Measuring Core Inflation in India: An Application of Asymmetric-Trimmed Mean Approach

Transcription

1 Measuring Core Inflaion in India: An Applicaion of Asymmeric-Trimmed Mean Approach Moilal Bicchal * Naresh Kumar Sharma ** Absrac The paper seek o obain an opimal asymmeric rimmed means-core inflaion measure in he class of rimmed means measures when he disribuion of price changes is lepokuric and skewed o he righ for any given period. Several esimaors based on asymmeric rimmed mean approach are consruced, and evaluaed by he condiions se ou in Marques e al. (2000). The daa used in he sudy is he monhly 69 individual price indices which are consiuen componens of Wholesale Price Index (WPI) and covers he period, April 994 o April 2009, wih as he base year. Resuls of he sudy indicae ha an opimally rimmed esimaor is found when we rim 29.5 per cen from he lef-hand ail and 20.5 per cen from he righ-hand ail of he disribuion of price changes. Key words: Core inflaion, underlying inflaion, asymmeric rimmed mean JEL Classificaion: C43, E3, E52 * Ph.D Research scholar and ** Professor a Deparmen of Economics, Universiy of Hyderabad, P.O Cenral Universiy, Hyderabad , India. Corresponding auhor: bmoilal@gmail.com We are very graeful o Carlos Robalo Marques of Banco de Porugal for providing he program for compuing he asymmeric rimmed-mean as well as for his valuable commens and suggesions. We also wish o hank Bandi Kamaiah for going hrough previous draf and for offering commens, which improved he presenaion of his paper.

2 . Inroducion Various approaches o measuring core inflaion have been discussed in he lieraure. Among hese, he Limied Influence Esimaor (LIE) approach has gained considerable aenion due o is saisical and economic raionale. Two esimaors are classified under he LIE approach: convenional symmeric rimmed mean and asymmeric rimmed mean. Use of symmeric rimmed mean and median as a core inflaion measure is jusified on he grounds of is efficiency when he disribuion of price change is symmeric hough lepokuric. However, when disribuion of price change is posiively skewed, symmeric rimmed mean esimaors are biased esimaors of measured inflaion. For eliminaing his sysemaic bias, Roger (997) pioneered asymmeric rimmed mean approach. Subsequenly, his approach has been applied by researchers in he various counries. Mohany e al (2000) were he firs o consruc a LIE based core inflaion measure for India. Some oher sudies have also used LIE mehod for measuring core inflaion for India. However, hese sudies have compued symmeric rimmed means. As noed above, and as we show laer, given skewnees in he disribuion of price changes in Indian daa, he symmeric rimmed mean esimaor will produce a core inflaion rae ha sysemaically underesimaes he headline inflaion rae. Consequenly, he symmeric rimmed mean may no be a very useful esimaor of underlying rend inflaion in India. The presen paper is aimed a finding opimal rimmed mean in case of India using asymmeric rimmed mean as an esimaor of core inflaion. Several esimaors based on asymmeric rimmed mean approach are consruced and esimaes generaed by hese for India are evaluaed by he condiions se ou in Marques e al (2000), in order o find he bes asymmeric rimmed mean-core inflaion measures for India. The paper is organized as follows: Secion 2 discusses he lieraure on limied influence esimaor (LIE) as a core inflaion measure, highlighing is saisical issues relaing o symmeric vs. asymmeric rimming. Secion 3 describes key characerisics of cross- Empirical evidence as summarized in Roger (2000), clearly suggess ha he disribuion of price changes in differen counries and ime periods are found o be lepokuric wih posiively skewed disribuion.

3 secional disribuion of price changes in India. Secion 4 deals wih he compuaion of he various rimmed means for India and an evaluaion of hese measures according o pre-specified crieria. The secion ends wih some comparisons beween symmeric and asymmeric rimmed mean core inflaion esimaes for India. The las secion offers some concluding observaion. 2. Limied Influence Esimaor: An Overview Limied influence esimaor (LIE) is an alernaive approach for convenional ex-food and ex-energy core inflaion measures. The basic idea of LIE approach o deriving core inflaion is ha i excludes cerain componens from cross-secion disribuion of price changes in each period on he basis of heir conribuion o noise in measured inflaion 2. I sysemaically excludes a percenage from each ail of he cross-secion disribuion of price changes and akes he weighed average of price changes for he res of componens in he aggregae price index. This process is followed in each period so ha a componen ha was exreme or an oulier in one period may or may no be an oulier in same or all of subsequen periods. The use of LIE for esimaing core inflaion rae is generally suppored boh in economic and saisical senses. The economic argumens are generally based on New-Keynesian models of price-seing behavior, in he presence of adjusmen coss while in saisical erms i is argued ha LIE is he bes esimaor of cenral endency, in he presence of non-normaliy in he disribuion of price change. 2. Symmeric vs. Asymmeric Trims Consider ha here are n commodiy groups. Le he proporionae price changes in a given period for hese commodiy groups be arranged in an increasing order (from lowes o he highes) as π, π 2,..., π n. Le he corresponding weighs of hese commodiy groups in he oal commodiy baske be w, w2,..., wn respecively. We denoe cumulaive weigh of commodiy groups, 2 i as W i ( = i w ) k = k. If α % is 2 Measured inflaion, headline inflaion and WPI inflaion are used here as inerchangeable erms. 2

