Topic 3:Index Numbers and Times Series

Transcription

1 Page Topc 3:Index Numbers and Tmes Seres Part I: Index Numbers: What s an number? Constructng Indces Choosng between Indces Usng Prce Indces The Canadan Prce Indces and the Cost of Lvng Reference: Chapter 6 We need to develop a measure that summarzes the characterstcs of data sets and s useful for perod to perod comparsons.

2 Page 2 An I number accomplshes ths by aggregatng nformaton nto a sngle measure that permts easy comparsons. Defnton: An INDEX NUMBER s a number whch summarzes peces of nformaton n a way, and forms the bass for comparatve judgements. An ndex number s a rato of 2 numbers expressed as a percentage. Used f we want to keep track of prce changes over tme or need to get an aggregate measure of output dfferent tems have the own unts. Examples: CPI (Consumer Prce Index); PPI (Producer Prce Index)

3 Page 3 Basc Consderatons for a good Index Number: () Include nformaton on tems. () Combne n way whch recognzes r mportance of each tem. () Need a bench mark value and perod to provde reference pont for comparsons. *Base perod requred

4 Page 4 Prce and Quantty Indces Prce Indces Two broad types: (and there are several varants of each) () A Indces () R Indces

5 Page 5 Lookng at (): Aggregatve Indces Example: Bundle of Two Goods Item Prce ($) (P) Quantty (Q) Prce ($) (P) Good Good 2 Quantty (Q) We want to summarze overall prce change from 990 to 200. (Base Year =990.) Must be the same goods n each year! (Assumpton)

6 Page 6 Type of Prce Index Prce of the th tem n the year of nterest. Base Year P S 0t = Year of nterest n = n = P P t 0 00 Prce of the th tem n the base year. The 00 ndcates that we have a base value of 00 n the base year.

7 Page 7 S Aggregatve Index: Let P t = Prce of the th tem at tme t. (=, 2,...,n; t=0,, 2,...,T) Constructon of the Smple Prce Index: P S 0t = n = n = P P t 0 * 00 The smple prce ndex from base year 0' to year t for tems to n.

8 Page 8 If t=0, ndex=00 (Base year value). Example: See prevous data for 990 and 200: Base Year =990 P P S 0t = n = n = S 0= 980, t = 2000 P P t 0 = = $ $ Untless Prces rose, overall, to be tmes hgher n 200 than n 990.

9 Page 9 Ths type of ndex only tells about changes n. Indvdual falls can be offset by other ndvdual rses. Wth the smple ndex, all goods are gven equal. So...

10 Page 0 W Aggregatve Index: P W 0t = n = n PW t PW 0 = where W =(fxed) weght gven to the th prce. 00 (Lmtaton of these ndces: qualty changes: Assumpton of dentcal tems.)

11 Page How to Choose the Weghts: Q of goods purchased. (Whch quantty?) Percentage of on good. (p q ) Both measure mportance, but may dffer over tme. So, when should they be measured?

12 Page 2 Possbltes: (I) L Index: P L 0t = n = n = Base perod quanttes= weghts pq t pq Dsadvantage: W get outdated over tme. Tends to overstate changes n prce for the prce ndex.

13 Page 3 (II) P s Index: P P 0t = n = n = pq t pq 0 t t 00 perod quanttes=weghts Dsadvantage: Weghts constantly changng, so ndex change may be partly due to quantty changes. (Tends to changes.)

14 Page 4 Some Compromses: (III) M -Edgeworth Index: P p q q p q q t ME t t n t n = + + = = Average of base perod and current perod are the weghts.

15 Page 5 (IV) F s Ideal Index: P F = P L P P 0t 0t 0t (Geometrc Mean of the L and P ndces) Example: Calculatons from earler data: Year S L P ME F P0 t P0 t P0 t P0 t P0 t

16 Page 6 Calculatons:(from data on page 2) If the base year s 990, wth a value of 00, then for t=200: Smple Aggregatve Index P t S 0 = = 360 Laspeyres : P t L 0 = 80. ( ) ( 4) 075. ( ) ( 4) 00 = 4588.

17 Page 7 Paasche: P t P = + + =. ( ). ( ). ( ). ( ). Marshall-Edgeworth: P p q q p q q t ME t t n t n = + + = = = =..... Fsher: P P P t F t L t P = = =.

