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Median-unbiased estimation of structural changemodels: an application to real exchange ratepersistenceHatice Ozer Ballia, Christian J. Murrayb & David H. Papellba Department of Economics and Finance, Massey University, Palmerston North, New Zealandb Department of Economics, University of Houston, Houston, TX 77204, USAPublished online: 24 Jun 2014.

To cite this article: Hatice Ozer Balli, Christian J. Murray & David H. Papell (2014) Median-unbiased estimation ofstructural change models: an application to real exchange rate persistence, Applied Economics, 46:27, 3300-3311, DOI:10.1080/00036846.2014.927570

To link to this article: http://dx.doi.org/10.1080/00036846.2014.927570

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Median-unbiased estimation of

structural change models: an

application to real exchange rate

persistence

Hatice Ozer Ballia, Christian J. Murrayb,* and David H. Papellb

aDepartment of Economics and Finance, Massey University, Palmerston North,New ZealandbDepartment of Economics, University of Houston, Houston, TX 77204, USA

Measuring deviations from purchasing power parity has been the subject ofextensive investigation. The most common practice in empirical research formeasuring real exchange rate persistence is to estimate univariate autoregressive(AR) time series models and calculate the half-life, defined as the number ofperiods for a unit shock to a time series to decay by 50%. In the presence ofstructural change, there are two potential biases in the parameter estimates of ARmodels: (1) a downward small sample median-bias and (2) an upward bias, whichoccurs when structural change is present and ignored. We conduct a variety ofMonte Carlo simulations and demonstrate that the existence of structural changecauses a substantial increase in the small sample bias documented in Andrews(1993). We then propose an extension of median-unbiased estimation, whichexplicitly accounts for structural change, and apply these methods to estimatehalf-lives of several long-horizon real exchange rates analysed by Lothian andTaylor (1996) and Taylor (2002). The upward bias from neglecting structuralchange dominates the downwardmedian-bias for these real exchange rates.Whenstructural change is present and accounted for, the median-unbiased half-livestowards a changing mean decrease and the confidence intervals tighten.

Keywords: real exchange rates; persistence; median-bias; structural change

JEL Classification: C01; C22; E37; F31

I. Introduction

Measuring deviations from purchasing power parity (PPP)has been the subject of extensive investigation. The mostcommon practice in empirical research for measuring realexchange rate persistence is to estimate univariate autore-gressive (AR) time series models and calculate the half-life, defined as the number of periods for a unit shock to atime series to decay by 50%. For stationary time series,

since all shocks eventually die out, the half-life is neces-sarily finite. For unit root process, the ‘whole-life’ isinfinite, and whether the half-life is infinite depends onthe particular data generating process.

There are two threads of research in the estimation ofpersistence in macroeconomic time series. First, estimat-ing AR models by least squares is known to produce adownward median-bias, which worsens as persistenceincreases. This median-bias causes half-lives to be

*Corresponding author. E-mail: cjmurray@uh.edu

Applied Economics, 2014Vol. 46, No. 27, 3300–3311, http://dx.doi.org/10.1080/00036846.2014.927570

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underestimated. Andrews (1993) and Andrews and Chen(1994) discuss this bias and show how to construct med-ian-unbiased estimates of the AR parameters and of thehalf-life. There are numerous papers that apply thismethod to real exchange rates (e.g. Murray and Papell,2002, 2005a, b; Rossi, 2005). Murray and Papell (2005a)estimate half-lives of real exchange rates using Lothianand Taylor (1996) data set. They find that correcting smallsample bias raises point estimates of half-lives from 5.78to 6.58 years for the dollar-sterling rate and from 2.73 to2.94 years for the franc-sterling rate.

A second line of research has examined persistence ofreal exchange rates in the presence of structural change.Neglecting structural change in the estimation of persis-tence will produce an upward bias, which can cause half-lives to be overestimated. Hegwood and Papell (1998,2002) estimate half-lives of real exchange rates and findfaster mean reversion after accounting for structuralchange. For example, the half-life of PPP deviations forthe Lothian and Taylor (1996) sterling-dollar rate fallsfrom 5.78 to 2.32 years. In a nonlinear framework,Lothian and Taylor (2008) allow for breaks in the equili-brium exchange rate for the real sterling-dollar exchangerate over a sample period that spans nearly two centuries.Astorga (2007) uses a new data set for the period 1900 to2005 to analyse the behaviour of real exchange rates in thesix largest economies in Latin America. After allowing fortrends and structural breaks, he finds that the half-life ofthe processes of the adjusted series ranges from 0.8 to2.5 years compared to 1.6–7.0 years for the unadjustedseries.

In this article, we extend median-unbiased estimation toaccount for structural change. We conduct a variety ofMonte Carlo simulations and demonstrate that the exis-tence of structural change causes a substantial increase inthe small sample median-bias documented in Andrews(1993). In a model allowing for structural change, wefind that the inherent median-bias in estimating the ARcoefficient is larger than the Andrews benchmark for agiven sample size. In addition to the standard results thatthe median-bias increases as persistence rises anddecreases as the sample size increases, we find that themedian-bias increases as the size of the break increases.

To illustrate our results, we apply these techniques tolong-horizon real exchange rate data and calculate half-lives of PPP deviations in the presence of structuralchange. We estimate the speed of mean reversion of thedollar-sterling and franc-sterling real exchange rates ana-lysed by Lothian and Taylor (1996) and the dollar realexchange rates for the Netherlands, the United Kingdomand Portugal studied by Taylor (2002) and Lopez et al.(2005) to account for both small sample bias and structuralchange. Using both exactly and approximately median-unbiased estimation techniques, we show that accountingfor structural breaks while neglecting small sample

median-bias will understate persistence, while accountingfor small sample median-bias while neglecting structuralbreaks will overestimate persistence.

After correcting both of the biases with our extendedmethod, we find that the upward bias from neglectingstructural change dominates the downward median-biasfor all of the real exchange rates studied in this article, sothat the half-lives of PPP deviations are much smaller thanwould be found from AR models that account for neithersmall sample bias nor structural change. In addition, theconfidence intervals for these estimates are narrowed con-siderably. When we incorporate structural breaks in thecontext of median-unbiased estimation for the realexchange rates from both data sets, we find that thespeed of mean reversion to a changing mean increasesconsiderably.

