Additional sources of bias in half-life estimation
ABSTRACT When the automobile was developed near the beginning of the last century, it was the relatively new fuel gasoline, not the familiar ethanol that became the fuel of choice. We examine the intersections of the early development of the automobile and the petroleum industry and consider the state of the agriculture sector during the same period. Through this process, we find a series of influences, such as relative prices and alternative markets, that help to explain how in the early years of automobile development, gasoline won out over the equally likely technical alternative ethanol. We also examine the industrial relations in the automobile industry that seem to have influenced the later adoption of leaded gasoline, rather than ethanol, as a solution to the problem of engine knock.
- [Show abstract] [Hide abstract]
ABSTRACT: This paper provides evidence that unemployment rates across US states are stationary and therefore behave according to the natural rate hypothesis. We provide new insights by considering the effect of key variables on the speed of adjustment associated with unemployment shocks. A highly-dimensional VAR analysis of the half-lives associated with shocks to unemployment rates in pairs of states suggests that the distance between states and vacancy rates respectively exert a positive and negative influence. We find that higher homeownership rates do not lead to higher half-lives. When the symmetry assumption is relaxed through quantile regression, support for the Oswald hypothesis through a positive relationship between homeownership rates and half-lives is found at the higher quantiles.Physica A: Statistical Mechanics and its Applications 11/2013; 392(22):5711-5722. · 1.72 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: This article investigates how the price indices of major cities of the US respond to the shock from a city and from monetary policy. We find that the crisis of Bretton Woods system in 1968 and the oil crisis in 1974 should be incorporated as structural breaks in monetary policy variables and price indices. Using cointegration technique with structural break in our aggregated data, we find that the average half-life is 1.75 years, which is closer to what some of others found in disaggregated data, and that the interest rate is an effective tool for controlling cities’ price in short run.Applied Economics 05/2012; 44(14):1849-1862. · 0.46 Impact Factor - SourceAvailable from: Jesus Otero[Show abstract] [Hide abstract]
ABSTRACT: This study examines the Prebisch and Singer hypothesis using a panel of 24 commodity prices from 1900 to 2010. The modelling approach stems from the need to meet two key concerns: (1) the presence of cross-sectional dependence among commodity prices; and (2) the identification of potential structural breaks. To address these concerns, the Hadri and Rao test (2008) is employed. The findings suggest that all commodity prices exhibit a structural break at different locations across series, and that support for the Prebisch and Singer hypothesis is mixed. Once the breaks are removed from the underlying series, the persistence of commodity price shocks is shorter than that obtained in other studies using alternative methodologies.Portuguese Economic Journal 01/2011; 12(1). · 0.12 Impact Factor
Page 1
This article was originally published in a journal published by
Elsevier, and the attached copy is provided by Elsevier for the
author’s benefit and for the benefit of the author’s institution, for
non-commercial research and educational use including without
limitation use in instruction at your institution, sending it to specific
colleagues that you know, and providing a copy to your institution’s
administrator.
All other uses, reproduction and distribution, including without
limitation commercial reprints, selling or licensing copies or access,
or posting on open internet sites, your personal or institution’s
website or repository, are prohibited. For exceptions, permission
may be sought for such use through Elsevier’s permissions site at:
http://www.elsevier.com/locate/permissionusematerial
Page 2
Author's personal copy
Computational Statistics & DataAnalysis 51 (2006) 2056–2064
www.elsevier.com/locate/csda
Additional sources of bias in half-life estimation
Byeongchan Seonga,∗,A.K.M. Mahbub Morshedb, Sung K.Ahna
aDepartment of Management and Operations, Washington State University, Pullman, WA 99164, USA
bDepartment of Economics, Mail Code 4515, Southern Illinois University, Carbondale, IL 62901, USA
Received 13 September 2005; received in revised form 18 December 2005; accepted 20 December 2005
Available online 10 January 2006
Abstract
Recently,anincreasingamountofattentionisbeingpaidtobiasesinthemeasurementoftimeseriesdynamicsbasedoncalculations
of half-life. In particular, this issue amplifies the controversy surrounding the purchasing power parity doctrine. Cross-sectional and
temporal aggregations, along with mis-specified models, were previously identified as sources of this bias. We identified several
other sources of bias, namely, sampling error, incorrect approximations, and structural breaks in time series. These sources should
also receive sufficient attention for a sound measurement of half-life.
