Does a Threshold Inflation Rate Exist? Quantile Inferences for Inflation and Its Variability

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1 Does a Threshold Inflaton Rate Exst? Inferences for Inflaton and Its Varablty WenShwo Fang Department of Economcs Feng Cha Unversty Tachung, TAIWAN Stephen M. Mller* Department of Economcs Unversty of Nevada, Las Vegas Las Vegas, Nevada, U.S.A stephen.mller@unlv.edu Chh-Chuan Yeh Department of Fnance The Overseas Chnese Insttute of Technology Tachung, TAIWAN Abstract: Usng quantle regressons and cross-sectonal data from 152 countres, we examne the relatonshp between nflaton and ts varablty. We consder two measures of nflaton the mean and medan and three dfferent measures of nflaton varablty the standard devaton, relatve varaton, and medan devaton. All results from the mean and standard devaton, the mean and relatve varaton, or the medan and the medan devaton support both the hypothess that hgher nflaton creates more nflaton varablty and that nflaton varablty rases nflaton across quantles. Moreover, hgher quantles n both cases lead to larger margnal effects of nflaton (nflaton varablty) on nflaton varablty (nflaton). We partcularly consder whether thresholds for nflaton rate or nflaton varablty exst before fndng such postve correlatons. We fnd evdence of thresholds for the effect of nflaton (nflaton varablty) on nflaton varablty (nflaton). That s, for low nflaton (nflaton varablty) countres, nflaton (nflaton varablty) does not affect nflaton varablty (nflaton). Fnally, a seres of robustness checks, ncludng a set of addtonal explanatory varables as well as controllng for potental endogenety wth nstrumental varables, leaves our fndngs generally unchanged. Keywords: JEL classfcaton: nflaton, nflaton varablty, nflaton targetng, threshold effects, quantle regresson C21; E31 * Correspondng author

2 1. Introducton Uncertanty emanates from the dffculty of knowng the future values of the varable of nterest. Hgher uncertanty reflects hgher volatlty of the varable s expected value or a hgher varablty of the varable around a gven mean. In hs Nobel lecture, Fredman (1977) suggests that hgher nflaton creates nomnal uncertanty, whch lowers welfare and output growth. Johnson (1967) and Okun (1971) argue that although desrable, achevng and mantanng steady nflaton proves problematc because of poltcal factors or polcy dfferences. That s, nflaton varablty s unavodable. Usng quantle regresson analyss, ths paper emprcally reexamnes the relatonshp between aggregate nflaton and ts varablty, especally the ssue of whether a threshold nflaton rate exsts. The lnkages, f any, between nflaton and nflaton varablty receved consderable attenton over the past forty years. Fredman (1977) outlnes an nformal argument regardng how an ncrease n nflaton rases nflaton varablty. Ball (1992) formulates Fredman s hypothess n a model of monetary polcy, where hgh nflaton creates uncertanty about future monetary polcy and, thus, hgher nflaton varablty. Ungar and Zlberfarb (1993) argue, however, that wth rsng nflaton agents may nvest more resources n forecastng nflaton, thus, reducng nflaton varablty. Cukerman and Meltzer (1986), on the other hand, consder the reverse lnkage. To wt, they argue that ncreases n nflaton uncertanty rase nflaton by ncreasng the ncentve for the polcy maker to create nflaton surprses to stmulate output growth n a game-theory framework. Thus, nflaton varablty leads to hgher nflaton. In contrast, Holland (1995) suggests that hgher nflaton varablty lowers nflaton, f the monetary authortes succeed n stablzng the economy. 2

3 Usng annual cross-secton data on 17 OECD countres for the perod 1951 to 1968, Okun (1971) reports a postve assocaton between the average nflaton rate and ts standard devaton, supportng the Fredman-Ball hypothess. In a comment, Gordon (1971) notes that the elmnaton of the data from the 1950s causes the sgnfcant postve correlaton to dsappear. Logue and Wllett (1976) fnd smlar results for 41 countres across the perod 1948 to 1970, but note that ths strong relatonshp breaks down when dsaggregatng the sample. Foster (1978) uses average absolute changes n the nflaton rate rather than the standard devaton as a measure of varablty for 40 countres from 1954 to 1975 and obtans results smlar to those of Okun (1971) and Logue and Wllett (1976). Davs and Kanago (1998) employ survey data for 44 countres over 20 years, fndng a robust, strong, postve relatonshp between nflaton and ts varablty across countres, but the support for Okun s hypothess weakens consderably for ntracountry data. Smlar fndngs emerge n Davs and Kanago (2000), who use squared forecast-errors from OECD nflaton forecasts for 24 countres. They fnd a sgnfcant, postve cross-secton relatonshp across countres between nflaton and nflaton uncertanty, but the tme-seres relatonshp wthn countres proves weak, at best. Regardng ths weak lnk at the ndvdual country level, Katsmbrs and Mller (1982) and Davs and Kanago (1996) fnd, on a country-by-country bass for OECD and hgh-nflaton countres, a less pervasve, postve relatonshp between the nflaton rate and ts varablty than suggested by Okun s (1971) orgnal fndngs. Most recent emprcal studes that examne the relatonshp between nflaton and ts varablty focus on tme-seres analyss of a specfc economy, snce Engle (1982, 1983) appled the autoregressve condtonal heteroskedastcty (ARCH) model to ths ssue. Ths approach, however, produces mxed evdence. For example, the Fredman-Ball hypothess receves support from Ball and Cecchett (1990), Grer and Perry (1998), Fountas (2001), Fountas et al. (2004), 3

