APPLIED STATISTICS. Economic statistics

Transcription

1 APPLIED STATISTICS Ecoomic saisics Reu Kaul ad Sajoy Roy Chowdhury Reader, Deparme of Saisics, Lady Shri Ram College for Wome Lajpa Nagar, New Delhi Ja-2007 (Revised 20-Nov-2007) CONTENTS Time series aalysis Compoes of Time series Models for Time series Deermiaio of Tred Growh curve Aalysis of Seasoal Flucuaios Cosrucio of seasoal idices Mehod of simple averages Raio o red mehod Raio o movig average mehod Lik relaive mehod (Pearso s mehod) Measureme of cyclic moveme Measureme of radom compoe Idex umbers Problems ivolved i compuaio of idex umbers Calculaio of idex umbers Price idex umbers Quaiy idex umber Value idex umer Idex umbers based o average of price-relaives Lik ad chai idices Tess for idex umbers Cosumer price idex umber Cosrucio of cosumer price idex umber Limiaios of idex umbers Pracice sessio Keywords Time Series; Addiive & muliplicaive models; Treds; Growh curves, Price idex umber; Quaiy idex umber; Value idex umber; Chai idices; Price relaives; Cosumer price idex

2 Time series aalysis A ime series is a se of saisical daa spread over ime. Example: he daily producio of milk i a milk pla, he weekly sales i a deparmeal sore, he mohly publicaio of he Cosumer Price Idex, he quarerly saeme of GNP, as well as he aual reveue of a firm. A umber of facors ifluece he value of he variable uder sudy. The aim of he ime series aalysis is o ideiy ad isolaes hese ifluecig facors for forecasig purposes. Time series aalysis provides kowledge abou he flucuaios i ecoomic ad busiess pheomea. The pas red is projeced io fuure red, o predic he chages, which are likely o occur i he ecoomic aciviy. Compoes of Time series The facors ifluecig he moveme of a ime series are called he compoes of a ime series. There are four compoes of a ime series, viz.:. Tred [T ]- Secular red or Log erm moveme 2. Seasoal variaios [S ] - Periodic chages or shor erm flucuaios 3. Cyclic variaios [C ] - Periodic chages or shor erm flucuaios 4. Irregular movemes [R ] A ay give ime, he value of ime series may be obaied by he combiaio of some or all of hese. The par of he ime series, which ca be explaied may be aribued o compoes:. Secular Tred 2. Seasoal Variaio ad 3. Cyclical Variaio. While he par which ca o be explaied may be aribued o he: 4. Radom Compoe. Models for Time series The priciple objecive of sudyig ime series aalysis is o ideify he various compoes of he ime series (some or all of which may be prese i a ime series) ad o measure each oe of hem separaely. However, he aalysis of ime series would deped o how hese compoes have bee combied. There are geerally wo mehods of combiig he effecs of various compoes. I he firs mehod, i is assumed ha he various compoes operae idepedely of each oher ad he value of he ime series is obaied by merely addig hem i.e. U = T + S + C + R () where, U represes he value of he variable uder sudy a ime, T is he red value, S is he seasoal variaio, C is he cyclic variaio ad R sads for he radom compoe. The model cosidered is called he Addiive Model. I his case, S will have some posiive values ad some egaive, depedig o he seaso of he year. Similarly C will have posiive ad egaive values depedig o wheher we are above ormal or below ormal phase of he cycle ad sum oal of hese values for ay year/ cycle will be zero. R will also have posiive ad egaive values ad sum oal of hese values over a log period of ime (R ) will be zero. If he ime series values are give aually, he S compoe will o appear. The very assumpio ha he various compoes operae idepedely of each oher becomes he major drawback of his model. I realiy he disic compoes operae i cojucio wih 2

3 each oher ad i is o possible o separae hem. I fac, oday s value is affeced by yeserday s value ad i ur affecs he fuure value o a large exe. I he secod mehod, i is assumed ha he various compoes operae proporioaely o he geeral value of he series. Thus i his case, S, C ad R, isead of assumig posiive ad egaive values, ake values below or above uiy so ha he geomeric mea of hese values i a year or i a cycle or over a log period of ime is uiy. Mos of he busiess ad ecoomic ime series follow he muliplicaive model. I his case we ake: U = T. S. C. R (2) However, if we ake logarihms of boh side i T-2, he above model reduces o: log U = log T + log S + log C + log R (3) i.e. a addiive model oly. NOTE: Someimes, he compoes of a ime series may be combied i a umber of differe ways such as: U = T + S. C. R U = T + S + C. R (4) U = T. C + S. R These are kow as Mixed models. Deermiaio of Tred By he secular red of a ime series we mea he overall or persise, smooh log erm moveme, which may be upward, dowward or cosa. This compoe of he ime series is basically sudied for predicive purposes. Also oe may wish o sudy red i order o isolae i ad he elimiae is effec o he ime series. The red ges affeced by: he iroducio of ew echology, populaio growh, marke flucuaio, ec. Is duraio is over several years. There are maily four differe mehods for he measureme of Tred.. Graphical Mehod This mehod does o ivolve ay mahemaical compuaios. I his mehod we plo a free had smooh curve based o ime series values U ad ime. The ploed curve depics oly he direcio of red ad does o predic he fuure values. The give ime series may have a icreasig or decreasig red. However, oe may come across cases where ime series values flucuae aroud a cosa erm. EXAMPLE - The followig able gives he producio of cars (i housads) by a car maufacurer i Idia over he years. Draw a graph o show he producio red. 3

4 Year () Producio (U ) SOLUTION: Producio (No. of cars i housads) Producio Tred Years CONCLUSION: The graph shows a upward red over he years showig a icrease i demad of cars. 2 Semi-average mehod I his mehod, he ime series daa U are divided io wo equal groups wih respec o ime. The firs group cosiss of he firs half periods (years) ad he secod group cosiss of he remaiig periods. If he ime series daa cosiss of a eve umber of periods, say 2 where is a ieger, he each group will cosis of periods. A simple arihmeic mea of U values is compued for each group ad is ploed agais he mid poi of he respecive ime period covered. I case is eve, oe of he groups has a mid-year ad hus he middle poi of he wo mid years will be he mid period where he average of each group is o be ploed. The lie joiig he wo pois gives he required red lie. I case he ime series cosiss of a odd umber of periods (years), say 2+, he he value correspodig o he mid period is omied o obai wo equal groups ad agai a sraigh lie is ploed agais he mid-values of each group. The semi average mehod does o depic he rue red lies. However, oe ca forecas he fuure value for ay give year by obaiig correspodig poi o he red lie. The semi-average mehod is affeced by he exreme values ad does o esure he elimiaio of shor erm ad cyclic variaios. EXAMPLE - 2 The followig able gives he aual producio of cerai ypes of crops culivaed i a mouai regio over a period of 5 years. Fi a red lie by he mehod of semi-averages. 4

5 Years Producio (i 000 Kg.) Years Producio (i 000 Kg.) SOLUTION Sice here are 6 observaios, he average of firs 8 observaios is ad he average of las 8 observaios is The red lie is as show below: CONCLUSION: The graph shows a upward red. This implies ha he producio of crops is icreasig over he period. 3 Mehod of Leas Squares I his mehod we fi a suiable mahemaical model o he give ime series daa. The selecio of he appropriae mahemaical model is made eiher by ploig he give ime series daa agais ime or by sudyig he aure of he variable ivolved i he series. The cosas of he mahemaical model are esimaed from he give series by he mehod of leas squares so ha he sum of squares of he deviaios of acual values from he correspodig esimaed values is miimum. For example, cosider a liear model 5

