Powerful unit root tests free of nuisance parameters∗ Mehdi Hosseinkouchack†and Uwe Hassler Goethe University Frankfurt‡

February 23, 2015

Abstract The application of standard unit root tests is plagued by having to handle short-run autocorrelation as a nuisance parameter. We therefore propose a variance ratio type test where the nuisance parameter cancels asymptotically both under the null of a unit root and a local-to-unity alternative. This scale-invariant test can be implemented for the model with a constant or with linear time trend. It relies on a Karhunen-Lo`eve representation under the null and the local alternative. The critical values can be computed using standard numerical techniques. Our test exhibits higher power compared to tests that share the virtue of being free of nuisance parameters. In fact, the local asymptotic power curves of our procedure get close to the power functions of the point optimal test, where the latter suffers from the drawback of having to correct for a nuisance parameter consistently. Keywords: Karhunen-Lo`eve, variance ratio, scale-invariance, local-to-unity. JEL classification: C12, C22. ∗

Acknowledgements: We thank In Choi, Ulrich M¨ uller, Morten Nielsen, Paulo Rodrigues and Robert Taylor for helpful comments. † Corresponding author. ‡ Goethe University Frankfurt, Grueneburgplatz 1, 60323 Frankfurt, Germany. Email: [email protected], [email protected].

1

1

Introduction

Consider the following time series model

y t = dt + u t ,

t = 1, . . . , T,

(1)

where dt is the deterministic component, which we allow to be either a constant or a linear trend, dt = β0 or dt = β0 + β1 t. Testing for an autoregressive unit root, we assume for the stochastic component ut that

ρT = 1 −

ut = ρT ut−1 + εt ,

c , c ≥ 0, T

(2)

and are interested in testing ρT = 1 against ρT < 1, i.e. c = 0 against c > 0. This is conventionally called a unit root test or a test of whether yt is integrated of order one, I(1), against a local alternative of integration of order zero, I(0). To perform such a test, one often employs the Augmented Dickey-Fuller (ADF) test according to the works by Dickey and Fuller (1979) and Said and Dickey (1984), the Phillips-Perron (PP) test introduced by Phillips (1987a) and Phillips and Perron (1988), or efficient tests according to Elliott, Rothenberg, and Stock (1996). The related literature is copious, due to the relevance of distinguishing between I(1) and I(0) processes in empirical work. The influential works of Nelson and Plosser (1982) in macroeconomics and Meese and Rogoff (1983) in international finance are preeminent examples. Generally, the innovations εt in (2) are allowed to be weakly dependent and even heteroskedastic. A typical assumption made in this literature is that they satisfy a functional central limit theorem (FCLT) scaled with ω, where the long-run variance ω 2 is given by 2

0 < ω := lim T T →∞

−1

E

T X

!2 εt

< ∞.

(3)

t=1

Since ω is a scale parameter, conventional unit root tests are not scale-invariant 2

without accounting for this so-called nuisance parameter. There are two routes to handle the unknown long-run variance. First, tests in the tradition of ADF rely on augmenting the test equation by lagged endogenous differences where the lag length has to grow with the sample size T ; for a discussion under more general conditions than in Said and Dickey (1984) see Chang and Park (2002). Elliott et al. (1996) proposed a “nearly efficient” test in this tradition, where the ADF regression is combined with an efficient removal of deterministics. Harvey, Leybourne, and Taylor (2009) discussed these easily implemented unit root tests under uncertainty over the deterministic component. In practice, one faces a conflict between correct size and low power when choosing the number of lags. Second, the PP tests build on a direct consistent estimation of ω 2 by spectral methods. This requires a choice of a kernel and of a bandwidth that has to grow more slowly than the sample size.1 Similar in spirit is the efficient unit root test by Elliott et al. (1996) in that it requires a consistent estimator of ω.2 In practice, the bandwidth choice for PP-type tests is as delicate an issue as the lag length selection for ADF-type tests. Moreover, works by Kiefer, Vogelsang, and Bunzel (2000) or M¨ uller (2007) show that the estimation of the long-run variance can have adverse effects on the testing procedures in general. In particular, M¨ uller (2007) reinforced the earlier arguments on “near observational equivalence” of I(1) and I(0) processes by Blough (1992) and Faust (1996). For these reasons M¨ uller (2008) stressed and focussed on the variance ratio type test by Breitung (2002). It has the desirable property of being scale-invariant without having to estimate or to account for the long-run variance that simply cancels from the ratio. This describes a third route to unit root testing that we call “free of nuisance parameters”. 1

Vogelsang and Wagner (2013) adopted the related, so-called fixed-b approach where the bandwidth, chosen as bT , is growing at the same rate as T with the limiting distribution depending on b as well as on the kernel. 2 Stock (1999) suggested a finite-sample correction of the PP tests. This class of so-called M tests suffers less from size distortions in finite samples, see Perron and Ng (1996). M tests have been further improved by again efficient removal of deterministics; in fact, Ng and Perron (2001) established this variant to be “nearly efficient”. A further “nearly efficient” unit root test building on the likelihood ratio principle has been introduced more recently by Jansson and Nielsen (2012).

