Mean 0.006 Standard deviation 0.154 Maximum 0.477 Minimum -0.364 -0.009 0.087 0.240 -0.189 In practice, it is advisable to use a parsimonious version of the estimator by including only as many lags as suggested by economic intuition and/or empirical evidence. For example, daily returns in international equity markets are likely to display some form of serial correlation because markets in different countries are open at different times. Suppose that new information becomes available at time t, when the U.S. market is open and the Japanese market is closed. Also assume that the news is expected to have a positive effect on all equity markets around the globe. The U.S. market will presumably incorporate the new information at time t, whereas prices in Japan can adjust only at time t + 1. This suggests that one should expect to observe nonnegligible correlation between returns that are one day apart. Of course, if the information is not immediately incorporated into prices (for example, because of lack of liquidity in parts of the market) then one may have to incorporate a higher order of serial correlation into the estimator. A formal analysis of the serial correlation of daily data can be useful at this stage.5 Estimation is performed by replacing the covariances in equation (16.2') with their sample counterparts:6 1 T covT [r; {d)Sj {d)\ = - J, rut (d)rh t (d) t=1 (16.3) T-k covT[ri(d),rh+k (d)] = - ^rKt(d)rUt+k (d) t=\ At this point, it is convenient to introduce some matrix algebra to write the estimator in a more compact form. If T daily return observations are available for N assets, then we can organize them in a matrix R(d). Each column in the matrix con- 5A description of techniques for the detection of serial correlation is beyond the scope of this chapter. The interested reader can find a discussion of this topic in any time-series textbook. Hamilton (1994) is a very thorough reference. sIn the formulas we assume that daily returns have a mean equal to zero. Although this is not necessarily the case, Merton (1980) points out that this approximation is often innocuous when dealing with high-frequency data, considering the amount of estimation error that characterizes average returns. If necessary the formula is easily generalized to incorporate the estimated mean of the returns.