Sample size for the estimate of consumer price subindices with alternative statistical designs 1

Transcription

1 Saple size for te estiate of consuer price subindices wit alternative statistical designs 1 Carlo De Gregorio * Abstract Tis paper analyses te saple sizes needed to estiate Laspeyres consuer price subindices under a cobination of alternative saple designs, aggregation etods and teporal targets. In a siplified consuer arket, te definition of te statistical target as been founded on te etodological fraework adopted for te Haronized Index of Consuer Prices. For a given precision level, saple size needs ave been siulated under siple and stratified rando designs wit tree distinct approaces to eleentary aggregation, founded on Carli, Jevons and Dutot forulae. Alternative teporal targets are also exained: te single ontly target, te wole sets of ontly and quarterly indices, te annual average and te annual link. Epirical evidence is finally provided, based on te elaboration of survey icrodata referred to eleentary aggregates - suc as air transport and package olidays - caracterized by ig volatility witin and between onts. Keywords: cained index; aggregation; saple size; sapling variance; air transports; package olidays. 1. Introduction In recent years tere as been a growing concern for a ore explicit use of te concepts and tools of statistical inference to produce estiates of consuer price indices (CPI) and, in particular, to define te targets of te estiates in te fasion typical of statistical survey etods. Altoug tis issue as never been at te core of CPI literature, te pioneering works on tis subject date back to Banerjee (1956) and Adelan (1958), wile systeatic researc on te sapling variance of te Laspeyres CPI index as been developed since id-eigties: see for exaple te session dedicated to tis issue at te 1987 joint ISI-IASS 1 I wis to tank Alexandre Makaronidis, forer ead of te HICP unit at EUROSTAT, Keit Hayes (ead of HICP etodology) and Jan Walscots, as well as te colleagues of te National Statistical Institutes participating to te EUROSTAT Task force on HICP sapling. I a also grateful to y colleagues at ISTAT: Paola Anitori for er elpful coents, criticis and suggestions, and Alessandro Brunetti for furter observations. I a as well grateful to te oter colleagues of te forer CPI unit, naely Carina Munzi (tis paper is dedicated to er eory), Patrizia Caredda, Stefania Fatello, Rosanna Lo Conte, Maurizio Massaroni, Stefano Mosca, Francesca Rossetti and Paola Zavagnini. I finally acknowledge te useful coents and advices of an anonyous referee. Neverteless, te views expressed ere are solely tose of te autor and do not necessarily reflect tose of ISTAT nor EUROSTAT. An earlier draft of tis paper as been presented at te 12 t eeting of te Ottawa group (International working group on price * indices), eld in Wellington (NZ) te 4-6 May Senior researcer (Istat), eail: cadegreg@istat.it ISTITUTO NAZIONALE DI STATISTICA 19

2 SAMPLE SIZE FOR THE ESTIMATE OF CONSUMER PRICE SUB-INDICES conference and in particular te works of Biggeri et al. (1987), Andersson et al. (1987) and Leaver et al. (1987), or te works by Kott (1984) and ore recently by Dalèn (2001). 2 Furter researc focused also on specific issues, suc as te properties of saple designs based on alternative forulas for eleentary aggregation (Dalèn 1992; Baskin et al. 1996; Fenwick 1998; ILO 2004, cap.5; Balk 2008, cap. 5). Stiuli for te adoption of ore developed statistical tecniques also cae along wit te innovations in price collection, especially in selected consuer arkets: tis appened wit scanner data (De Haan et al. 1997; Fenwick 2001; Koskiäki et al. 2003) and wit sources like e-coerce and adinistrative or private databases. In general, te availability of larger and ore flexible data sets of price quotes ade it necessary to set up generalized etods, fostering a greater attention on sapling issues. Te integration wit oter statistical sources also favoured te adoption of sapling based approaces: for instance, te availability of regularly updated business registers as been considered to iprove saple design in Biggeri et al. (2006). Neverteless, ost of te epirical approaces adopted to easure te variance of te estiates relied on te use of replication tecniques, since te data available for analysis derive ainly fro purposive saples and quite rarely fro probabilistic designs. Several autors dealt wit tis issue (Biggeri et al. 1987; Andersson et al. 1987; Balk 2008, p.176) and wit te need to provide a suitable statistical design (Kott 1984; Dalèn et al. 1995; Dalèn 1998, 2001; Ribe 2000; Dorfan et al. 2006). Te definition of bot te universe and te target paraeters appears by far te ost critical issue in a CPI sapling design and, ore in general, in te CPI itself (Dalèn 1998, 2001). Several factors connected to te rapid evolution of consuer arkets ipair tis definition: tey are related to products and outlets replaceents as well as to te canges in teir caracteristics, and tey sould be tackled, at least teoretically, in order to provide a solid foundation for te production of te estiates. Terefore, te need of a structured fraework of concepts and definitions as eerged in order to reduce coplexity, and to provide sufficiently general and operative solutions. 3 Ribe (2000) and te ost recent etodological developents of te cained Laspeyres index adopted in te EU Haronized Index of Consuer Prices (ereafter HICP) propose to structure te universe of transactions into oogeneous partitions based on te concepts of product-offer and consuption segent. Starting fro tis approac, tis paper proposes a definition of te target universe under soe assuptions on te functioning of consuer arkets (tey elp to siplify te identification of te statistical target), and provides a tool to easure te saple size needed to estiate te sub-indices of HICP under alternative designs. Te results obtained wit siple and stratified rando designs are copared, taking into account te use of alternative criteria for segentation, eleentary aggregation and teporal targets. Two case studies are also developed. Based on te annually cained Laspeyres forula used in te HICP, a definition of te statistical target for a ontly index is firstly provided (par. 2). Given te desired precision 2 Te literature on tis subject is briefly surveyed in Dalèn et al. (1995). See also Wilkerson (1967), Dippo et al. (1983), Leaver et al. (1987), Leaver et al. (1991), Baskin et al. (1996), Norberg (2004). For an overview of variance estiation approaces in selected countries see ILO (2004, cap.5). 3 Te consuer arket ultiately consists of an enorous (but finite!) nuber of transactions, were goods and services (products) are purcased by consuers. However, it is not feasible to copare transactions directly between periods. Like te pysicists wo divide atter successively into olecules, atos and nucleons, we ave to bring soe structure into our arket universe as a prerequisite for a easureent procedure. Dalèn (2001, p.3). 20 ISTITUTO NAZIONALE DI STATISTICA

