Testing for Stationarity at High Frequency

Transcription

1 esting for Stationarity at High Frequency Bibo Jiang School of Economics Fudan University Ye Lu Department of Economics Indiana University Joon Y. Park Department of Economics Indiana University and Sungkyunkwan University February 2, 25 Abstract his paper considers testing for stationarity using high frequency observations. We introduce a continuous time framework to analyze the frequency dependence of the commonly used statistic to test for the stationarity of a time series or the presence of cointegration in a set of nonstationary time series. Our analysis shows that the statistic diverges up to infinity as the sampling frequency increases, if the bandwidth of longrun variance estimate used in the test is selected within the usual discrete time framework. herefore, we may well be led to falsely reject the null hypothesis of stationarity or the presence of cointegration, if high frequency observations are used. We also show how to use the statistic derived in a discrete time framework to test for the stationarity of a continuous time process. he simulation shows that our test performs satisfactorily and, in particular, much more favorably than other existing tests. JEL Classification: C3, C22 Keywords: continuous time processes, high frequency observations, testing for stationarity, testing for cointegration, two-way asymptotics Shanghai, China jiangbibo@fudan.edu. Bloomington, IN lu33@indiana.edu. Bloomington, IN joon@indiana.edu.

2 . Introduction he test proposed in Kwiatkowski et al. 992, which is frequently referred as KPSS test, has been used extensively to test for stationarity of a time series. It is based on cumulation of squared partial sums of the demeaned and/or detrended series with a correction for autocorrelation using a nonparametric long run variance estimate. he MR/S test of Lo 99 is similar to KPSS test except that it is based on the cumulations of difference between max and min. Another variant of the KPSS test is the indicator KPSS test of de Jong et al. 27, who calculate the same statistic as KPSS but using data that is an indicator of whether the original data point is above or below the median. he indicator KPSS test is shown to be robust to fat tailed distributions. here are other variants of KPSS test existing in the literature, such as Giraitis et al. 23 and Choi 994. he nonparametric longrun variance estimator used in all these tests is well recognized to play a crucial role on the testing performance and the comparison of different estimators, which is essentially the comparison of different bandwidth selection schemes employed in nonparametric estimation, can be found in Lee 996, Choi 994, Hobijn et al. 24, Müller 25 and others. Shin 994 extended KPSS test to test for the presence of cointegration in a set of multiple time series. he asymptotic properties of the test is now well understood in the standard discrete time setup, where we fix the sampling interval and increase the sample size up to infinity. However, the performance of the test on high frequency data is largely unknown. In this paper, we study the properties of the test in a continuous time framework using asymptotics based on the sampling interval δ as well as the sample span. Our study is motivated by the fact that the test results from KPSS test and its variants are highly dependent upon the sampling interval. As illustrated in the paper, in particular, the value of KPSS test increases rapidly up to infinity as δ decreases down to zero, unless we appropriately choose the bandwidth for the longrun variance estimate used in the test. Of course, this implies that we may well be led to falsely reject the stationarity of the presence of cointegration, if the test is implemented using high frequency observations. We develop a novel continuous time framework to effectively analyze the frequency dependence of the test, and show that the test in fact asymptotically diverges to infinity as the sampling frequency increases, if the usual bandwidth choice is made within the discrete time framework. At high frequencies, KPSS test behaves as expected only when the bandwidth for the longrun variance estimate used in the test is appropriately chosen. If so, we may use it to test for the stationarity of a continuous time process or for the presence of cointegration in a set of nonstationary continuous time processes. In fact, KPSS test using a proper 2

3 continuous time scheme in the choice of bandwidth for the longrun variance estimator is widely applicable, since it does not make any specific assumptions on the structures of underlying continuous time processes. his is an important contribution to the literature of testing stationarity and cointegration of a continuous time process, where not much research has been done. Only recently, Bandi and Corradi 24 and Shin 2 each proposes a nonparametric testing approach. he method proposed in Bandi and Corradi 24 uses nonstationarity as the null hypothesis and is only applicable on nonstationary diffusions, an important but special class of continuous time nonstationary processes. As a test of stationarity for continuous time process, KPSS test is more comparable to that of Shin 2. he former, however, appears to have substantially better discriminatory powers than the latter. If only stationary processes are involved, the asymptotics of KPSS test used as a test for the stationarity of a continuous time process are exactly the same as its asymptotics in discrete time setup. In the presence of nonstationary continuous time processes, however, the relevant asymptotics of KPSS test diverge sharply from those in discrete time. In continuoustime, wemayallow foramuchmoregeneralclassofprocesses,whichyieldsavery diverse class of limit distributions. his is in contrast with the discrete time nonstationary asymptotics that always yield Brownian motion as the limit process. As a result, KPSS test for cointegration has limit distribution that is not free of nuisance parameters. Rather, it involves the limit processes of underlying continuous time nonstationary processes. he standard critical values tabulated in Shin 994 are therefore not applicable. o avoid this non-invariance of limit distributions, we propose to use a subsampling method using a modified KPSS test that does not involve the longrun variance estimate. he subsample test appears to work satisfactorily according to our simulations. he rest of the paper is organized as follows. In Section 2, as background and preliminaries, we introduce KPSS test with some commonly used bandwidth selection rules, behaviors of KPSS test for different choices of bandwidth and sampling intervals and some motivational empirical illustrations. Section 3 presents the continuous time version of KPSS test, introduces continuous time models and assumptions and presents asymptotics in continuous time. Subsequently, we analyze the frequency dependence of KPSS test in Section 4. Section 5 provides the formal asymptotics of KPSS test in our asymptotic setup within a continuous time framework. he subsample test for stationarity is considered in Section 6. Section 7 reports the simulation results. Some concluding remarks follow in Section 8. All detailed mathematical proofs are provided in Appendix. A word on notation. he standard notations such as p and d are used extensively to denote the convergence in probability and in distribution, respectively. Also, we write 3