© 2006 Elsevier B.V.All rights reserved.
Keywords: Impulse response function; Structural break
1. Introduction
The empirical evidence suggesting the high persistence of the deviation of real exchange rates from their long-
run equilibrium warranted a simple measure to capture this slow transitional dynamics.1Consequently, economists
borrowed the concept of “half-life” from the natural sciences. In the natural sciences, half-life is defined as the time
requiredfortheamountofradioactivitytodecreasebyone-half.Alongthesamelines,intherealexchangerateliterature,
we define half-life as the time required for the effects of a unit innovation to dissipate to one-half. Half-life is also
used in economics as a simple measure of time series dynamics such as the income and price levels. In particular, as
in Cecchetti et al. (2002) and Morshed et al. (2005), half-life is used to obtain information pertaining to the nature of
the observed persistence of the deviations of city consumer price indices (CPIs) from the common trend in prices, by
estimatingtherateatwhichameanreversionoccurs.Inthiscontext,informationabouthalf-lifewouldenablemonetary
policy makers to design an optimum monetary policy that can deal with the impact and persistence of regional inflation
divergence.
Empiricalstudiesonhalf-livesareoftensurroundedbycontroversiespertainingtotheaccuracyofhalf-lifeestimates.
Thisisbecause,comparedwiththecommonlyexpectedhalf-life,somestudiesover-estimatehalf-lifeandothersunder-
estimate it (for a detailed discussion, see Murray and Papell, 2002; Taylor, 2001). Various efforts have been made to
explore the sources of differences in half-life estimates: Basker and Hernandez-Murillo (2004), Choi et al. (2004),
∗Corresponding author. Tel.: +15093356819; fax: +15093357736.
E-mail address: bcseong@wsu.edu (B. Seong).
1The rate of convergence of the real exchange rate has been estimated to be roughly 15% (Froot and Rogoff, 1995; Cheung and Lai, 2000a).
0167-9473/$-see front matter © 2006 Elsevier B.V.All rights reserved.
doi:10.1016/j.csda.2005.12.016
Page 3
Author's personal copy
Usingtheaboveequation,wetabulatethe(approximate)coefficientofvariation(CV)ofˆhinTable1for0.8??1?0.95
andforsamplesizesn=100and200.TheCVvariesfrom33%to64%withintherangeof?1forthesampleofsize100;
this amounts to 100 years of annual data. More specifically, for an AR(1) process with ?1= 0.9, the half-life is 6.58.
With n=100, the CV ofˆh is 46% and the standard error ofˆh is 3.02. Therefore, for annual data, a half-life estimate of
B. Seong et al. / Computational Statistics & Data Analysis 51 (2006) 2056–2064
2057
Chen and Devereux (2003), and Imbs et al. (2005) investigated cross-sectional aggregation as a contributing factor;2
Chambers (2004) investigated temporal aggregation as a contributing factor of biased estimates, and Taylor (2001)
investigated temporal aggregation and mis-specified linear models as contributing factors. The expositions of the latter
two articles are within the context of the autoregressive process of order one,AR(1).
In an effort to add feasible explanations to purchasing power parity (PPP) puzzles, we explore other sources for the
differences in half-life estimates. These sources are the sensitivity of the half-life formula, an inappropriate formula
commonly used for half-life estimations, and mis-specified models that are attributable to structural breaks. Our simu-
lations revealed that the commonly used half-life formula is very sensitive to sampling errors even if the autoregressive
process is AR(1). The half-life formula can be quite inaccurate when the time series is a higher order (for example,
AR(2)) or a mixed process (for example,ARMA(1,1)). Moreover, when a structural break exists in time series, and this
is not taken into consideration, half-lives are over-estimated.
