Asymmetric inflation dynamics: Evidence from quantile regression analysis

Asymmetric inflation dynamics: Evidence from quantile regression analysis

Ching-Chuan Tsong

a,⇑

, Cheng-Feng Lee

b,1

a_{Department of Economics, National Chi Nan University, Nantou 545, Taiwan} b

Department of Business Administration, National Kaohsiung University of Applied Sciences, No. 415, Jiangong Rd., Sanmin District, Kaohsiung City 80778, Taiwan

a r t i c l e

i n f o

Article history: Received 9 January 2011 Accepted 1 August 2011

Available online 16 September 2011 JEL classification: C32 E31 Keywords: Inflation rate Quantile regression

a b s t r a c t

This paper applies the regression quantile approach developed byKoenker and Xiao (2004) to investigate the dynamic behavior of inflation in 12 OECD countries. By analyzing the behavior in a wide range of quantiles, this method allows us to quantify the influence of various sizes of shocks that hit the inflation, and is able to capture possible asymmetric adjustment of the inflation towards to its long-run equilibrium. It therefore sheds new lights on the inflation dynamics compared with the conventional unit root methodologies. Our results suggest that generally, the inflation rates are not only mean-reverting but also exhibit asymmetries in their dynamic adjustments, in which large negative shocks tend to induce strong mean reversion, and on the contrary, large positive shocks do not. Policy implications related to the empirical findings are also provided.

Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction

The behavior of macroeconomic variables has received considerable attention in the literature. In particular, inflation rate is one of the most important variables due to its key roles in many macroeconomic models (e.g.,Levin and Piger, 2003; Angeloni et al., 2006), and to the association with monetary policies implemented by the authorities whose objective is to deliver price stability (e.g.,Zhang and Clovis, 2010). In addition, as argued by, inter alia,Lee and Wu (2001), Ho (2009), andHenry and Shields (2004), the dynamic behavior of inflation rate has a number of economic implications.

In a large body of empirical literature, the time-series properties of inflation are investigated by using unit root testing procedure assuming constant dynamics, which focus on the conditional central tendency (e.g.,Rose, 1988; Ng and Perron, 2001; Tsong and Lee, 2010). Under such assumption, the speed of inflation adjustment towards its equilibrium would be constant no matter how far the inflation is above or below its long-run level or how big the negative or positive shock hitting the inflation. As a result, the aforementioned studies at most can only distinguish inflation between a unit-root and a sta-tionary process, lacking the ability to further elaborate on the inflation dynamics. Another important observation is that the distribution of inflation is often leptokurtic, differing significantly in shape from the Normal (Charemza and Hristova, 2005). In this case, as pointed out byKoenker and Xiao (2004), the commonly-used unit root tests can exhibit rather poor power performance, tending to bias test results in favor of a unit root.

In this paper, we seek to re-investigate the inflation dynamics by employing the quantile regression inference developed byKoenker and Xiao (2004), and to provide complementary information on the inflation adjustments. Such approach is used in recent empirical studies on real exchange rates (Nikolaou, 2008) and on nominal interest rates (Koenker and Xiao, 2004). The methodology has several advantages over the conventional counterparts. First, instead of only concentrating on the

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⇑ Corresponding author. Tel.: +886 49 291 0960x4662; fax: +886 49 291 4435. E-mail addresses:tcc126@ncnu.edu.tw(C.-C. Tsong),jflee@cc.kuas.edu.tw(C.-F. Lee).

1

Tel.: +886 7 381 4526x7310; fax: +886 7 396 1245.

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constant speed of adjustment with conditional mean function, this method allows for different and possibly asymmetric speed of adjustment at various quantiles. Consequently, the possibility of sign asymmetry, different adjustment mechanism towards the long-run equilibrium for positive or negative shocks, can be detected. Also, the tendency of mean reversion can be quantified based on the size of the shock hitting the inflation. As noted byBoldin (1999), taking into account possible asymmetric dynamics for a series can improve the performance of monetary policy.

Second, the quantile unit root tests can provide an alternative way to study local persistence in time series. To be specific, within certain quantiles, the inflation rate can display unit-root behavior, but within the others, its mean-reverting property may be sufficient to insure global mean reversion. By contrast, the inflation rate can merely be distinguished globally be-tween nonstationarity and mean reversion with the conventional counterparts.