3 level, we derive te saple size wit siple and stratified rando designs under Carli, Jevons and Dutot aggregation. For eac design, alternative teporal targets are exained suc as ontly, quarterly and annual indices, as well as te annual link of te caining sequence (par. 3). Te approaces are ten tested on an experiental ground, siulating artificial populations fro te icrodata relating to two sub-indices of te HICP - air transports and package olidays - bot caracterized by a ig volatility of price dynaics witin and between onts (par. 4). 2. Te statistical target 2.1. Soe aspects of te construction of te HICP Te HICP is a ontly Laspeyres index based on te average of a reference year (yr). 4 It is built as a cained index by linking togeter te ontly price indices I of te current year y based on te link ont of Deceber y-1 and te fixed base index H of Deceber y- 1. By iteration, in te reporting ont of year y te aggregate HICP is derived as te product of tree eleents: a fixed base index (H), te product of te (y-yr-1) annual links, and te link index of te reporting ont (). In forulas: 5 H y, H y1,12 I y, H yr,12 y 1 x,12 I x yr1 I y, (1). y Te link of te reporting ont is copiled as te weigted average of te subindices referred to an exaustive set of disjoint aggregates j of te total consuer expenditure in te weigt reference year: I, y, I I j j w (2) were te expenditure weigts w add up to unity and in principle cange every year since tey are referred to te consuption expenditure of year y-1. 6 Te construction of any HICP aggregate follows a ierarcical procedure: te aggregation of te link indices coes first, ten te result is cained to te fixed base index of te sae aggregate. Te sub-indices y I j, y, y j can be terefore interpreted as te priary coponents of te HICP and, as a consequence, eac of te represents a distinct statistical target (Ribe 2000, p.1): ereafter we sall refer to te proble of estiating tese sub-indices. Notice also tat expression (2) can be applied to any exaustive partition of te target consuption expenditure: we coose in particular to deal wit te partition realized troug te groups of COICOP-HICP classification at te lowest level of detail used for At present yr=2005. See EUROSTAT (2001, p ). To siplify notation, in wat follows te basis of all indices as been set to 1 instead of te usual 100. Furterore, te weigts are price-updated fro te weigt reference period (y-1) to te price reference period (Deceber y-1). See EUROSTAT (2001, p ), Hansen (2006), ILO (2004, cap.9). ISTITUTO NAZIONALE DI STATISTICA 21

4 SAMPLE SIZE FOR THE ESTIMATE OF CONSUMER PRICE SUB-INDICES HICP disseination. Tis partition concerns alost 100 sub-indices (EUROSTAT 2001, p ) Re-pricing of transactions and statistical target in a HICP perspective One of te ain advanceents in HICP etodology and legal basis regards te statistical definition of te HICP universe. 7 In particular, te target paraeter for te annual links of a ontly Laspeyres CPI corresponds to te ratio of two siulated consuption expenditures obtained by apping te universe of transactions in te weigt reference period (year y-1) onto te sets of available offers in te price reference period (Deceber y-1) and in te reporting period (ont of year y). In order to define tis re-pricing of transactions, te concept of product-offer was introduced in EC Regulation 1334/2007: product-offer eans a specified good or service tat is offered for purcase at a stated price, in a specific outlet or by a specific provider, under specific ters of supply, and tus defines a unique entity at any one tie. As a atter of fact, product-offers are te observation units in CPI sapling and tey deterine te partition of total transactions. Neverteless tey represent a rapidly canging stock: tey ay cange as te caracteristics of te goods and services evolve, as tey are replaced, as retail evolves, or siply as prices cange. In order to provide stable entities on wic price coparisons can be based, te sets of all te transactions and product-offers in te statistical universe are exaustively clustered into consuption segents, were eac segent identifies oogeneous product-offers wit regard to arketing targets, consuption purposes and caracteristics. In te HICP fraework, consuption segents represent te fixed objects to be followed by te Laspeyres index. 8. Te ecanis of re-pricing can be suarised on te basis of Figure 1, in wic full inforation is assued on consuer expenditure. Te squared area on te left suarises te total consuer expenditure in te weigt reference year y-1 witin a given consuption segent: te geoetric sapes (circles, lozenges and exagons) identify te product-offers, wile te black saller circles represent te actual transactions. Only a subset of te product-offers and transactions of year y-1 is in coon wit te price reference ont (tat is Deceber y-1) te upper central side of Figure 1. Given tis inforation, te consuer expenditure in year y-1 is siulated by eans of apping functions connecting te product-offers available in tat year wit tose available in te price reference ont. 9 An identical approac is adopted for te re-pricing based on te reporting ont (y, ): in tis case tere is no overlapping wit transactions and product-offers in year y-1. Te index 7 See bot Ribe, 2000 and Coission Regulation (EC) no. 1334/2007; a collection of te early HICP legislation can be 8 9 found in EUROSTAT (2001). In particular: ( ) consuption segent eans a set of transactions relating to product-offers wic, on te grounds of coon properties, are deeed to serve a coon purpose, in te sense tat tey: are arketed for predoinant use in siilar situations, can largely be described by a coon specification, and ay be considered by consuers as equivalent. (. ). Te notion of consuption segents by purpose is terefore central to sapling and to te eaning of quality cange and quality adjustent. However, an abiguity in tis concept concerns te level of aggregation at wic it is defined and applied. ( ) Te range of product-offers will cange over tie as products are odified or replaced by retailers and anufacturers. Te HICP requires te representation of all currently available productoffers witin te consuption segents by purpose selected in te reference period in order to easure teir ipact on inflation. Tis applies particularly to new odels or varieties of previously existing products. (EC Regulation 1334/2007). Te coon set of Deceber y-1 transactions is siply replicated (see te continuous border fors in te first row of te upper rigt square in Figure 1) wile te rest of y-1 expenditure is siulated (dotted border). 22 ISTITUTO NAZIONALE DI STATISTICA