This paper comprises five sections. In Section 2, we discuss the sensitivity of the commonly used half-life formula
obtained from an AR(1) model. Biases resulting from using the half-life formula for higher order autoregressive
processes and mixed processes are discussed in Section 3. Section 4 discusses the effects of structural breaks on
half-life calculations. Concluding remarks are in Section 5.
2. Sensitivity of the half-life formula
Based on the cumulative impulse response analysis of Campbell and Mankiw (1987), researchers define the moving
average (MA) coefficients of the MA representation of the process as impulse responses. More specifically, for a linear
processyt=?∞
material, the impulse response does not always decay monotonically. If ?jis not a monotonically decreasing function
of lag j, then the half-life is not well-defined (Cheung and Lai, 2000b; Choi et al., 2004).
In econometric literature, the commonly used formula for the half-life of a (stationary) time series yt is h =
−log2/log?1, where ?1is the autocorrelation of ytat lag one, i.e., ?1= corr(yt,yt−1). This formula is valid only
when ?1>0, and is correct if ytis anAR(1) satisfying
j=0?jεt−j,where?0=1andεtarei.i.d.randomvariables,thehalf-lifedenotedbyhissuchthat?h=1
this is the lag where the impulse response ?jbecomes half the initial impulse response. However, unlike radioactive
2;
yt= ?1yt−1+ ?t.
This is because for AR(1), ?j= ?j
positive and negative values; as a result, the half-life is not well-defined.
Given a sample of size n, the half-life of anAR(1) process is usually estimated by
(1)
1. If ?1<0 for an AR(1) process, then the impulse response ?joscillates between
ˆh = −log2
log ˆ ?1
,(2)
where ˆ ?1is the least squares estimator (LSE) of ?1. From the first order Taylor series expansion, we obtain
(log2)ˆ?
?1(log?1)2,
ˆh − h ≈
(3)
whereˆ? = ˆ ?1− ?1. It is well known that var(ˆ ?1) ≈ (1 − ?2
?
1)/n. Therefore,
var(ˆh − h) ≈
log2
?1(log?1)2
?21 − ?2
1
n
.(4)
2Chen and Engel (2005), however, showed that the cross-sectional aggregation bias might not be sufficiently large to explain the PPP puzzle.
Page 4
Author's personal copy
p
?
the impulse response ?jsatisfies the linear difference equation
2058
B. Seong et al. / Computational Statistics & Data Analysis 51 (2006) 2056–2064
Table 1
Approximate standard errors and coefficient of variations of the half-life estimates for the selectedAR(1) process with sample sizes of 100 and 200
?1
h
n = 100
?
1.0440
1.0857
1.1302
1.1777
1.2285
1.2830
1.3416
1.4047
1.4728
1.5465
1.6264
1.7133
1.8081
1.9117
2.0255
2.1508
2.2894
2.4433
2.6149
2.8073
3.0242
3.2700
3.5506
3.8732
4.2471
4.6846
5.2017
5.8205
6.5711
7.4965
8.6593
n = 200
?
