2008 Oxford Business &Economics Conference Program ISBN : 978-0-9742114-7-3
Jesús Otero 
Facultad de Economía
Universidad del Rosario
Bogotá, Colombia
February 2008
Abstract
This paper applies the Hadri (2000) tests for the null of panel stationarity to re-examine evidence on mean-reversion of inflation rates for a group of 13 OECD countries. Applying individual tests for stationarity, we find support for the view that inflation rates appear to be non-stationary processes. However, within a panel context the previous result is overturned, as we find evidence that the inflation rates are stationary or, in other words, exhibit mean-reversion. Our empirical findings appear consistent with the stable macroeconomic policy environment that has characterised the economies of the countries under consideration.
JEL Classification: C12; C15; C22; C23; E31.
Keywords: Heterogeneous dynamic panels, inflation, mean reversion, panel stationarity test.

Work on this paper began while I was a Visiting Fellow in the Department of Economics at the
University of Warwick. For this, I thank the Universidad del Rosario for its financial assistance. I would like to thank Jeremy Smith for useful comments and suggestions. Any remaining errors are my own responsibility.
Correspondence address:
Jesús Otero
Facultad de Economía
Universidad del Rosario
Calle 14 # 4-69
Bogotá, Colombia
(+571) 297 02 00 Ext. 661
(+571) 344 57 63 jotero@urosario.edu.co
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2008 Oxford Business &Economics Conference Program
1. Introduction
ISBN : 978-0-9742114-7-3
The question of whether inflation rates should be treated as stationary or non-stationary processes has received a great deal of attention ever since the publication of the paper by
Nelson and Plosser (1982). The main reason for this is that classifying inflation rates as one or the other has a number of important statistical and economic implications. Indeed, stationary series have a finite (non-zero) variance and exhibit temporary memory in the sense that the effect of a perturbation disappears as time passes. Non-stationary series, on the other hand, have an unbounded variance and exhibit a permanent memory. From an economic perspective, a stationary inflation rate may be viewed as the result of a stable macroeconomic policy environment that is in turn conductive to growth (see e.g. Fischer,
1993).
Applying individual augmented Dickey-Fuller (ADF) (1979, 1981) and
Kwiattowski, Phillips, Schmidt and Shin (KPSS) (1992) tests to different countries and time periods, Baillie (1989), Johansen (1992), Banerjee, Cockerell and Russell (2001) and
Juselius (2001), find that the level of prices are best described as I(2) statistical processes, hence inflation rates appear processes integrated of order one. By contrast, Ericsson,
Hendry and Prestwich (1998) find that UK inflation appears to be I(0) rather than I(1); see also the studies of Bårdsen (1992) and Johansen and Strøm (1997) for Norway providing support for the inflation rate being I(0).
In all the above papers, single-equation tests are applied to test mean reversion of inflation rates. However, it is well known that unit root tests applied to single series suffer from low power. Thus, in recent years a number of alternative procedures have been
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2008 Oxford Business &Economics Conference Program ISBN : 978-0-9742114-7-3 proposed to test for the presence of unit roots in panels. These tests combine information from the time-series dimension with that from the cross-section dimension, such that fewer time observations are required for these tests to have power. Perhaps the most commonly used unit root test applied to panels are Maddala and Wu (MW) (1999) and Im,
Pesaran and Shin (IPS) (2003), which test the null hypothesis of a unit root against the alternative of at least one stationary series, by using the ADF statistic across the crosssectional units of the panel. Culver and Papell (1997) and Lee and Wu (2001) apply both individual and panel unit root tests to test for mean reversion of inflation rates, using data for 13 OECD countries during the post-war period. These authors find mixed results in the sense that while the individual tests suggest that there appears to be a unit root in the inflation rates of the countries under consideration, panel tests provide more support against a unit root.
This paper re-examines the validity of mean reversion of inflation rates using updated data for the same 13 OECD countries analysed by Culver and Papell (1997) and
Lee and Wu (2001). The paper differs in one important aspect from Culver and Papell
(1997) and Lee and Wu (2001), and that is that we apply the tests developed by Hadri
(2000) for the null hypothesis that all of the individual series are stationary (either around a mean or around a trend), against the alternative of at least a single unit root in the panel.
The Hadri tests thus offer the advantage that if the null hypothesis is not rejected, then there would be evidence that all of the inflation rates in the panel are stationary. It should be noted that IPS (2003, p.73) warn that due to the heterogeneous nature of the alternative hypothesis in their test, one needs to be careful when interpreting the results, because the
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2008 Oxford Business &Economics Conference Program ISBN : 978-0-9742114-7-3 null hypothesis that there is a unit root in each cross section may be rejected when only a fraction of the series in the panel are stationary.
The plan of the paper is as follows. Section 2 briefly reviews the Hadri (2000) approach to test for stationarity in heterogeneous panels of data, also allowing for the likely case in which there is cross section dependence. Section 3 describes the data and presents the empirical analysis. Section 4 concludes.
2. Testing for stationarity in heterogeneous panel data
Hadri (2000) proposes an LM procedure to test the null hypothesis that all the individual series are stationary (either around a mean or around a trend) against the alternative of at least a single unit root in the panel. The two LM tests proposed by Hadri (2000) are panel versions of the test developed by Kwiatkowski, Phillips, Schmidt and Shin (KPSS) (1992).
Following Hadri (2000), consider the models: y it r it
 it
, (1a) and y it r it
 i t
  it
, (1b) where r is a random walk, it r it
 r it

