bootstrap seasonal unit root test under periodic variation · bootstrap seasonal unit root test...

Bootstrap seasonal unit root test under periodic variation

Nan Zoua, Dimitris N. Politisb

aDepartment of Statistical Sciences, University of Toronto, Toronto, ON M5S 3G3.bDepartment of Mathematics, University of California-San Diego, La Jolla, CA 92093.

Abstract

Both seasonal unit roots and periodic variation can be prevalent in seasonal data. When testingseasonal unit roots under periodic variation, the validity of the existing methods, such as the HEGYtest, remains unknown. This paper analyzes the behavior of the augmented HEGY test and theunaugmented HEGY test under periodic variation. It turns out that the asymptotic null distributionsof the HEGY statistics testing the single roots at 1 or −1 when there is periodic variation areidentical to the asymptotic null distributions when there is no periodic variation. On the otherhand, the asymptotic null distributions of the statistics testing any coexistence of roots at 1, −1, i,or −i when there is periodic variation are non-standard and are different from the asymptotic nulldistributions when there is no periodic variation. Therefore, when periodic variation exists, HEGYtests are not directly applicable to the joint tests for any concurrence of seasonal unit roots. As aremedy, bootstrap is proposed; in particular, the augmented HEGY test with seasonal independentand identically distributed (iid) bootstrap and the unaugmented HEGY test with seasonal blockbootstrap are implemented. The consistency of these bootstrap procedures is established. Thefinite-sample behavior of these bootstrap tests is illustrated via simulation and prevails over theircompetitors’. Finally, these bootstrap tests are applied to detect the seasonal unit roots in variouseconomic time series.

Keywords: Seasonality, Unit root, AR sieve bootstrap, Block bootstrap, Functional central limittheorem,

1. Introduction

As deterministic trend and unit root exist in time series, deterministic seasonality and seasonalunit root occur in seasonal time series. Intuitively, seasonal unit root process is a process withnon-stationary stochastic seasonality. The hypothesis testing for seasonal unit root dates back to[1, 2]. The most widely-used test may be the HEGY test proposed by [3]. Recent advances in thisvein include [4, 5, 6, 7].

In addition to non-stationary stochastic seasonality, the generating processes of seasonal timeseries may consist of periodically varying coefficients, for example, the process may be an AutoRe-gressive (AR) process with periodically varying AR parameters. Examples of such periodically

Email addresses: [email protected] (Nan Zou), [email protected] (Dimitris N. Politis)

Preprint submitted to Elsevier September 24, 2019

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varying time series include the consumption series in [8], the air pollutant series in [9], and theriver flows in [10]. Theoretical research on periodically varying processes includes, among others,[11, 12]. For more information on periodically varying time series, see [13, 14, 15].

Indeed, seasonal unit roots and periodic variation sometimes coexist in seasonal data. Forexample, the seasonal consumption in UK has been found periodically varying and seasonallyintegrated by [16] and by [3], respectively. As a result, it is important to design seasonal unit roottests that allow for periodic variation. In particular, consider quarterly data Y4t+s : t = 1, . . . ,T ,s = −3, . . . , 0 generated by

αs(L)Y4t+s = V4t+s, (1.1)

where αs(L) are seasonally varying AR filters, and Vt = (V4t−3, . . . ,V4t)′ is a weakly stationaryvector-valued process. If for all s = −3, . . . , 0, αs(L) have roots at 1, −1, or ±i, then respectivelyY4t+s has non-stationary stochastic components with period +∞, 2, or 4. The test for the seasonalroots at 1, −1, or ±i indeed precedes the removal of these non-stationary stochastic componentsand the inference on the detrended time series. To carry out this test for seasonal roots, [17] ap-plies Johansen’s method by [18], while [19] refer to the idea of likelihood ratio; however, bothapproaches limit scopes to finite order seasonal AR time series and cannot directly test the exis-tence of a certain root without first checking the number of seasonal unit roots. As a remedy, [20]design a Wald test that directly tests whether a certain root exists. However, the asymptotics of[20] is not totally correct according to [21], and the simulation in [20] shows the Wald test lesspowerful than the augmented HEGY test.

Can we directly apply the HEGY test in the periodic setting (1.1)? To the best of our knowl-edge, no literature has offered a satisfactory answer. [22] analyze the behavior of the augmentedHEGY test when only seasonal heteroscedasticity exists; [7] take into consideration the seasonalnon-stationary heteroscedasticity and the seasonal conditional heteroscedasticity but again limittheir scope to heteroscedasticity; [23] analyze the augmented HEGY test in the periodically inte-grated model, a model related to but different from model (1.1). No literature has ever touchedon the behavior of the unaugmented HEGY test proposed by [24], the important semi-parametricversion of the HEGY test. Since the unaugmented HEGY test does not assume the noise havingan AR structure, it may suit our non-parametric model (1.1) better.

To check the legitimacy of the HEGY test in the periodic setting (1.1), this paper derives theasymptotics of the unaugmented HEGY test and the augmented HEGY test. It turns out that,the asymptotic null distributions of the statistics testing the single roots at 1 or −1 are standard.More specifically, for each single root at 1 or −1, the asymptotic null distribution of the augmentedHEGY statistic is identical to that of Augmented Dickey-Fuller (ADF) test by [25], and the asymp-totic null distribution of the unaugmented HEGY statistic is identical to that of Phillips-Perron testby [26]. However, the asymptotic null distributions of the statistics testing any combination ofroots at 1, −1, i, or −i depend on the periodically varying coefficients, are non-standard and non-pivotal, and cannot be directly pivoted. Therefore, when periodic variation exists, the augmentedand the unaugmented HEGY tests can be applied to single roots at 1 or −1 but cannot be straight-forwardly applied to the coexistence of any roots.

As a remedy, this paper proposes the application of bootstrap. In general, bootstrap’s advan-tages are two fold. Firstly, bootstrap helps when the asymptotic distributions of the statistics of

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interest cannot be found or simulated. Secondly, even when the asymptotic distributions can befound and simulated, bootstrap method may enjoy second-order efficiency when these asymptoticdistributions are pivotal. For the aforementioned problem, bootstrap serves as an appealing so-lution. Firstly, it is hard to estimate the periodically varying parameters in the asymptotic nulldistributions, and it is hard to simulate these asymptotic null distributions. Secondly, it can beconjectured that the bootstrap seasonal unit root test inherits second order efficiency from thebootstrap non-seasonal unit root test when the asymptotic distributions are pivotal; see [27]. Themethodological literature we find on bootstrapping the HEGY test only includes [28, 7]. It willbe shown in Remark 3.9 that none of these bootstrap approaches is consistent under the generalperiodic setting (1.1).

To cater to the general periodic setting (1.1), this paper designs new bootstrap tests, namely1) the seasonal iid bootstrap augmented HEGY test, and 2) the seasonal block bootstrap unaug-mented HEGY test. When calculating the test statistics, the two tests run HEGY regression usingall data in order to preserve the orthogonal structure of the HEGY regression. On the other hand,when generating bootstrap replicates, both tests conduct season-by-season regressions to dupli-cate the periodic structure of the original data. In particular, the first test obtains residuals fromseason-by-season augmented HEGY regressions, and then applies the seasonal iid bootstrap tothe whitened regression errors, while the second test starts with season-by-season unaugmentedHEGY regressions, and then handles the correlated errors with the seasonal block bootstrap pro-posed by [29]. We establish the Functional Central Limit Theorem (FCLT) for both bootstrap testsand then demonstrates the consistency of both bootstrap procedures.

The paper proceeds as follows. Section 2 formalizes the settings, states the assumptions, andpresents the hypotheses. Section 3 gives the asymptotic null distributions of the augmented HEGYtest statistics, details the algorithm of the seasonal iid bootstrap augmented HEGY test, and es-tablishes the consistency of the bootstrap. Section 4 presents the asymptotic null distributionsof the unaugmented HEGY test statistics, specifies the algorithm of the seasonal block bootstrapunaugmented HEGY test, and proves the consistency of the bootstrap. Section 5 shows that insimulation our two bootstrap tests outperform their competitors, namely, the non-seasonal boot-strap augmented HEGY test by [28] and the Wald test by [20]. Section 6 applies our two bootstraptests to various economic time series. Appendix includes all technical proofs.

2. Periodically varying time series

Consider the real-valued quarterly data Y4t+s : t = 1, . . . ,T , s = −3, . . . , 0 generated by theseasonal model

αs(L)Y4t+s = V4t+s, (2.1)

where LY4t+s = Y4t+s−1 and αs(L) = 1 −∑4

j=1 α j,sL j. Suppose that for all s = −3, . . . , 0, the roots ofαs(L) are on or outside the unit circle. If for all s = −3, . . . , 0, αs(L) has roots on the unit circle,then suppose that for s = −3, . . . , 0, αs(L) share the same set of roots on the unit circle, this setof roots is a subset of 1,−1,±i, and Y−3 = Y−2 = Y−1 = Y0 = 0; otherwise, suppose that ourdata is a stretch of the process Y4t+s, t = . . . ,−1, 0, 1, . . . , s = −3, . . . , 0. Let V4t+s and α j,s be theprediction errors and coefficients of (2.1), respectively. More specifically, α j,s is defined such that

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for each s = −3, . . . , 0,V4t+s

de f= αs(L)Y4t+s

is orthogonal to Y4t+s−1, . . . ,Y4t+s−4. Let ε t = (ε4t−3, . . . , ε4t)′ and Bε t = ε t−1. Then B = L4. Denoteby AR(p) an AR process with order p, by MA Moving Average, by VMA(∞) a Vector MA processwith infinite moving average order, and by VARMA(p, q) a Vector ARMA process with AR orderp and MA order q. Let Re(z) be the real part of complex number z, bxc be the largest integersmaller or equal to real number x, and dxe be the smallest integer larger or equal to x.

Assumption 1.A. AssumeVt = Θ(B)ε t

where Θ(B) =∑∞

i=0ΘiBi; the ( j, k) entry of Θi, denoted by Θ( j,k)i , satisfies

∑∞i=1 i|Θi|

( j,k) < ∞ forall j and k; the determinant of Θ(z) has all roots outside the unit circle; Θ0 is a lower diagonalmatrix whose diagonal entries equal 1; ε t is a vector-valued white noise process with mean zeroand covariance matrix Ω; and Ω is diagonal.

Assumption 1.A assumes that Vt is VMA(∞) with respect to white noise innovations. This isequivalent to the assumption that Vt is a weakly stationary process with no deterministic part inthe multivariate Wold decomposition. The assumptions on Θ0 and the determinant of Θ(z) ensurethe causality and the invertibility of Vt and the identifiability of Ω.

Assumption 1.B. AssumeVt = Ψ(B)−1Λ(B)ε t ≡ Θ(B)ε t

where Ψ(B) =∑p

i=0ΨiBi; Λ(B) =∑q

i=0ΛiBi; the determinants of Ψ(z) and Λ(z) have all rootsoutside the unit circle; Ψ0 and Λ0 are lower diagonal matrices whose diagonal entries are 1; ε t isa vector-valued white noise process with mean zero and covariance matrix Ω; and Ω is diagonal.