4 Kontonkas (2004), Conrad and Karanasos (2005), Daal et al. (2005), Henry et al. (2007), Thornton (2007, 2008), Özdemr and Fsunoğlu (2008), and Chen et al. (2008) for a postve relatonshp for the G7 and other developed and emergng-market countres. Engle (1983), Cosmano and Jansen (1988), and Evans (1991) fnd no support for the hypothess, where they focus only on the US. Dfferng from the ARCH modelng approach, Bhar and Hamor (2004) adopt the Markov-swtchng heteroscedastcty model and fnd that hgh uncertanty assocates wth a sgnfcant postve shft n nflaton for Canada, Germany, and Japan n the long-run, for Germany and the US n the short-run, and a sgnfcant negatve shft n nflaton for Canada n the short-run. The Cukerman-Meltzer hypothess receves support from Balle et al. (1996) only for a few hgh nflaton countres. Grer and Perry (1998), Fountas et al. (2004), Daal et al. (2005), Henry et al. (2007), Thornton (2007), Özdemr and Fsunoğlu (2008), and Chen et al. (2008) report mxed evdence to support the hypothess and even uncover some support for Holland s counter hypothess n developed and developng countres. Hwang (2001) dscovers no statstcal evdence for a relatonshp n the US. In contrast to tme-seres tests n ndvdual countres, we apply quantle regressons to the nflaton and nflaton-varablty relatonshps for a cross-secton of 152 countres over 1993 to 2003, returnng to the cross-secton samplng approach of Okun (1971) and Gordon (1971). Our cross-secton analyss exhbts several dfferences from prevous studes. Frst, we use more sample countres. That s, we employ 152 countres as compared to 17 n Okun (1971), 41 n Logue and Wllett (1976), 40 n Foster (1978), 44 n Davs and Kanago (1998), or 24 n Davs and Kanago (2000). A larger sample sze can mnmze the chances of spurous results from relatvely few observatons. 4

5 Second, the sample perod of 1993 to 2003 provdes analyss for more recent data that captures several mprovements ncludes more cross-secton observatons, covers the perod of the Great Moderaton, and captures the adopton of targetng. One, we maxmze the number of countres wthn the sample wth more recent nflaton data. Two, the sample perod avods the ssue of potental structural change n nflaton varablty due to the Great Moderaton. That s, nflaton ncreased globally and became more volatle n the 1970s, but snce the 1980s, nflaton rates fell and became substantally less volatle, as a pattern across many countres. Three, nflaton targetng became an ncreasngly popular monetary polcy strategy, snce New Zealand s frst adopton n Now, over twenty countres (ndustral and emergng market) target nflaton wth other countres consderng the possblty. That s, lower nflaton, lower persstence, and lower volatlty exst n nflaton-targetng countres (Mshkn 1999, Kng 2002, Mshkn and Schmdt-Hebbel 2007a). Our sample perod captures the nflaton-targetng era. Thrd, we mplement quantle regresson analyss at dfferent levels of nflaton and varous degrees of varablty. Ths approach to the ssues consttutes an nnovaton n that pror studes examne Pearson product-moment correlatons or ordnary least squares (OLS) regresson analyss. Also, we examne both the Fredman-Ball (Okun) hypothess and the Cukerman-Meltzer hypothess. regresson permts dfferent (heterogeneous) response effects at dfferent parts of the nflaton or nflaton varablty dstrbutons. Fourth, n testng the Fredman-Ball and Cukerman-Meltzer hypotheses potental endogenety of nflaton and nflaton varablty may exst. To control for ths ssue, we mplement nstrumental-varable quantle regressons to examne the relatonshps between nflaton and ts varablty. Ffth, we use two measures of nflaton the mean and medan and three measures of ts 5

6 varablty the standard devaton, relatve varaton, and medan devaton to examne the robustness of the relatonshps, f any. The postve correlaton does prove robust across the dfferent measures of level and varablty. Fnally, Davs and Kanago (1998, 2000) note that some researchers (Logue and Wllet, 1976 and Hafer and Heyne-Hafer, 1981) fnd that the postve correlaton between nflaton and ts varablty does not hold for low nflaton countres. Logue and Wllett (1976) report nsgnfcant correlaton for hghly ndustralzed countres or for those wth modest nflaton rates between two to four percent over the perod Hafer and Heyne-Hafer (1981) conclude that the threshold level of nflaton above whch the postve correlaton emerges rose from around 4 percent for data n the 1950 to 1970 sample perod to around 9 percent for ther sample from 1970 to We splt the sample nto two dfferent sets of sub-samples. Frst, we splt the sample at the medan (.e., just over 6 percent) and show that, across countres, a sgnfcant postve relatonshp exsts between the mean nflaton rate and ts varablty, for both low and hgh nflaton countres. That s, we reject the noton of a threshold effect at 6 percent. Second, we splt the low nflaton sample (.e., countres less than the medan) at ts medan (.e., just under 3 percent), creatng low and moderate nflaton countres. In ths study, we show that low nflaton countres wth nflaton rates below 3 percent exhbt no sgnfcant effect of nflaton on nflaton varablty n the perod 1993 to Fnally, we mplement two robustness checks on our fndngs. One, we ntroduce a set of addtonal explanatory varables to augment our bvarate results. Two, we address the possble endogenety of the ndependent varables by mplementng nstrumental varables estmaton. 1 Gale (1981) reports that a clercal error may explan the nsgnfcant correlaton for ndustralzed countres n the 1949 to 1970 sample. 6

7 regresson, developed by Koenker and Bassett (1978) and popularzed by Buchnsky (1998), extends estmaton of ordnary least squares (OLS) of the condtonal mean to dfferent condtonal quantle functons. Condtonal quantle regressons mnmze an asymmetrcally weghted sum of absolute errors. Many areas of appled econometrcs -- such as nvestgatons of wage structure, earnng moblty, educatonal attanment, value at rsk, opton prcng, captal structure, and economc development now employ quantle regressons. Koenker (2000) and Koenker and Hallock (2001) provde an excellent dscusson of the ntuton behnd quantle estmators and varous emprcal examples. More recently, Chernozhukov and Hansen (2006) and Chernozhukov et al. (2007) extend Koenker and Bassett s (1978) quantle regresson model wth all exogenous varables to an nstrumental varable model to address endogenety. We provde the frst applcaton of the quantle regresson method to the cross-country relatonshp between nflaton and ts varablty. Our emprcal fndngs support both the Fredman-Ball and Cukerman-Meltzer hypotheses. Moreover, hgher nflaton and hgher nflaton varablty generally exhbt larger margnal effects and the postve correlaton between nflaton and ts varablty proves robust to alternatve defnton of nflaton and ts varablty. More mportantly, we fnd evdence of thresholds for the effect of nflaton (nflaton varablty) on nflaton varablty (nflaton). That s, for low nflaton (nflaton varablty) countres, nflaton (nflaton varablty) does not affect nflaton varablty (nflaton). Fnally, a seres of robustness checks, ncludng a set of addtonal explanatory varables as well as controllng for potental endogenety wth nstrumental varables, leaves our fndngs generally unchanged. The rest of the paper flows as follows. Secton 2 presents a bref revew of the quantle regresson method and ts propertes. Secton 3 dscusses the data and the results. Secton 4 consders the possblty of threshold effects. Secton 5 ncludes a set of addtonal explanatory 7

