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Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Why Frequency Matters for Unit Root Testing in

Financial Time Series

H. Peter Boswijk & Franc Klaassen

To cite this article: H. Peter Boswijk & Franc Klaassen (2012) Why Frequency Matters for Unit

Root Testing in Financial Time Series, Journal of Business & Economic Statistics, 30:3, 351-357, DOI: 10.1080/07350015.2011.648858

To link to this article: http://dx.doi.org/10.1080/07350015.2011.648858

Accepted author version posted online: 20 Dec 2011.

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Why Frequency Matters for Unit Root Testing

in Financial Time Series

H. Peter B

OSWIJK

Department of Quantitative Economics, University of Amsterdam, 1018 XE Amsterdam, The Netherlands (h.p.boswijk@uva.nl)

Franc K

LAASSEN

Department of Economics, University of Amsterdam, 1018 XE Amsterdam, The Netherlands (f.j.g.m.klaassen@uva.nl)

It is generally believed that the power of unit root tests is determined only by the time span of observations, not by their sampling frequency. We show that the sampling frequency does matter for stock data displaying fat tails and volatility clustering, such as financial time series. Our claim builds on recent work on unit root testing based on non-Gaussian GARCH-based likelihood functions. Such methods yield power gains in the presence of fat tails and volatility clustering, and the strength of these features increases with the sampling frequency. This is illustrated using local power calculations and an empirical application to real exchange rates.

KEY WORDS: Fat tails; GARCH; Mean reversion; Purchasing-power parity; Sampling frequency; Systematic sampling.

1. INTRODUCTION

Testing purchasing-power parity (PPP) is one of the main applications of unit root analysis. Rejecting a unit root in real exchange rates is evidence that PPP holds (in the long run). Quite often the unit root cannot be rejected. However, this is often not considered to be decisive evidence against PPP, but it is typically explained by the notoriously low power of unit root tests, together with the fact that a tendency toward PPP is so weak that it is not detected by conventional unit root tests. To solve this problem, researchers have tried to increase the power of these tests by obtaining more data, considering a longer time span (of a century or more) or a panel of exchange rates; see Frankel (1986), Frankel and Rose (1996), Lothian and Taylor (1996), and Taylor (2002).

Another way to obtain larger sample sizes is to consider the same time span, but by using data observed at a higher fre-quency. However, it is generally believed that this route will not lead to more power, because the power of unit root tests is mainly affected by time span, and much less by sampling frequency. This result was first derived by Shiller and Perron (1985), and repeated in the influential review paper by Campbell and Perron (1991). At an intuitive level, finding a significant mean reversion requires a realization of a process that indeed does pass its mean quite regularly within the sample; increasing the sampling fre-quency does not change this in-sample mean reversion, whereas a longer history of the time series will demonstrate more in-stances of passing the mean. The theoretical basis is provided by a continuous-record asymptotic argument. For finite samples, Choi and Chung (1995) nevertheless showed that increasing the data frequency can improve power significantly. Still, power de-pends much more on the span than on the frequency, as Perron (1991) demonstrated.

These results refer to systematic sampling, where the low-frequency observations are obtained by skipping intermediate

high-frequency observations. This type of sampling is relevant for stock variables, such as asset prices. Choi (1992) showed that temporal aggregation of flow variables, where the low-frequency observations are defined by cumulating the intermediate high-frequency observations, can have an effect on the power of unit root tests, because of the necessity to account for higher-order dynamics in the aggregated data. However, this is only a finite-sample effect, which vanishes asymptotically, as can be derived from the theoretical results of Chambers (2004,2008).

The present article focuses on stock data and asymptotic power [the working paper version of this article, Boswijk and Klaassen (2010), adds some finite-sample results]. It argues that the effect of systematic sampling on the asymptotic local power of unit root tests is not negligible when high-frequency innova-tions are fat-tailed and display volatility clustering, and when these properties are accounted for in the construction of the unit root tests. Our claim is based on two results. First, recent research by Lucas (1995), Rothenberg and Stock (1997), Seo (1999), Boswijk (2001), and Li, Ling, and McAleer (2002) has demonstrated that when the errors of an autoregressive (AR) process display these typical features of financial data (fat tails and persistent volatility clustering), then (quasi-) likelihood ra-tio tests within a model that takes these effects into account can be considerably more powerful than the conventional least-squares-based Dickey–Fuller tests, which do not explicitly in-corporate these properties.

Second, these typical features tend to be more pronounced in high-frequency (in particular, daily) data than in low-frequency (monthly or quarterly) data. This phenomenon is explained by the analysis of Drost and Nijman (1993), who showed © 2012American Statistical Association Journal of Business & Economic Statistics July 2012, Vol. 30, No. 3 DOI:10.1080/07350015.2011.648858

351

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352 Journal of Business & Economic Statistics, July 2012 that systematic sampling and temporal aggregation of GARCH

(generalized autoregressive conditional heteroscedasticity) pro-cesses decrease both the kurtosis of the unconditional distribu-tion and the persistence in the volatility process.

Combining these two results implies that the possibilities of obtaining unit root tests with higher power increase when one moves from low-frequency to high-frequency observations. In this sense, frequency may matter. In a different model (allowing for regime switching in the mean growth rate of a process), Klaassen (2005) found a similar effect of sampling frequency on the power of a test of the random walk null hypothesis. Hence, short-run features of the data may matter even when testing long-run properties.

The outline of the remainder of this article is as follows. In Section2, we review the most important results about testing for a unit root based on non-Gaussian GARCH likelihood functions. Section3demonstrates the theoretically possible power gain of increasing the sampling frequency. Section4applies the theory to real exchange rates in the post-Bretton Woods era. The final section, Section5, contains some concluding remarks.

2. UNIT ROOT TESTS WITH NON-GAUSSIAN GARCH ERRORS

Unit root testing based on a non-Gaussian but iid likeli-hood was considered by Cox and Llatas (1991), Lucas (1995), and Rothenberg and Stock (1997). Tests based on a Gaussian GARCH likelihood were analyzed by Ling and Li (1998,2003), Seo (1999), Boswijk (2001), and Li et al. (2002),inter alia. A combination of both approaches—that is, unit root inference based on a non-Gaussian GARCH likelihood—can be found in Ling and McAleer (2003). We refer to these articles and the references therein for a full derivation of these results, and only mention the most important aspects here. Moreover, some tech-nical results that are used here are derived in Appendix B of Boswijk and Klaassen (2010).

Consider the AR(1)-GARCH(1,1) model

Xt =δ+γ Xt−1+εt, (1a)

σ_t2 ₌ω₊αε_t2₋₁₊βσ_t2₋₁, (1b)

ηt =

εt

σt ∼

iidf(ηt), (1c)

where f(ηt) is a symmetric (possibly non-Gaussian, possibly

misspecified) density with zero mean and unit variance, soσ_t2

is the conditional variance ofεt. We assume thatα+β <1, so

{εt}has a finite unconditional varianceσε2=ω/(1−α−β).