Assumption 1.B restricts Vt to be VARMA(p, q) with respect to white noise innovation.Compared to the VMA(∞) model in Assumption 1.A, VARMA(p, q)’s main constraint is its expo-nentially decaying autocovariance. Again, the assumptions on Ψ0, Λ0 and the determinant of Ψ(z)and Λ(z) in Assumption 1.B ensure the causality and the invertibility of Vt and the identifiablityof Ω.

At this stage ε t is only assumed to be a white noise sequence of random vectors. In fact, ε t

needs to be weakly dependent as well; however, ε t needs not to be iid.

Assumption 2.A. (i) ε t is a fourth-order stationary, martingale difference vector-valued process.(ii) ∃K > 0, ∀ i, j, k, and l,

∑∞h=−∞ |Cov(εiε j, εk−hεl−h)| < K.

Assumption 2.B. (i) ε t is a strictly stationary, strong-mixing vector-valued process with finite4 + δ moment for some δ > 0. (ii) ε t’s strong mixing coefficient a(k) satisfies

∑∞k=1 k(a(k))δ/(4+δ) <

∞.

Notice that the assumption on the stationarity of the vector-valued process ε t is weaker thanan assumption on the stationarity of the scalar-valued process ε4t+s. In addition, the strong mixingcondition in Assumption 2.B actually guarantees (ii) of Assumption 2.A; see Lemma 3.

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Hypotheses. We tackle the following set of null hypotheses. The alternative hypotheses are thecomplements of the null hypotheses.

H10 : αs(1) = 0, ∀s = −3, . . . , 0.

H20 : αs(−1) = 0, ∀s = −3, . . . , 0.

H1,20 : αs(1) = αs(−1) = 0, ∀s = −3, . . . , 0.

H3,40 : αs(i) = αs(−i) = 0, ∀s = −3, . . . , 0.

H1,3,40 : αs(1) = αs(i) = αs(−i) = 0, ∀s = −3, . . . , 0.

H2,3,40 : αs(−1) = αs(i) = αs(−i) = 0, ∀s = −3, . . . , 0.

H1,2,3,40 : αs(1) = αs(−1) = αs(i) = αs(−i) = 0, ∀s = −3, . . . , 0.

Indeed, the alternative hypotheses can be written as one-sided hypotheses. Notice that for alls = −3, . . . , 0, αs(0) = 1, αs(·) is continuous, and the roots of αs(·) are either on or outside the unitcircle. By the intermediate value theorem, αs(1) , 0 implies that αs(1) > 0, αs(−1) , 0 impliesthat αs(−1) > 0, and αs(i) , 0 implies that Re(αs(i)) > 0.

To analyze the roots of αs(L), [3] propose the partial fraction decomposition

αs(L)1 − L4 = λ0,s +

λ1,s

1 − L+

λ2,s

1 + L+λ3,sL + λ4,s

1 + L2 ;

thusαs(L) = λ0,s(1 − L4)

+ λ1,s(1 + L)(1 + L2) + λ2,s(1 − L)(1 + L2)+ λ3,s(1 − L)(1 + L)L + λ4,s(1 − L)(1 + L).

(2.2)

Substituting (2.2) into (2.1), we get

(1 − L4)Y4t+s =

4∑j=1

π j,sY j,4t+s−1 + V4t+s, (2.3)

whereY1,4t+s = (1 + L)(1 + L2)Y4t+s, Y2,4t+s = −(1 − L)(1 + L2)Y4t+s,

Y3,4t+s = −L(1 − L2)Y4t+s, Y4,4t+s = −(1 − L2)Y4t+s,

π1,s = −λ1,s, π2,s = −λ2,s,

π3,s = −λ4,s, π4,s = λ3,s.

(2.4)

By (2.2) and (2.4), π j,s relates to the root of αs(z).

Proposition 2.1 ([3]).

αs(1) = 0 ⇐⇒ π1,s = 0, αs(1) , 0 ⇐⇒ π1,s < 0,αs(−1) = 0 ⇐⇒ π2,s = 0, αs(−1) , 0 ⇐⇒ π2,s < 0,αs(i) = 0 ⇐⇒ αs(−i) = 0 ⇐⇒ π3,s = π4,s = 0, αs(i) , 0 ⇐⇒ αs(−i) , 0 ⇐⇒ π3,s < 0.

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By Proposition 2.1, the test for the null hypotheses can be carried on by checking the cor-responding π j,s, where π j,s can be estimated by Ordinary Least Squares (OLS) regression. Toestimate π j,s by OLS, one might first attempt to implement the OLS season by season with season-specific coefficients. Unfortunately, this season-by-season regression indeed has a non-orthogonaldesign matrix; see [13], p. 158, and Lemma 1. On the other hand, since the non-periodic regres-sions (3.1) and (4.1) preserve the orthogonality, we will instead apply the non-periodic regressionequations (3.1) and (4.1).

When we regress Y4t+s with non-periodic regression equations (3.1) and (4.1), the periodi-cally varying sequence V4t+s is fitted in misspecified non-periodic AR models. Consider, as anexample, fitting V4t+s in a misspecified AR(1) model Vτ = φVτ−1+ ζτ. Then φ = γ(1)/γ(0)+op(1),where

γ(h) =14

0∑s=−3

E[V4t+sV4t+s−h]. (2.5)

Since γ(·) is positive semi-definite, we can find a weakly stationary sequence Vτ with mean zeroand autocovariance function γ(·). We call Vτ a misspecified constant parameter representation ofV4t+s; see also [30]. We will refer to Vτ in later sections.

3. Seasonal iid bootstrap augmented HEGY Test

3.1. Augmented HEGY test[3] assume that the π j,s and V4t+s in (2.3) do not depend on s. Consequently, they propose to

run the OLS regression equation

(1 − L4)Yτ =

4∑j=1

πAj Y j,τ−1 +

k∑i=1

φi(1 − L4)Yτ−i + ζAτ , (3.1)

where augmentations (1 − L4)Yτ−i, i = 1, 2, . . . , k, pre-whiten the time series (1 − L4)Yτ up to anorder of k. If k → ∞ as sample size T → ∞, the residual ζA

τ will be asymptotically uncorrelated.Let A stands for “Augmented”. Let πA

j be the OLS estimator in (3.1), tAj be the t-statistics

corresponding to πAj , and FA

3,4 be the F-statistic corresponding to πA3 and πA

4 . Other F-statistics FA1,2,

FA1,3,4, FA

2,3,4, and FA1,2,3,4 can be defined similarly. Where there is no periodic variation, [3] proposes

to reject H10 if πA

1 is too small, reject H20 if πA

2 is too small, reject H3,40 if FA

3,4 is too large, and rejectother composite hypotheses if their corresponding F-statistics are too large.

3.2. Augmented HEGY test under model misspecificationNow we apply the augmented HEGY test to periodically varying processes. Namely, we run

regression equation (3.1) with Y4t+s generated by (2.1). Our results show that when testing rootsat 1 or −1 separately, the t-statistics tA

1 , tA2 , and the F-statistics have pivotal asymptotic distributions.

On the other hand, when testing joint roots at 1 and −1, and when testing hypotheses that involveroots at ±i, the asymptotic distributions of the testing statistics are non-pivotal and cannot be easilypivoted.

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Theorem 3.1. Assume that Assumption 1.B and one of Assumption 2.A or 2.B hold. Further,assume T → ∞, k = kT → ∞, k = o(T 1/3), and ck > T 1/α for some c > 0 and α > 0. Then underH1,2,3,4

0 , the asymptotic distributions of tAj , j = 1, 2, and F-statistics are given by

tAj ⇒

∫ 1

0W j(r)dW j(r)√∫ 1

0W2

j (r)dr≡ ξ j, j=1,2,

FA1,2 ⇒

12

(ξ21 + ξ2

2), FA3,4 ⇒

12

(ξ23 + ξ2

4),

FA1,3,4 ⇒

13

(ξ21 + ξ2

3 + ξ24), FA

2,3,4 ⇒13

(ξ22 + ξ2

3 + ξ24),

FA1,2,3,4 ⇒

14

(ξ21 + ξ2

2 + ξ23 + ξ2

4), with

ξ3 =λ2

3

∫ 1

0W3(r)dW3(r) + λ2

4

∫ 1

0W4(r)dW4(r)√

(λ23 + λ2

4)( 12λ

23

∫ 1

0W2

3 (r)dr + 12λ

24

∫ 1

0W2

4 (r)dr),

ξ4 =λ3λ4(

∫ 1

0W3(r)dW4(r) −

∫ 1

0W4(r)dW3(r))√

(λ23 + λ2

4)( 12λ

23

∫ 1

0W2

3 (r)dr + 12λ

24

∫ 1

0W2

4 (r)dr),

where c1 = (1, 1, 1, 1)′, c2 = (1,−1, 1,−1)′, c3 = (0,−1, 0, 1)′, c4 = (−1, 0, 1, 0)′, λ j =√

c′jΘ(1)ΩΘ(1)′c j/4,

W j = c′jΘ(1)Ω1/2W/(2λ j), and W(·) is a four-dimensional standard Brownian motion.

Remark 3.1. The asymptotic distributions presented in Theorem 3.1 degenerate to the distributionsin [31] and [32] when V4t+s has neither periodic variation nor seasonal heteroscedasticity, and tothe distributions in [22] when V4t+s is a heteroscedastic, finite-order AR sequence with non-periodic AR coefficients.

Remark 3.2. Notice that each of W j is a standard Brownian motion. When V4t+s has no periodicvariation, W j’s are independent, so are the asymptotic distributions of tA

1 and tA2 . On the other

hand, when V4t+s has periodic variation, W j’s are in general dependent, so tA1 and tA

2 are in generaldependent, even asymptotically. Hence, when testing H1,2

0 , it is problematic to test H10 and H2

0separately and calculate the size of the test with the independence of tA

1 and tA2 in mind. Instead,

the test of H1,20 should be handled with FA

1,2.

Remark 3.3. Because of the dependence of tA1 and tA

2 , the asymptotic distribution of FA1,2 under

periodic variation is different from its non-periodic counterpart. More generally, the asymptoticdistributions of any aforementioned F-statistics under periodic variation is different from theirnon-periodic counterparts. Hence, the augmented HEGY test cannot be directly applied to testany coexistence of roots at 1, −1, i, or −i under potential periodic variation. (From another pointof view, the asymptotic distribution of any F-statistics does not solely depend on the distributionof Vτ, the misspecified constant parameter representation of V4t+s; hence the F-tests are trulyaffected by the periodic variation.)

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Remark 3.4. When V4t+s is only seasonally heteroscedastic, as in [22],Θ(1) does not occur in theasymptotic distributions of the F-statistics. On the other hand, when V4t+s has generic periodicvariation, Θ(1) impacts first the correlation between Brownian motions W3 and W4, and secondthe weights λ3 and λ4.Remark 3.5. As [22] point out, the dependence of the asymptotic distributions on weights λ3 andλ4 can be expected. Indeed, Y3,4t+s = Y4,4t+s−1 is the partial sum of −V4t+s−1,V4t+s−3, . . . , whileY3,4t+s+1 = Y4,4t+s is the partial sum of −V4t+s,V4t+s−2, . . . . Since these two partial sums differ intheir variances, both

∑s,t Y3,4t+s and

∑s,t Y4,4t+s involve two different weights λ3 and λ4.

