# Intraday Event Studies

Over recent decades, equity market dynamics have changed. High-frequency and other forms of algorithmic trading have increased the speed of price responses to new information. Further, social media and the Internet at large have led to an increase in the breadth of information processed by markets. Both developments suggest the need for a more fine-grained event study framework where an individual event' becomes more accurately located in time than by traditional daily event studies. Intraday event studies take the event study methodology one step further in this direction and contract the time unit the method analyses from days to minutes.

With the increasing availability of intraday price data, intraday event studies are quickly becoming the new standard, beginning with high-profile use cases such as securities litigation. There are two groups of significance test methods for intraday event studies, traditional methods, and novel jump detection techniques.

**Traditional ESM **

As its name suggests, the traditional approach applies the classic event study methodology to the intraday context. It tends to be complemented by a programmatic determination of the length of the event window - typically the amount of minutes after the event when the CAR ceases to be statistically relevant. The formulas for this approach read as follows:

\begin{equation}CAR_T = \sum\limits_{t=0}^T AR_t,\end{equation}

where $T$ denotes a period of time in minutes. The CAR is calculated both for each time period $0$ to $T$ of the control time frame (e.g. 30 days before the event) and for the period immediatelly followed the event. The event window is then determined data-driven by following procedure

- Calculate the CAR value for the first $m$ minutes (e.g. 5 Minutes) after the event. Denote this value by $CAR_0$.
- Assuming $CAR_0$ is found to be statistically significant. We continue calculating preceeding CAR values ($CAR_1, CAR_2, ...$) on the event day and test each of this CAR value on significance.
- The first CAR value that is not significant, defines the end of the event window, e.g. $T^*$. Our API will then deliver the sequence $CAR_1, CAR_2, ..., CAR_{T^*}$.

The CAR test statistics for intraday event studies differ from the ones of daily event studies. This is the case because volatility structurally differs throughout a trading day; notably, it is highest right after the opening of the market and shortly prior to its closing. To accommodate for this particularity, the following adjustments to the test statistics calculation have been suggested:

- Assume we have a control period of 60 days and a p value of 5%. Then we calculate for each day in the control period the corresponding $CAR_T$ value (mapped by the minutes). $CAR_T$ of the event day is then marked as significant iff this value is greater or less than the $0.975$- or $0.025$-percentile of the $CAR_T$ values from the control period.
- Step 1 is repeated till the event window CAR value is not anymore significant.

**Novel jump detection techniques**

Besides of the above-presented direct adaptation of the event study methodology, novel jump detection techniques are proposed. They differ from the traditional ESM as they leverage specific characteristics of intraday data and address issues common in intraday data, such as microstructure noise. Hereafter, we shortly describe the most common novel jump detection techniques, notably, the Ait-Sahilia model, the Bi-Power Variation model, and the Jiang-Omen Statistics. Each of these models has been widely used in financial econometrics for a variety of financial time series data, including stock prices, exchange rates, and interest rates.

The Ait-Sahalia model estimates the probability of high-frequency jumps in financial time series data. The model is based on the assumption that the underlying financial time series follows a jump-diffusion process, which means that it consists of a combination of continuous price changes (diffusion) and discrete price jumps. The Ait-Sahalia model specifies the probability of a jump occurring at each time point as a function of the time series data and a set of parameters that can be estimated using maximum likelihood or other estimation techniques. The model also allows for the estimation of the magnitude and frequency of the jumps, as well as the underlying diffusion process. The model has the advantage of allowing for the estimation of both the continuous and jump components of financial time series, which can be useful for understanding the underlying dynamics and risk of financial assets. However, the model is based on statistical assumptions and the results may be sensitive to the choice of model and assumptions. Therefore, it is important to carefully consider the suitability of the Ait-Sahalia model for a given financial time series and to carefully interpret the results.

The Bi-Power Variation (BPV) model measures the roughness of financial time series data. The BPV model is based on the concept of multifractality, which refers to the presence of multiple scaling exponents in a time series, indicating the presence of multiple levels of roughness or complexity. The BPV model estimates the roughness of a financial time series by decomposing it into a set of wavelets (small oscillating functions) at different scales, and then measuring the variance of the wavelets as a function of scale. The variance is then used to estimate the scaling exponent at each scale, which gives a measure of the roughness of the time series at that scale. The model has the advantage of allowing for the measurement of roughness at multiple scales, which can be useful for understanding the underlying complexity and risk of financial assets. However, the model is based on statistical assumptions and the results may be sensitive to the choice of model and assumptions. Therefore, it is important to carefully consider the suitability of the BPV model for a given financial time series and to carefully interpret the results.

The Jiang-Omen Statistics (J-O Statistics) is a statistical test to detect the presence of jumps in financial time series data. The J-O Statistics is based on the assumption that the underlying financial time series follows a jump-diffusion process, which means that it consists of a combination of continuous price changes (diffusion) and discrete price jumps. The J-O Statistics involves estimating the distribution of the returns of the financial time series and comparing it to the distribution of the returns of a benchmark process, such as a geometric Brownian motion (GBM) process. If the distribution of the returns of the financial time series is significantly different from the distribution of the returns of the benchmark process, this is taken as evidence of the presence of jumps in the financial time series. The test has the advantage of being simple and easy to implement, and it does not require the specification of a jump model or the estimation of jump parameters. However, the test is based on statistical assumptions and the results may be sensitive to the choice of benchmark process and the statistical significance level. Therefore, it is important to carefully consider the suitability of the J-O Statistics for a given financial time series and to carefully interpret the results.