The unit root hypothesis isH₀_:γ ₌0, which is to be tested against the alternative hypothesisH₁_:γ <0. UnderH₁ _(and provided thatγ >₋2), the process{Xt}t≥1is weakly stationary

with a constant meanµ_{= −}δ/γ. The parameterγ represents the strength of the tendency of the process to revert to this mean. UnderH₀_{, the mean of}_{Xt}t≥1is not defined, and instead, the

process{Xt}t≥1is stationary with a constant mean δ, which

means that{Xt}t≥1displays a linear trend ifδ=0. In this article,

we focus on the case where this trend under the null hypothesis is assumed to be zero, which is a realistic assumption for, for example, real exchange rates and interest rates. Although we do not consider this explicitly, the conclusion of this article

continues to hold in generalizations of (1a)–(1c), allowing for a linear trend under both the null and the alternative hypotheses. Furthermore, the analysis can be generalized to higher-order AR models, as discussed in Appendix A of Boswijk and Klaassen (2010).

Let θ₁₌₍γ , δ)′ _and Z_t ₌₍Xt−1,1)′, and let θ denote the

full parameter vector, containing the AR parameters θ₁_{, the}

GARCH parameters θ₂₌₍ω, α, β)′_{, and possibly additional}

parametersθ₃_{characterizing the distributional shape of}ηt. Since

the conditional density ofXtgiven the past isσt−1f((Xt−

θ′₁Z_t₎_/σ_t_{), the quasi-log-likelihood function has the following} form:

ℓ(θ₎₌

t=1

ℓt(θ)= n

t=1

−1

2logσ

2 t +logf

Xt−θ′1Zt

σt

Letθˆ_{be the quasi-maximum likelihood (QML) estimator,}

ob-tained by maximizingℓ(θ_{). The QML-based}t-statistic forγ ₌0 is given by

tQML=

varQML( ˆγ)

, (2)

where ˆγ is the first component ofθˆ_{, and}_var_QML_{( ˆ}γ) is the first diagonal element of

varQML(θˆ)=

∂2_ℓ₍_θˆ₎

∂θ∂θ′

−1 _n

t=1

∂ℓt(θˆ)

∂θ

∂ℓt(θˆ)

∂θ′

∂2_ℓ₍_θˆ₎

∂θ∂θ′

−1

. (3)

The use of (3) implies that thetstatistic is robust against possible misspecification off.

The asymptotic properties oftQMLare derived from a

second-order Taylor series expansion of the quasi-log-likelihood, and hence, from the first and second derivatives ofℓ(θ_{). Defining the}

score functionψ(η)= −dlogf(η)/dη, the partial derivative of the quasi-log-likelihood with respect toθ₁_{is given by}

∂ℓ(θ₎

∂θ₁ =

t=1

σt Z_tψ

Xt−θ′1Zt

σt

+ 1

2σ2 t

Xt−θ′1Zt

σt

Xt−θ′1Zt

σt −

∂σ_t2 ∂θ₁

(4) where the GARCH model implies that

∂σ2 t

∂θ₁ = −2α

t−1 i=1

βi−1Z_t₋_i₍Xt−i−θ′1Zt−i).

Under the null hypothesis, the score vector (4) (divided byn) is asymptotically equivalent to

t=1

σt

Z_t_{_ψ₍_η_t₎₊_[_ψ₍_η_t₎_η_t₋_1]_V_t_{} =} 1

t=1

Z_t_υ_t_, ₍₅₎

whereVt = −αti−=11βi−1εt−i/σt, andυtis implicitly defined.

Becausef is symmetric, it follows thatψ is an odd function, which in turn implies that{(εt, υt)}t≥1 is a vector martingale

difference sequence with finite variancesσ_ε2andσ_υ2. This leads to the following bivariate functional central limit theorem: as

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n_{→ ∞}, 1

√

⌊sn⌋ t=1

εt υt L −→

σεW(s)

συB(s)

, s_∈[0,1], (6)

where⌊x_⌋denotes the integer part of x,−→L denotes conver-gence in distribution, and [W(s), B(s)]′ _{is a bivariate standard}

Brownian motion with correlationρ₌cov(εt, υt)/(σεσυ).

The sample moment (5) converges in distribution, as n_→

∞, to the stochastic integralσυ 1

0 G(s)dB(s), whereG(s)=

[σεW(s),1]′. Together with a corresponding result for the

sec-ond derivative ofℓ(θ_{) with respect to}θ₁_{, and the block}

diago-nality of the information matrix, these results imply, underH₀ and asn_{→ ∞}, that

tQML L −→ 1 0 ˜

W(s)2ds

−1/2 ₁ 0

W(s)dB(s), (7)

where ˜W(s)=W(s)−01W(u)du.

For an iid Gaussian (quasi-) likelihood, it can be derived that

υt =εt/σε2, soρ=1. In general, however, 0< ρ≤1, where

(as will be shown in the next section) smaller values ofρ are associated with a larger degree of variation in the volatility and with fatter tails in the densityf(η). This implies that the limit-ing null distribution oftQMLdepends on the nuisance parameter

ρ₌corr(W(1), B(1)). This parameterρmay be estimated con-sistently by the sample correlation of ˆεtand ˆυt, and approximate

quantiles of the asymptotic null distributions for a given value ofρmay be obtained by simulating the asymptotic functionals or from the normal approximation of Abadir and Lucas (2000). Under a sequence of local alternativesH_n_:γ ₌c/n, where

c <0 is a noncentrality parameter, the results of Chan and Wei (1987) imply that asn_{→ ∞},

n−1/2X_⌊sn⌋ L

−→σεU(s),

where U(s)=0se

c(s−u)_dW₍_u_{), an Ornstein–Uhlenbeck}

pro-cess, which is the solution to the stochastic differential equation

dU(s)=cU(s)ds₊dW(s). This implies that underH_n_,

tQML L −→ 1 0 ˜

U(s)2ds

−1/2 ₁ 0

U(s)

dB(s)+c

ρU(s)ds

(8) where ˜U(s)=U(s)−01U(u)du.

These results simplify to the asymptotic distribution, under the null and local alternatives, of the Dickey–Fullerttest ˆτµ, if

we imposeα₌β₌0, and choose the standard normal density forf. As mentioned earlier, this impliesυt =εt/σε2 and hence

ρ₌1, so, underH₀_,

ˆ τµ L −→ 1 0 ˜

W(s)2ds

−1/2 ₁ 0

W(s)dW(s), (9) whereas under local alternativesH_n_,

ˆ τµ L −→ 1 0 ˜

U(s)2ds

−1/2 ₁ 0

U(s)[dW(s)+cU(s)ds]. (10) The probability that the right-hand side expressions in (8) and (10) are less than a given quantile of the null distributions in (7)

and (9), respectively, defines the asymptotic local power func-tions of the tests. These provide an approximation to the actual power oftQMLand ˆτµfor finite samples. As the noncentrality

pa-rametercbecomes more negative, the center of the distributions oftQMLand ˆτµwill shift to the left, leading to higher power.