4 P p Q to signify P Q+o p. However, P Q just implies that we approximate P by Q, and it does not have any precise mathematical meaning in regards to the proximity between P and Q. 2. Background and Preliminaries 2.. KPSS est o test for the stationarity of a time series, we routinely employ the test developed by Kwiatkowski et al. 992, which is commonly referred to as the KPSS test. For a given time series u i, the test relies on the statistic defined as λ n n 2 ω 2 n n i j u j 2, 2. where n is the sample size, and ω 2 n is the longrun variance estimate of u i given by ω 2 n j n j K γ n j, 2.2 b n where K is a kernel function and γ n is the sample autocovariance function given by γ n j n i u iu i j. he bandwidth parameter b n is a numerical sequence satisfying b n and b n /n as n. It plays a very important role in our subsequent analysis, and will be specified more explicitly below. In the paper, we also consider the KPSS test applied with the fitted residuals from regressions y i α+u i and y i α+βx i +u i, which we write as u ni y i n u ni y i n y i u i n n u i 2.3 n n y i x i xi 2 x i u i n u i x i n x 2 i x i 2.4 respectively. If based on u ni in 2.3 in place of u i, the statistic λ n can be used to test for the stationarity of a time series y i, while allowing it to have nonzero mean. Moreover, in case that both time series y i and x i are nonstationary with unit roots, we may use the statistic λ n with u i replaced by u ni in 2.4 to test for the presence of cointegration Here and elsewhere in the paper, we assume the summation in the definition of γ n runs only over the range where both indices i and i k are between and n. 4

5 between two nonstationary time series y i and x i. he KPSS test in this context was studied earlier by Shin 994. In what follows, all these different versions of the test, which we simply call the KPSS test, will be analyzed within a single framework. 2 For the kernel function K in 2.2, we make the following assumption. Assumption KF We assume that i K is symmetric with K and ιk 2 <, and it is continuous at and at all but a finite number of other points, and ii the characteristic exponent of K defined as the maximum of nonnegative integer k for which lim x Kx/ x k exists. he conditions in Assumption KF are standard and not stringent, and they are satisfied by virtually all kernels used in practical applications. In what follows, we denote by r the characteristic exponent of K, and define π r lim x Kx x r. For many commonly used kernel functions, the characteristic exponent is either r or 2. he reader is referred to, e.g., Andrews 99 for more detailed discussions on the longrun variance estimation. he choice of bandwidth b n in 2.2 plays a central role in our subsequent asymptotic analysis. As shown in, e.g., Andrews 99, for stationary time series u i, the optimal bandwidth b n minimizing the asymptotic mean squared error of the longrun variance estimator is given by b n c rn /2r+, where /2r+ c r rπr ιk 2 θ2 r 2.5 with θ r j j r γj j γj. he constants r,π r and ιk 2 are all fully determined by the choice of kernel function K. However, the constant θ r depends upon the unknown autocovariance function γj of the underlying time series u i. We consider three widely used selection schemes for the bandwidth parameter b n. i he scheme, called the rule of thumb R, sets b n cn a for some arbitrarily chosen constants c > and < a <. 2 hough we do not explicitly consider in the paper to save space, our subsequent theory is also applicable for the test in the presence of linear time trend with some obvious modifications. 5

6 he other two schemes considered in the paper set b n c rn /2r+, where c r is defined as in 2.5 with an estimate θ r 2 of θr. 2 ii he scheme, called the semiparametric method SP, assumes that u i follows the AR with the autoregressive coefficient ρ, in which case we have θ 2 4ρ 2 ρ 2 +ρ 2 and θ 2 2 4ρ2 ρ 4, and therefore, we may obtain the estimate ˆθ r 2 of θ2 r from the estimated AR coefficient ˆρ in the AR regression of u i. his was proposed and analyzed earlier by Andrews 99. iii he scheme, called the nonparametric method NP, uses a nonparametric estimate θ 2 r j m j r γ n j j m γ nj of θ 2 r with m cna for some arbitrarily chosen constant c > and < a <. his was considered earlier in Newey and West 994. In the subsequent development of our theory, the schemes introduced here will be simply referred to as R, SP and NP respectively Bandwidth Choice and Sampling Interval It is well known that the performance of the KPSS test depends crucially on the choice of bandwidth b n. Properties of the test employing different bandwidth selection schemes are thoroughly investigated in the literature of testing stationarity of discrete time processes. When a process is stationary but strongly autocorrelated, the test using R suffers from size distortion, see simulation results in Kwiatkowski et al. 992, Caner and Kilian 2 and others. Lee 996 conducted a simulation study to compare NP, SP and pre-whitening procedure by Andrews and Monahan 992. he results indicate that the test using SP has good size property but suffers from significant power loss. he inconsistency of the test using SP or pre-whitening procedure is also discussed in Choi 994. NP, on the other hand, is shown to be consistent, see Hobijn et al. 24. Müller 25 also investigates the size and power properties of stationarity tests when the underlying processes are strongly autocorrelated in a local-to-unity asymptotic framework. He points out that stationarity tests either fail to control size or are inconsistent against a nonstationary process, depending on which bandwidth selection approach is employed in the long-run variance estimation. Moreover, Müller 25 and others also noticed that the performance of the KPSS 6

