Event Studies

Abstract

One of the most common applications of asset pricing models is event studies. Today, event studies not only provide evidence on the question of market efficiency but how investors perceive different kinds of information that will impact stock prices. Asset pricing models are used to measure abnormal returns not explained by systematic risk factors. A large body of literature has evolved to investigate the abnormal returns of stocks in response to important news events, including both firm-level announcements and macroeconomic announcements. Short-run and long-run event study tests provide insights into abnormal stock returns as well as market efficiency in terms of how fast markets react to new market information.

Appendices

Questions

• 1. Explain the concepts of event date, event time, event window, and estimation window.

• 2. Explain normal return, abnormal return, and cumulative abnormal return.

• 3. Discuss pros and cons of standardized event study tests with relation to non-standardized tests.

• 4. Suppose that you want to test the null hypothesis of no mean effect when event days are completely clustered. The sample size is $n = 100$ firms and the test statistic applied is BMP due to its good sample properties. You estimate the value of $t_{\textrm{bmp}} = 2.7$ with $p = 0.007$ in two-sided testing. As a result, you infer that the event effect is highly statistically significant. Since the average cross-sectional correlation of the abnormal returns is $\bar{r} = 0.02$, you argue that the small average cross-sectional correlation will not materially change the test results. Is this right?

• 5. In the text it was noted that the BHAR approach in long-run event studies is vulnerable to cross-sectional correlation. Is it vulnerable to autcorrelation or heteroskedasticity of the returns too?

• 6. Consider a long-run event study of SEO events with abnormal returns defined as the difference between the returns of the event firm and its matched control firm. What would you consider to be a major problem with this definition of the abnormal return?

• 7. Discuss some of the major pros and cons of nonparametric rank tests in event studies (see Appendix A).

Problems

• 1. A key requirement for efficient capital markets is that prices fully and instantaneously reflect all available relevant information. Consider the event of an earnings announcement. Suppose that the current price of a stock is 100. Sketch a stock price process for $\pm 5$ days around the event (using end of day prices) to reflect:

  • a. an efficient capital market when the announcement is

    i.:

    a positive surprise with +5% price effect

    ii.:

    a negative surprise with −5% price effect

    iii.:

    neutral (i.e., according to analyst expectations) with no price effect

  • b. leakage of information before the event day when the surprise is

    i.:

    positive (+5%)

    ii.:

    negative (−5%)

  • c. a price process as in (a) when the full price adjustment takes place over a few days after the event.

• 2. The table below reports event day abnormal returns, standard deviations of the estimation window abnormal returns (i.e., OLS residuals of the market model), and rank number of the event day abnormal return for each stock relative to its estimation window abnormal returns in a sample of $n = 30$ stocks. The estimation window is $T_e = 200$ days so that $T = T_e + 1$ (i.e., estimation window returns $+$ event return) is the total number of returns for each stock. The smallest return for each stock assumes rank number 1 and the largest 201. Using this ranking approach, the rank number 164 for the first stock indicates that the event day abnormal return of 1.21% is the 164th largest among the 201 returns from the combined estimation period and the event day. Test the null hypothesis of zero abnormal return using each of the following statistics:

  a.:

  $t_{ AR }$

  b.:

  $z_{\textrm{patell}}$

  c.:

  $t_{\textrm{bmp}}$

  d.${}^*$:

  $z_{\textrm{sgn}}$

  e.${}^*$:

  $t_{\textrm{grank}}$.

${}^*$Optional: In the optional problem (d), assume that the $d_{it} = 1 / T_e$ in Eq. (A11.1). In optional problem (e), when computing Eq. (A11.4), you can use the theoretical standard error that in this case is:

$$\begin{aligned} s_{\bar{U}} = \sqrt{\frac{1}{nT}\sum _{i = 1}^T(i/(T + 1) - 1/2)^2}, \end{aligned}$$

where $T = T_e + 1 = 201$. Using this equation, the result is $s_{\bar{U}} = 0.0524$ (rounded to four decimal places).

Stock	AR	Std dev	Rank
1	1.21	1.04	165
2	0.27	1.01	134
3	1.01	1.13	162
4	−0.81	1.03	47
5	0.32	0.84	116
6	−0.87	0.98	60
7	−0.41	1.20	82
8	1.13	0.94	159
9	1.16	1.17	161
10	−0.20	0.98	84
11	0.89	1.17	153
12	0.97	1.08	150
13	1.68	1.14	180
14	1.13	0.99	171
15	1.79	1.05	186
16	0.42	1.18	111
17	−1.64	1.10	19
18	−0.71	0.96	56
19	0.91	1.20	154
20	0.29	1.15	124
21	2.10	1.18	187
22	0.22	1.06	114
23	0.19	1.03	116
24	1.32	1.06	172
25	2.29	0.92	199
26	−3.05	0.95	1
27	0.77	0.85	161
28	0.86	0.97	162
29	−1.38	0.97	14
30	1.32	0.99	175

Appendix A: Nonparametric Testing

Section 11.1.3 discussed parametric tests wherein data are assumed to be generated from a particular process like the normal distribution. If this assumption holds, the implied test statistics are optimal for statistical testing. Even if the normality of returns does not hold, under fairly general assumptions, the Central Limit Theorem (CLT) guarantees incrementally accurate test results as the sample size grows. Nonparametric approaches are free from any assumption about the specific distribution of the returns. In this respect, the sign test and the ranks test are the two major categories of nonparametric tests.