0.7382
0.7677
0.7991
0.8327
0.8687
0.9072
0.9487
0.9933
1.0415
1.0935
1.1500
1.2115
1.2785
1.3518
1.4322
1.5209
1.6189
1.7277
1.8490
1.9851
2.1384
2.3122
2.5106
2.7387
3.0032
3.3125
3.6782
4.1157
4.6465
5.3008
6.1231
var(ˆh − h)
CV (%)var(ˆh − h)
CV (%)
0.800
0.805
0.810
0.815
0.820
0.825
0.830
0.835
0.840
0.845
0.850
0.855
0.860
0.865
0.870
0.875
0.880
0.885
0.890
0.895
0.900
0.905
0.910
0.915
0.920
0.925
0.930
0.935
0.940
0.945
0.950
3.1063
3.1955
3.2894
3.3884
3.4928
3.6032
3.7200
3.8439
3.9755
4.1156
4.2650
4.4247
4.5958
4.7795
4.9773
5.1909
5.4223
5.6737
5.9480
6.2484
6.5788
6.9439
7.3496
7.8030
8.3130
8.8909
9.5513
10.3133
11.2023
12.2528
13.5134
33.61
33.98
34.36
34.76
35.17
35.61
36.07
36.54
37.05
37.58
38.13
38.72
39.34
40.00
40.69
41.43
42.22
43.06
43.96
44.93
45.97
47.09
48.31
49.64
51.09
52.69
54.46
56.44
58.66
61.18
64.08
23.77
24.02
24.29
24.58
24.87
25.18
25.50
25.84
26.20
26.57
26.96
27.38
27.82
28.28
28.78
29.30
29.86
30.45
31.09
31.77
32.50
33.30
34.16
35.10
36.13
37.26
38.51
39.91
41.48
43.26
45.31
3.6 years or less is as likely as a half-life estimate of 9.6 years or more. This illustrates that half-life estimates are very
sensitive to sampling errors.
3. Precision of the approximate formula
More often than not, the process under consideration is not just an AR(1) process. Rather, it is a higher order
AR process or a mixed process such as an autoregressive moving-average (ARMA) process. For such models, the
aforementioned half-life formula serves as an approximation, and the quality of this approximation requires further
investigation.
For ytfollowing an autoregressive process of order p,AR(p), satisfying
yt=
j=1
?jyt−j+ ?t,(5)
?j− ?1?j−1− ··· − ?p−1?j−(p−1)− ?p= 0, (6)
Page 5
Author's personal copy
?j= (? − ?)?j−1,
and the exact half-life h is
B. Seong et al. / Computational Statistics & Data Analysis 51 (2006) 2056–2064
2059
and the half-life h is obtained by solving ?h=1
roots of the auxiliary equation
2. It is well known that the impulse response ?jis obtained from the
mp− ?1mp−1− ··· − ?p−1m − ?p= 0.
Since ?jdoes not necessarily decay monotonically, the half-life is not always well-defined. The commonly employed
practice in economics literature is to approximate the half-life based on the formula
(7)
h = −
log2
log(1 + ?),(8)
regardless of the existence of the well-defined half-life, by obtaining the “convergence speed” ? from the following
error correction representation of theAR(p) model:
?yt= ?yt−1+
p−1
?
j=1
?∗
j?yt−j+ εt, (9)
where ? =?p
For ease of exposition, we assess the quality of this approximation using the followingAR(2) process:
j=1?j− 1 and ?∗
j= −?p
k=j+1?k. We note that for an AR(1) process, ? = ?1− 1 and the formula in
(8) is equivalent to that in (2).
yt= ?1yt−1+ ?2yt−2+ εt.
It is well known that the impulse response of this process is
?(1 + j)(?1/2)j
?
For theAR(2) process to be stationary, it is well known (see Box et al., 1994, p. 60) that theAR coefficients ?1and
?2lie in the triangular region
(10)
?j=
if ?2
if ?2
1+ 4?2= 0,
1+ 4?2?= 0,
?
c1?j
1+ c2?j
?
2
(11)
where ?1=
?1+
?2
1+ 4?2
?
/2, ?2=
?1−
?
?2
1+ 4?2
?
/2, c1= ?1/(?1− ?2), and c2= ?2/(?2− ?1).
?2+ ?1<1,
Within this triangular region, impulse response ?jdecreases monotonically only in the region for ?1>0 and ?2
4?2>0.Therefore,thehalf-lifeisnotwell-definedintheotherregion.However,solongas?2+?1>0,theapproximate
formula will yield a half-life. Even in the region where the half-life is well defined, the approximate formula can be
quiteinaccurate.TheshadedregionofFig.1representstheregionwherethedifferencebetweenthehalf-lifeby(11)and
theapproximatehalf-lifeof(8)isgreaterthan3.Fromtheshadedregion,weseethatthedifferenceismorepronounced
as the process approaches nonstationarity, i.e., ?1+ ?2= 1, where ?2>0. This is because formula (8) diverges to
infinity as the value of ?1+ ?2approaches one.