7 Fig.. Stationarity est on Forward Premium of US/UK Exchange Rates and -Bill Rates.2 Month Forward Premium Data Plot 6 Stationarity est for Forward Premium est Statistic CFN AND NW Sampling Interval 2 3 Month Bill Rates Data Plot 4 Stationarity est for Bill Rates 5 5 est Statistic CFN AND NW Sampling Interval Note: he solid horizontal line represents the 5% critical value given in Kwiatkowski et al he sampling interval δ ranges from.4 daily frequency to.25 quarterly frequency. test varies widely when sampling frequency changes. In the following section, we use tow applications to illustrate this sampling frequency dependency problem Empirical Illustrations In the first application, we apply the stationarity test on -month forward premium of US/UK exchange rate and 3-month US treasury bill rate, respectively. he series of forward premiumcoverstheperiodfromjanuary2, 979toApril6, 24andtheseriesof-billrate covers the period from January 4, 97 to March 3, 24. he test statistic is calculated using data collected at various frequencies from quarterly to daily. In Figure 3, the upper left panel presents the log transformed forward premium at daily frequency and the bottom left panel plots the daily -bill rates. We present the calculated test statistic verses sampling frequency on the right two panels. We can see that the test statistic employing SP scheme is stable across different frequencies for both series. However, the test statistic calculated using the other two schemes, R and NP, varies as sampling frequency increases. More specifically, the test statistic does not vary much from quarterly δ.25 to monthly δ.8, but increases substantially as sampling frequency further increases. It indicates 7

8 Fig. 2. Cointegration est on Spot and Forward US/UK Exchange Rates, US -bill and -bond Rates 2.5 Forward and Spot US/UK Exchange Rates Forward Rate Spot Rate Cointegration est of Forward and Spot Exchange Rates 6 CFN AND 5 NW 2.5 est Statistic Sampling Interval Long and Short Interest Rates Year Bond Rate 3 Month Bill Rate est Statistic Cointegration est of Long and Short Interest Rates 6 CFN AND 5 NW Sampling Interval Note: he solid horizontal line represents the 5% critical value given in Shin 994. he sampling interval δ ranges from.4 daily frequency to.25 quarterly frequency. that the test statistic calculated using R or NP is sensitive to data frequency, especially at high frequency region. In the second application, we apply the cointegration test on testing the cointegrating relationship between spot and -month forward US/UK exchange rates and between US 3-month treasury bill and -year treasury bond rates. he two series of exchange rates cover from January 2, 979 to April 6, 24 and the time span for the interest rates is from January 2, 962 to March 3, 24. Again, the test statistic is calculated using data at frequencies varying from quarterly to daily. he left upper and lower panels of Figure 4 present the two log-transform exchange rate series and the two interest rate series, respectively, both at daily frequency. he calculated test statistics are plotted again in the right two panels. Figure 4 reveals similar information as Figure 3. Although it is crystal clear from data evolution path that the two pairs of series are both cointegrated, the test employing R or NP increases as data frequency increases, which indicates that the cointegration test using these two schemes is sensitive to data frequency, especially when data are collected at weekly or higher frequency. Again, the test employing SP scheme does not show this pattern. 8

9 From these two applications, we can see that sampling frequency does play a role on the test employing two conventional bandwidth selection schemes, R and NP. he sampling frequency does not seem to affect the stationarity test and cointegration test using SP scheme. 3. esting for Stationarity in Continuous ime 3.. Stationarity est in Continuous ime o study the stationarity test at high frequency, we consider the test for stationarity of a continuous time process based on the continuous time version of the KPSS test introduced in 2.. Let U U t be a continuous time process observed for t, and define the test statistic Λ as Λ 2 2 t 2 U s ds dt 3. similarlyasin2., where 2 isanestimator forthelongrunvariance 2 ofu. Analogously as in 3.2, it is given by 2 s s K B Γ sds, 3.2 where K is the kernel function, B is the bandwidth parameter such that B and B / as, and Γ s U tu t s dt is the sample autocovariance function of U. he estimator 2 for the longrun variance of stationary process U in continuous time has been studied extensively in Lu and Park 24. In particular, the optimal bandwidth B minimizing the asymptotic mean squared error of the longrun variance estimator is given by B c r /2r+, where c r /2r+ rπr ιk 2 Θ2 r 3.3 with Θ r s r Γsds Γsds correspondingly as in 2.5. As discussed, the parameter Θ r is unknown and has to be estimated. In parallel to our earlier choice of bandwidth, we consider three choices of B. i he scheme, called CR, sets B c a for some constant c > and < a <. 9

10 ii he scheme, called CSP, assumes that U is generated as du t κu t dt+υ t dw t +dj t, where υ t is a general stochastic volatility, W is the standard Brownian motion and J is the jump process defined by dj t xndx,dt from the random Poisson measure N, in which case we have Θ r r 2 /κ r. herefore, we may obtain the estimate ˆΘ r of Θ r from an estimate ˆκ of κ, which we may use to set B ĉ r /2r+, where ĉ r with ˆΘ r r 2 /ˆκ r. is given as in 3.3 iii he scheme, called CNP, sets B c r /2r+, where c r is given as in 3.3 with Θ r s S s r Γ sds/ s S Γ sds, S c a for some arbitrarily chosen constants c > and < a <. he reader is referred to Lu and Park 24 for a more detailed discussion on the choice of bandwidth B. Note that CR, CSP and CNP can be regarded respectively as the continuous time analogues of R, CP and NP Continuous ime Models and Assumptions We consider two assumptions on the underlying process U introduced in 4.. In what follows, we denote by D[,] the space of cadlag functions on [,] endowed with the Skorohod topology. Assumption S For theprocessu Ut on [,] definedas Ut /2 t U s ds, we assume that U U in D[,] as, where U is Brownian motion with variance 2 given by 2 lim 2 E U tdt >. Moreover, assume U2 t dt p σ 2 > and [U] p τ 2 > as. Assumption NS For the process U U t on [,] defined as U t c U t with some normalizing sequence c such that c as, we assume that U U in D[,] as, where U is a nondegenerate stochastic process on [,]. Assumptions S and NS will be used as the basic regularity conditions respectively for stationary and nonstationary U in the subsequent development of our theory. Assumption S is satisfied for a broad class of zero mean stationary processes. It assumes that U satisfies the conventional invariance principle in continuous time, which corresponds to the same type of invariance principle for discrete stationary time series that we routinely invoke in the statistical analysis of models with unit roots and cointegration. Likewise,