18 Page 8 Changng The Base: Often one may want to change the base perod and /or base value of an ndex after t has been constructed. Suppose we have an ndex wth a base value of 00 n 970. But, we desre to have a base of 00 n 972: Year Old Index New Index Adjust 97 0 each entry by dvdng M M by M

19 Page 9 What numercal value multpled by 20 equals 00, such that the relatves are left unaltered? (Or equvalently 20 dvded by what numercal value equals 00?) 20 = 00 solvng for X: 20 = 2. X 00 or 20 X = 00 solvng for X: 00 = Base perod s changed by proportonal scalng r unaltered. To create the new ndex, dvde each value of the old ndex by.2 (or multply each value of the old ndex by 0.833). Overall % changes do not change.

20 Page 20 Quantty Indces: Same dea as prce ndces. Reverse the roles of prces and quanttes. Prces are now actng as the. S Quantty Index: Q S 0t qt = q0 00 Laspeyres Quantty Index: Base perod prces = weght Q L 0t = n = n = q p t q p

21 Page 2 P Quantty Index: Q q p q p t P t t n t n = = = Current perod prces are the weghts. Marshall Edgeworth: Q q p p q p p t ME t t n t n = + + = = F : Q Q Q t F t L t P =

22 Page 22 Choosng Between Indces: 3 TESTS () Tme Test: P t =prce of the th good at tme t. For the sngle good: P P t = P P 0 0 t I.e. Informaton about prce change s the same, regardless of reference date.

23 Page 23 Example: Gum: Prce n 2000 s $.5. Prce n 2005 s $ = ( 75. )( ) 5 ( 5 75 ) = =

24 Page 24 Ths motvates the dea of the Tme Reversal Test for ndces. If t passes: ( P * P ) where: P P 0t t 0 0t t0 = s base at 0 to tme t and s base at t to tme 0.

25 Page 25 Example: Laspeyres Index: ( ) ( ) P pq pq P pq pq P P P P L n n L n n L L L L = = = = = = = (assume the base value s not 00) Test: Fals f: Laspeyres generally fals the Tme Reversal Test.

26 Page 26 Usng the data from the prevous example: P L 990, 200 P L 200, 990 Test: = = n = n = n = n = p p p p q q q q L L ( P990, 200 P200, 990) = (.80)() + (2.70)(4) = = 2.6 (0.75)() + (0.50)(4) 2.75 = ( 075. )( 2) + ( 050. )( 3) ( 80. )( 2) + (. )( ) = = = ( )( ) = 748. Laspeyres Prce ndex fals the Tme Reversal test Generally: Tme Reversal Test > for Laspeyres and < for Paasche.

27 Page 27 (2) Factor Test: For a sngle good, prce rato s p p 0 and quantty rato s q q 0. The change n e s the product: pq pq 0 0 = p p 0 q q 0. The Factor Reversal test for an ndex checks: pq [ P * Q ] = 0 0 pq 0 0 Where P = prce ndex 0 Q 0 = quantty ndex

28 Page 28 Suppose P 0 shows a 20% rse and expendture has rsen by 50%, then Q 0 should show a 25% rse: pq 0 0 = pq (. 20) (. 25) = 50. [ P Q ] 0 0 Both Laspeyres and Paasche Indces the Factor Reversal Test.

29 Page 29 Example: Usng the prevous example: P L 0 = = n pq 0 pq L The test: P 0 n = 0 0 ( L Q ) 0 = Q L 0 n = = n = n = n = pq pq 0 0 q p 0 q p 0 0

30 Page 30 P Q L 990, 200 L 990, 200 = = n = n = n = n = p p q q q q p p ( L L ) 990, , 200 (.80)() + (2.70)(4) = = 2.6 (0.75)() + (0.50)(4) 2.75 = ( 2)( 075. ) + ( 3)( 05. ) = ( )( 075. ) + ( )(. ) = = LHS a P Q = ( )(. 0909) =

31 Page 3 RHS pq t pq t 0 0 a = ( 8. )( 2) + ( 27. )( 3) = 7. = ( 075. )( ) + ( 05. )( 4) = 275. pq t pq t = = Snce, , fals test. Marshall-Edgeworth tme-reversal test factor-reversal test Fsher s Ideal tme reversal test factor reversal test

32 Page 32 (3) Tme Perod Sequences: So far we have consdered only 2 ponts n tme. How do we measure prce changes from perod to perod 2? Ether: () Form P 2 drectly (usng perod as the base year) Or (2) take P 02 P =P 2 drectly. 0 Should get the same result ether way.