6 U = b0 + b + ε (5) 2 where, b ad b, are wo cosas ad ε, is he radom error erm disribued N (0, σ ). 0 Le a ad b be he esimaes of b 0 ad b. The ormal equaios for obaiig he values of a ad b are: U = a+ b = + U a b 2 (6) Thus he fied model will be of he form: Uˆ = a+ b (7) where, a, is he iercep which he esimaed lie cus o he axis of U, ad b, is he slope of he red lie. By his mehod ( U Uˆ 2 ) is miimum ad also his mehod gives ( U U ˆ ) = 0. The red lie obaied by his mehod o oly depics he red bu also eables oe o forecas he fuure values for ay give year. The mos commoly used curves for fiig red over ime are: Liear Model: U = b 0 + (b - ) Quadraic Model: U = b 0 + (b - ) + (b 2-2 ) Cubic Model: U = b 0 + (b - ) + (b - 2 ) + (b ) Logarihmic Model: U = b + b l() 0 (b) Expoeial Model U = b0e or l(u) = l(b 0) + (bl) Iverse Model U = b0 + (b / ) (b 0 + bl) Growh Model U = e or l(u) = b + (b ) 0 l P 0 0 l b ower Model U = b ( ) or l(u) = l(b ) + b l() (8) The esimaes obaied by he mehod of leas squares are ubiased ad have miimum variace. However, hey sill are o opimum. This is because i he above models we assume ha he variable U depeds oly o he facor (ime period), whereas i acual siuaios U also depeds o a umber of oher explaaory variables. EXAMPLE - 3 The followig able gives he producio of oil by a oil producig compay over a period of 0 years. Fi a liear red by he mehod of leas squares. 6

7 Year () Producio (U ) (i 000Gallos) SOLUTION Cosider he expressio (6), i.e. U ˆ = a+ b. For esimaig he cosas by he mehod of leas squares we cosruc he followig able: Years () Producio (U ) (i 000 gallos) u 2 Tred Values TOTAL O solvig he ormal equaios for esimaig a ad b we ge: a = ad b = 9. Hece he red equaio is: U ˆ = By givig differe values of, he esimaed red values for differe years are show i he las colum of he above able. The followig graph shows he red of he producio of oil. Fiig of Tred Lie by Leas Square Mehod Producio (U) ('000 gallos) Tred (' 000 gallos) Producio ('000 gallos) Years 7

8 CONCLUSION The upward red shows he icrease i producio from 990 o Movig average mehod Fiig of red by he mehod of movig averages is based o compuig a series of successive arihmeic averages over a fixed umber of years. This mehod smoohes ou he flucuaios of he give daa wih he help of a movig average. To obai red by movig average we cosider he observaios for he firs few years (say ) of he series i a group ad fid heir arihmeic average i.e. he arihmeic average of he firs values, is called he period of he movig average. Eer he average value of his group agais he mid poi of he group. Nex delee he s observaio from his group of years ad add he (+) h observaio o he group. Compue he average of his group. This gives he secod mea. Agai eer he average value of he group agais he mid poi of he group. Repea his procedure ill we exhaus all he observaios. Whe is odd, he movig averages correspod o a ime period i he give ime series. However, whe is eve hey are placed bewee he wo middle values. I his case, we furher calculae a movig average of period wo of hese movig averages ad place i bewee hem. The values ow correspod o a ime period i he give ime series. Fially, draw a graph ploig he movig average values agais ime. This gives he required red. The mehod of movig averages reduces he effec of exreme observaios i he series. If he cyclical variaios are regular boh i period ad ampliude he his mehod elimiaes he flucuaios o a grea exe provided he period of he movig average is equal o or a muliple of he period of cycles ad red is liear. However, his mehod does o provide a umber of red values for each ed of he series ad ca o be used for forecasig purposes. EXAMPLE 4 The followig able gives he cosumpio of rice i a paricular village of Wes Begal. Deermie he uderlyig red by a 3 year movig average ad a 5 year movig average ad comme. Year Cosumpio ( 000 Quials) Year Cosumpio ( 000 Quials) SOLUTION For fiig red b y he mehod of m ovig averages, followi g able has bee cosruced. 8

9 Year Cosumpio (i 000 quials) 3 year movig average 5 year movig average The followig graph shows he red. Fiig of Tred by he Mehod of Movig Averages Cosumpio (i '000 quaals) 3 year movig average 5 year movig average Y ears CONCLUSION I has bee observed ha he smoohess of he red depeds o he value of. The fied red shows a icrease i he cosumpio of rice over he period. 9

10 Growh curve M os of he ime series relaig o busiess ad ecoomic pheomea over log period of ime do o exhibi growh which is a a cosa rae ad i a paricular direcio over log period of ime, chroological series are o likely o show eiher a cosa amou of chage or a cosa raio of chage. The rae of growh is iiially slow, he i picks up ad becomes faser ad ges acceleraed, he becomes sable for some ime afer which i shows reardaio. The curves, which ca be fied o such daa are called growh curves. These asympoic growh curves are suiable for a spaially limied uiverse, i which a populaio grows ad are also useful i describig he growh of a idusry, where iiially he growh is slow durig he period of experimeaio, he he growh is rapid durig he period of developme ad he he growh is agai slow ad sable whe a period of sabiliy is reached. The followig are some of he impora growh curve, which are geerally used o describe he measuremes i a ime series. Modified Expoeial Curve 2 Gomperz Curve 3 Logisic or Pearl-Reed Curve NOT E: I all he hree growh curves meioed above, he umber of parameers exceeds he umber of variables ad hus he usual echique of leas squares o fi hese curves fails. Modified Expoeial Curve This curve is a modified form of he expoeial curve. This curve o oly describes a red i which he amou of growh declied by a cosa perceage, bu he curve also approaches a upper limi called asympoe. The gee ral equaio of his curve is: U=a+bc (9) where, a, b, c, are he hree cosas or parameers o be deermied, ad U ad, are he variables. Fiig of Modified Expoeial Curve There are followig wo mehods of fiig a Modified Expoeial Curve: Mehod of Three Seleced Pois 2 Mehod of Parial Sums. Mehod of Three Seleced Pois Cosider he hree ordiaes U,U,U,, such ha = correspodig o hree equidisa pois of ime Subsiu ig U, ; U, ; U, i he geeral equaio of modified expoeial curve we ge:

11 U a bc 2 2 = + U = a+ bc U3 = a+ bc 3 (0) 2 Thus, = [c 3 2 ] ad 2 U U bc U3 U2 = bc [c ] () 2 U U U U => = c 2 (2) => c U 2 3 U ( ) 2 = U2 U (3) U U => U U = bc U2 U (4) => (U U ) U U b =. (U3 2U2 + U ) U3 U2 (5) Similarly, = a U bc => UU a = U U U2 + U (6) Oce he values of a, b, c has bee compued he fied curve is obaied as: Û a bc = + (7) Mehod of Parial Sums I his mehod we divide he daa io hree equal pars, each par coaiig cosecuive values of U say =,2,,, he = +, +2,, 2 ad fially = 2+, 2+2,, 3. Le S,S,S be he parial sums of he hree pars give by S = U, S = U ad S = U 2 3 = = + = 2+ Subsiuig he value of U i he above equaios we ge: (8)