Remark 3.6. Theorem 3.1 presents the asymptotics when Y4t+s is generated under H1,2,3,40 , that is,

when Y4t+s has all roots at 1, −1, and ±i. When Y4t+s is generated under other null hypotheses inSection 2, that is, when Y4t+s has some but not all roots at 1, −1, and ±i, we let Uτ = (1 − L4)Yτ,Ut = (U4t−3,U4t−2,U4t−1,U4t)′, and define H(z) such that Ut = H(B)ε t. The asymptotic distri-butions under other null hypotheses has exactly the same form as those in Theorem 3.1, exceptthat Θ(1) is replaced by H(1). When there is no periodic variation, these asymptotic distributionsdegenerate to those in [33] and [34].Remark 3.7. The preceding results give the asymptotic behaviors of the testing statistics under thenull hypotheses. Under the alternative hypotheses, we conjecture that the power of the augmentedHEGY tests tends to one as the sample size goes to infinity. Indeed, if Y4t+s does not have acertain unit root at 1, −1, or ±i, then by the asymptotic orthogonality of regression equation (3.1),we can without loss of generality assume that Y4t+s has none of the unit roots at 1, −1, or ±i.If Y4t+s has none of the unit roots at 1, −1, or ±i, then it has a stationary misspecified constantparameter representation. Then, by [35], for j = 1, 2, 3, πA

j converge in probability; by Proposition2.1, the limits of πA

j , j = 1, 2, 3, are negative. See also Theorem 2.2 of [36].

3.3. Seasonal iid bootstrap algorithmTo accommodate the non-pivotal asymptotic null distributions of the augmented HEGY test

statistics, we propose the application of bootstrap. Specifically, we first pre-whiten the data seasonby season to obtain uncorrelated noises. Although these noises are uncorrelated, they are notidentically distributed due to seasonal heteroscedasticity. Hence, we second resample season byseason to generate bootstrapped noise, as in [28]. Finally, we post-color the bootstrapped noise.The detailed algorithm of this seasonal iid bootstrap augmented HEGY test is given below.

Algorithm 3.1. Step 1: calculate tA1 and tA

2 , the t-statistics corresponding to πA1 and πA

2 , and theF-statistics FA

B, B = 1, 2, 3, 4, 1, 3, 4, 2, 3, 4, 1, 2, 3, 4, from the augmented non-periodic

HEGY test regression

(1 − L4)Yτ =

4∑j=1

πAj Y j,τ−1 +

k∑i=1

φi(1 − L4)Yτ−i + ζAτ ;

Step 2: record OLS estimators πAj,s, φi,s and residuals ε4t+s from the season-by-season regression

(1 − L4)Y4t+s =

4∑j=1

πAj,sY j,4t+s−1 +

k∑i=1

φi,s(1 − L4)Y4t+s−i + ε4t+s;

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Step 3: let ε4t+s = ε4t+s −1T

∑Tt=bk/4c+1 ε4t+s. Store demeaned residuals ε4t+s of the four seasons

separately, then independently draw four iid samples from each of their empirical distributions,and then combine these four samples into a vector ε?4t+s, with their seasonal orders preserved;

Step 4: set all πAj,s corresponding to the null hypothesis to be zero. For example, set πA

3,s = πA4,s = 0

for all s when testing roots at ±i. Let Y?4t+s be generated by

(1 − L4)Y?4t+s =

4∑j=1

πAj,sY

?j,4t+s−1 +

k∑i=1

φi,s(1 − L4)Y?4t+s−i + ε?4t+s;

Step 5: calculate t?1 and t?2 , the t-statistics corresponding to π?1 and π?2 , and F-statistics F?B

fromthe non-periodic regression

(1 − L4)Y?τ =

4∑j=1

π?j Y?j,τ−1 +

k∑i=1

φ?i (1 − L4)Y?τ−i + ζ?τ ;

Step 6: repeat steps 3, 4, and 5 for B times to get B sets of t-statistics t?1 , t?2 , and F-statistics F?B

.Count separately the numbers of t?1 , t?2 , and F?

B, than which tA

1 , tA2 , and the F-statistics FA

Bare more

extreme. If these numbers are higher than B(1 − size), then we consider tA1 , tA

2 , and the F-statisticsFAB

extreme, and reject the corresponding hypotheses.

Remark 3.8. It is also reasonable to keep steps 1, 2, 3, 5, and 6 of the Algorithm 3.1, but changethe generation of Y?

4t+s in step 4 to

(1 − L4)Y?4t+s =

k∑i=1

φi,s(1 − L4)Y?4t+s−i + ε?4t+s. (3.2)

This new algorithm is in fact theoretically invalid for the tests of any coexistence of roots (seeRemark 3.3, 3.4, and 3.6), but it is valid for tests of any single roots at 1 or −1, due to the pivotalasymptotic distributions of tA

1 and tA2 in Theorem 3.1.

Remark 3.9. If we let steps 1, 3, 5, and 6 be the same as in Algorithm 3.1, but run non-periodicregression equations with non-periodic coefficients πA

j and φi in steps 2 and 4, then this version ofalgorithm is identical with [28]. However, the step 2 of this new version cannot fully pre-whitenthe time series, and consequently leaves the regression error ζA

τ serially correlated. When ζAτ

is bootstrapped by the seasonal iid bootstrap in step 3, this serial correlation structure is ruined.As a result, (1 − L4)Y?

4t+s differs from (1 − L4)Y4t+s in its correlation structure, in particular Θ(1),and consequently the conditional distributions of the bootstrap F-statistics F?

Bdiffer from the

distributions of the original F-statistics FAB

; see Remark 3.3 and 3.4. Similarly, the conditionaldistributions of the wild bootstrap F-statistics in [7] differ from the real-world distributions ofthese F-statistics.

9

3.4. Consistency of seasonal iid bootstrapNow we justify the seasonal iid bootstrap augmented HEGY test (Algorithm 3.1). Since the

derivation of the real-world asymptotic distributions in Theorem 3.1 calls on FCLT (see Lemma1), the justification of bootstrap approach also requires FCLT in the bootstrap world. From now on,let P, E, Var, S td, Cov be the bootstrap probability, expectation, variance, standard deviation,and covariance, respectively, conditional on our data Y4t+s.

Proposition 3.1. Suppose the assumptions in Theorem 3.1 hold. Let S ?T (u1, u2, u3, u4)

=1√

4T

( b4Tu1c∑τ=1

ε?τ /σ?1 ,

b4Tu2c∑τ=1

(−1)τε?τ /σ?2 ,

b4Tu3c∑τ=1

√2 sin

(πτ2

)ε?τ /σ

?3 ,

b4Tu4c∑τ=1

√2 cos

(πτ2

)ε?τ /σ

?4

)′,

where

σ?1 = S td

[ 1√

4T

4T∑τ=1

ε?τ

], σ?

2 = S td[ 1√

4T

4T∑τ=1

(−1)τε?τ],

σ?3 = S td

[ 1√

4T

4T∑τ=1

√2 sin

(πτ2

)ε?τ

], σ?

4 = S td[ 1√

4T

4T∑τ=1

√2 cos

(πτ2

)ε?τ

].

Then, no matter which hypothesis is true, S ?T ⇒ W? in probability as T → ∞, where W?(·) is a

four-dimensional standard Brownian motion.

By the FCLT given by Proposition 3.1 and the proof of Theorem 3.1, in probability the con-ditional distributions of t?j , j = 1, 2, and F?

Bconverge to the limiting distributions of tA

j , j = 1, 2,and FA

B, respectively. Indeed, since conditional on Y4t+s, Y?

4t+s is a finite-order seasonal ARprocess, the derivation of the conditional distributions of t?j , j = 1, 2, and F?

Bturns out easier than

that of Theorem 3.1, and in particular does not involve the fourth moments of Y?4t+s. Hence the

consistency of the bootstrap.

Theorem 3.2. Suppose the assumptions in Theorem 3.1 hold. Let PB be the probability measurecorresponding to the null hypothesis HB

0 . For example, P1,2 corresponds to the null hypothesisH1,2

0 . Then,

supx|P(t?j ≤ x) − P j(tA

j ≤ x)|p→ 0, j = 1, 2,

supx|P(F?

B ≤ x) − PB(FAB ≤ x)|

p→ 0, where B = 1, 2, 3, 4, 1, 3, 4, 2, 3, 4, or 1, 2, 3, 4.

4. Seasonal block bootstrap unaugmented HEGY test

4.1. Unaugmented HEGY testIn the preceding section our analysis focuses on the augmented HEGY test, an extension of

the ADF test to the seasonal unit root setting. An important alternative of the ADF test is thePhillips-Perron test ([26]). While the ADF test assumes an AR structure over the noise and thus

10

becomes parametric, its semi-parametric counterpart, Phillips-Perron test, allows a wide classof weakly dependent noises. The unaugmented HEGY test ([24]), as the extension of Phillips-Perron test to the seasonal unit root, inherits the semi-parametric nature and does not assume thenoise to be AR. Given periodic variation, it will be shown in Theorem 4.1 that the unaugmentedHEGY test estimates seasonal unit roots consistently under a very general VMA(∞) class of noise(Assumption 1.A), instead of a more restrictive VARMA(p, q) class of noise (Assumption 1.B),which we need for the augmented HEGY test.

Now we specify the unaugmented HEGY test. Consider the non-periodic regression equation

(1 − L4)Yτ =

4∑j=1

πUj Y j,τ−1 + Vτ, (4.1)

where U stands for “Unaugmented”. Let πUj be the OLS estimator in (4.1), Vτ be the OLS residual,

tUj be the t-statistic corresponding to πU

j , and FU3,4 be the F-statistic corresponding to πU

3 and πU4 .

Other F-statistics FU1,2, FU

1,3,4, FU2,3,4, and FU

1,2,3,4 can be defined analogously. Similar to the Phillips-Perron test, the unaugmented HEGY test can apply both πU

j and tUj when testing roots at 1 or −1.

As in the augmented HEGY test, we reject H10 if πU

1 (or tU1 ) is too small, reject H2

0 if πU2 (or tU

2 )is too small, and reject the joint hypotheses if the corresponding F-statistics are too large. Thefollowing results give the asymptotic null distributions of πU

j , tUj , j = 1, . . . , 4, and the F-statistics.

