NonSynchronous trading
This paper by Lo and MacKinlay analyze the effects of non synchronous trading on stochastic properties. The transaction data of any asset traded in an exchange is irregularly spaced. Homogeneous time series is an artifact. Non Homogeneous time series is the reality. For example, the daily prices of securities quoted in the news papers as “closing prices” are not the prices that are exactly traded at the very last second of the market close. Some exchanges aggregate the price data based on time/ volume/ type of asset etc. and report these prices.
Some stocks react to news faster than others. The nonsynchronous nature of trading might make one infer that there is cross correlation between assets when all that we are seeing is a delayed response to a common factor. So, a natural question arises ? How does one capture this aspect a model and what can the model say about the stochastic properties of univariate and multivariate time series ?
This paper starts with a very simple model where the data is censored based on Bernoulli random variable.. The authors assume an unobserved returns series for each security. Based on the censored random variable and unobserved series, an observed time series is created. The math behind the computations is tedious but not complex. The authors show that there is a spurious correlation induced in individual security returns which is proportional to the square of its expected return. The authors extend this setting to portfolio of securities and find that the effect is more pronounced. Here are the main findings from the model about the effects of Nonsynchronous trading :

Does not effect the mean of either individual or portfolio returns.

Increases the variance of individual security returns.

Decreases the variance of observed portfolio returns.

Induces geometrically declining negative serial correlation in individual security returns.

Induces geometrically declining positive serial correlation in observed portfolio returns.

Induces geometrically declining crossautocorrelation between returns of securities that have same sign for their betas.