4 rimmed from each of he ails of his disribuion, he inerval[ α 00, α 00] is called he unrimmed range. A commodiy group i is included in he inflaion measure, if W i falls wihin he unrimmed range. We define Ι α = {i α 00 W i α 00}. The α % rimmed mean is given as: π = α α 2 00 i Ι α w π i i Here α % is rimming from each of lef and righ ails of he disribuion and he 50 h percenile is he cenre of rimming. The mean and he median of he disribuion of price changes are hen as special cases of rimmed means namely, wih a rimming percenage (α ) of 0% and 50 % respecively from each ail. The saring hypohesis of LIE is ha under assumpion ha price change are disribued normally, sample mean of price change disribuion (measured inflaion rae) is an unbiased and efficien esimaor of he unknown populaion mean. Furher, median, mean and mode will coincide under he normaliy assumpion; herefore each of hese cenral endencies can represen underlying rend inflaion. If he disribuion of price changes is symmeric bu exhibis high kurosis, hen sample mean is an unbiased bu inefficien esimaor of sample mean (Roger 2000). Bryan and Cecchei (993) found such characerisics of price disribuion for he U.S daa. They proposed use of median as core inflaion measure. Laer Bryan e al (997) examined efficiency of inflaion esimae for purpose of moneary policy. They argued ha since he observed price changes exhibi high levels of kurosis, so simple averages of price daa no longer provide efficien esimaes of inflaion. Given his observaion, hey suggesed he symmeric rimmed-mean as a core inflaion measure. They also show ha he more lepokuric he price disribuion, he larger he ideal rim. The basic idea behind Bryan and Cecchei (993) Bryan e al (997) proposed median and symmeric mean is ha if he kurosis of empirical disribuion of price changes is larger han ha of a normal disribuion, hen i can be shown ha an esimaor for he mean ha pus more weigh on cenral price changes, is more efficien han he sample mean (Marques and 3

5 Moa 2000). This is wha symmeric mean and median do bu no sample mean of price change disribuion which gives equal imporance o each observaion. Bryan e al (997) also provide a mehod of deermining he opimal rimming percenage. They examine he enire range of rimmed means, wih he rim from each ail going from zero o 50 per cen. The rimmed means are hen compared wih he 36- monh cenered moving average of acual CPI inflaion, which is supposed o represen he rend (or core) inflaion and is herefore used as benchmark. The aim is o find he rimming percenage ha minimizes he deviaion gap beween rimmed means and benchmark measure where he deviaion gap is measured by RMSE and he rim ha minimized he RMSE was chosen as he opimal rimming percenage. The resuls of he paper shows ha 9% rim from each ail is mos efficien esimae of inflaion in he sense ha i reduces RMSE by around one quarer for CPI han he sandard Mean of CPI. Roger (995, 997) found ha disribuion of price changes in New Zealand showed a high degree of kurosis and chronically righ skewness. He noed ha sample mean is unbiased bu relaively inefficien in he case of lepokuric disribuion, while all symmeric rimmed means of limied-influence esimaors (including median) are relaively efficien, bu hey are sysemaically biased due o chronic skewness. In case of a posiively skewed disribuion we know ha mode < median < mean. In such case, he effec of α % larges price changes on he measured inflaion would be higher han he effec of α % lowes prices changes, because as noed by Marques and Moa (2000), he observaions in he righ hand ail of he disribuion are furher away from he mean compared o hose in he lef hand ail. So, If we rim he same percenage from boh he ails, he resuling rimmed mean series hen has endency o underesimae he measured inflaion (sample mean of price disribuion) in a sysemaic way ( and herefore also populaion mean which can be represened by sample mean). For eliminaing his bias, Roger (997) proposed he 57 h percenile as a measure of core inflaion for New Zealand daa. I was robus and unbiased in case of he lepokuric disribuion wih posiive skewnees for New Zealand daa. This can also be inerpreed as a case of asymmeric rimming ha is 50 per cen rimmed mean cenered on he average mean percenile (57 h 4

6 percenile for New Zealand). More generally, if he disribuion of price changes is posiively skewed, hen in order o ge rimmed mean ha is no sysemaically biased relaive o measured inflaion rae, we should rim α % cenred on he righward of 50 h percenile. For example, for α % asymmeric rimmed-mean cenered on c h percenile (c >50) we may rim ( c +α 50) % from lef ail and ( α + 50 c) % from he righ ail of he disribuion 3. Thus, he asymmeric rimming approach has he combinaion propery of being unbiased in he case of posiive skewness wih he efficiency propery in he case of lepokuric disribuions. A number of researchers have found cross-secion inflaion disribuion o be skewed. They herefore followed Roger (997) pioneered asymmeric rimmed mean approach-for insance in case of Ausralia (Kearns 998), Ireland (Meyler, 999), England (Bakhshi and Yae, 999), Belgium (Aucremanne 2000) and Porugal (Marques and Moa 2000). I should furher be noed ha Kearns (998) and Meyler (999) used Bryan e al mehodology as described above o deermine simulaneously he opimal rim and he asymmery of he rimming procedure. Kearns (998) compued asymmeric rimmed mean wih ceners lying beween he 40 h and 60 h perceniles, and Meyler (999) wih ceners lying beween he 40 h and he 70 h perceniles. They hen seleced opimal asymmeric rimmed mean ha minimizes he deviaion gap measured by RMSE or MAD relaive o reference rend series-a moving average of measured inflaion. While Aucremanne (2000) compued he rimmed means by choosing cenre beween 50h and 60 h perceniles and as a firs sep, he seleced he opimal rimmed means as he ones for which he null hypohesis of normaliy was no rejeced according o he Bera-Jarque saisic. Among hese, he hen seleced he opimum rimmed mean ha minimizes he average absolue error relaive o he inflaion rae. Marques e al (2000), on he oher hand, criicized use of benchmark-reference rend series as a device o search opimal rimmed mean series. They argued ha rend reference measure such as cenre moving 3 There is a rade-off beween efficiency and unbiasedness. Bakhshi and Yae (999), and Meyler (999) jusify rimming more from he righ side of he disribuion in order o ge minimum variance of resuling rimmed mean series. Marques and Moa (2000) and ohers sress on he finding rimmed means ha is no sysemaically biased relaive o measured inflaion rae and subsequenly hey look rimmed means wih minimum relaive variance among unbiased rimmed mean series. Roger (997) examine efficiency of he 57h percenile core measure relaive o he sample mean. 5