When anAR(1) process at a higher frequency is aggregated and observed at a lower frequency, this observed process
becomes anARMA (1,1) process,
?2− ?1<1,
−1<?2<1.(12)
1+
yt= ?yt−1+ εt− ?εt−1,
see Wei (1996) and Chambers (2004). The impulse response ?jis obtained by
(13)
(14)
h = −log2
log?−log(? − ?)
log?
+ 1,(15)
Page 6
Author's personal copy
which is obtained by solving (? − ?)?h−1= 1/2, provided ?>? and ?>0. Since the lag one autocorrelation of the
ARMA(1,1) process is
2060
B. Seong et al. / Computational Statistics & Data Analysis 51 (2006) 2056–2064
00.51 1.52
-1
-0.5
0
0.5
1
?1
?1 + ?2 = 1
?2
Fig. 1. The region where the difference between the half-life by (11) and the approximate formula of (8) is greater than 3.
00.5
?
100.5
?
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Approx. by AR(1)Approx. by AR(2)
??
Fig. 2. The region where the difference between the half-life by (15) and the approximate formula of (17):AR(1) or (18):AR(2) is greater than 3.
?1=(1 − ??)(? − ?)
(1 − 2?? + ?2)
,(16)
Page 7
Author's personal copy
0.4 0.950.83
(175)(0.03)(0.07)
B. Seong et al. / Computational Statistics & Data Analysis 51 (2006) 2056–2064
2061
the approximate formula (based on anAR(1) model) yields a half-life of
?(1 − ??)(? − ?)
In addition, an approximated model of AR(2) can be taken into consideration instead of an AR(1) model. In such a
case, the approximate formula based on anAR(2) model yields a half-life of
?(1 − ??)(? − ?)
the proof is provided in theAppendix.
In order to illustrate the inaccuracies of the approximate formulae, the shaded region in Fig. 2 represents the region
where the difference between the half-life by (15) and the approximate half-lives of (17) or (18) is greater than 3,
even when the parameters are known. Similar to Fig. 1, this difference becomes larger as the process approaches
nonstationarity, i.e., ? approaches one.This large difference is attributable to the fact that the half-lives in (17) and (18)
as well as in (15) diverge to infinity as the value of ? increases to one. There is a higher degree of inaccuracy when
models are estimated; however, this has not been discussed in the present study.
Therefore,whenweestimatethehalf-lifeforahigherorderARoramixedprocess,itisrecommendedthatresearchers
adopt the exact half-life formula from the impulse response function. The half-life formulae in (6), (11), and (15) are
based on the impulse response functions of their respective models.
−log2/log
(1 − 2?? + ?2)
?
. (17)
−log2/log
1 − (? + 1)? + ?2
?
,(18)
4. The effect of structural breaks
It is well known that the Dickey–Fuller unit root test lacks the power, when a true process is trend stationary with
structural breaks; see Perron (1989). This implies that the LSE (of the Dickey–Fuller type) of ?1in (1) or ? in (9) is
Table 2
Averages and standard deviations of the estimated half-lives and other statistics, depending on the structural change
??