Comparing (10) with (8), we observe two differences. First, the termdW(s) has been replaced by dB(s) in the stochastic integrals, and second, the noncentrality parameterchas been replaced by c/ρ. The former has a relatively minor effect on the local power of the test, but the effective noncentrality pa-rameter c/ρ implies that large power gains of tQML over ˆτµ

are obtained for cases whereρ is relatively small. In the next section, we will show how large these power gains are, and how they are affected by the observation frequency of financial data.

3. SAMPLING FREQUENCY AND LOCAL POWER

In this section, we first analyze, in Section3.1, the effect of sampling frequency on the noncentrality parameterc and the correlation parameterρ. Next, in Section3.2, we quantify the local power gain for realistic parameter values. As indicated in Section 1, we focus on stock data, and hence systematic sampling.

3.1 Sampling Frequency and Model Parameters

Suppose first that the model (1) applies to the high-frequency data{Xt}nt=0. The low-frequency data are defined by {X

(m)

j =

Xmj}njm=0, where nm = ⌊n/m⌋ and the positive integer m

de-notes the sampling period of the low-frequency data. Writing the model as Xt−µ=φ(Xt−1−µ)+εt, whereµ= −δ/γ

andφ₌1+γ, the implied model for the low-frequency ob-servations isX(m)_j ₋µ₌φm₍_X(m)

j−1−µ)+ε (m)

j , whereε (m)

j =

m−1 i=0 φ

i_ε

mj−i, so{ε (m)

j }j≥1 is a serially uncorrelated process.

In other words, we find

X(m)_j ₌δm+γmX (m) j−1+ε

(m) j ,

where the low-frequency mean-reversion parameter satisfies

γm=φm−1=(1+γ)m−1,

andδm= −µγm=δγm/γ. Under the null hypothesis γ =0,

this clearly impliesγm=0, so the low-frequency observations

follow a random walk. Under local alternatives γ ₌c/n, and assuming for convenience that n is a multiple of m so that

nm =n/m, a Taylor series expansion gives

γm=(1+c/n)m−1=

mc n +o(n

−1

)=_nc

m +

o(n−1).

Therefore, up to an asymptotically negligible term, the low-frequency mean-reversion parameterγmhas the same

represen-tation as the high-frequency parameter γ, with the same non-centrality parameterc. This, together with the results in Section 2, immediately implies that the local power of the Dickey– Fuller test based on n high-frequency observations is exactly the same as the local power based on thenmlow-frequency

ob-servations in the same time span (see also Stock1994, p. 2776). This motivates the commonly held view that frequency does not matter.

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354 Journal of Business & Economic Statistics, July 2012 Consider now the effect of sampling frequency on the

corre-lation parameterρ, in the model (1a)–(1c). It will be convenient first to consider the model under the null hypothesis γ ₌0, so ε_j(m) ₌m_i₌−₀1εmj−i. This implies that the temporal

aggre-gation results of Drost and Nijman (1993) forflowdata apply to{εt}nt=1 and{ε

(m) j }

j=1, becauseεt =Xt is the first

differ-ence of a stock variable. Drost and Nijman (1993) showed that if{εt}nt=1 is a GARCH(1,1) process, with unconditional

kurto-sisκε, then{ε (m) j }

j=1 is a weak GARCH(1,1) process, withαm

andβmsatisfyingαm+βm=(α+β)mand with unconditional

kurtosis

κε,m=3+

κε−3

m +6(κε−1)

× [m−1−m(α+β)+(α+β)

m_]_α_[1

−β(α₊β)]

m2₍₁₋_α₋_β₎2₍₁₋_β2₋₂_βα₎ .

This implies that as m increases (and hence we move from high- to low-frequency data), the GARCH persistence parameter

αm+βmdecreases, and the excess kurtosisκε,mdecreases (for

parameter valuesα andβ implying a finite kurtosis). For the individual αm and βm parameters, Drost and Nijman (1993)

showed that βm decreases with m, and that depending on the

kurtosis,αmmay initially increase but then will also decrease

withm.

Under the alternativeγ <0, the above results will not hold exactly, and the implied process is weak ARMA-GARCH of a higher order; however, because we consider local alternatives so thatεtis close toXt, we suspect that an AR(1)-GARCH(1,1)

model will also be adequate for the low-frequency data in these cases.

The results of Lucas (1995) imply that the correlation pa-rameterρdecreases with the unconditional kurtosisκε; because

high-frequency data have a higher kurtosis than low-frequency observations, this implies a lower value ofρ(and hence a larger power potential for unit root tests) for high-frequency data. Simi-larly, Boswijk (2001) showed thatρdecreases with the volatility persistenceα₊β, and in particular with the parameterα, which determines the degree of variation in the volatility. This implies once more that high-frequency data will correspond to a lower value ofρ, and hence a larger power gain potential from the

tQML-based test.

3.2 Local Power Functions

The effects of the model parameters onρ, and ofρon the local power functions cannot be evaluated analytically, although the latter effect has been established numerically by Hansen (1995). Therefore, we evaluate the dependence ofρand the correspond-ing asymptotic local power of the tests by simulation. As the data-generating process (DGP) for the high-frequency data, we consider the AR(1)-GARCH(1,1)-t(ν) model, where the inno-vationsηt follow a standardizedt(ν) distribution. Inspired by

typical parameter values for daily financial returns, we set

α₌0.07, β ₌0.9, ν₌5,

implying that the volatility is persistent (α₊β ₌0.97), and both the conditional and the unconditional distributions have a

high (but finite) kurtosis of

κη =E

η_t4₌3ν−2

ν₋4 =9 and

κε=κη

1−(α₊β)2

1−(α₊β)2₋₍_κ

η−1)α2 =

26.7,

respectively; see He and Ter¨asvirta (1999) for moments of GARCH processes. For the noncentrality parameter, we choose

c_{∈ {}0,₋5,₋10,₋15,₋20,₋25,₋30}. Note that the analysis is invariant toµandω, which only determine the location and the scale ofXt but not its dynamic properties or distributional

shape.

To study the effect of observation frequency, we take a sam-pling period ofm₌20. Thus, the low-frequency observations

{X(m)_j _}j≥0 may be thought of as the end-of-month versions of

{Xt}t≥0, assuming 20 trading days in a month. The results

of Drost and Nijman (1993) cited earlier imply that ifc₌0,

{X(m)_j _}j≥0follows a weak GARCH(1,1) process with

αm=0.16, βm=0.38, κε,m=12.75.

