The file taq-aa-t-june 72010 . txt contains the tick-by-tick trading data of Alcoa stock on June 7, 2010. It has seven columns, namely, date, hour, minute, second, price, and volume. Focus on the transactions that occurred during the normal hours.

(a) Obtain the nonzero time durations between trades. What are sample mean and variance of the series.

(b) Remove the diurnal pattern using

\[ f\left(t_{i}\right)=\exp \left[d\left(t_{i}\right)\right], \quad d\left(t_{i}\right)=\beta_{0}+\beta_{1} f_{1}\left(t_{i}\right)+\beta_{2} f_{1}^{2}\left(t_{i}\right)+\beta_{3} f_{3}\left(t_{i}\right) \]

where \(f_{1}\left(t_{i}\right)\) is defined as in Equation (6.31) and \(f_{3}\left(t_{i}\right)=\ln \left(t_{i}\right)\). Write down the fitted model for diurnal pattern.

(c) Build an EACD model for the adjusted duration and check the fitted model.

(d) Build a WACD model for the adjusted duration and check the fitted model.

(e) Compare the prior two duration models.