Declaration of Professor Steven Grenadier

Summary

Exhibit 1 to Document 583 in In re: January 2021 Short Squeeze Trading Litigation, Case No. 21-2989-MDL-ALTONAGA/Damian (Case 1:21-md-02989-CMA, Document 583-1), entered on the FLSD docket June 28, 2023: the 16-page "Declaration of Professor Steven Grenadier In Response to the Declaration of Dr. Adam Werner In Response to Defendants' Daubert Motion" dated June 28, 2023. Grenadier gives Opinions 1, 2 and 3: that Werner's new test does not rebut the Grenadier Report's showing that price movements in the week before the proposed Class Period were highly abnormal; that Werner's definition of "news" does not identify new, value-relevant information; and that Werner's binomial test is unreliable. A closing Fisher's Exact Test on January 21-27, 2021 finds 8 of 14 news days and 13 of 21 non-news days statistically significant, with a p-value of 100.00%.

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Full text

Exhibit 1
Case 1:21-md-02989-CMA   Document 583-1   Entered on FLSD Docket 06/28/2023   Page 1 of 16

UNITED STATES DISTRICT COURT
SOUTHERN DISTRICT OF FLORIDA

CASE NO. 21-2989-MDL-ALTONAGA/Damian

In re: JANUARY 2021 SHORT SQUEEZE
TRADING LITIGATION

Declaration of Professor Steven Grenadier
In Response to the Declaration of Dr. Adam Werner
In Response to Defendants’ Daubert Motion

June 28, 2023
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i
Table of Contents

I.
Introduction ..........................................................................................................................1
II.
Opinion 1:  Dr. Werner Fails to Show That the Market Prices for the At-Issue Stocks in
the Week Before the Proposed Class Period Behaved Similarly to the Rest of the
Previous Year and Therefore Does Not Rebut the Statistical Tests in the Grenadier
Report Showing That the Price Movements in the Week Prior to the Proposed Class
Period Were Highly Abnormal ............................................................................................3
A.
The Grenadier Report Provides Statistical Tests That Demonstrate the Extreme
Price Movements for the At-Issue Stocks and Structural Breaks in the
Relationships Between the Returns of the At-Issue Stocks and the Market and
Industry Returns Starting the Week Prior to the Proposed Class Period ................ 3
B.
Dr. Werner’s Breusch-Pagan Test for Heteroskedasticity Cannot Test Whether It
Is Appropriate to Assess Market Efficiency by Examining the Year Prior to the
Start of the Proposed Class Period .......................................................................... 5
III.
Opinion 2: Dr. Werner Fails to Address That His Definition of “News” Does Not Identify
New, Value-Relevant Information That Might Cause an Investor to Revise Expectations
of a Company’s Cash Flows or the Risks to Those Cash Flows .........................................8
A.
Dr. Werner’s Assertion That His Definition of “News” Is a “More Objective
Selection Methodology” Does Not Demonstrate That It Is Helpful in Identifying
New, Value-Relevant Information .......................................................................... 8
B.
Dr. Werner’s Analysis in Table 2 Is Not Reliable As He Does Not Provide Any
Criteria for His New “Adjustments” ....................................................................... 8
IV.
Opinion 3:  Dr. Werner’s Binomial Test Is Unreliable Even Under His Own Stated
Methodology Because It Fails to Test If the Stock Price Reacted Differently on Days with
New, Potentially Value-Relevant Information Compared to Days without New,
Potentially Value-Relevant Information ..............................................................................9
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I.
Introduction
1.
I have been asked by counsel for Robinhood Markets, Inc., Robinhood Financial LLC,
and Robinhood Securities, LLC (together, “Robinhood,” “the Company,” or “Defendants”) to
assess certain issues concerning market efficiency and a potential class-wide damages
methodology in this litigation.  I submitted an expert report on February 16, 2023 (the
“Grenadier Report”).1  I responded to the Declaration of Adam Werner dated February 16, 2023
(“Werner Declaration”)2 in a report submitted on March 28, 2023 (the “Grenadier Rebuttal
Report”).3  On March 28, 2023, Dr. Werner submitted a report responding to the Grenadier
Report (“Werner Rebuttal Report”),4 and on June 21, 2023, Dr. Werner submitted a declaration
in response to Defendants’ Daubert Motion (“Werner Daubert Declaration”).5
2.
Counsel for Defendants has asked me to evaluate and respond to opinions put forth in the
Werner Daubert Declaration.6, 7  Based on my review of the Werner Daubert Declaration, I
conclude the following:
a. In the Grenadier Rebuttal Report, I explained why examining market efficiency
for the full year prior to the proposed Class Period is unreliable, given the highly
unusual “meme stock” episode that occurred shortly before the start of the
proposed Class Period.  As a part of my analysis, I conducted a number of
statistical tests showing that the pattern of stock price movements starting the