33 Page 33 Note: P P 02 0 P 2 P = = P 0 P 0 P2 P 0 P P 0 P P 2 0 rearrangng: * P P 0 P2 = = P P Suppose: Prces rse by 20% from perod 0 to. Prces rse by 0% from perod to 2. Overall rse from perod 0 to perod 2 s (.2.) =.32 (32% rse) Index Values: (0) () (2) 2

34 Page 34 Ths motvates the C Test: Index passes ths test f: P = ( 02 P 0 P ) 2 For example, consder the Laspeyres : P P L 0 L 2 = = But: n = n = n = n = ( pq 0) ( pq 0 0) ( p2q) ( pq ) n ( p2q0) L = L L ( P 02 ) = n [ P0 P2 ] ( pq 0 0) =

35 Page 35 Laspreyres ndex the crcularty test. Is ths mportant? YES!! the CPI s essentally a Laspreyres - type of ndex. Fals tests and tends to -state prce rses. All ndces mentoned so far fal ths test, except for the weghted aggregatve ndex wth fxed weghts.

36 Page 36 We have only consdered Aggregatve Indces up to ths pont. Now consder: (II) Averages of Indces A prce relatve s: Pt P 0. The assumpton s that each commodty s used as much. But, an ndex should take nto account the dfferng quanttes of the tems used. Weghted prce ndces permt the consderaton of the relatve mportance of the commodtes n the basket of goods. I.e. use quanttes as weghts.

37 Page 37 A weghted average of prce relatves ndex weghts the varous prce relatves n a market basket by the total amount s on that commodty. Weghted Arthmetc Mean of Prce Relatves v P t P wm 0 P0 t = * 00 v wm Note: f v = p q, then P = P ( ) L 0 0 0t 0t (. e. base year weghts exp endture) v represents the weght. P pq pq wm t 0 0t 0 0 = = P L 0t.

38 Page 38 Geometrc Mean of Prce Relatves P G 0t = p p t 0 n can also be weghted. Harmonc Mean of Prce Relatves v P t H P 0 P = 0t v H Note: f v = p q, then P = P ( ) t t 0t 0t (. e. current year weghts exp endture) P

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40 Page 40 Usng Prce ndces: Prce D It s common practce to dstngush between nomnal and real values for a seres. I.e. current dollar terms versus constant dollar terms. Recall: the real value s obtaned by: Real Value Nomnal Value = 00 Index Number Example: nvest money for a year at 7% p.a. nterest. Now suppose prces also rse by 2%.

41 Page 4 Real Rate of Interest: = % pa.. You can purchase approxmately 5% more goods wth each dollar than was possble a year ago. Convert a varable nto a real varable by deflatng dvde nomnal value by prce ndex to get real value. Prce Index used as a prce d. Must choose the approprate prce ndex for the purpose of prce deflator. Real ncome s commonly referred to as the purchasng power of the money ncome.

42 Page 42 In comparng ncomes, wages, rents, GNP, and personal ncome per capta of dfferent countres, the use of an approprate deflator s common practce. Hence, the real value s more easly recognzed. Real Exports Real Consumpton Real Payroll Nomnal Exports = 00 Export Prce Index Nomnal Consumpton = 00 CPI Nomnal Value of Payroll = 00 Wage Rate Index Obvously, ncreases/decreases n nomnal seres may be n the same/ opposte drecton to those n correspondng real seres.

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46 Page 46 Lnkng ( Splcng ) Indces L Suppose we have a Laspeyres prce ndex, P0 t for the perod t=963,...,973. The weghts have become outdated over ths tme perod. You decde to form new weghts n 973 and construct a separate P t L 0 seres for t=973,974,975,...,990.

47 Page 47 Although these are dfferent ndces, you may need a contnuous prce tme seres, so combne the two seres nto one lnked seres: Year Seres Seres 2 Lnked M M M (00).6 = M M (85).6 To make the 00 n year 973 be equal to 60, smply multply (00)(.6)=60. Relatve prces are preserved over tme.