5 for te consuption segents is finally obtained as te ratio of te value of te two sets of siulated transactions. Figure 1 - Te re-pricing of transactions Legend Consuption segent Repriced y-1 transactions based on (y-1, 12) product offers Product-offer Transaction Product-offers in te price reference ont (y-1,12) Mapping function Transactions and product offers in te weigt reference period (year y-1) Product-offers in te reporting ont (y,) Repriced y-1 transactions based on (y, ) product offers Re-pricing as te advantage of providing a general fraework for te provision of suitable solutions for te statistical treatent of inflation estiates. Mapping functions iplicitly or explicitly incorporate various aspects of consuer beaviour odelling, and consequently define statistical iputation tecniques, non response treatent, quality adjustent. Tey represent te etodological core of te estiates: teir coplexity directly depends on te rapidity of te canges tat occur in te set of product-offers bot generated by canges in te price level and by te range of te goods supplied to consuers. It is easily understandable tat apping functions are open to ost several alternative ypoteses. As a atter of fact, te wole fraework for te definition of HICP statistical universe is a teoretical tool open to a wide range of possible solutions, wile etodological and epirical researc is still needed in order to test its applicability as a statistical tool. Anoter key point is given by te definition of consuption segents. HICP regulation itself recognizes tat abiguities still concern te level of aggregation wit wic consuption segents are defined and applied. It is likely tat consuption ISTITUTO NAZIONALE DI STATISTICA 23

6 SAMPLE SIZE FOR THE ESTIMATE OF CONSUMER PRICE SUB-INDICES segents need to be specified case-by-case and tis fosters te strategic role of consuer arkets analysis, suc as for exaple te structure of supply and deand, te arketing approaces and te segentations adopted by producers and dealers. Tis part of te job strictly interacts wit te definition of apping functions Te definition of te statistical target Given te Laspeyres forula, and following te approac set up in Ribe (2000) and in HICP legal basis, te starting point for te definition of eac target sub-index is given by te set of all te transactions in te weigt reference year y-1 concerning te COICOP- HICP group j. We assue a perfect knowledge of all inforation necessary to copile te indices. In particular, eac transaction in te weigt reference period is tracked; it concerns te purcase of a product-offer, and eac product-offer is attributed to a specified consuption segent. Product-offers are defined by te cobination of two sets of caracteristics. A first set consists of a vector g i of variables describing te product, te outlet and te corresponding consuption segent (): as a sortcut, we sall refer to suc a vector wit te ter product. Naing wit G g i i 1,..., N te set of all available products for te consuption purpose j, it is exaustively divided in M disjoint consuption segents G, wit =1,,M. Tis partition ( M ) can be expressed as follows (to econoize notation, ereafter we oit te suffix j): M G, 1,..., M, were G G and G ' G " 0 for every ' ". In order to siplify te definition of te universe and of te statistical target, soe assuptions on te available product-offers are ere introduced: te objective is to liit to price canges te possible sources of canges in te reference universe, and to provide a siplified fraework for te definition of te statistical target (Dalèn 2001; Balk 2008, cap.5). Te eleents of te set G are assued to be fixed and tie-invariant: in oter words, te nuber and te caracteristics of te available offers do not cange, and outlets and providers reain also uncanged (ypoteses A). In tis static environent, eac vector g i in te set G is associated to a second set of caracteristics wic describes te sequences of price spells and te corresponding tie t intervals of validity ( p i ). Fro te definition of product-offer - recalled in par. 2.2 eac cobination ( g i, product-offers associated to eac product i t p i ) describes a single product-offer. In order to control te nuber of g we assue tat discrete ontly pricing policies are adopted, were te prices of eac eleent of G are eventually canged only at te very beginning of eac ont (ypoteses B) Tis ypotesis ay appear quite restrictive and not realistic, since ontly policies are quite rare. Neverteless it is needed ere in order to siplify notation and foralisation: te results can anyway be easily generalised, at least conceptually, to take account of intra-ont policies. 24 ISTITUTO NAZIONALE DI STATISTICA

7 Figure 2 describes te siplified fraework derived fro ypoteses A and B: it sould be copared wit te ore general case reported in Figure 1. Product-offers can now be easily apped given te invariance of G and te regularity of te sequence of price canges. Figure 2 - A siplified fraework for re-pricing Legend Consuption segent Product-offers in te price reference ont (y-1,12) Repriced y-1 transactions based on (y-1, 12) product offers Product-offer Transaction Mapping function Transactions and product offers in te weigt reference period (year y-1) Product-offers in te reporting ont (y,) Repriced y-1 transactions based on (y, ) product offers Given te ypoteses A and B, te generic eleent of te (Nx13) atrix product-offers in available year y-1 is given by: y1,1 y1,12 i... pi g ; p. i 1 y of all te Te consuption expenditure (E) in te weigt reference year y-1 can be expressed as follows: E y 1 y1 p y1 y 1 T (3) y 1, were T labels te nuber of transactions and y1 y1, T p p y 1 T average price actually paid for transactions T in te consuption segent. is te annual ISTITUTO NAZIONALE DI STATISTICA 25