8 varables n the quantle regressons as well as quantle regressons wth nstrumental varables to explore the robustness of our fndngs. Secton 6 concludes. 2. regressons n nflaton and nflaton varablty regresson s outlned as follows: β y = x + u and (1) ( y x ) = x β, (2) where y equals the dependent varable (.e., nflaton or nflaton varablty) of country, x equals a vector of ndependent varables (.e., nflaton varablty or nflaton, respectvely) of country, β equals the vector of parameters assocated wth the th quantle (percentle), and u equals an unknown error term. Unlke ordnary least squares (OLS), the dstrbuton of the error term u remans unspecfed n equaton (2). We only requre that the condtonal th quantle of the error term equals zero, that s, ( u x) = 0. ( y x) = x β equals the th condtonal quantle of y gven x wth (0,1). By estmatng β, usng dfferent values of, quantle regresson permts dfferent parameters across dfferent quantles of nflaton or nflaton varablty. In other words, repeatng the estmaton for dfferent values of between 0 and 1, we trace the dstrbuton of y condtonal on x and generate a much more complete pcture of how explanatory varables affect the dependent varable. Furthermore, nstead of mnmzng the sum of squared resduals to obtan the OLS (mean) estmate of β, the th quantle regresson estmate β solves the followng mnmzaton problem: mn 2 y x β + 2(1 ) y x β. (3) β {: y x β } {: y< x β } That s, the quantle approach mnmzes a weghed sum of the absolute errors, where the weghts depend on the quantle estmated. Thus, the estmated parameter vector remans less senstve to 8

9 outler observaton on the dependent varable than the ordnary-least-squares method. The soluton nvolves lnear programmng, usng a smplex-based algorthm for quantle regresson estmaton as n Koenker and d Orey (1987). The medan regresson occurs when = 0.5 and the coeffcents of the absolute values both equal one. 2 When = 0.75, for example, the weght on the postve errors equals 1.5 and the weght on the negatve errors equals 0.5, mplyng a much hgher weght assocates wth the postve errors and leads to more negatve than postve errors. In fact, the optmzaton leads to 75-percent (25-percent) of the errors less (greater) than zero. One addtonal comment dstngushes quantle regresson from wthn quantle OLS regressons. That s, some analysts thnk that results smlar to quantle regresson occur when one segments the dependent varable s uncondtonal dstrbuton and then uses OLS estmaton on these subsamples. Koenker and Hallock (2001) argue that such truncaton on the dependent varable generally fals precsely because of the sample selecton ssues rased by Heckman (1979). To conduct parameter tests, we employ the desgn matrx bootstrap method to obtan estmates of the standard errors, usng STATA, for the parameters n quantle regresson (Buchnsky, 1998). In every case, we use 10,000 bootstrap replcatons. Ths method performs well for relatvely small samples and remans vald under many forms of heterogenety. More convenently, these bootstrap procedures can deal wth the jont dstrbuton of varous quantle regresson estmators, allowng the use of the F-statstc to test for the equalty of slope parameters across varous quantles (Koenker and Hallock, 2001). 2 That s, the least or mnmum absolute devaton (LAD or MAD) estmator occurs wth = 0.5. We nsert the twos so that the value of the functon equals the LAD or MAD functon value when = 0.5. Some references exclude the twos, snce the estmates prove nvarant to ts ncluson or excluson. 9

10 We estmate the followng two smple lnear quantle regresson models: 3 V γ δ ν = + + and (4) = α + β V + u, (5) where V equals the measure of the nflaton-rate varablty of country the standard devaton, relatve varaton, or medan devaton -- over 1993 to 2003, equals the measure of the nflaton rate of country mean or medan -- over 1993 to 2003, γ, δ, α, and β equal unknown parameters that are estmated for dfferent values of, and ν and u equal the random error terms. By varyng from 0 to 1, we trace the entre dstrbuton of nflaton varablty (or nflaton), condtonal on nflaton (or nflaton varablty). Fredman and Ball predct that δ > 0 and Cukerman and Meltzer, that β > Data and emprcal results Annual nflaton rates equal the percentage change n the logarthm of the consumer prce ndex (base year n 2000) gathered from the Internatonal Monetary Fund (IMF) Internatonal Fnancal Statstcs for 152 countres from 1993 to We proxy the nflaton-rate varablty by the standard devaton, relatve varaton, or medan devaton of the nflaton rate. 4 Average and medan values of the nflaton rates and the three measures of the nflaton rate varablty n each country comprse 152 sample observatons. Table 1 presents the summary statstcs as well as statstcs for the fve countres wth the hghest and lowest means and standard devatons of the 3 We also nclude a number of other potental explanatory varables. See below. 4 We note that nflaton varablty measured by the standard devaton wll equal nflaton uncertanty, when the expected nflaton rate of the sample perod equals the average nflaton rate over that perod. That s, nflaton uncertanty typcally equals the varablty of the actual nflaton rate around ts expected value. So, f average nflaton equals the expected nflaton, then the standard devaton of the nflaton rate wll equal the nflaton uncertanty as well. 10