3

The same principle was carried to the framework of fractional (co-) integration by Nielsen (2010), and has been applied by Taylor (2005) when testing for seasonal unit roots. The scale-invariance of the Breitung test comes at a price that has been quantified in terms of power by an asymptotic comparison in Hosseinkouchack (2014). Inspired by Breitung (2002) is the variance ratio type test by Nielsen (2009) with better asymptotic and small sample properties. M¨ uller and Watson (2008) suggested another ratio based unit root test that is not plagued by the nuisance parameter ω 2 . While directing power against stationary autoregressive alternatives, this procedure focusses at frequencies lower than the business cycle at the same time. Hence it comes with the caveat of a loss of information and efficiency relative to procedures involving all frequencies. This is where our paper comes in. We adopt a testing principle recently proposed by Hassler and Hosseinkouchack (2014). It is applicable whenever data or transformations thereof are asymptotically normal upon appropriate normalization, which is the case in the present framework according to the mentioned FCLT, see Assumption 1 below. The test relies on an orthonormal expansion using the Karhunen-Lo`eve (KL) theorem, which is used to construct weighted averages of the data. Following Hassler and Hosseinkouchack (2014) (in short HH from now on), the test statistic is a ratio of quadratic forms of these averages, and hence free of nuisance parameters by construction: The long-run variance cancels without having to be estimated. Critical values and asymptotic local power functions can be calculated along the lines in HH. All that is required to apply the principle by HH is the eigenstructure of the limiting Gaussian process, namely the eigenvalues and eigenfunctions of the KL expansion. The limiting process resulting from (1) under (2) is the demeaned or detrended Ornstein-Uhlenbeck process (OUP); its eigenstructure is provided in the present paper. The resulting test turns out to be asymptotically more powerful than the comparable tests by Breitung (2002), M¨ uller and Watson (2008) or Nielsen (2009). In fact, we observe numerically that we get close to the power function of

4

the point optimal test by Elliott et al. (1996). The plan of the paper is as follows. Section 2 becomes precise on the assumptions and the testing principle behind HH. The third section gives the required OUP eigenstructure for the test. The asymptotic power comparison is carried out in Section 4. Conclusions are offered in Section 5. Mathematical proofs are relegated to the Appendix. A final word on notation. Throughout this paper, integrations are taken from √ 0 to 1, unless stated otherwise. The letter I represents −1. Im (z) and Re (z) return the imaginary and real parts of z ∈ C, respectively. Further, ⇒ stands for weak convergence as the sample size T diverges, and bxc denote the largest integer smaller than or equal to x ≥ 0, x ∈ R.

2

Assumptions behind the KL based test

Let uit with i = µ, τ represent the demeaned and detrended process yt in (1), where demeaning and detrending are performed by means of ordinary least squares (OLS):

uµt = yt − y

and uτt = yt − βb0 − βb1 t.

Under certain technical conditions spelled out in Chan and Wei (1987) or Phillips (1987b) we may maintain the following assumption. Assumption 1 Let uit under (2) with (3) satisfy as T → ∞ T −1/2 uibrT c ⇒ ωJci (r) , 0 ≤ r ≤ 1, i = µ, τ ,

5

(4)

where Jci is a demeaned or detrended Ornstein-Uhlenbeck process, respectively: Jcµ Jcτ

Z (r) := Jc (r) − (r) :=

Jcµ

Jc (s) ds Z

(r) − 12 (r − 1/2)

Here, the standard OUP is given by

Jc (r) =

(s − 1/2) Jc (s) ds. Rr 0

e−c(r−s) dW (s) for a standard

Wiener process W . Since Jci (r) is Gaussian and continuous in quadratic mean it satisfies Hassler and Hosseinkouchack (2014, Ass. 1) and is endowed with an orthonormal KarhunenLo`eve (KL) decomposition:

Jci

(r) =

∞ X

λij,c


−1/2

i fj,c (r) ζj ,

i ∈ {µ, τ } ,

(5)

j=1

i (r) and λij,c , j = 1, 2, ..., are the eigenfunctions and eigenvalues of the where fj,c

covariance function kci (s, t) of Jci , and ζj , j = 1, 2, ..., are independent standard normals. For notational simplicity we will often suppress that the eigenvalues λj and eigenfunctions fj (r) (also called “eigenstructure” for brevity) depend in general on c and i through k(s, t) = kci (s, t). To provide the KL representation for Jci we need to find the eigenvalues and eigenfunctions. As discussed in Lo`eve (1978, p. 144), these are in fact the nontrivial solutions of the following Fredholm integral equation Z f (t) = λ

k (s, t) f (s) ds,

(6)

where the eigenfunctions form an orthonormal basis of L2 , that is Z fj (r) fk (r) dr = δjk ,

(7)

with δjk being the Kronecker delta. In the next section, we determine the eigenstructure of the demeaned and detrended OUP, which is all one requires to construct 6

a test following HH. Therefore, we now briefly review their proposal. We test the null hypothesis against one value c1 > 0, where the choice of c1 will depend on the model (dt in (1)) and is discussed below:

H0 : c = c0 = 0 vs. H1 : c = c1 .

For this purpose, we define q weighted sums under the null and under the alternative hypotheses (h ∈ {0, 1}). They are built from the weights

i wj,c h ,t

Z

t/T

:= (t−1)/T

i fj,c (s) ds, h

t = 1, . . . , T, j = 1, . . . , q .

(8)

We suppress the dependence on the model, i = µ, τ , and denote the weighted sums i as XT,j,ch = XT,j,c , j = 1, . . . , q: h

XT,j,ch := T

−1/2

T X

i wj,c ui . h ,t t

t=1

They are collected in a q-dimensional vector XT,ch that follows a limiting normal distribution, see Hassler and Hosseinkouchack (2014, eq. (8)): XT,ch := (XT,1,ch , . . . , XT,q,ch )0 ⇒ ω Ych (c) ,

Ych (c) ∼ Nq (0, Ωch (c)) .