8 SAMPLE SIZE FOR THE ESTIMATE OF CONSUMER PRICE SUB-INDICES y I j, In order to define te true value of te target sub-index by adopting te consuption segents as te fixed objects defined in te HICP frae, 11 we need to siulate by eans of re-pricing te total consuer expenditure of te weigt reference year (y-1) on te basis of te product-offers available in te reporting ont of year y (identified by te couple (y, )) and in te price reference ont (Deceber y-1, conventionally labelled wit (y,0)). By applying (3) we obtain: I y, E E y 1 y 1 y, y 1 y, p T y,0 y,0 y1 p T I y, w (4) were w y y, y,0 y,0 y1 p T y,0 y1 p T is te noralized value weigt of consuption segent and y, p I is its price relative. 12 Average prices are derived on te basis of a apping of p 1 te set of product-offers y y,0 y, into te sets and available in te price reference and reporting onts. Te ypoteses A and B relating to set G ake it possible to assue te existence of a one-to-one correspondence troug apping functions connecting product-offers: for eac transaction involving product g i in year y-1, te corresponding y,0 y product-offers in te base and reference years are ( g i, p i ) and ( g i, p, i ) respectively. Different versions of te target paraeter defined in (4) can now be provided adopting alternative aggregation etods to calculate average prices. Two alternative approaces are proposed ere, naely te weigted aritetic ean: p y, p y, y1, i Ti gig y1, Ti gig (5) and te geoetric ean: p y, y1, y, T i ln pi gig exp y (6) 1, Ti gig 11 See EC Regulation 1334/ Non zero average prices by segent in te base ont (y,0) are ere assued; on te treatent of zero prices in te HICP see EUROSTAT (2001, p ). 26 ISTITUTO NAZIONALE DI STATISTICA

9 Eac approac iplies specific assuptions on consuers elasticity to price canges (see below, section 3.1). Te ai, ten, is to copare te properties of alternative standard y approaces to sapling in order to provide an estiate for I, Saple size wit alternative designs and aggregation forulas 3.1. Siple rando sapling (SRS) Assue tat a siple rando saple S of n products is drawn fro G and to collect te prices of te corresponding product-offers in te price reference ont and in a generic reporting ont. Given te ypoteses A and B, tis is equivalent to drawing an identical y,0 y, saple of products fro te sets of available product-offers and. No oter inforation is available on te universe of product-offers, consuption segents and transactions. Different estiates of I can be produced 14 depending on te approac followed to aggregate te sapled quotes and to produce te target index estiates. Tree alternative types of frequently used unweigted eans are ere copared: Carli (aritetic ean of price relatives, labelled wit C ), Dutot (ratio of ean prices, D ) and Jevons (geoetric ean, J ), respectively: pi 0 Ii C i S pi i S Iˆ, (7) n n pi 0 is Ii pi D is Iˆ, n (8) 0 0 p p ˆ, J I is i n ln I is exp n Te relative convenience of aggregation forulas as been deeply debated in literature and it as been evaluated on te basis of teir econoic properties and on te caracteristics of te underlying distributions of price canges. Eac approac entails in fact specific assuptions on consuers elasticity to price canges, wic are reflected on 13 Tis etodological fraework is clearly open to a larger set of different approaces to aggregation. Even te stocastic approac can be considered, altoug it as been largely criticised, ainly for its weak econoic foundations. Particular conditions concerning te distribution of price canges igt anyway spur te adoption of tis approac. 14 Hereafter, we drop te suffix labelling te year. is i i (9) ISTITUTO NAZIONALE DI STATISTICA 27

10 SAMPLE SIZE FOR THE ESTIMATE OF CONSUMER PRICE SUB-INDICES te iplicit weigting of transactions (Ilo 2004, cap.9; Leifer 2002, 2008; Viglino 2003; Balk 2003, 2008, cap. 5; Silver et al. 2006): Carli aggregation iplies equal value weigts for te product-offers and, for eac product-offer, a constant expenditure in bot te price reference and in te reporting ont. 15 Te Dutot forula iplies equal and tie-invariant quantities for eac product-offer, wilst te Jevons forula assues tat te expenditure sares of te price reference ont do not cange wen relative prices cange so tat soe substitution due to te cange in relative prices is terefore iplied. Wit respect to te Carli forula, te Dutot approac assigns a iger weigt to te product-offers wit a iger price level in te price reference ont and te Jevons approac assigns a iger weigt to te product-offers wit a lower price dynaics. Witout going into te issue of te coice of te rigt forula, we want to discuss ere soe of teir statistical properties in ters of precision witin different sapling designs. 16 Forula (7) provides an unbiased estiator of (4)-(5) only if te probability of selection is proportional to te weigt of eac product (Adelan 1958). Te sae applies to (9) wit respect to te target set by expressions (4) and (6). For te Dutot forula (8) to be unbiased wit respect to (4)-(5) it is necessary to add te condition tat te price relatives be independent fro te price levels in te price reference ont (Balk 2008, cap.5). Given a confidence level and a relative error expressed as a sare of te saple, q ean ( r I, were q C, D, J), te adoption of tese tree etods iplies soe differences in te saple sizes needed to produce an error lower tan te % of te true value of te paraeter wit a probability of %. Tese differences depend on te standard errors of te tree types of saple eans and on te for of teir distributions. By adopting standard siple rando sapling teory (Cocran 1977, cap. 4-6) separately for te tree aggregation forulas, te necessary saple size in ont in te case of te Carli forula can be expressed as follows: n, C SRS 2, C 2, t s t, C I, C Iˆ 2 C were t is te corresponding value of te t-student distribution, of te saple Carli ean and I (Cocran 1977, sect. 4.6; Ilo 2004, cap.5). 17 For te Dutot forula we obtain: (10) Iˆ C C s, I ˆ is te standard error C, is te coefficient of variation of te Carli index 2, D 2 nsrs p p 0 0, p 0 t 2, C C 2 C C p (11) 15 Notice tat te systeatic use of te Carli forula as been banned for te estiates of te HICP (Coission Regulation (EC) No 1749/96, Art.7; EUROSTAT (2001, p.129, )). 16 For discussions on tis issue see for exaple Fenwick (2008) and Baskin et al. (1996). 17 Hereafter, te saple fraction correction is not considered. 28 ISTITUTO NAZIONALE DI STATISTICA