48 Page 48 We could lnk wth any other base year: Year Seres Seres 2 Lnked M M (60).6 = M In each case, the lnked seres suggests prces are.96 tmes hgher n 990 than n = =

49 Page 49 Constructng the CPI The CPI s based on prce for each tem n each locaton. Then they are aggregated to get regonal ndces and ndces for groups. Eventually, you get to an All groups ndex for Canada. The constructon of the CPI nvolves the weghted average of prce relatves approach. The weghts are (base perod), so ths s the Laspeyres Index. Base year updated regularly -- used to be every 4 years.

50 Page 50 Weghts revsed to reflect changng expendture patterns (2009 s the latest.) Note: Laspeyres: weghted aggregatve ndex, wth base perod as weghts. CPI: weghted average of prce relatves, wth base perod e as weghts.

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54 Page 54 The Consumer Prce Index The CPI measures changes n consumer prces facng the Canadan consumer. It s derved by comparng the cost of a basket of commodtes purchased by consumers over tme. The basket s assumed to contan commodtes of unchangng quantty and and hence s beleved to reflect only pure prce movements.

55 Page 55 Between 93 and926 both the Department of Labour and the Domnon Bureau of Statstcs (Statstcs Canada) each produced separate CPI s. The present CPI seres descended from these sources. Some component ndexes n the CPI begn wth the date they were ntroduced nto the CPI. The CPI s wdely used as an ndcator of the changes n the rate of nflaton. Hence, consumers can montor changes n ther personal ncome.

56 Page 56 The CPI s generally employed n four ways: ) Evaluate and escalate the p g power over tme of: -wages -rents -leases -chld or spousal support allowance -prvate and publc penson programs (Old Age Securty and Can. Penson) -personal ncome tax deductons -government socal payments 2) Used to current dollar estmates to obtan constant dollar estmates. 3) Settng and evaluatng economc polces - Bank of Canada s monetary polces - Assessment of publc polcy food prces, etc.

57 Page 57 4) Economc analyss and research by economsts - explore effects of nflaton - causes of nflaton - regonal dspartes n prce movements CPI s based upon prces (ncludng taxes) pad by consumers n prvate retal outlets, government stores, offces and other consumer servce establshments. CPI consders a constant of goods over tme prces n ctes greater than or equal to,. Prce movements of the goods and servces represented n the CPI are weghted accordng to the relatve mportance of commodtes n the total expendture of consumers.

58 Page 58 CPI basket weghts are collected from multple types of relatng to a specfc year. The weghts are currently based on 2009 consumer expendture data. The current tme base of the ndex s 2002 = 00. Note: The CPI does not measure changes n the cost of lvng the latter s the change n ncome needed to keep a consumer as well off depends on ndvdual preferences.

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60 Page 60 Hghlghts of the Canadan CPI () Relates to prces n urban centres. () Based on prces of over 300 tems. () 8- Groups of goods: Food Shelter Household Operatons and Furnshngs Clothng and Footwear Transportaton Health and Personal Care Recreaton, Educaton and Readng Tobacco and Alcoholc beverages

61 Page 6 There are separate ndces plus an All Groups category. (v) Index computed monthly regonal ndces also avalable. (v) Weght gven to each ndvdual prce change s based on expendture as % of total household expendture. The prmary sources of expendture data on consumer goods and servces are from the Survey of Household Spendng and the Food Expendture Survey. (v) Prces of each group/servce sampled at several outlets: 0,000 prce quotatons each month. The All-tems ndex at the Canada level s based on an annual sample of over 500,000 prces quotes.

62 Page 62 (v) Some prces quoted twce/month; some once/quarter; some twce/year; some annual. EG. Food: 2 / month; Harcuts: / 3 months; Car nsurance: b-annual (v) The extent and qualty of the sample s constraned by budgets and the nformaton avalable on whch to base the samplng prces.

63 Page 63 Cost of Lvng It s reasonable to measure the change n the cost of lvng of an ndvdual between two perods as the change n ther money ncome whch wll be necessary for the ndvdual to mantan hs or her orgnal standard of lvng no more or no less. It must be ponted out that f all prces change n the same proporton, that proporton wll measure the change n the cost of lvng and there wll be no problem of measurement. However, all prces do not change n the same.

