EXAMINATIONS OF THE HONG KONG STATISTICAL SOCIETY

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1 EXAMINATIONS OF THE HONG KONG STATISTICAL SOCIETY HIGHER CERTIFICATE IN STATISTICS, 2013 MODULE 7 : Tme seres and ndex numbers Tme allowed: One and a half hours Canddates should answer THREE questons. Each queston carres 20 marks. The number of marks allotted for each part-queston s shown n brackets. Graph paper and Offcal tables are provded. Canddates may use calculators n accordance wth the regulatons publshed n the Socety's "Gude to Examnatons" (document Ex1). The notaton log denotes logarthm to base e. Logarthms to any other base are explctly dentfed, e.g. log 10. Note also that ( n r ) s the same as n C. r 1 HC Module Ths examnaton paper conssts of 8 prnted pages. Ths front cover s page 1. Queston 1 starts on page 2. RSS 2013 There are 4 questons altogether n the paper.

2 1. () A 3 3 symmetrc movng average s defned as a smple 3-pont movng average of a smple 3-pont movng average. Calculate the weghts assocated wth such a movng average. What problem would you face f usng such a movng average to smooth a tme seres? (6) () Use the symmetrc 5-term Henderson movng average (weghts: 0.073, 0.294, 0.558, 0.294, 0.073) and the 3 3 symmetrc movng average to estmate the trend of the tme seres n the table. Present your results to the nearest whole number. (6) Year 1 Q1 154 Year 1 Q2 166 Year 1 Q3 159 Year 1 Q4 165 Year 2 Q1 165 Year 2 Q2 164 Year 2 Q3 160 Year 2 Q4 163 () Assumng there s a turnng pont durng the eght perods n the table, where does t occur accordng to (a) (b) the 3 3 symmetrc movng average, the Henderson movng average? (2) (v) Brefly specfy three other ways of trend estmaton or removal n tme seres, and gve for each one a reason why t mght be consdered. (6) 2

3 2. () Descrbe the tme seres plotted below n terms of ts trend, ts seasonalty and any other noteworthy features. (3) Jan 2006 Jan 2007 Jan 2008 Jan 2009 Jan 2010 Jan () () (v) Adjustments for outlers are often made pror to the seasonal adjustment process. Why mght ths be necessary when movng averages are used for seasonal adjustment? (3) Descrbe what s meant by the phrases permanent pror adjustment and temporary pror adjustment n the context of seasonal adjustment. In general, should pror adjustments for outlers be made temporarly or permanently? Justfy your answer. (4) Suggest two methods, other than plottng the raw data, for dentfyng possble outlers n a tme seres. (2) (v) Brefly descrbe a statstcal method for estmatng the sze of an outler. (3) (v) Outlne the nferences you mght draw from the followng extract of computer output taken from a seasonal adjustment run of the seres llustrated n part () above. Based solely on ths output, what acton would you take when seasonally adjustng ths seres? What other nformaton mght you lke to have before takng acton? (5) Regresson Model Parameter Standard Varable Estmate Error t-value Automatcally Identfed Outlers: 2008.Jul

4 3. Consder a country enterng and then recoverng from an economc downturn, as quantfed n the table below. For smplcty, assume that consumers purchase only two types of tem, luxures and necesstes, wth equal expendture on each n The relatves shown n the table all have 2007 as the base perod Prce relatve of luxures Volume relatve of luxures Prce relatve of necesstes Volume relatve of necesstes The followng notaton s used: v(t, ) s the expendture on any tem n any perod t; R(s, t, ) s the prce relatve for any tem comparng any year t wth any year s; R * (s, t, ) s the volume relatve for any tem comparng any year t wth any year s. () Explan why v(t, ), the expendture on tem (ether luxures or necesstes) n any year t, can be expressed as R(07, t, ) R(07, t, ) vt (,) = v(07,) * where v(07, ) s the expendture on tem n 2007, R(07, t, ) s the prce relatve for tem comparng year t wth 2007, R * (07, t, ) s the volume relatve for tem comparng year t wth (1) () The Laspeyres volume ndex Q L for perod t wth perod s as the base perod can be wrtten Q L ( s, t) = R * ( s, t, ) v( s, ). vs (, ) Prove that ths can be expressed as Q L ( st, ) = (,, ) (07,, ) (07,, ) R s R s. (6) * * R s t R s R s * (07,, ) (07,, ) () (v) (v) For each year n the table above, calculate the chan-lnked Laspeyres volume ndex, ntroducng a new base perod n each of 2008, 2009, 2010 and Use 2007 as the reference perod throughout. (10) Wthout calculaton, gve the Laspeyres volume ndex for 2011 usng 2007 as the base perod. State why no calculaton s necessary. (2) Consderng your answer to part (v), state why the chan-lnked Laspeyres volume ndex for 2011 calculated n part () s undesrable. (1) 4

5 4. () There are varous ways n whch ndex numbers can be appled to motor fuel consumpton. One such way s to treat dstance travelled per ltre of fuel as beng analogous to prces for economc ndex numbers, and the amount of fuel consumed as analogous to quantty. What varable s analogous to value (the product of prce and quantty)? (1) () () State a formula for the Laspeyres "dstance per ltre" ndex, and a formula for the Paasche "dstance per ltre" ndex, n each case wth perod 0 as the base perod and perod t as the current perod. (4) At the end of 2011 a company, wshng to reduce ts expendture on fuel, sent ts employees on an "effcent drvng" course. The table below gves data on the use of fuel and dstance travelled by company employees on offcal busness. Calculate the Laspeyres, Paasche and Fsher "dstance per ltre" ndces for 2012, usng 2011 as the base perod. What do these ndces tell us about the overall fuel effcency of the company's drvers snce attendng the course? (15) Fuel type Dstance travelled n 2011 (thousands of km) Amount of fuel consumed n 2011 (thousands of ltres) Dstance travelled n 2012 (thousands of km) Amount of fuel consumed n 2012 (thousands of ltres) Petrol (standard) Petrol (hgh octane, used n hgh performance supercars only) Desel fuel

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