As expected, we observe that both the volatility persis-tenceαm+βm=0.54 and the unconditional kurtosisκε,m are

substantially lower for the low-frequency observations. Al-though the low-frequency data do not follow an exact AR(1)-GARCH(1,1)-t(ν) process, even if c₌0, assuming such a model withν₌5 also for the low-frequency data would yield virtually the same kurtosis, so we will takeν ₌5 to be the pseudo-true value for the low-frequency observations.

Figure 1displays the asymptotic local power functions of the following four QML unit root tests: DF, the Dickey– Fuller test; QML-t, based on an iid t(ν) likelihood (ignor-ing the GARCH(1,1) property); QML-G, based on a Gaussian GARCH(1,1) likelihood (ignoring the conditionalt(ν) distribu-tion); and ML, based on a GARCH(1,1)-t(ν) model. The mo-tivation for studying QML-G and QML-t is to investigate how much power may be gained by taking into account only volatility clustering or fat-tailedness. These tests are based on a misspec-ified likelihood function, which is corrected in the construction of the standard error from (3). The test based on the

GARCH-tlikelihood is actually only a true maximum-likelihood-based test for the high-frequency data; for the monthly observations, the actual DGP will not exactly be an AR(1)-GARCH(1,1)-t(ν) process, so the test in this case is also a QML-based test.

The values ofρfor the high- and low-frequency observations have been obtained by simulating a large number of the errors

εtand “scores”υt, the latter evaluated at the pseudo-true values

of the parameters, and by calculating their sample correlation (we use 1,000,000 observations, after deleting 100 starting val-ues). All numerical results in this section have been obtained using Ox 6 (see Doornik2009). Next, the local power curves have been obtained by discretizing the stochastic integrals (us-ing 2,000 time points), and by simulat(us-ing 500,000 replications of the asymptotic representations of the tests. The differences between the dashed curves and the dotted curve (indicating the local power function of the Dickey–Fuller test) reflect the power gains from using a non-Gaussian and/or GARCH-based likeli-hood for low-frequency observations. The differences between

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Figure 1. Asymptotic local power functions (as function of₋c; 5% nominal level) of four unit root tests for low- and high-frequency data. The online version of this figure is in color.

the solid and the dashed curves reflect the additional power gains from using high-frequency observations instead of low-frequency observations.

The figure shows that using high-frequency data and exploit-ing its features in the test can lead to substantial improvements in power. For example, forc_{= −}10, the ML test based on high-frequency data has a power of about 64%, more than twice the power of the Dickey–Fuller test. We observe that the largest power gains from high-frequency observations are obtained by the ML and QML-G tests. This is also reflected in the values of the correlation parameterρ, which are given by 0.93, 0.92, and 0.90 for the low-frequency versions of QML-t, QML-G, and ML, respectively, and which decrease to 0.89, 0.83, and 0.80 for the corresponding high-frequency based tests.

The local power curves also allow us to compare the effects of sampling frequency and time span on the power of the tests. Con-centrating on the ML test, we have seen that for the high (daily) frequency, a noncentrality parameter ofc_{= −}10 is sufficient to obtain a power of about 64%. When using a low (monthly) frequency, a noncentrality of aboutc_{= −}13 is needed to ob-tain the same power. Suppose thatc_{= −}10 corresponds to 8 years of daily data (n₌2000), with daily mean-reversion pa-rameterγ ₌c/n_{= −}0.005. The corresponding monthly mean-reversion parameter isγ20=(1−0.005)20−1= −0.095, so

about 11 years of monthly observations (nm=136) are needed

to obtain the desired noncentrality parameterc_{= −}13. In other words, in this specific case, 11 years of monthly data contain about the same information on the mean reversion of the process as 8 years of daily data.

In summary, the local power curves illustrate the theoretical predictions: sampling at a higher frequency increases the degree of fat-tailedness and volatility clustering in the data, and hence, the possibility of obtaining more power from tests that take these properties into account. In the next section, we explore whether

this also has consequences for the empirical analysis of real exchange rates.

4. EMPIRICAL APPLICATION

In this section, we apply the unit root tests studied in this article to real exchange rate data in the post-Bretton Woods era (using data from April 1978 through October 2009). We focus on the three real exchange rates between the United States, Germany, and the United Kingdom (for the euro period, we use the fixed EUR/DM exchange rate to convert euro rates to D-mark rates). Daily nominal exchange rates have been converted to real exchange rates using daily producer price indices, constructed by log-linear interpolation from the monthly price indices taken from IMF’s International Financial Statistics. From these daily real exchange rates, we define the end-of-week, end-of-month, and end-of-quarter series by systematic sampling. The question of interest is whether PPP holds in the long run, that is, whether a unit root in real exchange rates can be rejected.

Inspection of the daily exchange rate returns reveals that they display volatility clustering and excess kurtosis, so the theory developed in the previous sections applies.Table 1displays the

p-values of the Dickey–Fuller test ˆτµ, and thetQMLtest based on

(1) withηt ∼iidt(ν), for the three different real exchange rates

and four different frequencies. For ease of comparison, we only consider first-order AR models. For the daily data, this some-times leads to some residual autocorrelation, but accounting for this by including lagged differences hardly affects the results inTable 1. The GARCH-t(ν) model has only been estimated when this model provides a significant improvement over the Gaussian iid model for the disturbances, which is the case for all series at the weekly and daily frequencies, and for two se-ries at the monthly frequency. All results in this section have

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356 Journal of Business & Economic Statistics, July 2012

Table 1. Unit root testp-values and estimated parameters for the real exchange rate returns

Frequency p( ˆτµ) p(tQML) γˆ×102 αˆ αˆ+βˆ νˆ ρˆ

U.S. dollar versus German mark

Quarterly 0.296 ₋ ₋6.327 ₋ ₋ ₋ ₋

Monthly 0.336 ₋ ₋1.944 ₋ ₋ ₋ ₋

Weekly 0.350 0.806 ₋0.189 0.096 0.925 11.17 0.920

Daily 0.350 0.861 ₋0.016 0.055 0.990 5.640 0.841

U.S. dollar versus British pound

Quarterly 0.155 ₋ ₋7.740 ₋ ₋ ₋ ₋

Monthly 0.146 0.809 ₋1.395 0.037 0.987 7.716 0.926

Weekly 0.167 0.853 ₋0.124 0.089 0.952 8.725 0.862

Daily 0.187 0.860 ₋0.014 0.050 0.994 5.909 0.807

German mark versus British pound

Quarterly 0.082 ₋ ₋9.240 ₋ ₋ ₋ ₋

Monthly 0.076 0.061 ₋2.854 0.193 0.906 5.599 0.780

Weekly 0.092 0.020 ₋0.687 0.111 0.972 5.747 0.796

Daily 0.103 0.012 ₋0.127 0.075 0.997 5.277 0.752

NOTE: Results based on the estimation period from April 1978 through October 2009.p(·) denotes thep-value of the argument. The test statistics ˆτµandtQMLare defined in Section2.