1 All references to the “Grenadier Report” are to my corrected report filed February 24, 2023 (Corrected Expert
Report of Professor Steven Grenadier, In re: January 2021 Short Squeeze Trading Litigation, February 24, 2023).
2 Declaration of Dr. Adam Werner, In re: January 2021 Short Squeeze Trading Litigation, February 16, 2023.
3 Rebuttal Expert Report of Professor Steven Grenadier, In re: January 2021 Short Squeeze Trading Litigation, March
28, 2023.
4 Rebuttal Report of Dr. Adam Werner, In re: January 2021 Short Squeeze Trading Litigation, March 28, 2023.
5 Declaration of Dr. Adam Werner in Response to Defendants’ Daubert Motion, In re: January 2021 Short Squeeze
Trading Litigation, June 21, 2023.
6 My work in this matter is ongoing.  The opinions presented in this report are the result of the information available to
me as of the report date.  I reserve the right to supplement or modify my opinions if new information comes to light
and to respond to any additional report(s) or opinions offered by other experts.  I am being compensated at my
standard billing rate of $1,250 per hour.  I have been assisted in this matter by staff of Cornerstone Research, who
worked under my direction.  I receive compensation from Cornerstone Research based on its collected staff billings
for its support of me in this matter.  Neither my compensation in this matter nor my compensation from Cornerstone
Research is in any way contingent or based on the content of my opinion or the outcome of this or any other matter.
7 In addition to the documents listed in Appendix C of the Grenadier Report and Appendix A of the Grenadier Rebuttal
Report, I considered the following additional documents when forming my opinions in this report: Werner Rebuttal
Report; Werner Daubert Declaration; J. Stock and M. Watson (2015), Introduction to Econometrics, 3rd ed., Boston:
Pearson Education, Inc; T.S. Breusch; A.R. Pagan (1979), “A Simple Test for Heteroscedasticity and Random
Coefficient Variation,” Econometrica, 47(5), pp. 1287–1294; and A. King et al. (2017), “Econometric Analysis” in
Litigation Services Handbook: The Role of the Financial Expert, R. L. Weil, D. G. Lentz and E. A. Evans, 6th ed.,
Hoboken, NJ: John Wiley & Sons, Inc.
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week prior to the proposed Class Period was highly abnormal.  Dr. Werner
presents a test in the Werner Daubert Declaration that does not address changes
over time and therefore does not rebut the results presented in the Grenadier
Report.
b. Dr. Werner’s claim that the definition of “news” used in his report is “more
objective,” even if true, would not make his tests appropriate or reliable, as his
definition is unhelpful in identifying new, value-relevant information that is
critical to assess the “cause and effect” relationship under the fifth Cammer factor
that he purports to test.  Furthermore, in response to issues raised in the Grenadier
Rebuttal Report, Dr. Werner’s new analysis in Table 2 of his Daubert Declaration
purports to filter out “articles with potentially unrelated news” and “discussions of
price movements only.”  However, he does not provide any criteria or
methodology that would be required for one to evaluate or replicate his analysis.
c. Dr. Werner’s binomial test is unreliable even under his own stated methodology
because it fails to test if the stock price reacted differently on days with new,
potentially value-relevant information compared to days without new, potentially
value-relevant information.  When I tested if there was a difference in the
frequency of statistically significant residual returns between “news” days and
“non-news” days8 in the Grenadier Report Appendix E, I found that there was no
statistically significant difference.

