## Volatiliy Event Study

We provide a single day tes statistics that simultaneously tests the effects, on both, the mean and the conditionally volatility function on the time series (Balaban and Constantinous, 2006). Furthermore, we offer a Wilcoxon Signed test for comparing pre-event and post-event volatilities (Agrawal, 2003). The time series is utilized as a whole, such that no event window has to be selected. For estimating abnormal return and time-varying volatility, we use a market model with GARCH errors (Bollerslev (1986) and an indicator variable on event day in both, the mean and the volatility function, namely

$$R_{it} = \alpha_i + \beta_i \cdot R_{mt} + \gamma_i \cdot D_{i, t} + \varepsilon_{it}.$$

The conditional variance may be written as:

$$\sigma^2_{t} = \omega + \gamma_{i, 1} \cdot \varepsilon^2_{t-1} + \delta_1 \cdot \sigma^2_{t-1} + \delta_i \cdot D_{i, t}$$

with $D_{i, t}$ is modeled as dummy variable such that $D_{i, t} = 1$ on event day and $0$ otherwise. Event study tools provides for this model both abnormal return and abnormal volatility test statistics based on the widely used cross-sectional test statistic developed by Brown and Warner (1980) adapted for the market model with GARCH error estimates by Balaban and Constantinous (2006).

Abnormal Volatility Test Statistics

 Cross-Sectional Test for Abnormal Volatility

$$t_{AV,BMP, t} = \frac{\bar{\delta}}{\left(\frac{1}{n\cdot(n-1)} \sum\limits_{i=1}^N \left( \delta_i - \bar{\delta}_i \right)\right)^{0.5}}$$

where $\bar{\delta} =\frac{1}{N}\sum\limits_{i=1}^N \gamma_i$

 Corrected Cross-Sectional Test for Abnormal Volatility

$$t_{AV,cBMP, t} = \frac{\bar{S}}{\left(\frac{1}{n\cdot(n-1)} \sum\limits_{i=1}^N \left( S_i - \bar{S}_i \right)\right)^{0.5}}$$

where $\bar{S} =\frac{1}{N}\sum\limits_{i=1}^N S_i$ and $S_i = \frac{\delta_i}{h_{i, 0}}.$

 Wilcoxon Signed Test for comparing pre-event and post-event volatility

We estimate the volatility of returns, both firm $\sigma_i$ and market $\sigma_m$, on a pre-event window $[-L, 0]$ and post-event window $[0, L]$. Next, the pre-event and post-event volatility ratio

$$\lambda^{pre}_i= \sqrt{\frac{\sigma^{pre}_i}{\sigma^{pre}_m}} \text{ and } \lambda^{post}_i = \sqrt{\frac{\sigma^{post}_i}{\sigma^{post}_m}}$$

are calculated for each firm $i$. Then we apply a Wilcoxon Signed Test for testing pre-event volatility ratios versus post-event volatility ratios.

Abnormal Returns Test Statistics

 Cross-Sectional Test for Abnormal Returns

$$t_{AR,BMP, t} = \frac{\bar{\gamma}}{\left(\frac{1}{n\cdot(n-1)} \sum\limits_{i=1}^N \left( \gamma_i - \bar{\gamma}_i \right)\right)^{0.5}}$$

where $\bar{\gamma} =\frac{1}{N}\sum\limits_{i=1}^N \gamma_i$

 Corrected Cross-Sectional Test for Abnormal Returns

$$t_{AR,cBMP, t} = \frac{\bar{S}}{\left(\frac{1}{n\cdot(n-1)} \sum\limits_{i=1}^N \left( S_i - \bar{S}_i \right)\right)^{0.5}}$$

where $\bar{S} =\frac{1}{N}\sum\limits_{i=1}^N S_i$ and $S_i = \frac{\gamma_i}{h_{i, 0}}.$