ˆ ?1
0.67
(0.08)
0.81
(0.05)
0.93
(0.02)
ˆ ?2
0.55
(0.09)
0.55
(0.09)
0.55
(0.09)
ˆh1
ˆh2
MSE1
MSE2
0.6 0.11.83
(0.54)
3.58
(1.02)
10.65
(2.89)
1.23
(0.33)
1.23
(0.33)
1.21
(0.32)
0.52
(0.86)
5.99
(5.72)
94.68
(70.95)
0.13
(0.18)
0.12
(0.18)
0.13
(0.16)
0.2
0.4
0.80.10.78
(0.07)
0.86
(0.05)
0.94
(0.02)
0.74
(0.07)
0.74
(0.07)
0.74
(0.07)
3.22
(1.25)
5.24
(2.20)
13.51
(5.80)
2.57
(0.89)
2.56
(0.87)
2.51
(0.90)
1.58
(3.78)
9.41
(23.91)
141.94
(217.81)
1.09
(1.54)
1.06
(1.46)
1.17
(1.98)
0.2
0.4
(11)
0.90.1
(1)
0.2
(10)
0.85
(0.06)
0.89
(0.06)
0.83
(0.06)
0.83
(0.06)
5.29
(2.90)
7.52
(4.72)
16.03
(9.27)
4.54
(2.23)
4.54
(2.26)
4.59
(3.08)
10.10
(41.04)
23.13
(100.53)
175.23
(377.13)
9.12
(22.82)
9.28
(21.26)
13.43
(64.79)
Note: 1. ˆ ?1and ˆ ?2are the estimators of ? in model (19) by the estimated models (20) and (21), respectively.
2.ˆhj= −log2/log ˆ ?jfor j = 1, 2 denotes the estimator of half-life (not adjusted to integers) and MSEj= (ˆhj− h0)2for j = 1, 2, where h0is
the true half-life, 1.36, 3.11, and 6.58 corresponding to ? = 0.6, 0.8, and, 0.9, respectively.
3. The parentheses in the second column denote the number of cases where ˆ ?1> = 1 or ˆ ?2> = 1. We do not consider these cases in the results
because the corresponding half-lives cannot be calculated. The parentheses in the other columns denote the corresponding standard deviations.
Page 8
Author's personal copy
yt= ?2Dt+ ?2yt−1+ ˜ et.
We assume that T0is known so that the comparison is unaffected by the estimation of the break point, T0.
InTable 2, we compare the results of the estimation from both models (20) and (21). In the fourth and sixth columns,
it is observed, similar toAndrews (1993) and Murray and Papell (2005), that all the estimators ˆ ?2are biased downward.
Therefore the half-life estimatorsˆh2are all under-estimated even though the structural breaks are considered.
2062
B. Seong et al. / Computational Statistics & Data Analysis 51 (2006) 2056–2064
0%
10%
20%
30%
40%
(iii)
0.40.6 0.8
1
0.40.60.81
0%
10%
20%
30%
40%
(i)
0 20406080
0%
20%
40%
60%
80%
(iv)
0 20406080
0%
20%
40%
60%
80%
(ii)
Fig. 3. Histograms for ˆ ?1,ˆh1, ˆ ?2, andˆh2, which correspond to (i), (ii), (iii), and (iv), respectively, when (?,?) = (0.9, 0.2). Note: 1. The y-axis in
each histogram denotes the relative frequency. 2. The arrows denote the existing range of the histograms.
over-estimated. Macro-economic data, such as price indices and exchange rates, often have structural breaks in the
trend (or level). Therefore, an analysis that does not incorporate such breaks yields over-estimated half-lives.
In order to assess the effect of a structural break in the trend (at a single point in time) on the estimation of half-lives,
we conduct a small Monte Carlo experiment. We generated 10,000 replications of a series {yt} of length T = 100,
defined by
yt= ?Dt+ ?yt−1+ et,
whereDt=t−T0ift >T0,and0otherwise,representingastructuralbreakinthetrendatT0.Forsimplicity,weassume
that T0= 50, and the innovations etare i.i.d. N(0,1). For various values of ? and ?, we consider ? = 0.6,0.8,0.9 and
? = 0.1,0.2,0.4. For ? = 0.6,0.8,0.9, the corresponding half-lives are 1.36, 3.11, and 6.58.
In order to estimate half-life when a structural break is not considered, we computed half-lifeˆh1= −log2/log ˆ ?1
based on the following model:
(19)
yt= ?1+ ?1t + ?1yt−1+ ˜ et.