64 Page 64 Suppose n perod 0 an ndvdual spends hs by purchasng q 0 of varous commodtes at prce p 0. Assumng he saves nothng, hs total ncome equals hs total expendture: pq 0 0. In perod, prces change to p. How much does he need n perod to make hm as well off as he was n perod 0? If the quanttes of goods whch he would need to buy to leave hm exactly as well off as he was n perod 0 are q, then wth an ncome of pq he would be exactly as well off.

65 Page 65 It follows that the change n the cost of mantanng hs orgnal (perod 0) standard of lvng wll be gven by C pq =. 0 0 pq 0 0 The superscrpt (0) ndcates that the change n hs cost of lvng s beng measured n terms of hs perod 0 standard of lvng. Unfortunately we do not know the q s because they are not the quanttes he actually buys n perod, but only what he would need to buy to be as well off as before.

66 Page 66 Consequently, we do not know the aggregate pq. However, we do know the aggregate pq 0. Ths s the amount of he would requre n perod to enable hm to purchase the quanttes he purchased n perod 0. It can be shown that pq 0 wll be greater than pq, provded hs tastes have remaned unchanged.

67 Page 67 If he dd have the ncome pq 0 n perod, he would not buy the same quanttes as he bought n perod 0, (the q 0 quanttes) because he would take advantage of the changes n relatve prces and would alter hs allocaton of, buyng relatvely more of those goods whose prces had fallen relatvely more or rsen relatvely less. The fact that he would do ths n preference to buyng the q 0 quanttes ndcates that an ncome of pq 0 would make hm better off than the orgnal of pq 0 0 and, hence better off than an of pq whch s ts equvalent.

68 Page 68 We have accordngly,. pq 0 > pq Ths assumpton that hs tastes must have remaned unchanged s vtal because f they have changed, we cannot conclude that the quanttes he would buy n perod wth an of pq 0 are preferred to those he actually dd buy n perod 0. Ths can be llustrated, by means of ndfference curves, n the case of an ndvdual spendng hs on two goods.

69 Page 69 We call the two goods A and B. Let the prce of A n perod 0 be p 0 A and of B be p 0 B. If the ndvdual s money n perod 0 s M 0, we shall have: where q 0 A A A B B M = p q + p q 0 and q 0 B are the quanttes of A and B respectvely whch he can purchase under these crcumstances.

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71 Page 7 On the dagram, q 0 0 A s measured on the X-axs and q B on the Y-axs. The ndvdual can take up any poston on the lne hs budget lne, M 0. It has a slope gven by the prce of A relatve to that of B. 0 p A 0 p Slope: The ndvdual wll move to the pont Q 0, where quanttes of 0 0 A and B, q A and q B, meet on the budget lne, at whch pont he reaches hs hghest ndfference curve. The ndvdual wll 0 0 purchase q A and q B n perod 0. B

72 Page 72 Suppose, n perod prces begn to change. (Prce of A ncreases.) How much wll the ndvdual now need to be as well off as he was before? The ndvdual wll need suffcent to enable hm to take up a poston just on the ndfference curve. We now draw the lne M wth a slope gven by touches the ndfference curve. p p A B such that t just

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74 Page 74 Wth an correspondng to ths lne, he would take up the poston Q, purchasng q A of A and q B of B. Ths would requre : M = paqa + pbqb, as aganst the old ncome of M0 = paqa + pbqb. But the ndvdual s ndfferent between the postons Q 0 and Q, so that the rato M M 0 measures the change n hs old ncome necessary to make hm as well off n perod as he was n perod 0.

75 Page 75 Suppose that the ndvdual s gven n perod suffcent to enable hm to purchase the same quanttes as he dd n perod 0. Ths must be suffcent to make the budget lne for the new prces pass through Q 0 and must be equal to 0 0 M = paqa + pbqb. Ths s shown as the broken lne n the dagram, and t must le parallel to but to the rght of M, so that: 0 0 M = paqa + pbqb > M = paqa + pbqb.