(9)

Of course, the limit depends on the true parameter value c entering through the data by Assumption 1, and on the value ch under the hypothesis employed to compute the weighted sums. In general, the typical (j,k) element of the covariance matrix Ωch (c) is Z Z fj,ch (s) fk,ch (r) kc (r, s)drds .

(10)

HH observe due to the orthonormality (7) that the covariance matrix becomes di-

7

agonal if the null or if the alternative is true (ch = c)
−1 Ωch (ch ) = diag λ−1 1,ch , . . . , λq,ch .

(11)

The test hence amounts to discriminate between two normal distributions:

H0 : XT,c0 ∼ Nq (0, ω 2 Ωc0 (c0 )) vs. H1 : XT,c1 ∼ Nq (0, ω 2 Ωc1 (c1 )) as T → ∞ . (12)

Following King (1987), one might consider a likelihood ratio in order to obtain a point optimal test. This approach, however, is plagued by the nuisance parameter ω in general. Therefore, HH propose the following variance ratio type statistic:

VRT,q

Pq 2 j=1 λj,c0 XT,j,c0 , = Pq 2 j=1 λj,c1 XT,j,c1

(13)

The limiting distribution of this test statistic is free of ω, and becomes for a true value c VRT,q ⇒ VRq (c) :=

Yc00 (c)Ωc0 (c0 )−1 Yc0 (c) . Yc01 (c)Ωc1 (c1 )−1 Yc1 (c)

(14)

Remember that all entities like VRT,q or VRq as well as Ωch (c) and c1 are implicitly understood to carry the superscript i = µ, τ to distinguish the demeaned from the detrended case. For any x ≥ 0 Hassler and Hosseinkouchack (2014, Theo. 1, Rem. 2) show how to compute the probability Pr(VRq (c) ≤ x) in terms of Ωch (ch ) and Ωch (c). They reduce VRq (c) ≤ x to a quadratic form in normal variates, obtain the characteristic function, such that the usual inversion formula by GilPelaez (1951) or Imhof (1961, eq. (3.1)) allows to compute probabilities for c = 0 or c = c1 > 0. Hence, critical values under the null hypothesis or limiting power under local alternatives are readily available for the test building on VRT,q . To put this variance ratio test into work, c1 and q must be chosen. First, for c1 HH follow the familiar approach by King (1987): Given q and a type one error probability of α, choose c1 so that the asymptotic power under this alternative is 8

50%. Second, for q it is important to note, that this choice does not affect the asymptotic size. q is not a tuning parameter similar to a bandwidth that would have to grow with the sample size at an appropriate rate. We rather have a family of tests indexed by q, and any choice of q results in an asymptotically valid test. The choice of q is solely an issue of power, see Section 4.

3

Eigenstructure of Ornstein-Uhlenbeck processes

The eigenstructure of a standard OUP Jc with covariance kernel kcJ (s, t) =

1 2c

e−c|s−t| − e−c(s+t)

is contained as a special case in Hassler and Hosseinkouchack (2014, Prop. 2). For a demeaned OUP, Jcµ , standard calculations show k µ (s, r) = k J (s, r) − g (r) − g (s) + δ0

where

g (s) =

e−c − 2 −cs e−c cs 1 e − 2e + 2 , 2 2c 2c c

δ0 = −

3 − 2c + e−2c − 4e−c . 2c3

In the Appendix we prove the following result. Proposition 1. For the eigenstructure of k µ (s, r) from Jcµ it holds that λµj,c are
the roots of ρµ λµj,c = 0, where 

 ρµ (λ) = e−c 2cλ − c 2λ + cυ 2 cos υ + υ λ − c3 sin υ /υ 4 ,

(15)

and µ fj,c

with υj,c =

(r) =

cµ2

  λµj,c cos υj,c r + βµ sin υj,c r + 2 κ , υj,c

(16)

q λµj,c − c2 . The constants κ, βµ , cµ2 depend on c and are given in the

Appendix.

9




Proof. See Appendix. Remark 1. For c → 0 the process Jcµ turns into a demeaned Wiener process. We observe that limc→0 ρµ (λ) =

√ sin √ λ λ

whose roots are j 2 π 2 for j ∈ N. Therefore, we

have µ lim fj,c (t) =

c→0

√ 2 cos jπt.

(17)

This equation reproduces the result by M¨ uller and Watson (2008, Theo. 1) as a special case of Prop. 1. We now turn to the detrended OUP, Jcτ . By elementary means we find for the covariance kernel     1 1 k (s, r) = k (s, r) − 3 r − h (s) − 3 s − h (r) 2 2        1 1 1 1 +3ω1 r − + 3ω1 s − + 6ω2 s − r− , 2 2 2 2 τ

µ

where h (s) =

 e−c  (2 + c + 2cec ) e−cs − (2 + c) ecs + 2cec (2s − 1) , 3 c e−2c (ec − 1) [2 + c + (c − 2) ec ] , ω1 = 4 c

and ω2 =


e−2c [−3 (c + 2)2 − 12c (2 + c) ec + 12c − 9c2 + 2c3 + 12 e2c ]. 5 c

In the Appendix we show the following result. Proposition 2. For the eigenstructure of k τ (s, r) from Jcτ it holds that λτj,c are
the roots of ρτ λτj,c = 0, where 
ρτ (λ) = e−c 4λ c2 (c + 3) υ 2 + 6λ (c + 1)
+ c4 λ4 + 8c3 υ 2 λ − 24 (c + 1) λ2 cos υ

 +υ c5 υ 2 − 4c2 3 + 3c + υ 2 λ − 12 (1 + c) λ2 sin υ /υ 8 , 10

(18)

and τ fj,c (r) = cτ1 sin υj,c r + cτ2 cos υj,c r +

with υj,c =

γ2 γ1 + 2 r, 2 υj,c υj,c

(19)

p τ λj,c − c2 . The constants cτ1 , cτ2 , γ1 , γ2 depend on c and are given in

the Appendix. Proof. See Appendix. Remark 2. For c → 0 the process Jcτ turns into a detrended Wiener process √ √ √  √  with limc→0 ρτ (λ) = 48λ−2 sin 2λ sin 2λ − 2λ cos 2λ whose roots are j 2 π 2 for √

even j ∈ N and the other set of roots are characterized by sin

λ 2

√

−

λ 2

√

cos

λ 2

= 0.