11 nflaton rates. 5 Both the mean and the medan exhbt hghly rght-skewed dstrbutons wth outlers, as evdenced by a larger mean than the medan. regresson proves robust to departures from normalty wth skewed tals. Geometrcally, the mean of a varable equals ts center of gravty. In Table 1, the mean nflaton ranges from percent n Japan to percent n Turkey, and responds sgnfcantly to extreme values. The hghly skewed dstrbuton (skewness=2.3257) suggests that the medan may provde a better alternatve to measure central locaton. The medan, a postonal value, dvdes the observatons on the nflaton rate nto two equal parts. It does not equal the mean, and does not respond to extreme values. Dfferent measures of nflaton and ts varablty, that s, the mean and standard devaton versus the medan and medan devaton, should not nfluence the relatonshp between the two varables for a robust relatonshp. Addtonally, n Table 1, the fve countres wth the hghest nflaton rates (standard devatons) face hgher standard devatons (nflaton rates), whle countres wth the lowest nflaton rates (standard devatons) face lower standard devatons (nflaton rates). The mean value and ts standard devaton appear postvely related. Ths appearance, however, seems to dsappear for countres wth the lowest nflaton rates and the lowest standard devatons. Compare Japan to the US, for example, a lower nflaton rate does not mean a lower standard devaton, or vce versa. Thus, low-nflaton countres may exhbt dfferent patterns between nflaton and ts varablty from hgh-nflaton countres. To consder a unt-free measure, we also consder a relatve measure of varaton, the standard devaton dvded by one plus the mean nflaton, as suggested by Davs (1989) and Davs and Kanago (1992), to 5 Tables A1 and A2 n the Appendx report the average nflaton rate and the standard devaton, respectvely, for all 152 countres. 11

12 measure varablty n our analyss. 6 Table 2 presents results of estmatng the quantle regressons, usng the mean and standard devaton of the nflaton rate, for = 0.05, 0.25, 0.50, 0.75 and 0.95, an OLS regresson, and F-statstcs testng for equalty of the estmated slope parameter between varous quantles. The homogenety test consders whether the fve slope coeffcents equal each other across the fve quantles. Such tests provde a robust alternatve to conventonal least-squares-based test of heteroskedastcty, because we can construct them to reman nsenstve to outlyng response observatons. Panel A1 n Table 2 reports the results of estmatng the Fredman-Ball hypothess. The OLS regresson generates postve and sgnfcant coeffcent of nflaton at the 1% level. The fve-quantle regresson estmates of nflaton, condtonal on nflaton varablty, all prove postve and sgnfcant at the 1% level. These results support the Fredman-Ball hypothess that nflaton creates nflaton varablty. Moreover, the quantle regresson results llustrate that the margnal effect of nflaton on nflaton varablty ncreases as one moves from lower to hgher nflaton varablty quantles. That s, at hgher nflaton varablty quantles, nflaton exerts a larger effect on nflaton varablty. Ths evdence suggests that potental nformaton gans assocate wth the estmaton of the entre condtonal dstrbuton of nflaton varablty, as opposed to the condtonal mean only. In the bottom of Panel A, sgnfcant F-statstcs ndcate a statstcally sgnfcant dfference n the effect of nflaton across the dstrbuton of nflaton varablty, except between the 0.50 th and 0.95 th, and 0.75 th and 0.95 th quantles. The homogenety 6 We orgnally used the coeffcent of varaton as a measure of relatve uncertanty. An anonymous referee suggested usng the measure proposed by Davs and Kango (1992), who use a theoretcal model of Drffll, Mzon, and Ulph (1990) and the ntutve example from Davs (1989) to argue that researchers use relatve varablty (.e., the standard devaton dvded by one plus the mean) to measure nflaton uncertanty. An earler verson of ths paper provdes the results for the coeffcent of varaton. See Unversty of Connectcut Workng Paper # at 12

13 test rejects the null hypothess that all fve slope coeffcents equal each other. Inflaton exhbts a larger effect on nflaton varablty for the upper tal dstrbuton of nflaton varablty than the lower tal. The ntercept term does not dffer sgnfcantly from zero, except at the 0.25 th and 0.50 th quantles. Panel B of Table 2 reports the results of estmatng the Cukerman-Meltzer hypothess. All estmates of nflaton varablty prove postve and sgnfcant at the 1% level. The margnal effects of nflaton varablty on nflaton rse sgnfcantly across quantles except at the 0.95 th quantle tal, as the F-statstcs, testng for equalty of slope estmates across quantles, demonstrate. The homogenety test, once agan, rejects the null hypothess that all fve slope coeffcents equal each other. The ntercept term, whch proves sgnfcantly postve except nsgnfcantly postve at the 0.95 th quantle, ncreases across quantles. The evdence supports the Cukerman-Meltzer hypothess. Table 3 reports the OLS and quantle estmates for the Fredman-Ball and Cukerman-Meltzer hypotheses, usng the mean and the relatve measure of varaton of the nflaton rate. The OLS regressons fnd a sgnfcant relatonshp between nflaton and ts varablty, or vce versa. The constant terms also prove sgnfcant. Examnng the quantle results, nflaton postvely affects nflaton varablty sgnfcantly at the 0.05 th, 0.25 th, and 0.50 th quantles n Panel A, but the coeffcents prove small n magntude. Inflaton varablty postvely affects the nflaton rate sgnfcantly except at the 0.05 th and 0.95 th quantles n Panel B. 7 The constant terms rse across the quantles and prove sgnfcant. Thus, the use of the relatve measure 7 The F-statstcs testng for the equalty of the slope coeffcents across quantles cannot reject equalty, except between 0.95 th and each of the lower quantles n the Fredman-Ball model and between 0.05 th and each of the hgher quantles n the Cukerman-Meltzer model, at the 10-percent level. The homogenety tests stll reject the null hypothess that all fve slope coeffcents equal each other. 13

