With, μτ = E[CARτ], is the expected value of the cumulative abnormal returns over the period of length τ.
The generalized rank t-statistic (GRANK-T), with T-2 degrees of freedom, for testing hypothesis (19) is defined as:
𝑡
𝑔𝑟𝑎𝑛𝑘= 𝑍(
𝑇−2𝑇−1−𝑍2
)
1⁄2(
20)
Where,
𝑍 =
𝑈̅0𝑆𝑈̅
(
21)
With,
𝑆
𝑈̅= √
1𝑇
∑
𝑛𝑡𝑛
𝑈 ̅
𝑡2𝑡∈
(
22)
&
𝑈 ̅
𝑡=
1𝑛𝑡
∑
𝑛𝑖=1𝑡𝑈
𝑖𝑡(
23)
𝑛𝑡is the number of valid 𝐺𝑆𝐴𝑅𝑖𝑡, available at time point 𝑡, with 𝑡 ∈ T, T = {T0+1,…,T1,0}.
T=T1-T0+1 equals to the numbers of observations (estimation window length +1 observation).
𝑈 ̅
is the mean𝑈 ̅
𝑡 at time t= 0 (the cumulative abnormal return).This testing methodology allows the GRANK procedure to outperform previous rank tests of CARs and was robust to abnormal return serial correlation and event-induced
volatility (Kolari & Pynnonen, 2011). Additionally, the GRANK procedure exhibits superior empirical power relative to popular parametric tests.
4.1. Data
Table I.
Data sources
This table summarizes which variable is represented by which factor, and where the data is sourced from:
Seven variables are necessary to conduct the event study, six are presented in Table 1. The seventh is the date variable, which is attributed to all other variables, as they are all of the time aggregation type. The definitions of the presented variables stay consistent with the literature, except if mentioned otherwise. The Market return is defined as the S&P500 return, as all firms in the sample are US-based. The Kenneth R. French database provides the SMB and HML factors. The individual stock’s return was found in the CRSP Daily Stock databank. In order to make prices and returns easier to interpret all stocks that experienced stock splits are adjusted to a base value. The base price was defined at the stock price within the estimation window. If a stock had a “1 to 2 split” after the estimation window all prices after the split were multiplied by two. This avoids including daily returns that are unrepresentative of the actual market value change. The number of shares shorted is found through the ‘Shares held short as of Settlement Date – Adjusted’ variable provided by the COMPUSTAT database. This variable is defined as the number of shares held short on the 15th and final business day of each month. It gives insight on the number of short positions on a specific stock in a bi-monthly fashion.
It should be noted that CAAR is the dependent variable of interest for testing the hypotheses. The KP-test and GRANK test are performed on each CAR and CAAR. They are found through transformation of the 𝑅𝑖𝑡 variable, as described by the methodology section and are standardised for time series correlation and cross-sectional correlation. As the table states, all the variables above it, Rmt, SMBt, HMLt, Rit are necessary to compute the value of CAR and CAAR. 𝑅𝑖𝑡 is a dependent variable in the context of the two preliminary estimation models, the market model and FF3F-model.
Role Variable in model / Name Factor in model Intervals Source
Independent Market return Rmt Daily NASDAQ based on S&P500 index
Independent Small-minus-Big SMB Daily Kenneth R. French database (2021)
Independent High-minus-Low HML Daily Kenneth R. French database (2021)
Dependent Stock return Rit Daily CRSP Daily Stock
Dependent Cumulative average abnormal return CAAR Over event window All of the above
Descriptive Number of shares shorted - Bimonthly NASDAQ, NYSE via Compustat
4.2. Confirmation of short squeeze Table II.
Confirmation of short squeeze
This table provides information on the selected stocks, the specific event date for each, the sector in which the company is active and in what country the company is based. The last two columns provide information regarding the IHS MARKIT assumptions, helping this paper to confirm if a short squeeze took place. Condition 1 is defined as the stock’s return, over 1 (up to 3) trading days, divided by the standard deviation of the stock’s returns of its previous 60 trading days. It is defined as fulfilling the condition if the multiplication factor is above 3. Condition 2 is defined as the percentage change between the closest settlement date before and after the short squeeze of short interest. It is defined as fulfilling the condition if the percentage change is below 0.