11 Assumption NS just requires that under an appropriate normalization U has a well defined limit distribution, which holds for a very general class of nonstationary processes. As is well known, all discrete nonstationary time series of unit root type satisfy this condition if we embed them into continuous time processes taking constant values between their observation intervals. he reader is referred to Park 24 for more related discussions. It is possible to replace the high-level assumptions in Assumptions S and NS with more primitive conditions, if we let the underlying continuous time process U in 4. be given by some function of continuous time markov process. More explicitly, we let U be defined as U t φv t 3.4 for each t, where V V t is a markov process on state space D and φ is a real-valued function defined on D. All markov processes we consider in the paper are homogeneous. A homogeneous markov process is specified typically by its transition P t or its infinitesmal generator A lim t P t /t, which are both operators defined on appropriate classes of functions on D. he reader is referred to, e.g., Chapter III of Revuz and Yor 999 for more detailed discussions on markov processes. he process U defined as in 3.4 itself may or may not be a markov process. See, e.g., Exercise.7 in Chapter III of Revuz and Yor 999 to see when U itself becomes a markov process. For any Harris recurrent markov process V with transition P t, there exists a unique up to constant multiples invariance measure m such that m is σ-finite and mp t m for all t. A Harris recurrent markov process V with invariant measure m is called positive recurrent or ergodic if md <, null recurrent if md. A markov process V becomes stationary if it is positive recurrent, and nonstationary if it is null recurrent. he marginal distribution of a positive recurrent process V is time invariant and given by m/md, whereas a null recurrent process V does not have such a time invariant marginal distribution. Bhattacharya 982 shows that the invariance principle holds for a wide class of U generated as in 3.4 from positive recurrent markov process V. In particular, if V is ρ- mixing and U is generated from V with φ L 2 D,dm such that mφ, 3 the invariance principleis shown to holdfor U. Ontheother hand, thelimit theory for additive functionals of U generated as in 3.4 from null recurrent markov process V is well developed for the case that φ is m-integrable, i.e., mφ <, for the details of which the reader is referred to, e.g., Höpfner and Löcherbach hroughout the paper, we denote by mf the integral of a function f with respect to m on D, whenever it exists.

12 If a markov process V is given more explicitly as a diffusion by dv t µv t dt+σv t dw t, 3.5 where µ and σ are real-valued functions on D, called drift and diffusion functions respectively, and W is standard Brownian motion, we may obtain more explicit results regarding the conditions in Assumptions S and NS in terms of the scale function s and speed density m of V. he derivative s of scale function of a diffusion is defined as x y s 2µz x exp σ 2 z dz dy w for any w in the interior of D. Note that the scale function s is defined uniquely only up to increasing affine transformations, i.e., as+b with a > and < b <, since w may be chosen arbitrarily in the definition of s. If µ and V becomes driftless, s reduces to identity, in which case we say that V is in natural scale. It follows immediately from Ito s formula that sv is driftless and in natural scale, since s is a solution to the differential equation µs +σ 2 s /2. If a diffusion V is defined with µ and σ such that σ 2 x > and both /σ 2 x and µx /σ 2 x are locally integrable at every x D x,x, as is virtually always the case in practical applications, then V is recurrent if and only if sx and sx. See, e.g., Proposition 5.22 of Karatzas and Shreve 99. he speed density m of a diffusion is defined as mx w σ 2 s x. his is indeed the density with respect to Lebesgue measure of the markov invariant measure of a diffusion, which is often called the speed measure. 4 herefore, a diffusion V with speed measure m is positive recurrent if md <, in which case V becomes stationary with marginal distribution given by density m/md, and null recurrent if md, in which case V becomes nonstationary with marginal distribution changing over time. he invariance principle in Assumption S holds widely for U generated as in 3.4 from a positive recurrent diffusion V with φ such that mφ. Indeed, Van Der Vaart et al. 25 shows that the invariance principle holds for U in this case, provided its longrun variance given by 2 4mD D x 2 φymdy s xdx is finite. Furthermore, Kim and Park 24 show that Assumption NS holds for general 4 Following the usual convention, we use m to denote both the speed density and speed measure of a diffusion. 2

13 U defined from null recurrent diffusion V in natural scale with regularly varying φ. If, for general null recurrent diffusion V in natural scale, we define V as Vt λ V t with λ given by λ 2 mλ, then we have V d V, where V is a skew Bessel process in natural scale. See, e.g., Watanabe 995 for detailed discussions on skew Bessel processes. However, it follows that U t φλ φv t φλ φλ V φλ t φv t d φv t, and Assumption NS holds for U with normalizing sequence c φλ and limit process U φv. he reader is referred to Kim and Park 24 for the rigorous development of the asymptotics discussed here Asymptotics in Continuous ime he asymptotics for the continuous time version of the KPSS test in 3. may be easily deduced from our continuous time models and assumptions introduced above. For stationary U, we consider three choices of B, assuming B / p, and provide the asymptotics of Λ for each of the three choices. Lemma 3.. Let Assumption KF hold. Under Assumption S, we have a 2 p 2, if B p b 2 p Ks/ BΓsds, if B p B c B 2 p ιkσ 2, if B p as. Proposition 3.2. Let Assumption KF hold. Under Assumption S, we have a Λ d W2 t dt, if B p b Λ d 2 / Ks/ BΓsds W2 t dt, if B p B c B Λ p 2 / ιkσ 2 W2 t dt, if B p as. Under Assumption S, all three schemes, CR, CSP and CNP, of bandwidth choice we consider in continuous time yield B. If B p, the statistic Λ has a well defined limit distribution free of nuisance parameter as shown in Part a of Proposition 3