7 average of inflaion does no guaranee ha i is he bes proxy for rue rend of inflaion series on number of accouns 4. Empirical findings of Luc Aucremanne (2000), Healh e al (2004) and Dolmas (2005) ec. also show ha he opimal rim varies wih smoohness of moving average and also using differen proxy reference measures for rend series. Marques e al (2000) and Marques and Moa (2000) raher proposed a new se of crieria according o which hey do find opimal rimmed mean series. They found posiive skewness in he Poruguese price disribuion. In such case o find opimal rimmed means, hey se wo seps, hey are described below. Firs, hey se 50h percenile as a lower limi o compuing rimmed means. In case of posiively skewed disribuions, if compued rimmed is cenered on a percenile below 50 h, he resuling rimmed means will be sysemaically biased downwards relaive o headline inflaion hus resuling in sysemaic underesimaes. Leas amoun of rimming was decided as 5%. Thus 50h percenile and 5 % rimming se he lower limi for searching unbiased rimmed means. While hey se upper limi as average mean percenile (in he Poruguese case i was 56h percenile) and highes level rimming was 50%. Any rimmed mean ha is cenered on above he average mean percenile, will be sysemaically biased upwards (overesimae) relaive o headline inflaion. They searched level of rimming higher han 5 % and lower han 50% i.e. hey calculaed rimmed mean wih rim varying from 5 o 50 per cen in seps of five per cen in he open inerval percenile beween 50 h mean percenile and average mean percenile. They found ha asymmeric rimmed means do no saisically exhibi a sysemaic bias relaive o measured inflaion rae. Second, hey hen evaluaed unbiased asymmeric rimmed means by he condiions se ou in Marques e al (2000) o find bes asymmeric rimmed mean among he unbiased asymmeric rimmed means. The assumpion of ime-invarian opimal rim, implici in he above discussion, is open o quesion and scruiny. Since rimming parameer depends on he values of he momens of he cross-secional disribuion so even if we found opimal rimmed mean in 4 See Marques e al (2000) for deailed argumen. 6

8 one period, i may change in anoher period wih changes in he sample disribuion of price changes. Therefore, robusness associaed of opimal rimmed mean needs o be esablished. Moreover, rimming parameers are also sensiive o changes in he degree of disaggregaing of price componens. One possible soluion o former is o check he asymmeric behavior of price disribuion over sample period. This can be done by esing he saionariy propery of he mean percenile. 3. Cross-Secional Disribuion of Price Changes in India The purpose of his secion is o examine he key characerisics of he cross secional disribuion of prices changes in he WPI and heir implicaions for compuing core inflaion measures in India. The daa used in he sudy is he monhly 69 individual price indices which are consiuen componens of Wholesale Price Index (WPI) and covers he period, April 994 o April 2009, wih as he base year. Despie various shorcoming of WPI index, we focus on he WPI mainly because RBI bases is definiion of price sabiliy in erms of his price index. The weighs and daa for each componen of WPI index are colleced from RBI daa warehouse websie. The inflaion rae of each individual componen is he rae of change of ha individual index. These in urn provide a cross-secional disribuion of price changes a a given poin of ime. To circumven he seasonal effec on individual prices, we use year on year inflaion rae saisics. Subsequenly, he momens of cross-secional disribuion of price changes are calculaed by he ime varying weighs. Le P sand for price level in period, which is defined as follows: n P = w i= P i0 i () where P i is he price index for good i in period, and w i0 is he weigh of good i in he price index fixed for a base year, wih 7

9 n i= w i0 = Wih monhly ime series daa on prices, inflaion for each commodiy group i is defined as, P π = i P i i, 2 00 Inflaion for all commodiies, likewise, is defined as: P π = P 2 *00 n ( Pi Pi, 2 ) Pi, 2 = w i0 * * 00 P P i= i, 2 2 Thus, π = n i= w π i i (2) where Pi, 2 w i = wi0 is he ime-varying weigh of group i in monh P 2 The higher order, defined as: h k weighed cenral momens of a cross-secion disribuion are hen m n = k w i i= ( π π ) k i (3) In paricular, he skewness ( S ) and kurosis ( K ), which can be expressed as: ( m ) S = m (4) K = ( m ) { } 2 m 3 (5) 4 2 The coefficien of skewness ( S ) for a disribuion is a measure of asymmery of he disribuion of he series around is mean. The posiive skewness coefficien implies ha 8