Therefore, one may show that

τ lim fj,c (r) =

c→0

√

   r  

√

√

2

λj

λj −sin

√

λj

2 cos (j + 1) πr p  (−1)(j+2)/2 sin λj (r − 1/2)

j ∈ N and odd j ∈ N and even (20)

√

λ 2

where λj ’s are root of sin

√

−

λ 2

√

cos

λ 2

= 0. The equation (20) hence contains the

result by M¨ uller and Watson (2008, Theo. 1) or Ai, Li, and Liu (2012, Theo. 1) as a special case of Prop. 2. For a fixed value of q, the choice of c1 has briefly been mentioned in the previous section following King (1987): For a given value of q and for a type one error of α, we determine c1 such that     Pr VRi (c0 ) ≥ xi q q,1−α = α,  1   Pr VRi (c ) ≥ xi = , q

1

q,1−α

(21)

2

where xiq,1−α are the corresponding critical values. For values of q = 20, 40, 60, 80, 100, 200, and 400 and for α = 0.05, we solve the system (21) for the constant mean and linear trend models. The resulting critical values and corresponding values for c1 are reported in Table 1. The null hypothesis is rejected for too large values. With growing q the value c1 decreases which reflects an increase in power since a 11

50% rejection probability is achieved closer to the null hypothesis. A more detailed power study is reserved for the next section. Table 1: Critical values of VRiq (c0 ) at α = 0.05.

q c1 xµq,1−α c1 xτq,1−α

i = µ (demeaned) 20 40 60 80 100 10.3 8.2 7.7 7.5 7.4 0.7761 0.9057 0.9404 0.9562 0.9653 i = τ (detrended) 19.8 15.5 14.6 14.2 14.0 0.6514 0.8486 0.9027 0.9283 0.9431

200 400 7.2 7.1 0.9830 0.9916 13.6 13.5 0.9721 0.9860

Note: For each value of q, c1 and corresponding xiq,1−α solve (21) for α = 0.05. To set up the test statistic VRiT,q does not only require the eigenvalues λij,c that could easily be tabulated. A further ingredient are the eigenfunctions, or more precisely the vector of weighted sums from (9). The simplest way to implement our test hence is to use the matlab code provided on the homepage by the first author.

4

Power comparison

To see how the proposed variance ratio test compares with other unit root tests, we picked three competitors. The first one is the test by Breitung (2002) briefly introduced in the Introduction. Second, we consider the OLS-based test building on the statistic T (ρbi − 1). Here, ρbi , i = µ, τ , is the OLS estimator of the demeaned or detrended AR(1) model. Third, we include the point optimal test by Elliott et al. (1996); it is denoted as Pµ for the model with constant and Pτ for the trend model, and it tests against the specific values c¯ = 7 and c¯ = 13.5, respectively, that correspond to c1 in our notation. While Breitung’s test is free of nuisance parameters, we assume for the latter two tests unrealistically the long-run variance to be known, although the tests are in practice plagued by the nuisance parameter ω. The limiting power of Breitung and OLS is taken from Hosseinkouchack (2014).

12

Closed-form expressions of the characteristic functions under local-to-unity for Pµ and Pτ were given in Elliott and Stock (2001, Prop. 4). We found it computationally more robust to open their expressions along similar lines as in Tanaka (1996, Ch. 4). Proposition 3. Under (1) with (2) and Assumption 1 the characteristic functions of the limiting distributions of Pµ and Pτ are  −1/2 φPµ (θ; c, c) = e(β−c)/2 1 − (β − c + 2cθI) eβ sinh β/β , p  φPτ (θ; c, c) = e(β−c)/2 6β 5 3cβ 4 + 3β 5 − 3cβ 4 cos 2β + 3β 5 cosh 2β
+8c2 eβ θ2 3 + 3c − 6 (c + 1) λ + (3 + (c + 3) c) λ2 ×

−3β cosh β + 3 + β 2 sinh β


−12Ieβ θ (c + 1) 3 + β 2 (λ − 1)2 3 + β 2 sinh β − 3β cosh β +12Ieβ θc (λ − 1) (3 (λ − 1) − c (3 − c − (c + 3) λ)) ×

−3β cosh β + 3 + β 2 sinh β

+12βIeβ θc2 −3 + β 2 + 3 + β 2 λ2 cosh β  
2 −4Ieβ θc2 3 + β 2 λ2 − 9 sinh β −1/2 −3cβ 4 sinh 2β + 3β 5 sinh 2β ,

where λ = (1 + c) / (1 + c + c2 /3) and β =

√

c2 − 2Ic2 θ with c¯ = 7 and c¯ = 13.5

for the cases of demeaning and detrending, respectively. Proof. See Appendix. Remark 3. To calculate the power curves of Pi , we use the following inversion formula given in Gil-Pelaez (1951), 1 1 Pr {Pi ≤ bi } = − 2 π