14 of varaton, the same as the standard devaton, generally fnds support for the Fredman-Ball and Cukerman-Meltzer hypotheses n the OLS and quantle specfcatons. The wdely agreed postve assocaton between nflaton and ts dsperson proves robust to the relatve measure. The fndngs, however, provde much less support for dfference n responses across quantles, as seen for the standard devaton (see Table 2). Table 4 reports the estmates for the Fredman-Ball and Cukerman-Meltzer hypotheses, usng the medan and medan devaton of the nflaton rate. The OLS regressons fnd a sgnfcant postve relatonshp between nflaton and ts varablty. Inflaton sgnfcantly and ncreasngly affects nflaton varablty at each of the quantles. 8 The constant terms prove sgnfcantly postve at the hgher tals of 0.75 th and 0.95 th. Smlarly, nflaton varablty sgnfcantly affects the nflaton rate at each of the quantles. 9 Thus, the use of the medan and the medan devaton produces a postve correlaton, supportng the Fredman-Ball and Cukerman-Meltzer hypotheses and matchng the fndngs for the mean and standard devaton as well as the mean and the relatve varaton. The relatonshp between the mean and relatve varaton dffers from the other two sets of results because dfferences n responsveness changes across quantles appears only at the 0.05 th and 0.95 th. 4. Does a Threshold Inflaton Rate Exst? Most researchers fnd a postve relatonshp between nflaton and ts varablty across countres, as we report n Secton 3. A few authors, however, do fnd that for low nflaton countres, the 8 The F-statstcs testng for the equalty of the slope coeffcents across quantles rejects equalty for the 0.75 th and 0.95 th quantles relatve to the 0.05 th, 0.25 th, and 0.50 th quantles n the Fredman-Ball model at least at the 5-percent level. The homogenety test rejects the null hypothess that all fve slope coeffcents equal each other. 9 The F-statstcs testng for the equalty of the slope coeffcents across quantles rejects equalty for the 0.75 th and 0.95 th quantles relatve to the 0.05 th and 0.25 th quantles n the Cukerman-Meltzer model at the 1-, 5-, or 10-percent levels. The homogenety test, once agan, rejects the null hypothess that all fve slope coeffcents equal each other. 14

15 postve relatonshp does not prove sgnfcant (e.g., Logue and Wllet, 1976 and Hafer and Heyne-Hafer, 1981). Ths secton revsts the ssue of whether a threshold level of nflaton exsts before fndng the postve correlaton between nflaton and ts varablty n the nflaton targetng era. The analyss of ths secton consders the Fredman-Ball and Cukerman-Meltzer hypotheses usng only the mean and standard devaton of the nflaton rate. A summary of the results for other specfcatons appears at the end of ths secton. The Appendx Table reports the average annual nflaton over 1993 to 2003 for all 152 countres. Generally, developed and successful developng countres exhbt low average nflaton rates, other countres, whch do not develop over tme, exhbt hgh nflaton rates. We splt the full sample nto two sub-samples, low-nflaton countres and hgh-nflaton countres, at the medan nflaton (.e., percent), to examne the relatonshp between mean nflaton and ts standard devaton for each sub-group (76 countres n each group). Panel A of Table 5 presents the results of estmatng the Fredman-Ball hypothess from the hgh- and low-nflaton country samples, respectvely. All slope parameters of nflaton n the OLS and quantle regressons prove postve and sgnfcant and they rse as we move from lower to hgher quantles. That s, for the Fredman-Ball hypothess, the same basc pattern of effects occurs across the quantles for the hgh and low nflaton country samples. Ths decomposton of our 152-country sample at the medan nflaton rate shows that nflaton varablty and the level of nflaton postvely relate across countres n each group. Thus, no evdence of a threshold effect emerges. When we estmate and test the Cukerman-Meltzer hypothess, we frst splt the full sample at the medan standard devaton (.e., percent) nto two sub-samples, hgh- and low-nflaton-varablty countres. Panel B of Table 5 presents the estmated results. Snce both 15

16 sample countres exhbt the postve correlaton between nflaton and ts varablty sgnfcantly, no threshold effect emerges. We note, however, that all sgnfcant postve slope parameters n the hgh-nflaton-varablty countres prove less than those n the low-nflaton-varablty countres. Ths consstent pattern may reflect Holland s (1995) vew that the monetary authortes n hgh-nflaton-varablty countres actvely stablze the economy. As a result, nflaton turns out to be less senstve to nflaton varablty, even f not lower, on average. Polcymakers may want to know the nflaton rate above whch sgnfcant ncreases n varablty occur, lowerng welfare and output growth. Barro (1995, 1996) fnds a negatve relatonshp between nflaton and economc growth, Bruno and Easterly (1996), however, argue that the evdence for ths negatve relatonshp s weak at low nflaton rates. Mshkn and Schmdt-Hebbel (2007b) clam that fndng an emprcal drect relatonshp between nflaton and growth wll not help to dscrmnate between dfferent nflaton goals under 10 percent. They suggest choosng the long-run nflaton target that establshes prce stablty. Prevous studes provde only lmted and mxed evdence on the senstvty of nflaton varablty to ts level n hgh-, low-, or moderate-nflaton regmes. Logue and Wllett (1976) fnd nsgnfcant correlaton for countres wth moderate nflaton between two to four percent. Hafer and Heyne-Hafer (1981) dscover the upper bound of the threshold ncreases sharply from four to nne percent n the 1970s. Ram (1985) argues that although the average nflaton rate rses durng the 1970s, the level-varablty correlaton falls n the 1970s. Moreover, a sgnfcant postve correlaton emerges only when nflaton rates exceed eght percent n the 1960 to 1970 sample and twenty percent n the 1972 to 1981 sample. Edmonds and So (1993) dscover sgnfcant relatonshps for a group of hgh- and low-nflaton countres, but not for a group of moderate-nflaton between sx and ten percent. Hess and Morrs (1996), on the other hand, 16