γ, ˆα, ˆβ, and ˆνare maximum likelihood estimators in the AR(1)-GARCH(1,1)-t(ν) model, in cases where this model is estimated; in the other cases, ˆγis the least squares estimate. ˆρis the sample correlation between residuals ˆεtand scores ˆυt, where the latter are defined in (5).

been obtained using the GARCH module within PcGive 13; see Doornik and Hendry (2009).

Consider first the p-values of the Dickey–Fuller tests. We observe that for all three series, these p-values display little variation over the different frequencies. None of these tests lead us to conclude that there is a significant mean reversion in the real exchange rates at the 5% significance level, for any sampling frequency. This may be seen as an illustration of Shiller and Perron’s (1985) conclusion that sampling frequency matters very little for the power of the Dickey–Fuller test.

For the two real exchange rates vis-`a-vis the U.S. dollar, we find that using thetQMLtest does not lead to a different outcome:

regardless of the sampling frequency, the unit root in these series cannot be rejected. In fact, the p-values of this test are often higher than the Dickey–Fuller p-values, and generally do not decrease with the sampling frequency. This result holds despite the fact that the conditions for more powerful tests at higher frequencies are satisfied: as the sampling frequency increases, the estimated volatility persistence ˆα₊βˆ increases, and the estimated degrees of freedom ˆνdecreases [as predicted by the analysis of Drost and Nijman (1993)], which together leads to a decrease in the estimated correlation ˆρ.

For the real exchange rate of the German mark vis-`a-vis the British pound, on the other hand, we do observe that the higher power pays out. At the monthly frequency, the tQML test has

a p-value already closer to 5% than the Dickey–Fuller test, and as we move to weekly or daily observations, thep-values drop to about 1%, indicating evidence against the unit root in this series. Note that for this series, the estimated mean-reversion parameter closely follows the theoretical predictions. For example, the daily mean-reversion parameter of −0.0013 would imply a weekly mean-reversion parameter of−0.0065, quite close to the estimated−0.0069, and the same applies to the monthly and quarterly mean-reversion parameters implied by the weekly and monthly estimates, respectively. This highlights that the stronger evidence in favor of mean reversion is not a

consequence of a larger estimated effect, but of a more efficient estimate.

In summary, we conclude that in one out of three real ex-change rates, the use of high-frequency data in combination with a GARCH-t(ν)-based test changes the evidence against the unit root hypothesis from insignificant to significant, as predicted by the theoretical results in the previous sections. An obvious ex-planation of the difference between the mark-pound rate and the other two rates is that the close geographical proximity of the UK and Germany, and their joint membership of the European Union, leads to more effective goods-market arbitrage between these two countries. Another, more technical explanation is that the dollar exchange rates display more prominent swings in the first decade than the intra-European exchange rates. This might indicate that the possible mean-reverting behavior of the dollar-mark and dollar-pound real exchange rates over this period is not adequately described by a linear AR process, but requires, for example, threshold effects or switching regimes. As the models considered here do not allow for such nonlinear effects, this leads to an estimate ofγ that is biased toward zero. Increasing the data frequency does not solve this problem, so it is difficult to find significant evidence of PPP in these cases. The results for the mark-pound real exchange rate, however, demonstrate that for series where the linear AR specification seems more appropriate, the power gains from increasing the frequency are empirically relevant.

5. CONCLUDING REMARKS

The main conclusion of this article is that the common belief, that only time span matters for the power of unit root tests, is incorrect for financial data, where high-frequency observations display properties that may be exploited for obtaining tests with higher power. Clearly, the alternative approaches to obtaining more power, such as longer time series, panel data restrictions, or alternative treatments of the constant and trend are useful as

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well, and could be combined with the approach presented here. Similar power gains from non-Gaussian or GARCH likelihood analysis may be obtained in a multivariate cointegration context; see Boswijk and Lucas (2002), Li, Ling, and Wong (2001), and Seo (2007).

ACKNOWLEDGMENTS

Helpful comments from the editor, an associate editor, and four anonymous referees are gratefully acknowledged.

[Received March 2010. Revised September 2011.]

REFERENCES

Abadir, K. M., and Lucas, A. (2000), “Quantiles for t-Statistics Based on M-Estimators of Unit Roots,”Economics Letters, 67, 131–137. [353] Boswijk, H. P. (2001), “Testing for a Unit Root With Near-Integrated Volatility”

[on-line], Discussion Paper No. 01-077/4, Tinbergen Institute. Available at

http://www.tinbergen.nl/discussionpapers/01077.pdf. [351,352,354] Boswijk, H. P., and Klaassen, F. (2010), “Why Frequency Matters for Unit Root

Testing” [on-line], Working Paper, University of Amsterdam. Available at

http://www1.fee.uva.nl/pp/bin/1186fulltext.pdf. [351,352]

Boswijk, H. P., and Lucas, A. (2002), “Semi-Nonparametric Cointegration Test-ing,”Journal of Econometrics, 108, 253–280. [357]

Campbell, J. Y., and Perron, P. (1991), “Pitfalls and Opportunities: What Macroeconomists Should Know About Unit Roots,” inNBER Macroeco-nomics Annual(Vol. 6), eds. O. J. Blanchard and S. Fisher, Cambridge, MA: The MIT Press. [351]

Chambers, M. J. (2004), “Testing for Unit Roots With Flow Data and Varying Sampling Frequency,”Journal of Econometrics, 119, 1–18. [351] ——— (2008), “Corrigendum to: Testing for Unit Roots With Flow Data

and Varying Sampling Frequency,”Journal of Econometrics, 144, 524– 525. [351]

Chan, N. H., and Wei, C. Z. (1987), “Asymptotic Inference for Nearly Nonsta-tionary AR(1) Processes,”The Annals of Statistics, 15, 1050–1063. [353] Choi, I. (1992), “Effects of Data Aggregation on the Power of Tests for a Unit

Root,”Economics Letters, 40, 397–401. [351]

Choi, I., and Chung, B. S. (1995), “Sampling Frequency and the Power of Tests for a Unit Root: A Simulation Study,”Economics Letters, 49, 131–136. [351]

Cox, D. D., and Llatas, I. (1991), “Maximum Likelihood Type Estimation for Nearly Nonstationary Autoregressive Time Series,”The Annals of Statistics, 19, 1109–1128. [352]

Doornik, J. A. (2009), An Object-Oriented Matrix Programming Language—OxTM_{6, London: Timberlake Consultants Press and Oxford:} www.doornik.com. [354]

Doornik, J. A., and Hendry, D. F. (2009), Econometric Modelling— PcGiveTM13(Vol. III), London: Timberlake Consultants Press. [356]