4.2. Unaugmented HEGY test under model misspecificationTheorem 4.1. Assume that Assumption 1.A and one of Assumption 2.A or Assumption 2.B hold.Then under H1,2,3,4

0 , as T → ∞,

(4T )πUj ⇒

λ2j

∫ 1

0W j(r)dW j(r) + Γ( j)

λ2j

∫ 1

0W2

j (r)dr, for j = 1, 2,

(4T )πU3 ⇒

λ23

∫ 1

0W3(r)dW3(r) + λ2

4

∫ 1

0W4(r)dW4(r) + Γ(3)

12 (λ2

3

∫ 1

0W2

3 (r)dr + λ24

∫ 1

0W2

4 (r)dr),

(4T )πU4 ⇒

λ3λ4(∫ 1

0W3(r)dW4(r) −

∫ 1

0W4(r)dW3(r)) + Γ(4)

12 (λ2

3

∫ 1

0W2

3 (r)dr + λ24

∫ 1

0W2

4 (r)dr),

tUj ⇒

λ2j

∫ 1

0W j(r)dW j(r) + Γ( j)√γ(0)λ2

j

∫ 1

0W2

j (r)dr≡ D j, for j = 1, 2,

tU3 ⇒

λ23

∫ 1

0W3(r)dW3(r) + λ2

4

∫ 1

0W4(r)dW4(r) + Γ(3)√

γ(0)12 (λ2

3

∫ 1

0W2

3 (r)dr + λ24

∫ 1

0W2

4 (r)dr)≡ D3

tU4 ⇒

λ3λ4(∫ 1

0W3(r)dW4(r) −

∫ 1

0W4(r)dW3(r)) + Γ(4)√

γ(0)12 (λ2

3

∫ 1

0W2

3 (r)dr + λ24

∫ 1

0W2

4 (r)dr)≡ D4

11

FU1,2 ⇒

12

(D21 + D2

2 ), FU3,4 ⇒

12

(D23 + D2

4 ),

FU1,3,4 ⇒

13

(D21 + D2

3 + D24 ), FU

2,3,4 ⇒13

(D22 + D2

3 + D24 ),

FU1,2,3,4 ⇒

14

(D21 + D2

2 + D23 + D2

4 ),

where c1 = (1, 1, 1, 1)′, c2 = (1,−1, 1,−1)′, c3 = (0,−1, 0, 1)′, c4 = (−1, 0, 1, 0)′, λ j =√c′jΘ(1)ΩΘ(1)′c j/4, W j = c′jΘ(1)Ω1/2W/(2λ j), W(·) is the same four-dimensional standard

Brownian motion as in Theorem 3.1, γ( j) are defined in (2.5), Γ(1) =∑∞

j=1 γ( j), Γ(2) =∑∞

j=1(−1) jγ( j),Γ(3) =

∑∞j=1 cos(π j/2)γ( j), and Γ(4) = −

∑∞j=1 sin(π j/2)γ( j).

Remark 4.1. The results in Theorem 4.1 degenerate to the asymptotics in [22] and [31] whenV4t+s is serially uncorrelated, to the asymptotics in [24] when V4t+s has no periodic variation,and to the asymptotics in [37] when V4t+s is seasonally heteroscedastic.

Remark 4.2. When V4t+s has no periodic variation, as in [24], the asymptotic distributions of(πU

1 , tU1 ) and (πU

2 , tU2 ) are independent. On the other hand, when V4t+s has periodic variation,

(πU1 , t

U1 ) and (πU

2 , tU2 ) are dependent, as what we have seen for the augmented HEGY test in Remark

3.3. Hence, when testing H1,20 , it is problematic to test H1

0 and H20 separately and calculate the size

of the test with the independence of (πU1 , t

U1 ) and (πU

2 , tU2 ) in mind. Instead, the test of H1,2

0 shouldbe handled with FU

1,2.

Remark 4.3. The parameters λ j have the same definition as in Theorem 3.1. Since λ21 =

∑∞j=−∞ γ( j),

and λ22 =

∑∞j=−∞(−1) jγ( j), the asymptotic distributions of πU

j and tUj , j = 1, 2, only depends on

the autocorrelation function of Vτ, a misspecified constant parameter representation of V4t+s

whose autocovariance function is given by (2.5). Since Vτ can be considered as a non-periodicversion of V4t+s, we can conclude that the asymptotic behaviors of the tests for H1

0 and H20 are

not affected by the periodic variation in V4t+s. On the other side, the asymptotic distributions ofthe F-statistics do not solely depend on the distribution of Vτ. Hence, the tests for all hypothesesother than H1

0 and H20 are affected by the periodic variation.

Remark 4.4. To remove the nuisance parameters in the asymptotic distributions, we notice thatthe asymptotic behaviors of πU

j and tUj , j = 1, 2, have identical forms as in [26]. In light of their

approach, we can construct pivotal versions of πUj and tU

j , j = 1, 2, that converge in distributionto standard Dickey-Fuller distributions in [25]; see also [37]. More specifically, for j = 1, 2, byTheorem 4.1 we have

(4T )πUj −

12 (λ2

j − γ(0))

(4T )−2 ∑4Tτ=1 Y2

j,τ−1

⇒

∫ 1

0W j(r)dW j(r)∫ 1

0W2

j (r)dr,

√γ(0)λ j

tUj −

12 (λ2

j − γ(0))

λ2j

√(4T )−2 ∑4T

τ=1 Y2j,τ−1

⇒

∫ 1

0W j(r)dW j(r)√∫ 1

0W2

j (r)dr.

where λ2j and γ(0) can by substituted by their consistent estimators.

12

Remark 4.5. However, there is no easy way to construct pivotal statistics for πU3 , tU

3 , πU4 , tU

4 , andF-statistics such as FU

3,4. The difficulties are two-fold. Firstly the denominators of the asymptoticdistributions of these statistics contain weighted sums with unknown weights λ2

3 and λ24; secondly

W3 and W4 are in general correlated standard Brownian motions as in Theorem 3.1.Remark 4.6. The result in Theorem 4.1 can be generalized. Suppose Y4t+s is not generatedby H1,2,3,4

0 , and only has some of the seasonal unit roots. Let Uτ = (1 − L4)Yτ, and Ut =

(U4t−3,U4t−2,U4t−1,U4t)′. Define H(z) such that Ut = H(B)ε t. The asymptotic distributions ofπU

j , tUj , j = 1, 2, and the F-statistics have the same forms as those in Theorem 4.1, with Θ(1)

substituted by H(1), and γ based on Uτ.Remark 4.7. Under one of the alternative hypotheses, we conjecture that for j = 1, 2, 3, the OLSestimators πU

j in (4.1) converge in probability to π j, the prediction coefficient of the misspecifiedconstant parameter representation of Y4t+s. Since under the alternative hypotheses we can withoutloss of generality assume Y4t+s is stationary, we have π j < 0. Hence, as a result of this conjecture,the power of the unaugmented HEGY tests tends to one as the sample size goes to infinity.

4.3. Seasonal block bootstrap algorithmSince many of the asymptotic distributions delivered in Theorem 4.1 are non-standard and non-

pivotal and cannot be easily pivoted, we propose the application of bootstrap. Since the regressionerror V4t+s of (4.1) has periodic structure, we may apply the seasonal block bootstrap of [29].The algorithm of the seasonal block bootstrap unaugmented HEGY test is illustrated below.

Algorithm 4.1. Step 1: get the OLS estimators πU1 , πU

2 , t-statistics tU1 , tU

2 , and the F-statistics FUB

,B = 1, 2, 3, 4, 1, 3, 4, 2, 3, 4, and 1, 2, 3, 4, from the unaugmented HEGY regression

(1 − L4)Yτ =

4∑j=1

πUj Y j,τ−1 + ζU

τ , τ = 1, . . . , 4T ;

Step 2: record residual V4t+s from regression

(1 − L4)Y4t+s =

4∑j=1

πUj,sY j,4t+s−1 + V4t+s;

Step 3: let V4t+s = V4t+s −1T

∑Tt=1 V4t+s, choose a integer block size b, and let l = b4T/bc. For

t = 1, b + 1, . . . , (l − 1)b + 1, let

(V∗t , . . . ,V∗t+b−1) = (VIt , . . . , VIt+b−1),

where It is a sequence of iid uniform random variables taking values in t − 4R1,n, . . . , t − 4, t, t +

4, . . . , t + 4R2,n with R1,n = b(t − 1)/4c and R2,n = b(n − b − t + 1)/4c;

Step 4: set the πUj,s corresponding to the null hypothesis to be zero. For example, set πU

3,s = πU4,s = 0

for all s when testing roots at ±i. Generate Y∗4t+s by

(1 − L4)Y∗4t+s =

4∑j=1

πUj,sY∗j,4t+s−1 + V∗4t+s;

13

Step 5: get OLS estimates π∗1, π∗2, t-statistics t∗1, t∗2, and F-statistics F∗B

from regression

(1 − L4)Y∗τ =

4∑j=1

π∗jY∗j,τ−1 + ζ∗τ , τ = 1, . . . , 4T ;

Step 6: repeat steps 3, 4, and 5 for B times to get B sets of statistics π∗1, π∗2, t∗1, t∗2, and F∗B

. Countseparately the numbers of π∗1, π∗2, t∗1, t∗2, and F∗

Bthan which πU

1 , πU2 , tU

1 , tU2 , and FU

Bare more extreme.

If these numbers are higher than B(1 − size), then consider πU1 , πU

2 , tU1 , tU

2 and FUB

extreme, andreject the corresponding hypotheses.

4.4. Consistency of seasonal block bootstrapProposition 4.1. Let S ∗T (u1, u2, u3, u4)

=1√

4T

( b4Tu1c∑τ=1

V∗τ/σ∗1,

b4Tu2c∑τ=1

(−1)τV∗τ/σ∗2,

b4Tu3c∑τ=1

√2 sin

(πτ2

)V∗τ/σ

∗3,

b4Tu4c∑τ=1

√2 cos

(πτ2

)V∗τ/σ

∗4

)′,

where

σ∗1 = S td[ 1√

4T

4T∑τ=1

V∗τ], σ∗2 = S td

[ 1√

4T

4T∑τ=1

(−1)τV∗τ],

σ∗3 = S td[ 1√

4T

4T∑τ=1

√2 sin

(πτ2

)V∗τ

], σ∗4 = S td

[ 1√

4T

4T∑τ=1

√2 cos

(πτ2

)V∗τ

].

If b → ∞, T → ∞, b/√

T → 0, then no matter which hypothesis is true, S ∗T ⇒W∗ in probability,where W∗(·) is a four-dimensional standard Brownian motion.

By the FCLT given by Proposition 4.1, the proof of Theorem 4.1, and the convergence ofthe bootstrap standard deviation σ∗j in [29], we have that the conditional distributions of t∗j , π

∗j,

j = 1, 2, and F∗B

in probability converges to the limiting distributions of πUj , tU

j , j = 1, 2, and FUB

,respectively. Hence the consistency of the bootstrap.

Theorem 4.2. Suppose the assumptions in Theorem 4.1 hold. Let PB be the probability measurecorresponding to the null hypothesis HB

0 . For example, P1,2 corresponds to the null hypothesisH1,2

0 . If b→ ∞, T → ∞, b/√

T → 0, then

supx|P(π∗j ≤ x) − P j(πU

j ≤ x)|p→ 0, j = 1, 2,

supx|P(t∗j ≤ x) − P j(tU

j ≤ x)|p→ 0, j = 1, 2,

supx|P(F∗B ≤ x) − PB(FU

B ≤ x)|p→ 0, where B = 1, 2, 3, 4, 1, 3, 4, 2, 3, 4, or 1, 2, 3, 4.