12 Similarly, = + = + c S (a bc ) a bc = c 2 c c ) a bc + S (a b 2 = + = + = + c 3 2+ c S 3 = (a + bc ) = a + bc = 2+ c (9) => S S c c S S c c = = c + 2 (20) => c S 3 S 2 = S2 S (2) Also, => + c S2 S = b[c c] (22) c 3 c (S ) 2 S b =. (23) 2 c (S 2S + S ) 3 2 (S2 S ) Also, S2 S = a +. 2 c (S 2S + S ) (c ) 3 c (c ) 3 2 (24) => => a = (S S ) 2 2 SS 3 S 2 a = S3 2S2 + S 2 (S2 S ) S 2S + S 3 2 (25) (26) Subsiuig he values of a, b, c we ge he fied modified expoeial curve. Gomperz Curve The Gomperz curve amed afer Befami Gomperz, is a ype of a mahemaical model for a ime series where growh is he slowes a he sar ad ed of he ime period. The Gomperz curve describes a red i which he growh icremes of he logarihms are decliig by a cosa perceage. Thus he aural values of he red would show a decliig raio of icrease, bu he raio does o decrease by eiher a cosa amou or a cosa perceage. 2

13 The geeral equaio of he Gomperz curve is: U c = ab ( 27) where, a = he upper asympoe c = he growh rae b = cosa (b ad c ca be egaive) The equaio ca also be wrie as: U c be = ae (28) Fi ig of Gomperz Curve I he Gomperz curve, as i he case of modified expoeial curve, he umber of parameers exceeds he umber of variables. Thus he mehod of leas squares fails. Cosider he equaio of Gomperz curve U This ca be wrie as: Y = A+ Bc ( 3) where, Y = log U ; A = log a; B = log b The above equaio is comparable o he equaio of modified expoeial curve. We ca hu s fi a Gomperz curve by he same pricipal as ha used i modified expoeial cur ve i.e. b y he mehod of hree seleced pois ad he mehod of parial sums. Logisic Curve / Pearl-Reed Curve c = ab (29) Takig log o boh sides we ge: log U = log a + c log b (30) A logisic fucio or logisic curve models he S-shaped curve growh of some se P. The iiial sage of growh is approximaely expoeial, he as compeiio arises, he growh slows ad a mauriy growh sops. The curve was firs developed by P. F. Verhuls i 838. Furher, Raymod Pearl ad Lowell J. Reed idepedely developed his curve i 920. I is ow frequely referred o as he Pearl-Reed Curve. The Verhuls equaio is a ypical applicaio of logisic equaio ad is a commo model for populaio growh which saes ha he rae of reproducio is proporioal o exisig populaio, all else beig equal ad ha he rae of reprodu cio is proporioal o amou of available resources all else beig equal. The geeral form of he logisic curve is: k U =, b>0 a b + e + (32) 3

14 wher e, k, a, b are he parameers of he curve ad k = max (U )i.e. he maximum value which he variable ca ake over all values of ime. If he give daa follows logisic law of growh he heir reciprocals follows modified expoeial law:.[ e a+ b ] U = + (33) k or a e b = +.(e ) (34) U k k if e A ; B= ad C=e k k a = b we ge (35) A BC U = + (36) or Y = A+ BC (37) which is he equaio of a modified expoeial curve. Fiig of Logisic Curve There are several mehods of fiig a logisic curve, viz: ) Mehod of Three Seleced Pois 2) Yule s Mehod 3) Hoellig s Mehod 4) Nair s Mehod 5) Mehod of Successive Approximaios 6) Mehod of Sum of Reciprocals 7) Rhodes Mehod We describe below some of he commo ly used mehods. Mehod of Three Seleced Pois Cosider he equaio of logisic curve: k U =, b>0 a b + e + (38) I his mehod we selec hree ordiaes U,U 2,U 3 correspodig o hree equidisa pois of ime, 2, 3 such ha 3 2 = 2 Now a b k + e + = U (39) 4

15 k k k log = a + b ;log = a + b ;log = a + b (40) => 2 3 U U2 U3 O simplificaio we ge: k U2 U log. = b( 2 ) (4) U2 k U similarly, k U U = 3 2 log. b(3 2) U3 k U2 (42) as 3-2 = 2 -, we ge: k U2 U k U3 U2 log. = log. U2 k U U k U 3 2 (43) Afer simplificaio we ge: U(U+ U) 2UUU k = U2 UU3 (44) k U2 U b= log. 2 U2 k U ad (45) k k U2 U a = log log. U U k U 2 2 (46) Usig he values of a, b ad k we ge he bes fied equaio. Yule s Mehod Agai cosider he logisic curve as: a+ b k + e = (47) U Le us suppose he value of k is kow, say k he a b k + e + = (48) U k => a b log Y (say) U + = = (49) => Y = a+ b (50) 5

16 which is he equaio of sraigh lie havig wo parameers ad wo variables. Thus he ormal equaios by he mehod of leas squares are: Y = a+ b (5) Y = a + b 2 (52) Thus he values of a ad b will give he bes fied curve. Hoelig s Mehod Agai cosider he logisic curve as: a+ b k + e = (53) U differeiaig wih respec o we ge: a b k. du b.e + = (54) 2 U d du d 2 bu = e k a +b (55) du U = b U d k (56) => Furher, if he ierval of differecig is o oo large he we ca approximae U du d by U U U b = > = b + U (57) U k Le U b = (58) Y ; A=-b; B= U k => Y = A+ BU (59) A gai A ad B ca be es imaed u sig leas square mehod ad hece we ca fid b ad k. Furher a ca be obai ed o assumig ha he curve passes hrough mea o f U ad mea of. A alysis of seasoal flucuaios I may busies s ad ecoomic pheomea he seasoal paers are commoly observed due o social ad religious cusom s, chage s i weaher codiios highlighig he effec of seasos o daa. The duraio of such flucua ios is wihi a period of 2 mohs based o mohly, quarerly, weekly, daily or hourly daa. 6

17 The sudy of seasoal variaios is ecessary for wo reasos:. Oe may be ieresed i forecasig some fuure mohly or quarerly movemes 2. Oe may be ieresed i isolaig ad elimiaig he effec of red, seasoal variaios, irregular flucuaios so as o sudy he effec of cycles. These flucuaios are regular i aure ad ed o repea hemselves year afer year. Cosrucio of seasoal idices The differe mehods for measurig seasoal variaios are: Mehod of simple averages I his mehod we arrage he daa by years ad mohs or quarers (depedig o wheher mohly or quarerly daa is available). Nex we compue he average for each moh/quarer for all he years i.e. we co mpue he average x i (i =,2,...,, = 2 if mohly daa is available ad = 4 if quarerly d aa is available) over all he years. We he compue he avera ge of all he average s i.e. x= xi i= Fially, he seasoal idex h x i for he i moh/quarer is compued as 00 i.e. by expressig x he respecive averages as perceage of he overall average. NOTE: The sum of he seasoal idices is 200 for mohly daa ad 400 for quarerly daa. This mehod is based o he assumpio ha he give ime series is idepede of he red ad cyclic variaios. EXAMPLE 5 The followig able shows he mohly cosumpio of sugar i Idia for 4 years. Compue he seasoal idices by simple averages mehod. SOLUTION Year Moh JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Mohs Year oal AVG. (Mohly) seasoal idex JAN FEB MAR

18 Mohs Year oal AVG. (Mohly) seasoal idex APR MAY JUN JUL AUG SEP OCT NOV DEC TOTAL AVG EXAMPLE 6 The quarerly cosumpio of elecriciy over a period of 7 years is give. Compu e he s easoal idex for each quarer. SOLUTION YEAR QUARTER Ja - Mar Apr - Ju Jul - Sep Oc - Dec Quarerly Toal Quarerly Average Seasoal Idex Average of Quarerly Average = Raio o red mehod This mehod is applicable uder he assumpio ha he seasoal variaio for ay give moh/quarer is cosa facor of he red. I Raio o red mehod before compuig he seasoal variaios we firs fid he yearly averages/oals for all he years. We he fi a mahemaical model, usually liear, quadraic or expoeial ec. o he yearly averages/oals ad obai he red values by he mehod of leas squares. This gives aual red values. The mohly/quarerly red values are obaied by suiably adjusig he red equaios. Nex, by assumig a muliplicaive model, red elimiaed values are obaied by expressig 8

