"Frontmatter". In: Analysis of Financial Time Series

176 HIGH-FREQUENCY DATAof the results obtained. In particular, we discuss nonsynchronous trading, bid-askspread, duration models, price movements that are in multiples of tick size, andbivariate models for price changes and time durations between transactions associatedwith price changes. The models discussed are also applicable to other scientificareas such as telecommunications and environmental studies.5.1 NONSYNCHRONOUS TRADINGWe begin with nonsynchronous trading. Stock tradings such as those on the NYSEdo not occur in a synchronous manner; different stocks have different trading frequencies,and even for a single stock the trading intensity varies from hour to hourand from day to day. Yet we often analyze a return series in a fixed time interval suchas daily, weekly, or monthly. For daily series, price of a stock is its closing price,which is the last transaction price of the stock in a trading day. The actual time of thelast transaction of the stock varies from day to day. As such we incorrectly assumedaily returns as an equally-spaced time series with a 24-hour interval. It turns outthat such an assumption can lead to erroneous conclusions about the predictabilityof stock returns even if the true return series are serially independent.For daily stock returns, nonsynchronous trading can introduce (a) lag-1 crosscorrelationbetween stock returns, (b) lag-1 serial correlation in a portfolio return,and (c) in some situations negative serial correlations of the return series of a singlestock. Consider stocks A and B. Assume that the two stocks are independent andstock A is traded more frequently than stock B. For special news affecting the marketthat arrives near the closing hour on one day, stock A is more likely than B to showthe effect of the news on the same day simply because A is traded more frequently.The effect of the news on B will eventually appear, but it may be delayed until thefollowing trading day. If this situation indeed happens, return of stock A appearsto lead that of stock B. Consequently, the return series may show a significant lag-1 cross-correlation from A to B even though the two stocks are independent. Fora portfolio that holds stocks A and B, the prior cross-correlation would become asignificant lag-1 serial correlation.In a more complicated manner, nonsynchronous trading can also induce erroneousnegative serial correlations for a single stock. There are several models available inthe literature to study this phenomenon; see Campbell, Lo, and MacKinlay (1997)and the references therein. Here we adopt a simplified version of the model proposedin Lo and MacKinlay (1990). Let r t be the continuously compounded returnof a security at the time index t. For simplicity, assume that {r t } is a sequence ofindependent and identically distributed random variables with mean E(r t ) = µ andvariance Var(r t ) = σ 2 . For each time period, the probability that the security is nottraded is π, which is time-invariant and independent of r t .Letrto be the observedreturn. When there is no trade at time index t, wehaverto = 0 because there isno information available. Yet when there is a trade at time index t, wedefinerto asthe cumulative return from the previous trade (i.e., rt o = r t + r t−1 + ··· +r t−kt ,where k t is the largest non-negative integer such that no trade occurred in the periods