17 demonstrate a sgnfcant postve relaton for countres wth low- and moderate-nflaton less than ffteen percent a year. Davs and Kanago (1996) fnd a sgnfcant postve relaton n ten hgh nflaton countres, however, the coeffcents are no longer sgnfcant when Davd and Kanago (2000) restrct the sample to OECD countres wth nflaton rates under eght percent. More recently, Kley (2007) argues that moderate-to-hgh nflaton at levels around four percent per year assocate wth nflaton volatlty. How low (moderate or hgh) s a low (moderate or hgh) nflaton rate? No theory or emprcal analyss gves a defnte answer. That s, although sample dates, countres, measures of varablty, and sources of data may lead to dfferent results, the relevant polcy queston for most ndustralzed countres and many emergng market countres n recent years concerns the benefts from reducng nflaton from hgh or moderate levels to low levels. Our sample perod, , encompasses the nflaton-targetng era and the perod of the Great Moderaton. Thus, we search for a threshold level of nflaton, f any, based on the nflaton targets adopted by nflaton-targetng countres. Inflaton targetng provdes an operatonal framework for monetary polcy to attan prce stablty. Typcally, nflaton targets correspond to an annual rate of nflaton n the low sngle dgts (Bernanke et al. 1999, Batn and Yates 2003). Table 6 (Internatonal Monetary Fund 2005) lsts 21 countres that use nflaton targets, ther nflaton-targetng adopton years and ther current nflaton targets. The Table ncludes 8 ndustral countres and 13 emergng market countres. The numercal nflaton target typcally reflects an annual rate for the CPI n the form of a range, such as one to three percent (e.g., New Zealand and Canada). Alternatvely, the nflaton rate target equals a pont target wth a range, such as a two-percent target plus or mnus one percent (e.g., Sweden) or a pont target wthout any explct range, such as a two-percent target (e.g., the Unted Kngdom). For ndustral countres, the targets range between zero and three 17

18 percent. For emergng market countres, they all adopt a target range or a pont target wth a range. The mddle of the range or the pont target generally exceeds that n the ndustral countres. The range runs from zero and sx percent (except for seven percent n Brazl), whch nearly matches the range from zero to medan nflaton rate ( percent) n our sample. We saw n Panel B of Table 5 that nflaton varablty postvely and sgnfcantly relates to the nflaton rate for the sample of nflaton rates between zero and percent. The practce of nflaton targetng n the world leaves open the queston of whether nflaton varablty dffer n hgh or low nflaton-targetng regmes, even at the already lower level of nflaton. Thus, we further break our sample at a lower nflaton rate to look for a threshold. An examnaton of our sample data, the medan nflaton of our 76 low-nflaton countres (or, equvalently, the 25 percent of our 152 countres) equals percent, whch matches the edge of the three percent rate target for the ndustral countres. The specfcaton of the optmal long-run nflaton goal remans an unsolved ssue that s central to nflaton-targetng regmes. Mshkn and Schmdt-Hebbel (2007b) suggest that any nflaton target between 0 and 3 percent seems approprate for prce stablty (p.426). We, thus, splt our 76 low-nflaton countres at ts medan nflaton nto two groups (38 countres n each) low and moderate nflaton rate countres. Panel A of Table 7 presents the estmaton results from the moderate- and low-nflaton country samples, respectvely. For the moderate-nflaton countres, all slope parameters of nflaton n the OLS and quantle regressons prove postve and sgnfcant, except at the 0.50 th quantle. That s, the Fredman-Ball hypothess holds n countres wth moderate nflaton rates. For the low-nflaton countres, however, all slope parameters n the OLS and quantle regressons appear nsgnfcant, where margnal effects of nflaton prove much lower than the smlar effects n the moderate countres. In sum, dfferent effects occur across quantles for the moderate and low 18

19 nflaton country samples. Consderable evdence exsts that nflaton and ts varablty postvely correlate across countres. Our fndngs demonstrate that a threshold level of nflaton does exst before the postve correlaton emerges. The threshold occurs around the three percent nflaton rate. Countres wth nflaton rates below the threshold, such as those ndustral countres adoptng and achevng nflaton targets of less than three percent, generally fnd no assocaton between nflaton and ts varablty. Countres that acheve ther nflaton rate targets above the threshold, such as most emergng market countres, face the fact that hgher nflaton assocates wth hgher nflaton varablty. Ths evdence suggests that f the authortes want to elmnate the uncertanty of nflaton, then nflaton targets must not exceed the threshold of three percent. We further examne whether a threshold level of nflaton varablty exsts n low-varablty countres. We splt the 76 low-nflaton-varablty countres at ts standard devaton (.e., percent) nto two sub-samples for the moderate- and low-nflaton-varablty countres. Panel B of Table 7 presents the estmated results for the two subsamples separately. The sgnfcant OLS estmate of the slope proves less n the moderate-nflaton-varablty countres than n the low-nflaton-varablty countres, the latter s sgnfcant at the 10-percent level. The quantle regressons provde dverse, non-systematc results. At low quantles (.e., 0.05 th, 0.25 th, and 0.50 th ), the sgnfcant postve slope parameters suggest that the moderate-nflaton-varablty countres exhbt hgher margnal effects of nflaton varablty than low-nflaton-varablty countres. The stuaton reverses at hgh quantles (0.75 th and 0.95 th ), however. Hgher margnal effects emerge n the low-nflaton-varablty countres. The evdence of a threshold level of nflaton varablty n the Cukerman-Meltzer model proves weaker than that of the Fredman-Ball model. 19

20 We also examne the threshold effect, f any, for the mean and relatve varaton and the medan and medan devaton for the two hypotheses. For the Fredman-Ball hypothess, the use of the medan and the medan devaton dentfes a threshold effect. That s, n those countres wth nflaton rates below the medan nflaton rate (.e., the 25 percent of our 152 countres), the relatonshp between nflaton and ts varablty proves nsgnfcant. The use of the mean and the relatve varaton, however, exhbts no evdence of a threshold effect. For the Cukerman-Meltzer hypothess, nether of the two sets of measures dsplays a threshold effect. As a comparson, usng the mean and standard devaton n the text, we fnd a threshold effect below the nflaton rate (or, equvalently, the 25 percent of our 152 countres) for the Fredman-Ball hypothess, whle no evdence of a threshold effect emerges for the Cukerman-Meltzer hypothess. In sum, our emprcal evdence of the threshold effect for the mean and standard devaton and the medan and medan devaton does not prove robust to the relatve measure. Thus, ths study rases one ssue that deserves further attenton: whch measure more approprately captures varablty absolute or relatve measure. 5. Robustness Checks: Addtonal Explanatory Varables and Instrumental Estmaton Ths secton consders the robustness of our fndngs by conductng a multple regresson model ncludng a set of addtonal explanatory varables and an nstrumental varable estmaton. Intally, we consder those factors assocated wth the level of nflaton and ts varablty. Then, we mplement an analyss, usng of nstrumental varables. In ther study evaluatng quanttatve goals of monetary polcy for 42 countres from 1960 to 2000, Fatás et al. (2007) ndentfy four factors that sgnfcantly lower nflaton -- an nflaton targetng dummy varable, openness measured by exports plus mports as a percentage of GDP, the 20