II. Median-Unbiased Estimation in thePresence of Structural Change

In this section, we extend the median-unbiased techniqueof Andrews (1993) to correct for median-bias when thetime series in question contains structural change. Toassess the effect of structural breaks on the estimation ofthe persistence parameter, we conduct a number of MonteCarlo simulations. We consider four different time seriesmodels. As in Andrews (1993), our benchmark model isthe AR(1) model with a constant term:

yt ¼ μþ αyt�1 þ εt (1)

We refer to this asModel 1. α determines the persistencein the time series and is assumed to lie within the unitinterval. The innovations are assumed to be iid standardnormal, μ ¼ 0 and y0 ¼ 0. We extend this basic model toallow for structural change. The basic idea is to adddummy variables to the conventional Dickey–Fullerregressions corresponding to known break dates. For oursimulations, we extend Model 1 by adding either one ortwo breaks ranging from 1 to 3 SDs of the error term.Models 2–4 are as follows:

yt ¼ μþ γDUþ αyt�1 þ εt (2)

yt ¼ μþ γ1DU1 þ γ2DU2 þ αyt�1 þ εt (3)

yt ¼ μþ γ1DU1 þ γ2DU2 þ αyt�1 þ εt : γ1 þ γ2 ¼ 0

(4)

In Model 2, we add one dummy variable, DU, toaccount for one structural break. Let TB be the date ofthe break in the mean. DU ¼ 1 if t > TB and 0 otherwise.Throughout this article, the date of break is assumed to be

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known. Models 3 and 4 allow for two structural breaks.The structural change is unrestricted in Model 3, whereasin Model 4, the breaks offset each other. This type ofrestricted structural change was used to analyse long-runPPP in Papell and Prodan (2006).

We measure persistence with the half-life, the numberof periods for a unit shock to a time series to decay by50%. It is a nonlinear function of α, computed as ln(0.5)/ln(α). It is well-known that α is biased downward in finitesamples. A standard method of correcting the bias is touse median-unbiased estimation proposed by Andrews(1993). This is particularly well-suited for correctingbias in half-lives because median-unbiased estimation isinvariant to monotonic nonlinear transformation. As aresult, the half-life based on the median-unbiased estimateof α will be median-unbiased, and confidence intervalswill have proper coverage probabilities.1

When we augment Model 1 by including dummy vari-ables to incorporate structural change, the least squares

estimate of α will also be downward median-biased. Tocorrect for this, we construct median-unbiased estimatorsfor Models 2–4 using Andrews (1993) methodology. Wecreate artificial data with structural breaks, run 100 000Monte Carlo trials and record the median values of theleast squares estimates of the AR coefficients for each truevalue of α. We use this recorded table to correct our OLSestimator of α. For example, if the OLS estimator for α is0.8, we find the true value of α such that the median of theleast squares estimate is 0.8. This true value of α is themedian-unbiased estimate.

Median-bias in the presence of structural change

Table 1 presents simulation results for Models 1 and 2. Werecord the median value of the least squares estimate of theAR coefficient for a range of values of α and for each breaksize, using a sample size of 120 with 100 000 replications.We consider values of α between 0.1 and 1.0 and breaks of

Table 1. Simulation of models: one break of sizes 1σ, 2σ and 3σ with N = 120

No structural break Structural breaks

Andrews (1993) 1σ 2σ 3σ

α/Quantile 0.025 0.5 0.975 0.025 0.5 0.975 0.025 0.5 0.975 0.025 0.5 0.975

1 0.8633 0.9639 1.0039 0.8212 0.9383 0.9962 0.8126 0.9355 0.9951 0.7997 0.9315 0.99410.995 0.8557 0.9580 1.0002 0.8130 0.9323 0.9922 0.8036 0.9292 0.9913 0.7927 0.9247 0.98950.99 0.8492 0.9534 0.9970 0.8064 0.9268 0.9894 0.7974 0.9243 0.9885 0.7847 0.9189 0.98620.985 0.8426 0.9492 0.9941 0.8006 0.9227 0.9867 0.7920 0.9197 0.9852 0.7783 0.9137 0.98350.98 0.8369 0.9452 0.9914 0.7941 0.9184 0.9833 0.7864 0.9153 0.9825 0.7719 0.9095 0.98030.975 0.8308 0.9412 0.9888 0.7905 0.9147 0.9800 0.7800 0.9110 0.9795 0.7672 0.9048 0.97710.95 0.8028 0.9200 0.9745 0.7631 0.8942 0.9646 0.7539 0.8897 0.9631 0.7379 0.8825 0.96060.925 0.7740 0.8970 0.9591 0.7378 0.8731 0.9482 0.7273 0.8678 0.9458 0.7107 0.8588 0.94330.9 0.7449 0.8734 0.9427 0.7108 0.8506 0.9315 0.7003 0.8445 0.9293 0.6834 0.8347 0.92570.875 0.7164 0.8496 0.9257 0.6822 0.8274 0.9142 0.6753 0.8213 0.9108 0.6565 0.8101 0.90630.85 0.6874 0.8253 0.9080 0.6579 0.8041 0.8956 0.6459 0.7965 0.8930 0.6287 0.7849 0.88620.825 0.6593 0.8010 0.8898 0.6304 0.7805 0.8776 0.6184 0.7729 0.8732 0.6011 0.7601 0.86690.8 0.6317 0.7767 0.8711 0.6043 0.7565 0.8592 0.5919 0.7485 0.8545 0.5747 0.7355 0.84630.7 0.5205 0.6791 0.7932 0.4971 0.6603 0.7803 0.4844 0.6510 0.7738 0.4670 0.6364 0.76360.6 0.4133 0.5809 0.7109 0.3913 0.5632 0.6978 0.3809 0.5528 0.6905 0.3662 0.5390 0.67780.5 0.3087 0.4829 0.6259 0.2900 0.4660 0.6132 0.2808 0.4579 0.6053 0.2677 0.4437 0.5920

Notes: We estimate the following models:

Yt ¼ μþ αYt�1 þ εt

Yt ¼ μþ γDUþ αYt�1 þ εt

where DU = 1 if t > TB and 0 otherwise. TB is the date of break. The sample size is 120 and the number of simulations is 100 000. Yt hasone break of sizes 1σ, 2σ and 3σ in the middle of the sample.

