```
##    Location Year  P   Q Ql Peoria_D Milwaukee_D Madison_D
## 1   Chicago 2003 75 2.0 10        0           0         0
## 2   Chicago 2004 85 1.8 10        0           0         0
## 3    Peoria 2003 50 1.0  5        1           0         0
## 4    Peoria 2004 48 1.1  5        1           0         0
## 5 Milwaukee 2003 60 1.5  7        0           1         0
## 6 Milwaukee 2004 65 1.4  7        0           1         0
## 7   Madison 2003 55 0.8  6        0           0         1
## 8   Madison 2004 60 0.7  6        0           0         1
```

---
class: middle

.content-box-green[**Fixed effects estimation (alternative way)**]

```r
feols(Q ~ P + Peoria_D + Milwaukee_D + Madison_D, data = massage_data_d)
```

```
## OLS estimation, Dep. Var.: Q
## Observations: 8 
## Standard-errors: IID 
##              Estimate Std. Error   t value   Pr(>|t|)    
## (Intercept)  3.562338   0.221059  16.11488 0.00051976 ***
*## P           -0.020779   0.002755  -7.54247 0.00483179 ** 
## Peoria_D    -1.494156   0.088759 -16.83378 0.00045649 ***
## Milwaukee_D -0.813636   0.053933 -15.08599 0.00063230 ***
## Madison_D   -1.617532   0.066534 -24.31141 0.00015255 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.014804   Adj. R2: 0.997324
```

Note that the coefficient estimate on `P` is exactly the same as the one we saw earlier when we regressed `Q_dev` on `P_dev`.

---
class: middle

.content-box-green[**What does this tell us?**]

By including individual dummies (individual fixed effects), you are effectively eliminating the between (inter-city) variations and using only the clean within (within-city) variations for estimation.

<br>

.content-box-red[**Very Important**]

More generally, including dummy variables of a categorical variable (like city in the example above), eliminates the variations <span style = "color: red;"> between </span> the elements of the category (e.g., different cities), and use only the variations <span style = "color: blue;"> within </span> each of the element of the category.

---
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# Let's go back to the avovado example

This was how the avocado makert worked:

<span style = "color: blue;"> At the beginning of each month, avocado suppliers make a plan for what avocado prices will be each week in that month, and never change their plans until the next month. </span>

We agreed that we should use only the within-month varaitions instead of between-month variations.

We have three months of avocado purchase and price observed weekly.

<br>

.content-box-green[**Question**]

What should we do?

<br>

.content-box-green[**Answer**]

Include month dummy variables

---
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We have two years of avocado purchase and price observed weekly.

<br>

.content-box-green[**Question**]

What should we do?

<br>

.content-box-green[**Answer**]

+ Include <span style = "color: blue;"> month-year </span> dummy variables.

+ Including <span style = "color: blue;"> month </span> dummy variables will not do it. Because the observations in the same month in two different years are considered to be belong to the same group. That is, variations between two different years of the same month will be used for estimation. (e.g., January in 2014 and January in 2015)

---
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# Fixed Effects Estimation in Practice Using R

.content-box-green[**Advice**]

+ Do not within-transform the data yourself and run a regression
+ Do not create dummy variables yourself and run a regression with the dummies

<br>

.content-box-red[**In practice**]

We will use the `fixest` package.

<br>

.content-box-green[**Syntax**]

```r
feols(dep_var ~ indep_vars | FE, data)
```

+ `FE`: the name of the variable that identifies the cross-sectional units that are observed over time (`Location` in our example) 
+ `dep_var`: (non-transformed) dependet variable
+ `indep_vars`: list of (non-transformed) independent variables

---
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.content-box-green[**Data**]

```r
massage_data_2p
```

```
##    Location Year  P   Q Ql
## 1   Chicago 2003 75 2.0 10
## 2   Chicago 2004 85 1.8 10
## 3    Peoria 2003 50 1.0  5
## 4    Peoria 2004 48 1.1  5
## 5 Milwaukee 2003 60 1.5  7
## 6 Milwaukee 2004 65 1.4  7
## 7   Madison 2003 55 0.8  6
## 8   Madison 2004 60 0.7  6
```

.content-box-green[**Example**]

```r
feols(Q ~ P | Location, data = massage_data_2p)
```

```
## OLS estimation, Dep. Var.: Q
## Observations: 8 
## Fixed-effects: Location: 4
## Standard-errors: Clustered (Location) 
##    Estimate Std. Error t value   Pr(>|t|)    
*## P -0.020779   0.001159 -17.923 0.00037879 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.014804     Adj. R2: 0.997324
##                  Within R2: 0.949907
```

---
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# Random Effects (RE) Model

+ Can be more efficient than FE

+ If `\(\alpha_i\)` and independent variables are correlated, then RE estimators are biased

+ Unless `\(\alpha_i\)` and independent variables are not correlated (which does not hold most of the time unless you got data from controlled experiments), `\(RE\)` is not an attractive option

+ You almost never see this estimation method used in papers that use non-experimental data

<br>

.content-box-green[**Note**]

We do not cover this estimation method as you almost certainly would not use this estimation method.