11 were 0, is te Pearson s correlation coefficient of te price levels in te price reference and in te coparison ont, wile C and C 0 are te coefficients of variation of te price series in te two onts (Cocran 1977, sect ; Ilo 2004, cap.5). In te case of te Jevons aggregation, te following expression is derived: p p n, J SRS t, 2 logx 2 log( I ) s 2 2 t 2 C, J (12) 2log x I 2 4 were x, s log( I ) is te standard deviation of te logarits of te 2 individual indices I i and it is equal to te coefficient of variation of te Jevons ean C I, (Cocran 1977, sect. 4.6; Ilo 2004, cap.5). Te expression for x J can be derived by applying te SRS forula for confidence interval to te log transfored variable and J ten transforing back and resolving by n. Following Norris (1940), I ˆ, slog( I ) corresponds to an estiate of te standard deviation of te geoetric ean. Expressions (10)-(12) derive saple size fro te product between two eleents: one dependent on and, and te oter one is based on te coefficients of variation of indices and in te case of Dutot aggregation - price levels. For reasonably low values of (e.g., lower tan 10%), te coparison aong tese forulas can be liited to tis last eleent. In general, wen te variability of prices and indices is very sall, te tree approaces lead to very siilar saple sizes. On te contrary, soe iportant differences igt eerge wen te variability indices and price levels is relatively large. Te Dutot index needs a iger saple size wen tere is a strong eterogeneity in price levels wit negative or low positive correlation between price levels, and in particular wen te largest price canges are associated wit goods wit a iger price level in te price reference ont. In te case of te Jevons forula, te saple size tends to be relatively iger if te distribution of te price canges is negatively skewed wile te opposite appens wit a positive skewness Stratified rando sapling (STRS) It is reasonable to expect tat a partition in consuption segents can potentially isolate oogeneous product-offers and, once adopted as a stratification criterion, ay consequently reduce differences in te aggregation forulas. Neverteless, in principle partitioning in consuption segents ay not represent a good stratification criterion, and in any case tis ay not be te best way to control te variability of price canges. Tese two concepts are clearly conceptually distinct but ay nearly coincide if te econoic criteria adopted to define consuption segents eet also, as a by-product, te objective of isolating clusters of products caracterised by oogeneous pricing policies. Consuption segents sould be based - according to te HICP legal based recalled in section 2.2 on supply and deand side arket analysis, and it is very likely ten tat tey can target well te variability of price canges. Te definition of te border between segentation and ISTITUTO NAZIONALE DI STATISTICA 29

12 SAMPLE SIZE FOR THE ESTIMATE OF CONSUMER PRICE SUB-INDICES stratification very uc depends on te abiguity reinded also in HICP sapling regulation - on te level of aggregation, tat is on ow deep te segentation is run. Suc an issue deserves deeper case studies necessarily based also on survey icrodata. Here we iplicitly cose to collapse te two concepts and to copare te perforance of different degrees of aggregation, fro no stratification (segentation) at all to deeper stratification. We assue tat ore inforation is available concerning te consuption expenditure in te weigt reference year: te true weigting structure ( w ) of a partition in consuption segents ( M ) is known, altoug no oter inforation is available witin eac segent concerning te expenditure sares of te product-offers. It is iportant to notice ere tat weigts are not identified ere as a potential source of errors: tis practice is coon to ost of te approaces to te easureent of te statistical error in CPI estiates (Biggeri et al is a eaningful exception). In tis work we adopt tis sae ypotesis, altoug we are perfectly conscious tat additional work needs to be done in order to join tis analysis of price and price indices variability wit tat of te precision of weigting: te latter is of paraount iportance in order to evaluate te effectiveness of deeper stratifications. If a stratified rando design is adopted, te estiate ay be obtained as a valueweigted aritetic ean of te indices of eac segent (stratu): ˆ, ˆ. q, q I STRS I w Te standard deviations witin eac stratu, for te tree alternative forulas, will be given by: s, C s i Ii Iˆ n, C 2 2, D 1 2 D D s Iˆ,, s 2Iˆ s 0 n p p p p p s, J, J Iˆ s log( I ) Independently of te type of eleentary aggregation, total saple size wit optial allocation can be expressed as follows: n, q STRS 2, 2 q s w t, ˆ, q (13) I w were C s, is te standard deviation witin stratu (Cocran 1977, sect ) As in te case of SRS, te sapling fraction correction as been skipped. 30 ISTITUTO NAZIONALE DI STATISTICA

13 Following te optial allocation per strata (i.e. proportional to te standard deviation), saple size in eac stratu can be expressed as follows: n, q, nstrs, q STRS,,, q s w, q s w (14) Expression (13) suggests tat stratified designs can reduce te source of discrepancies aong aggregation forulas, depending on te ability of te forer to reduce te variance witin strata by eans of a clustering approac able to isolate te criteria used to define pricing policies Alternative teporal targets In te siplified fraework under ypoteses A and B, expressions (10)-(12) and (13)- (14) ave been for te oent referred to a generic reporting ont. Tey fix te nuber of product-offers wose price ust be collected in te price reference ont (Deceber y-1) and in te generic reporting ont in order to acieve te desired precision level for te estiate of te price index in. Neverteless, if our ai is to produce coplete annual series of ontly estiates, ten a nuber of consequences do eerge, depending on te way we approac tis task. Te saple size needed for te estiates referred to ont is in fact in general different fro te one needed to arrange te sae precision for anoter ont '. Tis appens because te nature of price dynaics possibly canges fro ont to ont in a way wic ay depend on te specific deand and supply caracteristics of eac consuer arket. 19 Montly saples can differ substantially, especially in te case of seasonal goods or services. In any case te price collection in Deceber y-1 provides te base for te annual link, and terefore its role is crucial for all te ontly estiates tat we are targeting: if we target a iniu precision level in every ont, te saple in tis price reference ont as to be drawn in order to satisfy te size requireents of all te twelve following onts. In particular, in SRS designs, saple size in te price reference ont ust be equal to te axiu size needed in te twelve onts: SRS n ( ontly) ax ( n (, )) (15) SRS If on one side te price collection in Deceber y-1 is te largest one, it igt be not necessary to activate a ontly price collection extended to all tis saple for te entire sequence of twelve ontly estiates. It is in fact possible to odulate price collection according to te actual ontly needs based on expressions (10)-(12). If we know tat in a given ont te expected variability is very low and tat te desired precision can be acieved wit a saple wic is alf te one drawn for te base according to (15), tan we 19 We ay ave to do wit a very eterogeneous set of consuer arkets - seasonal products, igly copetitive arkets, oligopolistic or onopolistic arkets, arkets igly dependent on external influences (arkets for international coodities, weater, natural events) or even adinistered prices, and so on -, all beaving in quite different ways and wit a variety of pricing policies. For a classification of price index dynaics witin te HICP see De Gregorio (2011). ISTITUTO NAZIONALE DI STATISTICA 31