Z 0

∞

 e−Iθbi Im φPi (θ; c, c) dθ, θ 

(22)

where bi = bi (c) is the five percent critical value of Pi for i = µ, τ . We used the 13

1 0.9 0.8 0.7

power

0.6

ERS q=400 q=200 q=100 q=20 OLS BR

0.5 0.4 0.3 0.2 0.1 0 0

5

10

15 c

20

25

30

Figure 1: Asymptotic power at 5% level when demeaning. BR, OLS and ERS stand for Breitung, T (ρbµ − 1) and Pµ ; results of VRµq for q = 20, 100, 200, 400 critical values obtained from Proposition 3 for c = 0, which differ slightly from those provided by simulation in Elliott et al. (1996). Remark 4. When calculating integrals of the form (22) we employ a simple Simpson’s rule. As the characteristic functions are entire functions with infinitely many roots we use the Euler’s transformation method to calculate the integrals, for which we set an error value of 10−6 .3 Furthermore, we use a change of variable of the form θ = u2 to achieve a faster convergence rate while easing the calculation of the value of the integrand at zero. Let us first look at Figure 2 where the results are very clear-cut. The lowest power curve is for Breitung’s test, followed by the variance ratio test for q = 20; both tests are dominated by the simple OLS test. From q = 100 on, the power curve of VRτq is well above that of OLS, and this is all the more true the larger q is. For VRτ400 the power curve is very close to that one of the point optimal Pτ (ERS). In Figure 1 for the case of demeaning the power is higher for all tests as expected. Further, the ranking of the power functions is the same with one exception: the 3

See Tanaka (1996, Ch. 6) for further details.

14

1 0.9 0.8 0.7

power

0.6

ERS q=400 q=200 q=100 q=20 OLS BR

0.5 0.4 0.3 0.2 0.1 0 0

5

10

15 c

20

25

30

Figure 2: Asymptotic power at 5% level when detrending. BR, OLS and ERS stand for Breitung, T (ρbτ − 1) and Pτ ; results of VRτq for q = 20, 100, 200, 400 curves of OLS and VRµ20 cross. From both figures we observe that power gains for VRiq are small when increasing q beyond 100, so the value q = 100 may be recommended for applied purposes. Without producing power curves we may also compare VRiq with the low-frequency unit root test by M¨ uller and Watson (2008). This test is constructed in such a way, that a rejection probability of 50% is achieved at c¯ = 14 and c¯ = 28 for the demeaned and detrended cases, respectively (M¨ uller and Watson, 2008, p. 992). From Figures 1 and 2 we learn that already VRi20 has more power at these points; see also Table 1: c1 < c¯. The same comparison is possible with the test by Nielsen (2009). He suggested a family of tests indexed with a fractional differencing parameter d, and recommended the most powerful variant for d = 0.1 in connection with GLS demeaning or detrending along the lines of Elliott et al. (1996). Nielsen (2009, Table 2) contains values c¯ yielding 50% power from simulations for T = 500; they are 9.4 and 15.1 for the case of demeaning and detrending, respectively. Figures 1 and 2 show that already VRi100 is more powerful at these points. Similarly, from Table 1

15

we have c1 < c¯ for q = 100.

5

Conclusion

In this paper we provide a Karhunen-Lo`eve representation of a (demeaned or detrended) Ornstein-Uhlenbeck process. These are the ingredients required to set up a variance ratio test as offered by Hassler and Hosseinkouchack (2014). More precisely, we obtain a family of tests depending on a parameter q that does not matter for the control of size but affects power. For any q our test enjoys the useful property that the nuisance parameter plaguing the more traditional unit root tests cancels and thus does not have to be estimated. In that sense we call our test “free of nuisance parameters”. The theory by Hassler and Hosseinkouchack (2014) allows us to compute asymptotic local power curves for our new test. It turns out that it is more powerful than the variance ratio type competitors by Breitung (2002), M¨ uller and Watson (2008) or Nielsen (2009) as long as q is large enough, for instance q = 100. In fact, with larger values of q the power curve of the point optimal test by Elliott et al. (1996) is nearly reached.

Appendix Proof of Proposition 1 We first add the following piece of notation for the function g showing up in the covariance kernel:

g (s) =

1 e−c − 2 −cs e−c cs e − 2 e + 2 ≡ δ1 e−cs + δ2 ecs + δ3 . 2 2c 2c c

16

The Fredholm integral equation (6) for k µ (s, t) is equivalent to the following boundary condition differential equation

f 00 (t) + υ 2 f (t) = λ

where υ =

√

λ − c 2 , a1 =

R




 1 − c2 δ0 + c2 δ3 a1 + c2 δ1 a2 + c2 δ2 a3

f (s) ds, a2 =

R

e−cs f (s) ds and a3 =

R

(23)

ecs f (s) ds with

the following boundary conditions

f 0 (0) = λa2 − λg 0 (0) a1 , f 0 (1) =

(24)

λe−c λe−c a2 − a3 − λg 0 (1) a1 . 2 2

The solution to (23) can be written as f (t) = cµ2 cos υt + cµ1 sin υt +

(25)