Drost, F. C., and Nijman, Th. E. (1993), “Temporal Aggregation of GARCH Processes,”Econometrica, 61, 909–927. [351,354,354,356]

Frankel, J. A. (1986), “International Capital Mobility and Crowding Out in the U.S. Economy: Imperfect Integration of Financial Markets or of Goods Markets?,” inHow Open is the U.S. Economy?, ed. R. Hafer, Lexington, MA: Lexington Books. [351]

Frankel, J. A., and Rose, A. K. (1996), “A Panel Project on Purchasing Power Parity: Mean Reversion Within and Between Countries,”Journal of Inter-national Economics, 40, 209–224. [351]

Hansen, B. E. (1995), “Rethinking the Univariate Approach to Unit Root Test-ing: Using Covariates to Increase Power,”Econometric Theory, 11, 1148– 1171. [354]

He, C., and Ter¨asvirta, T. (1999), “Properties of Moments of a Family of GARCH Processes,”Journal of Econometrics, 92, 173–192. [354]

Klaassen, F. (2005), “Long Swings in Exchange Rates: Are They Really in the Data?,” Journal of Business and Economic Statistics, 23, 87– 95. [352]

Li, S., Ling, W. K., and McAleer, M. (2002), “Recent Theoretical Results for Time Series Models With GARCH Errors,”Journal of Economic Surveys, 16, 245–269. [351,352]

Li, S., Ling, W. K., and Wong, H. (2001), “Estimation for Partially Nonstationary Multivariate Autoregressive Models With Conditional Heteroskedasticity,” Biometrika, 88, 1135–1152. [357]

Ling, S., and Li, W. K. (1998), “Limiting Distributions of Maximum Likelihood Estimators for Unstable Autoregressive Moving-Average Time Series With General Autoregressive Heteroskedastic Errors,”The Annals of Statistics, 26, 84–125. [352]

——— (2003), “Asymptotic Inference for Unit Root Processes With GARCH(1,1) Errors,”Econometric Theory, 19, 541–564. [352]

Ling, S., and McAleer, M. (2003), “On Adaptive Estimation in Nonstationary ARMA Models With GARCH Errors,”The Annals of Statistics, 31, 642– 674. [352]

Lothian, J., and Taylor, M. P. (1996), “Real Exchange Rate Behavior: The Recent Float From the Perspective of the Past Two Centuries,”Journal of Political Economy, 104, 488–509. [351]

Lucas, A. (1995), “Unit Root Tests Based onM-Estimators,”Econometric The-ory, 11, 331–346. [351,352,354]

Perron, P. (1991), “Test Consistency With Varying Sampling Frequency,” Econo-metric Theory, 7, 341–368. [351]

Rothenberg, T. J., and Stock, J. H. (1997), “Inference in a Nearly Integrated Au-toregressive Model With Nonnormal Innovations,”Journal of Econometrics, 80, 269–286. [351,352]

Seo, B. (1999), “Distribution Theory for Unit Root Tests With Conditional Heteroskedasticity,”Journal of Econometrics, 91, 113–144. [351,352] ——— (2007), “Asymptotic Distribution of the Cointegrating Vector Estimator

in Error Correction Models With Conditional Heteroskedasticity,”Journal of Econometrics, 137, 68–111. [357]

Shiller, R. J., and Perron, P. (1985), “Testing the Random Walk Hypothesis: Power Versus Frequency of Observation,”Economics Letters, 18, 381– 386. [351]

Stock, J. H. (1994), “Unit Roots, Structural Breaks and Trends,” inHandbook of Econometrics(Vol. IV), eds. R. F. Engle and D. F. McFadden, Amsterdam: North-Holland. [353]

Taylor, A. M. (2002), “A Century of Purchasing-Power Parity,”Review of Eco-nomics and Statistics, 84, 139–150. [351]

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that systematic sampling and temporal aggregation of GARCH (generalized autoregressive conditional heteroscedasticity) pro-cesses decrease both the kurtosis of the unconditional distribu-tion and the persistence in the volatility process.

2. UNIT ROOT TESTS WITH NON-GAUSSIAN GARCH ERRORS

Consider the AR(1)-GARCH(1,1) model

Xt =δ+γ Xt−1+εt, (1a) σ_t2 ₌ω₊αε_t2₋₁₊βσ_t2₋₁, (1b)

ηt = εt σt ∼

iidf(ηt), (1c)

where f(ηt) is a symmetric (possibly non-Gaussian, possibly

misspecified) density with zero mean and unit variance, soσ_t2

is the conditional variance ofεt. We assume thatα+β <1, so

{εt}has a finite unconditional varianceσε2=ω/(1−α−β).

The unit root hypothesis isH₀_:γ ₌0, which is to be tested against the alternative hypothesisH₁_:γ <0. UnderH₁ _(and

provided thatγ >₋2), the process{Xt}t≥1is weakly stationary

with a constant meanµ_{= −}δ/γ. The parameterγ represents the strength of the tendency of the process to revert to this mean. UnderH₀_{, the mean of}_{Xt}t≥1is not defined, and instead, the

process{Xt}t≥1is stationary with a constant mean δ, which

means that{Xt}t≥1displays a linear trend ifδ=0. In this article,

Let θ₁₌₍γ , δ)′ _and Z_t ₌₍Xt−1,1)′, and let θ denote the full parameter vector, containing the AR parameters θ₁_{, the}

GARCH parameters θ₂₌₍ω, α, β)′_{, and possibly additional} parametersθ₃_{characterizing the distributional shape of}ηt. Since

the conditional density ofXtgiven the past isσt−1f((Xt− θ′₁Z_t₎_/σ_t_{), the quasi-log-likelihood function has the following} form:

ℓ(θ₎₌ n

t=1

ℓt(θ)= n

t=1

−1

2logσ

t +logf

Xt−θ′1Zt σt

Letθˆ_{be the quasi-maximum likelihood (QML) estimator,}

ob-tained by maximizingℓ(θ_{). The QML-based}t-statistic forγ ₌0 is given by

tQML=

varQML( ˆγ)

, (2)

where ˆγ is the first component ofθˆ_{, and}_var_QML_{( ˆ}γ) is the first diagonal element of

varQML(θˆ)=

∂2_ℓ₍_θˆ₎

∂θ∂θ′

−1 _n

t=1

∂ℓt(θˆ) ∂θ

∂ℓt(θˆ) ∂θ′

∂2_ℓ₍_θˆ₎

∂θ∂θ′

−1 . (3) The use of (3) implies that thetstatistic is robust against possible misspecification off.

The asymptotic properties oftQMLare derived from a

second-order Taylor series expansion of the quasi-log-likelihood, and hence, from the first and second derivatives ofℓ(θ_{). Defining the}

score functionψ(η)= −dlogf(η)/dη, the partial derivative of the quasi-log-likelihood with respect toθ₁_{is given by}

∂ℓ(θ₎ ∂θ₁ =

t=1

σt

Z_tψ

Xt−θ′1Zt σt

+ 1

2σ2 t

Xt−θ′1Zt σt

Xt−θ′1Zt σt −

∂σ_t2 ∂θ₁

(4) where the GARCH model implies that

∂σ2 t

∂θ₁ = −2α

t−1

i=1

βi−1Z_t₋_i₍Xt−i−θ′1Zt−i).