In order to calculate half-life when a structural break is considered, we computed half-lifeˆh2= −log2/log ˆ ?2based
on the following model:
(20)
(21)
Page 9
Author's personal copy
Previous researchers have identified a number of sources of bias in half-life estimation, namely, cross-sectional ag-
gregation,temporalaggregation,andmis-specifiedmodels.However,inthispaper,wehaveidentifiedseveraladditional
sources of instability in the conventional half-life estimation. We found that even for an AR(1) process, the sampling
bias cannot be ignored. For higher order or mixed time series process, the biases resulting from the use of conventional
B. Seong et al. / Computational Statistics & Data Analysis 51 (2006) 2056–2064
2063
JAN81JAN85 JAN89JAN93
DATE
JAN97 JAN01DEC04
4.4
4.6
4.8
5
5.2
5.4
log (CPI)
Fig. 4. Logarithm of the monthly CPI during the period January 1981–December 2004 (T = 288); the dotted vertical line denotes January 1991.
In the third and fifth columns, it is observed that the estimators of ˆ ?1are biased upward with the exceptions of
(?,?)=(0.8,0.1), (0.9, 0.1), and (0.9, 0.2), and all the estimators of half-lifeˆh1are over-estimated with the exceptions
of (?,?) = (0.9,0.1). Further, from the last two columns, it is observed that all the mean squared errors (MSEs) ofˆh1
are larger than those ofˆh2. In the cases of (?,?) = (0.8,0.1), (0.9, 0.1), and (0.9, 0.2),ˆh1has a larger MSE thanˆh2,
although the corresponding ˆ ?1is less biased than ˆ ?2. This can be explained from Fig. 3, which shows the distributions
of ˆ ?1, ˆ ?2,ˆh1, andˆh2when (?,?)=(0.9,0.2). The distribution of ˆ ?1has a higher concentration near one than that of ˆ ?2,
which makes the right tail ofˆh1longer than that ofˆh2.
Thisover-estimationphenomenonisnotsurprisingbecause ˆ ?1’sarereadytoconvergetooneassamplesizebecomes
larger regardless of the value of ?; see Perron (1989). Therefore, when there is a doubt regarding the existence of
structural breaks, it is desirable to consider a model that incorporates structural breaks.
As an empirical example, we consider the monthly CPI of the U.S. during the period January, 1981–December,
2004 (T = 288), obtained from the Bureau of Labor Statistics, U.S. Department of Labor. The time series plot of the
logarithm of the CPI is shown in Fig. 4. From a visual inspection, it is determined that a decline in the slope occurred
in January 1991. This decline is attributed to low energy and food prices,3and the sustained low energy prices during
the entire 1990s are manifested in the CPI estimates.Therefore, assuming T0=120 (December, 1990), we estimate the
half-livesfrommodels(20)and(21).Frommodel(20),weobtain ˆ ?1=0.988(t-ratio=209.97)andˆh1=57.415,thatis
approximately 57 months or 4.8 years, while from model (21), we obtain ˆ ?2=0.962 (t-ratio=98.07) andˆh2=17.892,
that is approximately 18 months or 1.5 years. The difference in both the half-lives is approximately 3.3 years, and this
illustrates the over-estimation of half-life from a model without any structural break.
5. Conclusions
3Stewart (1992) provides a detailed discussion of the reasons for this drop in the slope of the CPI.
Page 10
Author's personal copy
Murray, C.J., Papell, D.H., 2002. The purchasing power parity persistence paradigm. J. Internat. Econom. 56, 1–19.
Murray, C.J., Papell, D.H., 2005. The purchasing power parity puzzle is worse than you think. Empirical Econom. 30, 783–790.
Perron, P., 1989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361–1401.
Stewart, K.J., 1992. Energy, food prices helped slow inflation in 1991. Monthly Labor Review, May, 3–5.
