Machine Learning and Predicted Returns for Event Studies in Securities Litigation

# Machine Learning and Predicted Returns for Event Studies in Securities Litigation
### Andrew C. Baker and Jonah B. Gelbach
### Stanford University and UC Berkeley
### 2019 JLFA Conference, NYU, September 6, 2019

---

# Event Studies and Securities Litigation

<span style="color:red"> - A lot of money changes hands in securities litigation </span>

---
class: center
# Largest Securities Class Action Settlements

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Company </th>
   <th style="text-align:left;"> Amount </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Enron Corporation </td>
   <td style="text-align:left;"> $ 7.2 B </td>
  </tr>
  <tr>
   <td style="text-align:left;"> WorldCom Inc. </td>
   <td style="text-align:left;"> $ 6.1 B </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Tyco International Ltd. </td>
   <td style="text-align:left;"> $ 3.2 B </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cendant Corporation </td>
   <td style="text-align:left;"> $ 3.2 B </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Petroleo Brasileiro S.A. - Petrobras : American Depositary Shares </td>
   <td style="text-align:left;"> $ 3.0 B </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Nortel Networks Corporation </td>
   <td style="text-align:left;"> $ 2.9 B </td>
  </tr>
  <tr>
   <td style="text-align:left;"> AOL Time Warner Inc. </td>
   <td style="text-align:left;"> $ 2.5 B </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Bank of America Corporation: Merger with Merrill Lynch </td>
   <td style="text-align:left;"> $ 2.4 B </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Household International Inc. </td>
   <td style="text-align:left;"> $ 1.6 B </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Koninklijke Ahold NV : Royal Ahold Corporation Securities </td>
   <td style="text-align:left;"> $ 1.1 B </td>
  </tr>
</tbody>
</table>
---
class: center
# Aggregate Settlement Value By Year

---

# Event Studies and Securities Litigation

<span style="color:red"> - A lot of money changes hands in securities litigation </span>

--
- Single-firm event studies play a critical role

- Typical: daily return model to estimate abnormal return for 1 date

- Problems due to single-firm nature of event study

- Non-normality of returns complicates inference (really an effect?)
  
--
  - Low power/high variance
  
--

- Enter this paper: Can we do a better job at measuring event effect?

---

---

---

---
# Event Studies as a Prediction Problem

Suppose event reveals fraud. Effect on daily return:
  
`$$\color{red}{r_{i, event} \Rightarrow (r_{i, event} - \text{event effect})}$$`
--

`$\hspace{2cm}$`

Estimate event effect: predict counterfactual value `$\color{red}{r_{i, event}}$` using `$\color{blue}{\widehat{r}_{i, event}}$`

`$\hspace{2cm}$`

Then construct predicted residual:

`$$\color{blue}{\widehat{\zeta}_{i, event}} \equiv \color{red}{r_{i, event}} - \color{blue}{\widehat{r}_{i, event}}$$`
---
# Event Studies as a Prediction Problem

One oft-used daily return model for firm `$i$` in absence of event:

`$$\color{red}{r_{it} = \alpha + \beta_1 mkt_{it} + \beta_2 peer_{it} + \zeta_{it}}$$`

<br>
<br>
Typically estimate using `$\color{blue}{OLS}$`

<br>
<br>
<span style="color:purple"> How much can we improve performance with alternative approaches? </span>

---

# This paper's contributions

<br>

1. Observation: SecLit event studies can be viewed as prediction problems

2. Offer flexible, data-driven approach to predicting event-date return

--
<br>

`$$r_{it} = g^k(X_{it}) + \zeta_{it}$$`

--
<br>
`$$\color{blue}{\Rightarrow \hat{r}_{it} = \hat{g}^k(X_{it})}$$`

In particular, construction of peer index variable `$\color{red}{peer_{it}}$`.