21 budget surplus as a percentage of GDP, and real GDP per capta. 10 Tables 8, 9, 10, 11, 12, and 13 report estmaton results from ncludng ths set of regressors for our full sample of countres. The coeffcents δ and β capture the effect of nflaton on nflaton varablty and the effect of nflaton varablty on nflaton, respectvely, n the two models. They are sgnfcantly postve and generally correspond to the estmates wthout these condtonng varables n Tables 2, 3, and 4. Our fndngs seem robust. Caveats certanly exst, however. Almost all of the auxlary regressors are not sgnfcant. Fatás et al. (2007) use OLS estmaton. They also fnd no effect of the nflaton targetng dummy varable on nflaton varablty, once they control for the level of nflaton as shown n Panels A. For the Cukerman-Meltzer hypothess, we fnd that only nflaton varablty affects nflaton sgnfcantly. Inflaton targetng affects nflaton postvely or negatvely, but all coeffcents, save one, are nsgnfcant n Panels B. In Fatás et al. (2007) when they use post-1982 data estmatng ther nflaton regresson model, they fnd that only nflaton targetng and real GDP per captal exhbt sgnfcance. 11 Tables 14, 15, and 16, respectvely, report the senstvty of the results wth respect to ncluson of the four factors for the hgh-, moderate-, and low-nflaton countres and hgh-, moderate-, and low-nflaton-varablty countres The Fredman-Ball regresson model reports nsgnfcant coeffcents of nflaton, δ, for the OLS and each of the quantles only n the 10 The authors also nclude the dfference between real GDP growth and average GDP growth as a busness cycle varable n ther study, whch proves nsgnfcant n many cases. We drop ths varable n ths study. 11 We fnd that real GDP per captal becomes sgnfcant negatve n 25-percent of the coeffcent estmates and the budget surplus to GDP becomes sgnfcantly postve for three coeffcents n the Fredman-Ball specfcaton (never sgnfcant n the Cukerman-Meltzer specfcaton. Exports plus mports to GDP never acheves a sgnfcant coeffcent. 12 To repeat, hgh-nflaton (varablty) means the nflaton rate (varablty) above (4.3639); moderate-nflaton (varablty) denotes the nflaton rates (varablty) between (1.7284) and (4.3639); and low-nflaton (varablty) represents the nflaton rate (varablty) below (1.7284). 13 Note that n Table 6, ndustral countres adopt nflaton targets below 3 percent and most of those emergng market economes adopt nflaton targets between 3 to 6 percent. 21

22 low-nflaton countres (Table 16). The concluson of no assocaton between nflaton and ts varablty n countres wth nflaton rates below three percent also holds when we nclude the nflaton targetng dummy varable (IT) and the other three covarates. The Cukerman-Meltzer model, ncludng the four varables n the OLS regresson, reports sgnfcant coeffcents on nflaton varablty, β, at least at the 10 percent level n the three Tables as n Table 7. All the effects n quantle regressons, however, appear nsgnfcant n the low-nflaton-varablty countres (Table 16). Accordng to ths evdence, a threshold level of nflaton varablty at percent n the Cukerman-Meltzer model exsts. For the low-nflaton and low-nflaton-varablty countres n Table 16, the polcy dummy varable (IT) does not sgnfcantly nfluence the nflaton rate or nflaton varablty. Ball and Sherdan (2005) and Ln and Ye (2007) also fnd nsgnfcant effects of the nflaton targetng monetary polcy on nflaton or nflaton varablty for developed countres, whch generally experence low nflaton rates and low nflaton varablty. Tables 17, 18, and 19 use nstrumental varable estmaton to account for possble endogenety n each of the two equatons for the full sample. Barro (1995, 1996) proposes the use of lagged nflaton to solate exogenous varaton n nflaton. We extend our coverage to nclude the nflaton targetng dummy varable, snce nflaton targetng countres exhbt, on average, low nflaton and low nflaton varablty, as well as lagged values of openness, the budget surplus to GDP, and real GDP per capta. Thus, n the frst stage of our two-stage estmaton, we estmate the reduced form equatons for nflaton and nflaton varablty, where nflaton (nflaton varablty) relates to lagged nflaton (nflaton varablty) over the fve years ( ) pror to the sample perod ( ) as an nstrument along wth the nflaton targetng dummy varable and the lagged values openness, the budget surplus to GDP, and real GDP per capta. We estmate the unknown 22

23 parameters n the reduced form equatons by OLS. In the second stage, we estmate equatons (4) and (5) by OLS and quantle regressons, except that we replace nflaton ( varablty ( V ) wth ther predcted values from the frst stage regresson. Π ) and nflaton Tables 17, 18, and 19 report estmates, δ and β, from the bvarate regresson models, snce the addtonal explanatory varables generally prove nsgnfcant n the multvarate regressons, for the four groups of sample countres. The nstrumental varables of nflaton and nflaton varablty seem to work well. For all the sample countres except the low-nflaton ones, the estmates, δ and β, for the Fredman-Ball and Cukerman-Meltzer hypotheses are sgnfcantly postve and consstent wth the fndngs of our models when we use actual mean and medan nflaton rates and the standard devaton, relatve varablty, and medan devaton n estmatons. The use of the nstrumental varables suggests that the estmated relaton between nflaton and nflaton varablty does not represent reverse effects of nflaton varablty on nflaton or of nflaton on nflaton varablty. It remans true that the nsgnfcant nfluence of nflaton on nflaton varablty, δ, shows up only n the low-nflaton countres for the Fredman-Ball hypothess. In addton, we fnd mxed results for the effect of nflaton varablty on nflaton, β, n the low-nflaton-varablty countres for the Cukerman-Meltzer hypothess. Tables 20, 21, and 22 present the estmaton results from the hgh-, moderate- and low-nflaton country samples, respectvely. For the hgh- and moderate-nflaton countres, all slope parameters of nflaton n the OLS and quantle regressons prove postve and sgnfcant. That s, the Fredman-Ball hypothess holds n countres wth hgh and moderate nflaton rates. For the low-nflaton countres, however, all slope parameters n the OLS and quantle regressons appear nsgnfcant, where margnal effects of nflaton prove much lower than the smlar effects n the hgh- and moderate-nflaton-rate countres. In sum, dfferent effects occur across quantles 23