8 Note that Dr. Werner mischaracterizes the identification of “news” days in my report.  I identify potentially value-
relevant pre-Class Period news in Appendix E of my report, which he treats as a “news” day.  For the purposes of my
Opinion 3, I use Dr. Werner’s definition of “news” and “non-news” days, despite my disagreement with this
categorization.
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II.
Opinion 1:  Dr. Werner Fails to Show That the Market Prices for the At-Issue
Stocks in the Week Before the Proposed Class Period Behaved Similarly to the Rest
of the Previous Year and Therefore Does Not Rebut the Statistical Tests in the
Grenadier Report Showing That the Price Movements in the Week Prior to the
Proposed Class Period Were Highly Abnormal
A.
The Grenadier Report Provides Statistical Tests That Demonstrate the
Extreme Price Movements for the At-Issue Stocks and Structural Breaks in
the Relationships Between the Returns of the At-Issue Stocks and the Market
and Industry Returns Starting the Week Prior to the Proposed Class Period
3.
In the Werner Declaration, Dr. Werner examines market efficiency of the At-Issue Stocks
over a full year before the beginning of the proposed Class Period.  As discussed in the
Grenadier Rebuttal Report, such an approach is unreliable because it masks the highly unusual
market conditions directly before the proposed Class Period, including evidence of coordinated
trading, short squeezes, attempted short squeezes, and short-sale constraints.9  Dr. Werner
erroneously claims that “Defendants offer zero evidence other than anecdotes and speculation
that there is even a distinction between this so-called ‘general data set’ and the ‘narrower data
set’.”10  Contrary to his claim of “zero evidence,” in the Grenadier Report I performed five
different sets of statistical tests to show such a distinction.
4.
In particular, I examined the extreme nature of the stock price movements leading up to
and, separately, during the proposed Class Period by running statistical tests comparing (i) the
stock price movements and residual stock price movements for the At-Issue Stocks to those of
other stocks on the NYSE/Nasdaq; (ii) the stock price movements for the At-Issue Stocks to their
own historical stock price movements over the past 25 years; and (iii) the stock price movements
of the At-Issue Stocks as a group to the historical performance of a broad set of stocks over the
past 25 years.11, 12  These tests showed that in the week leading up to the proposed Class Period,
the At-Issue Stocks experienced highly unusual returns, including some of the largest returns in
the past 25 years, let alone the past year.

9 Grenadier Rebuttal Report, Section III.C.
10 Werner Daubert Declaration, ¶ 14.
11 Grenadier Report, ¶¶ 74–91, Figures 2–6, Exhibits 3–4, Appendix D.
12 I ran these tests using cumulative returns and additional tests comparing raw price movements for the At-Issue
Stocks to their own historical stock price movements using daily returns (Grenadier Report, ¶¶ 81–91).
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5.
In addition, in a separate set of analyses in the Grenadier Report, I established that there
was a statistically significant structural break in the relationship between the stock returns of the
At-Issue Stocks and market and industry returns when comparing the 120 day period ending
December 31, 2020 and the period of January 21 to February 4, 2021.13  This means that, in the
period starting January 21, 2021, the manner in which the prices of the At-Issue Stocks
fluctuated relative to the market and industry movements changed when compared to the period
before December 31, 2020.
6.
Finally, I established that there was a statistically significant change in the variance of
residual returns when comparing the same two periods.14  To conduct this test, I calculated
residual returns as the difference between returns of each of the At-Issue Stocks and returns
predicted by a regression model using the market and industry indices.  I used the Brown-
Forsythe test to determine whether the variances of residual returns were equal across the two
periods.15  Hypothetical Illustration 1 below shows an example of equal variances across two
time periods while Hypothetical Illustration 2 below shows variances changing across two time
periods.16  Like in Hypothetical Illustration 2, I found statistically significant changes in the
variances of residuals of all the At-Issue Stocks between the 120 day period ending December
31, 2020 and the period of January 21 to February 4, 2021.  This means that, in the period
starting January 21, 2021, the variance of the residual returns for the At-Issue Stocks was not the
same as the variance of the residual returns for the At-Issue Stocks in the period before
December 31, 2020.

