The file mmm9912-adur . txt contains an adjusted intraday trading duration of 3M stock in December 1999. There are thirty-nine 10-minute time intervals in a trading day. Let \(d_{i}\) be the average of all \(\log\) durations for the \(i\) th 10 minute interval across all trading days in December 1999. Define an adjusted duration as \(t_{j} / \exp \left(d_{i}\right)\), where \(j\) is in the \(i\) th 10 -minute interval. Note that more sophisticated methods can be used to adjust the diurnal pattern of trading duration. Here we simply use a local average.

(a) Is there a diurnal pattern in the adjusted duration series? Why?

(b) Build a duration model for the adjusted series using exponential innovations. Check the fitted model.

(c) Build a duration model for the adjusted series using Weibull innovations. Check the fitted model.

(d) Build a duration model for the adjusted series using generalized gamma innovations. Check the fitted model.

(e) Compare and comment on the three duration models built before.