1An alternative method to correct for small sample bias of least squares estimators is mean-unbiased estimation, as in Kilian (1998),where the expected value of an estimator is equal to the true parameter value. For estimating the half-life of an AR (1), however, mean-unbiased techniques will not be unbiased under the monotonic nonlinear transformation. In addition, median-unbiasedness is more usefulthan mean-unbiasedness when the parameter space is bounded or when the distributions of estimators are skewed and/or kurtotic(Andrews, 1993).

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1, 2 and 3 SDs (σ) of the innovations, which are iidstandard normal. We also report the upper and lower2.5% quantiles so that 95% confidence intervals can beconstructed. To provide a benchmark for our results, anexact replication of Andrews (1993), where there is nostructural change present in the data, for the same samplesize appears in column (1). The median column (labelled0.5) displays two well-known results. For an AR(1)model, the median-bias of αLS (the least squares estimatefor α) is always negative and the bias worsens as the seriesbecomes more persistent.

To learn how the presence of structural change affectsmedian-bias, we start with Model 2 in which one dummyvariable is added to the AR(1). The single break is speci-fied to occur in the middle of the sample and to be cor-rectly estimated. We perform Monte Carlo simulations forModel 2 and report the results in the columns labelled0.5 under the heading of structural breaks. Equation 2 isestimated for values of α between 0.1 and 1.0 and breaksranging from 1σ to 3σ. As in the benchmark AR(1) model(column 1), the median-bias is downward. As is expectedfrom econometric intuition, the median-bias is larger(more negative) for Model 2 than Model 1, since we areadding an extra deterministic regressor. As in Model 1, themedian-bias is larger as the series becomes more persis-tent. It is also of interest that the median-bias increasesas the size of the break increases. For example, if α ¼ 0:9,a 1σ break almost doubles the median-bias in Model 1increasing from 0.0268 to 0.0494. Breaks of 2σ and3σ increase the median-bias to 0.0533 and 0.0656,respectively.2

One of the objectives of this article is to quantify theimpact of median-bias in the presence of structural changeon the speed of convergence, as measured by the half-life.For example, suppose that the true persistence parameterα ¼ 0:9, so that the true half-life is equal to ln(0.5)/ln(0.9) = 6.58 years. The median of the least squares half-life, however, is 5.11 years for Model 1, which is under-estimated by 1.47 years. With a 1σ break, the median ofthe least squares half-life falls to 4.28 years. Breaks of 2σand 3σ lower the median least squares half-lives to 4.11and 3.83 years, respectively. This demonstrates that evenif structural change is accounted for properly, least squareshalf-lives in Model 2 can be severely median-biased.

To examine the effect of sample size, we replicateTable 1 with a sample size of 240 and present the resultsin Table 2. As before, we provide a benchmark by repli-cating the results from Andrews (1993) without structuralchange in the first column. As is well known, the median-bias decreases with the larger sample size for every valueof α when there is no structural change. This same pattern

emerges when structural change is present. Comparing thecolumns labelled 0.5 in Tables 1 and 2, the median-bias issmaller for the larger sample size for every value of α andall break sizes considered. With a true autocorrelation of0.9 and a 1σ break, the size of the median-bias decreasesfrom 0.0494 (120 observations) to 0.0227 (240 observa-tions). In terms of the half-life, the median least squaresestimate of 5.29 years for T ¼ 240 is closer to the truehalf-life of 6.58 years than the estimate of 4.28 yearsfor T ¼ 120.

Two general patterns regarding confidence intervalsemerge from Tables 1 and 2. Using Andrews (1993)models as a benchmark, the lower bounds are uniformlyhigher and increase with the sample size. The upperbounds are also uniformly higher but decrease with thesample size. Except for values of α very close to or equalto unity, the confidence intervals for Model 2 are widerthan Andrews’ (1993) benchmark for any given samplesize and break size.

We now proceed to investigate median-bias in the con-text of multiple structural breaks. Table 3 contains resultsfor two structural breaks of the same size in both the sameand opposite directions. In column (2), we report estimatesof Model 3, where we do not impose any restrictions onthe coefficients of the break dummies. We add two 1σbreaks of the same sign, the first after one-third and thesecond after two-thirds of the sample, and compare theresults with the one break case in Table 1. For a truepersistence parameter of 0.9, the median least squareshalf-life with two breaks is equal to 3.63 years, consider-ably lower than the half-life of 4.28 years with one 1σbreak or the half-life of 4.11 years with one 2σ breaks inTable 1. Not only does the bias increase with the numberof breaks (holding the size of the breaks constant), but thebias with two 1σ breaks is larger than the bias with one 2σbreak, even though the total size of the structural change isthe same in both cases. We find this latter result to be thecase in general. For a given total size of structural change,a larger number of breaks will increase the median-bias.We conjecture that this is due to the increased number ofdeterministic regressors.

An interesting question that arises in the presence ofmultiple structural breaks is what happens if two breaksoffset each other. When the two breaks are of the samesign, changes to the mean are permanent and the series isnonstationary. However, when the breaks offset eachother, the time series will revert to its long-run mean aslong as α < 1. The last column in Table 3 reports theresults for two breaks, of sizes � 1σ, located after one-third and two-thirds of the sample. Within the simulations,Model 4 is estimated with the restriction that the breaks

2The break dates in our Monte Carlo simulations are specified to occur at the actual break date in the true DGP. If we were to endogenizethe break dates, then our results could change if the estimated break dates were to differ from the actual break dates. However, this articleis not concerned with testing for structural change.

