# 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.

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**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. · 1.68 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 - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper evaluates price variability and price convergence in Indonesia. Using price indices of 35 products in 45 cities from January 2002 to April 2008, this study shows that, during the observed period, prices in Indonesia converged to the ‘relative’ law of one price. The price variability of one product across cities is found to be smaller than the price variability of all products within a city. Transportation costs and the level of development matter to price variability. This study also reveals that the average speed of convergence, which is measured by the half-life, for perishable goods is about 9 months, non-perishable goods 32–36 months, and services 18–19 months, while the median of the half-life of all products is about 16–17 months. The speed of convergence depends on the initial price difference, but not the distance between cities.Journal of Asian Economics. 01/2009;

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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

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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

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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.

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p

?

the impulse response ?jsatisfies the linear difference equation

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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)

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?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)

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which is obtained by solving (? − ?)?h−1= 1/2, provided ?>? and ?>0. Since the lag one autocorrelation of the

ARMA(1,1) process is

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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)

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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.

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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.

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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)

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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.