Under the null hypothesis, the score vector (4) (divided byn) is asymptotically equivalent to

n n

t=1

σt

Z_t_{_ψ₍_η_t₎₊_[_ψ₍_η_t₎_η_t₋_1]_V_t_{} =} 1

n n

t=1

Z_t_υ_t_, ₍₅₎

whereVt = −αti−=11βi−1εt−i/σt, andυtis implicitly defined.

Becausef is symmetric, it follows thatψ is an odd function, which in turn implies that{(εt, υt)}t≥1 is a vector martingale

difference sequence with finite variancesσ_ε2andσ_υ2. This leads to the following bivariate functional central limit theorem: as

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n_{→ ∞}, 1 √

⌊sn⌋

t=1

εt υt

L −→

σεW(s) συB(s)

, s_∈[0,1], (6)

where⌊x_⌋denotes the integer part of x,−→L denotes conver-gence in distribution, and [W(s), B(s)]′ _{is a bivariate standard} Brownian motion with correlationρ₌cov(εt, υt)/(σεσυ).

The sample moment (5) converges in distribution, as n_→

∞, to the stochastic integralσυ

0 G(s)dB(s), whereG(s)=

[σεW(s),1]′. Together with a corresponding result for the

sec-ond derivative ofℓ(θ_{) with respect to}θ₁_{, and the block}

diago-nality of the information matrix, these results imply, underH₀

and asn_{→ ∞}, that

tQML_−→L

W(s)2ds

−1/2 ₁ 0

W(s)dB(s), (7) where ˜W(s)=W(s)−01W(u)du.

For an iid Gaussian (quasi-) likelihood, it can be derived that

υt =εt/σε2, soρ=1. In general, however, 0< ρ≤1, where

ρ₌corr(W(1), B(1)). This parameterρmay be estimated con-sistently by the sample correlation of ˆεtand ˆυt, and approximate

c <0 is a noncentrality parameter, the results of Chan and Wei (1987) imply that asn_{→ ∞},

n−1/2X_⌊sn⌋

−→σεU(s),

where U(s)=0se

c(s−u)_dW₍_u_{), an Ornstein–Uhlenbeck}

pro-cess, which is the solution to the stochastic differential equation

dU(s)=cU(s)ds₊dW(s). This implies that underH_n_, tQML_−→L

U(s)2ds

−1/2 ₁ 0

U(s)

dB(s)+c

ρU(s)ds

(8) where ˜U(s)=U(s)−01U(u)du.

These results simplify to the asymptotic distribution, under the null and local alternatives, of the Dickey–Fullerttest ˆτµ, if

we imposeα₌β₌0, and choose the standard normal density forf. As mentioned earlier, this impliesυt =εt/σε2 and hence ρ₌1, so, underH₀_,

τµ L −→

W(s)2ds

−1/2 ₁ 0

W(s)dW(s), (9) whereas under local alternativesH_n_,

τµ L −→

U(s)2ds

−1/2 ₁ 0

U(s)[dW(s)+cU(s)ds]. (10) The probability that the right-hand side expressions in (8) and (10) are less than a given quantile of the null distributions in (7)

and (9), respectively, defines the asymptotic local power func-tions of the tests. These provide an approximation to the actual power oftQMLand ˆτµfor finite samples. As the noncentrality

pa-rametercbecomes more negative, the center of the distributions oftQMLand ˆτµwill shift to the left, leading to higher power.

are obtained for cases whereρ is relatively small. In the next section, we will show how large these power gains are, and how they are affected by the observation frequency of financial data.

3. SAMPLING FREQUENCY AND LOCAL POWER

3.1 Sampling Frequency and Model Parameters Suppose first that the model (1) applies to the high-frequency data{Xt}nt=0. The low-frequency data are defined by {X

(m)

j =

Xmj}njm=0, where nm = ⌊n/m⌋ and the positive integer m

de-notes the sampling period of the low-frequency data. Writing the model as Xt−µ=φ(Xt−1−µ)+εt, whereµ= −δ/γ

andφ₌1+γ, the implied model for the low-frequency ob-servations isX(_jm)₋µ₌φm₍_X(m)

j−1−µ)+ε

(m)

j , whereε (m)

j =

m−1

i=0 φ

i_ε

mj−i, so{ε

(m)

j }j≥1 is a serially uncorrelated process.

In other words, we find

X(_jm)₌δm+γmX (m)

j−1+ε

(m)

j ,

where the low-frequency mean-reversion parameter satisfies

γm=φm−1=(1+γ)m−1,

andδm= −µγm=δγm/γ. Under the null hypothesis γ =0,

this clearly impliesγm=0, so the low-frequency observations

follow a random walk. Under local alternatives γ ₌c/n, and assuming for convenience that n is a multiple of m so that

nm =n/m, a Taylor series expansion gives γm=(1+c/n)m−1=

mc n +o(n

−1

)=_nc

m +

o(n−1).

Therefore, up to an asymptotically negligible term, the low-frequency mean-reversion parameterγmhas the same

ob-servations in the same time span (see also Stock1994, p. 2776). This motivates the commonly held view that frequency does not matter.

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Consider now the effect of sampling frequency on the corre-lation parameterρ, in the model (1a)–(1c). It will be convenient first to consider the model under the null hypothesis γ ₌0, so ε_j(m) ₌_im₌−₀1εmj−i. This implies that the temporal

aggre-gation results of Drost and Nijman (1993) forflowdata apply to{εt}nt=1 and{ε

(m)

j }

j=1, becauseεt =Xt is the first

differ-ence of a stock variable. Drost and Nijman (1993) showed that if{εt}nt=1 is a GARCH(1,1) process, with unconditional

kurto-sisκε, then{ε (m)

j }

j=1 is a weak GARCH(1,1) process, withαm

andβmsatisfyingαm+βm=(α+β)mand with unconditional

kurtosis

κε,m=3+ κε−3

m +6(κε−1)

× [m−1−m(α+β)+(α+β)

m_]_α_[1

−β(α₊β)]

m2₍₁₋_α₋_β₎2₍₁₋_β2₋₂_βα₎ .

This implies that as m increases (and hence we move from high- to low-frequency data), the GARCH persistence parameter

αm+βmdecreases, and the excess kurtosisκε,mdecreases (for

parameter valuesα andβ implying a finite kurtosis). For the individual αm and βm parameters, Drost and Nijman (1993)

showed that βm decreases with m, and that depending on the

kurtosis,αmmay initially increase but then will also decrease

withm.

model will also be adequate for the low-frequency data in these cases.