Finally, it offers a more flexible and refined analysis of the inflation dynamics by relaxing the assumption that the infla-tion rate follows a particular distribuinfla-tion. Moreover, the shocks analyzed are actual, whose magnitudes are determined endogenously by the data. Due to better data descriptions accommodating potential heavy-tailed behavior in inflation, this method can lead to substantial power gains over the conventional least squared-based counterparts as argued byKoenker and Xiao (2004). As a result, strong evidence in favor of mean-reverting inflation rate, as expected, is more likely to be uncovered.

Our empirical results suggest that the inflation rates in 12 OECD countries are globally meaning-reverting based on the quantile inference, in sharp contrast to the counterparts from the univariate unit root tests focusing on the average behavior. Moreover, both the sign and size of the shock, in general, have marked impact on the speed of inflation adjustment towards its long-run equilibrium. Specifically, the negative inflation shocks with large magnitude can induce strong mean-reverting tendencies, while the positive counterparts found at medium or extreme quantiles could have infinite lives for inflation rates. As a result, asymmetric mean reversion towards the equilibrium for inflation rates is prevalent. This implies that for effectively curbing inflation, policymakers should keep alert to monitoring potential inflation increase and take precau-tionary measures to anchor inflation expectations. When the inflation is hit by negative shocks, however, the authorities may prefer not to intervene in the inflation dynamics such that the inflation could revert back to its long-run equilibrium level eventually.

The remainder of the paper is organized as follows. In Section2, we review the studies on the inflation dynamics. Section

3presents the methodology used in this study. The empirical results are collected in Section 4. Section 5 concludes the paper.

2. Literature review

Since the pioneer work ofNelson and Plosser (1982), the issue that whether most macroeconomic time series can be de-scribed as a unit root process has motivated a large body of literature. The empirical studies, however, still cannot reach a consensus. With conventional univariate unit root or stationarity tests,Evans and Lewis (1995), Nelson and Schwert (1977), Crowder and Wohar (1999), Camarero et al. (2000), MacDonald and Murphy (1989), Ball and Cecchetti (1990), andCrowder and Hoffman (1996)fail to find strong evidence in favor of mean reversion in inflation rates. By employing the state-of-the-art univariate unit root tests with good asymptotic size and power properties,Ng and Perron (2001)still cannot uncover sufficient evidence supporting that shocks to inflation are short-lived. However,Rose (1988)rejects the unit-root null in US inflation, andEdwards (1988)reports strong persistence in Latin American inflation rates. In addition,Baillie et al. (1996)

find that inflation rates in developed countries are fractionally integrated and mean-reverting.

It is well-documented in the literature that traditionally univariate unit root tests have low power against persistent alternatives, especially for small sample sizes encountered in empirical studies. To alleviate this problem, the panel-type tests and covariate tests are used in recent studies. The former tests exploit cross-sectional information to increase power, while the latter tests incorporate related covariates which contain valuable information to boost power. For example,Culver and Papell (1997)implement the panel unit root test ofLevin et al. (2002)to the data of 13 OECD countries, and reject the unit-root null in inflation rate. By taking into account the cross-sectional dependency among individual countries,Lee and Wu (2001)provide strong empirical evidence to support the mean reversion of inflation rate with the test proposed byIm et al. (2003). In addition,Basher and Westerlund (2008)apply a battery of panel tests to the same data set inCulver and Papell (1997), and their results suggest that inflation rates are stationary after controlling for cross-sectional dependency and structural break. However, with the nonlinear IV panel unit root test suggested byChang (2002) and Ho (2009)finds that the inflation rates may accelerate. In other words, shocks to inflation appear to be infinitely persistent. By carrying out a battery of covariate tests proposed byHansen (1995), Elliott and Jansson (2003),Jansson (2004), and Tsong and Lee (2010)provide strong evidence supporting that the inflation process in 15 OECD countries displays mean reversion.