13 Grenadier Report, fn. 140.
14 Grenadier Report, fn. 140.
15 Grenadier Report, fn. 140.
16 Illustrations are not based on actual data.
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B.
Dr. Werner’s Breusch-Pagan Test for Heteroskedasticity Cannot Test
Whether It Is Appropriate to Assess Market Efficiency by Examining the
Year Prior to the Start of the Proposed Class Period
7.
Instead of addressing the tests described above, Dr. Werner claims that the Breusch-
Pagan test for heteroskedasticity can test whether the “use of the one-year estimation period prior
to the Class Period is ‘improper’ because it ‘commingles a general data set with a narrower
one.’”17  However, contrary to Dr. Werner’s assertion, his Breusch-Pagan heteroskedasticity test
does not test differences between the one year period over which he claims to test market

17 Werner Daubert Declaration, ¶ 14.
-80%
-60%
-40%
-20%
0%
20%
40%
60%
80%
Residual  Returns
Hypothetical Illustration 1:
Variance of Residual Returns Is EQUAL Across Two Comparison Periods
12/31/20
1/21/21
TIME
-80%
-60%
-40%
-20%
0%
20%
40%
60%
80%
Residual  Returns
Hypothetical Illustration 2:
Variance of Residual Returns Is NOT EQUAL Across Two Comparison Periods
12/31/20
1/21/21
TIME
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efficiency and the narrow “meme stock” period preceding the proposed Class Period, because it
does not compare results across different time periods.18
8.
Dr. Werner’s Breusch-Pagan heteroskedasticity test takes the residual returns from his
regression model and tests whether the variance of these residual returns differs across different
levels of the independent variables, which in Dr. Werner’s model are market and industry
returns.19  Hypothetical Illustrations 3 and 4 below show a graphical representation of
heteroskedasticity, the data pattern that Dr. Werner’s Breusch-Pagan test is supposed to discern.20
As shown in the Hypothetical Illustration 3 below, the variance of the plotted residual returns can
be the same for different levels of market (and industry) returns, resulting in the Breusch-Pagan
test finding of no heteroskedasticity, or different (see Hypothetical Illustration 4), resulting in the
Breusch-Pagan test finding of heteroskedasticity.

18 Note that Dr. Werner incorrectly defines heteroskedasticity by claiming that “[h]eteroskedasticity means that the
residual price movements of a particular stock are not normally distributed” (Werner Daubert Declaration, ¶ 15).  The
term “normally distributed,” refers to the underlying data following a normal distribution, which is based on certain
statistical properties (See J. Stock and M. Watson (2015), Introduction to Econometrics, 3rd ed., Boston: Pearson
Education, Inc., available at https://www.amazon.com/Introduction-Econometrics-3rd-Addison-wesley-
Economics/dp/0138009007/ref=tmm_hrd_swatch_0?, pp. 36–37).  This proposed definition is not a definition of
heteroskedasticity and is not consistent with the definition Dr. Werner quotes from the Litigation Services Handbook
in ¶ 15 of the Werner Daubert Declaration (See A. King et al. (2017), “Econometric Analysis” in Litigation Services
Handbook: The Role of the Financial Expert, R. L. Weil, D. G. Lentz and E. A. Evans, 6th ed., Hoboken, NJ: John
Wiley & Sons, Inc., pp. 1–62 at p. 7).  On the contrary, residual price movements can be heteroskedastic and
normally distributed, just as they can be homoskedastic and not normally distributed.  My criticisms of Dr. Werner’s
application of the Breusch-Pagan test for heteroskedasticity are independent of this definitional error, however.
19 I have replicated the results in Table 1 of the Werner Daubert Declaration and verified that the independent
variables Dr. Werner included in the Breusch-Pagan test in Table 1 were the market and industry returns.  For a
definition of the Breusch-Pagan test, see T.S. Breusch and A.R. Pagan (1979), “A Simple Test for Heteroscedasticity
and Random Coefficient Variation,” Econometrica, 47(5), pp. 1287–1294.
20 Illustrations are not based on actual data.
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III.
Opinion 2: Dr. Werner Fails to Address That His Definition of “News” Does Not
Identify New, Value-Relevant Information That Might Cause an Investor to Revise
Expectations of a Company’s Cash Flows or the Risks to Those Cash Flows
A.
Dr. Werner’s Assertion That His Definition of “News” Is a “More Objective
Selection Methodology” Does Not Demonstrate That It Is Helpful in
Identifying New, Value-Relevant Information
11.
In his opening report, Dr. Werner identified “news” days for his fifth Cammer factor
analysis by obtaining a list of articles from Factiva21 and categorizing days as either in the “high
information flow group” or “low information flow group” based on a count of the number of
articles on each day.22  Dr. Werner conceded that this methodology “captures all news regardless
of whether it is ‘valuation relevant.’”23  In the Werner Daubert Declaration, Dr. Werner attempts
to justify his definition of “news” using Factiva article counts by describing the definition as
being “a more objective selection methodology”24 than “typical collective test designs” used in
his past reports.25  However, a test being “more objective” does not make it appropriate.  As
discussed in the Grenadier Rebuttal Report, this determination of “news” days is fatally flawed
because it does not assess whether there was new, value-relevant information that might cause an
investor to revise expectations of the company’s cash flows or the risks to those cash flows.26
B.
Dr. Werner’s Analysis in Table 2 Is Not Reliable As He Does Not Provide
Any Criteria for His New “Adjustments”
12.
In the Werner Daubert Declaration, Dr. Werner appears to attempt to address issues that I
raised in the Grenadier Rebuttal Report about his inclusion of non-value relevant information in
his definition of “news.”27  Dr. Werner conducts a new analysis in which he claims to “filter[] for