14 3.2, analogously as in the discrete time asymptotics. Our continuous time asymptotics in Parts b and c of Proposition 3.2 are also similar to those in discrete time in cases B p B and B p, and we will use them to analyze the effect of sampling frequency on the KPSS test in the next section. FornonstationaryU, weconsidertwochoicesofb andpresenttheasymptoticsofλ for each of the two choices. Below we let U is the limit process of U and Γ s U t U t sdt for s. Lemma 3.3. Let Assumption KF hold. Under Assumption NS, we have a c 2 B 2 p ιk U 2 t dt, if B / p b c 2 2 p as. Ks/B Γ sds, if B / d B a.s. Proposition 3.4. Let Assumptions KF hold. Under Assumption NS, we have t 2dt / a B /Λ d U sds ιk U 2 t dt, if B / b Λ d t U s ds 2dt / Ks/B Γ sds, if B / d B a.s. as. Under Assumption NS, we have B / p for CR and CNP, and B / d B with some B a.s. for CSP. his will be shown below. For CR, it is easy to see that B /. For CNP and CSP, we have Lemma 3.5. Let Assumptions KF hold. Under Assumption NS, we have a +2ar/+2r B p rπ 2 r c 2r/ +r 2 ιk 2 /+2r, if CNP is used b B d r 3 πr/ιk 3 2 /+2r / 2r/+2r, U 2 t dt U t dut if CSP is used as. We may now easily obtain the asymptotics of Λ for all three schemes CR, CSP and CNP. he asymptotics of Λ are given for CR and CNP by Part a of Proposition 3.4, and for CSP by Part b of Proposition 3.4 with B given by Part b of Lemma

15 4. Analyzing Frequency Dependence of KPSS est 4.. Continuous ime Approximation o study how sampling frequency affects the test statistic λ n in 2. calculated using high frequency observations, we conduct asymptotic analysis of the test statistic in continuous time framework. For our analysis, we assume that the samples u i are collected from the underlying continuous time process U U t at sampling interval δ over time span, nδ, i.e., u i U iδ 4. for i,...,n. Our asymptotic analysis in the paper relies on δ and. In our asymptotics, it is allowed that δ and simultaneously. However, we let δ sufficiently fast relative to. his is to obtain the asymptotics more relevant for applications on high frequency observations. Note that in both cases, we have n, as in the usual asymptotics in discrete time framework. In this section, we assume Assumption CA Define δ, U sup s,t sup t s δ U t U s, and ι U sup t U t. Assume 3 ι U δ, U p. his is a technical assumption which insures that the test statistic given in 2. has a continuous time approximation as δ with fixed or. he assumption is satisfied as long as the sampling interval δ goes to fast enough relative to the increase of time span. In case the time series u i is generated from a continuous time process as in 4., we may well expect that the conventional KPSS test for stationarity in 2. becomes closely related to its continuous time version introduced in 3.. In fact, we have n i δ 3 j u j 2 n i δ δu j j 2 t 2 U s ds dt 4.2 for δ small enough compared with. Moreover, we have γ n j n n u i u i j n δu iδ U iδ jδ U s U s jδ ds Γ jδ uniformly in j,...,n, if δ is sufficiently small relative to, and therefore, it follows 5

16 that δ j K j n b n γ n j jδ δk jδ s Γ jδ K Γ sds 4.3 b n δ s B n,δ with B n,δ b n δ 4.4 as long as δ is small enough for. with Let Λ n,δ 2n,δ t 2 2 2n,δ U s ds dt 4.5 s s K B n,δ Γ sds, 4.6 correspondingly as Λ defined with 2 in 3. and 3.2, where B n,δ is introduced in 4.4. hen it is well expected from 4.2 and 4.3 that Lemma 4.. Let Assumptions KF and CA hold. We have λ n p Λ n,δ as δ and n. It is now clear from Lemma 4. that the effect of δ on the KPSS test is entirely determined byhow B n,δ b n δ dependsuponnandδ. Inparticular, ifwechooseb n suchthat B n,δ b n δ depends only upon asymptotically as δ and, then λ n p Λ and the KPSS test will no longer be dependent upon δ at least asymptotically Dependence of Bandwidth on Sampling Frequency As shown above, the frequency dependency of the KPSS test is mainly due to the dependency of B n,δ b n δ on δ. herefore, it will be useful to write B n,δ explicitly as a function of δ and for three schemes, R, SP and NP, of bandwidth choices that are commonly used. For R, we have B n,δ cn a δ cδ a a for some c > and < a <. For SP and NP, Lu and Park 24 establish under Assumption S that Lemma 4.2. Let Assumptions KF and CA hold. Under Assumption S, we have /+2rδ a B n,δ p rπrc 2 2r/ +r 2 ιk 2 2 ar/+2r +2ar/+2r, if NP is used 6