---
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name: select

# Year Fixed Effects

---
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# Year Fixed Effects

.content-box-green[**Definition**]

Just a collection of year dummies, which takes 1 if in a specific year, 0 otherwise.

```
##    id year income educ FE_2015 FE_2016 FE_2017
## 1   1 2015     77   12       1       0       0
## 2   1 2016     82   13       0       1       0
## 3   1 2017     84   14       0       0       1
## 4   1 2015    110   18       1       0       0
## 5   2 2016    120   19       0       1       0
## 6   2 2017    131   20       0       0       1
## 7   2 2015     56   10       1       0       0
## 8   2 2016     60   11       0       1       0
## 9   3 2017     61   12       0       0       1
## 10  3 2015     70   13       1       0       0
## 11  3 2016     71   14       0       1       0
## 12  3 2017     74   15       0       0       1
```
  
---
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.content-box-green[**What do year FEs do?**]

They capture anything that happened to .red[all] the individuals for a specific year relative to the base year

<br>

.content-box-green[**Example**]

Education and wage data from `\(2012\)` to `\(2014\)`,

`\(log(income) = \beta_0 + \beta_1 educ + \beta_2 exper  + \sigma_1 FE_{2012} + \sigma_2 FE_{2013}\)`

+ `\(\sigma_1\)`: captures the difference in `\(log(income)\)` between `\(2012\)` and `\(2014\)` (base year)

+ `\(\sigma_2\)`: captures the difference in `\(log(income)\)` between `\(2013\)` and `\(2014\)` (base year)

<br>

.content-box-green[**Interpretation**]
 
`\(\sigma_1=0.05\)` would mean that `\(log(income)\)` is greater in `\(2012\)` than `\(2014\)` by `\(5\%\)` on average for whatever reasons with everything else fixed.

---
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.content-box-green[**Recommendation**]

It is almost always a good practice to include year FEs if you are using a panel dataset with annual observations.

<br>

.content-box-green[**Why?**]

+ Remember year FEs capture .blue[anything] that happened to all the individuals for a specific year relative to the base year

+ In other words, .blue[all the unobserved factors] that are common to all the individuals in a specific year is .blue[controlled for (taken out of the error term)]

<br>

.content-box-green[**Example**]

Economic trend in:

`\(log(income) =  \beta_0 + \beta_1 educ + \sigma_1 FE_{2012} + \sigma_2 FE_{2013}\)`

+ Education is non-decreasing through time

+ Economy might have either been going down or up during the observed period

+ Without year FE, `\(\beta_1\)` may capture the impact of overall economic trend.

---
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.content-box-green[**R implementation**]

In order to include year FEs to individual FEs, you can simply add the variable that indicates year like below:

```r
feols(I(log(income)) ~ educ | id + year, data = year_fe) %>%
  tidy()
```

```
## # A tibble: 1 × 5
##   term  estimate std.error statistic p.value
##   <chr>    <dbl>     <dbl>     <dbl>   <dbl>
## 1 educ    0.0801   0.00931      8.60  0.0132
```

---
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.content-box-red[**Caveats**]

+ Year FEs would be perfectly collinear with variables that change only across time, but not across individuals.

+ If your variable of interest is such a variable, you cannot include year FEs, which would then make your estimation subject to omitted variable bias due to .red[other] unobserved yearly-changing factors.

---
class: inverse, center, middle
name: select

# Standard Error Estimation for Panel Data Methods

---
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.content-box-green[**Heteroskedasticity**]

Just like we saw for OLS using cross-sectional data, heteroskedasticity leads to biased estimation of the standard error of the coefficient estimators if not taken into account

---
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.content-box-green[**Serial Correlation**]

Correlation of errors over time, which we call .blue[serial correlation]

<br>

.content-box-green[**Consequences of serial correlation**]

+ just like heteroskedasticity, serial correlation could lead to biased estimation of the standard error of the coefficient estimators if not taken into account

+ do not affect the unbiasedness and consistency property of your estimators

<br>

.content-box-red[**Important**]

+ Taking into account the potential of serial correlation when estimating the standard error of the coefficient estimators can dramatically change your conclusions about the statistical significance of some independent variables!!

+ When serial correlation is ignored, you tend to underestimate the standard error (why?), inflating `\(t\)`-statistic, which in turn leads to over-rejection that you should.

---
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.content-box-green[**Bertrand, Duflo, and Mullainathan (2004)**]

+ Examined how problematic serial correlation is in terms of inference via Monte Carlo simulation

* generate a fake treatment dummy variable in a way that it has no impact on the outcome (dependent variable) in the dataset of women's wages from the Current Population Survey (CPS)

* run regression of the oucome on the treatment variable

* test if the treatment variable has statistically significant effect via `\(t\)`-test

+ They rejected the null `\(67.5\%\)` at the `\(5\%\)` significance level!!

---
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.content-box-green[**SE robust to heteroskedasticity and serial correlation**]

+ You can take into account .blue[both] heteroskedasticity and serial correlation by clustering by individual (whatever the unit of individual is: state, county, farmer)

+ Cluster by individual allows correlation within individuals (over time)

<br>

.content-box-green[**R implementation**]

The last partition is used for clustering standard error estimation by variable like below.

```r
feols(I(log(income)) ~ educ | id + year, cluster = ~id, data = year_fe) %>% tidy()
```

```
## # A tibble: 1 × 5
##   term  estimate std.error statistic p.value
##   <chr>    <dbl>     <dbl>     <dbl>   <dbl>
## 1 educ    0.0801   0.00931      8.60  0.0132
```