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Table 2. Simulation of models: one break of sizes 1σ, 2σ and 3σ with N = 240

No structural break Structural breaks

Andrews (1993) 1σ 2σ 3σ

α/Quantile 0.025 0.5 0.975 0.025 0.5 0.975 0.025 0.5 0.975 0.025 0.5 0.975

1 0.9307 0.9819 1.0018 0.9092 0.9693 0.9979 0.9074 0.9687 0.9979 0.9028 0.9673 0.99720.995 0.9232 0.9766 0.9984 0.9009 0.9636 0.9946 0.8990 0.9628 0.9943 0.8948 0.9614 0.99380.99 0.9170 0.9725 0.9956 0.8966 0.9593 0.9916 0.8927 0.9584 0.9913 0.8885 0.9569 0.99090.985 0.9107 0.9684 0.9929 0.8903 0.9553 0.9887 0.8874 0.9544 0.9881 0.8838 0.9523 0.98770.98 0.9049 0.9641 0.9901 0.8845 0.9511 0.9856 0.8813 0.9501 0.9852 0.8771 0.9483 0.98460.975 0.8992 0.9597 0.9872 0.8785 0.9470 0.9825 0.8763 0.9457 0.9819 0.8716 0.9438 0.98130.95 0.8692 0.9362 0.9713 0.8511 0.9249 0.9660 0.8479 0.9231 0.9650 0.8423 0.9202 0.96380.925 0.8393 0.9120 0.9539 0.8231 0.9013 0.9483 0.8195 0.8993 0.9470 0.8133 0.8957 0.94560.9 0.8101 0.8874 0.9356 0.7948 0.8773 0.9296 0.7903 0.8745 0.9283 0.7840 0.8704 0.92620.875 0.7806 0.8629 0.9165 0.7671 0.8531 0.9100 0.7620 0.8499 0.9088 0.7551 0.8449 0.90580.85 0.7521 0.8382 0.8969 0.7385 0.8287 0.8903 0.7335 0.8253 0.8885 0.7264 0.8199 0.88510.825 0.7236 0.8136 0.8769 0.7102 0.8041 0.8706 0.7047 0.8003 0.8678 0.6969 0.7946 0.86440.8 0.6955 0.7888 0.8564 0.6832 0.7792 0.8494 0.6775 0.7752 0.8469 0.6693 0.7690 0.84290.7 0.5850 0.6898 0.7721 0.5736 0.6807 0.7651 0.5690 0.6762 0.7616 0.5603 0.6688 0.75590.6 0.4778 0.5905 0.6843 0.4676 0.5822 0.6777 0.4627 0.5775 0.6739 0.4544 0.5693 0.66650.5 0.3726 0.4915 0.5948 0.3643 0.4838 0.5881 0.3588 0.4788 0.5843 0.3513 0.4718 0.5761

Notes: We estimate the following models:Yt ¼ μþ αYt�1 þ εt

Yt ¼ μþ γDUþ αYt�1 þ εt

where DU = 1 if t > TB and 0 otherwise. TB is the date of break. The sample size is 240 and the number of simulations is 100 000. Yt hasone break of sizes 1σ, 2σ and 3σ in the middle of the sample.

Table 3. Models with two breaks of size (1σ, 1σ) and (1σ, −1σ)

No structural break Structural breaks

Andrews (1993) Unrestricted (1σ, 1σ) Restricted (1σ, −1σ)

α/Quantile 0.025 0.5 0.975 0.025 0.5 0.975 0.025 0.5 0.975

1 0.8633 0.9639 1.0039 0.7807 0.9131 0.9854 0.8314 0.9560 1.00580.995 0.8557 0.9580 1.0002 0.7741 0.9069 0.9813 0.8231 0.9474 1.00260.99 0.8492 0.9534 0.9970 0.7673 0.9013 0.9779 0.8153 0.9407 1.00010.985 0.8426 0.9492 0.9941 0.7615 0.8973 0.9756 0.8093 0.9352 0.99730.98 0.8369 0.9452 0.9914 0.7565 0.8928 0.9724 0.8004 0.9298 0.99400.975 0.8308 0.9412 0.9888 0.7506 0.8885 0.9695 0.7972 0.9253 0.99120.95 0.8028 0.9200 0.9745 0.7264 0.8687 0.9534 0.7681 0.9006 0.97350.925 0.7740 0.8970 0.9591 0.7020 0.8481 0.9366 0.7383 0.8768 0.95480.9 0.7449 0.8734 0.9427 0.6795 0.8264 0.9186 0.7113 0.8529 0.93620.875 0.7164 0.8496 0.9257 0.6520 0.8047 0.9011 0.6834 0.8282 0.91660.85 0.6874 0.8253 0.9080 0.6266 0.7816 0.8832 0.6551 0.8043 0.89790.825 0.6593 0.8010 0.8898 0.6014 0.7588 0.8647 0.6298 0.7800 0.87880.8 0.6317 0.7767 0.8711 0.5758 0.7345 0.8450 0.6010 0.7556 0.85890.7 0.5205 0.6791 0.7932 0.4722 0.6412 0.7673 0.4932 0.6584 0.77820.6 0.4133 0.5809 0.7109 0.3715 0.5456 0.6843 0.3887 0.5609 0.69640.5 0.3087 0.4829 0.6259 0.2712 0.4502 0.5999 0.2884 0.4640 0.6108

Notes: We estimate the following models:

Yt ¼ μþ γ1DU1 þ γ2DU2 þ αYt�1 þ εtYt ¼ μþ γ1DU1 þ γ2DU2 þþαYt�1 þ εt subject to γ1 þ γ2 ¼ 0

where DUi = 1 if t > TBi for i = 1, 2 and 0 otherwise. TB is the date of break. The sample size is 120 and the number of simulations is100 000. Yt has two breaks of sizes (1σ, 1σ) and (1σ, −1σ).

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sum to zero. For α ¼ 0:9, the median least squares half-lifeis 4.35 years, which is downward biased by 2.23 years.While the bias is smaller than for the unrestricted modelwith two 1σ breaks in the same direction, it is very largewhen compared to the benchmark model. These resultssuggest that, even when breaks offset each other so thatlong-run mean reversion is achieved, omitting them fromthe regression equation will result in downward median-biased half-life estimates.

The upper and lower bounds of the confidence intervalsfor Model 3 are uniformly higher than that for Andrews’(1993) model and are almost always wider. Whenrestricted structural change is imposed as in Model 4,there is no distinct pattern in the lower bounds, but theupper bounds are always higher than in Model 1, so thatthe confidence intervals are wider. Comparing Model 3and Model 4 directly, both lower and upper bounds ofModel 4 are lower than those for Model 3, with net effectbeing that the intervals for Model 4 are wider than thosefor Model 3.