17 b B n,δ p 2r 6 π 2 rσ 8/ ιk 2 τ 8 /2r+ /+2r, if SP is used as δ and. Of course, the asymptotic behaviors of B n,δ under schemes SP and NP would be totally different if U is nonstationary. hey are given below. Lemma 4.3. Let Assumptions KF and CA hold. Under Assumption NS, we have a B n,δ p rπ 2 rc 2r/ +r 2 ιk 2 /+2rδ 2 ar/+2r +2ar/+2r, if NP is used b B n,δ p r 3 πr 2 ιk 2 as δ and. U 2 t dt / 2 /+2r U t dut, if SP is used From Lemma 4.3, we can see that under Assumption NS, sampling interval has no effect on the test statistic if SP is used. However, if R or NP is used, B n,δ / p and B n,δ is a function of δ as well as. In particular, B n,δ p as δ. i.e., for any given, 2n,δ decays to as δ. Hence, the test statistic diverges as δ even when sampling span is fixed. 5. Asymptotics of KPSS est 5.. Asymptotics of Stationarity est In this section, we extend the results derived in Section 3 to test for statistionarity of Y, with Y t α+u t using the fitted residuals from regression y i α+u i, where U t satisfies Assumption S under the null hypothesis or Assumption NS under the alternative. Notice that the residulas u ni u i n n u i δu i nδ n δu i U t U t dt. We derive the primary asymptotics of the test as: Lemma 5.. Let Assumptions KF and CA hold. We have λ n 2 t U s U ds 2dt K s S Γ sds +o p, 5. 7

18 Motivated by the theoretical results in previous sections, we consider choosing bandwidth using b n B /δ, such that B and B / as. he asymptotics of the test statistic can then be derived. heorem 5.2. Let Assumptions KF and CA hold and. Under Assumption S, we have λ n d W t tw 2 dt. As in discrete time framework, the test using residuals from a regression with only constant converges to a distribution given as an integral of Brownian bridge. herefore the critical values provided in discrete time are still valid. heorem 5.3. Let Assumptions KF and CA hold and.. Under Assumption NS, we have a B λ n d t U s U ds2 dt Ksds U t U 2 dt, if B / as b λ n d t U s U ds2 dt Ks/B Γ sds, if B / B a.s., where U U t dt and Γ s Ut U Ut s U dt. Lemma 5.4. Let Assumptions KF and CA hold and. Under Assumption NS, we have [ a +2κr +2r B p c 2r rkr 2 +r 2 K 2 sds ] /2r+, if CNP is used b B r 3 Kr 2 /2r+ 2r/2r+ d K sds 2 U t U 2 dt, if CSP is used. U t U dut Since CNP and CR satisfies the condition that B /, the asymptotics of the corresponding test under Assumption NS is given by heorem 5.3 a. On the other hand, the asymptotics of the test using CSP under the alternative is given by heorem 5.3 b. It is obvious that the test using CNP or CR is consistent, while the test using CSP is not Asymptotics of Residual Based Cointegration est One natural extension of the stationarity test is to use it as a residual based cointegration test for testing cointegration among related time series. In this section, we analyze the asymmptotics of the cointegration test statistic obtained from high frequency observations of two nonstationary continuous time processes. 8

19 5.2. Model and Assumptions We make the following assumptions on the underlying continuous time processes Y t and X t : Assumption 5.. For the processes Y Yt and X Xt on [,] defined respectively as Yt a Y t and Xt b X t, with some normalizing sequences a and b such that a, b as, we assume that Y Y and X X jointly as, where both Y and X are nondegenerate stochastic processes on [,]. Assumption C Y t and X t satisfy Assumption 5. with a b, and there exists someβ such that continuous time processu t definedas U t Y t βx t satisfies Assumption S and U t is independent of X t. Assumption NC- Y t and X t satisfy Assumption 5. and a b. Assumption NC-2 Y t and X t satisfy Assumption 5. with a b. For any value of β, continuous time process U t defined as U t Y t βx t satisfies Assumption NS. Assumption C requires that Y t and X t are at the same level of nonstationarity and some linear combination of the two processes is stationary. he level of nonstationarity of a continuous nonstationary process is measured by its normalizing sequency a or b. In discrete time framework, nonstationarity is usually represented as a unit root process, which converges to Brownian Motion in continuous time framework. he normalizing sequence for Brownian motion is. If Assumption NC- holds, Y t and X t are of different levels of nonstationarity, model Y t βx t + U t is meaningful only when β, i.e. Y U. Under Assumption NC-2, Y t and X t are of the same level of nonstationarity and any linear combination of the two processes can at most reduce some persistency of component processes without completely eliminating nonstationarity Continuous ime Approximation As in the asymptotic analysis of the stationarity test, we first derive the primary asymptotics for the residual based cointegration test. Note that, the residuals u ni u i u n n j x j x n u j n j x j x n 2 x i x n. We derive the continuous time approximation of the covariance matrix of u ni in the following lemma, which will be used in deriving the continuous time approximation of the 9

20 test. Lemma 5.5. Under Assumption CA, we have where R s D 2 for < s <, with D Γ n j p Γ jδ+r jδ, for j n X t X X t s X dt 2D Xt X U tdt Xt X 2 dt. X t X U t s U dt 5, he lemma states that when sampling interval δ is small enough relative to time span, the sample auto-covariance Γ n j j n of the fitted residual u ni can be approximated as the summation of the corresponding auto-covariance Γ kδ of the underlying process U t and R kδ, a component resulting from estimation error of β. In case that a b, i.e., Y and X are of different levels of nonstationarity, we have β, hence U Y and the continuous approximation of the estimation error becomes D Xt X Y tdt Xt X 2 dt. Using Lemma 5.5, we can derive the primary asymptotics of the test as: Lemma 5.6. Let Assumptions KF and CA hold. We have λ n p where R t D t X s X ds. 2 t U s U ds R t 2dt K s S Γ s+r sds, 5.2 By comparing Lemma5.6 with Lemma 5., we can see that the continuous time approximation oftheresidualbasedcointegration test hastwoextraterms: R t inthenumeratorand R s in the denominator. It is worth noting that both terms would disappear if D. his is intuitive since D implies that there is no estimation error and the true β is known. u i then becomes directly observable and the test turns into the stationarity test discussed in Section 3. However, estimation error always exists and D. If the two terms resulting from D do not vanish asymptotically, the asymptotics of the test will be different from that of the stationarity test discussed in the previous section Continuous ime Asymptotics In this section, we investigate the asymptotic properties of the cointegtation test 5.2 by letting. More specifically, the asymptotic distribution of 5.2 is derived both under 5 Here we use the convention that integration is on t, s.t. < t,t s <. 2

