Time Endogeneity in Financial Sampling and Market Data
Summary
The document explains time endogeneity in financial sampling as dependence between observation times and the value of the process being observed. It notes that correlation alone is too narrow a definition because it measures only linear dependence. In the cited framing, observation times are stopping times that are not independent of process values.
Examples include tick-by-tick prices, where price changes trigger new observations, and illiquid bonds that are observed irregularly. The response contrasts this with exogenous sampling, such as choosing a regular time grid independently of the process. It also describes how bid-ask spreads measured just before trades can differ from spreads measured on a fixed grid, because trade timing may depend on spread size. The explanation is conceptual and illustrative; it does not develop statistical estimators or quantify the effect.
Key ideas
- Time endogeneity concerns dependence between observation times and the values of the process being sampled.
- A nonzero correlation is not required as the definition because correlation captures only linear dependence.
- Price changes can trigger observations in tick-by-tick data, making sampling endogenous.
- A regular time grid is an example of exogenous sampling when it is set independently of process values.
- Sampling spreads before trades can produce different averages from sampling them at fixed intervals.
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Full text
# In how many ways can time endogeneity be defined? # In how many ways can time endogeneity be defined? In the literature many papers such as Wenhao Cui take into consideration the problem of time endogeneity. Is it correct to define time endogeneity uniquely as having nonzero correlation between sampling times and observed prices? Or are there other ways to define time endogeneity? ## Answer by lehalle (score 5, accepted) https://quant.stackexchange.com/a/78474 Rather than using correlation (that is a linear measure of dependence), I would define time endogenetity as in Li, Yingying, Per A. Mykland, Eric Renault, Lan Zhang, and Xinghua Zheng. "Realized volatility when sampling times are possibly endogenous" Econometric theory 30, no. 3 (2014): 580-605. It is when the observation times (that are a stopping time) are not independent of the value of the process. In finance, they are multiple occasion to have this dependence: for instance at a tick by tick level, we only observe a new price when it changes (i.e. when the "fair value" from the viewpoint of liquidity consumers is different by one tick to the previous observed price). Same for illiquid assets like high yield corporate bonds. In general, you can have in mind that it is a change in the value of the process that triggers its observation. On the opposite, exogenous sampling of time is when one decide the observation time an arbitrary way that is not considering the value of the process (a regular grid is a typical example). Here is an example: it is well-known in high-frequency finance that the average bid-ask spread sampled just before a transaction is smaller than the average bid-ask spread sampled on a fixed time grid (for instance every 1 seconds, or 10 seconds). See C-A L and Sophie Laruelle. Market microstructure in practice. World Scientific, 2018 (2nd Edition) for details. It is because traders and algos have a tendency to accept to cross the bid ask spread when it is small (it is perceived as ``less costly''). Clearly the choice of the stopping time (the occurence of a trade) is indeed influenced by the observed variable (the bid-ask spread), hence the obtained statistic is different from the one observed if one samples independently.
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