Median-bias and structural change bias

Up to this point, we have assumed that, if present in thedata, structural change is correctly accounted for by theeconometrician. In practice, however, persistence is oftenmeasured without investigating structural change, even inlong-horizon time series where structural changes are morelikely to be present. In this case, ignoring structural changewhen estimating persistence may produce an upward biasof the AR parameter.3 The intuition here is straightforward.Structural changes represent permanent shifts in the meanof the series. If these shifts are ignored by not includingdummy variables, persistence will be incorrectly measuredtowards an unchanging, rather than to a changing, mean.These permanent breaks will then be accounted for by theAR term α, causing it to be upward biased.4

To assess the combined effects of median and structuralchange biases, we generate data according to Model 2where a single break occurs in the middle of the sample.As before we consider breaks from one to three innovationSDs. We then estimate the misspecified Model 1 by leastsquares. We record the median value of the least squaresestimate of the AR coefficient for each size of break andvalue of α, using a sample size of 120 with 100 000replications. We calculate the bias by subtracting the truevalue of α from the estimated median value of the ARparameter: medðαLSÞ � α. Since the estimated α correctsfor neither median-bias nor structural change, the calcu-lated bias represents the combined effects of both biases.

Table 4 presents the combined consequences of failingto account for both median-bias and structural change.

The combined bias is neither always negative, as wouldbe the case for only median-bias, nor always positive, aswould be the case for only structural change bias. Whilemedian-bias dominates for more persistent series, struc-tural change bias dominates for less persistent series. Thecombined bias starts out negative for large values of α,eventually becoming positive for less persistent time ser-ies. The value of α such that the bias switches fromnegative to positive increases with the size of the break.For a 1σ break, the switch occurs around α ¼ 0:775, for a2σ break around α = 0.90, and around α ¼ 0:925 for a 3σbreak. In contrast with the context investigated byAndrews (1993) where neglecting median-bias alwaysreduces the estimates of half-lives, neglecting both med-ian-bias and structural change bias can yield either underor over-estimated half-lives, depending on the particulardata generating process.

Table 4. Bias for models with one break of size 1σ, 2σ and 3σ

Bias

1σ 2σ 3σ

α α′-α α′-α α′-α

1 −0.0360 −0.0358 −0.03510.995 −0.0368 −0.0361 −0.03520.99 −0.0364 −0.0352 −0.03400.985 −0.0354 −0.0342 −0.03230.98 −0.0342 −0.0328 −0.02980.975 −0.0330 −0.0309 −0.02810.95 −0.0286 −0.0240 −0.01690.925 −0.0247 −0.0161 −0.00440.9 −0.0214 −0.0079 0.00970.875 −0.0179 0.0016 0.02430.85 −0.0143 0.0114 0.04030.825 −0.0102 0.0225 0.05700.8 −0.0057 0.0339 0.07370.7 0.0139 0.0852 0.14670.6 0.0373 0.1424 0.22360.5 0.0622 0.2017 0.30220.4 0.0907 0.2628 0.38100.3 0.1168 0.3233 0.45940.2 0.1440 0.3822 0.53660.1 0.1691 0.4396 0.6113

Notes: We estimate the following model:

Yt ¼ μþ αYt�1 þ εt

The sample size is 120 and the number of simulations is100 000. Yt has one break of 1σ, 2σ and 3σ in the middle ofthe sample. When there is a structural break and we runModel 1, we call the median values of the least squaresestimates of AR coefficient for each true value of α as α’.Columns 2–4 contain the related biases when we do notaccount for the structural changes.

3 Seong et al. (2006) discusses upward bias caused by structural breaks.4 The intuition does not apply to breaks of equal and opposite sign, which do not cause permanent changes in the mean.

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To better understand how these two biases interact, wedraw a decomposition of the opposing bias factors.Figure 1 shows the two biases for a single break of 1σand 3σ. The true value of α is plotted on the horizontalaxis, and the median estimated values of α’s for variousmodels are plotted on the vertical axis. We consider Model1 and both a correctly and incorrectly specified Model 2.The figure clearly shows how the upward bias due toneglecting structural change decreases and the downwardmedian-bias increases as the series becomes more persis-tent. In addition, as the size of break increases, the rangeforα, say ½v; 1�where the combined bias is negative, with vrepresenting the switching point, becomes smaller.

Table 5 shows where the switching point occurs forbreaks ranging in size from 0:25σ to 4σ and sample sizesof 120 and 240. Panel A shows that, consistent withmedian-bias falling with a larger sample, the switching

point increases if the sample size is larger, holding thebreak size constant.5 Panel A also shows that, consistentwith structural change bias increasing with the break size,the switching point is higher if the break size is larger,holding the sample size constant.

We also investigate whether the switching pointchanges with the number and direction of the breaks.Panel B shows that, for two breaks of same sign, theswitching point increases as the size of the breaks and/orthe number of observations increase. If each of the twobreaks in Panel B is the same size as the single break inPanel A, so that the total magnitude is double, the switch-ing point is larger with two breaks than with one break. If,however, each of the two breaks in Panel B is one-half thesize of the single break in Panel A, so that the totalmagnitude is equal, the switching point is larger withone break than with two breaks.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

True Persistence Parameter

Andrews (1993)

Do Not Account for Structural Change

Do Account for Structural Change

Downward smallsample bias

Upward bias dueto neglectingstructural breaks

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1(a)

(b)

0.2 0.30 0.1 0.4 0.5 0.6 0.7 0.8 0.9 1α

0.2 0.30 0.1 0.4 0.5 0.6 0.7 0.8 0.9 1

α

Quantile: 0.5

True Persistence Parameter

Andrews (1993)

Do Not Account for Structural Change

Do Account for Structural Change

Downward smallsample bias

Upward bias due toneglecting structural breaks

Quantile: 0.5

Fig. 1. Biases of estimators with artificial data. (a) One break of 1σ. (b) One break of 3σ

5We also set the timing of break at one-third or two-thirds of the sample and the results did not change. The timing of the break does notaffect the switching point.

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Panel C depicts the results for two breaks with equaland opposite signs. As in the cases of one break and twobreaks of the same sign, the switching point increases asthe size of the breaks and/or the number of observationsincrease. Holding the break size and number of observa-tions constant, the switching point is smaller than with onebreak and considerably smaller than for two breaks of thesame sign.