14

5. Simulation

5.1. Data generating processWe focus on the hypothesis testing for root at 1, i.e., H1

0 against H11 , roots at ±i, i.e., H3,4

0 againstH3,4

1 , and roots at 1, −1, and ±i, i.e., H1,2,3,40 against H1,2,3,4

1 . In the first two hypothesis tests, weequip one sequence with all nuisance unit roots, and the other with none of the nuisance unit roots.The detailed data generating processes are listed in Table 1. In an unreported simulation we havesimulated the hypothesis test for root at −1, i.e., H2

0 against H21 , but found the simulation result to

a large extent similar to the result of root at 1.To produce power curves, we let parameter ρ = 0.00, 0.02, 0.04, 0.06, 0.08, and 0.10. Notice

that ρ is set to be seasonally homogeneous for the sake of simplicity. Further, we generate six typesof innovations V4t+s according to Table 2, where εt ∼ iid N(0, 1). The values of φs in Table 2 areassigned so that the misspecified constant parameter representation (see Section 2) of the “arper”sequence has almost the same AR structure as the “arpos” sequence. Notice that in the “maper”setting in Table 2, V4t+s = (1 − θsL)−1(1 − θsθs−1L2)ε4t+s; the values of θs are assigned such that apotential seasonal unit root filter (1 + L2) is partially cancelled out by the MA filter (1 − θsθs−1L2)above.

Table 1: Data generation processes

Data GeneratingProcesses

Nuisance RootNo Yes

Root1 (1 − (1 − ρ)L)Yτ = Vτ (1 + L)(1 + L2)(1 − (1 − ρ)L)Yτ = Vτ

±i (1 + (1 − ρ)L2)Yτ = Vτ (1 + L)(1 − L)(1 + (1 − ρ)L2)Yτ = Vτ

1,−1,±i (1 − (1 − ρ)L4)Yτ = Vτ

Table 2: Types of noises

NoiseType

iid Vτ = ετ

heterV4t+s = σsε4t+s,

σ1 = 10, σ2 = σ3 = σ4 = 1arpos Vτ = ετ + 0.5Vτ−1

maneg Vτ = ετ − 0.5ετ−1

arperV4t+s = ε4t+s + φsV4t+s−1,

φ1 = 0.2, φ2 = 0.45, φ3 = 0.65, φ4 = 0.8

maperV4t+s = ε4t+s + θsε4t+s−1,

θ1 = 0.5, θ2 = −1.8, θ3 = 0.5, θ4 = −1.8

5.2. Testing procedureHere we give additional implemental details for the algorithms of the seasonal iid bootstrap

augmented HEGY test described in Algorithm 3.1, the seasonal block bootstrap unaugmentedHEGY test described in Algorithm 4.1, the non-seasonal bootstrap augmented HEGY test by [28],and the Wald test by [20].

15

5.2.1. Seasonal iid bootstrap augmented HEGY testTo improve the empirical performance of seasonal iid bootstrap algorithm (Algorithm 3.1), we

select stepwise, truncate the coefficient estimators, and apply (3.2) when testing roots at 1 or −1.Firstly, a stepwise selection procedure is applied to the regression in step 2 of Algorithm 3.1. Tobegin with, we choose a maximal order of lag kmax. kmax may be chosen by AIC, BIC, or modifiedinformation criterion by [38] (for further discussions, see [5]). In our simulation we fix kmax = 4for simplicity. Afterward, we apply a backward stepwise selection with Variance Inflating Factor(VIF) criterion to solve the multicollinearity between the regressors. In this selection, we locate theregressor with the largest VIF, remove this regressor from the regression if its VIF is larger than 10,and rerun the regression. Then we implement another stepwise selection on lags (1 − L4)Y4t+s−i,i = 1, 2, . . . , k, by iteratively removing lags of which the absolute values of the t-statistics aresmaller than 1.65; see also [28]. Then the estimated coefficients of the deleted regressors are set tobe zero, while the estimated coefficients of the remaining regressors are recorded and used in step2 and 4. The backward stepwise selection of the lags based on their t-statistics is also applied tostep 1 and 5.

Secondly, notice that in step 2, the true parameters π j,s, j = 1, 2, 3, are smaller or equal tozero under both null and alternative hypotheses. However, the OLS estimators πA

j,s, j = 1, 2, 3,are often positive, especially when π j,s = 0. This positivity not only renders the estimation of π j,s

inaccurate, but also makes the equation in step 4 of Algorithm 3.1 non-causal, and the bootstrapsequence Y?

4t+s explosive. The solution of this problem is to truncate the OLS estimator. LetπA

j,s = min(0, πAj,s), j = 1, 2, 3. Immediately we get |πA

j,s − πAj,s| ≤ |π

Aj,s − π j,s|. After we substitute πA

j,s

for πAj,s in step 4, the empirical performance of seasonal iid bootstrap improves significantly.

Thirdly, by Assumption 1.B, intuitively the true parameters φi,s in step 2 should make the rootsof polynomial φs(z) B 1−φ1,sz−φ2,sz2−· · ·−φk,szk staying outside the unit circle. On the other hand,the roots of polynomial φs(z) B 1 − φ1,sz − φ2,sz2 − · · · − φk,szk are sometimes on or inside the unitcircle. To correct φs(z), suppose that φs(z) can be factored out as φs(z) = (1−r1,sz)(1−r2,sz) · · · (1−rk,sz), where r j,s, j = 1, 2, . . . , k, are complex numbers. Let r j,s = (r j,s/|r j,s|) · min(1/1.1, |r j,s|),j = 1, 2, . . . , k. Then 1 − ˜φ1,sz − ˜φ2,sz2 − · · · − ˜φk,szk B ˜φs(z) B (1 − r1,sz)(1 − r2,sz) · · · (1 − rk,sz)has all roots outside the unit circle. After we substitute ˜φi,s for φi,s in step 4, the simulation resultimproves.

Fourthly, we apply the original step 4 of Algorithm 3.1 when testing roots at ±i, but applythe alternative step (3.2) to the test of the root at 1 or −1. (When apply the alternative step (3.2),we select the lags and truncate the coefficients similarly.) Unpublished simulation result showsan advantage of (3.2) when testing root at 1 or −1. This advantage occurs especially when allnuisance roots occur, or equivalently when all of the true π j,s’s are zero, since in this case theinclusion of Y?

j,4t+s−1 in the original step 4 becomes redundant.

5.2.2. Seasonal block bootstrap unaugmented HEGY testTo improve the empirical performance of the seasonal block bootstrap algorithm (Algorithm

4.1), we truncate the coefficient estimators, taper the blocks, and optimize the block size. Firstly,as in the seasonal iid bootstrap algorithm, we let πU

j,s = min(0, πUj,s), j = 1, 2, 3, and substitute πU

j,s

for πUj,s in step 4.

16

Secondly, it is known that the bootstrapped data around the edges of the bootstrap blocks arenot good imitations of the original data. To reduce this “edge effect”, we apply tapered seasonalblock bootstrap proposed by [39], which puts less weight on the bootstrapped data around theedges. In our simulation the weight function is set identical to the function suggested by [39].

Thirdly, both test statistics πUj and tU

j can be employed to run the seasonal block bootstrapunaugmented HEGY test. So do various block sizes. In an unreported simulation we check theimpact of test statistics and block sizes on the empirical size and power. It turns out that, first, thechoice of statistics and block sizes does not affect the empirical size and the power very much;second, the distortion of the empirical size becomes the worst when testing root at −1 with thepresence of nuisance roots and mapos noise; third, the bootstrap test based on the t-statistics andblock size four gives the best result in the aforementioned worst scenario. Hence, we base ourtest on the t-statistics and let the block size be four in the succeeding simulations. For a thoroughdiscussion on an optimal block size, see [40].

5.2.3. Non-seasonal bootstrap augmented HEGY testFor a brief description of the non-seasonal bootstrap augmented HEGY test by [28], see Re-

mark 3.9. To improve its empirical performance, as in the seasonal iid bootstrap algorithm,we apply a backward stepwise selection of the lags based on their t-statistics and correct φ j,j = 1, 2, . . . , k with a polynomial factorization.

5.2.4. Wald testWe find it necessary to pass the data through a (1 − L4) filter before sending it to the Wald test

by [20]; otherwise the nuisance roots in our data will result in a non-stationary noise sequence inthe regression of the Wald test and a ill-behaved test statistic; see also [41, 21]. When selectingthe order of lag of the regression, we refer to the AIC and set the largest possible order of lag tobe four.

5.3. ResultsNow we present in Figure 1, 2, 3, 4, and 5 the main simulation result. The simulation includes

five types of data generating processes (see Table 1) and six types of noises (see Table 2). Insimulation we let sample size be T = 30 or T = 120, number of bootstrap replicates B = 500,number of iterations N = 2400, and nominal size α = 0.05.

5.3.1. Root at 1Figure 1 gives the simulation result when our data has a potential root at 1 but no other nuisance

roots at −1 or ±i. In this scenario, the seasonal block bootstrap test and the non-seasonal bootstraptest suffer from a slight size distortion in (f) and (l), where the seasonal iid test enjoys moreaccurate size. Except that, the power curves of the three bootstrap tests almost overlap; they startat the correct size and tend to one when ρ departs from zero, get higher when the sample sizegrows from T = 30 to T = 120, and are far above the curves of the Wald test in all of (a)-(l) but(b) and (h), where the Wald test suffers a upward size distortion.

Figure 2 gives the result when data has a potential root at 1 and all nuisance roots at −1 and±i. Notice that the size of the seasonal block bootstrap unaugmented HEGY test is distorted in

17

(b) and (h) in Figure 2; this may result from the errors in estimating π j,s and the need to recoverY4t+s with the estimated π j,s. Moreover, the size of both the seasonal block bootstrap test and thenon-seasonal bootstrap test is distorted in (d) and (j); this is in part due to the fact that the unit rootfilter (1 − L) is partially cancelled by the MA filter (1 − 0.5L). See also [42].

In contrast, the seasonal iid bootstrap augmented HEGY test has less size distortion when datahas nuisance roots. This is partially because the seasonal iid bootstrap test recovers Y4t+s usingthe true values of π j,s, namely zero, instead of using the estimated values. Moreover, compared tothe Wald test, the seasonal iid bootstrap augmented HEGY test has much higher power. Therefore,when testing the root at 1, the seasonal iid bootstrap augmented HEGY test is recommended.

5.3.2. Root at ±iFigure 3 and Figure 4 present the results of the simulation when data has potential roots at ±i

but has no or all nuisance roots at 1 and −1, respectively. In both Figure 3 and Figure 4, it turns outthat all the three bootstrap tests have size distortions in (f) and (l), where, as discussed in Section5.1, the seasonal unit root filter (1 + L2) is partially cancelled out by the MA filter (1 − θsθs−1L2).Other than that, the bootstrap tests overall achieve the correct size. Since in Figure 3 and Figure 4the seasonal block bootstrap test overall has higher power than other bootstrap tests and than theWald test, we recommend it for testing roots at ±i.

5.3.3. Root at 1, −1, and ±iFigure 5 illustrates the result when we test the concurrence of roots at 1, −1, and ±i. Notice

that all the three bootstrap tests have distorted sizes in (f) and (l), where the seasonal unit root filter(1 + L2) is partially cancelled out by the MA filter (1− θsθs−1L2). In addition, when the sample sizeT = 30, all the three bootstrap tests suffer size distortion in (c), (d), and (e). However, when thesample size rises to T = 120, the seasonal iid bootstrap augmented HEGY test restores the correctsize; see (i), (j), and (k). Since overall in Figure 5 the seasonal iid bootstrap test prevails over theWald test, we recommend it for testing joint roots at 1, −1, and ±i.