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77 Page 77 Prce Indces and the Cost of Lvng Can a prce ndex, such as the CPI, measure a change n the cost of lvng? Indvduals are affected by prce changes. Defnton: A change n the cost of lvng s the change n ($) ncome needed to mantan orgnal standard of lvng. Ths depends on ndvdual preferences. Let q be the quantty of good that I need n perod " to mantan the standard of lvng I enjoyed n perod 0.

78 Page 78 Cost of Lvng Index s: C pq 0 0 p0q0 = 0 Compare C 0 wth prce ndces we have used. Assume: () One ndvdual (2) preferences

79 Page 79 q b M = p q + p q A A B B M* = p q + p q A A0 B B0 q B Q q B0 Q 0 M = p q + p q 0 A0 A0 B0 B0 q A q A q A0

80 Page 80 The prce of A ncreases: Relatve prces change: A s now more. Budget lne shfts slope ncreases Consumer s ndfferent between: So, C pq 0 0 p0q0 = = M M 0. ( q,q ) A0 B0 ( q,q ) A B

81 Page 8 To purchase the orgnal bundle, consumer needs ncome of : ( p q p q ) + (the green lne). M * A A0 B B0 Note: pq > pq 0 ( M * ) ( M ) If we dvde both sdes by : pq 0 0 base perod expendure :

82 Page 82 pq 0 p q 0 0 > or L P > C 0 pq p q Laspeyres Prce Index o rses n true cost-of-lvng, and understates falls n the true cost-of-lvng.

83 Page 83 Can also defne an alternatve ndcator of the change n cost of lvng: the ndvdual s ncome n perod " as a rato of what would have been needed n perod 0", at perod 0" prces to be as well off : C pq 0 p0q0 = where q 0 = quantty of good n perod 0" needed to mantan welfare experenced n perod ". the superscrpt on C.O.L. ndex refers to the utlty base perod. = M M 0

84 Page 84 q b M = p q + p q A A B B q B Q M * = p q + p q 0 A0 A B0 B M = p q + p q 0 A0 A0 B0 B0 q B0 Q 0 q A q A q A0

85 Page 85 Start at the blue lne: M = p q + p q A A B B Suppose prce s lower s the prevous tme perod: M = ( p q + p q ) A A B B M = ( p q + p q ) 0 A0 A0 B0 B0 Now:( p q + p q ) > ( p q + p q ) or A0 A B0 B A0 A0 B0 B0 green lne black lne * M M 0 p q > p q

86 Page 86 If we dvde by pq : p0q pq > p q 0 pq 0 nvert both sdes & reverse the nequalty sgn pq pq Paasche prce ndex p0q < p0q0 C 0 understates rses & overstates falls n true C.O.L. ndex. P < C P 0 0 A Paasche prce ndex rses and overstates falls n the true cost of lvng ndex!

87 Page 87 Note: 2 dfferent C.O.L. ndces. Each ndex nvolves quanttes. Cannot tell extent of under/over-statement. Each ndvdual has own preferences. A prce ndex takes no account of preferences cannot measure changes n true C.O.L. (Or of welfare ).

88 Page 88 Topc 3 Part II Analyss of Economc Tme Seres Data Recordng observatons of a varable that s a functon of tme, results n a set of numbers called a tme seres.

89 Page 89 Tme Seres Data: a sequence of data values, gather over tme usually at some fxed nterval. Snce data have a natural order, we can analyse a tme seres of data n terms of 4 basc components: () T : Long term drecton of the seres, over many years. Drecton and shape of the seres s mportant to us.

90 Page 90 () C : Represents a pattern repeated over tme perods of dfferent length, usually longer than a. Wave-lke pattern (3-5 duraton) eg. Busness cycles.

91 Page 9 () S : represents fluctuatons that repeat themselves wthn a fxed perod of one. Same pattern each year.

92 (v) I : fluctuaton that s unpredctable, or takes place by chance or randomly. Random nose n the data. Non-systematc. Page 92

93 Page 93 The Tme Seres Model: It s convenent to model a tme seres usng these components. Tme seres models fall nto 2 major categores: (A) A : Y=T+C+S+I assume components are ndependent of one another. (B) M : Y=T*C*S*I 4 components are related to each other, yet result from dfferent basc causes. **(Multplcatve model s more mportant and connecton between the two.)