Taylor, A.M., 2001. Potential pitfalls for the purchasing-power-parity puzzle? Sampling and specification biases in mean-reversion tests of the law
of one price. Econometrica 69, 473–498.
Wei, W.W.S., 1996. Time SeriesAnalysis.Addison-Wesley, NewYork.
2064
B. Seong et al. / Computational Statistics & Data Analysis 51 (2006) 2056–2064
formula are quite large.The presence of structural breaks in time series creates additional noise in half-life calculation.
Thus, to ensure a more accurate calculation of half-life, more attention must be paid to these issues.
Acknowledgements
The authors thank two anonymous referees for their helpful comments that lead to a significant improvement of
this paper. Byeongchan Seong’s research was supported by the Post-doctoral Fellowship Program of Korea Science &
Engineering Foundation (KOSEF). Sung K. Ahn’s research was supported by the Korea Research Foundation Grant
(KRF-2005-070-C00022) funded by the Korean Government (MOEHRD).
6. Appendix. Proof of (18)
Assume AR(2) to be an approximate model for ARMA(1,1). By a property of the partial autocorrelation function,
we can find the coefficients (?12,?22) ofAR(2) using
??12
Then we obtain
1
1 − ?2
= (1 + ?)
1 + ?1
=
1 − (? + 1)? + ?2
since ?2= ??1and ?1= (1 − ??)(? − ?)/1 − 2?? + ?2. Therefore, we can deduce that
−log2
log(?12+ ?22)= −log2/log
?22
?
=
?1
?1
1
?1
?−1??1
?2
?
.
?12+ ?22=
1
(?1− ?1?2+ ?2− ?2
?1
1)
(1 − ??)(? − ?)
?(1 − ??)(? − ?)
1 − (? + 1)? + ?2
?
.
References
Andrews, D.W.K., 1993. Exactly median-unbiased estimation of first order autoregressive/unit root models. Econometrica 61, 139–165.
Basker, E., Hernandez-Murillo, R., 2004.A further look at distance. Mimeo.
Box, G.E.P., Jenkins, G.M., Reinsel, G.C., 1994. Time series analysis: forecasting and control. Prentice-Hall, New Jersey.
Campbell, J.Y., Mankiw, N.G., 1987.Are output fluctuations transitory? Quart. J. Econom. 102, 857–880.
Cecchetti, S.G., Mark, N.C., Sonora, R.J., 2002. Price index convergence among United States cities. Internat. Econom. Rev. 43, 1081–1099.
Chambers, M.J., 2004. The purchasing power parity puzzle, temporal aggregation, and half-life estimation. Econom. Lett. 86, 193–198.
Chen, L.L., Devereux, J., 2003. What can US city price data tell us about purchasing power parity? J. Internat. Money and Finance 22, 213–222.
Chen, S., Engel, C., 2005. Does ‘aggregation bias’explain the PPP puzzle? Pacific Econom. Rev. 10, 49–72.
Cheung,Y.W., Lai, K.S., 2000a. On cross-country differences in the persistence of real exchange rates. J. Internat. Econom. 50, 375–397.
Cheung,Y.W., Lai, K.S., 2000b. On the purchasing power parity puzzle. J. Internat. Econom. 52, 321–330.
Choi, C., Mark, N., Sul, D., 2004. Unbiased estimation of the half-life to PPP convergence in panel data. NBER Working Paper No. 10614.
Froot,K.A.,Rogoff,K.,1995.PerspectivesonPPPandlong-runrealexchangerates.In:Grossman,G.,Rogoff,K.(Eds.),HandbookofInternational
Economics, vol. III. North-Holland,Amsterdam, pp. 1647–1688.
Imbs, J., Mumtaz, H., Ravn, M.O., Rey, H., 2005. PPP strikes back: aggregation and the real exchange rate. Quart. J. Econom. 120, 1–44.
Morshed,A.K.M.,Ahn, S.K., Lee, M., 2005. Price index convergence among Indian cities: a cointegration approach.Working paper, Department of
Economics, Southern Illinois University.