19 he give ime series values as perceage of he red values. These perceages coai he seasoal, cyclic ad irregular variaios. Now he cyclic ad irregular variaios ca be elimiaed by averagig he perceages for differe mohs/quarers. Fially, he seasoal idices are adju sed o a oal of 200 for mohly daa ad 400 for quarerly daa by muliplyig hem by a correcio faco r c give as: c = 200 (f Toal of he idices or m ohly d aa) 400 c = (for quarerly daa) Toal of he idices EXAMPLE 7 The followig able gives he quarerly producio of coo fabric (i 000 ms.) by a maufacurer for he years 200 o Compue he seasoal idices by raio o red mehod. YEAR QUARTER Ja - Ma r Apr - Ju Jul - Sep Oc - Dec SOLUTION To obaied he yearly red values we firs fi a liear model g ive by a expressio U = a + b, o he yearly averages. The followig able shows he yearly red values obaied by he mehod of leas squares. The values of cosas are: a = 49.4 ad b = 33.3 Year Ja-Mar Apr-Ju Jul-Sep Oc-Dec oal Average T = UT T 2 Yearly () (U) Tred Value From he above cosas he value of he yearly icreme = 33.3 => he value of he quarerly icreme is 33.3/4 = Thus usig he above yearly red values he quarerly red values are show i he followig able: 9

20 YEAR Quarerly Tred Value Ja - Mar Apr - Ju Jul - Sep Oc - Dec The above able has bee compued as follows: The red value for he secod quarer for ay year is o baied by subraci g half of he quarerly icreme from he red value of he correspodig year. Similarly, red value for he hird qu arer is obaied by addig h alf of he quarerly icreme o he red value of ha year. The red value for he firs quarer is obaied by subracig quarerly icreme from he red value for he secod quarer ad fially he red value for fourh quarer is obaied by addig quarerly icreme o he red value for he hird quarer. Fially, he red elimiaed red values obaied are as follows: YEAR Tred Elimiaed Values Ja Mar Apr Ju Jul Sep Oc Dec Toal Average Seasoal Toa l = 400 Idex For compuig he values of seasoal idex he value of h e correcio facor c = EXAMPLE - 8 The daa give i he followig able shows he mohly producio of wool (i.000 os) by sae idusries from Ja 200 o Dec Compue he mohly seasoal idices by raio o red mehod. Year JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

21 SOLUTION We firs fi a liear red o he yearly averages by he mehod of leas squares. year () yearly oal yearly average (U) T = UT TT Yearly Tred Values Toal where, a = ; ad b= Thus he red equa io is: U T = T ad Mohly Icreme = 3.35 Mo hly T red Values Year JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Tred Elimiaed Values Ye ar JAN FEB MAR APR MAY JUN JUL AU G SE P O CT NOV DEC Toal Avg SI (Adj) Correcio Facor = Raio o movig average mehod This mehod is based o calculaig movig averages by cosiderig = 2 for mohly daa ad = 4 for quarerly daa. For mohly daa firs calculae he successive averages for he groups of size 2 ad he ake a 2-poi movig avera ge of hese averages. The resula movig averages will give he esimaes of he combied effec s of re d ad cyclical variaios. Leavig he firs 6 mohs ad he las 6 mohs (as = 2), cover he give 2

22 daa series as he perceages of he 2-poi movig average values i.e. ((give daa)/(2-poi movig average)*00). These perceages would ow represe seasoal variaios alog wih radom compoes. Furher, he radom compoe is elimiaed by averagig hese mohly perceages. Sice he sum oal of hese seasoal idices is o equal o 200 (for mohly daa) ad 400 (for quarerly daa) so, fially adjused seasoal idices are compued o make he sum of he idices 200 or 400 by muliplyig hem hrougho u by a correcio facor c defied earlier. EXAMPLE 9 The Airpor auhoriies co lleced daa o he umber of aircrafs, which could o fly o ime from Delhi airpor due o adverse weaher codiios from 200 o The umber of aircrafs per mo h for he period are sho w i he followig able. Compue seasoal idices by raio o movig average mehod. Year Moh JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC No. of Aircrafs Year Moh JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC No. of Aircrafs Year Moh JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC No. of Aircrafs Year Moh JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC No. of Aircrafs SOLUTION To compue he seasoal idices by 2 poi raio o movig avera ge mehod followig wo ables have bee cosruced: Year Moh No. of Aircrafs 200 JAN FEB MAR APR MAY 53 Toal 2-poi Movig Average 2-poi Movig Average Raio o Movig Average 22

23 Year Moh No. of Aircrafs Toal 2-poi Movig Average 2-poi Movig Average Raio o Movig Average 200 JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB

24 Year Moh No. of Aircrafs Toal 2-poi Movig Average poi Raio o Movig Movig Average Average 2003 MAR APR MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB M AR APR MAY JUN JUL AUG SEP OCT NOV DEC 32 24

25 Moh 200 Year Seasoal Idices Adjused Seasoal Idices JAN FEB MAR APR MAY JUN JU L AUG SEP OCT NOV DEC TOTAL C = EXAMPLE 0 The followig daa gives he quarerly sale of umbrellas i a commuiy marke. Compue seasoal idices by ra io o movig average mehod. Quarer Ja-Mar Apr-Ju Jul-Sep Oc-Dec SOLUTION Year Quarer S ale 4-quarer Toal of wo 4-quarer Raio o (U) movig oal 4-quarer movig movig movig oal average average 200 Ja-Mar 5876 Apr-Ju Jul-Sep Oc-Dec Ja-Mar

26 Year Quarer Sale (U ) 4-quarer movig oal Toal of wo 4-quarer movig oal 4-quarer movig average Raio o movig average 2006 Apr-Ju Jul-Sep Oc-Dec Ja-Mar Apr-Ju Jul-Sep Oc-Dec Ja-Mar Apr-Ju Jul-Sep 4633 Oc-Dec 5985 Tred Elimiaed Values Year Ja-Mar Apr-Ju Jul-Sep Oc-Dec Toal Average Adj. Seasoal Idices Toal = 400 Toal of Average = = > he v alue of c = Lik relaive mehod (Pearso s mehod) This mehod is based o he cocep of averagig he lik relaives. Lik relaive for ay seaso (moh/quarer) is defied as: 26

27 Li Value of he i seaso alue of he ( i-) seaso h k relai h ve for i seaso = 00 V h (60) (lik relaive for he firs seaso ca o be defied) Firs we cover he give daa (mohly/q uarerly) for all yea rs i erms of lik relaives by he above expressio. Nex compue aver age lik relaives for each moh/quarer over he years. Cover he average li k relaives i o chai rela ives o h e basis of he expressio: h Chai rel aive for i seaso = Averag h h e lik relaive for i seaso chai relaive of (i-) seaso 00 (6) wh ere, he chai relaive for he firs seaso is ake o be 00. Now a ew chai relaive fo r he firs seaso is obaie d o he basis of chai relaive of he las seaso. This value i geeral will o be equal o 00 (as assumed by us) due o maily red. We hus adjus he chai relaives for red by subracig c, 2c,., c from February, March,.., December values (assumig a liear red). Thus h h The adjused chai relaive of i seaso = chai relaive of i seaso - (i-) where New chai relaive for firs seaso -00 c= ad = 2 for mohly daa ad 4 for quarerly daa. Fially, he ad jused seasoal i dices are compued a s: (62) c (63) Adjused seaso h h Adjused chai relaive for i seaso al id ex for i seaso = 0 0 Average of adjused chai relaives (64) EXAMPLE - The followig able gives he quarerly producio of coffee (i 000 kg) maufacured by XYZ coffee idusry durig he years Compue he seasoal idices by he mehod of lik relaives. ` Year Ja-Mar Apr-Ju Jul-Sep Oc-Dec