1
 u it
, and

and it u are mutually independent it normal distributions. Also,

and it u it i i d across i and over t , with E
  
0,
E
 

2 
0,
  
0, 2

2 u

0, t

1,..., T and i

1,..., N .
Let it
  it
be the residuals from the regression of y it
on an intercept, for model
(1a), (on an intercept and a linear trend term, for model (1b)). Let
 ˆ
2
 
 
be a
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2008 Oxford Business &Economics Conference Program ISBN : 978-0-9742114-7-3 consistent estimator of the error variance from the appropriate regression, which is given by:
 ˆ
 l
2

1
N T
N T 
, i

1 t

1
 ˆ it l 2
Also, let
S l it
be the partial sum process of the residuals,
S it l  j t 

1
 ˆ , ij l
  
.
l
  
.
Then the LM statistic is:
LM l

1
N


N i

1


T
T  1
2
 t

1
S it
 ˆ
 i
2 l 2 l




.
l
  
.
It should be noted that the LM statistic is based on averaging the individual KPSS test statistics. In order to obtain a consistent estimator of
 ˆ
 i
2 l
which is efficient in the presence of residual serial dependence, we follow Hobijn et al . (2004) who suggest applying the Newey and West (1994) automatic bandwidth selection procedure for the
Quadratic Spectral kernel.
Finally, Hadri (2000) considers the standardised statistics:
Z





 



N
  and
Z






 



N
 
The asymptotic mean and the variance of the random variable Z

are



1
6
and
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2008 Oxford Business &Economics Conference Program ISBN : 978-0-9742114-7-3

2


1
45
, respectively, while the asymptotic mean and the variance of the random variable
Z

are



1
15
and


2 
11
6300
, respectively. Subsequently, Hadri and Larsson (2005) find the exact formulae for the two finite-sample moments of the KPSS statistic.
The Monte Carlo experiments of Hadri (2000) demonstrate that these tests have good size properties for T and N sufficiently large. However, as noted by Giulietti et. al.
(2008), even for relatively large N and T , the Hadri tests suffer from severe size distortions in the presence of cross-sectional dependence, the magnitude of which increases as the strength of the cross-sectional dependence increases. This finding is in line with the results obtained by Strauss and Yigit (2003) and Pesaran (2007) for the IPS and MW panel unit root tests. To correct the size distortion caused by cross-sectional dependence, Giulietti et. al.
(2008) apply the bootstrap method and find that the bootstrap
Hadri tests are approximately correctly sized.
3. Data and empirical analysis
The data set, obtained from the International Financial Statistics of the International
Monetary Fund, consists of seasonally unadjusted monthly observations on consumer price indices for the following 13 OECD countries: Belgium, Canada, Finland, France,
Germany, Italy, Japan, Luxemburg, the Netherlands, Norway, Spain, the United Kingdom and the United States. These are the countries used by Culver and Papell (1997) and by
Lee and Wu (2001). The sample period is 1957m1-2004m12 and the inflation rates are calculated by differencing the logarithm of the consumer price indices, so that the sample period used for estimation effectively starts in 1957m2.
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2008 Oxford Business &Economics Conference Program ISBN : 978-0-9742114-7-3
Table 1 presents the results of applying the KPSS stationarity test to the inflation rates of the 13 OECD countries listed above (based on the model with intercept only). To obtain a consistent estimator of
 ˆ