λ κ υ2

with κ =

(2c2 δ3 − c2 δ0 ) a1 + c2 δ1 a2 + c2 δ2 a3 . Using this solution, we write out a1 , a2 and a3 as κc2 λ + cµ1 υ − cµ1 υ cos υ + cµ2 υ sin υ , υ2 κcλ (1 − e−c ) cµ1 e−c (ec υ − υ cos υ − c sin υ) + = υ2 λ µ −c c c e (ce − c cos υ + υ sin υ) + 2 , λ κcλ (ec − 1) cµ1 ec (e−c υ − υ cos υ + c sin υ) = + υ2 λ cµ2 ec (−ce−c + c cos υ + υ sin υ) + . λ

a1 = a2

a3

Substituting κ in the expressions for a1 , a2 and a3 we can write a = (a1 , a2 , a3 )0 in terms of cµ1 and cµ2 , i.e. a = Mµ−1 Bµ ≡ cµ1 Mµ−1 B1,µ + cµ2 Mµ−1 B2,µ ,

17

(26)

with appropriate definitions for Mµ , Bµ , B1,µ and B2,µ . Plugging (26) in (24) and (25), we obtain the following system of equation in cµ1 and cµ2 :      cµ1


cµ1 v1 Mµ−1 B1,µ − υ + cµ2 v1 Mµ−1 B2,µ = 0

v2 Mµ−1 B1,µ − υ cos υ + cµ2 v2 Mµ−1 B2,µ + υ sin υ = 0

(27)

where v1 = λ (−g 0 (0) , 1, 0) and v2 = λ (−g 0 (1) , e−c /2, −e−c /2). On the other hand we can write κ = cµ1 δMµ−1 B1,µ + cµ2 δMµ−1 B2,µ ≡ cµ1 κ1 + cµ2 κ2 with δ = (2c2 δ3 − c2 δ0 , c2 δ1 , c2 δ2 ). And using the first equation from (27), we obtain cµ1 =

v1 Mµ−1 B2,µ cµ υ−v1 Mµ−1 B1,µ 2

βµ cµ2 with −c2 ec υ + (ec − 1) λ sin υ + cυ (c cos υ − υ sin υ) βµ = . λ + ec λ (cos υ − 1) − λ cos υ − cυ 2 cos υ − c2 υ sin υ Putting all these together we have

f (t) =

cµ2

  λ λ cos υt + βµ sin υt + 2 βµ κ1 + 2 κ2 . υ υ

Some algebra also shows that

κ1 =


e−2c υ  × −λ + 4ec λ − e2c 3λ + 2c3 + c2 (−1 + ec )2 υ sin υ 3 2c λ
 + λ − 4ec λ + (3 + 2c) e2c λ − 2cec υ 2 (1 + sinh c) cos υ

and

κ2 =

e−2c υ  2 c 2c e (−1 + ec ) υ + c2 (−1 + ec )2 υ cos υ 3 2c λ

 −2ec sin υ c3 − cλ − 2λ + (2 + c) λ cosh c + λ + c3 sinh c .

18

≡

Finally

R

f (t)2 dt = 1 results in



4 (κ2 + βµ κ1 )2 λ + 8βµ (κ2 + βµ κ1 ) λυ + 2υ 3 βµ + υ + βµ2 υ
−1/2 −2υ (βµ cos υ − sin υ) 4 (κ2 + βµ κ1 )2 + υ 2 (cos υ + βµ sin υ) .

cµ2 = υ 2



Now we need to determine the eigenvalues, λ, so that the eigenfunctions of k µ (s, t) be identified. Using Tanaka (1996, Chapter 5), λ should be chosen such that (27) possesses nonzero solutions. This is achieved by v1 Mµ−1 B2,µ v1 Mµ−1 B1,µ − υ e D (λ) = v M −1 B − υ cos υ v M −1 B + υ sin υ 2 µ 1,µ 2,µ 2 µ

= 0.

With some simple but tedious algebra we show that −c 2 3 e (λ) = e sinh c (2cλ − c (2λ + cυ ) cos υ + υ (λ − c ) sin υ) . D c υ

Note that ρµ (λ) =

e cD(λ) , υ 3 sinh c

which after some simplifications can also be found in

Tanaka (1996, p. 337, eq. (9.49)), satisfies the conditions of Nabeya (2000, Theo. 4), and hence is the Fredholm determinant of k µ (s, t). This completes the proof. 

Proof of Proposition 2 The proof is structured similarly to the previous one. The function h showing up in k τ (s, t) is written as

h (s) =

 e−c  × (2 + c + 2cec ) e−cs − (2 + c) ecs + 2cec (2s − 1) ≡ x1 e−cs +x2 ecs +x3 (s − 1/2) . 3 c

The Fredholm integral equation for k τ (s, t) is equivalent to the following differential equation f 00 (t) + υ 2 f (t) = γ1 + γ2 t,

19

(28)

with 

γ1

γ2

   3 2 3 2 2 2 = a1 λ + c λω1 − c λ (δ0 − δ3 ) + a2 c λδ1 − c λx1 2 2     3 2 3 2 2 2 2 +a3 c λδ2 − c λx2 + a4 3c λω2 − c λx3 − 3c λω1 − 6λ , 2 2
2 2 2 = −3c λa1 ω1 + 3c λa2 x1 + 3c λa3 x2 + a4 12λ + 3c2 λx3 − 6c2 λω2 ,

and with the following boundary conditions

f (0) = λ (δ0 a1 − (δ1 a2 + δ2 a3 + δ3 a1 )) − λg (0) a1

(29)

3 3 + λ (x1 a2 + x2 a3 + x3 a4 ) − 3λh (0) a4 − ω1 λa1 + 3ω1 λa4 − 3ω2 λa4 , 2 2

f 0 (0) = λ (3ω1 − g 0 (0)) a1 + λ (1 − 3x1 ) a2 − 3λx2 a3

(30)

+3λ (2ω2 − x3 − h0 (0)) a4 ,

where a1 =

R

f (s) ds, a2 =

R

e−cs f (s) ds, a3 =

R

ecs f (s) ds and a4 =

R

s−

1 2




f (s) ds.