21 Assumption CI and under Assumption NC- or NC-2 with a proper bandwidth B, such that B and B / as when the underlying processes are cointegrated. heorem 5.7. Let Assumptions KF and CA hold and. Under Assumption C, If B and B / as, we have λ n d W t tw X s X dw s X s X 2 ds t X s X d s 2 dt. heorem 5.7 states that even with an appropriate scheme, the test statistic τ n converges to a distribution involving nuisance parameters under Assumption C. In particular, the limit distribution is given as a function of standard Brownian motion W and X, the limit process of X on [,]. his limit distribution is different from the null distribution of the cointegration test derived in discrete time framework. In discrete time setting, stochastic trending process is modeled only through unit root process which converges to Brownian motion in the limit. i.e., X is Brownian motion in discrete time setting. When X is indeed a Brownian motion. he asymptotic distribution becomes identical to the one derived in Shin 994 for the corresponding case in discrete time framework. In continuous time framework, however, Brownian motion only represents a special type of nonstationary process and other types of nonstationary processes, such as nonstationary diffusions with stochastic drift term or stochastic diffusion term, converge to processes which can be quite different from Brownian motion. It implies that the critical values given in Shin 994 are generallynotapplicableincontinuoustimeexceptforthecasewherex isbrownianmotion. Blindly using those critical values is likely to lead to severe size distortion. In fact, because the null distribution of the test depends on the limit process X, conventional asymptotic test is not feasible and we propose using the modified subsampling test introduced in Section 3. Now we investigate the asymptotics of the test under Assumption NC- or NC-2. heorem 5.8. Let Assumptions KF and CA hold. Under Assumption NC- or NC-2, we have a B λ n d t Us U ds R t 2 dt U t U 2 dt+r Ksds, if B /, b λ n d t Us U ds R t 2 dt Ks/B Γ s+r sds, if B / B a.s. where D X s X U sds X s X 2 ds, R t D t X s X ds, and R D 2 X t X 2 dt 2D X t X U t U dt. Note that U Y under Assumption NC-. 2

22 Lemma 5.9. Let Assumptions KF and CA hold. Under Assumption NC- or NC-2, we have [ a +2κr c +2r B 2r rk /2r+, r 2 p +r 2 K sds] if CNP is used 2 b B r 3 Kr 2 /2r+ 2r/2r+ d K sds 2 U t U R t 2 dt, if CSP is used U t U R t du t he results in heorem 5.8 are analogous to those in heorem 5.3. he test statistic does not diverge when CSP is employed, but diverges under CR or CNP. However, because asymptotic test is not applicable here and we need use subsampling to perform the test, the divergence rate of the test statistics no longer represents the power. he testing power is determined by the relative divergence rate of the test calculated from entire sample verses that from subsamples under Assumption NC- or NC-2. Indeed, as we show in the following section, the bandwidth selection scheme no longer affects size or power of the test. 6. Stationarity est Using Subsampling Approach o conquer the limit process dependent problem in the null distribution specified in heorem 5.7, we propose using a modified subsampling method to conduct the test. he crux of test using conventional subsampling approach is to recompute test statistic on smaller blocks, or subsamples with size s n, such that s n and s n /n as n. he empirical distribution of the subsample test statistic is used to approximate the sampling distribution of the statistic calculated using entire sample. hen quantiles, say 95% quantile, of the empirical distribution can be used to approximate the 5% critical value of the distribution under the null. Compared to conventional subsampling approach, our modified subsampling algorithm is different in the fowllowing two aspects. First, similar to the selection of continuous time bandwidth, we set subsample size based on sample span instead of sample size n, i.e., we set s n S /δ, where s α with α being a constant and < α <. In this way, we can insure that the subsample span b satisfies the condition that S and S / as. he second modification is that instead of calculating the test statistic using 2., we calculate a modified test statistic λ s n using the following formula: λ s n n j 2 n 2 u ni. 6. j 22

23 he difference between 6. and 2. is that 2. uses longrun variance estimator as a scaling factor, but 6. does not. his modification is made because longrun variance of the underlying process is a nuisance parameter, which enters the test statistic simply as a positive scaling factor. herefore, whether scaling the test statistic with longrun variance or not shall not affect the relative ordering of the full-sample statistic among the statistics calculated based on subsamples, i.e. the testing result under the null shall not be affected. On the other hand, under the alternative, the modified subsampling approach has power gain compared to conventional subsmapling algorithm for not having to estimate the longrun variance. he asymptotics of the test statistic specified in 6. as be easily derived as follows. Corollary 6.. Let Assumptions KF and CA hold,. a Under Assumption C, λ s n d b Under Assumption NC- or NC-2, λs n d t U s U ds R t 2dt t 2dt U s U t ds R Based on Corollary 6.3 i, we can conclude that the modified full-sample test statistic λ s n and subsample test statistic λ s s n follow the same asymptotic distribution since S α as, given < α <. Similarly, the asymptotic distributional equivalence between conventional full-sample statistic λ n and subsample statistic λ sn follows directly from heorem 5.2 assuming that bandwidth is properply chosen. he asymptotic distributional equivalence of the two pairs of test statistics validates both conventional and modified subsampling algorithms under Assumption C. Because the asymptotic power of a test based on subsampling is determined by the relative divergence rate of full-sample test statistic with respect to the divergence rate of subsample test statistic, we can derive the asymptotic power of conventional subsampling test and modified subsampling test from the results in heorem 5.8 and Corollary 6.3, respectively. Corollary 6.2. Let Assumptions KF and CA hold, s n S /δ with S α and α being a constant satisfying < α < and. he power of the modified subsampling test is of order α and the power of conventional subsampling test is α η, with η being a constant determined by bandwidth selection scheme, and < η 6. It is worth mentioning that the modified subsample test is also applicable to the stationarity test. 6 η for CSP, η κ for CR and η +2rκ +2r for CNP, r,2 and < κ < 23

