A Brief Discussion of Event-Driven Strategy Testing: Counterfactual Inference, Risk Factor Models, and Convolution Integrals
This article discusses how to test and exploit events in quantitative investing, including event factors and event strategies.
Counterfactual Framework
Event testing can be built on a counterfactual analysis framework.
Suppose we have a market event \(E\) that occurs at time \(t_0\). We want to evaluate the event’s impact on an asset return \(R\) over some interval, such as from \(t_1\) to \(t_2\). In the world where the event \(E\) occurs, the asset return can be written as \(R_{t_1:t_2}^{(E)}\). In the counterfactual world where the event does not occur, the return can be written as \(R_{t_1:t_2}^{(\neg E)}\).
The causal effect of event \(E\) on returns, \(\Delta R\), is the difference between the realized return and the counterfactual return:
$$ \Delta R_{t_1:t_2} = R_{t_1:t_2}^{(E)} - R_{t_1:t_2}^{(\neg E)} $$
Risk Model
Estimating \(R_{t_1:t_2}^{(\neg E)}\) is challenging, but under a multi-factor framework we can at least use existing results and strip out common style exposures such as size and industry through a risk model. In a factor-based risk model, realized returns are decomposed into a style-factor-driven component and a residual component:
$$ R_{t_1:t_2}^{(E)} = \beta \lambda_{t_1:t_2} + \epsilon_{t_1:t_2} $$
In an ideal case with no firm-specific information, expected returns should equal a linear combination of style-factor returns. In the real A-share market, roughly 20% to 30% of returns can be explained by style factors. That means residual returns can be viewed as closer to the counterfactual return.
$$ ||\epsilon_{t_1:t_2} - R_{t_1:t_2}^{(\neg E)} ||_2 \le ||R_{t_1:t_2}^{(E)} - R_{t_1:t_2}^{(\neg E)} ||_2 $$
Convolution Model
One advantage of multi-factor theory is that factor testing and factor usage can be handled inside the same formula. Events, by contrast, are too sparse to be tested cross-sectionally through regression, and identical events that happen close together can stack their impact onto the same return series. For that reason, event testing should center on a convolution model, shifting from tests on T cross sections to tests on N time series, and then computing expected excess returns through the linear superposition of responses.
In circuit theory, the superposition of responses caused by stimuli at different times is usually expressed through the idea of a linear time-invariant system (LTI). This superposition relies mainly on the superposition principle and the application of the convolution integral.
For a linear system, if the response to a single input signal is known, then the total response to multiple input signals is the linear sum of the responses to each input taken separately. If the response to input \(x_1(t)\) is \(y_1(t)\) and the response to input \(x_2(t)\) is \(y_2(t)\), then the response to input \(x_1(t) + x_2(t)\) is \(y(t) = y_1(t) + y_2(t)\).
For an LTI system, the response to the unit impulse signal \(\delta(t)\) is called the impulse response and is denoted by \(h(t)\). Once the impulse response is known, the response to any input signal can be computed through convolution. For input signal \(x(t)\), the system output \(y(t)\) is
$$ y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau $$
where \(*\) denotes convolution and \(\tau\) is the integration variable.
We can treat the intensity of event \(E\) as the signal and the causal effect of event \(E\) on returns, \(\Delta R\), as the system response.
$$ \Delta R_t = \Sigma_{i=1}^{k} {x(t-i)h(i)} $$
Under this framework, the event-strategy researcher mainly needs to focus on how to define the signal itself. The more generic parts, such as choosing the convolution kernel, hypothesis testing, and parameter estimation, can then be handled uniformly by the framework.
A Brief Discussion of Event-Driven Strategy Testing: Counterfactual Inference, Risk Factor Models, and Convolution Integrals