2t The solution to (28) is f (t) = cτ1 sin υt + cτ2 cos υt + γ1 +γ which we use to write out υ2

a = (a1 , a2 , a3 , a4 )0 in terms of cτ1 and cτ2 , i.e. we have a = cτ1 Mτ−1 Bτ,1 + cτ2 Mτ−1 Bτ,2 where Mτ , Bτ,1 and Bτ,2 are identified taking the same steps as for part a. Using a, γ1 and γ2 are identified too. First note that γ1 = v1 a and γ2 = v2 a with

v1

v2

 3 3 3 = λ + c2 λω1 − c2 λ (δ0 − δ3 ) , c2 λδ1 − c2 λx1 , c2 λδ2 − c2 λx2 , 2 2 2  3 3c2 λω2 − c2 λx3 − 3c2 λω1 − 6λ , 2
2 = −3c λω1 , 3c2 λx1 , 3c2 λx2 , 12λ + 3c2 λx3 − 6c2 λω2 .

Hence we have γ1 = cτ1 v1 Mτ−1 Bτ,1 +cτ2 v1 Mτ−1 Bτ,2 ≡ γ11 cτ1 +γ12 cτ2 and γ2 = cτ1 v2 Mτ−1 Bτ,1 + cτ2 v2 Mτ−1 Bτ,2 ≡ γ21 cτ1 + γ22 cτ2 . Define v3 = λ(3ω1 − g 0 (0), 1 − 3x1 , −3x2 , 6ω2 − 3x3 −

20

3h0 (0)). Using (30) and the solution for (28) we have

cτ1 =

Therefore using

R

     A=   

and

υ 2 v3 Mτ−1 Bτ,2 − γ22 cτ2 ≡ βτ cτ2 . 2 2 −1 γ22 + υ − υ v3 Mτ Bτ,1

(31)

p f (t)2 dt = 1 we find cτ2 = 1/ Trace (AB) where βτ2

βτ

βτ (βτ γ11 +γ12 ) υ2

βτ (βτ γ21 +γ22 ) υ2

βτ

1

βτ γ11 +γ12 υ2

βτ γ21 +γ22 υ2

βτ (βτ γ11 +γ12 ) υ2

βτ γ11 +γ12 υ2

(βτ γ11 +γ12 )2 υ4

βτ γ11 +γ12 βτ γ21 +γ22 υ2 υ2

βτ (βτ γ21 +γ22 ) υ2

βτ γ21 +γ22 υ2

βτ γ11 +γ12 βτ γ21 +γ22 υ2 υ2

(βτ γ21 +γ22 )2 υ4

     B=   

sin2 υ 2υ

1−cos υ υ

−υ cos υ+sin υ υ2

sin2 υ 2υ

2υ+sin 2υ 4υ

sin υ υ

−1+cos υ+υ sin υ υ2

1−cos υ υ

sin υ υ

1

1 2

−υ cos υ+sin υ υ2

−1+cos υ+υ sin υ υ2

1 2

1 3

1 2

−

sin 2υ 4υ

     ,   

     .   

Some algebra shows that

γ11 =

1  2λ (3 + c) {−υ (3 + c + (3 + 2c) cos υ) + 3 (2 + c) sin υ} e−2c c5
−2 (2c − 3) 6c + c2 + 3 λυ

+6 5c2 − 7c + c3 − 6 λ sin υ − 4ce−c c3 λ − 9cλ − c5 − 21λ sin υ
−16ce−c (3 + c) λυ − 4ce−c υ 9λ + 5cλ − c4 + c2 λ cos υ
+4c3 υ 3 − 2 6c2 − 9c + c3 − 9 λυ cos υ  −2e−c c2 υ 2 (3 + c) (c cosh c sin υ − υ cos υ sinh c) ,

21

γ12 =

2  3 −2c −c e (3 + c) ec υ 2 cos υ cosh c 5 c +e−2c (3 + c) λ (3 (2 + c) (−1 + cos υ) + (3 + 2c) υ sin υ) −3 (6 + c) (−1 + (−1 + c) c) λ + 2c4 υ 2 +3 (6 + c) (−1 + (−1 + c) c) λ cos υ + (−9 + c (−9 + c (6 + c))) λυ sin υ
−6ce−c (7 + 3c) λ + c2 ce−c (3 + c) υ 2 + 2ce−c 21λ + 9cλ − c3 υ 2 cos υ
 +2υce−c 9λ + 5cλ + c2 υ 2 sin υ − cce−c (3 + c) υ 3 sin υ sinh c ,

γ21 =

6  −2c e λυ (2 + c) (3 + c + (3 + 2c) cos υ) c5

+υ c5 − 7cλ − 6λ + 5c2 λ + 4c2 − 5c + c3 − 6 λ cos υ
+e−c (2c (7 + 3c) λυ) + c2 e−c c2 λ − 2c3 − c4 − 12λ cosh c sin υ

+e−c c4 λ − c6 − 12c2 λ − 24cλ − 2λ 3c2 − 12c + c3 − 12 sinh c sin υ
 +ce−c υ cos υ 10λ + 6cλ + c2 υ 2 − c (2 + c) υ 2 sinh c ,

γ22 =

6  −2c e λ (2 + c) (3c + 6 − (3 + 2c) υ sin υ) c5 +c6 − 12cλ − 12λ + 9c2 λ + 2c3 λ − c4 λ + e−c c (2 + c) 12λ − c2 υ 2
− 4c2 − 5c + c3 − 6 λυ sin υ
+e−c c4 λ − c6 − 12c2 λ − 24cλ cos υ
+c2 e−c c2 λ − 2c3 − c4 − 12λ cosh c cos υ
−2λ 3c2 − 12c + c3 − 12 sinh ce−c cos υ
 +e−c c −2 (5 + 3c) λ − c2 υ 2 + c (2 + c) υ 2 sinh c υ sin υ .