---

# This paper's contributions

<br>

1. Observation: SecLit event studies can be viewed as prediction problems

2. Offer flesible, data-driven approach to predicting event-date return

3. Investigate performance of alternative estimation approaches

- Out-of-sample prediction variance: `$\color{blue}{\widehat{V}(\widehat{\zeta}_{it})}$`
  
--

- Significance tests - relevant for liability

---
# Summary of Findings

1. ML can improve performance of event-date abnormal return estimation

2. ML-based improvements in variance don't seem all that big
  - <span style="color:blue"> Data-driven </span> `$\color{red}{peer_{it}}$` is best, but ...

--
  - ... we do pretty well as long as we use _some_ peer index.
  
--

3. Same basic result for significance test performance
  - All specifications do poorly with parametric Normal/Student *t* testing

--
  - All do better with robust SQ test
  
--
  - ML methods with <span style="color:blue"> Data-driven </span> `$\color{red}{peer_{it}}$` improve on others modestly.
  
--

<span style="color:green"> Takehome point: ML methods help modestly, but not overwhelmingly </span>

---
# Turning it over to Andrew to Explain:

* Substantive importance of data-driven peer index
<br>

* The modifications of standard practice we consider
<br>

* Details of our simulation method and performance metrics
<br>

* What we're planning to add to this paper
<br>

--
<br>
Take it away, Andrew...

---

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> Firm </th>
   <th style="text-align:right;"> Mkt </th>
   <th style="text-align:right;"> Peer A </th>
   <th style="text-align:right;"> Peer B </th>
   <th style="text-align:right;"> Peer C </th>
   <th style="text-align:right;"> Peer D </th>
   <th style="text-align:right;"> Peer E </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> 0.22 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.03 </td>
   <td style="text-align:right;"> -0.01 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;"> -0.06 </td>
   <td style="text-align:right;"> -0.10 </td>
   <td style="text-align:right;"> -0.09 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.02 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.16 </td>
   <td style="text-align:right;"> 0.01 </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;"> -0.18 </td>
   <td style="text-align:right;"> 0.08 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.02 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> -0.06 </td>
   <td style="text-align:right;"> 0.16 </td>
   <td style="text-align:right;"> -0.01 </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;"> -0.16 </td>
   <td style="text-align:right;"> -0.05 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.25 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.05 </td>
   <td style="text-align:right;"> 0.08 </td>
   <td style="text-align:right;"> 0.24 </td>
   <td style="text-align:right;"> -0.07 </td>
   <td style="text-align:right;"> 0.09 </td>
   <td style="text-align:right;"> -0.03 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.06 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.14 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;"> -0.19 </td>
   <td style="text-align:right;"> 0.00 </td>
   <td style="text-align:right;"> 0.03 </td>
   <td style="text-align:right;"> 0.09 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.01 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.01 </td>
   <td style="text-align:right;"> -0.06 </td>
   <td style="text-align:right;"> 0.17 </td>
   <td style="text-align:right;"> 0.07 </td>
   <td style="text-align:right;"> 0.06 </td>
   <td style="text-align:right;"> 0.17 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> 0.11 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.19 </td>
   <td style="text-align:right;"> -0.19 </td>
   <td style="text-align:right;"> 0.21 </td>
   <td style="text-align:right;"> -0.17 </td>
   <td style="text-align:right;"> -0.10 </td>
   <td style="text-align:right;"> 0.17 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.06 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> -0.10 </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;"> -0.02 </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;"> 0.12 </td>
   <td style="text-align:right;"> 0.08 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.09 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> -0.13 </td>
   <td style="text-align:right;"> -0.06 </td>
   <td style="text-align:right;"> -0.21 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;"> 0.21 </td>
   <td style="text-align:right;"> 0.14 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.16 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.06 </td>
   <td style="text-align:right;"> -0.15 </td>
   <td style="text-align:right;"> 0.03 </td>
   <td style="text-align:right;"> 0.34 </td>
   <td style="text-align:right;"> -0.06 </td>
   <td style="text-align:right;"> -0.09 </td>
  </tr>
</tbody>
</table>
---
class: center
# Selecting Peer Firms

