Chain-linking and seasonal adjustment of the quarterly national accounts

Transcription

1 Sascs Denmark Naonal Accouns 6 July 00 Chan-lnkng and seasonal adjusmen of he uarerly naonal accouns The mehod of chan-lnkng he uarerly naonal accouns was changed wh he revsed complaon of daa hrd uarer 007 whch was publshed 6 January 008. Mehod of chan-lnkng Indrec sesasonal adjusmen In earler complaons he uarerly chan-lnked daa was calculaed wh fourh uarer of he prevous year as pon of reference. Ths mehod s called uarerly overlap. The mehod used now calculaes he chan-lnked daa wh he annual daa of he prevous year as pon of reference. Ths mehod s called annual overlap. Wh hs mehod s no longer necessary o benchmark he uarerly chanlnked daa o he annual chan-lnked daa because he sum of he uarers euals he annual chan-lnked value when he annual overlap mehod s used. In hs paper s shown how mulas aggregaon of and conrbuon o growh uarerly chan-lnked seres can be derved from he mula annual overlap chan-lnkng. When he annual overlap mehod s used s possble o use ndrec seasonal adjusmen he chan-lnked seres. The seasonally adjused chan-lnked aggregaes can be calculaed as an aggregae of he seasonally adjused componens usng he above menoned mula aggregaon of annual overlap chan-lnked seres. For daa n curren prces ndrec seasonal adjusmen s also used n he dansh uarerly naonal accouns. Conac: Jonas Dan eersen jop@ds.dk

2 Aggregaon and conrbuon o growh Annual overlap chan-lnkng Chan-lnked daa n he dansh uarerly naonal accouns are calculaed usng he annual overlap mehod. The mula an annual overlap chanlnked value n he uarer n year s D where D s he value n prevous years prces s curren prces and 4 X X Aggregaon of chan-lnked seres can be rewren o D Because he normal rules of addvy apples daa n prevous years prces he aggregae D euals he sum of s componens D D 3 where s an ndex of he subcomponens. When s combned wh 3 4 where

3 3 Formula 4 can hen be used o aggregae chan-lnked uarerly seres wh use of he prce ndces from he prevous year. Ths mehod of aggregaon s used o creae ndrecly seasonally adjused chan-lnked seres by aggregaon of he seasonally adjused subcomponens. Quarerly conrbuons o growh An expresson he conrbuon o growh ( from a subcomponen can be derved from he above mula 4. The resul wll depend on he uarer because he growh from fourh o frs uarer depends on prce deflaors wo dfferen years. An expresson he conrbuon o growh he second hrd and fourh uarer wll heree be derved a frs and aferwards wll be derved he frs uarer. To calculae he uarerly conrbuon o growh he second hrd and fourh uarer (34 mula 4 s used as sarng pon ( ( The above euaon shows ha he growh n an aggregae (lef sde can be seen as he sum of conrbuons from each of he subcomponens. The uarerly conrbuon o growh from he subcomponen ( can heree be calculaed as 34 5

4 4 The uarerly conrbuon o growh he frs uarer can n a smlar manner be derved usng mula 4 ( ( Agan an euaon s derved where he growh n an aggregae (lef sde euals a sum of conrbuons from each of he subcomponens. The uarerly conrbuon o growh n he frs uarer from he subcomponen ( can heree be calculaed as 6 Boh of he mulas conrbuon o growh 5 and 6 looks a lo lke normal conrbuon o growh calculaons. The dfference s ha he conrbuons s weghed wh he relaonshp beween he prce ndces he subcomponen and he aggregae n he prevous year. The dfference beween 5 and 6 s ha he annual prce ndex he prevous year changes from - o when he growh from fourh o frs uarer s consdered n 6. The above mehod o calculae conrbuons o growh was derved usng he mula aggregaon 4 where he chan-lnked seres s aggregaed usng prce ndces from he prevous year. Ths was derved usng he addvy of daa n prevous years prces bu an expresson a chan-lnked aggregae as a funcon of s subcomponens can also be derved wh he use of addvy n curren prces 7

5 Because 8 7 can be rewren o Usng 8 expressons he conrbuon o growh chan-lnked uarerly seres can be derved wh use of he same mehods ha were used o derve 5 og 6. These expressons are shown below n 9 and In hese alernave calculaons of conrbuon o growh he prce ndces he specfc uarer s used as weghs. Theree he wo erms n he numeraor s always weghed usng dfferen prce ndces whch was only he case he conrbuon he frs uarer growh wh he mulas derved usng prevous years prces. The wo procedures usng respecvely he addvy of daa n prevous years prces and addvy n curren prces can also be used n combnaon. For example. Usng he above expressons an expresson he conrbuon o growh he frs uarer could be derved where he chan-lnked values n he frs uarer would be weghed wh he annual prce ndex - whle he values n he fourh uarer of - would be weghed wh he prce ndex of ha fourh uarer. Only when usng addvy n prevous years prces boh uarers an expresson he conrbuon o growh wh he same prce ndex n boh uarers can be obaned as n mula 5. Ths s why mula 5 and 6 has been chosen o calculae he conrbuons o growh he uarerly seasonally adjused growh n he dansh uarerly naonal accouns. 5

6 Annual conrbuons o growh Expressons conrbuons o uarerly growh n an aggregae was calculaed above. In hs secon conrbuons o annual growh n a uarerly aggregae wll be calculaed. Tha s he growh from uarer n year - o uarer n year. Ths corresponds o he above dervaons of uarerly conrbuons o growh he frs uarer excep ha he same s used n year and -. Usng 6 he expresson below s derved he annual conrbuon o growh a uarer n year Alernavely 0 can be used nsead In boh and prce ndces wo dfferen years are used n he calculaon whch resuls n ha hese mulas annual conrbuons o growh are nconssen wh he mula calculang conrbuons o growh annual chan-lnked daa as seen below n 3 3 Bu f a combnaon of he wo approaches are used as was shown n he las secon a page 5 a mula annual conrbuons o growh uarerly daa can be calculaed as n 4 4 In 4 wo dfferen prce ndces are sll used he wo uarers bu now hey are from he same year. Anoher mporan propery of 4 s ha because he frs prce ndces are he whole year and he ohers are he specfc uarer mula 4 s conssen wh 3. When calculang annual conrbuons o growh he sum of all uarers n a year usng 4 resuls n he same as usng 3. I s sragh ward o show hs algebrac by replacng (and he corrrespondng prce ndces wh n 4. Because of hs conssency 4 has been chosen o calculae he annual conrbuons o growh uarerly daa n he dansh uarerly naonal accouns. 6