Another strand of literature further concentrates on the changes in the degree of inflation persistence, instead of only using unit root tests to distinguish the inflation process as either an I(0) or I(1) process.Taylor (2000)estimates the largest autoregressive root (LAR) and suggests that the US inflation persistence has been declined after the Volcker–Greenspan era. Considering a possible structural break,Levin and Piger (2003)showed that high inflation persistence is not intrinsic in industrialized countries. Similarly, with the estimation of a Bayesian VAR model,Cogley and Sargent (2001, 2005)claimed that the US inflation persistence has experienced a significant decline.Kumar and Okimoto (2007)investigate the dynamics of inflation persistence using long memory approach and find that there has been a marked decrease in US inflation persis-tence over the past two decades. By employing an ARMA model with time-varying autoregressive parameter,Beechey and C.-C. Tsong, C.-F. Lee / Journal of Macroeconomics 33 (2011) 668–680 669

Österholm (2009)show that inflation persistence has fallen remarkably in the Euro area after January 1999 and inflation no longer displays non-stationary behavior. In addition, Chinese inflation over the post-1997 period tends to revert more quickly to its long-run level than over the pre-1997 period as is exhibited inZhang and Clovis (2010).

On the contrary,Stock (2001)estimated the LAR with rolling window estimation method, and concluded that there is no indication of a clear reduction in US inflation persistence. Similar conclusions are drawn byPivetta and Reis (2007), but with both Bayesian and rolling window approach to the estimation of the LAR. With the employment of rolling regressions on split samples,Batini (2006)finds that European inflation is rather inertial; more importantly, the inflation persistence seems to have varied only marginally over the past 30 years. Similar results are presented inO’Reilly and Whelan (2005)as well, in which the estimates of persistence parameter in Euro area are generally close to one and rather stable over time.

3. Empirical methodology

In this section we briefly describeKoenker and Xiao’s (2004)quantile regression framework employed in the present pa-per, which enjoys power gains over the augmented Dickey–Fuller (ADF) test when the shock exhibits heavy-tailed behavior. More importantly, this testing procedure enables us to explore the speed of mean reversion for a series under different mag-nitudes and signs of the shock. To be precise, this methodology is capable of unveiling possibly different mean-reverting pat-terns by explicitly testing for a unit root at different quantiles, in which the underlying series is hit by a shock with various sizes and signs.

Let us consider the ADF regression model given by:

yt¼

a

1yt1þ Xq

j¼1

a

jþ1

D

ytjþ ut; t ¼ 1; 2; . . . ; n; ð1Þ

where yt=

p

t

l

, with

p

tand

l

denoting the inflation rate and its long-run equilibrium value, respectively; utis iid random

variable with zero mean and constant variance. In this model, the AR coefficient

a

1measures the persistence of yt. If

a

1= 1,

then ytfollows a unit root process, and if |

a

1| < 1, then the behavior of ytdisplays mean reversion. FollowingKoenker and Xiao (2004)and based on Eq.(1), the

s

th conditional quantile of yt, conditional on the past information set It1, can be

ex-pressed as a linear function of yt1and lagged values ofDytas follows:

Qytð

s

jIt1Þ ¼ x

0

t

a

ð

s

Þ; ð2Þ

where xt= (1, yt1,Dyt1, . . . ,Dytq)0 and

a

(

s

) = (

a

0(

s

),

a

1(

s

), . . . ,

a

q+1(

s

))0, with

a

0(

s

) denoting the

s

th quantile of ut. It is

important to note that

a

1(

s

) measures the speed of mean reversion of ytwithin each quantile, and is dependent on the

s

th quantile under investigation.

For a given

s

, the parameter vector

a

(

s

) in Eq.(2)is estimated by minimizing sum of asymmetrically weighted absolute deviations:

minX n

t¼1

ð

s

 Iðyt<x0t

a

ð

s

ÞÞÞjyt x0t

a

ð

s

Þj; ð3Þ

where I denotes an indicator function, i.e., I = 1 if yt<x0t

a

ð

s

Þ, and I = 0, otherwise. Given the solution of Eq.(3), denoted by

^

a

ð

s

Þ,Koenker and Xiao (2004)suggest testing the time-series properties of ytwithin the

s

th quantile by using the following t

ratio statistic: tnð

s

Þ ¼ ^_{f ðF}1 ð

s

ÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s

ð1 

s

Þ p ðY01PXY1Þ 1=2 ð^

a

1ð

s

Þ  1Þ; ð4Þ

where ^fðF1ð

s

ÞÞ is a consistent estimator of f(F1₍

_s

_{)), with f and F representing the density and distribution function of u} tin

Eq.(1), Y1is the vector of lagged dependent variables (yt1), and PXis the projection matrix onto the space orthogonal to