94 Page 94 Focus frst on the decomposton of Y nto components. Why? (I) Components are useful n own rght. (II) Ads n forecastng of Y. For the purpose of estmatng each of the components of a tme seres, models normally treat S, C, and I as devatons from the. Trend s usually estmated n unts of Y, and the other components are measured n form, wth values greater than 00 ndcatng a devaton above the trend value and values below00 ndcatng a movement below the trend. I.e. Represent cycle and seasonal as fracton of trend: > 00% above trend; < 00% below trend

95 Page 95 The Steps of Decomposton of a tme seres nto ts components for a multplcatve model: () Isolate Seasonal component by the rato-to-movng average method. Determne seasonal ndex wth a base value of 00. (2) Remove Seasonalty from Y, leavng (T C I). Dvde the values of Y by the seasonal ndex S, and multply by 00 to obtan: 00 Y = T C I S

96 Page 96 (3) Determne trend component, and remove, leavng (C I) dvde T C I by the trend value $ Y to obtan (C I). (4) Smooth remander to elmnate rregular component leaves cycle. Once all of the components of a tme seres are dentfed, forecasts of the value of the tme seres can be made by frst estmatng the value of the trend component at that pont n tme n the future, and then modfyng ths trend value by an adjustment that takes nto account S and C components.

97 Page 97 Seasonalty We are nterested n the seasonal component and n t from Y. (.e. Seasonally adjustng the data). Basc procedure nvolves smoothng the data usng movng averages: Y t = { } 2 Perod MA: } } } } = 5 = 4 = 35. = More terms n average smoother Choose terms to span exactly one year. Quarterly data 4 terms; Monthly data 2 terms Need to take account of tmng of data. 2 2

98 Example: How To determne A Seasonally Adjusted Seres (Multplcatve Model) Page 98 Year Quarter Y t 4-Qtr. M.A. 994 Q 40 Q2 60 Q3 20 Q Q 50 Q2 70 Q3 30 Q Q 40 Q Centred M.A. (Tt*Ct ) Rato:S I = 52.5 Q Q t Yt (T C ) t t

99 Page 99 Obtan Seasonal Indces: (Average) S 2 = = 86. 2% ( ) ( ) S 2 2 = = 3. 9% ( ) S 3 2 = = 48. 0% ( ) S 4 2 = = 33. 8%

100 Page 00 Obtan the geometrc mean of S s: ( ) S GM = = % Snce the geometrc mean does not equal 00%, we must dvde each S by 92.44: * S = 00 = * 3. 9 S 2 = 00 = * 48 S 3 = 00 = * S 4 = 00 =

101 Page 0 S GM* ( GM = ) ( ) = % Apply same S s to each year. To seasonally adjust the data: Y S t Yt = S 00 * : 4 Y t = T t * C t *S t * I t Y S t Yt = ( ) ( S ) * * 00

102 Page 02 Year Quarter Y t S * Seasonally Adjusted Seres 994 Q Q Q Q Q Q Q Q Q Q Q Q

103 Page 03 Dfferences Between Addtve and Multplcatve Models Multplcatve Model ) Seasonal factors should average to or 00%: ( Π ) GM = s = Example: GM = ( s s s s ) )When adjustng for seasonalty: Dvde by ndex: y s = y * s Addtve Model ) Seasonal factors should average to 0: s = 0 Example: x = s + s + s + s )When adjustng for seasonalty: Always subtract: y s = y s * = 0

104 Page 04 Seasonalty and Addtve Model: There are only two dfferences: S * () Always nstead of dvde: Y = ( Y S ) (2) Seasonal factors should average to (not 00). S + S + S + S = 0 t t

105 Page 05 Multplcatve or Addtve? Multplcatve: assume fxed proporton of Y t s seasonal. Example: retal trade turnover (December Versus September) (generally reasonable for trended data.) Addtve: assumes fxed amount of Y t s seasonal. (May be approprate f no trend; real terms.)

106 Page 06 Refnements to Rato-to-Movng-Average Method () Tradng Adjustments Dfferent months have dfferent number of tradng days. May dffer year after year. (2) O : May need to trm data to avod dstortng seasonal factors. Example: ntroducton of G.S.T. (3) Evolvng Pattern S * s usually allowed to change cyclcally over years. (4) Pont Problems Calculaton of S * s may be senstve toy t values at end (s) of seres.