28 SOLUTION The followig able shows he calculaios for compuig seasoal idices by he mehod of lik relaives. Quarerl y Lik Relaives Year Ja-Mar Apr -Ju Jul-Sep Oc-Dec Toal Average Chai Relaives Adj. Chai Relaives (correced for re d) Adj. Seasoal Idices Toal = 400 Correcio Facor = EXAMPLE - 2 The daa give i he followig able represes he mohly cosumpio of milk (i 000 ls.) by ice-crèam idusries from Ja o Dec Compue Seasoal idices by he mehod of lik relaives. Cosumpio of Milk (i 000 ls.) Year Ja Feb Mar Apr May Ju Jul Aug Sep Oc Nov Dec

29 SOLUTION Lik Relaives Year Ja Feb Mar Apr May Ju Jul Aug Sep Oc Nov Dec Toal Avg CR ACR ASI Correcio Facor = 0.3 CR = Chai Relaives ACR = Adjused Chai relaives ASI = Adjused Seaso Idices Measureme of cyclic moveme A very simple mehod of measurig he cyclic variaios of a ime series is he residual mehod. I his mehod, we divide he give ime series values by red values ad he seasoal idices, by assumig a muliplicaive model. The resula will give us he cyclic ad he irregular compoe. The irregular compoe is furher elimiaed by usig movig average of a appropriae period. NOTE: The compoes may be elimiaed i ay order. Measureme of radom compoe There does o exis a sophisicaed mehod of deermiig he radom compoe of a ime series geerally he o-radom compoes are deermied ad whaever is lef uaccoued for by hese compoes cosiues he radom compoe of he series. Idex umbers Prices of various commodiies keep chagig over a period of ime. While he prices of some of he commodiies may icrease, hose of ohers may decrease. Also differe commodiies are measured i differe uis. For example whea ad rice i kilograms, milk, oil ec. i liers, cloh i meers. Thus o have a geeral idea abou he collecive chage i he prices of a group of relaed commodiies, we have o reach a a sigle represeaive figure called a idex umber. A idex umber is a saisical device desiged o measure he relaive chage i a variable or a group of variables over a period of ime, over differe places, professios ec. The variable may refer o prices of commodiies, heir quaiies cosumed or sold or impored or expored, scores obaied by a sude i differe ess ec. The period wih which he comparisos are made is called he base period. Idex for he base period is 29

30 always ake as 00. Also his period should be a period of ecoomic sabiliy ad should be close o he period of compariso, so ha compariso are o he period of compariso, so ha comparisos are o uduly affeced by chagig ases, habis, echology ec. Idex umbers have become icreasigly impora o ecoomiss, policy makers ad busiess ad goverme orgaizaios. These decisio makers sudy he direcio as well as he magiude of he price movemes i a ecoomy, o formulae ecoomic policies, ake execuive decisios, decide abou he dearess allowace ec. Wholesale price idex umbers are used i he measureme of chage i he geeral price movemes i a ecoomy. Whereas cosumer price idex umbers serve as a measure of reail price movemes. Idex umbers are also used i ime-series aalysis. Through he sudy of log erm reds, seasoal variaios ad cyclical developmes, busiess decisio makers ad beer acquaied wih he chagig ecoomic codiios. Thus hey are able o ake he righ decisios. Idex umbers are used i measurig he purchasig power of moey. By purchasig power of moey we mea he quaiy of goods ha a fixed quaiy of moey ca buy. The reciprocal of a price idex umber is used o obai purchasig power of moey. Oe has ofe heard people sayig, Thiry years back I was earig oly a housad rupees per moh ad was much beer off ha oday whe I am earig Rs. 30,000/- a moh. O face of i, i seems ha oe should be beer off as moey icome has icreased hiry imes. Bu wha abou he real icome? To kow he real icome, oe has o deflae he moey icome by dividig by a price idex. By doig so we are able o express he uis of moey i erms of he purchasig power i he base period. The cocep of deflaio is also used for deflaio of icome ad value series i aioal accous. Similarly, sales of a deparmeal sore may be deflaed by cosumer price idex. Idex umbers are ow a days also beig used by psychologiss, sociologiss, educaiosiss ad public healh auhoriies. Problems ivolved i compuaio of idex umbers Before cosrucig a idex umber followig pois should be kep i mid: Purpose of he Idex: The purpose of he idex umber should be clearly saed as may of he relaed problems ge auomaically solved. For example, if oe was o cosruc cos of leavig idex for idusrial workers of Agra he he problem of selecio of commodiies is auomaically solved. Also, i is clear ha oe has o collec reail prices of he goods ad services beig used by his class. 2 Choice of Commodiies: Oce he purpose of he idex umber has bee defied, oe should ake a suiable sample of oly releva commodiies. Thus, i he above example of cosrucio of a cos of leavig idex for idusrial workers of Agra, oly hose commodiies should be ake ha are beig used by his class. I is bes o divide he commodiies io homogeous groups ad he ake a sample represeaive of he group. Umos care should be ake i keepig he qualiy of he goods he same i differe periods of ime. 3 Dae for Idex umbers: The dae for he cosrucio of a idex umber will deped o he ypes of idex ieded. For example, if oe was o measure he chages i he volume of goods maufacured he iformaio o his should be ake direcly from 30

31 maufacures or from sadard publicaios. Similarly, for he cosrucio of whole sale price idex, wholesale prices of commodiies should be ake from wholesale dealers. Wha ever be he case, as far as possible daa colleced should be accurae ad represeaive of he se of commodiies. 4 Choice of he Basic Period: The period wih which comparisos are made is called he base period. The idex for h e base period is always ake as 00. I should be a period of ecoomic sabiliy free from boom, depressios, war, famies, srikes ec. I should be close o he period of compariso. As wih he passage of ime ow commodiies keep eerig he marke while old oce may disappear hus he comparisos made will o be valid. Also base period should eiher be oo log or should be oo small. 5 Type of Average o be used for Combiig Daa: As we wa o represe he price chages or quaiy chages i a group of relaed commodiies by meas of a simple represeaive figure, various averages such as arihmeic mea, geomeric mea or harmoic mea may be used. Ou of hese, o heoreical grouds, geomeric mea is he bes as i is o uduly affeced by he exreme observaios. Also he idices based o geomeric mea are reversible. However, arihmeic mea is mos frequely used because of is ease i calculaios. 6 Choice of Appropriae Weighs: Suppose we wish o cosruc cosumer price idex he ou of he eire se of commodiies such as whea, rice, pulses, ghee, fruis, vegeables, ea, clohes, fuel ec. o all of hem are of equal imporace o us. Thus appropriae weighs should be aached o he commodiies o brig ou heir relaive imporace. Geerally prices are weighed by he quaiies ad price relaives by values. The prices or quaiies of he commodiies used for weighig may refer o eiher base period or give period or some ierveig period. The quaiies of commodiies used may be he quaiies produced, sold, impored, expored ec. By usig differe ypes of weighs, differe idex umbers have bee developed i lieraure. Calculaio of idex umbers There are differe mehods for calculaig idex umbers. The mehods are as follows: Simple Agregaive mehod This is a simple average mehod where he relaive imporace of various commodiies is o ake io accou. The price idex umber for he give period () as compared o base period (0) is give by: P pk = 00 p k= 0 k= 0k (65) where pk is he price of he k h commodiy i he give period where k =,.., NOTE: Here he prices of all he commodiies are simply added irrespecive of he uis i which hey have bee measured. Also he releva imporace of various commodiies is o ake io accou. 3