X = (1,Dyt1, . . . ,Dytq). According to Koenker and Xiao (2004), ^fðF1ð

s

ÞÞ can be written as ^fðF1ð

s

ÞÞ ¼ ð

s

i

s

i1Þ=

x0

tð^

a

ð

s

iÞ  ^

a

ð

s

i1ÞÞ with

s

i

e

C. We chooseC= {0.1, 0.2, . . . , 0.9} in our empirical study. By employing the test statistic tn(

s

),

we can examine the unit root properties of the series by looking at its behavior in each quantile. In other words, this not only enables us to take a closer look at the dynamics of the series, but also to investigate possibly different mean reverting behavior when the series is hit by different magnitudes and signs of shock at different quantiles. By contrast, the widely-used unit root tests only focus on the conditional central tendency and lack the ability to elaborate on such behavior. In addition, with ^

a

1ð

s

Þ we calculate the half-lives (HLs) of a shock hitting the inflation rate within the quantile with the formula

lnð0:5Þ= lnð^

a

1ð

s

ÞÞ.2

2

As noted inMurray and Papell (2005), OLS and the median-unbiased method can be used to estimate the AR coefficienta1in Eq.(1). Then HLs can be

calculated with the formula lnð0:5Þ= lnð^a1Þ or through impulse response function. Though the median-unbiased method is more robust in conditional mean

specification, to the best of our knowledge, its property is still unknown for quantile autoregression models. Therefore, this method is not employed in this paper. We thank the referee for this insightful comment.

Another approach to generally analyze the unit root behavior based on the quantile framework involves examining the nonstationary properties over a range of quantiles, instead of only focusing on the selected quantile. For this purpose, Koen-ker and Xiao (2004)recommend the quantile Kolmogorov–Smirnov (KS) test, which is given by

QKS ¼ supjtnð

s

Þj; ð5Þ

where tn(

s

) is the t ratio statistic defined in Eq.(4). In practice, we calculate tn(

s

) at

s

e

C, and thus the QKS test can be

con-structed by taking maximum overC.

The limiting distributions of the tn(

s

) and QKS tests are nonstandard, and depend on nuisance parameters.Koenker and Xiao (2004)suggest using a re-sampling procedure to approximate their small-sample distributions as described below.

(1) Fit the following q-order autoregression withDytby ordinary least squares (OLS):

D

yt¼ Xq

j¼1 ^

bj

D

y_tjþ ^ut; ð6Þ

and obtain estimates ^bjfor j = 1, 2, . . . , q, as well as the residuals ^ut.

(2) Draw a bootstrap sample of u

t with replacement from the empirical distribution of the centered residuals

~

ut¼ ^ut ðn  qÞ1Pnt¼qþ1u^t.

(3) Generate the bootstrap sample ofDy

t recursively using the fitted autoregression given by

D

y t ¼ Xq j¼1 ^ bj

D

ytjþ ut; ð7Þ

with ^bjbeing OLS estimates in Eq.(6), and initial valuesDyj ¼Dyjfor j = 1, 2, . . . , q.

(4) A bootstrap sample of y

t can be obtained based on

y

t¼ yt1þ

D

yt; ð8Þ

with y 1¼ y1.

(5) With the re-sample y

t, compute the bootstrap counterparts of ^

a

0ð

s

Þ, ^

a

1ð

s

Þ, the tn(

s

) and QKS tests, denoted by ^

a

0ð

s

Þ,

^

a



1ð

s

Þ, tnð

s

Þ and QKS

_{, respectively.}

(6) Repeat Steps 2 to 5 NB times. NB is 500 in this paper.

(7) Compute the empirical distribution function of the NB values of ^

a



0ð

s

Þ, ^

a

1ð

s

Þ, the tnð

s

Þ and QKStests, and use these

empirical distribution functions as an approximation to the cumulative distribution functions of the respective tests under the null.

(8) Using the bootstrap p-value to make inference. Also, the bootstrap confidence intervals for ^

a

0ð

s

Þ and ^

a

1ð

s

Þ can be

accordingly obtained from the empirical distribution functions of ^

a



0ð

s

Þ and ^

a

1ð

s

Þ, respectively.