22




Finally, for v3 Mτ−1 Bτ,1 and v3 Mτ−1 Bτ,2 which appear in the definition of βτ in (31) we have

v3 Mτ−1 Bτ,1 = −


e−c  2 3 2c 3c − 18c + c − 42 λυ c5 υ 2


+υc 12λ − 36cλ − 6c4 + c6 + 4c3 λ − c4 λ cosh c
+2λυc c3 − 3c2 − 18c − 6 cosh c cos υ
−6c2 2c2 λ − 2c3 − c4 − 12λ cosh c sin υ
+υ 72λ + 72cλ − 6c5 + c7 − 24c2 λ + 4c4 λ − c5 λ sinh c
+ 144cλ + 6c6 − c8 + 72c2 λ − 12c3 λ − 12c4 λ + c6 λ sin υ
−6λ 24c − 6c2 − 2c3 + c4 + 24 sinh c sin υ
+ 6c5 − 60cλ − c7 − 36c2 λ + 4c4 λ + c5 λ υ cos υ
 +2υ 36λ + 36cλ − 6c4 − 3c5 + c4 λ sinh c cos υ

v3 Mτ−1 Bτ,2 =


e−c  c 72cλ + 6c5 − c7 + 12c2 λ − 12c3 λ + c5 λ cosh c 5 2 cυ
+6c2 2c2 λ − 2c3 − c4 − 12λ cosh c cos υ
+2λυc c3 − 3c2 − 18c − 6 cosh c sin υ
+c 144λ + 72cλ + 12c4 + 6c5 − 24c2 λ − 12c3 λ
+c c7 − 72cλ − 6c5 − 144λ + 12c2 λ + 12c3 λ − c5 λ cos υ
+υc 6c4 − 36cλ − 60λ − c6 + 4c3 λ + c4 λ sin υ
+ 6c6 − 144cλ − 144λ − c8 + 36c2 λ + 12c3 λ − 12c4 λ + c6 λ sinh c
+6λ 24c − 6c2 − 2c3 + c4 + 24 sinh c cos υ
 +2υ 36λ + 36cλ − 6c4 − 3c5 + c4 λ sinh c sin υ

Now with cτ1 = βτ cτ2 from (31) we have

f (t) =

cτ2



 γ11 βτ + γ12 γ21 βτ + γ22 + t . βτ sin υt + cos υt + υ2 υ2 23

Therefore, for given eigenvalues, λ, the eigenfunctions of k τ (s, t) are identified. As for Proposition 1, using Tanaka (1996, Chapter 5), λ should be chosen so that the system of equations in cτ1 and cτ2 , which is formed by manipulating (29) and (30), possesses nonzero solutions. This together with the conditions of Theorem 4 in Nabeya (2000) lead to finding the Fredholm determinant, ρτ (λ), of k τ (s, t) whose roots, by definition, characterize the sought after eigenvalues. ρτ (λ) can also be found in Tanaka (1996, p. 341). 

Proof of Proposition 3 First, consider the constant mean model. For simplicity, denote the limit of this R point optimal test as Pµ with Pµ = c2 Jc2 (s) ds + cJc2 (1). It holds

φPµ (θ; c, c) = E exp {IθPµ }  Z Z 2 2 2 = E exp Iθc Jβ (s) ds + IθcJβ (1) + (c − β) Jβ (s) dJβ (s)  Z
2 2 2 −0.5 c − β Jβ (s) ds ,

where have used the Girsanov’s Theorem (see Girsanov, 1960) to change the measure from that of Jc (t) to Jβ (t) where dJβ (t) = βJβ (t) dt + dW (t). Choosing β = √ c2 − 2Ic2 θ we have 

 φPµ (θ; c, c) = E exp (1) + (c − β) Jβ (s) dJβ (s) ,  β   e sinh β 2 c−β (β−c)/2 = e E exp + Iθc Z , 2 β Z

IθcJβ2

where Z is a standard normal random variable. Now, given that Z 2 is χ2(1) the result can be easily established. Second, for the linear trend case, denote the limit of the point optimal test by ElR liott et al. (1996) as Pτ , with Pτ = c2 Vc2 (s, c) ds+(1 + c) Vc2 (1, c), where Vc (s, c) =
R Jc (s) − s λJc (1) + 3 (1 − λ) xJc (x) dx with λ = (1 + c) / (1 + c + c2 /3). To ob24

tain φPτ (θ; c, c) we use Girsanov’s Theorem and after doing some simplification we find that φPτ (θ; c, c) = e(β−c)/2 E exp {X 0 AX} ,
0 R where X = Jβ (1) , sJβ (s) ds and A is symmetric with

A11 =

Iθc2 λ2 c−β + Iθ (1 + c) (1 − λ)2 + , 3 2

A12 = A21 = −Iθc2 λ2 − 3Iθ (1 + c) (1 − λ)2 ,
A22 = 3Iθ c2 + 3 + 3c (1 − λ)2 − 6Iθc2 (1 − λ) .

This in turn results in φPτ (θ; c, c) = e(β−c)/2 |I2 − 2AΣ|−1/2 ,

where Σ = var (X). It is now straightforward, but tedious, to establish the result. 

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