2

that is efficient in the presence of residual serial correlation, Hobijn et. al. (2004) suggest applying the Newey and West (1994) automatic bandwidth selection procedure for the Quadratic Spectral kernel.
On the basis of the country-by-country KPSS tests, the null hypothesis of stationarity is rejected for nine countries when the sample period analysed by Culver and
Papell (1997), that is 1957m2-1994m9, is applied. For one country rejection is at the 1% significance level and for eight more countries rejection is at the 5% level.
1 These findings are in line with those obtained by Culver and Papell (1997) for the KPSS test. When the sample period is extended until 1999m4, as in Lee and Wu (2001), the null hypothesis of stationarity is rejected for two countries at the 1% significance level and for the eleven remaining countries at the 5% level. The results of the ADF test reported by Lee and Wu
(2001) also support the existence of a unit root on the inflation rates of most countries, except for France, Japan and the Netherlands. Finally, using updated data until 2004m12, the null hypothesis of stationarity is rejected for seven countries at the 1% significance level and for the six remaining countries at the 5% level. In summary, the country-bycountry results tend to suggest that the inflation rates of the OECD countries under consideration can be best described as I(1) processes.
Next, we apply the Hadri tests to the inflation rates under consideration. The main motivation for testing stationarity in a panel of data instead of individual time series is that
1 For the four remaining countries rejection is only at the 10% level.
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2008 Oxford Business &Economics Conference Program ISBN : 978-0-9742114-7-3 it has been noted that the power of the tests increases with the number of cross-sections in the panel. To allow for potential cross section dependence, we apply the bootstrap method to the Hadri tests by resampling the residuals from either a regression of y on a constant i for the Z

test, or on a constant and a trend for the Z

test. As suggested by Maddala and
Wu (1999, p.646), we resample the residuals with the cross-section index fixed, so that we preserve the cross-correlation structure of the error term.
With dependent data, a further refinement in the bootstrap described above can be obtained by applying the idea of bootstrapping overlapping blocks of residuals rather than the individual residuals, also known as the moving block bootstrap approach.
2
This approach requires the researcher to choose the block size, i.e. the number of contiguous residuals to be resampled with replacement. The choice of the block size is based on the values suggested by the inspection of the correlogram of the series, which involves identifying the smallest integer after which the correlogram becomes negligible, as suggested by Künsch (1989; p.1226). In particular, the results shown in Table 2 are based on 2,000 bootstrap replications used to derive the empirical distribution of the Z

statistic
(since we focus on the model with intercept only), for alternative block sizes
BS