10 he disribuion is skewed o he righ and vice-versa. The coefficien of kurosis ( K ) measure excess kurosis relaive o he normal disribuion. Any value above zero indicaes lepokuric disribuion of prices changes. Figure plos he coefficiens of Skewness ( S ), which demonsrae ha over he enire sample period he coefficien of skewness is mosly posiive: i is posiive for 50 monhs ou of 69. This finding suggess ha here is persisen posiive skewness in he disribuion of WPI price changes. The doed line in he figure is he average value of skewness and i is equal o.34. This finding of posiive skewnees is consisen wih empirical evidences for oher counries - for insan, skewness was found o be 0.70 for New Zealand by Kearns (998), for Porugal i was 0.83 (Marques and Moa, 2000), for Indonesia 2.24 (Kacaribu, 2002), and for Ukraine.23 (Mykhaylychenko and Wozniak, 2004), ec. Figure 2 0 Skewness Average value of Skewness Anoher measure of asymmery of he disribuion is he mean percenile. The mean percenile is nohing bu percenile score of he sample mean of he disribuion. As previously noed, he normal disribuion indicaes mode=median ( 50 h percenile) 9

11 =mean and he posiively skewed disribuion indicaes mode < median < mean. Therefore, if he disribuion of price changes is posiively skewed hen on average, mean percenile will lie above 50 h percenile (i.e., he value of mean percenile will be greaer han 50). Accordingly, sample mean of price disribuion is also expeced o lie above 50 h percenile. Figure 2 plos empirical mean percenile for he price change disribuion over he sample period. The resul suggess ha he mean perceniles lie above 50 h mean percenile in 53 imes ou of he 69 monh disribuions. This finding provides furher empirical evidence for he srong chronic posiive skewness in he disribuion of price changes. The doed line in he figure 2 is he average value of mean percenile scores or average mean percenile, which is obained by averaging he monhly empirical mean perceniles over he sample period. For he sample period, he average mean percenile is as shown by doed line in he figure. This indicaes ha he sample means of price change disribuion, on average are conained a 58 h percenile. Figure Mean Percenile Average Mean Percenile Finally, Figure 3 plos kurosis of coefficien ( K ) and he respecive average value over he sample period. The average value of kurosis is 4.8, indicaing ha he empirical disribuion of he price changes is srongly lepokuric. The coefficien of kurosis for enire sample disribuion is always greaer han zero. There is sharp peak for period 0

12 For mos oher years (barring 995 and 998) he disribuion are mildly lepokuric. This resul can be clearly seen in he figure. Figure Kurosis Average value of Kurosis Overall, he price change disribuion in India exhibis lepokuric and a persisenly chronic righ skewness. This disincive characerisic of WPI daa is consisence wih he findings for oher counries daa and ime periods. The resul would herefore sugges for applicaion of asymmeric rimmed mean approach o Indian daa for deriving core inflaion measures. 4. Trimmed Mean Measures for India This secion compues various rimmed mean measures for India and subsequenly evaluaes hem according o pre-specified crieria o find ou an opimal rimmed mean core inflaion measure for India. 4. Asymmeric Trimmed Mean Inflaion Esimaors Before going on o compue rimmed means, i is imporan o address he issue of ime series behavior of asymmery (skewness) of price change disribuion. This can be checked by esing for he saionariy of he mean percenile. The second row of able

13 presens resuls of uni roo es for mean percenile. The uni roo ess saisics show ha mean percenile is saionary in he sample period. The basic idea of esing saionariy of mean percenile is ha if mean percenile is saionary hen here is no problem of ime variabiliy of skewnees (Marques and Moa, 2000). Consequenly, he degree of asymmery can be assumed as consan. Furher, if we compue rimmed mean under he assumpion of consan asymmery, when in fac i is ime varying hen he compued rimmed mean core inflaion will no be co-inegraed (if inflaion, is I()) wih headline inflaion or saisically will no have he same mean (if inflaion, is I(0)) as he headline inflaion. Trimmed means are compued by choosing various values of lef rim and righ rim based on disribuion of price changes. One way of represening such a disribuion for any given period is o express all commodiy groups according o heir percenile scores (ranging from 0 o 00). Now any rim scheme can be represened by a cenre ( c) and a rim ( α ) as fallows: suppose lef rim is a l percenile and he righ rim is a r percenile, i.e. he range of price changes o be included is given by percenile inerval[ l, r]. hen, he cenre l + r r l c = and rimα = 50, and we represen i 2 2 l + r r l as TM, 50. Thus a TM ( c, α ) represen he percenile inerval, 2 2 [( c +α 50), ( c α + 50)]. Whenc = 50, we have a case of symmeric rim. TM (50, 0), for example, denoes 0 per cen rimmed mean cenred on he 55 h percenile, is shor for percenile inerval of [0, 90], which is nohing bu rimming symmerically 0 per cen of o he smalles and 0% of he larges price changes or 0 per cen from each ail of he price change disribuion. A TM (57, 5) denoes 5 per cen rimmed mean cenred on he 57 h percenile, gives percenile inerval of [22, 92], which is obained by asymmerically rimming he smalles 22 per cen and he larges 8 per cen price change. Likewise a TM (45, 20) represens inerval [5, 75], bu, since disribuions of price changes are posiively skewed on average, we ignore he case like he las example. This mehod of represenaion has he advanage of explicily showing where he percenile 2