The sign test assumes that the (cumulative) abnormal returns are cross-sectionally independent and, under the null hypothesis of no event effect, the positive and negative sign of the CAR is equally likely. That is, $P( CAR \le 0) = P( CAR > 0) = 1/2$. Defining dummy variables $D_i = 0$ when $CAR_i < 0$ and $D_i = 1$ when $CAR_i \ge 0$, and denoting $\bar{D} = \sum _{i = 1}^nD_i / n$, then under the null hypothesis of no event effect $E(\bar{D}) = 1/2$ and $\textrm{var}(\bar{D}) = 1/(4n)$. Subsequently, we can define a test statistic

$$\begin{aligned} z_{\textrm{sgn}} = 2 \sqrt{n} (\bar{D} - 1/2), \end{aligned}$$

(A11.1)

which under the null hypothesis of no event effect is asymptotically N(0, 1) distributed by the CLT.

The basic assumption of the sign test is that the return distribution is symmetric. For this reason, it is vulnerable to skewness of the distribution, in which case under the null distribution the probabilities of negative and positive signs deviate from one half.

Corrado (1989) developed a test based on ranks of returns. Kolari and Pynnönen (2011) refined the approach to cover both single day and cumulative abnormal returns. They defined estimation window standardized returns as $SAR _{it} = AR _{it} / s_i$, in which $AR _{it}$s are estimation window OLS residuals, and $s_i$ is the respective OLS standard error. For an event window, they defined standardized cumulative abnormal returns from event day $\tau _1$ to $\tau _2$ as

$$\begin{aligned} SCAR _{i}(\tau _1, \tau _2) = \frac{ CAR _i(\tau _1, \tau _2)}{s_i(\tau _1, \tau _2)}, \end{aligned}$$

where $s_i(\tau _1, \tau _2)$ is the prediction error corrected OLS standard deviation of the cumulative abnormal return (see Kolari et al. 2018). Re-standardizing $SCAR _i(\tau _1, \tau _2)$ by their cross-sectional standard deviation helps to account for possible event-induced variance. The re-standardized SCARs are defined as

$$\begin{aligned} SCAR _i^*(\tau _1, \tau _2) = \frac{ SCAR _i(\tau _1, \tau _2)}{s(\tau _1, \tau _2)}, \end{aligned}$$

where

$$\begin{aligned} s(\tau _1, \tau _2) = \sqrt{\frac{1}{n-1}\sum _{i = 1}^n( SCAR _i(\tau _1, \tau _2) - \overline{ SCAR }_{\tau _1, \tau _2})^2}. \end{aligned}$$

Using these together with the estimation window abnormal returns, Kolari and Pynnonen defined the following generalized standardized abnormal returns (GSARs):

$$\begin{aligned} GSAR _{it} = \left\{ \begin{array}{ll} SAR _{it}, &{} \text {for the estimation window returns}\\ SCAR _i^*(\tau _1, \tau _2), &{} \text {for the cumulative event window return}. \end{array} \right. \end{aligned}$$

(A11.2)

GSARs are homogeneous in that, under the null hypothesis of no event (mean) effect, they have zero mean and unit variance. Using these results, rank transformations can be defined as

$$\begin{aligned} U_{it} = \text {rank}( GSAR _{it}) / (T + 1) - 1/2 \end{aligned}$$

(A11.3)

for $i = 1, \ldots , n$, where T indicates the number of estimation window returns plus the event window SCAR observation so that T refers to the rank number associated with $SCAR _i^*(\tau _1, \tau _2)$. As such, these ranks capture both single day abnormal returns as well as cumulative abnormal returns. For testing the null hypothesis of no event effect, the authors derived the following rank test statistic:

$$\begin{aligned} t_{\textrm{grank}} = Z\left( \frac{T - 2}{T - 1 - Z^2}\right) ^{\frac{1}{2}}, \end{aligned}$$

(A11.4)

where

$$\begin{aligned} Z = \frac{\bar{U}_{T}}{s_{\bar{U}}} \end{aligned}$$

with

$$\begin{aligned} s_{\bar{U}} = \sqrt{\frac{1}{T} \sum _{t = 1}^T \frac{n_t}{n} \bar{U}_{t}^2}, \end{aligned}$$

$$\begin{aligned} \bar{U}_t = \frac{1}{n_t} \sum _{i = 1}^{n_t}U_{it}, \end{aligned}$$

and $n_t$ is the number of available cross-section returns at time $t = 1, \ldots , T$.

Under the assumption of cross-sectional independence, the asymptotic null distribution of $t_{\textrm{grank}}$ is Student’s t-distribution with $T - 2$ degrees of freedom. In addition to event-induced variance, $t_{\textrm{grank}}$ accounts for cross-sectional correlation of abnormal returns when the event days are clustered. Similar to Kolari et al. (2018), Pynnonen (2022) developed an adjustment for $t_{\textrm{grank}}$ to account for cross-sectional dependence in cases where the event windows are partially clustered.

The major attractiveness of nonparametric tests is their robustness to non-normality. Also, rank tests are invariant to the usage of simple returns or log-returns as log-transformation preserves the order of observations. Many researchers conduct both nonparametric and parametric tests to check the robustness of the results.