Is it always true thatThey are a lot of ways to compute an "estimated bid-ask spread". The most straightforward one is to sample the bid-ask spread sampled beforeon a transaction is smaller thanregular time grid (for instance every second), but that for you need all the spread sampled during any other instant betweenquote changes (a quote is the time duration of these 2 trades?best bid and ask, price and quantity).
This may be justified byUsually it is easier (and cheaper) to get the quotes sampled just before the transactions of the day. Hence it is natural the fact that traders are more willing to accept crossingcompute the average bid ask-ask spread when iton such a database. Unfortunately this sample time is perceived to be less expensivea stopping time and not a deterministic good that is known "a priori".
for example assume Worst than that we have a transactionit is probable that (occurrence ofthe occurence of a trade is not independent of the bid-ask spread: most probably traders or algorithms are making a trade ) at instant $T_5$. Is-off between the spread observed at this time always smaller thancost of crossing the spread observed during the time frame following the trade $T_4$ and that until the occurrencesome predictors or a waiting cost. Because of that the bid-ask spread compute just before a trade $T_5$? Isis in general smaller than the sample spread observed"average bid-ask spread" $T_5$ the smallest among(on a regular time grid).
In this context, the sampled spread observed between $T_4$ and $T_5$original question is: "is the bid-ask spread sampled just before a trade always lower than the bid-ask spread sampled on a regular time grid?"