18

(a) noise=iid, T=30 (b) noise=heter, T=30 (c) noise=arpos, T=30

(d) noise=maneg, T=30 (e) noise=arper, T=30 (f) noise=maper, T=30

(g) noise=iid, T=120 (h) noise=heter, T=120 (i) noise=arpos, T=120

(j) noise=maneg, T=120 (k) noise=arper, T=120 (l) noise=maper, T=120

Figure 1: Power as a function of ρ when testing roots at 1 with no nuisance root. Blue dotted curve with triangle knotis for the seasonal iid bootstrap test. Red solid curve with circle knot is for the seasonal block bootstrap test. Blackdotted curve with “+” knot is for the non-seasonal bootstrap test. Green dashed curve with square knot is for the Waldtest. In (a)-(f) sample size T = 30. In (g)-(l) sample size T = 120.

19





Figure 2: Power as a function of ρ when testing roots at 1 with all nuisance roots. Blue dotted curve with triangle knotis for the seasonal iid bootstrap test. Red solid curve with circle knot is for the seasonal block bootstrap test. Blackdotted curve with “+” knot is for the non-seasonal bootstrap test. Green dashed curve with square knot is for the Waldtest. In (a)-(f) sample size T = 30. In (g)-(l) sample size T = 120.

20





Figure 3: Power as a function of ρ when testing roots at ±i with no nuisance root. Blue dotted curve with triangle knotis for the seasonal iid bootstrap test. Red solid curve with circle knot is for the seasonal block bootstrap test. Blackdotted curve with “+” knot is for the non-seasonal bootstrap test. Green dashed curve with square knot is for the Waldtest. In (a)-(f) sample size T = 30. In (g)-(l) sample size T = 120.

21





Figure 4: Power as a function of ρ when testing roots at ±i with all nuisance roots. Blue dotted curve with triangleknot is for the seasonal iid bootstrap test. Red solid curve with circle knot is for the seasonal block bootstrap test.Black dotted curve with “+” knot is for the non-seasonal bootstrap test. Green dashed curve with square knot is forthe Wald test. In (a)-(f) sample size T = 30. In (g)-(l) sample size T = 120.

22





Figure 5: Power as a function of ρ when testing roots at 1, −1, and ±i. Blue dotted curve with triangle knot is forthe seasonal iid bootstrap test. Red solid curve with circle knot is for the seasonal block bootstrap test. Black dottedcurve with “+” knot is for the non-seasonal bootstrap test. Green dashed curve with square knot is for the Wald test.In (a)-(f) sample size T = 30. In (g)-(l) sample size T = 120.

23

6. Real Data Application of Seasonal Unit Root Test

6.1. DatasetsHere we present the result of the seasonal unit root tests on four quarterly economic time series

that have not been seasonally adjusted. The first dataset contains gas consumption in millions oftherms in United Kingdom from quarter one, 1960 to quarter four, 1986. The second dataset givesthe E-commerce retail sales as a percent of total sales in United States from quarter four, 1999to quarter three, 2016. The third dataset presents the owned and securitized outstanding studentloans in billions of dollars in United States from quarter one, 2006 to quarter four, 2016. Thefourth includes the logarithms of the earnings per Johnson&Johnson share in dollars from quarterone, 1960 to quarter four, 1980. The deterministic linear and quadratic trends and the deterministicseasonal component of these time series are first estimated with OLS and then removed from thedata. The detrended and deseasonalized time series are presented in Figure 6. Since [8, 43] haveindicated possible periodic structure in economic time series, when investigating the stochasticseasonality of these time series we include tests catering to periodicity. Specifically, we implementthe seasonal iid bootstrap augmented HEGY test (SIB), the seasonal block bootstrap unaugmentedHEGY test (SBB), the non-seasonal bootstrap augmented HEGY test (NSB) by [28], and the Waldtest (WALD) by [20].

(a) Gas (b) E-Commerce (c) Student Loan (d) Johnson&Johnson

Figure 6: Quarterly time series with deterministic trend and seasonal component removed

24

6.2. Results

Table 3: P-values of seasonal unit root tests on economic data

Gas E-CommerceH1

0 H20 H3,4

0 H1,2,3,40 H1

0 H20 H3,4

0 H1,2,3,40

SIB 0.068 0.000 0.944 0.020 0.322 0.162 0.290 0.540SBB 0.038 0.000 0.876 0.026 0.544 0.014 0.252 0.306NSB 0.042 0.000 0.988 0.208 0.476 0.192 0.508 0.668

WALD 0.719 0.013 0.440 0.108 0.438 0.967 0.473 0.027Student Loan Johnson&Johnson

H10 H2

0 H3,40 H1,2,3,4

0 H10 H2

0 H3,40 H1,2,3,4

0SIB 0.136 0.002 0.000 0.000 0.226 0.012 0.002 0.000SBB 0.128 0.000 0.000 0.000 0.092 0.000 0.002 0.000NSB 0.110 0.000 0.000 0.000 0.286 0.036 0.002 0.006

WALD 0.362 0.204 0.979 0.611 0.513 0.028 0.011 0.002

6.2.1. Gas ConsumptionFirst, we investigate the p-values of the gas consumption time series. Overall, from Table 3, we

observe that the seasonal iid bootstrap test, which is recommended for testing the concurrence ofroots at 1, −1, and ±i, rejects at the size of 5% the hypothesis that the gas consumption series hasall roots at 1, −1, and ±i. In a root-by-root analysis, we found that at the size of 5%, all the testsunanimously reject root at −1, but on the other hand none of the tests rejects root at ±i. Hence,the gas consumption time series may possess roots at ±i. Notice that in the gas consumption timeseries, the sample size T = 27 is fairly small. In the test of root at 1, if the time series possessesnuisance roots and the sample size is small, the Wald test loses power; see Figure 2. Hence, whentesting root at 1, we consider the high p-value from the Wald test unreliable. Since, in testing theroot at 1, the p-values of all tests other than the Wald test are around 5%, at the size of 5% weconclude that the gas consumption process may have roots at ±i, may not have a root at −1, andmay or may not have a root at 1.

6.2.2. E-Commerce SalesAt a size of 5%, none of the tests, except for the Wald test, can reject the hypothesis that the

e-commerce sales series has all roots at 1, −1, and ±i. Notice that in the e-commerce sales context,the sample size T = 18 is fairly small. When testing jointly the roots at 1, −1, and ±i and whenthe sample size is small, the Wald test suffers severe upward size distortions, see Figure 5. Hence,we ignore the small p-value of the Wald test when testing jointly the roots at 1, −1, and ±i andconclude that the e-commerce sales series may simultaneously have roots at 1, −1, and ±i. Thisconclusion is consistent with the high p-values of the root-by-root tests.

6.2.3. Student LoansWhen analyzing the student loans series, we focus on the p-values of the three bootstraps tests,

whose superiority over the Wald test has been illustrated in the simulation. We observe that all the

25

bootstrap tests reject at the size of 5% the hypothesis that the student loans series has all roots at1, −1, and ±i. In a root-by-root analysis, all the bootstrap tests unanimously reject the root at −1and ±i, but fail to reject the root at 1. Hence, we conclude that the student loans series may have aroot at 1, but may not have roots at −1 or ±i.

6.2.4. Johnson&Johnson EarningsAccording to Table 3, all of the tests reject, at the size of 5%, the hypothesis that the John-

son&Johnson earnings series has all roots at 1, −1, and ±i. In a root-by-root analysis, all of thetests reject the roots at −1 and ±i, but fail to reject the root at 1. Hence, we conclude that theJohnson&Johnson earnings series may have a root at 1, but may not have roots at −1 or ±i.

7. Conclusion

In this paper we analyze the augmented and the unaugmented HEGY tests in the periodicallyvarying setting. For root at 1 or −1, the asymptotic distributions of the testing statistics are stan-dard. However, for any combinations of roots at 1, −1, i, and −i, the asymptotic distributions arenot standard, not pivotal, and cannot be easily pivoted. Therefore, when periodic variation exists,the HEGY test can be applied to test any single real roots, but cannot be directly applied to anycombinations of roots.

Bootstrap proves to be an effective remedy for the HEGY test in the periodically varyingsetting. The two bootstrap approaches, namely 1) the seasonal iid bootstrap augmented HEGY testand 2) the seasonal block bootstrap unaugmented HEGY test, turn out to be theoretically solid.In the simulation study, we compare these two bootstrap tests with the non-seasonal bootstrapaugmented HEGY test by [28] and the Wald test by [20]. It turns out that the seasonal iid bootstrapaugmented HEGY test has the best performance when we test root at 1, −1 and when we test theconcurrence of roots at 1, −1, and ±i; on the other hand, the seasonal block bootstrap unaugmentedHEGY test prevails when we test roots at ±i. Real data application shows the importance of ourbootstrap approaches in constructing powerful tests.

Acknowledgment

We are grateful to the Associate Editor and two anonymous referees for their insightful feed-back.

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28

Appendix to: “Bootstrap seasonal unit root test under periodic variation”

By Nan Zou and Dimitris N. Politis

University of Toronto and University of California-San Diego

The appendix includes the proofs of the theorems in the main manuscript. We first present theproof for the asymptotics of the unaugmented HEGY test, then the asymptotics of the augmentedHEGY test, then the consistency of the seasonal iid bootstrap augmented HEGY test, and finally theconsistency of the seasonal block bootstrap unaugmented HEGY test. Thoughout the appendix, letYt = (Y4t−3,Y4t−2,Y4t−1,Y4t)′, Γ j = E[VtV′t− j],

∫WdW′ denotes

∫ 10 W(r)dW(r)′, and

∫WW′ denotes∫ 1

0 W(r)W(r)′dr.

Appendix A. Proof of Theorem 4.1.

Lemma 1. Suppose one of Assumption 1.A and Assumption 1.B and one of Assumption 2.A andAssumption 2.B hold. Then under H1,2,3,4

0 ,

T−1T∑

t=1

Yt−1V′t ⇒ Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′ +

∞∑j=1

Γ′j ≡ Q1,

T−2T∑

t=1

Yt−1Y′t−1 ⇒ Θ(1)Ω1/2

∫WW′Ω1/2Θ(1)′ ≡ Q2,

T−1T∑

t=1

VtV′t− jp→ Γ j.

Proof. See [44] (Proposition 18.1, pp. 547-548) for the proof with iid innovations, [45] for theproof under Assumption 2.A, and [46] for the proof under Assumption 2.B.

Lemma 2. Let XU, j = (Y j,0, . . . ,Y j,4T−1)′, XU = (XU,1, XU,2, XU,3, XU,4), where U stands for unaug-mented HEGY, and Y j,4t+s be defined by (2.4). Let V = (V1, . . . ,V4T )′ and Υ be a 4 × 4 matrixsuch that

Υi j =

(Γ0)i j if i < j,0 if i ≥ j.

Then, under H1,2,3,40 ,

(a)

(4T )−2(X′U XU)11 ⇒14

c′1Q2c1 ≡ η1,

(4T )−2(X′U XU)22 ⇒14

c′2Q2c2 ≡ η2,

29

(4T )−2(X′U XU)33 ⇒18

(c′3Q2c3 + c′4Q2c4) ≡ η3,

(4T )−2(X′U XU)44 ⇒18

(c′3Q2c3 + c′4Q2c4) ≡ η3,

(4T )−1(X′U XU)i jp→ 0, for i , j.