We summarize the results as follows. First, median-bias worsens with structural change. While this biasincreases as we increase the number and size ofbreaks, it gets smaller as we increase the samplesize. Second, neglecting structural breaks producesupward bias in the estimation of persistence.Therefore, when there is structural change that is not

accounted for, the combined bias from least squarescan be either positive or negative. The direction andseverity of the bias depends on the characteristics ofthe structural breaks as well as the size of AR coeffi-cient. We therefore suggest an extension of Andrews(1993) median-unbiased estimator to explicitly allowfor the presence of structural change.

III. Half-Lives of Real Exchange RatesCorrecting for Median and StructuralChange Bias

An important empirical application involving persistenceis the question of half-lives of deviations from PPP. In thissection, we compute median-unbiased half-life estimatesand confidence intervals, explicitly allowing for structuralchange. We use two long-horizon data sets to apply ourmethodology and to make comparisons. The first data setis from Lothian and Taylor (1996), which consists ofannual observations of nominal exchange rates for theUK sterling in terms of US dollars (1791–1990) andFrench francs (1802–1990) and wholesale price indicesfor the United States, the United Kingdom and France.Hegwood and Papell (1998) find that both of these realexchange rates are stationary, with and without allowingfor structural change. The second data set is from Taylor(2002) and is updated by Lopez et al. (2005). It includes16 annual dollar denominated real exchange rates fordeveloped countries, measured in domestic currencyunits per US dollar, and price indices measured as con-sumer price deflators (or when they are not available, GDPdeflators) starting between 1870 and 1892 and ending in1998. According to the results in Papell and Prodan(2006), we select from these real exchange rates thosewhere the unit root null is rejected, and there is evidenceof structural change in the form of a changing mean. Thisleaves three real exchange rates: the Netherlands, Portugaland the United Kingdom.

The real exchange rate with the US dollar (Taylor 2002extended) or the UK sterling (Lothian and Taylor, 1996) asthe numeraire currency is calculated as

qt ¼ et þ pt� � pt (5)

where qt is the logarithm of the real dollar (or sterling)exchange rate, et is the logarithm of the nominal dollar (orsterling) exchange rate, pt and p�t are the logarithms of thedomestic CPI and US CPI (or UK sterling CPI),respectively.6

Table 5. The switching point with one and two breaks

V

Break size N = 120 N = 240

(A) One break0.25 0.20 0.410.5 0.55 0.681 0.76 0.831.5 0.84 0.882 0.87 0.912.5 0.90 0.933 0.91 0.943.5 0.92 0.954 0.93 0.95

(B) Two breaks with the same sign0.25, 0.25 0.46 0.620.5, 0.5 0.72 0.801, 1 0.85 0.891.5, 1.5 0.90 0.932, 2 0.92 0.942.5, 2.5 0.94 0.953, 3 0.95 0.963.5, 3.5 0.95 0.964, 4 0.96 0.97

(C) Two breaks with opposite sign0.25, −0.25 0.14 0.360.5, −0.5 0.53 0.661, −1 0.74 0.821.5, −1.5 0.82 0.872, −2 0.86 0.902.5, −2.5 0.88 0.923, −3 0.90 0.933.5, −3.5 0.9 0.944, −4 0.9 0.94

Notes: The number of simulations is 100 000. N is the samplesize and V is the cut-off point. Decreasing from α = V to α = 0, thebias is upward and increasing (α is the true AR unit root). Thesame simulations are conducted choosing a break with the oppo-site direction; the results did not change.

6We analyse half-lives of PPP deviations with real exchange rates constructed using aggregate price indexes. The use of moredisaggregated price data produces shorter half-lives.

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Exactly median-unbiased estimation with and withoutstructural change

To establish a benchmark, we begin by estimating thespeed of mean reversion of real exchange rates usingAndrews’ (1993) exactly median-unbiased estimator,ignoring possible structural change. When the errorterms are serially uncorrelated, we run Dickey–Fullerregressions:

qt ¼ cþ αqt�1 þ ut (6)

Since αLS (the least squares estimate for α) is downwardmedian-biased, we remove the bias by replacing theleast squares estimate with αMU (the median-unbiasedestimator for α). The median-unbiased half-life is thuslnð0:5Þ=lnðαMUÞ. When we want to account for median-bias in the presence of structural change, we estimate thespeed of mean reversion (to a possibly changing mean) ofreal exchange rates using our extended median-unbiasedtechnique by adding dummy variables to the Dickey–Fuller regressions. For the two data sets, we take asgiven the estimated break dates reported by Hegwood

and Papell (1998) (for Lothian and Taylor, 1996 data)and Papell and Prodan (2006) (for the extended Taylor,2002 data).7When there is no serial correlation, we run theequation below:

qt ¼ cþXnb

i¼1

γiDUiþαqt�1 þ ut (7)

where nb is the number of breaks.Least squares and exactly median-unbiased half-lives

and 95% confidence intervals are presented in Panel A ofTable 6 for the Lothian and Taylor data. The estimates forModel 1 are reported in the upper half of Panel A forcomparison and are reproduced from Lopez et al.(2005). The estimates of Model 3 are reported in thelower half of Panel A. Accounting for median-biasincreases the half-lives by 0.19 years from 2.75 to2.94 years for the franc-sterling rate and by 0.80 yearsfrom 5.78 to 6.58 years for the dollar-sterling rate. Both ofthese real exchange rates contain two breaks. For thedollar-sterling real exchange rate, the breaks occur in1863 and 1929 and have size −0.48 and −0.57 times the

Table 6. Exactly median-unbiased half-lives in Dickey–Fuller regressions

N αLS HLLS αMU 95% CI HLMU 95% CI

(A) Lothian and Taylor (1996) sterling real exchange ratesModels without structural breaksFrance–UK 189 0.78 2.75 0.79 [0.70, 0.89] 2.94 [1.94, 5.95]US–UK 200 0.89 5.78 0.90 [0.83, 0.97] 6.58 [3.72, 22.76]