The results of Lucas (1995) imply that the correlation pa-rameterρdecreases with the unconditional kurtosisκε; because

tQML-based test.

3.2 Local Power Functions

typical parameter values for daily financial returns, we set

α₌0.07, β ₌0.9, ν₌5,

implying that the volatility is persistent (α₊β ₌0.97), and both the conditional and the unconditional distributions have a

high (but finite) kurtosis of

κη =E

η_t4₌3ν−2

ν₋4 =9 and

κε=κη

1−(α₊β)2

1−(α₊β)2₋₍_κ

η−1)α2 =

26.7,

respectively; see He and Ter¨asvirta (1999) for moments of GARCH processes. For the noncentrality parameter, we choose

c_{∈ {}0,₋5,₋10,₋15,₋20,₋25,₋30}. Note that the analysis is invariant toµandω, which only determine the location and the scale ofXt but not its dynamic properties or distributional

shape.

To study the effect of observation frequency, we take a sam-pling period ofm₌20. Thus, the low-frequency observations {X(_jm)_}j≥0 may be thought of as the end-of-month versions of

{Xt}t≥0, assuming 20 trading days in a month. The results

of Drost and Nijman (1993) cited earlier imply that ifc₌0, {X(_jm)_}j≥0follows a weak GARCH(1,1) process with

αm=0.16, βm=0.38, κε,m=12.75.

As expected, we observe that both the volatility persis-tenceαm+βm=0.54 and the unconditional kurtosisκε,m are

The values ofρfor the high- and low-frequency observations have been obtained by simulating a large number of the errors

εtand “scores”υt, the latter evaluated at the pseudo-true values

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Figure 1. Asymptotic local power functions (as function of₋c; 5% nominal level) of four unit root tests for low- and high-frequency data. The online version of this figure is in color.

the solid and the dashed curves reflect the additional power gains from using high-frequency observations instead of low-frequency observations.

−1_{= −}0.095, so about 11 years of monthly observations (nm=136) are needed

this also has consequences for the empirical analysis of real exchange rates.

4. EMPIRICAL APPLICATION

Inspection of the daily exchange rate returns reveals that they display volatility clustering and excess kurtosis, so the theory developed in the previous sections applies.Table 1displays the p-values of the Dickey–Fuller test ˆτµ, and thetQMLtest based on

(1) withηt ∼iidt(ν), for the three different real exchange rates

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Table 1. Unit root testp-values and estimated parameters for the real exchange rate returns

Frequency p( ˆτµ) p(tQML) γˆ×102 αˆ αˆ+βˆ νˆ ρˆ

U.S. dollar versus German mark

Quarterly 0.296 ₋ ₋6.327 ₋ ₋ ₋ ₋

Monthly 0.336 ₋ ₋1.944 ₋ ₋ ₋ ₋

Weekly 0.350 0.806 ₋0.189 0.096 0.925 11.17 0.920

Daily 0.350 0.861 ₋0.016 0.055 0.990 5.640 0.841

U.S. dollar versus British pound

Quarterly 0.155 ₋ ₋7.740 ₋ ₋ ₋ ₋

Monthly 0.146 0.809 ₋1.395 0.037 0.987 7.716 0.926

Weekly 0.167 0.853 ₋0.124 0.089 0.952 8.725 0.862

Daily 0.187 0.860 ₋0.014 0.050 0.994 5.909 0.807

German mark versus British pound

Quarterly 0.082 ₋ ₋9.240 ₋ ₋ ₋ ₋

Monthly 0.076 0.061 ₋2.854 0.193 0.906 5.599 0.780

Weekly 0.092 0.020 ₋0.687 0.111 0.972 5.747 0.796

Daily 0.103 0.012 ₋0.127 0.075 0.997 5.277 0.752

NOTE: Results based on the estimation period from April 1978 through October 2009.p(·) denotes thep-value of the argument. The test statistics ˆτµandtQMLare defined in Section2.

been obtained using the GARCH module within PcGive 13; see Doornik and Hendry (2009).

For the two real exchange rates vis-`a-vis the U.S. dollar, we find that using thetQMLtest does not lead to a different outcome: regardless of the sampling frequency, the unit root in these series cannot be rejected. In fact, the p-values of this test are often higher than the Dickey–Fuller p-values, and generally do not decrease with the sampling frequency. This result holds despite the fact that the conditions for more powerful tests at higher frequencies are satisfied: as the sampling frequency increases, the estimated volatility persistence ˆα₊βˆ increases, and the estimated degrees of freedom ˆνdecreases [as predicted by the analysis of Drost and Nijman (1993)], which together leads to a decrease in the estimated correlation ˆρ.

For the real exchange rate of the German mark vis-`a-vis the British pound, on the other hand, we do observe that the higher power pays out. At the monthly frequency, the tQML test has a p-value already closer to 5% than the Dickey–Fuller test, and as we move to weekly or daily observations, thep-values drop to about 1%, indicating evidence against the unit root in this series. Note that for this series, the estimated mean-reversion parameter closely follows the theoretical predictions. For example, the daily mean-reversion parameter of −0.0013 would imply a weekly mean-reversion parameter of−0.0065, quite close to the estimated−0.0069, and the same applies to the monthly and quarterly mean-reversion parameters implied by the weekly and monthly estimates, respectively. This highlights that the stronger evidence in favor of mean reversion is not a

consequence of a larger estimated effect, but of a more efficient estimate.

5. CONCLUDING REMARKS

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ACKNOWLEDGMENTS

Helpful comments from the editor, an associate editor, and four anonymous referees are gratefully acknowledged.

[Received March 2010. Revised September 2011.]

REFERENCES

Abadir, K. M., and Lucas, A. (2000), “Quantiles for t-Statistics Based on

M-Estimators of Unit Roots,”Economics Letters, 67, 131–137. [353] Boswijk, H. P. (2001), “Testing for a Unit Root With Near-Integrated Volatility”

[on-line], Discussion Paper No. 01-077/4, Tinbergen Institute. Available at

http://www.tinbergen.nl/discussionpapers/01077.pdf. [351,352,354] Boswijk, H. P., and Klaassen, F. (2010), “Why Frequency Matters for Unit Root

Testing” [on-line], Working Paper, University of Amsterdam. Available at

http://www1.fee.uva.nl/pp/bin/1186fulltext.pdf. [351,352]

Boswijk, H. P., and Lucas, A. (2002), “Semi-Nonparametric Cointegration Test-ing,”Journal of Econometrics, 108, 253–280. [357]

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07350015%2E2011%2E648858

Journal of Business & Economic Statistics

Why Frequency Matters for Unit Root Testing in

Financial Time Series

H. Peter Boswijk & Franc Klaassen

Why Frequency Matters for Unit Root Testing

in Financial Time Series

H. Peter B

Franc K

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