4. Empirical investigation

4.1. Data description and preliminary results

Quarterly observations of consumer price indices (CPI) are retrieved from the International Monetary Fund’s IFS data base for a period ranging from 1957Q1 to 2010Q1. The 12 OECD countries considered include Australia, Denmark, France, Ger-many, Ireland, Italy, Japan, New Zealand, Spain, Sweden, the United Kingdom (UK), and the United States (US). The CPI is transformed into annual inflation rates

p

twith

p

t¼ 400ðlnðCPItÞ  lnðCPIt1ÞÞ: ð9Þ

Table 1reports the first four sample moments of the inflation rates and the results of the Jarque–Bera (JB) normality tests. Spain has the largest long-run mean for inflation (7.157), while Germany has the smallest counterpart (2.740). The largest and smallest sample standard deviations are 6.407 and 2.513 for Ireland and Germany, respectively. Note also that the sam-ple correlation between these inflation means and standard deviations is as high as 0.815, conforming to the results that high inflation may accompany high inflation uncertainty (e.g.,Okun, 1971; Davis and Kanago, 1988; Daal et al., 2005, to name a few). More importantly, all the inflation rates exhibit fat tail and non-normality since the JB test overwhelmingly rejects the null hypothesis of normality with extremely small p-values, consistent with the results inCharemza and Hristova (2005). The significant evidence of non-normality in the inflation rates lends strong support to the employment of the quantile regression approach in this study as emphasized byKoenker and Xiao (2004).

Though our focus is on the results of quantile regression, the counterparts of the conventional unit root tests such as DF-GLS and MZa-GLS are also included for the sake of comparison. Since it is well documented in the literature that the

inflation rate might be described as an MA process with a negative root, to reduce serious size distortions with more correct model specification, the lag length for the two tests are selected by the modified Akaike information criterion (MAIC) proposed byNg and Perron (2001)with maximum lag set at 16.

Table 2collects these testing results. In general, the two univariate unit-root tests fail to uncover broad evidence in favor of stationary inflation rates, in line with previous studies such asRapach and Weber (2004)andMacDonald and Murphy (1989), among others. The DF-GLS test rejects the unit-root null at the 5% level for Australia and Germany, with one addi-tional rejection for the US at the 10% level. For the MZa-GLS test, only Australia and Germany are rejected at the 5% and 10% C.-C. Tsong, C.-F. Lee / Journal of Macroeconomics 33 (2011) 668–680 671

may prefer not to intervene in the inflation through monetary policies since it can return to its long-run equilibrium level eventually.

Acknowledgments

The authors thank the editor and the anonymous referee for insightful comments and suggestions. All errors are ours. This paper is a part of the research project financially sponsored by the National Science Committee of Taiwan (NSC 100-2410-H-151-018).

References

Angeloni, I., Aucremanne, L., Ehrmann, M., Gali, J., Levin, A., Smets, F., 2006. New evidence on inflation persistence and price stickness in the euro area: implications from macro modelling. Journal of the European Economic Association 4, 562–574.

Baillie, R.T., Chung, C.F., Tieslau, M.A., 1996. Analysing inflation by the fractionally intergrated ARFIMA-GARCH model. Journal of Applied Econometrics 11, 23–40.

Ball, L., Cecchetti, S.G., 1990. Inflation and uncertainty at short and long horizons. Brookings Papers on Economic Activity 1, 215–254.

Basher, S.A., Westerlund, J., 2008. Is there really a unit root in the inflation rate? More evidence from panel data models. Applied Economics Letters 15, 161– 164.

Batini, N., 2006. Euro area inflation persistence. Empirical Economics 31, 977–1002.

Beechey, M., Österholm, P., 2009. Time-varying inflation persistence in the Euro area. Economic Modelling 26, 532–535.

Boldin, M.D., 1999. Should policy makers worry about asymmetries in the business cycle? Studies in Nonlinear Dynamics and Econometrics 3, 203–220. Camarero, M., Esteve, V., Tamarit, C., 2000. Price convergence of peripheral European countries on the way to the EMU: a time series approach. Empirical

Economics 25, 149–168.

Caner, M., Hansen, B.E., 2001. Threshold autoregression with a unit root. Econometrica 69, 1555–1596.

Chang, Y., 2002. Nonlinear IV unit root tests in panels with cross-sectional dependency. Journal of Econometrics 110, 261–292. Charemza, W.W., Hristova, D., 2005. Is inflation stationary? Applied Economics 37, 901–903.