14 inerval (used for calculaing core inflaion) is cenred and wha he average rimming from he wo ails is. Trims ( α' s) are a inerval of 5 percenile poins from 0 percenile o 45 percenile and we choose all ceners ( c' s) beween 50 h percenile and 60 h percenile a he inerval of 0.5 percenile poins. Thus, a oal 68 rimmed means are compued over he sample period 995m04 o 2009m04. Noe he symmeric rimmed means are a special case in his procedure, when we chose c = The Opimal Asymmeric Trimmed Means: Evaluaion Crieria and Resuls This subsecion evaluaes differen rimmed means in order o find opimal asymmeric rimmed means as core inflaion measures. For his purpose, Marques e al (2000) inroduced hree economeric evaluaion crieria. Those rimmed means ha pass hese hree evaluaion ess possess some nice economeric properies, and hence can be used as useful core inflaion measuress. The hree ess and he resuls based on hese es are discussed below: Tes : Unbiased Propery of Core Inflaion: If headline inflaion, π is I (), hen core inflaion, π should be I () as well and boh of hem are coinegraed wih coefficien, i.e. ε = ( π -π ) should be a saionary variable wih zero mean. If headline inflaion, π is I (0), hen i is sufficien if E ( π -π ) = 0 holds. The firs row of able shows ha headline inflaion in India is saionary, I (0). Therefore, headline inflaion can no be co-inegraed wih core inflaion. In such case, i is sufficien ha E ( π -π ) = 0 should saisfy i.e. headline and core inflaion series should have equal uncondiional mean. We es his condiion by resricion a = 0; β 0 = in he saic regression: π = a 0 + βπ + u (6) 3

15 Core inflaion measures ha pass his es are unbiased esimaors. The OLS esimaion of regression (6) exhibis srong auo correlaion herefore he sandard error for regressions are compued using Newey-Wes (987) procedure wih 4 lag. Table 2 repors he resuls of p-values from F-saisics for 68 rimmed means. The resuls indicae ha among 68 rimmed means, 43 pass his es. All hese 43 rimmed means are asymmeric rimmed means. Tes 2: Aracor Propery of Core Inflaion: This is based on he error correcion mechanism, which given by z = ( π π ) for Δ π, m n = a jδπ j + β jδπ j γ ( π π ) ε j= j= Δπ + (7) where m respecively. and n represen number of lags for headline inflaion and core inflaion This second condiion implies ha core inflaion, π, is an aracor of he headline inflaion, π, and requires an error-correcion mechanism ha describe he long-erm causaliy relaionship from core o headline. The condiion is hus o es aracor propery of core inflaion by simply esing he null hypohesis of no aracion, γ = 0, using - es saisic. The pracical quesion in he esimaion of equaion (7) would be selecing he number of lags for m Informaion Crierion (SIC). and n. We se he number of lags based on Schwarz The hird column of able 3 repors he resuls of p-values for es: 2. The resuls sugges ha he null hypohesis of γ = 0 is rejeced for 8 asymmeric rimmed means ou of 43 unbiased asymmeric rimmed means a 5 per cen significance level. This means ha he 8 unbiased asymmeric rimmed means have passed his es and so can be as leading indicaors of headline inflaion 4

16 Tes 3: Exogenous Propery of Core Inflaion: π has o be srongly exogenous for he parameers in he equaion (7). This implies ha in he error correcion model forπ : r s δ jδπ j + θ jδπ j λ( π π ) j= j= Δπ = + η (8) and he hypohesis λ θ =... = θ 0 should be acceped. In he above equaion, r and = s = s represen number of lags for core inflaion and headline inflaion respecively. This hird condiion guaranees ha he movemen in core inflaion, deermined by pas headline inflaion, Δ π, is no Δπ j. As in Marques and Moa (2000), we es boh for weak exogeneiy ( λ = 0) and srong exogeneiy ( λ θ =... = θ 0 ) in he = s = above equaion. We again here use Schwarz Informaion Crierion (SIC) o se he number of lags. Third column of able 4 presens resuls of he firs par of es: 3, namely p-values of he -es for he λ =0 in equaion (8) i.e. weakly exogenous propery of core inflaion. The es resuls show ha all of he 8 asymmeric rimmed means ha passed es: 2 also pass he weak exogeneiy es. However, he resuls of he second par of Tes: 3, namely ha ( λ θ =... = θ 0 ) in equaion (8) in he fourh column of able 4 shows ha among = s = he 8 asymmeric rimmed means (ha passed he weak exogeneiy es ( λ =0)), he null hypohesis of srong exogeneiy is saisfied only for 5 asymmeric rimmed means a 5 per cen level of significance. These 5 asymmeric rimmed means are TM (55, 20), TM (56, 20), TM (54.5, 25) TM (55.5, 20) and (56.5, 20). I should be noiced ha in case of TM (54.5, 25), he p-value of Wald es is 0.22 and he p-values for TM (55, 20), TM (56, 20), TM (55.5, 20) and (56.5, 20) are 0.050, 0.054, and 0.05 respecively (see fourh column of able 4). 5