14 SAMPLE SIZE FOR THE ESTIMATE OF CONSUMER PRICE SUB-INDICES can save resources for price collection and concentrate te, for exaple, for te ost critical onts. A odular approac to price collection is ten possible, and it gives te possibility to cange te saple size every ont in order to assure a given precision target in presence of eterogeneous variability patterns observed across onts. In te case of seasonal products, for instance, tis approac requires te largest effort in price collection in te price reference ont, a ig activation rate of te saple during peak onts and lower off-peak rates. 20 Wit stratified designs soe furter coplications ay arise since allocation is also a relevant factor. Te saple size in te price reference ont derives, in fact, fro te su of te largest ontly size of eac stratu: n STRS ( ontly) ax( nstrs, (, )) (16) wic ay be uc larger tan te axiu overall ontly size derived fro expression (14). Tis appens, in particular, wenever peaks in variability ave distinct tie patterns across strata, suc as in arkets caracterized by seasonal pricing were peak onts generally sow iger variability: te tiing of seasonal peaks, in fact, igt differ across strata and tis ere fact induces te need of larger saples in te price reference ont. Soeting siilar igt appen in sectors caracterized by igly irregular patterns. Te sub-indices wit a relatively large variability or tose caracterized by seasonal beaviours are indeed only a part of te wole set of HICP sub-indices. It as been estiated tat witin te euro zone between 2004 and 2008 about 25% of HICP four-digit sub-indices sowed a relatively strong ontly dynaics wilst about 7.3% sowed a clear seasonal pattern (De Gregorio 2011). For wat concerns te reaining indices, tey were referred to arkets were, price canges were quite regular and very slow, at least in periods of low inflation. In suc cases, in te first onts of te year wic are nearer to te base of Deceber y-1 - ost observations are concentrated in te no-cange zone: as a consequence, te distribution of price canges in tose onts is positively skewed. Tis asyetry progressively loses ground as one oves away fro te price reference ont towards te final part of te year. If on one side te inertia of price indices in te first onts reveals a very low variability and ence lower saple size needs, on te oter side it igt generate coplications since te ypotesis of norality could not apply. In any case, due to inertia, te last onts of te year igt be tose in need of te largest saples, and te adoption of te annual link of Deceber as a priary target for te estiates appears extreely reasonable: its iportance relies in fact on te peranent effect tat te link as on te cained index H. 21 In te case of te two types of design discussed above we obtain: 12 n( link) n (, ) (17) Tis forula bears relevant gains in saple size wit respect to expressions (15) and (16) only if te variability of price canges is diluted during te year and it is not 20 For an application of tis odular approac to seasonal products see De Gregorio, Munzi et al. (2008). 21 See expression (1); Fenwick (1999) exaines te issue of te coice of te price reference ont, epasizing te probles tat ay arise in te coice of te aggregation forula in case of large variability of price dynaics. 32 ISTITUTO NAZIONALE DI STATISTICA

15 concentrated in te final ont. Alternative reasonable targets igt be set on quarterly or yearly averages: and Q Q n n( quarterly) ax ( (, )) (19) Y n( yearly) n (, ) (20) In particular, for quarterly targets in stratified designs, allocation effects ust also be considered as in te case of ontly estiates. It is also possible to use cobined targets, for instance to guarantee te precision level on quarterly and annual link estiates. 4. Two case studies 4.1. Artificial populations In order to test te cobined effect of saple design and aggregation forulas on te variance of te estiates, we generated two artificial target populations starting fro a selection of te icrodata collected by ISTAT for te 2007 cycle of te HICP, and we iterated te extraction of saples fro tese populations in order to estiate te target paraeter defined in Section 2. In particular, two case studies are ere presented. 22 Tey are referred to price series caracterized by ig variability and eterogeneous beaviours: te first case regards European air transports, were te ig volatility of price canges is partly explained by seasonal patterns; te oter one regards package olidays, strongly affected by overlapping seasonal peaks wit soe inertia in te first onts of te year. 23 More forally, following te siplified approac outlined in par. 2.3, eac set of icrodata is interpreted as if it was a rando saple drawn fro te product-offers y available in year y ( Z ). Eac record is caracterized by a product identifier ( g i ) and by a vector of 13 price quotes - fro ont 0 (te price reference ont, naely Deceber 2006) to ont 12 (Deceber 2007). For eac arket, a detailed and exaustive partition of te goods in M 0 disjoint sets of consuption segents is ten given: M 0, 1,..., M. G 0 22 Official icrodata ave been treated ere wit a different purpose fro tat pursued by ISTAT; it follows tat results cannot be copared at any rate wit te official figures currently disseinated. 23 Fligts and package olidays are bot identified in De Gregorio (2011) as te sub-indices wit te ost eterogeneous beaviours across te countries of te euro zone, possibly needing furter aronization. For a etodological overview of te etods actually adopted by ISTAT to estiate tese indices, see ISTAT (2009) and De Gregorio, Fatello et al. (2008). ISTITUTO NAZIONALE DI STATISTICA 33

