# Volatiliy Event Study

The EST AVyC research app provides statistics that test the effects of an event on both the mean and the conditional volatility function of the time series (Balaban and Constantinous, 2006). Furthermore, the app offers 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 $S_i = \frac{\gamma_i}{h_{i, 0}}.$