Models with structural breaksFrance–UK 189 0.71 2.02 0.74 [0.64, 0.86] 2.30 [1.55, 4.60]US–UK 200 0.74 2.31 0.77 [0.67, 0.88] 2.65 [1.73, 5.42]

(B) Taylor (2002) dollar real exchange ratesModels Without Structural BreaksNetherlands–US 129 0.93 8.92 0.95 [0.88, 1.00] 13.51 [5.42, ∞]Portugal–US 109 0.88 5.37 0.91 [0.81, 1.00] 7.35 [3.29, ∞]UK–US 129 0.85 4.23 0.87 [0.77, 0.98] 4.98 [2.65, 34.31]

Models with structural breaksNetherlands–US 129 0.84 3.85 0.88 [0.77, 1.00] 5.42 [2.65, ∞]Portugal–US 109 0.77 2.63 0.83 [0.70, 0.99] 3.72 [1.94, 68.97]UK–US 129 0.72 2.14 0.78 [0.65, 0.93] 2.79 [1.61, 9.55]

Notes: We estimate the following models:

qt ¼ cþ αqt�1 þ ut

qt ¼ cþXnb

i¼1

γiDUiþαqt�1 þ ut

where DUi = 1 if t > TBi for i = 1, 2 and 0 otherwise. TB is the date of the break. nb is the number of breaks. The break dates are taken fromHegwood and Papell (1998) and Papell and Prodan (2006) for the Lothian and Taylor (1996) and Taylor (2002) data. The half-lifeestimated by least squares is denoted by HLLS and the median-unbiased half-life is denoted by HLMU.

7 In these papers, 10% trimming is used to avoid finding spurious breaks at the beginning and at the end of samples.

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SE of the residual, respectively.8 For the franc-sterling realexchange rate, the first break of � 0:51σ̂ occurs in 1942,while the second break of 1:01σ̂ occurs in 1980.

Correcting least squares for median-bias suggests thatthe dollar-sterling rate is much more persistent than thefranc-sterling rate. The same message emerges from theconfidence intervals. The lower and upper bounds for thefranc-sterling rate are 1.94 years and 5.95 years, muchsmaller than the lower and upper bounds for the pound-sterling rate of 3.72 years and 22.76 years, respectively.

What is most interesting about Panel A is the juxtaposi-tion of the median-unbiased half-lives and confidenceintervals in the upper and lower halves. When we accountfor structural change in addition to median-bias, the per-sistence of the two real exchange rates appears to be muchmore similar. Accounting for structural change reduces theModel 1 median-unbiased half-lives by 0.642.30 years forthe franc-sterling rate and by 3.932.65 years for the dollar-sterling rate. The lower bounds for the median-unbiasedconfidence intervals are1.55 years for the franc-sterlingrate and 1.73 years for the dollar-pound rate, with theupper bounds 4.60 years for the franc-sterling rate and5.42 years for the dollar-pound rate. This demonstrates theimportance of accounting for structural change when esti-mating persistence. Failing to account for the two rela-tively small breaks in the dollar-pound rate causes the ARparameter to receive too much credit in accounting for thepersistence of the series.

Panel B of Table 6 contains the same information for theNetherlands, Portugal and the UK dollar denominated realexchange rates for the updated Taylor data. The breakdates are taken from Papell and Prodan (2006).9 TheNetherlands has one break in 1970 of 0:82σ̂. ThePortuguese real exchange rate has a break in 1916 of� 0:79σ̂ and a break in 1986 of 0:81σ̂. For the UnitedKingdom, there is a break in 1944 of � 0:99σ̂ and a breakin 1972 of 0:80σ̂. The breaks are mostly larger than in theLothian and Taylor data set. Accounting for structuralchange results in a significant drop in estimated persis-tence. The median-unbiased half-life estimates fall from13.51 to 5.42 for the Netherlands, from 7.35 to 3.72 forPortugal and from 4.98 to 2.79 for the United KingdomThis is also true for the lower bounds of the confidenceintervals. The lower bound for the Netherlands drops from

5.42 to 2.65, while the other two rates have lower boundsof less than 2 years. In terms of upper bounds of theconfidence interval, not much changes qualitativelywhen structural change is modelled. Two of the threeupper bounds decrease, but still indicate very persistentreal exchange rates.

Approximately median-unbiased estimation with andwithout structural change

While the results from the previous subsection are illus-trative, they are not realistic since they do not allow forserially correlated errors. One standard method ofaccounting for serial correlation is to run augmentedDickey–Fuller (ADF) regressions,

qt ¼ cþ αqt�1 þXk

j¼1

ψjΔqt�jþ ut (8)

with k lagged first-difference of the dependent variable.Andrews and Chen (1994) demonstrate how to computean approximately median-unbiased estimator of α and theimpulse response function when k > 0.10 We augment ourModels 2–4 to allow for serial correlation by estimatingmodels of the form

qt ¼ cþXnb

i¼1

γiDUiþαqt�1 þXk

j¼1

ψjΔqt�jþut (9)

where nb is the number of breaks and the lag length ischosen by the general-to-specific method with a maximumof eight lags allowed.11

Panel A of Table 7 reports approximately median-unbiased half-lives and confidence intervals for theLothian and Taylor (1996) data. For the franc-sterlingrate, the chosen lag is zero, so there is no change in theresults in Table 6. When structural change is allowed, thepoint estimate and confidence intervals bounds dropslightly. For the dollar-sterling real exchange rate, fourlags are chosen, the approximately median-unbiased half-life is 7.28 years, the lower bound of the 95% confidenceinterval is 3.08 years, and the upper bound is 19.39 years.When we account for structural change, the half-life

8 The 1863 break occurred during the American Civil War and the 1929 break is due to Great Depression. Even though the breaks arefairly small, they provide the impact of one larger negative break since they are in the same direction.9 Papell and Prodan (2006) calculate break dates including the serial correlation. We assume the dates of breaks are same with or withoutserial correlation.10When k > 0, the half-life must be computed from the impulse response function, not solely from α. In every case we considered, theestimated impulse response function decreases monotonically. As in Andrews and Chen (1994), the median-unbiased estimator of the ARcoefficients, the IRF and the half-life in Table 7 are no longer exact, but approximate.11 The only exception is the model without structural change for the Lothian and Taylor dollar-sterling rate. As shown by Murray andPapell (2005a), the half-life with k = 5, the lag length chosen by the general-to-specific method, is not an appropriate measure ofpersistence because the impulse response function is not monotonic. We therefore report results with k = 4, the largest lag length with amonotonic impulse response function.

