Case Western Reserve University
Weatherhead School of Management

MSFI 435 – Empirical Finance
Fall 2022

Assignment 8
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Part I - Concepts
1. Let’s reexamine the concepts of type I and type II error in a hypothesis testing
framework.
Recall from our discussion in section 6.5 that a type I error occurs when the null hypothesis
is incorrectly rejected; that is, the null is correct but is rejected in the test.
In the context of a hypothesis testing for a regression coefficient
H0: βk = βk,0
Ha: βk ≠ βk,0
we’ve seen (see section 6.4) that the t-statistic for such test,

follows a standard normal distribution, t~N(0,1). Notice that the t-statistic depends on the
null hypothesis, so we can say if the null is correct, t will follow a standard normal. If we
set a significance level α=0.05, we establish a rejection region as shown in the shaded area
of Figure 1. This means there is a 5% chance of the t-statistic falling in the rejection region,
or a 5% chance of rejecting the null when it is in fact true. Thus, the probability of a type I
error is the significance level that one establishes for the hypothesis testing.

Figure 1. Rejection region, α=0.05

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ββˆ
t
k
,kk 0
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We drew parallels between the justice system and hypothesis test. The system works as a
test of hypothesis in which the null is that the defendant is not guilty, or:
H0: Defendant not guilty
Ha: Defendant is guilty
a) Now, (finally!) a question for you. If one sets a significance level for testing the
defendant hypothesis equal to α=0.1, and the system handles 150 innocent defendants per
year, how many of them will be deemed guilty on average per year?
b) The American judicial system is comprised of two different types of cases: civil and
criminal. Criminal cases are harsher and can take you to jail; civil cases, on the other hand,
usually bring penalties in terms of monetary compensation. According to the Findlaw
website, “criminal offenses and civil offenses are generally different in terms of their
punishment. Criminal cases will have jail time as a potential punishment, whereas civil
cases generally only result in monetary damages or orders to do or not do something. Note
that a criminal case may involve both jail time and monetary punishments in the form of
fines.”
Still according to Findlaw website, “The standard of proof is also very different in a
criminal case versus a civil case. Crimes must generally be proved "beyond a reasonable
doubt", whereas civil cases are proved by lower standards of proof such as "the
preponderance of the evidence" (which essentially means that it was more likely than not
that something occurred in a certain way). The difference in standards exists because civil
liability is considered less blameworthy and because the punishments are less severe.”
What can you say regarding the probability of type I error that the judicial system needs to
impose on criminal vs. civil offenses? Which type of case should have a lower significance
level? Please explain.
c) We now turn to the type II error. A type II error occurs when the null hypothesis is false
but we fail to reject it.

Figure 2. Acceptance region, α=0.05
To compute the probability of a type II error, we need to know the alternative hypothesis
that holds when the null hypothesis is false. Take our previous example that under the null
the t-statistic for a test on the coefficient of a regression follows a standard normal. We will
fail to reject the null if the t-statistic falls in the shaded region in Figure 2.
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Now assume we know the null is false, and that in fact the t-statistic follows a normal
distribution with µ=1 and σ=1, or t~N(1,1). Then we can compute the probability of a type
II error by getting Pr(–1.96
Figure 3. Computing the probability of a type II error
Compute the probability in the shaded area of Figure 3. (Recall you can use the
NORM.DIST function in Excel.)
d) There is a trade-off between the probability of getting a type I error and the probability
of getting a type II error. Reducing the probability of getting a type I error means reducing
the significance level of the test, but that comes at the expense of increasing the probability
of getting a type II error.
Let’s see that in our example in part c). Suppose you reduce your significance level from
α=0.05 to α=0.01, so that the new cutoff values for the rejection region in Figure 2 (and for
the acceptance region in Figure 3) become –2.57 and +2.57. Recompute the probability of
a type II error in part c) and verify that it has increased from the number you got in part c).
2) We will examine a phenomenon that came to be known by academics and investment
advisors as the post–earnings announcement drift (PEAD). The PEAD—as defined in
Wikipedia—is the “tendency for a stock’s cumulative abnormal returns to drift in the
direction of an earnings surprise for several weeks (even several months) following an
earnings announcement.”
We all know and tested how investors strongly react to the news in earnings announcement,
with event reactions (let’s say, cumulative abnormal returns over the announcement day
and the day after the announcement) being positive when the earnings carry good news and
negative when the earnings carry bad news.
The PEAD implies that, despite this strong event reaction, investors do not react enough!
Consequently, abnormal returns still correlate with the type of news in the earnings
announcement long after the earnings announcement date. For example, if earnings carry
good news, then abnormal returns will keep on being positive on average for weeks after
the earnings announcement. Therefore, PEAD exists when the surprise at the earnings
announcement and the post-earnings abnormal returns are positively correlated.
In order to test for the possible manifestation of PEAD, a sample of quarterly earnings
announcements is collected, and the following regression model is run:
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Here is a brief description of the variables measured for each earnings announcement:
 CAR_2_60: the cumulative daily abnormal return from day +2 to day +60, where
day=0 is the earnings announcement date (that is, from 2 trading days after the
announcement to 60 trading days after the announcement). Daily abnormal return
is defined as the firm’s stock return minus the market return in that day.
 SURPRISE: the gap between the actual earnings and the analyst’s expectation
about the earnings right before the announcement date.
 MOM: the average daily return of the firm announcing earnings, where the average
is computed over the 60 calendar days preceding the earnings announcement (that
is, from calendar day –60 to calendar day –1, where day 0 is the earnings
announcement date).
 LASSETS: the logarithm of total assets, where total assets is measured in the year
prior to the earnings announcement date.
 BEME: the ratio of book value of equity to market value of equity (variable ME),
both measured in the year prior to the earnings announcement date. Both BE and
ME appear in the accounting dataset.
The sample used in this study appears in Figure 4.