The statistical analysis finds that all firms in the sample pass the first condition, while one fails the second condition. It can be observed that the examined short squeeze events span from the 26th of October 2011 until the 27th of January 2021, over an almost 20-year period.
Except for Chesapeake Energy Corp and Rexx Energy Corp, all short squeezes qualified for the price increase Condition (1) with a 1-post-event day return. The latter two achieved the minimum 3x threshold after two trading days, which is still in accordance with the definition set up by IHS Markit. The highest measured one-day return is attributed to AMC with a one day return 27.72 times its usual standard deviation. At the same time, the lowest is EP Energy Corp, with 3.23 times its usual standard deviation.
All, except JC Penney, passed the Condition (2) check. As described above, due to data access restrictions of daily short positions held, this paper looked at bimonthly data. This data helps to gain insight if the short positions on a stock decrease after the short squeeze but is not a perfect proxy. Every short squeeze in the sample was researched in academia and in financial media publications. There is evidence favouring that the price spike for all presented firms, Security name Sector Date of short squeeze Condition 1 Condition 2
AMC Entertainment Holdings Inc Consumer services 27/01/2021 27.72871161 -15.57%
Bank Atlantic Corp Consumer Services 01/11/2011 19.61753164 -8.56%
Chesapeake Energy Corp Energy 03/03/2016 5.375532345 -2.19%
Cliffs Natural Resources Inc Materials 31/05/2016 4.614666921 -13.65%
Entero Medics Inc Health Care Equipment 05/01/2017 5.499605155 -88.97%
EP Energy Corp Energy 03/03/2016 3.22994931 -1.52%
Gamestop Consumer services 25/01/2021 3.096697544 -65.35%
JC Penney Retailing 27/02/2014 5.989783483 1.83%
Quidel Corp Health Care Equipment 26/10/2011 4.25289121 -1.55%
Rexx Energy Corp Energy 02/03/2016 5.85466668 -0.48%
Tesla Motors Inc Automobiles 09/05/2013 6.549403007 -19.33%
(Condition (1)) was followed by a decrease in short interest, Condition (2). JC Penney’s change in short positions does not seem to confirm the expected relationship between pre- and post-event short position held datapoints. For this one specifically the interpretation value of the proxy method used is weak, as the short event is exactly one day after the data point, on the 28th of February 2014. As such, the market participants only had one trading day to exit their short positions, which is inferior to the five consecutive trading days necessary to measure Condition (2). If one looks at the next short position data point, the 14th ofMarch 2014, a decrease of 6.68% can be observed in respects to the 28th of February 2014. This indicates a negative trend regarding the short positions held, which provides evidence in favour of fulfilling Condition (2). The analysis concludes that there is sufficient empirical evidence that all firms in the sample experienced a short squeeze.
4.3. Descriptive statistics
Table III.
Descriptive statistics
The table provides information on the numerical variables used by this study. It provides detailed information regarding the number of observations, mean of a given variable, standard deviation, and minimum and maximum value of each variable. The returns were found by dividing the price of stock i at time t by the price of stock i at time t-1 subtracted by 1. Some of the company’s stock prices acquired via CSRP databank had to be adjusted, as big spikes were present in stock prices due to stock splits. In order to adjust for this the stock prices were normalized to the price level during the estimation window of the event study. Displayed is the return of all 11 firms’ stock, the return of the market, proxied with the S&P500 index return, the Fama French SMB and HML factors, as well as the date of all the previously named variables and the event dates of the dataset. All % values are indicating until their 4th decimal.