17 To check he robusness of he resuls for srong exogeneiy es, he Tes: 3 was also conduced for shorer sample periods for 8 asymmeric rimmed means ha passed he weak exogeneiy es. In paricular, we esimaed equaion (8) wih various numbers of lagged values of headline and asymmeric rimmed means and for differen sample periods. The findings confirmed he earlier full sample resuls ha he five asymmeric rimmed means namely: TM (55, 20), TM (56, 20), TM (54.5, 25), TM (55.5, 20), TM (56.5, 20), are fulfilling srong exogenous propery of core inflaion. The resuls are repored in Table 5 for hese five rimmed mean series for he sample period 999m04 o 2008m04. All of he five asymmeric rimmed means ha passed he hree properies of core inflaion can be used as core inflaion measures. Each of hese 5 core inflaion measures are saisically equal. For insance, he Figure 4 plos he TM (55, 20) and TM (54.5, 25). As figure demonsraes, hese wo asymmeric rimmed means overlap each oher, hey display vary similar movemens over he sample period. Neverheless, o furher selec among five core measures, we need an addiional crierion. Following again Marques and Moa (2000), we choose a core inflaion measure ha exhibis smalles variance among five alernaive measures of core inflaion. This addiional crierion shows ha he seleced core inflaion indicaor exhibis a small shor-erm volailiy, and herefore makes i a good rend indicaor of headline inflaion. Variance of core inflaion measures and headline inflaion are repored in Table 6. Variance (shor erm volailiy) is measured by he quoien beween he variance of he firs difference of each core inflaion measure and variance of he firs difference of headline inflaion. This crierion can also be viewed as relaive efficiency of core inflaion vis-à-vis headline inflaion. Firs row of he able shows ha he variance of each core measure is lower han he variance of headline inflaion. Among he five measures, he variance of TM (54.5, 25) is he smalles, which is herefore he opimal core inflaion indicaor in he class of he rimmed mean measures. The TM (54.5, 25) is he 25 per cen rimmed mean cenered on he 54.5 h percenile i.e. he percenile inerval of [29.5, 79.5].This is he weighed asymmeric rimmed mean obained by rimming 29.5 per cen from he lef-hand ail and 20.5 per cen from he righ-hand ail of he price changes disribuion. 6

18 4.3 Symmeric vs. Asymmeric Trimmed Mean Core Measures in India As has been discussed previously, when he disribuion is posiively skewed, he mean is greaer han he median and, herefore all symmeric rimmed means including median underesimae he measured inflaion rae in a sysemaic way. In Indian conex, some effor has been made o consruc core inflaion using symmeric rimmed mean esimaors. Among hese, Mohany e al (2000) were he firs o consruc rimmed means in India. They calculaed hree symmeric rimmed means (5, 0 and 5% rim from each ail) over he period April 983 o March 999. Following Bryan e al (997) recommended RMSE approach as an evaluaion crierion, hey found 0 per cen symmeric rimmed mean as a good core inflaion measure for India. Subsequenly, similar resuls are refleced in Joshi and Rajpahak (2004). Recenly, Das e al (2009) calculaed median and symmeric rimmed mean ha rim 8 per cen from each ail of he price change disribuion. The graphs based on hese measures, ha show core inflaion as well as WPI for period 2000:0 o 2007:2, clearly esablish ha core inflaion hroughou he period lies below WPI, hus indicaing ha such core inflaion measures ends o sysemaically underesimae WPI inflaion. Kar (2009) compued differen saisical measures of core inflaion and proposed 57 h percenile measure as an indicaor of core inflaion for India. Given ha disribuion of price changes in India exhibis chronic righ skewness, i is imperaive o undersand how symmeric rimmed means sysemaically underesimae he WPI inflaion rae. Figure 5 plos, for example, 20 per cen symmeric rimmed mean (TM (50, 20) and WPI inflaion over he sample period. As can be seen, symmeric rimmed mean series TM (50, 20) is mos of he ime below he WPI inflaion rae. The graph uncovers he fac he symmeric rimmed mean is no a very useful rend inflaion indicaor of WPI inflaion as i fails o esimae rue level of core inflaion. This is also rue for any symmeric rimmed mean of LIE, as Marques and Moa (2000) showed ha simply changing he oal amoun of rimming in a symmeric way can change only he 7

19 expeced value of he esimaor. The resuls in he previous sub- secion provide evidence ha none of he compued symmeric rimmed means saisfied he unbiased mean es Conclusion This paper applied he asymmeric rimmed mean approach o measuring core inflaion in India. I compued several rimmed mean mesures of core inflaion and subsequenly evaluaed hem according o condiions specified in Marques e al. (2000), in order o find he bes measure in a class of he rimmed means measures. For his purpose, he paper firs analyzed he key characerisics of price change disribuions in India. This provided empirical evidence o jusify use of asymmeric rimmed mean esimaors as he appropriae esimaors of core inflaion in India. Among he several rimmed means, five asymmeric rimmed means saisfied all he hree necessary evaluaion crieria of core inflaion. Therefore, hey can be used as core inflaion indicaors for India. The final suggesed core inflaion measure was one wih he smalles relaive variance. This is asymmeric rimmed mean TM (55.4, 25), corresponding o percenile inerval [ 29.5, 79.5], wih 29.5 per cen rim from he lefhand ail and 20.5 per cen rim from he righ-hand ail of he disribuion of price changes. The Paper also provided he mehod of rimmed mean expression in erms of percenile score o show precisely where he percenile inerval used for calculaing core inflaion is cenred and wha he average percenage of rimming from boh side of he ails. Given asymmeric price change disribuion in India, he paper also graphically demonsraed ha he symmeric rimmed mean was sysemaically downward biased relaive o WPI inflaion as i was always below he WPI inflaion rae over he sample period. This highlighs he limiaion of symmeric rimmed means and he imporance of asymmeric rimmed mean for capuring underlying inflaion for India. 5 This is also rue for any rimmed mean ha pu relaively more weigh on righ hand ail disribuion. 8