Figure 4: Sample used in the study

Take for example, the first observation of the sample. It is for IBM (PERMNO=12490)
and for the earnings released in April 16th, 2003. For this observation, the post-earnings
abnormal returns for IBM was 3.66% (CAR_2_60=3.66); the earnings surprise was 10
cents per share (SURPRISE=0.10); IBM’s performance prior to the earnings
announcement—measured from the 60th day prior to April 16th, 2003 through the day
before April 16th, 2003—was an average daily return of 0.023% (MOM=0.023); IBM’s
total assets, measured in 2002, was $113 billion (LASSETS=log(146,000,000,000)=25.7);
and book-to-market ratio—also measured in 2002—was 0.8 (BEME=0.8).

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The results of the regression testing for PEAD appear in Figure 5. Recall that the dependent
variable is CAR_2_60.



Figure 5: OLS regression results

For each coefficient, we show the coefficient’s estimate, its standard error, the t-statistic,
and the p-value. (Some of these data can be redundant; for example, remember that the p-
value is obtained from the value of the t-stat.)
Based on these results please answer the following questions.
a) Is PEAD happening in the sample examined in the study? Formulate your hypothesis
clearly. Then, assess the effect of earnings surprise on the firm performance in the post-
earnings period.
b) Interpret the coefficient on LASSETS. Does total assets matter for the post-earnings
returns? How so?
c) Your colleague argues that it is irrelevant to control for LASSETS in the regression
model—that is, that you should remove this variable from the model—since you are
interested mostly on the effect of SURPRISE. Please explain some conditions under
which removing LASSETS from the model makes the inferences on SURPRISE
problematic.
d) Suppose you observe a specific earnings announcement, with the following values of
the explanatory variables: SURPRISE is 4 cents per share, MOM is 0.011%, total assets
measured the year before earnings is $46 billion, and book-to-market measure is 0.50.
What is the model’s predicted value for the post-earnings abnormal returns?
e) Construct a 95% confidence interval for the coefficient on MOM. Then use the
confidence interval to conclude whether past performance (MOM) is related to post-
earnings abnormal returns.
f) Suppose you want to test the hypothesis that the PEAD phenomenon is exacerbated for
the sample of firm in the tech industry. Explain how you would proceed (for example,
somehow expanding the regression model).
You can assume you have data that allows you to create a dummy variable, TECH, that
is equal to one if the firm announcing earnings belongs to the tech industry.
g) After running the regression model whose results appear in Figure 5, you were shown
the following pattern the errors in the regression.
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Figure 6: Errors from regression model explaining firm performance