(b)

(4T )−1X′U,1V ⇒14

(c′1Q1c1 + c′1Υc1) ≡ ξ1,

(4T )−1X′U,2V ⇒14

(c′2Q1c2 + c′2Υc2) ≡ ξ2,

(4T )−1X′U,3V ⇒14

(c′3Q1c3 + c′4Q1c4 + c′3Υc3 + c′4Υc4) ≡ ξ3,

(4T )−1X′U,4V ⇒14

(c′3Q1c4 − c′4Q1c3 + c′3Υc4 − c′4Υc3) ≡ ξ4.

Proof. For the proof of part (a), see the Lemma 3.2(a) of [22] and its proof. For part (b), we onlypresent the proof of the first statement. Other statements are proven in similar ways. By Lemma1,

(4T )−1X′U,1V = (4T )−1T∑

t=1

0∑s=−3

Y1,4t+s−1V4t+s

= (4T )−1T∑

t=1

0∑s=−3

(c′1Yt−1 +

s∑i=−2

V4t−1+i)V4t+s

= (4T )−1T∑

t=1

c′1Yt−1V′t c1 + (4T )−1T∑

t=1

0∑s=−3

s∑i=−2

V4t−1+iV4t+s

⇒14

(c′1Q1c1 + c′1Υc1).

Proof of Theorem 4.1. Under H1,2,3,40 , we have (1 − L4)Yτ = Vτ. Let π = (πU

1 , πU2 , π

U3 , π

U4 )′, t =

(tU1 , t

U2 , t

U3 , t

U4 )′, and σ2 = (4T )−1(V − XU π)′(V − XU π). Then by Lemma 2,

(4T )π = (X′U XU)−1X′UV ⇒ [diag(η1, η2, η3, η4)]−1(ξ1, ξ2, ξ3, ξ4)′.

Hence, π = op(1). By Lemma 1 and 2,

σ2 = (4T )−1V′V + op(1) = tr(Γ0)/4 + op(1).

Hence,

t = σ−1[diag(X′U XU)−1]−1/2(X′U XU)−1X′UV⇒ (tr(Γ0)/4)−1/2[diag(η1, η2, η3, η4)]−1/2(ξ1, ξ2, ξ3, ξ4)′.

Further, Lemma 2 (a) indicates an asymptotic orthogonality in the design matrix. Hence, asymp-totically the F-statistics equal the averages of the squares of the corresponding t-statsitics, e.g.,FU

3,4 −12 ((tU

3 )2 + (tU4 )2)

p→ 0.

30

Appendix B. Proof of Theorem 3.1.

The proof follows the lines of [47] and contains two parts. Firstly, we show when T → ∞ andk = kT → ∞ simultaneously, the statistic of interest approximates a quantity free of k, and then weprove this quantity tends to a certain distribution as T → ∞.

To begin with, notice that when k → ∞, the error term of regression (3.1) tends to a limit.Surprisingly, this limit is in general not ετ, because the regression (3.1) falsely assumes non-periodic coefficients and thus in general cannot find the correct residuals ετ. To find the limit,recall that Vτ is defined as a misspecified constant parameter representation of V4t+s. UnderAssumption 1.B, the spectral densities of Vτ are finite and positive everywhere, so Vτ hasAR(∞) and MA(∞) expressions

ψ(L)Vτ = ζτ and Vτ = θ(L)ζτ, (B.1)

where ψ(z) = 1 −∑∞

i=1 ψizi, θ(z) = 1 +∑∞

i=1 θizi. Let

ζ(k)τ = Vτ −

k∑i=1

ψiVτ−i, (B.2)

and ζτ = Vτ −∑∞

i=1 ψiVτ−i. Since a misspecified constant parameter representation of ζτ is Vτ −∑∞i=1 ψiVτ−i, which is exactly ζτ defined in (B.1), no ambiguity arises. We can straightforwardly

show that14

0∑s=−3

Cov(ζ4t+s−i, ζ4t+s) = 0, ∀i = 1, 2, . . . , (B.3)

and14

0∑s=−3

Cov(V4t+s−i, ζ4t+s) = 0, ∀i = 1, 2, . . . . (B.4)

Now we show when T → ∞ and k → ∞ simultaneously, the statistics of interest approximatesquantities free of k. Let X be the design matrix of regression equation (3.1), β = (πA

1 , πA2 , π

A3 , π

A4 , φ1, . . . , φk)′

be the estimated coefficient vector of regression equation (3.1), and β = (0, 0, 0, 0, ψ1, . . . , ψk)′,where ψi is defined in (B.1). Let ζ(k) = (ζ(k)

1+k, . . . , ζ(k)4T )′ and ζ = (ζ1+k, . . . , ζ4T )′. Define a

(4 + k) × (4 + k) dimensional scaling matrix DT = diag((4T − k)−1, (4T − k)−1, (4T − k)−1, (4T −k)−1, (4T − k)−1/2, . . . , (4T − k)−1/2). Then

D−1T (β − β) = (DT X′X DT )−1 DT X′ζ(k).

Let ‖ · ‖ be the L2 induced norm of matrices. Now we define a diagonal matrix R such that‖DT X′X DT − R‖ converges to 0 in probability. Specifically, let

R = diag(R1,R2,R3,R4, Γ),

where

R1 =c′1Θ(1)

∑4Tτ=k+1 SτS′τΘ(1)′c1

(4T − k)2

31

R2 =c′2Θ(1)


(4T − k)2

R3 =c′3Θ(1)

∑4Tτ=k+1 SτS′τΘ(1)′c3 + c′4Θ(1)


2(4T − k)2

R4 = R3, Sτ =

τ∑i=k+1

ε i, Γi, j = γ(|i − j|).

The definition of R j, j = 1, 2, 3, 4, follows from the multivariate Beveridge-Nielson Decomposi-tion; see [44], pp. 545-546. The definition of Γ is due to the fact that (4T − k)−1 ∑4T

τ=1+k Vτ−iVτ− j

converges in probability to the seasonal average of autocovariance of Vτ of lag |i − j|.Following the definition of R, we make the following decomposition:

D−1T (β − β) = (DT X′X DT )−1 DT X′ζ(k)

= [(DT X′X DT )−1 − R−1]DT X′ζ(k) + R−1 DT X′(ζ(k) − ζ) + R−1 DT X′ζ.(B.5)

Notice the last term in the right hand side summation, R−1 DT X′ζ, is free of k. Later we will findout its asymptotic distribution as T → ∞. But now we need to prove the first two terms in the righthand side of (B.5) converge to zero as T → ∞ and k → ∞. To do so, it suffices to show

‖(DT X′X DT )−1 − R−1‖ = op(k−1/2), (B.6)

‖DT X′(ζ(k) − ζ)‖ = op(1), (B.7)

‖DT X′ζ‖ = Op(k1/2), (B.8)

‖R−1‖ = Op(1). (B.9)

Equation (B.6) can be proven straightforwardly; see [47]. For (B.7), notice

E‖DT X′(ζ(k) − ζ)‖2

=E[(4T − k)−24∑

j=1

(4T∑

τ=k+1

Y j,τ−1(ζ(k)τ − ζτ))

2 + (4T − k)−1k∑

i=1

(4T∑

τ=k+1

Vτ−i(ζ(k)τ − ζτ))

2].

Notice that ζ(k)τ − ζτ =

∑∞i=k+1 ψiVτ−i. Under Assumption 1.B, V4t+s is a VARMA sequence with

finite orders, thus Vτ has an ARMA expression with finite orders; see [30]. Hence, ψ(L) hasexponentially decaying coefficient ψi. It follows straightforwardly that E‖DT X′(ζ(k) − ζ)‖2 → 0.For (B.8), notice that

E‖DT X′ζ‖2 = E[(4T − k)−24∑

j=1

(4T∑

τ=k+1

Y j,τ−1ζτ)2 + (4T − k)−1k∑

i=1

(4T∑

τ=k+1

Vτ−iζτ)2].

32

By (B.3), (B.4), and the stationarity of ετ,

E[(4T − k)−1(4T∑

τ=k+1

Vτ−iζτ)2]

=14

0∑s=−3

∞∑h=−∞

Cov(V4t+s−iζ4t+s,V4t+s−h−iζ4t+s−h) + o(1)

=14

0∑s=−3

∞∑h=−∞

Cov(Vs−iζs,Vs−h−iζs−h) + o(1).

(B.10)

Without loss of generality we can let i = 1 and s = 0 in (B.10). By Assumption 2.A, 2.B, and(B.2), we can write Vτ and ζτ as linear combinations of ετ. By doing so, we can straightforwardlyget

∞∑h=−∞

Cov(V−1ζ0,V−h−1ζ−h) ≤ const. supi1, j1,i2, j2

∞∑h=−∞

|Cov(εi1−1ε j1 , εi2−h−1ε j2−h)|. (B.11)

The right hand side of this inequality is assumed to be bounded under Assumption 2.A. On theother hand, the right hand side is also bounded under Assumption 2.B, by the lemma below.

Lemma 3. Under Assumption 2.B, there exists K > 0 such that for all i1, i2, j1, and j2,

∞∑h=−∞

|Cov(εi1ε j1 , εi2−hε j2−h)| < K.

Proof. Without loss of generality, assume that εtnt=1 is a strictly stationary strong mixing time

series, and εt’s strong mixing coefficient α(h) satisfies∑∞

h=1 αδ/(4+δ)(h) < ∞. By Lemma A.0.1 of

[48],

|Cov(εi1ε j1 , εi2−hε j2−h)|

≤ const. α(min(|i1 − (i2 − h)|, |i1 − ( j2 − h)|, | j1 − (i2 − h)|, | j1 − ( j2 − h)|)))1− 1(4+δ)/2−

1(4+δ)/2 .

Hence,

∞∑h=−∞

|Cov(εi1ε j1 , εi2−hε j2−h)|

≤ const.∞∑

h=−∞

(α(min(|i1 − (i2 − h)|, |i1 − ( j2 − h)|, | j1 − (i2 − h)|, | j1 − ( j2 − h)|)))δ

4+δ

≤ const.∞∑

h=−∞

(α(|i1 − (i2 − h)|)δ

4+δ + α(|i1 − ( j2 − h)|)δ

4+δ + α(| j1 − (i2 − h)|)δ

4+δ + α(| j1 − ( j2 − h)|)δ

4+δ )

≤ const.∞∑

h=−∞

α(|h|)δ

4+δ < ∞.

33

By (B.10), (B.11), Assumption 2.A, 2.B, and Lemma 3, we have E[(4T−k)−1(∑4Tτ=k+1 Vτ−iζτ)2] =

O(1). Similarly, E[((4T − k)−1 ∑4Tτ=k+1 Y j,τ−1ζτ)2] = O(1). Hence, (B.8) follows. Now justify (B.9).