20 Figure 4: Asymmeric Trimmed Means 2 0 TM(54.5,25) TM(55,20) Figure 5: Symmeric Trimmed Mean and WPI Inflaion 4 2 WPI TM(50,20)

21 Figure 6: Asymmeric Trimmed Mean and WPI Inflaion 4 2 WPI Inflaion TM(54.5,25) Table : Uni Roo Tess for WPI Inflaion and Mean Percenile ADF Tes PP Tes KPSS Tes WPI Inflaion -3.28(-2.88) -3.79(-3.47) 0.(0.74) Mean Percenile -4.60(-3.47) -6.34(-3.47) 0.2(0.35) Noes: Figure in parenheses are criical values of es saisics wih inercep. Lag lengh are chosen basis on SIC. Wih 5 % significance level, he null hypohesis of ADF uni roo for WPI inflaion can be rejeced Wih % significance level, he null hypohesis of PP uni roo for WPI inflaion can be rejeced Wih 0 % significance level, he null hypohesis of KPSS saionary es for WPI inflaion can no be rejeced Wih % significance level, he null hypohesis of ADF and PP uni roo for Mean Percenile can be rejeced and Wih 0 % significance level, he null hypohesis of KPSS saionary es for Mean Percenile can no be rejeced Table 2: Tes -Unbiased Propery of Core Inflaion Trimmed Means p-values Trimmed Means p-values Trimmed Means p-values TM (50,45) 0.00 TM (57,45) 0.00 TM (53.5,45) 0.00 TM (50,40) 0.00 TM (57,40) 0.00 TM (53.5,40) 0.00 TM (50,35) 0.00 TM (57,35) 0.00 TM (53.5,35) 0.50* TM (50,30) 0.00 TM (57,30) 0.00 TM (53.5,30) 0.20* TM (50,25) 0.00 TM (57,25) 0.03 TM (53.5,25) 0.0 TM (50,20) 0.00 TM (57,20) 0.57* TM (53.5,20)

22 TM (50,5) 0.00 TM (57,5) 0.83* TM (53.5,5) 0.00 TM (50,0) 0.00 TM (57,0) 0.42* TM (53.5,0) 0.00 TM (5,45) 0.6* TM (58,45) 0.00 TM (54.5,45) 0.00 TM (5,40) 0.00 TM (58,40) 0.00 TM (54.5,40) 0.00 TM (5,35) 0.00 TM (58,35) 0.00 TM (54.5,35) 0.03 TM (5,30) 0.00 TM (58,30) 0.00 TM (54.5,30) 0.66* TM (5,25) 0.00 TM (58,25) 0.00 TM (54.5,25) 0.23* TM (5,20) 0.00 TM (58,20) 0.06* TM (54.5,20) 0.03 TM (5,5) 0.00 TM (58,5) 0.65* TM (54.5,5) 0.00 TM (5,0) 0.00 TM (58,0) 0.97* TM (54.5,0) 0.00 TM (52,45) 0.00 TM (59,45) 0.00 TM (55.5,45) 0.00 TM (52,40) 0.22* TM (59,40) 0.00 TM (55.5,40) 0.00 TM (52,35) 0.02 TM (59,35) 0.00 TM (55.5,35) 0.00 TM (52,30) 0.00 TM (59,30) 0.00 TM (55.5,30) 0.* TM (52,25) 0.00 TM (59,25) 0.00 TM (55.5,25) 0.70* TM (52,20) 0.00 TM (59,20) 0.00 TM (55.5,20) 0.34* TM (52,5) 0.00 TM (59,5) 0.09* TM (55.5,5) 0.08* TM (52,0) 0.00 TM (59,0) 0.64* TM (55.5,0) 0.02 TM (53,45) 0.00 TM (60,45) 0.00 TM (56.5,45) 0.00 TM (53,40) 0.03 TM (60,40) 0.00 TM (56.5,40) 0.00 TM (53,35) 0.45* TM (60,35) 0.00 TM (56.5,35) 0.00 TM (53,30) 0.04 TM (60,30) 0.00 TM (56.5,30) 0.00 TM (53,25) 0.00 TM (60,25) 0.00 TM (50.5,25) 0.8* TM (53,20) 0.00 TM (60,20) 0.00 TM (56.5,20) 0.78* TM (53,5) 0.00 TM (60,5) 0.00 TM (56.5,5) 0.53* TM (53,0) 0.00 TM (60,0) 0.09* TM (56.5,0) 0.8* TM (54,45) 0.00 TM (50.5,45) 0.00 TM (57.5,45) 0.00 TM (54,40) 0.00 TM (50.5,40) 0.00 TM (57.5,40) 0.00 TM (54,35) 0.20* TM (50.5,35) 0.00 TM (57.5,35) 0.00 TM (54,30) 0.5* TM (50.5,30) 0.00 TM (57.5,30) 0.00 TM (54,25) 0.06* TM (50.5,25) 0.00 TM (57.5,25) 0.00 TM (54,20) 0.00 TM (50.5,20) 0.00 TM (57.5,20) 0.24* TM (54,5) 0.00 TM (50.5,5) 0.00 TM (57.5,5) 0.90* TM (54,0) 0.00 TM (50.5,0) 0.00 TM (57.5,0) 0.74* TM (55,45) 0.00 TM (5.5,45) 0.0 TM (58.5,45) 0.00 TM (55,40) 0.00 TM (5.5,40) 0.05* TM (58.5,40) 0.00 TM (55,35) 0.00 TM (5.5,35) 0.00 TM (58.5,35) 0.00 TM (55,30) 0.4* TM (5.5,30) 0.00 TM (58.5,30) 0.00 TM (55,25) 0.54* TM (5.5,25) 0.00 TM (58.5,25)