32 Similarly he quaiy idex umber for he give period () as compared o base period (0) is give by: q = 00 k k= Q0 q (66) k= 0k where qk is he price of he k h commodiy i he give period. Weighed average mehod I pracice, commodiies are measured i differe uis. Furher, o all of hem are of equal imporace o us. Thus weighs (ω k ) are assiged o hem o reflec heir relaive imporace i he idex umber. These weighs are usually cosidered as quaiies cosumed i he base period or give period or i ay paricular period. Thus he weighed average price idex is give by: P ωkpk = 00 ω p k= 0 k= k 0k (67) These weighs plays a impora role i he calculaio of idex umbers as hey reflec he releva imporace of he commodiies cosumed. Based o differe ypes of weighs differe idex umbers have bee developed. Price idex umbers Laspeyre s idex umber (Base period mehod) Here weighs are ake as quaiies cosumed i he base period i.e. ω =q. k 0k p q = 00 k 0k La k= 0 p0kq0k k= P (68) Noe: Sice he weighs are based o base per iod quaiies cosumed so hey are compued oly oce. Secodly, he quaiy of a commodiy cosumed depeds o is price. I is geerally observed people purchase less quaiy of he commodiy whose price has icreased over he period as compared o he base period. Thus, his formula usually eds o overesimae he price chages. 32

33 2 Paasche s idex umber (Give period mehod) Here weighs are he quaiies co sumed of he give period i.e. ω. k=qk P Pa 0 pkqk k= = 00 p q k= 0k k (69) Noe: Sice he weighs are based o he give period quaiies cosumed so hey are compued for each period uder compariso. Secodly, as discussed earlier, his idex umber usually uderesimaes he price chages. 3 Drobish-Bowley idex umber This price idex umber idex umbers. is compued as he arihmeic mea of Laspeyre s ad Paasche s p q p q 0 = p0kq0k p0kq k k= k= k 0k k k DB k= k= P (70) 4 Marshall-Edgeworh idex umber (Base ad give period mehod) Here weighs are cosidered as he arihmeic mea of he base period quaiies ad he ( q 0k +qk ) give period quaiies i.e. ω k =. 2 P ME 0 p k(q 0k +q k )/2 p k(q 0k +q k ) k= k= = 00= 00 p (q +q )/2 p (q +q ) 0k 0k k 0k 0k k k= k= (7) 5 Walsch price idex umber Here weighs are ake as geomeric mea (isead of he arihmeic mea) of base quaiies ad he give period quaiies i.e. ω ( ) = q q k 0k k period P Wa 0 pk q0kqk k= = 00 p q q k= 0k 0k k (72) 33

34 6 Irvig Fisher s Idex Number I is compued as he geomeric mea of Laspeyre s ad Paasche s idex umbers. P F 0 La Pa =(P P ) 2 pkq0k pkqk F k= k= => P0= 00 p0kq0k p0kq k k= k= (73) Quaiy idex umber Similarly, he various quaiy idex umbers may be compued as follows: Laspeyre s idex umber (Base period mehod) ω k = p 0k Q La o q p = 00 k 0k k= q0kp0k k= (74) 2 Paasche s idex umber (Give period mehod) ω. k=pk q p = 00 k k pa k= Q0 q p (75) k= 0k k 3 Drobish-Bowley idex umber Q DB 0 qkp0k qkpk k= k= = q0kpok q0kpk k= j= (76) 34

35 4 Marshall-Edgeworh idex umber (Base ad give period mehod) ω = k ( p +p ) 0k 2 k ME Q0 = 0k q k ( p0k k k= q (p 0k + p k k= +p ) 00 (77) ) 5 Walsch idex umber ( ) ω = p p k 0k k q p p Q Wa 0 = 0 q p p k 0k k k= 0 k= 0k 0k k (78) 7 Irvig Fisher s idex umber Q F 0 2 qkp0k qkpk k= k= = 00 q0kp0k q0kp k k= k= (79) EXAMPLE - 3 A cosrucio compay is ieresed i kowig he price ad quaiy idex umbers for he releva cosrucio maerial used over a period. The followig able gives he prices ad quaiies of he maerials used i 2002 (base period) ad 2006 (give period) Base Period Give Period Commodiy Ui Quaiy Price Quaiy Price Ceme - kg Seel - kg Bricks - o Pipes - rf

36 Cosruc various Price ad Quaiy Idex Numbers by usig differe weighs ad averages. SOLUTION: The various Price ad Quaiy Idex Numbers have bee calculaed as follows: Base Period Give Period Commodiy Ui q 0 p 0 q p p 0 q 0 p 0 q p q p q 0 Ceme - kg Seel - kg Bricks - o Pipes - rf TOTAL P La = Q La = 7.55 P Pa = Q Pa = 9.68 P ME = Q ME = 8.6 P F = Q F = 8.6 Value idex umer This idex umber is o commoly used. I is compued as he aggregae expediure for give period expressed as a perceage of he same i he base period. p q V = 00 (80) k k k= 0 p0kq0k k= EXAMPLE - 4 A idusry is maufacurig four differe producs. Keepig he average price same for he year 2000 ad 2005 he quaiy produced is as follows: Producio Commodiy Avg. Price Shirs Jeas T-shirs Jackes

37 Cosruc a value idex umber. SOLUTION Producio Value Commodiy Avg. Price Shirs Jeas T-shir s Jackes TOTAL Value Idex Nu mber = Idex umbers based o average of price-relaives Idex umbers may also b e compued by averagig he price relaives usig suiable measures of ceral edecy such as arihmeic mea or geomeric me a or harmoic mea. Le pk p 0k 00 (k =,,) be he price relaive i perceage form for each commodiy, he price idices based o arihmeic mea ad geomeric mea are give by: p k P (A.M.) = 0 00 k= p0k (8) P0 (G.M.) = p k 00 (82 k= p ) 0k Furher, if we cosider he relaive imporace of each commodiy ad assig appropriae weighs he he idices based o weighed average of he price relaives are compued as: P P 0 0 p k ωk k= 0k (A.M.) = p 00 ω k= k ωk ωk p k (83) (G.M.) = 00 (84) k= p 0k NOTE: We may similarly defie quaiy relaives ad obai idex umbers based o quaiy relaives. 37

38 EXAMPLE - 5 Through a household survey a agecy colleced he followig iformaio o expediure icurred by households durig he year 2000 ad Compue he idex umber for 2005 wih 2000 as base by usig (i) he weighed arihmeic mea of price relaives (ii) he weighed geomeric mea of price relaives SOLUTION: Price (i Rs.) Expediure o weighs Food Clohig Eeraime Educaio Coveyace Price (i Rs.) Expediure o Weighs (ω k ) Price Relaive p k /p 0k ω k p k /p 0k ( ) ωk Food Clohig Eeraime Educaio Coveyace TOTAL P 0(A.M.) = 02. P (G.M.) = p k p 0k Lik ad chai idices Till ow we have discussed various formulae based o fixed base period. However, he daa for he wo periods uder compariso may o be homogeous due o differece i ime period. Also some commodiies may leave he marke while o hers may eer, he qualiy of he commodiies may udergo a chage due o echical advacem e. Moreover, he ases ad habis of people may chage alerig heir preferece for various commodiies. Therefore, o obai homogeeiy he base period should be as close o he give period as possible ad his is bes obaied by akig wo adjace periods. So isead of fixig he base period oe should compue a series of idex umbers for each period wih previous period as he base period ad hece cosruc lik relaives. The chai idex umber (CI) is obaied by he successive muliplicaios of he lik relaives (LR) obaied. 38

