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I have a specific but interesting problem at hand:

I know that the price of a stock falls after a negative shock (in t-1). Now it is the day after the shock (t) and I want to know whether the price will likely rebound to its former level or stay at the lower level in the next x periods (t+1...t+x).

This event has happened in several different companies at several times in the past -> These are my Inputs (the event date is known).

This is not a problem of estimating the impact of the shock itself, it is about the effect after the impact on the price.

Alternatively this may be thought of as a problem of estimating the persistance/absorption of the time series or as the lasting effect of a shock or even as the an estimation of the parameters of a mean-reverting process.

Thoughts on what kind/class/category of problem I have here would be much appreciated. Because I really would like to be able to at least articulate my problem

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You can detect this kind of bounce with a moving average (MA) model. A moving average model is based on predicting $Y_{t}$ by using the past errors in estimation:

$Y_{t}=Y_{t-1}+\theta_{1}\epsilon_{t-1}+...+\theta_{q}\epsilon_{t-q}+\epsilon_{t}$

Where $\epsilon_{t-1}$ was the error in estimating $Y_{t-1}$. If there is a shock in period t-1 and that causes an unexpected fall in the value of a companies stock then you'll have a positive error $\epsilon$ in time period t-1. If you have rebound (or otherwise called a dead cat bounce) in period t you'd expect $\theta_{1}$ to have a positive coefficient, i.e. overestimating $Y_{t-1}$ would lead to a correction in estimating $Y_{t}$.

A moving average model of order one, MA(1), looks at just the error in the previous period. A moving average model of order MA(q) would look at the errors in predictions of the last q periods.

For more details have a look at ARIMA modelling (the MA standing for moving average), here's a good place to start:https://www.otexts.org/fpp/8

Technical Disclaimer: the above formulae is actually more than just a moving average as it in effect includes a first order difference term, $Y_{t-1}+\epsilon_{t}$, assuming your data is stationary after being differenced. The following model would be actually be considered a ARIMA(0,1,1) model:

$Y_{t}=Y_{t-1}+\theta_{1}\epsilon_{t-1}+\epsilon_{t}$

I only mention this as knowing when to difference your time-series is a really important part of ARIMA analysis.

https://www.otexts.org/fpp/8

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