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> Firm </th>
   <th style="text-align:right;"> Mkt </th>
   <th style="text-align:right;"> Peer A </th>
   <th style="text-align:right;"> Peer B </th>
   <th style="text-align:right;"> Peer C </th>
   <th style="text-align:right;"> Peer D </th>
   <th style="text-align:right;"> Peer E </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> 0.22 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.03 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.01 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.05 </td>
   <td style="text-align:right;"> -0.06 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.10 </td>
   <td style="text-align:right;"> -0.09 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.02 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.16 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.01 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.02 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.18 </td>
   <td style="text-align:right;"> 0.08 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.02 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> -0.06 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.16 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.01 </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.16 </td>
   <td style="text-align:right;"> -0.05 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.25 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.05 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.08 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.24 </td>
   <td style="text-align:right;"> -0.07 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.09 </td>
   <td style="text-align:right;"> -0.03 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.06 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.14 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.05 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.19 </td>
   <td style="text-align:right;"> 0.00 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.03 </td>
   <td style="text-align:right;"> 0.09 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.01 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.01 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.06 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.17 </td>
   <td style="text-align:right;"> 0.07 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.06 </td>
   <td style="text-align:right;"> 0.17 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> 0.11 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.19 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.19 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.21 </td>
   <td style="text-align:right;"> -0.17 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.10 </td>
   <td style="text-align:right;"> 0.17 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.06 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> -0.10 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.02 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.02 </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.12 </td>
   <td style="text-align:right;"> 0.08 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.09 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> -0.13 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.06 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.21 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.21 </td>
   <td style="text-align:right;"> 0.14 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.16 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.06 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.15 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.03 </td>
   <td style="text-align:right;"> 0.34 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.06 </td>
   <td style="text-align:right;"> -0.09 </td>
  </tr>
</tbody>
</table>
---
class: center
# Selecting Peer Firms

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> Firm </th>
   <th style="text-align:right;"> Mkt </th>
   <th style="text-align:right;"> Peer A </th>
   <th style="text-align:right;"> Peer B </th>
   <th style="text-align:right;"> Peer C </th>
   <th style="text-align:right;"> Peer D </th>
   <th style="text-align:right;"> Peer E </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> 0.22 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.03 </td>
   <td style="text-align:right;"> -0.01 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.05 </td>
   <td style="text-align:right;"> -0.06 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.10 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.09 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.02 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.16 </td>
   <td style="text-align:right;"> 0.01 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.02 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.18 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.08 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.02 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> -0.06 </td>
   <td style="text-align:right;"> 0.16 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.01 </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.16 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.05 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.25 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.05 </td>
   <td style="text-align:right;"> 0.08 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.24 </td>
   <td style="text-align:right;"> -0.07 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.09 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.03 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.06 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.14 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.19 </td>
   <td style="text-align:right;"> 0.00 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.03 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.09 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.01 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.01 </td>
   <td style="text-align:right;"> -0.06 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.17 </td>
   <td style="text-align:right;"> 0.07 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.06 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.17 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> 0.11 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.19 </td>
   <td style="text-align:right;"> -0.19 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.21 </td>
   <td style="text-align:right;"> -0.17 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.10 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.17 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.06 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> -0.10 </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.02 </td>
   <td style="text-align:right;"> 0.02 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.12 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.08 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.09 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> -0.13 </td>
   <td style="text-align:right;"> -0.06 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.21 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.21 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.14 </td>
  </tr>
  <tr>
   <td style="text-align:right;background-color: pink !important;border-right:1px solid;"> -0.16 </td>
   <td style="text-align:right;background-color: lightgreen !important;"> 0.06 </td>
   <td style="text-align:right;"> -0.15 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> 0.03 </td>
   <td style="text-align:right;"> 0.34 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.06 </td>
   <td style="text-align:right;color: white !important;background-color: black !important;"> -0.09 </td>
  </tr>
</tbody>
</table>
---

# .pull.center[Selecting Peer Firms]

-  Ordinary Least Squares

`$$r_{it} \equiv X_i \beta + \zeta_{it}$$`
`$$\widehat{\beta} \equiv \text{argmin}_\beta \sum_{i = 1}^N (r_{it} - X_t \beta)^2$$`
--