39 If P rs represes he price idex umber wih r as base period ad s as give period he by muliplyig he successive liks, various chai idices may be obaied as follows: P =firs lik ha is, P = P P P = P 2 P 03 P P =P P... P 0s 0 2 (s-)s = P P 0(s-) (s-)s (85) Chai Idex = Give period LR Previous period CI 00 (86) Oe ca also obai he value of Fixed base idex umber (FI) from Chai base idex umber (CI) by usig followig expressio: Give period FI = Give period CI Previous period FI 00 (87) EXAMPLE - 6 The average wholesale prices of six differe groups of commodiies are give i he followig able for he period Compue Chai Base Idex Numbers. Commodiy C C C C C C

40 SOLUTION Commodiy C C C C C C Toal of LR Average LR Chai Idices Tess for idex umbers As discussed earlier, idex umbers ca be compued by usig differe ypes of weighs. Due o hese choices some of hese idex umbers saisfy cerai properies/ ess. There are four differe ess: a) Ui Tes b) Time Reversal Tes c) Facor Reversal Tes d) Circular Tes Ui Tes: This es requires he idex umber o be idepede of he uis i which he prices ad quaiies of various commodiies are give. All idex umbers excep simple aggregae mehod saisfy his es. Time Reversal Tes: Irvi Fisher defied ime reversal es as a es of cosisecy for a good idex umber. He remarked he idex umber mus maiai ime cosisecy by workig boh forward ad backward w.r.. ime. For example, cosider Laspeyre s idex umber wihou he facor 00: P La 0 = p q p q k 0k 0k k k= La k= P 0 = p0kq0k pkqk k= k= (88) pkq0k p0kq P k La P La k= 0 0 = k= p q p q 0k 0k k k k= k= (89) 40

41 Hece Laspeyre s idex umber does o saisfy ime reversal es. Whereas if we cosider Fisher s ideal idex umber wihou he facor 00, 2 P k 0k k k k= k= 0 = p q p q F p0kq0k p0kq k k= k= P F 0 p q = p q 2 p q 0k k 0k 0k k= k= p q k k k 0k k= k= (90) F P 0 P F 0 = Hece Fisher s ideal idex umber saisfies ime reversal es. Facor Reversal Tes To es he cosisecy of a idex umber, Fisher defied aoher es called facor reversal es. Accordig o his es if we ierchage he price facor ad he quaiy facor i h e price idex fo rmula, he quaiy idex s o obaied whe muliplied by he price idex should give he value idex, ha is, 0 0 = P Q k= k= p k 0k q p q k 0k ( 9) Fisher s ideal idex um ber is he oly idex ha saisfies his es. Circular Tes This es is a exesio of ime reversal es by shifig ba se. This es is saisfied by he idices b ased o geomeri c mea o f he price relai ves ad Kelly s fixed weigh mehod. Accordig o his es: P P P oi ik ko =,i k o (92) For example, cosider he idices based o geomeric mea of price relaives (wihou facor 00), he accordig o circula r es: P.P.P p k p 2k p 0k =.. = k= P0k k= Pk k= P2k (93) Hece circular es is saisfied. Also cosider Kelly s fixed base idex (wihou facor 00) 4

42 P.P.P k k k 2k k 0k wp wp wp =.. = wp k 0kwp k kwp k 2k (94) saisfyig circular es. NOTE: I he verificaio of hese ess, facor 00 is o o be ake. EXAMPLE - 7 Show ha Laspeyre s ad Paasche s idex umbers do o saisfy ime reversal es ad facor reversal es whereas Fisher s idex umber saisfies boh he ess. For compuaio, cosider he followig daa: SOLUTION Base Period Give Period Commodiy Ui Quaiy Price Quaiy Price Whie Ceme - Kg Taps - No Tiles - Sq. F Tile Border - Rf Base Period Give Period Commodiy Ui q 0 p 0 q p p 0 q 0 p 0 q p q p q 0 Whie Ceme - Kg Taps - No Tiles - Sq. F Ti le Border - Rf TOTAL Time Reversal Tes La P 0 =.3922 Pa P 0 =.3724 F P 0 =.3823 Similarly, Q =.3733 La 0 La P 0 = Pa P 0 = F P 0 = La Q 0 = P P La =.044 La 0 Pa 0 0 P P Pa = F 0 0 P P = La 0 F 0 Q Q La =

43 Pa Q =.3538 Q = Q Pa = Pa Pa Q 0 F Q 0 =.3635 F Q 0 = F Q 0 Q F 0 = This shows ha Laspeyre s ad Paasche s idex umbers do o saisfy ime reversal es whereas Fisher s idex umber saisfies his es. Facor Reversal Tes Value Idex Number V 0 =.8848 La La P Q =.99 V P Q = Pa Pa 0 0 F P V 0 F Q =.8848 = V 0 This shows ha Laspeyre s ad Paasche s idex umbers do o saisfy facor reversal es whereas Fisher s idex umber saisfies his es. Sice Fisher s idex umber saisfies boh ime reversal as well as facor reversal ess herefore, i is called a Ideal idex umber. Cosumer price idex umber Cos of livig idex umber is compued whe specific ype of commodiies are seleced which are beig cosumed by a paricular group of people. This is also kow as cosumer price idex umber. This idex umber measures he chage i he cos of maiaiig he same sadard of livig as i he base period for a paricular group of people. The cos of livig idex umber helps us o ake socio-ecoomic decisios such as fixig dearess allowace o basic salary or o deermie real wages. I also helps us o sudy he purchasig power of moey. The ype of goods cosumed by differe groups of people is maily classified io groups like: food, housig, clohig, fuel ad ligh ec. Weighs are assiged i proporio o he cosumpio expediure icurred o a commodiy belogig o a group. Before compuig cos of livig idex umber i is esseial o divide he populaio io differe homogeeous classes o he basis of heir icome - lower, middle or higher icome group, populaio group - goverme employees, privae or idusrial workers, as well as geographical regio ciy, sae, village ec. The mos impora is ha he class of seleced people should be homogeeous as far as possible wih respec o icome. Afer selecig he class of people for whom he idex is ieded, a family budge equiry is coduced. Through his equiry, iformaio is colleced o he ype of commodiies (heir quaiy ad qualiy) used by he class of people ad expediure icurred o various commodiies. The reail prices of he commodiies cosumed are colleced from he local markes. The differe price quoaios of a commodiy are ake for is various brads from a umber of markes. Fially, he average price is ake io accou. 43

44 Cosrucio of cosumer price idex umber There are wo mehods for he cosrucio of cosumer price idex umber: Aggregae expediure mehod Here weighs assiged o various commodiies, cosumed by he seleced group, are proporioal o he quaiies cosumed i he base period. Thus Toal expediure i give period Cos of Livig Idex = 00 Toal expediu re i base period k k p k 0k q 0k = 00 p q 0k (95) 2 Family Budge Mehod Here we cosider weighed average of price relaives where he weighs are ake as he values of commodiies cosumed i he base period. Cos of Livig Idex = ω P k k k= ω k k= (96) pk where, ω k=p 0kq0k ad Pk = 00 p 0k EXAMPLE - 8 A marke research orgaizaio coduced a house hold surve y for a paricul ar group of people o sudy he cos of livig idex. The followig are he observaios: 44