21 Dr. Werner also determines when the articles were published and with what trading day each article would be
associated (Werner Declaration, ¶ 70).
22 Werner Declaration, ¶ 70.
23 Werner Daubert Declaration, ¶ 31.
24 Werner Daubert Declaration, ¶ 28.
25 Werner Daubert Declaration, ¶ 20.
26 This description of what constitutes new, value-relevant information is also consistent with the language Dr. Werner
has used in prior declarations, where he stated: “The efficient market hypothesis forms the foundation of modern
finance theory.  Finance theory holds that the market price of a stock reflects the discounted value of expected future
cash flows to the stockholder.  As a result, new information that causes the market to significantly alter its
expectations of future cash flows will cause a repricing of the security to reflect these new expectations.”  See
Declaration of Dr. Adam Werner, In re: Hi-Crush Partners L.P. Securities Litigation, April 15, 2014, ¶ 13.  Dr. Werner
cites E.F. Fama (1991), “Efficient Capital Markets: II,” Journal of Finance, 46(5), pp. 1575–1617.
27 Grenadier Rebuttal Report, Section III.C.
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‘noise’” and remove “articles with potentially unrelated news, discussions of price movements
only, and duplicate articles discussing the same information.”28  He reruns his “collective test”29
on this updated set of “news” and concludes that this does “not materially change [his]
findings.”30
13.
However, it is not clear from the analysis in Dr. Werner’s Daubert Declaration whether or
how his “adjustment” addressed my criticisms in the Grenadier Rebuttal Report, because he did
not provide any description of how his new methodology (which would necessarily require some
subjective decision-making) was implemented.  To identify and filter out “articles with
potentially unrelated news, discussions of price movements only, and duplicate articles
discussing the same information,” Dr. Werner presumably would have had to review the content
of articles and decide which articles to exclude.31  Unlike the documentation I provided in my
analysis in the Grenadier Report,32 Dr. Werner does not provide any criteria for how he filtered
his articles or how he defined “potentially unrelated news.”
14.
Dr. Werner’s classification of what constitutes “unrelated news” and “discussions of
price movements only” is particularly relevant given the media coverage of the “meme stock”
episode in the period immediately prior to the proposed Class Period.  For example, Dr. Werner
does not explain how he treated articles with information about retail investors coordinating over
social media, short squeezes, attempted short squeezes, or short-sale constraints in his “filtering
of noise.”33  As discussed in my reports, in an efficient market, these types of information do not
represent value-relevant information that might cause an investor to revise expectations of the
company’s cash flows or the risks to those cash flows.34
IV.
Opinion 3:  Dr. Werner’s Binomial Test Is Unreliable Even Under His Own Stated
Methodology Because It Fails to Test If the Stock Price Reacted Differently on Days