- Regularized Regression (Elastic Net - ENR)

`$$\widehat{\beta} \equiv \text{argmin}_\beta \sum_{i = 1}^N (r_{it} - X_t \beta)^2 + \lambda_2 \beta^2 + \lambda_1 |\beta|$$`

---
# .center.pull[Basic Linear Specification]

<br>

`$$r_{it} = \alpha + \beta Mkt_t + \gamma Peer_t + \zeta_{it}$$`
--

- This is estimated on a `$t = 1:250$` "estimation window"

- Fitted coefficients used out of sample

**Included Regressors**

- **(All)** Market Index (SP500)

- **(Some)** Peer Firm *Index*

- **(Some)** Peer Firms Individually

- **(Some)** Fama-French/Carhart (FFC)

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# .center.pull[Approaches]

1. OLS with various covariate sets
  - MM, MMPI

2. Quantile regression models

3. Regularized models (ENR) and non-linear regression

4. Data driven selection of peer firms

5. Alternative cross-validation methods

6. Inclusion of FFC Factors

---

# .pull.center[Simulation Exercise]

1. CRSP universe 2009 - 2019 with typical restrictions.

4. Randomly select `$b$` = 10,000 stock-date combinations ("event dates").

5. Select peers firms based on SIC industry.

6. Estimate 32 specifications with 250-date estimation window.

7. Calculate the event effect `$\widehat{\zeta}_{it}^k$` for each date/model combination.

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`$\hspace{2cm}$`

# Test Statistic 1

`$$\Large \widehat{R}_{het}^k = \frac{1}{10,000} \sum_{b = 1}^{10,000} \left( \frac{(\widehat{\zeta}_b^k)^2}{\widehat{MSE}_b^{MM}} \right)$$`
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# Test Statistic 1

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# Test Statistic 1

<img src="JLFA_Slides_files/figure-html/unnamed-chunk-12-1.png" width="864" />
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# Test Statistic 1

<img src="JLFA_Slides_files/figure-html/unnamed-chunk-13-1.png" width="864" />
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# Test Statistic 1

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`$\hspace{2cm}$`

# Test Statistic 2

`$$\Large \widehat{R}_{oos}^k \equiv \frac{\widehat{MSE}_{oos}^k}{\widehat{MSE}_{oos}^{MM}} \equiv$$` 
--

`$$\Large \frac{\frac{1}{10,000} \sum_{b = 1}^{10,000} \left(\widehat{\zeta}_b^k\right)^2}{\frac{1}{10,000} \sum_{b = 1}^{10,000} \left(\widehat{\zeta}_b^{MM}\right)^2}$$`

---
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# Test Statistic 2

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# Significance Tests (*t* Critical Values)

`$$\widehat{T}_b^k \equiv \frac{\widehat{\zeta}_{b, 251}^k}{RMSE_b^k} \leq z_{\alpha} \approx -1.28$$`
--
`$$RMSE_b^k \equiv \sqrt{\frac{\sum_{t = 1}^{250} \left(\widehat{\zeta}_{bt}^k \right)^2}{250}}$$`
--

`$$r_{b, 251}^{\delta} = r_{b, 251} - \delta$$`
`$$\delta \in \{0.01, 0.02, 0.03, 0.05, 0.10 \}$$`

---
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# Significance Tests (*t* Critical Values)

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# .center.pull[Significance Tests (SQ Test)]

<br>

- Parametric tests are underpowered and have distorted size.

<br>

- Use SQ Test.

<br>

- Sample .10 quantile is 25th most negative `$\widehat{\zeta}_{it}^k$` in estimation window.

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# Significance Tests (SQ Test)

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# Bias

---
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# Bias

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# .pull.center[Next Steps]

`$\hspace{2cm}$`

1. Extend analysis to financial crisis period.

2. Multiple firm, same date regressions.

3. Test whether the magnitudes of these differences are material.