45 Commodiies Quaiy cosumed i he base period q 0 Ui Price (i Rs.) per ui Base Give Period Period p 0 p Whea 3 Quial 5 20 Rice 5 Quial 8 25 Chaa Quial Bajra Quial Maize 0.5 Quial Tea 2 Kg Coffee 4 Kg Oil 0 Lier Sal 8 Kg. 8 2 Sugar 20 Kg Buer 5 Kg Milk 200 Lier 2 4 Gas Cylider 0 No SOLUTION Commodiies Quaiy cosumed i he base period q 0 Ui Price (i Rs.) per ui Base Period p 0 Give Period p P 0 q 0 p q 0 Whea 3 Quial Rice 5 Quial Chaa Quial Bajra Quial Maize 0.5 Quial Tea 2 Kg Coffee 4 Kg O il 0 Lier Sal 8 Kg Sugar 20 Kg Buer 5 Kg Milk 200 Lier Gas Cylider 0 No TOTAL Cos of Livig Idex =

46 Limiaios of idex umbers I spie of a large umber of applicaios idex umbers have a umber of limiaios oo:. Idex umbers are ofe o represeaive of chages i prices ad quaiies of commodiies as hey are based o limied daa. 2. Selecio of base period is a criical facor as i should be a ormal period 3. Idex umbers are compued by cosiderig he quaiy of he produc cosumed ad is price bu qualiy of he produc should also be cosidered. 4. The cosrucio of idex umber depeds o he purpose of sudy. Thus idex umber compued fo r oe purpose may o be appropriae f or he o her purpose. 5. The selecio of a few commodiies ou of a oal lis of commodiies plays a impora role. A iappropriae selecio of commodiies may creae a error. 6. Idex umbers are based o averages ad each ype of average has is relaive limiaios hus creaig a error i calculaios. 7. Idex umber are based o daa collece d from primary source s. Hece suffer from samplig/ daa collecio errors Pracice sessio Quesio - Plo a graph for he followig ime series daa o show he red. Year () Producio of seel (i 000 os) (U ) Year () Producio of see l (i 000 os) (U ) Quesio - 2 The followig able gives he aual producio of cerai ype of crops culivaed i he mouai regio over he period of 5 yea rs. Fi a red lie by he mehod of semi-averages. Years Producio (i 000 Kg.) Years Producio (i 000 Kg.) Quesio - 3 The followig able gives he producio of peroleum by a oil compay over a period of 8 years. Fi a liear red by he mehod of leas squares. 46

47 Years () Produc io (U ) ( i Gallos) Quesio - 4 The followig able gives he cosumpio of grai i souh. Fi a red lie by he mehod of 4 years ad 5 years movig average. Year Cosumpio (i 000 Quials) Year Cosumpio (i 000 Quials) Quesio 5 The followig able shows he moh wise cosumpio of whea i Norher pars of Idia for las 4 years. By usig simple averages mehod compue seasoal idex. Year Moh JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Quesio 6 The quarerly cosumpio of woole cloh (i 000 No.) over he period of 7 years is give. Compue seasoal idex for each quarer. 47

48 YEAR QUAR TER ( Fig. i 000 No.) Ja - Mar Apr - Ju Jul - Sep Oc - Dec Quesio 7 The followig able gives he quarerly producio of coo fa brics ( i 000 ms. ) by a maufacurer for he years 200 o Compue seasoal idex by raio o red mehod. YEAR QUARTER Ja - Mar Ap r - Ju Jul - Sep Oc - Dec Quesio - 8 The daa give i he followig able shows he mohly producio of coo (i.000 os) by Souher pars of Idia from Ja 200 o Dec Compue seasoal idex by raio o red mehod. Year JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Quesio 9 The Tourism Deparme colleced daa o he umber of ouris visied Taj per moh from 2002 o The umber of ouris visied Taj are show i he followig able. Compue seasoal idices by raio o movig average mehod. 48

49 Year Moh JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC No. of Touriss Year Moh JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC No. of Touriss Year Moh JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC No. of Touriss Year Moh JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC No. of Touriss Quesio 0 The followig daa gives he quarerly sale of rai coa i a local ma rke. Compue seasoal idices by movig average mehod. Quarer Year Ja-Mar Apr-Ju Jul-Sep Oc-Dec Quesio - The followig able gives he producio of chocolae packes (i 0000 o.) maufacured by coco idusry durig he year Compue seasoal idices by he mehod of chai relaives. Producio of Chocolae Packe s ( 000 o.) Year Ja-Mar Apr-Ju Jul-Se p Oc-Dec

50 Quesio - 2 The daa give i he followig able represes he mohly cosumpio of coco (i,000 kg.) by ice-crème idusries from Ja o Dec Compue Seasoal idices by he mehod of lik relaives mehods. Cosumpio of coco (i,000 kg.) Year Ja Feb Mar Apr May Ju Jul Aug Sep Oc Nov Dec Quesio -3 A saioery maufacurig compay is ieresed o kow he price idex umber for he releva saioery for he year 2004 akig 999 as he base period. The followig able gives he prices ad quaiies of he maerials used. Quaiy Price Commodiy Ui Paper Rim Pe No CD Box Carbo Box Cosruc a) Laspeyre s price idex umber b) Passche s price idex umber c) Drobish-Bowley price idex umber d) Marshall-Edgeworh price idex umber e) Walsch price idex umber f) Fisher s price idex umber Quesio - 4 Cosruc various quaiy idex umbers usig differe weighs for he followig se of daa akig 995 as he base period. Price Quaiy Price Quaiy Price Quaiy Commodiy Ui Compuer No. 45, , , Prier No. 9, , , UPS No Mouse No Modem No

51 Quesio 5 Cosruc a value idex umber for he followig daa (akig 2000 as base period) Commodiy Price Quaiy Price Quaiy (i Rs.) (i Kg.) (i Rs.) (i Kg.) Poaoes Oios Carros Tomaoes Radish Quesio - 6 Compue weighed average of relaives price idex umber usig arihmeic mea for he followig daa akig 2000 as he base period. Price (i Rs.) Expediure o weighs Food Clohig Eeraime Educaio Coveyace Quesio 7 The average reail prices of eigh differe groups of commodiies are give i he followig able for he period Compue Chai Base Idex Numbers. Commodiy G G G G G G G G Quesio 8 For he followig daa show ha Marshell-Edgeworh ad Fisher s idex umbers saisfy ime reversal es whereas facor reversal es is oly saisfied by Fisher s idex umber, akig 2000 as he base period. 5

52 Commodiy Quaiy (i quial) Price (i Rs.) Quaiy (i quial) Price (i Rs.) Plaser of Paris Ceme Sad Quesio 9 A marke research orgaizaio coduced a house hold survey for a paricular group of people o sudy he cos of livig idex. The quaiy cosumed by a group of 2 households per year is give i he able alog wih he price. Compue cos of livig idex. Commodiy Quaiy cosumed i he base period Price (i Rs.) per ui Ui Base Period Give Period Whea 3 Quial Rice 5 Quial Masoor Quial Dalia Quial Moog 8 Quial Tea 220 Kg Coffee 40 Kg Sal 80 Kg Sugar 200 Kg Jeera 25 Kg Milk Powder 2 Kg Coriader 25 Kg Red Chilli 0 Kg Gas Cylider 00 No Refereces. Bereso, M.L. ad Levie, D. M. (992), Basic Busiess Saisics Coceps ad Applicaios, Preice Hall 2. Croxo, F. E., Cowde, D.J. ad Bolch, B.W. (974), Pracical Busiess Saisics, Preice Hall of Idia Privae Ld. 3. Gupa S.C. ad Kapoor V.K. (987), Fudameal of Applied Saisics, Sula Chad & Sos. 4. Goo, A.M. Gupa, M.K. & Dasgupa, B. (986), Fudameals of Saisics, Vol. 2, The World Press Privae Limied, Calcua. 5. Medhi, J. (992) Saisical Mehods. A Iroducory Tex, Wiley Easer Ld. 52