28 Werner Daubert Declaration, ¶ 33.
29 Dr. Werner “employ[s] a Fisher’s exact test to compare the frequency of statistically significant stock price reactions
on high information flow days as compared to lesser information flow days” (Werner Declaration, ¶ 75).
30 Werner Daubert Declaration, ¶ 32.
31 Werner Daubert Declaration, ¶ 33.  I understand that Factiva has an option to exclude duplicate articles in
searches, but I am not aware of an automated approach to identify articles with “potentially unrelated news” or
“discussions of price movements only.”
32 See Grenadier Report, Appendix E and Appendix F.
33 Werner Daubert Declaration, ¶ 32.
34 Grenadier Rebuttal Report, ¶44.
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with New, Potentially Value-Relevant Information Compared to Days without New,
Potentially Value-Relevant Information
15.
In the Werner Daubert Declaration, Dr. Werner attempts to justify the binomial test
analysis he performed in the Werner Rebuttal Report.35  Dr. Werner claims that he provided a
“testable hypothesis” and tested that hypothesis with the binomial test.36  The binomial test Dr.
Werner performed combines days on which Prof. Fischel and I identified potentially value-
relevant information into a set of what he calls “news” days.37  Based on his binomial test, Dr.
Werner concluded that he could “reject the null hypothesis that the companies’ security prices
behave no differently on days with company-specific news than on days without company-
specific news.”38  However, the binomial test conducted by Dr. Werner does not and cannot
assess whether the At-Issue Stocks reacted differently on “news” days than on “non-news” days,
as price reactions on days without any “news” are not considered in the test at all.  To
demonstrate this limitation, if there were statistically significant residual returns for a stock on
every day of the period examined by Dr. Werner, and on half of those days there was “news” and
on half there was not, using the binomial test Dr. Werner would conclude that the market for that
stock was efficient even though it reacted equally on days with and without “news.”
16.
This flaw in Dr. Werner’s use of a binomial test is highlighted by Dr. Werner’s own
statements emphasizing the importance of comparing the frequency of statistically significant
returns on days with and without “news,” which the binomial test does not do.39  In order to
demonstrate this flaw, in considering the Werner Rebuttal Report in advance of my deposition, I
performed an alternative version of Dr. Werner’s test using the “non-news” days in Appendix E
of the Grenadier Report, and tested whether “the frequency of statistically significant price
movements on days [without] Company-specific news is too unlikely to be considered

35 Werner Daubert Declaration, Section III.C and Section III.D.
36 Werner Daubert Declaration, ¶ 50.
37 Note that Dr. Werner mischaracterizes the identification of “news” days in my report.  I identify potentially value-
relevant pre-Class Period news in Appendix E of my report, which he treats as a “news” day.  For the purposes of my
Opinion 3, I use Dr. Werner’s definition of “news” and “non-news” days, despite my disagreement with this
categorization.
38 Werner Rebuttal Report, ¶ 51.
39 For example, Dr. Werner states that an appropriate test “determines how frequently the security price exhibits a
statistically significant response on ‘news’ days and ‘non-news’ days” and that “[b]ased on these frequencies, a
statistical test is conducted to determine whether there is a statistically significant difference between the reaction
frequency of ‘news’ dates and ‘non-news’ dates” (Werner Daubert Declaration, ¶ 38).
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random.”40  I understand this was provided to Plaintiffs’ counsel.  I found that four of the seven
At-Issue Stocks (BB, BBBY, EXPR, GME) had too many statistically significant price
movements on “non-news” days to be considered random.  In summary, an alternative version of
Dr. Werner’s own test shows that “non-news” days (on which there was no new, potentially
value-relevant information) are associated with statistically significant stock price movements,
which is inconsistent with market efficiency.41
17.
In order to further demonstrate the flaw in Dr. Werner’s binomial test, I also previously
ran a test to compare the frequency of statistically significant residual returns on “news” days
and “non-news days.”42  To address Dr. Werner’s concern of a small sample size,43 I analyzed the
seven At-Issue Stocks (for which Dr. Werner claimed that the market was efficient) in a single
test to demonstrate the stock prices of the At-Issue Stocks collectively did not react more
frequently on “news” days than “non-news” days.  I found that there was no statistically
significant difference between the two sets of days, and in fact, days without new, potentially
value-relevant information were more likely to have statistically significant residual returns than
days with new, potentially value-relevant information.44  In other words, days on which I identify
new, potentially value-relevant information are no more likely to experience statistically
significant residual returns than days on which I do not identify new, potentially value-relevant
information.45
Executed this 28th of June, 2023

___________________________________

Steven Grenadier, Ph.D.