Variable Observations Mean Standard Devation Min Max
Return AMC Entertainment Holdings Inc 1,911 0.2737% 8.8784% -56.6332% 301.2097%
Return Bank Atlantic Corp 2,372 0.0980% 4.4961% -23.6735% 110.9705%
Return Chesapeake Energy Corp 2,387 -0.0679% 6.7005% -66.0326% 181.9355%
Return Cliffs Natural Resources Inc 2,517 0.0275% 4.4072% -24.2719% 39.4137%
Return Entero Medics Inc 2,011 -0.4799% 8.3947% -79.8904% 131.4861%
Return EP Energy Corp 1,345 -0.1826% 5.9847% -29.1513% 74.0458%
Return Gamestop 2,656 0.2404% 6.1871% -60.0000% 134.8358%
Return JC Penney 2,527 -0.0717% 5.8280% -35.6250% 128.5714%
Return Quidel Corp 2,517 0.1397% 2.8493% -28.1496% 31.7465%
Return Rexx Energy Corp 1,830 -0.1348% 5.9463% -39.6313% 62.5899%
Return Tesla Motors Inc 2,517 0.2189% 3.8078% -77.4862% 24.4029%
Return SP500 2,705 0.0514% 1.0814% -11.9841% 9.3828%
SMB 2,705 0.0000% 58.8426% -360.0000% 554.0000%
HML 2,705 -1.2510% 74.1612% -495.0000% 675.0000%
Date 2,705 20591.9 1133.662 18630 22553
Event Dates 11 20428.55 1142.938 18926 22307
Some variables have more observations than others. This is due to not all companies have data points at all the dates in the time series. The date variable is part of the time series aggregation of the other variables (excluding the event date variable). Each return, as well as the SMB, HML factors, are attributed to a date. The data set starts on the 1st of January 2011, displayed as the minimum of the date variable in the table as 18630. This is due to Stata16 using the 1st of January 1960 as a baseline for dates in order to convert them from string variables into numerical numbers. The 1st of January 2011 is thus the 18630th day after the baseline date. Similarly, the last date of any variable in the sample is the 30th of September 2021, the 22307th day after the baseline. Similar to the dates, the event date variable is also formulated in STATA16’s format, it can be observed that it contains 11 event dates, fitting with the 11 firms in the dataset.
It must be noted that some of the firms delisted within the timeframe or went public only after the start date of the sample. None of the companies have a return variable for all 2705 dates. The Fama French Factors, as well as the S&P500 returns, have the most observations as they span all along the timeframe of the dataset in order to be used as independent variables in the estimation models.
Over half of the sample experienced positive mean returns over their public listing during the covered years. 63.63% of the sample had positive returns, while the remaining 36.36%
experienced negative returns. AMC and Tesla lead the pack with 0.2737% and 0.2198% mean daily returns respectively, while the worst performer by far was Entero Medics with a mean daily return of -0.4799%. Out of the 5 firms which experienced a negative mean daily return, all were either delisted and/or went bankrupt or were target of an M&A deal.
All companies in the sample seem to be very volatile compared to the benchmark index.
One can observe that all stocks experienced maximum and minimum returns far exceeding their mean return. AMC Entertainment Holdings Inc’s returns have the highest standard deviation and maximum value out of the sample. The most volatile of the firms seems to be Entero Medics Inc and AMC Entertainment Holdings Inc, as both had standard deviations of returns of over 8%, the least volatile seems to have been Quindel Corp with around 2.8%.
GameStop’s return also mirrors this high volatility. AMC’s and GameStop’s returns include the high volatility period of the COVID-19 crisis, which could explain their high variation relative to the sample in returns. Coincidently all firms that experienced long term negative returns have been delisted as of the 1st of December 2021. Chesepeake Energy Corp, Entero
Medics Inc, JC Penney and Rexx Energy Corp had long term negative returns over their public existence within the timeframe, while the S&P500 returned 0.0514% on average daily.
One can observe that the S&P500 had relatively low volatility, as well as extreme values compared to the sample. Its maximum one-day loss of 11.98% of S&P500, as well as the 9.38%
increase are the former in reaction and the latter in recovery of the 2020 COVID-19 Crisis.
The remaining variables’ statistical characteristics should be interpreted separate from the previous stock market returns. Their values are less clearly interpreted as their formulisation is not as clear. The FF3F model specific SMB and HML factors have by far the highest standard deviation and extreme values out of all variables. This tells us that within the time window of the dataset the return of small versus big firms varies strongly and can reach highly different levels. The difference in returns of high book-to-value versus low book-to-value firms also seems to fluctuate strongly within the data sample. Both can be explained to the highly different reactions to market events of firms being attributed to either of these groups.
To sum up, the firm returns behave as expected and are highly volatile with high extrema.
The benchmark index is less volatile and experienced its relatively low extrema during the COVID-19 crisis in 2020.