Cogley, T., Sargent, T.J., 2001. Evolving post-World War II U.S. inflation dynamics. NBER Macroeconomics Annual 16, 331–373.

Cogley, T., Sargent, T.J., 2005. Drifts and volatilities: monetary policies and outcomes in the post WWII US. Review of Economic Dynamics 8, 262–302. Table 5

Results for quantile unit root test on inflation rates for eurozone countries.

Country s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 France a0(s) 2.680 1.471 1.023 0.559 0.155 0.217 0.723 1.243 2.332 p-value 0.000 0.000 0.000 0.000 0.154 0.095 0.001 0.000 0.000 a1(s) 0.603 0.775 0.806 0.868 0.916 0.927 0.974 0.956 1.114 p-value 0.002 _0.002 _0.000 _0.010 _{0.200 0.188 0.742 0.604 0.992} Half-lives 1.371 2.713 3.219 4.905 7.924 9.122 26.342 15.427 1

QKS/p-value 4.048/0.000 _{Start date for euro}

launch = 1999Q1 Germany a0(s) 2.934 1.624 1.208 0.721 0.164 0.640 1.039 1.543 2.942 p-value 0.000 _0.000 _0.000 _0.000 _{0.261 0.001} _0.000 _0.000 _0.000 a1(s) 0.301 0.350 0.344 0.352 0.385 0.387 0.360 0.309 0.191 p-value 0.000 _0.000 _0.000 _0.000 _0.000 _0.000 _0.000 _0.000 _0.000 Half-lives 0.577 0.660 0.649 0.663 0.726 0.730 0.678 0.590 0.419 QKS/p-value 6.570/0.000

Start date for euro launch = 1999Q1 Ireland a0(s) 4.212 2.896 1.960 1.355 0.268 0.517 1.342 3.084 4.966 p-value 0.000 0.000 0.000 0.000 0.242 0.069 0.001 0.000 0.000 a1(s) 0.554 0.623 0.680 0.756 0.818 0.939 1.021 1.187 1.176 p-value 0.006 0.000 0.000 0.002 0.014 0.572 0.902 1.000 0.986 Half-lives 1.172 1.467 1.794 2.480 3.443 11.082 1 1 1

QKS/p-value 5.402/0.000 _{Start date for euro}

launch = 1999Q1 Italy a0(s) 2.784 1.866 1.176 0.552 0.016 0.347 0.813 1.711 2.856 p-value 0.000 0.000 0.000 0.009 0.469 0.071 0.006 0.000 0.000 a1(s) 0.795 0.792 0.839 0.862 0.959 0.944 1.014 1.066 1.042 p-value 0.002 0.002 0.006 0.006 0.442 0.314 0.916 0.984 0.926 Half-lives 3.022 2.976 3.949 4.675 16.556 12.002 1 1 1

QKS/p-value 3.859/0.012 _{Start date for euro}

launch = 1999Q1 Spain a0(s) 3.934 2.775 1.826 1.247 0.507 0.376 1.170 1.919 5.121 p-value 0.000 _0.000 _0.000 _0.000 _0.078 _{0.170 0.006} _0.001 _0.000 a1(s) 0.726 0.787 0.733 0.782 0.829 0.885 0.886 0.921 1.012 p-value 0.042 _0.008 _0.000 _0.006 _0.034 _{0.170 0.252 0.642 0.882} Half-lives 2.162 2.900 2.227 2.815 3.686 5.660 5.745 8.435 1 QKS/p-value 4.147/0.004

Start date for euro launch = 1999Q1 Notes: See notes belowTable 3.

Crowder, W., Hoffman, D., 1996. The long run relationship between nominal interest rates and inflation: the Fisher equation revisited. Journal of Money, Credit, and Bank 28, 102–118.

Crowder, W., Wohar, M., 1999. Are tax effects important in the long run Fisher relationship: evidence from the municipal bond market. Journal of Finance 54, 307–317.

Culver, S.E., Papell, D.H., 1997. Is there a unit root in the inflation rate? Evidence from sequential break and panel data models. Journal of Applied Econometrics 12, 436–444.

Daal, E., Naka, A., Sanchez, B., 2005. Re-examining inflation and inflation uncertainty in developed and emerging countries. Economic Letters 89, 180–186. Davis, G., Kanago, B., 1988. High and uncertain inflation: results from a new data set. Journal of Money, Credit and Bank 30, 218–230.