16 SAMPLE SIZE FOR THE ESTIMATE OF CONSUMER PRICE SUB-INDICES Alternative but less detailed partitions M i igt be obtained by ierarcical aggregation of te subsets of. For eac partition a vector of noralized weigts is accordingly defined: M 0 WM w, 1,..., M i w 1 i. Microdata in eac set Z actually derive fro stratified saples wic ave not been selected wit probabilistic rules (ISTAT 2009; De Gregorio, Fatello et al. 2008, p. 20, 28-32). Neverteless, tey are treated ere as if tey were derived fro rando selections and te eleent of eac set are expanded proportionally to te weigting structure W M 0 in order to for an infinite population. K siple rando saples, eac of n product-offers, are finally drawn fro tese infinite artificial populations. Te yearly series of te ontly q estiates Iˆ, k (k=1,,k) are derived fro eac saple, adopting alternatively te Jevons, Dutot or Carli aggregation (expressions (7)-(9)). An inductive estiation of te saple ean variance is ten produced and, consequently, an estiate of te saple size by eans of te forulas derived in te preceding sections is provided. An identical approac is used to estiate te saple size for stratified designs based on alternative partitions of te target population. All te siulations for te arkets under scrutiny ave been ade by extracting iteratively 300 saples of 500 products eac. Given a 1% error and a 95% confidence level, distinct teporal targets ave been separately considered. Tables 1 and 2 (see par. 4.4 below) describe a relative easure of te saple size calculated as a ultiple of a bencark size (te one needed to estiate te yearly average wit Carli aggregation and SRS). In particular te saple sizes ave been deterined in order to obtain te desired precision level for alternative teporal targets: i.e. separately for eac single ont, te cuulative target extended to te wole set of onts (adopting expression (15) and (16)), te quarterly and yearly averages (expressions (18) and (19)). Te desired precision target as been finally set on te link ont of Deceber, wic - given te caining procedure - affects peranently te fixed base series (expression (17)). For te construction of te artificial population, in te case of European air transports we ave used data fro te original saple of N=328 product-offers, concerning as any European return fligts connecting te country of origin (national) wit te oter countries (foreign). Eac return fligt is defined by a national and a foreign airport area (for instance, Roe and Frankfurt). Four distinct partitions are used to provide alternative exaustive segentations of te target population. An eleentary stratification 51 (51 strata) establises an exaustive segentation by national and foreign regions (sub-national) and by type of carrier (low cost vs. full service carriers). A less detailed partition collapses te regions of eac foreign country ( 38 ); a furter aggregation of consuption segents uses only te country of destination and te type of carrier ( 15 ), and te less detailed partition only te country of destination ( 11 ). An eleentary consuption segent can identify, for exaple, te low cost fligts fro te region A1 in country A (national) to te region B1 in te foreign country B; less detailed partitions identify, orderly, all te low cost fligts fro region A1 to B, all te low cost fligts fro A to B and all te fligts fro A to B. 34 ISTITUTO NAZIONALE DI STATISTICA

17 In te case of package olidays 24 we ave used te data fro an original ontly saple of N=246 records. Eac package is defined by a region of destination. Two distinct partitions provide exaustive segentations of te target population. A ore detailed stratification 43 splits te universe into countries and type of oliday (i.e.: sea, ountain, city, etc.), wile a less detailed segentation 12 adopts only te splitting by country. As an exaple, an eleentary segent could be te arket for package olidays for type of oliday A1 in country A; a less detailed partition would concern all te packages for olidays for country A Design effect for independent teporal targets Cart 1 plots te saple sizes needed wit SRS for eac ont and quarter, and for te yearly average (in all te carts and tables, te saple size needed to estiate te yearly average wit SRS and Carli aggregation is used as a bencark and as been set equal to 100). Bot arkets sow large differences in te variability witin eac ont; Jevons forula delivers te best perforance and Carli te worst, altoug eterogeneity in price levels seriously ipairs te perforance of Dutot aggregation during seasonal peaks; quarterly and yearly targets are far less deanding, altoug inertia effects ay require larger saples in te last quarters. In particular, for te separate estiates of te ontly indices of air transports saller saples are needed at te beginning and at te end of te year (and in June) wilst te largest sizes are found in association wit seasonal peaks in May and August (several ties iger ten te bencark): in tese onts, in fact, te distributions of bot price levels and price canges are positively skewed. Jevons aggregation is relatively less deanding, since it requires in May a saple size laying between two and tree ties te bencark; in te sae ont te Dutot forula delivers by far te worst result (seven ties te bencark). Carli aggregation needs te largest size in eleven onts out of twelve (wit te edian ontly size ore tan 35% iger tan Jevons ). Dutot generates lower saple sizes in ost of te off-peaks onts (first and fourt quarter) due to a ore appreciable oogeneity in price levels. 24 Only foreign travels were considered. ISTITUTO NAZIONALE DI STATISTICA 35

18 SAMPLE SIZE FOR THE ESTIMATE OF CONSUMER PRICE SUB-INDICES Cart 1 - Saple size wit SRS by sub-index, teporal target and type of aggregation (Indices. Base: size for yearly target wit SRS and Carli aggregation = 100) 800 European air transport Carli Jevons Dutot Year Q1 Q2 Q3 Q4 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Package olidays Carli Jevons Dutot Year Q1 Q2 Q3 Q4 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 36 ISTITUTO NAZIONALE DI STATISTICA