What does that illustrate as a problem with respect to the assumptions required to run a
regression model? If this is indeed a problem, what procedure can you use to remedy the
problem?
Part II – Exploring Executive Compensation (Continued)
We will continue the exploration of the topic of executive compensation carried out in
module 9.
Your starting point is a modified version of the regression explaining cashpay (which
equals salary plus bonus). The model is

which resembles a mix of the regressions shown in columns 3 and 4 of Figure 9-4, but with
the caveat that we are not controlling for the LMB here. This is the starting point for many
of the models used in this assignment.
Also, we are using the same time period as in module 9. So, you should download the code
used in module 9, work on the necessary adjustments of library locations, etc, and then run
a regression as shown above. (To make sure your code is running properly, you may
compare the results of the regressions in that code with the ones shown in Figure 9-4. You
do not need to show this comparison, though, in your write-up. Also be aware of very tiny
differences between your new results and the ones printed in the lecture notes. The results
in the lecture notes were generated a few years ago, and there might have been updates in
the WRDS database since then.)
Please run the regression model above: generate and show the results of the regression in
a formatted output similar to the ones we have used in past assignments (that is, an output
showing the estimated coefficients, their t-statistics, then then number of observations in
the regression, and the adjusted-R2).
Note: This is one of those rare assignments when you do not need to show summary
statistics. This is because the assignment is deemed a continuation of module 9.
-15
-10
-5
0
5
10
15
20
25
30
-6 -4 -2 0 2 4 6
Residual
Predicted
CAR_2_60
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Let’s first examine the residuals:.
For the model shown above, generate the residuals. Then test whether the residuals are
different from zero. That is, treat the residual as a random variable and test whether the
expected value of the residuals is equal to zero (your null). For this you can run a PROC
MEANS or PROC TTEST on the residual variable.
Let’s now examine the robustness of our results to the concerns regarding
heteroscedasticity.
For the model shown above, generate a plot of residuals vs. predicted value of CASHPAY.
Does heteroscedasticity stand out visually from the graph?
Then, run the regression with the White test for heteroscedasticity. Does the test reject the
null of homoscedasticity?
If so, rerun the regression asking for t-statistics (and standard errors) that are robust to
heteroscedasticity. Create and show a new output for the regression results, and comment
on whether the inferences are robust to this change. That is, for each hypothesis testing on
a regression coefficient, does the inference you had before change after the correction for
heteroscedasticity?)
We then examine the robustness of our results to the concerns regarding correlation of
residuals.
First, make sure the data is sorted by the company identifier, that is, first run the code
proc sort data=d;
by permno;
run;
then run the regression asking for the Durbin-Watson test for serial correlation and
comment whether serial correlation seems to be a problem for this regression specification.
Now test for the Durbin-Watson again, but, before running the regression, sort the data by
BONUS, that is, run the statement:
proc sort data=d;
by bonus;
run;
Finally, test for the Durbin-Watson again after sorting the data by the STOCKOPTION
variable, or:
proc sort data=d;
by stockoption;
run;
Comment on the results you get for the Durbin-Watson test. (Hint: this is a test about
correlation of consecutive residuals, so it depends on sorting the data…)
Let’s now explore collinearity.
You will add to your dataset another measure of firm size, defined as the log of market
value of equity as of December of two years ago. (Since past performance is measured over
FNCE 435 Fall 2022 Assignment 8
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the previous years, if we collected firm size as of the end of the previous year we would by
construction create a correlation between firm size and past performance.) Since we have
compensation data for year 2000, you need to collect market value of equity as of
December 1998. Market value of equity is defined as the price of each stock (variable PRC
in the MSF dataset on CRSP) times the number of shares outstanding (variable SHROUT
in the dataset MSF on CRSP). Also, due to the fact that sometimes CRSP records stock
price as a negative number, you need to use the absolute value of the stock price measure.
Thus, if PRC and SHROUT are collected, you define LMVE as
log(abs(prc)*shrout)
where ABS(a) is a SAS function that returns the absolute value of a.
Then run the 4 regressions shown in Figure 7 below. That is, run regressions separately
with LSALE, then LAT, then LMVE; then add them all in the same specification.

Figure 7. Regression results

Generate and show Figure 7 with the regression results. Interpret the effect of LMVE on
cashpay.
Do you see any issues with the inferences, regarding the proxies for firm size (LSALE,
LAT, and LMVE)? Can you get a hint of what is going on? You can use a PROC CORR
to help you in your analysis.
Our last task is to examine predictions.
Using the model for cashpay as

first compute the predicted cashpay for AT&T (PERMNO=10401) in year 2000. That is,
write down the 95% confidence interval for the cashpay of a CEO of a specific firm exactly
like AT&T.
Second, compute the 95% confidence interval for the mean cashpay across all firms that
look like AT&T.
Discuss the two confidence intervals, and why they are different.
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