24 Corollary 6.3. Let Assumptions KF and CA hold,. i Under Assumption S, λ s t 2dt n d U s Uds ii Under Assumption NS, t 2dt λs n d U s Uds he comparison of asymptotic power among asymptotic test, convetional subsampling test and modified subsampling test is summarized in the following table. able. Asymptotic power comparison CSP CNP CR Asymptotic est Not Consistent +2κr +2r κ Conventional Subsampling Not Consistent a +2κr +2r a κ Modified Subsampling a According to able, the conventional subsampling test is overly dominated by the modified subsampling test since < +2κr +2r,κ <. It is also dominated by the asymptotic test if CNP or CR is employed due to the fact that < α <. When CSP is used, both asymptotic test and conventional subsampling test are not consistent. he asymptotic power comparison between the asymptotic test employing CNP or CR and the modified subsampling test depends on bandwidth selection parameters and subsample size selection parameter. More specifically, if +2κr +2r < a for CNP or κ < a for CR, the asymptotic test using the corresponding scheme has better asymptotic power, otherwise, the modified subsampling test has better asymptotic power. We will further investigate the finite sample performance of these two tests through simulations. 7. Simulations In this section, we first investigate how sampling frequency affects various stationarity test statistics and cointegration test statistics through simulations. hen we study the finite sample size and power of the modified subsampling approach and compare it to the asymptotic test employing continuous time bandwidth. 24

25 7.. Data generating process 7.2. Sampling frequency effect We investigate six bandwidth selection approaches studied in Section 3. he asymptotic theories imply that test statistics employing any of the three continuous time bandwidth selection schemes CR, CNP, CSP or discrete time SP are relatively stable across different frequencies. However, test statistics using R or NP stay relatively stable at low frequency region and become sensitive to sampling frequency after sampling frequency reaches a certain level. o illustrate the behavior of the sationarity test statistics under the null, we generate data based on the following Ornstein-Uhlenbeck OU process: du t κµ U t dt+σdw t, 7. where model parameters κ,µ,σ ,.353,.623 and W is standard Brownian Motion. he model parameters are obtained by fitting the OU model 7. to the daily forward premium of the US/UK exchange rates, since the forward premium is known to be stationary. Indeed, the mean reversion parameter κ > indicates that the process is stationary and the stationary distribution of the OU process is Nµ;σ 2 /2κ. We also consider processes with mean reverting parameter κ and κ / to illustrate the performance of the test on stationary processes with different levels of persistency. o illustrate the behavior of the test statistics under the alternative, we generate data using model du t σ dw t, with σ , which is estimated parameter after fitting the model to the daily -month forward US/UK exchange rates. o investigate the cointegration test statistic, under the null hypothesis, we generate X t and Y t as dx t du t and dy t du t +du t, with U t and U t defined as before. he correlation between the innovations of the two process is.299, which is the estimated correlation between first difference of forward exchange rates and the series generated by taking difference between spot exchange rates and forward exchange rates. Since U t is stationary, the simulated series X t and Y t are cointegrated. o generate two series which are not cointegrated, we use generate Y t as dy t dx t +db t, where B is standard Brownian motion independent of W. Again, for stationary process U t, we consider three mean reverting parameters, κ, κ and κ / to represent different level of persistency of cointegration regression error. Each process is simulated times and discrete samples are collected at frequency varying from quarterly to half-day frequency. We calculated the test statistics by employing 25

26 the six bandwidth selection methods mentioned above, respectively. 2 is used as the sampling span. he results are reported in Figure 3 and Figure 4. Figure 3 presents the simulation results for stationarity test and Figure 4 illustrates the reaction of the cointegration test statistics to the change of sampling frequencies both under the null and the alternative. Consistent with what predicted by theory, the test statistics generated using R or NP scheme is sensitive to sampling frequency change both under the null and alternative. Under the null, when sampling frequency increases, the statistics is relatively stable up to a certain frequency and then diverges. he turning point frequency depends on underlying process. When U t shows more persistency, the turning point appears at relatively low frequency. Fig. 3. Rejection Probabilities of Stationarity ests Using Variant Bandwidths for the LRV Estimator OU κ κ OU κ κ.35.9 R R CR.3 CR.8 SP SP Rejection Probability CSP NP CNP Rejection Probability CSP NP CNP Sampling Interval Sampling Interval OU κ κ / BM Rejection Probability R CR SP CSP NP CNP Rejection Probability R CR SP CSP NP CNP Sampling Interval Sampling Interval 7.3. Modified subsampling aproach vs. asymptotic test In this section, we investigate finite sample size and power of the modified subsampling approach and compare it to asymptotic test with various continuous time bandwidth selection schemes. A practical issue in conducting sumsampletest is to choose a propersubsamplespan S, which determines subsample size s n. Indeed, for s n too close to n, all subsample statistics 26