A twenty-foot equivalent unit (TEU) is a standard measurement of volume in container shipping. The majority of containers are either 20 or 40 feet in length. A 20-foot container is 1 TEU; a 40-foot container is 2 TEUs. Although the height of containers can also vary, this does not affect the TEU measurement. The following data are the total number of containers shipped from the port of Los Angeles for the years 1995 through 2007.

Date	Total TEUs	Date		Total TEUs	Date	Total TEUs
Jan-95	1,23,723	May-99		1,82,975	Sep-03	3,50,476
Feb-95	99,368	Jun-99		1,69,682	Oct-03	3,72,112
Mar-95	1,18,549	Jul-99		1,85,017	Nov-03	3,38,379
Apr-95	1,23,411	Aug-99		1,88,281	Dec-03	3,06,984
May-95	1,14,514	Sep-99		1,87,081	Jan-04	3,45,412
Jun-95	1,14,468	Oct-99		2,08,163	Feb-04	2,47,710
Jul-95	1,25,412	Nov-99		1,84,662	Mar-04	3,40,748
Aug-95	1,22,866	Dec-99		1,78,493	Apr-04	3,45,339
Sep-95	1,15,473	Jan-00		1,94,180	May-04	3,67,128
Oct-95	1,21,523	Feb-00		1,75,890	Jun-04	3,47,056
Nov-95	1,04,880	Mar-00		1,88,438	Jul-04	3,65,901
Dec-95	1,03,821	Apr-00		2,20,157	Aug-04	3,44,109
Jan-96	1,11,494	May-00		2,17,749	Sep-04	3,24,346
Feb-96	99,785	Jun-00		2,20,071	Oct-04	3,52,718
Mar-96	96,906	Jul-00		2,43,695	Nov-04	3,40,051
Apr-96	1,11,204	Aug-00		2,50,551	Dec-04	2,83,268
May-96	1,15,513	Sep-00		2,27,848	Jan-05	3,05,102
Jun-96	1,19,422	Oct-00		2,60,469	Feb-05	2,94,022
Jul-96	1,29,984	Nov-00		2,10,209	Mar-05	2,62,173
Aug-96	1,34,296	Dec-00		2,03,021	Apr-05	3,36,087
Sep-96	1,34,657	Jan-01		2,12,323	May-05	3,19,472
Oct-96	1,44,430	Feb-01		1,63,332	Jun-05	3,40,582
Nov-96	1,28,521	Mar-01		2,17,284	Jul-05	3,56,716
Dec-96	1,22,428	Apr-01		2,21,465	Aug-05	3,49,655
Jan-97	1,27,065	May-01		2,13,860	Sep-05	3,56,912
Feb-97	1,12,733	Jun-01		2,43,053	Oct-05	3,75,051
Mar-97	1,13,063	Jul-01		2,50,344	Nov-05	3,32,037
Apr-97	1,29,797	Aug-01		2,61,705	Dec-05	3,28,244
May-97	1,36,712	Sep-01		2,75,559	Jan-06	3,27,009
Jun-97	1,40,220	Oct-01		2,74,954	Feb-06	2,51,812
Jul-97	1,43,756	Nov-01		2,41,730	Mar-06	3,45,401
Aug-97	1,43,389	Dec-01		2,25,886	Apr-06	3,70,171
Sep-97	1,43,700	Jan-02		2,20,810	May-06	3,68,864
Oct-97	1,44,425	Feb-02		2,44,167	Jun-06	3,87,957
Nov-97	1,31,877	Mar-02		2,29,954	Jul-06	4,13,357
Dec-97	1,34,315	Apr-02		2,76,373	Aug-06	4,14,004
Jan-98	1,25,930	May-02		2,84,385	Sep-06	4,31,283
Feb-98	1,22,976	Jun-02		3,01,447	Oct-06	4,21,694
Mar-98	1,54,947	Jul-02		2,71,933	Nov-06	3,90,209
Apr-98	1,54,522	Aug-02		3,39,690	Dec-06	3,65,591
May-98	1,67,204	Sep-02		3,30,967	Jan-07	3,67,096
Jun-98	1,59,638	Oct-02		2,65,218	Feb-07	3,58,601
Jul-98	1,58,948	Nov-02		3,01,333	Mar-07	3,23,472
Aug-98	1,71,152	Dec-02		3,06,099	Apr-07	3,75,512
Sep-98	1,57,267	Jan-03		2,76,482	May-07	3,68,874
Oct-98	1,69,364	Feb-03		2,74,740	Jun-07	3,93,187
Nov-98	1,58,255	Mar-03		2,98,495	Jul-07	3,87,573
Dec-98	1,40,165	Apr-03		3,26,709	Aug-07	3,79,027
Jan-99	1,42,116	May-03		3,48,276	Sep-07	4,07,915
Feb-99	1,42,080	Jun-03		3,05,892	Oct-07	3,93,948
Mar-99	1,41,926	Jul-03		3,31,741	Nov-07	3,83,241
Apr-99	1,53,559	Aug-03		3,60,046	Dec-07	3,46,140

a). Plot the series. What can you learn from examining this plot?

b). Calculate and display the first 24 autocorrelations for the series. What do the ACF and PACF suggest about the series?

c). Suggest and estimate an optimal set of differencing to use with the series.

e). Estimate the ARIMA model that you believe to be a good candidate for forecasting container shipments. It may help to specify the seasonality as “12.” Test the Ljung-Box statistic and report your findings. Finally, plot the first 24 autocorrelations of the residuals to your best model.