19 Moreover, altoug in air transports strong seasonal fluctuations ide any effect related to te tie-distance between te reporting and te reference onts, in te case of package olidays seasonal and inertia effects are cobined: after te suer peaks (in July and August) saple size reains in fact quite large as copared to te first onts of te year. Tis is te effect of te inertia of price dynaics, since price levels in te first onts tend to range closer to teir reference level and te estiates are less callenging. Te annual link, in particular, sees to need a large saple as opposed to air transports were te link ont was one of te easiest targets. In package olidays te worst perforance is provided by Dutot aggregation: it works relatively well at te beginning of te year, but soon becoes by far te less appropriate (in ters of saple size) in te reaining onts. Tis is due, probably, to te ig eterogeneity of price levels, since tey vary considerably across arkets and sow soe likely correlation wit price canges. Te figures for quarterly targets partially confir tis picture, altoug tey are quite sooter for air transports were saple size never doubles te bencark: Q1 requires te sae saple as January or te yearly average separately for eac aggregation etod; Q2 and Q3 are ore deanding, altoug tey never double te saple size needed to estiate te yearly average. Package olidays on te contrary deand larger efforts in te last two quarters, due to te inertia effects, and confir te inadequacy of Dutot aggregation, wile Carli and Jevons require nearly te sae saple size for all te quarters and for te yearly average. Cart 2 reports te effects on saple size deriving fro te adoption of STRS at te ost detailed level of stratification. 25 Te effects of stratification are quite ipressive: saple size is strongly reduced, te seasonal peaks are considerably sooted and te differences aong aggregation forulae tend to disappear. In air transports te saple size necessary to eet te yearly target is sligtly ore tan 20% iger tan te bencark; quarterly saples and ontly saples are strongly reduced to 25-30% of te corresponding need in a SRS frae. Suc decrease is particularly strong in peak onts, especially in May. Te effect of stratification is stronger wit te Carli forula were te perforance in ters of saple size iproves considerably (saple size is only 10% iger tan Jevons, in edian). Wit te introduction of stratification, Jevons aggregation is still te one systeatically requiring saller saples: neverteless, wit stratification te differences in saple size due to alternative approaces to aggregation tend to srink considerably. Tis particular aspect is also evident in package olidays; stratification reoves seasonal effects and only inertia plays a ajor. For te first tree onts saples are less te 20% of te bencark, in June tey pass 100% and te link ont is te ore deanding (nearly 170%). Quarters beave siilarly, and tis effect plays a key role in sustaining also te size of te saple required to eet te yearly target. 25 Please, notice te different scale of tis cart as copared to Cart 1. ISTITUTO NAZIONALE DI STATISTICA 37

20 SAMPLE SIZE FOR THE ESTIMATE OF CONSUMER PRICE SUB-INDICES Cart 2 - Saple size wit STRS by sub-index, teporal target and type of aggregation (Indices. Base: size for yearly target wit SRS and Carli aggregation = 100) 200 European air transport Carli Jevons Dutot Year Q1 Q2 Q3 Q4 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Package olidays Carli Jevons Dutot Year Q1 Q2 Q3 Q4 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 38 ISTITUTO NAZIONALE DI STATISTICA

21 4.3. Cobined teporal targets and odular price collection In te previous sections, we discussed ontly or quarterly targets were eac tie span was considered independently fro any oter. Wat is ore interesting is to see ow large te saple as to be in order to eet at te sae tie all te ontly targets or all te quarterly targets. For SRS te solution is trivial: it is in fact sufficient to adopt te axiu ontly or quarterly saple sizes. On te contrary, wit stratified designs strata allocation effects igt coplicate te atter (see par. 3.3): te relative efficiency of Jevons aggregation looses part of its advantage as copared to Dutot and Carli wen cobined targets are pursued, since Jevons allocation tends to sow a iger eterogeneity in te saple size needed eac ont in eac stratu. It can be said tat te adoption of cobined targets and stratified designs brings towards a reduction in te differences in efficiency due to te aggregation etod. More specifically, for te wole set of ontly targets of air transports wit SRS te use of a Carli aggregation would need nearly 5.54 ties te bencark (Table 1). Te Dutot forula delivers an even worse result (7.12), due to te ig eterogeneity in price levels. Te Jevons approac (2.79) needs alf te saple size as copared wit Carli. Suc large saples derive fro te ig volatility observed for price levels and indices in peak onts. If we reduce te SRS target to quarterly estiates, te saple sizes srink drastically (between 1.30 to 1.81 ties te bencark) and te differences aong te etods also are strongly reduced. Te yearly estiates need nearly alf te saple used for te quarterly target, wile te annual link of Deceber is placed between te quarterly and te yearly target. As long as te yearly target is concerned, Dutot equals Jevons perforance. Te effects of stratification are confired for cobined targets: sarp reduction in saple size and ore oogeneous results across te tree aggregation approaces. Te introduction of te first two levels of stratification brings large iproveents, in particular for te Carli forula. Te partition in 38 strata is extreely fruitful for all types of forulae, wile te ost detailed partition brings a coparatively inor reduction in saple size. In te passage fro SRS to te ost detailed stratified design tere takes place a reduction of alost 70% of te necessary saple size. Te teporal patterns of variability witin strata are quite differentiated across onts: consequently, te allocation effect induces appreciable differences between te saple size needed to target te wole set of ontly prices and te axiu size for separate ontly targets. If we consider te wole set of ontly targets (see Cart 1 and forula (16)), te saple size for Carli and Dutot aggregation is nearly 20% iger tan te axiu size sown in Cart 1; Jevons forula, altoug it is in general ore efficient, needs a saple nearly 30% iger tan te respective axiu (151 vs. 117). For te quarterly indices, te size increase needed to eet all te ontly targets is sligtly above 20%. Te yearly target, independently of te design, requires about 15-20% of te saple size needed for te ontly targets, and te link deands nearly 30%. In general Jevons aggregation perfors better, wit soe exceptions were Dutot appears less deanding. Carli generally iplies larger saples, altoug te differences collapse as stratification runs deeper. Te eterogeneity of price levels daages te perforance of te Dutot forula, especially were stratification is absent or liited, wile te saple size derived fro te Jevons forula appears less influenced by te presence of larger prices. If we consider package olidays, te irregularity of te ontly variability witin strata appears less pronounced as copared to air transports (Cart 1). Te SRS saple size needed to target te wole set of twelve onts is in fact only 5% iger tan ISTITUTO NAZIONALE DI STATISTICA 39