40 Werner Rebuttal Report, ¶ 51.  I understand that Plaintiffs were provided this analysis prior to my deposition in
response to Plaintiffs’ notice of my deposition, in which they requested documents that I "considered ... in forming
[my] opinions and findings.” I have included it as Exhibit 1.
41 Grenadier Report, ¶ 64.
42 I ran a Fisher exact test, which Dr. Werner used in the Werner Declaration in order to conduct his “news vs. no-
news test” (Werner Declaration, ¶¶ 68–75).
43 In Dr. Werner’s Daubert Declaration, he stated that “it is generally advised for scientific studies to examine at least
30 observations so that the size of the sample is large enough to lead to robust conclusions” (Werner Daubert
Declaration, ¶ 47).
44 The Fisher test found that 62% of days without new, potentially value-relevant information had statistically
significant returns, while 57% of days with new, potentially value-relevant information had statistically significant
returns.
45 I understand that Plaintiffs were provided this analysis prior to my deposition, and I have included it as Exhibit 2.
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Applying Dr. Werner’s Binomial Test Using Days without News
with New, Potentially Value-Relevant Information[1]
1/21/21 – 1/27/21
Affected
Company
Days without News with
New, Potentially Value-
Relevant Information
Statistically Significant Days without
News with New, Potentially Value-
Relevant Information[2]
Binomial Test
P-Value[3]
Cumulative Distribution
P-Value[4]
AMC
1
0
95.00%
100.00%
BB
3
2
0.71%
0.73%
BBBY
2
2
0.25%
0.25%
EXPR
5
4
0.00%
0.00%
GME
5
4
0.00%
0.00%
NOK
1
0
95.00%
100.00%
TRVG
4
1
17.15%
18.55%
Source:  Grenadier Report, Figure 7;  Werner Rebuttal Report, Exhibit 4, Table 7
Note:
[1] "Days without News with New, Potentially Value-Relevant Information" refer to days that are not marked as "News Identified in
Grenadier Report" in the Werner Rebuttal Report, Exhibit 4.
[2] Significant days are based on the Grenadier Report.  The significant days in Table 7 in the Werner Rebuttal Report match the
significant days in the Grenadier Report.
[3] The "Binomial Test P-Value" tests the probability that exactly "X" days are significant out of a total of "Y" days.  Dr. Werner includes this
test in Table 7 of his rebuttal report.
[4] The "Cumulative Distribution" tests the probability that at least "X" days are significant out of a total of "Y" days.
Analysis in Response to Werner Rebuttal Report Table-7
Exhibit 1
Case 1:21-md-02989-CMA   Document 583-1   Entered on FLSD Docket 06/28/2023   Page 15 of
16

Applying a Fisher Exact Test Using Days with and without
News with New, Potentially Value-Relevant Information[1]
1/21/21 – 1/27/21
Days with News with New, Potentially Value-Relevant Information
Days without News with New, Potentially Value-Relevant Information
Affected
Companies
Total Days
Statistically
Significant Days[2]
% of
Significant Days
Total Days
Statistically
Significant Days[2]
% of
Significant Days
Fisher's
Exact Test
P-Value[3]
AMC, BB, BBBY, EXPR,
GME, NOK, TRVG
14
8
57.14%
21
13
61.90%
100.00%
Source:  Grenadier Report, Figure 7;  Werner Rebuttal Report, Exhibit 4, Table 7
Note:
[1]  "Days with and without News with New, Potentially Value-Relevant Information" refer to days that are marked (or are not marked) as "News Identified in Grenadier Report" in the Werner Rebuttal Report, Exhibit 4.
[2]  Significant days are based on the Grenadier Report.  The significant days in Table 7 in the Werner Rebuttal Report match the significant days in the Grenadier Report.
[3]  Fisher's Exact Test p-values are statistically significant at the 95% confidence level if they are less than 5%.
Analysis in Response to Werner Rebuttal Report Table-7
Exhibit 2
Case 1:21-md-02989-CMA   Document 583-1   Entered on FLSD Docket 06/28/2023   Page 16 of
16

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