23 TM (55,20) 0.2* TM (5.5,20) 0.00 TM (58.5,20) 0.0 TM (55,5) 0.02 TM (5.5,5) 0.00 TM (58.5,5) 0.30* TM (55,0) 0.00 TM (5.5,0) 0.00 TM (58.5,0) 0.94* TM (56,45) 0.00 TM (52.5,45) 0.00 TM (59.5,45) 0.00 TM (56,40) 0.00 TM (52.5,40) 0.9* TM (59.5,40) 0.00 TM (56,35) 0.00 TM (52.5,35) 0.6* TM (59.5,35) 0.00 TM (56,30) 0.0 TM (52.5,30) 0.00 TM (59.5,30) 0.00 TM (56,25) 0.50* TM (52.5,25) 0.00 TM (59.5,25) 0.00 TM (56,20) 0.65* TM (52.5,20) 0.00 TM (59.5,20) 0.00 TM (56,5) 0.24* TM (52.5,5) 0.00 TM (59.5,5) 0.0 TM (56,0) 0.06* TM (52.5,0) 0.00 TM (59.5,0) 0.29* Noes: Tes saisics were consruced using he Newey-Wes (987) covariance marix esimaor. p-values - a0 = 0; β = * indicae es of unbiasedness is saisfied Table 3: Tes 2- Aracor Propery of Core Inflaion Unbiased Asymmeric Trimmed Means -Core Inflaion Measures Tes : Unbiased propery of Core inflaion p-value, a = ; β 0 0 = Tes 2: Aracor Propery Of Core Inflaion p-value, γ = 0 TM (5,45) TM (52,40) TM (53,35) TM (54,35) TM (54,30) TM (55,30) TM (55,25) TM (55,20) 0.2 *0.02 TM (56,25) TM (56,20) 0.65 *0.03 TM (56,5) 0.24 *0.0 TM (56,0) 0.06 *0.0 TM (57,20) TM (57,5) 0.83 *0.03 TM (57,0) 0.42 *0.0 TM (58,5) 0.65 *0.05 TM (58,0) 0.97 *

24 TM (59,5) TM (59,0) 0.64 *0.05 TM (60,0) TM (52.5,40) TM (52.5,35) TM (53.5,35) TM (53.5,30) TM (54.5,30) TM (54.5,25) 0.23 *0.05 TM (55.5,30) TM (55.5,25) TM (55.5,20) 0.34 *0.02 TM (55.5,5) 0.08 *0.0 TM (50.5,25) TM (56.5,20) 0.78 *0.05 TM (56.5,5) 0.53 *0.02 TM (56.5,0) 0.8 *0.0 TM (57.5,20) TM (57.5,5) 0.90 *0.04 TM (57.5,0) 0.74 *0.02 TM (58.5,5) TM (58.5,0) 0.94 *0.04 * indicae es of aracion is saisfied Table 4: Tes 3- Exogenous Propery of Core Inflaion Tes 2: Aracor Propery of Core Inflaion Tes 3: Exogenous Propery of Core Inflaion (i) Weak Exogeneiy p-value, λ =0 Tes 3: Exogenous Propery of Core Inflaion (ii) Srong Exogeneiy p-value, λ θ =... = θ 0 = s = TM (55,20) 0.08 *0.86 **.050 TM (56,20) *0.65 **0.054 TM (56,5) 0.04 * TM (56,0) * TM (57,5) * TM (57,0) 0.04 * TM (58,5) * TM (58,0) * TM (59,0) *

25 TM (54.5,25) *0.650 **0.222 TM (55.5,20) *0.74 **0.053 TM (55.5,5) 0.02 * TM (56.5,20) *0.490 **0.05 TM (56.5,5) 0.09 * TM (56.5,0) 0.0 * TM (57.5,5) * TM (57.5,0) * TM (58.5,0) * * indicae es of weak exogenous is saisfied ** indicae es of srong exogenous is saisfied Table 5: Tes 3 - Exogenous Propery of Core Inflaion Esimaion Sample: 999m m04 Tes 3: Exogenous Propery of Core Inflaion Srong Exogeneiy Weak Exogeneiy p-value, p-value, λ =0 λ = θ =... = θ s TM (55,20) 0.76* 0.07** TM (56,20) 0.67* 0.08** TM (54.5,25) 0.77* 0.20** TM (55.5,20) 0.72* 0.08** TM (56.5,20) 0.63* 0.09** * indicae es of weak exogenous is saisfied ** indicae es of srong exogenous is saisfied Table 6: Relaive Variance of Core Inflaion Indicaors = WPI TM (55,20) TM (56,20) TM (54.5,25) TM (55.5,20) TM (56.5,20) Variance Relaive variance

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