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estimate drops to 1.40 years, the lower bound of the con-fidence interval is 0.95 years, and the upper bound of theconfidence interval is 1.94 years. Accounting for structuralchange dramatically decreases the persistence of the dollar-sterling rate. The estimated half-life falls bymore than four-fifths, and the lower bound of the confidence interval with-out structural change is greater than the upper bound of theconfidence interval with structural change.

Panel B of Table 7 reports approximately median-unbiased half-lives and confidence intervals for theextended Taylor data. The approximately median-unbiasedhalf-life estimates for the dollar real exchange rates withoutstructural breaks are 3.92 years for the United Kingdom,8.34 years for Portugal and 9.19 years for Netherlands, andall three confidence intervals have infinite upper bounds.Correcting for structural change again significantly lowersreal exchange rate persistence. The half-life point estimatesare 2.80 for the United Kingdom, 2.82 for Portugal and4.10 years for Netherlands, and the confidence intervals aremuch tighter. The narrowing of the intervals is caused bothby allowing for serial correlation and accounting for struc-tural change and is almost entirely in the form of loweringthe upper bound. When we control for structural changeand correlation in the errors, the upper bounds are 3.47,

4.20 and 6.23 for the United Kingdom, Portugal andNetherlands, respectively. The half-life point estimates aremuch lower and the confidence intervals are much tighterthan the approximately median-unbiased confidence inter-vals without accounting for structural change. If structuralchange is present and one only corrects for median-bias, thehalf-life estimates will be overstated and their correspond-ing confidence intervals too wide.

IV. Conclusions

How should persistence be measured in long macroeco-nomic time series involving structural change? Previouswork can be categorized into two main areas. One strandof research accounts for downward median-bias butneglects structural change. An important example of suchwork is the median-unbiased estimation of Andrews (1993)and Andrews and Chen (1994). Another strand of researchexamines persistence in the presence of structural changebut neglects median-bias. In this article, we unite these twostrands of research by analysing median-unbiased estima-tion in the presence of structural change.

Table 7. Approximately median-unbiased half-lives in augmented Dickey–Fuller regressions

N k αLS HLLS αMU 95% CI HLIRF, MU 95% CI

(A) Lothian and Taylor (1996) sterling real exchange ratesModels without structural breaksFrance–UK 189 0 0.78 2.75 0.79 [0.70, 0.89] 2.94 [1.94, 5.95]US–UK 200 4 0.89 5.74 0.91 [0.82, 0.98] 7.28 [3.08, 19.39]

Models with structural breaksFrance–UK 189 0 0.71 2.03 0.74 [0.64, 0.86] 2.36 [1.55, 4.60]US–UK 200 8 0.54 1.12 0.55 [0.49, 0.60] 1.40 [0.95, 1.94]

(B) Taylor (2002) dollar real exchange ratesModels without structural breaksNetherlands–US 129 1 0.90 6.9 0.92 [0.85, 1.00] 9.19 [4.43, 36.66]Portugal–US 109 5 0.88 5.55 0.91 [0.81, 1.00] 8.34 [2.85, ∞]UK–US 129 4 0.85 4.34 0.88 [0.76, 1.00] 3.92 [3.13, 11.71]

Models with structural breaksNetherlands–US 129 1 0.80 3.02 0.81 [0.74, 0.89] 4.10 [2.93, 6.23]Portugal–US 109 1 0.74 2.26 0.74 [0.70, 0.78] 2.82 [2.33, 3.47]UK–US 129 3 0.65 1.61 0.70 [0.56, 0.82] 2.80 [1.62, 4.20]

Notes: We estimate the following models:

qt ¼ cþ αqt�1 þXk

j¼1

ψjΔqt�jþ ut

qt ¼ cþXnb

i¼1

γiDUiþαqt�1 þXk

j¼1

ψjΔqt�jþut

where DUi = 1 if t > TBi for i = 1, 2 and 0 otherwise. TB is the date of the break. nb is the number of breaks. The lag length k is chosen bythe general-to-specific criterion. The break dates are taken fromHegwood and Papell (1998) and Papell and Prodan (2006) for the Lothianand Taylor (1996) and Taylor (2002) data. The half-life estimated by least squares is denoted by HLLS and median-unbiased half-life,estimated from the impulse function, is denoted by HLIRF, MU.

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We conduct Monte Carlo simulations for a variety ofsample sizes and structural breaks and draw two mainconclusions. First, the inclusion of structural breakscauses median-bias to increase substantially compared tothe Andrews (1993) benchmark model without structuralchange. The median-bias increases with both the size andnumber of breaks and decreases with the sample size. Thisbias is also important in the case of restricted structuralchange, with breaks of equal size and opposite sign.Second, neglecting structural change produces an upwardbias in the estimation of persistence. When there is astructural change and the econometrician does not accountfor it, the total of the two biases can be either downward orupward. Downward median-bias dominates for highlypersistent data with smaller breaks, while upward struc-tural change bias dominates for less persistent data withlarger breaks. Our proposed extension of median-unbiasedestimation allows us to account for both structural changeand median-bias.

We analyse convergence of long-run real exchangerates to a possibly changing mean from two differentlong-horizon data sets, Lothian and Taylor (1996) andan extended version of Taylor (2002). The upward biasfrom neglecting structural change dominates the down-ward median-bias for all of the real exchange rates stu-died in this article. Using exactly or approximatelymedian-unbiased estimation as our benchmark, weshow that when structural change is properly accountedfor, median-unbiased half-live estimates decrease andtheir confidence intervals substantially narrow, althoughconvergence is to a shifting mean.

Acknowledgements

We are grateful to Mehmet Caner, Lutz Kilian, DonggyuSul, Rebecca Thornton and to participants at the 20thMeetings of the New Zealand Econometric Study Group,the 2010 AEA Meetings, the Southern EconomicAssociation, the University of Houston, and MasseyUniversity, for helpful comments and discussions.

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Uni

vers

ity]

at 1

0:23

06

Oct

ober

201

4