24 for the hgh- and moderate-nflaton versus the low-nflaton country samples. More specfcally, the ndvdual slope coeffcents ncrease monotoncally across quantles. Moreover, these ndvdual coeffcents dffer sgnfcantly from each other across all pars of quantles, except for the 0.05 th and 0.25 th quantles for the hgh-nflaton countres. Consderable evdence exsts that nflaton and ts varablty postvely correlate across countres. Our fndngs demonstrate that a threshold level of nflaton does exst before the postve correlaton emerges. The threshold occurs around the three percent nflaton rate. Countres wth nflaton rates below the threshold, such as those ndustral countres adoptng and achevng nflaton targets of less than three percent, generally fnd no assocaton between nflaton and ts varablty. Countres that acheve ther nflaton rate targets above the threshold, such as most emergng market countres, face the fact that hgher nflaton assocates wth hgher nflaton varablty. Ths evdence suggests that f the authortes want to elmnate the uncertanty of nflaton, then nflaton targets must not exceed the threshold of three percent. We further examne whether a threshold level of nflaton varablty exsts n low-varablty countres. As noted above, we splt the 76 low-nflaton-varablty countres at ts medan standard devaton (.e., percent) nto two sub-samples for the moderate- and low-nflaton-varablty countres. Table 20, 21, and 22 present the estmated results for the three sub-samples separately. The sgnfcant OLS estmate of the slope proves less n the low-nflaton-varablty countres than n the hgh- and moderate-nflaton-varablty countres. The quantle regressons fnd nsgnfcant slope coeffcents for low-nflaton-varablty countres, except at the 0.95 th quantle. The hgh- and moderate-nflaton-varablty countres all experence sgnfcant postve slope coeffcents. Moreover, the magntudes of the effect for the moderate-nflaton-varablty countres exceed that for the hgh-nflaton-varablty countres. 24

25 The evdence of a threshold level of nflaton varablty n the Cukerman-Meltzer model proves only margnally weaker than that of the Fredman-Ball model. The ndvdual slope coeffcents for the hgh-nflaton-varablty countres ncrease monotoncally from the lowest to the hghest quantles. Moreover, the ndvdual slope coeffcents dffer from each other across all pars of quantles. None of the ndvdual slope coeffcents dffer across all quantle pars for the moderate-nflaton-varablty countres. We also examne the threshold effect, f any, for the mean and relatve varaton and the medan and medan devaton for the two hypotheses. For the Fredman-Ball hypothess, the use of the mean and relatve varaton or the medan and medan devaton dentfes a threshold effect. That s, n those countres wth nflaton rates below the medan nflaton rate (.e., the 25 percent of our 152 countres), the relatonshp between nflaton and ts varablty proves nsgnfcant (see Tables 25 and 28). For hgh- and moderate-nflaton countres, we fnd a postve and sgnfcant relatonshp (see Tables 23, 24, 26, and 27), except for the 0.05 th quantle for hgh-nflaton countres n Table 23. For the Cukerman-Meltzer hypothess, both sets of measures dsplay a threshold effect. That s, no sgnfcant slope coeffcent exsts across the quantles n Tables 25 and 28. In sum, our emprcal evdence of the threshold effect for the mean and standard devaton proves robust to the mean and relatve varaton as well as the medan and medan devaton measures. 6. Concluson Usng cross-sectonal data on 152 countres over the perod 1993 to 2003 our emprcal results support both hypotheses of Fredman-Ball and Cukerman-Meltzer from the parametrc quantle model when we use the mean and standard devaton, the mean and relatve varaton, or the medan and medan devaton of the nflaton rate to measure nflaton and ts varablty. Frst, 25

26 nflaton and nflaton varablty postvely relate to each other across quantles. Second, hgher nflaton assocates wth more nflaton varablty, supportng the Fredman-Ball hypothess. Thrd, nflaton varablty rases nflaton, supportng the Cukerman-Meltzer hypothess. The results for the mean and the relatve varaton as well as the medan and medan devaton specfcatons provde nearly the same support for both hypotheses. Gven the postve relatonshp between nflaton and ts varablty across countres, we explore the possblty of threshold effects. We fnd evdence of a threshold effect n the Fredman-Ball hypothess. That s, for nflaton rates under 3 percent, hgher nflaton does not assocate wth hgher nflaton varablty. Ths fndng s robust n a multvarate regresson that ncludes a set of addtonal explanatory varables or n nstrumental varables regressons to correct for any potental endogenetes. Ths fndng proves consstent wth those of Logue and Wllett (1976) and Hafer and Heyne-Hafer (1981), who fnd threshold nflaton rates of 4 and 9 percent, respectvely. Gven dfferences n average nflaton rates n the dfferng sample perods, our 3 percent threshold seems n the ballpark for the sample perod that ncludes the Great Moderaton. Ths evdence also supports Mshkn and Schmdt-Hebbel s (2007b) conjecture that the long-run nflaton target between 0 and 3 percent s reasonable, operatonal, and consstent wth prce stablty. Kley (2007) shows that many nflaton targeters n developng countres pursue moderate to hgh (above 4 percent annually) targets and that ths may contrbute to nflaton nstablty. Lower target nflaton rates may contrbute to macroeconomc stablty. (p.196). Ball and Sherdan (2005) and Ln and Ye (2007) fnd nsgnfcant effects of the nflaton targetng monetary polcy on nflaton varablty for developed countres, whch all target the nflaton rate below 3 percent as shown n Table 2. We also fnd smlar evdence of a threshold effect for nflaton varablty n the Cukerman-Meltzer hypothess. 26