Edwards, S., 1988. Two crises: inflationary inertia and credibility. The Economic Journal 108, 680–702.

Elliott, G., Jansson, M., 2003. Testing for unit roots with stationary covariates. Journal of Econometrics 115, 75–89.

Evans, M.D., Lewis, K.K., 1995. Do expected shifts in inflation affect estimates of the long run Fisher relation? The Journal of Finance 50, 225–253. Hansen, B.E., 1995. Rethinking the univariate approach to unit root testing: using covariates to increase power. Econometric Theory 11, 1148–1171. Henry, O.T., Shields, K., 2004. Is there a unit root in inflation? Journal of Macroeconomics 26, 481–500.

Ho, T.-W., 2009. The inflation rates may accelerate after all: panel evidence from 19 OECD economies. Empirical Economics 36, 55–64. Im, K., Pesaran, H., Shin, Y., 2003. Testing for unit roots in heterogeneous panels. Journal of Econometrics 115, 53–74.

Jansson, M., 2004. Stationarity testing with covariates. Econometric Theory 20, 56–94.

Koenker, R., Xiao, Z., 2004. Unit root quantile autoregression inference. Journal of the American Statistical Association 99, 775–787.

Kumar, M.S., Okimoto, T., 2007. Dynamics in inflation persistence in international inflation rates. Journal of Money, Credit and Bank 39, 1457–1479. Lee, H.Y., Wu, J.L., 2001. Mean reversion of inflation rates: evidence from 13 OECD countries. Journal of Macroeconomics 23, 477–487.

Levin, A., Lin, C.F., Chu, C.S., 2002. Unit root tests in panel data: asymptotic and finite-sample properties. Journal of Econometrics 108, 1–24. Levin, A., Piger, J., 2003. Is inflation persistence intrinsic in industrial economies? Working Paper. Federal Reserve Bank of St. Louis.

MacDonald, R., Murphy, P.D., 1989. Testing for the long run relationship between nominal interest rates and inflation using cointegration techniques. Applied Economic 21, 439–447.

Mishkin, F.S., 2008. Challenges for inflation targeting in emerging market countries. Emerging Markets Finance & Trade 44, 5–16. Murray, C.J., Papell, D.H., 2005. The purchasing power parity puzzle is worse than you think. Empirical Economics 30, 783–790.

Nelson, C.R., Plosser, C.I., 1982. Trends and random walks in macroeconomic time series: some evidence and implications. Journal of Monetary Economics 10, 139–162.

Nelson, C.R., Schwert, G.W., 1977. Short-term interest rates as predictors of inflation: on testing the hypothesis that the real rate of interest is constant. The American Economic Review 67, 478–486.

Ng, S., Perron, P., 2001. Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554. Nikolaou, K., 2008. The behaviour of the real exchange rate: evidence from regression quantiles. Journal of Banking & Finance 32, 664–679. Okun, A., 1971. The mirage of steady inflation. Brookings Papers on Economic Activity 2, 485–498.

O’Reilly, G., Whelan, K., 2005. Has euro-area inflation persistence changed over time? Review of Economics and Statistics 87, 709–720. Perron, P., 1989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361–1401.

Pivetta, F., Reis, R., 2007. The persistence of inflation in the United States. Journal of Economic Dynamics and Control 31, 1326–1358. Pétursson, T.G., 2004. Formulation of inflation targeting around the world. Monetary Bulletin 1, 57–84.

Rapach, D.E., Weber, C.E., 2004. Are real interest rates really nonstationary? New evidence from tests with good size and power. Journal of Macroeconomics 26, 409–430.

Rose, A.K., 1988. Is the real interest rate stable? Journal of Finance 43, 1095–1112.

Stock, J., 2001. Comment on evolving post World War II U.S. inflation dynamics. NBER Macroeconomics Annual 16, 379–387. Taylor, J.B., 2000. Low inflation pass-through, and pricing power of firms. European Economic Review 44, 1389–1408.

Tsong, C.C., Lee, C.F., 2010. Testing for stationarity of inflation rates with covariates. South African Journal of Economics 78, 344–362. Zhang, C., Clovis, J., 2010. China inflation dynamics: persistence and policy regimes. Journal of Policy Modeling 32, 373–388. 680 C.-C. Tsong, C.-F. Lee / Journal of Macroeconomics 33 (2011) 668–680