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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?

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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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  • $\begingroup$ thank you very much for your answer. your answer raised another question : if we construct an equispaced grid (at a frequency of 1 minute ,say) from the initial tick by tick sampling, and we obtain the observed prices by previous tick interpolation, can we say that this sampling scheme could also be endogenous? Does the deterministic nature of the time grid prevent us from saying so? I had this question because this sampling scheme has a predefined time duration between returns but the prices are not exactly observed in those points $\endgroup$
    – XY0
    Feb 26 at 17:42
  • $\begingroup$ @XY0 can you define what you have in mind by "previous tick interpolation"? $\endgroup$
    – lehalle
    Feb 27 at 18:47
  • $\begingroup$ I establish a regular time grid with a 5-minute frequency, resulting in 84 observations within one trading day, given a 7-hour trading period (12*7). Utilizing previous tick interpolation, I derive prices at each 5-minute interval by considering the last available price. Is this sampling schemes esogenous by construction? Does the deterministic nature of the time grid makes it esogenous? $\endgroup$
    – XY0
    Feb 27 at 21:24
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    $\begingroup$ @XY0 it is not exogenous, because you choose your grid before knowing the arrival of trades (I am adding an intraday example). $\endgroup$
    – lehalle
    Feb 28 at 12:39

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