48, 54, 60, 66, 72 residuals. The different choices of the block size are based on values suggested by the inspection of the correlogram of the series, as recommended by
Künsch (1989).
The results of the Hadri test using the moving block bootstrap approach are
2 For a discussion of the moving block bootstrap see Künsch (1989), Maddala and Kim (1998) and
Berkowitz and Kilian (2000). Lee and Wu (2001) use this approach within the context of the IPS panel unit
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2008 Oxford Business &Economics Conference Program ISBN : 978-0-9742114-7-3 reported in Table 2. For the three sample periods the results show that the null hypothesis of stationarity is not rejected, independently of the block size considered.
3
These findings provide support that the inflation rates of the countries under consideration are meanreverting.
4. Concluding remarks
This paper applies the Hadri (2000) tests for panel stationarity to re-examine evidence on mean-reversion of inflation rates for a group of 13 OECD countries. Unlike standard panel unit root tests, the Hadri tests offer the advantage that if the null hypothesis is not rejected, then there would be evidence that all of the series in the panel are stationary.
Based on evidence from individual tests for stationarity, we find support for the view that inflation rates can be best described as non-stationary processes. However, within a panel context the previous result is overturned, as we find evidence that the inflation rates are stationary or, in other words, exhibit mean-reversion. Our empirical findings appear consistent with the stable macroeconomic policy environment that has characterised the economies of the countries under consideration. root test. Details on the implementation of the moving block bootstrap can be found in these references, and so are not presented here to save space.
3 If, for the sample period 1957m1-1994m9, one considers a panel that excludes the inflation rates of Finland,
France, Germany and Spain, which appear to be stationary based on the individual KPSS test results (see
Table 1), our findings are qualitatively the same.
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2008 Oxford Business &Economics Conference Program
References
ISBN : 978-0-9742114-7-3
Baillie, R.T. (1989). Commodity prices and aggregate inflation: Would commodity price rule be worthwhile? Carnegie Rochester Conference Series on Public Policy, 31,
185-240.
Banerjee, A., Cockerell, L., & Russell, B. (2001). An I(2) analysis of inflation and the markup. Journal of Applied Econometrics, 16, 221-240.
Bårdsen, G. (1992). Dynamic modeling of the demand for narrow money in Norway.
Journal of Policy Modeling, 14, 363-393.
Berkowitz, J., & Kilian, L. (2000). Recent developments in bootstrapping time series.
Econometric Reviews, 19, 1-48.
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, 435-444.
Dickey, D.A., & Fuller W.A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74, 427-
431.
Dickey, D.A., & Fuller W.A. (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica, 49, 1057-1072.
Ericsson, N.R., Hendry, D.F., & Prestwich, K.M. (1998). The demand for broad money in the United Kingdom, 1878-1993. Scandinavian Journal of Economics, 100, 289-
324.
Fischer, S. (1993). The role of macroeconomic factors in growth. Journal of Monetary
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2008 Oxford Business &Economics Conference Program
Economics, 32, 485-512.
ISBN : 978-0-9742114-7-3
Giulietti, M., Otero, J., & Smith, J. (2008). Testing for stationarity in heterogeneous panel data in the presence of cross section dependence. Journal of Statistical
Computation and Simulation, forthcoming.
Hadri, K. (2000). Testing for stationarity in heterogeneous panels. The Econometrics
Journal, 3, 148-161.
Im, K., Pesaran, M.H., & Shin, Y. (2003). Testing for unit roots in heterogeneous panels.
Journal of Econometrics, 115, 53-74.
Johansen, K., & Strøm, B. (1997). Wages, prices and politics: Evidence from Norway.
Oxford Bulletin of Economics and Statistics, 59, 511-522.
Johansen, S. (1992). Testing weak exogeneity and the order of integration in UK money demand data. Journal of Policy Modeling, 14, 313-334.
Juselius, K. (2001). European integration and monetary transmission mechanisms.
Journal of Applied Econometrics, 16, 341-358.
K ü nsch, H.R. (1989). The jackknife and the bootstrap for general stationary observations.
Annals of Statistics, 17, 1217-1241.
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of
Econometrics, 54, 159-178.
Lee, H.-Y., & Wu, L.-J. (2001). Mean reversion of inflation rates: Evidence from 13
OECD countries. Journal of Macroeconomics, 23, 477-487.
Maddala, G.S., & Kim, I. M. (1998). Unit roots, cointegration, and structural change.
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2008 Oxford Business &Economics Conference Program
Cambridge: Cambridge University Press.
ISBN : 978-0-9742114-7-3
Maddala, G.S., & Wu, S. (1999). A comparative study of unit root tests with panel data and a new simple test. Oxford Bulletin of Economics and Statistics, 61, 631-652.
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.
Newey, W.K., & West, K.D. (1994). Automatic lag selection in covariance matrix estimation. Review of Economic Studies, 61, 631-653.
Pesaran, M.H. (2007). A simple panel unit root test in the presence of cross section dependence. Journal of Applied Econometrics, 22, 265-312.
Strauss, J., & Yigit, T. (2003). Shortfalls of panel unit root testing. Economics Letters, 81,
309-313.
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2008 Oxford Business &Economics Conference Program
Table 1. KPSS test for mean stationarity
ISBN : 978-0-9742114-7-3
Country
Belgium
Canada
Finland
France
Germany
Italy
Japan
1957m2 - 1994m9 1957m2 - 1999m4 1957m2 - 2004m12 l

 test l

 test l

 test
15 0.494 *
15 0.687 *
14 0.421
16 0.497 *
16 0.579 *
15 0.655 *
16 0.625 *
16 0.645 *
16 0.992 **
15 0.398
7 0.404
15 0.757 **
12 0.663 *
15 0.696 *
8 0.476 *
16 0.581 *
13 0.958 **
16 1.026 **
9 0.755 **
17 0.597 *
14 1.407 **
Luxemburg
Norway
Netherlands
14 0.527 *
11 0.679 *
7 0.705 *
Spain 14 0.437
United Kingdom 14 0.554 *
United States 15 0.618 *
14 0.500 *
12 0.662 *
8 0.935 **
15 0.554 *
14 0.532 *
16 0.488 *
13 0.581 *
12 0.978 **
7 1.215 **
15 0.867 **
15 0.621 *
17 0.489 *
For the individual KPSS tests (

 ), the 5 and 1 per cent critical values are 0.463 and
0.739, respectively; see Table 1 in KPSS (1992). * and ** indicate 5 and 1 per cent levels of significance, respectively. The lag truncation parameter ( l ) needed to calculate the test statistic is chosen using the Newey-West (1994) data-based automatic method weighting with a Bartlett window.
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2008 Oxford Business &Economics Conference Program ISBN : 978-0-9742114-7-3
Table 2. The bootstrap Hadri test for inflation rates p
 value Sample period
1957m2 - 1994m9
Z

Block size
(in months)
9.600
48
54
60
66
72
48
54
0.211
0.238
0.255
0.290
0.299
0.169
0.197
1957m2 - 1999m4 11.029
1957m2 - 2004m12 16.022
60
66
72
48
54
60
66
72
0.215
0.239
0.269
0.098
0.102
0.129
0.136
0.169
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