107 Page 07 The Trend T An mportant step n analyzng a tme seres s an estmaton of the trend component, T. One possblty s to use a movng average to smooth out (T t C t ) to leave trend. Not favoured because the length of M.A. (and choce of weghts ) arbtrary. Can be dstortve. The preferred approach s to dentfy a tme seres trend, and then model the trend components: {Increasng / Decreasng} {Lnear / Non-lnear}

108 Page 08 One prncple we can use to ft a trend lne to de-seasonalzed data s the L Squares prncple. Let: Y = th observaton on seasonally adjusted seres. X = th observaton on tme. Example: X =, 2, 3, 4,... X= 00, 0, 20, 30,... X=-3, -2, -, 0,, 2, 3,... Each observaton for data gves us an (X, Y ) pont (plot Y aganst X ). Try to ft a lne through so t s close to as many data ponts as possble.

109 Page 09 (Use all data.) Y=tme $ seres Y = a + bx a $ X=tme Error of ft b = slope Y 0 X

110 Page 0 Relatonshp s Y = α + β X (Not exact due to other components.) Ftted lne s Y $ = a + bx Error of ft=( Y Y) $ (+/-) Place the lne so as to mnmze: Sums to Zero!! T $ = ( Y Y) 2 (Choose a and b).

111 Page T = Mn ( Y a bx ) That s: ab, 2 Soluton: a = Y bx b = T = (X X)(Y Y) T = (X X) 2 = T = T = XY X 2 TXY 2 TX where T=# of observatons.

112 Page 2 Example: T=7 Y X X = 0 Y = X = 28 XY = b = = a = (.429)(0) = Y $ = X

113 Page 3 X Y $ Y ( Y Y) $

114 Page 4 Some Propertes of Least Squares () The predcted lne passes through the of X and Y, (X,Y): but, so, when Y$ = a + bx = a + bx a = Y bx ( ) Y$ = Y bx + bx = Y X = X. $Y Y X

115 Page 5 (2) Scalng each X by a constant leaves $ Y. Begn wth: Form: Then: where: So: Y$ = a + bx * X = CX. b X b * * [ (X X )(Y Y) ] * * 2 [ (X X ) ] * * = = CX; = [ (CX CX)(Y Y) ] 2 [ (CX CX) ] *, = C b

116 Page 6 Smlarly: a = Y b X * * * = Y C bc(x) = Y bx = a So: * * * * Y$ = a + b X ( ) = a + bc(x) ( ) a bx = + C = Y$ same ftted lne.

117 Page 7 (3) Addng a constant to each X leaves $ Y. Begn wth: Y$ = a + bx Then form: * X = (X + k); for some ' k'. b where: * X = (X + k) So, b b = * * [ ( X X )( Y Y) ] ( X * X * ) 2 * = [ ] [ ( X + k X k)( Y Y) ] 2 [ ( X + k X k) ] [ ( X X)( Y Y) ] = b 2 ( X X) * = [ ] *

118 Page 8 So: a = Y b X * * * = Y bx = Y b(x + k) = ( Y bx) - bk = (a - bk) * Y$ * * * * = a + b X = (a bk) + b(x + k) = a bk + bx + bk = (a + bx ) = Y$ same ftted lne. We are free to choose the X values just as we wsh the ftted lne s unaffected by ths choce.

119 Page 9 Often, a trend s clearly not approprate. There are several non-lnear possbltes. Consder an Exponental Trend: Y b > 0 X $Y = ab X 0 < b <

120 Page 20 To make ths trend lnear n parameters, take the natural log: lny$ = lna + X ln b or Y$ * * * = a + b X lnear n parameters Obtan a* and b* by least squares, usng Y * ( ln Y) = and X data. Then, the ftted trend lne s: [ ] Exp(Y $ * ) = Exp ln(y $ ) = Y$

121 Page 2 Example: T=7 X Y * Y = Y ( ln )

122 Page 22 X = 0 * Y = * X = 28; X Y = b = [ * XY TXY] * X TX 2 2 * * * a = Y b X = = =

123 Page 23 So, Y$ * = X or Y * 4448 Y$ = Exp( X ) X Y $ = (4.8399) (.4409) a b $ X Y Y $ ( Y Y $ ) =a

124 Page 24