By Assumption 1.B, the determinant of Ψ(z) has all its roots outside unit circle. Hence, Vτ isinvertible; hence, ‖Γ−1

‖ = O(1). In addition,

c′jΘ(1)Ω1/2 ∑4Tτ=k+1 SτS′τΩ

1/2Θ(1)′c j

(4T − k)2 ⇒ c′jΘ(1)Ω1/2∫

WW′Ω1/2Θ(1)′c j. (B.12)

SinceP(c′jΘ(1)Ω1/2

∫WW′Ω1/2Θ(1)′c j = 0) = 0,

we have that for all ε > 0, there exists Mε > 0, such that P(c′jΘ(1)Ω1/2∫

WW′Ω1/2Θ(1)′c j <Mε) < ε. (B.9) follows from the definition of Op(1).

Combining (B.5), (B.6), (B.7), (B.8), and (B.9), we have

D−1T (β − β) = R−1 DT X′ζ + op(1).

Now we find the asymptotic distribution of R−1 DT X′ζ. In a straightforward way, the limitingdistribution of R−1 can be derived by (B.12). Since

DT X′ζ = (4T − k)−14T∑

τ=k+1

Y j,τ−1ζτ + (4T − k)−14T∑

τ=k+1

Vτ−iζτ,

we can derive the limiting distribution of (4T − k)−1 ∑4Tτ=k+1 Y j,τ−1ζτ by the lemma below, and the

limiting distribution of (4T − k)−1 ∑4Tτ=k+1 Vτ−iζτ in a similar way.

Lemma 4.

14T

4T∑τ=1

Y1,τ−1ζτ ⇒ Var(ζτ)θ(1)∫ 1

0W1(r)dW1(r),

14T

4T∑τ=1

Y2,τ−1ζτ ⇒ Var(ζτ)θ(−1)∫ 1

0W2(r)dW2(r),

(1

4T

4T∑τ=1

Y3,τ−1ζτ)2 + (1

4T

4T∑τ=1

Y4,τ−1ζτ)2

⇒Var(ζτ)[ 1

4 c′4Θ(1)ΩΘ(1)′c4

∫W4(r)dW4(r) + 1

4 c′3Θ(1)ΩΘ(1)′c3

∫W3(r)dW3(r)]2

14 (c′4Θ(1)ΩΘ(1)′c4 + c′3Θ(1)ΩΘ(1)′c3)

+Var(ζτ)[

√14 c′4Θ(1)ΩΘ(1)′c4

14 c′3Θ(1)ΩΘ(1)′c3(

∫ 1

0W3(r)dW4(r) −

∫W4(r)dW3(r))]2

14 (c′4Θ(1)ΩΘ(1)′c4 + c′3Θ(1)ΩΘ(1)′c3)

.

34

Proof of Lemma 4. Firstly we focus on the convergence of 14T

∑4Tτ=1 Y1,τ−1ζτ. The convergence of

14T

∑4Tτ=1 Y2,τ−1ζτ can be proven analogously. Define Ψ(z) such that ζ t = Ψ(B)Vt. Let ξτ = ψ(L)Yτ,

ξ1,τ = ψ(L)Y1,τ, ξt = (ξ4t−3, ξ4t−2, ξ4t−1, ξ4t)′, ζ t = (ζ4t−3, ζ4t−2, ζ4t−1, ζ4t)′. Then Bξt = ζ t, and

14T

4T∑τ=1

Y1,τ−1ζτ

= θ(1)1

4T

T∑t=1

0∑s=−3

ξ1,4t+s−1ζ4t+s (by Beveridge-Nielson Decomposition, up to op(1))

= θ(1)1

4T

T∑t=1

[c′1ξt−1ζ′t c1 +

0∑s=−3

s−1∑k=−3

ζ4t+kζ4t+s]

⇒14θ(1)c′1Ψ(1)Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′Ψ(1)′c1 (by (B.3), (B.4), and FCLT)

+14θ(1)[

0∑s=−3

s−1∑k=−3

Eζ4t+kζ4t+s + c′1∞∑

i=1

Eζ t−iζ′t c1]

=14θ(1)c′1Ψ(1)Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′Ψ(1)′c1 (by (B.3), ζτ is white noise)

=14θ(1)(ψ(1))2c′1Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′c1 (since c′1Ψ(1) = ψ(1)c′1)

= Var(ζτ)θ(1)∫ 1

0W1(r)dW1(r)

(by [30], p. 378,14

c′1Θ(1)ΩΘ(1)′c1 = Var(ζτ)θ(1)2).

Secondly we show the convergence of ( 14T

∑4Tτ=1 Y3,τ−1ζτ)2 + ( 1

4T

∑4Tτ=1 Y4,τ−1ζτ)2. Let ξ3,τ = ψ(L)Y3,τ,

ψa = (ψ(i)+ ψ(−i))/2, ψb = (ψ(i)− ψ(−i))/2i, θa = (θ(i)+ θ(−i))/2, and θb = (θ(i)− θ(−i))/2i. Then

14T

4T∑τ=1

Y3,τ−1ζτ

=1

4T

4T∑τ=1

(θaξ3,τ−1 − θbξ4,τ−1)ζτ

(by Beveridge-Nielson Decomposition, up to op(1))

=1

4T

T∑t=1

θa[c′3ξt−1ζ′t c3 + c′4ξt−1ζ

′t c4 −

−2∑s=−3

ζ4t+sζ4t+s+2]

−1

4T

T∑t=1

θb[c′3ξt−1ζ′t c4 − c′4ξt−1ζ

′t c3 −

−1∑s=−3

ζ4t+sζ4t+s+1 + ζ4t−3ζ4t]

⇒14θa[c′3Ψ(1)Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′Ψ(1)′c3

35

+c′4Ψ(1)Θ(1)Ω1/2∫

WdW′Ω1/2Θ(1)′Ψ(1)′c4]

−14θb[c′3Ψ(1)Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′Ψ(1)′c4

−c′4Ψ(1)Θ(1)Ω1/2∫

WdW′Ω1/2Θ(1)′Ψ(1)′c3]

(by (B.3), (B.4), and FCLT, the covariances of ζ4t+s cancel out since ζτ is white noise)

=14θa|ψ(i)|2[c′4Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′c4 + c′3Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′c3]

−14θb|ψ(i)|2[c′3Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′c4 − c′4Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′c3]

(since c′3Ψ(1) = ψbc′4 + ψac′3, c′4Ψ(1) = ψac′4 − ψbc′3, and ψ2a + ψ2

b = |ψ(i)|2).

Similarly,

14T

4T∑τ=1

Y4,τ−1ζτ

=14θb|ψ(i)|2[c′4Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′c4 + c′3Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′c3]

+14θa|ψ(i)|2[c′3Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′c4 − c′4Θ(1)Ω1/2

∫WdW′Ω1/2Θ(1)′c3]

The lemma follows from |ψ(i)|2 = |θ(i)|−2 and the fact that, by [30],

Var(ζτ)|θ(i)|2 =14

(c′4Θ(1)ΩΘ(1)′c4 + c′3Θ(1)ΩΘ(1)′c3).

As mentioned, the asymptotic distribution of β can be derived straightforwardly from Lemma4. Now we come to the asymptotic distribution of the t-statistics and the F-statistics. Notice,

tAj = σ−1[[(X′X)−1] j j]−1/2[(X′X)−1X′ζ(k)] j

= σ−1[[[(4T − k)−2(X′X)−1] j j]−1/2 − [[R−1] j j]−1/2](4T − k)[(X′X)−1X′ζ(k)] j

+σ−1[[R−1] j j]−1/2((4T − k)(X′X)−1X′ζ(k) − R−1(4T − k)−1X′ζ) j

+σ−1[[R−1] j j]−1/2(R−1(4T − k)−1X′ζ) j

= σ−1[[R−1] j j]−1/2(R−1(4T − k)−1X′ζ) j + op(1).

By the consistency of β, we have σ2 p→ Var(ζτ). The asymptotic distributions of the t-statistics

follows straightforwardly from Lemma 4. Further, the asymptotic distributions of the F-statisticsare identical with the asymptotic distributions of the averages of the squares of the correspondingt-statistics because of the asymptotic orthogonality of the regression. Hence, the proof of Theorem3.1 is complete.

36

Appendix C. Proof of Proposition 3.1.

Define φi,s and ε(k)4t+s as the prediction coefficients and errors of the predictive equation

(1 − L4)Y4t+s =

4∑j=1

π j,sY j,4t+s−1 +

k∑i=1

φi,s(1 − L4)Y4t+s−i + ε(k)4t+s.

More specifically, φi,s are defined such that for each s = 1, 2, 3, 4, ε(k)4t+s

de f= (1 − L4)Y4t+s −∑4

j=1 π j,sY j,4t+s−1 −∑k

i=1 φi,s(1− L4)Y4t+s−i is orthogonal to Y j,4t+s−1, j = 1, 2, 3, 4 and (1− L4)Y4t+s−i,i = 1, . . . , k. Define iτ and It such that ε?τ = εiτ and ε?4t+s = ε4It+s. By Algorithm 3.1, iτ is asequence of independent but not identical random variables, while It is a sequence of iid randomvariables. Let

υ(1)T,τ = (ε(k)

iτ− Eε(k)

iτ)/S td(ε(k)

iτ)

υ(2)T,τ = (−1)τ(ε(k)

iτ− Eε(k)

iτ)/S td((−1)τε(k)

iτ)

υ(3)T,τ =

√2 sin(

πτ

2)(ε(k)

iτ− Eε(k)

iτ)/S td(

√2 sin(

πτ

2)ε(k)

iτ)

υ(4)T,τ =

√2 cos(

πτ

2)(ε(k)

iτ− Eε(k)

iτ)/S td(

√2 cos(

πτ

2)ε(k)

iτ)

Let R?T be the partial sum of υT,τ above. Formally,

R?T (u1, u2, u3, u4) = (

1√

4T

b4Tu1c∑τ=1

υ(1)T,τ,

1√

4T

b4Tu2c∑τ=1

υ(2)T,τ,

1√

4T

b4Tu3c∑τ=1

υ(3)T,τ,

1√

4T

b4Tu4c∑τ=1

υ(4)T,τ)′.

Let ‖ · ‖ denote the L2 norm. To justify Proposition 3.1, it suffices to show

‖S ?T − R?

T ‖p→ 0 uniformly in u1, u2, u3 and u4, (C.1)

and R?T ⇒W? in probability, (C.2)

because the unconditional convergence in (C.1) implies that in probability the conditional distribu-tion of ‖S ?

T −R?T ‖ given Y4t+s converges to zero. To prove (C.1), we can without loss of generality

focus on the uniform convergence of the first coordinate, that is, uniformly in u1,

|1√

4T

b4Tu1c∑τ=1

ε?τ /σ?1 −

1√

4T

b4Tu1c∑τ=1

υ(1)T,τ|

p→ 0.

37

Notice uniformly in u1,

1√

4T

b4Tu1c∑τ=1

ε?τ

=1√

4T

0∑s=−3

bTu1c∑t=1

ε?4t+s + op(1)

=1√

4T

0∑s=−3

bTu1c∑t=1

(ε4It+s −1T

T∑t=1

ε4t+s) + op(1)

=1√

4T

0∑s=−3

bTu1c∑t=1

(ε(k)4It+s −

1T

T∑t=1

ε(k)4t+s)

−1√

4T

0∑s=−3

bTu1c∑t=1

4∑j=1

(πAj,s − π j,s)(Y j,4It+s−1 −

1T

T∑t=1

Y j,4t+s−1)

−1√

4T

0∑s=−3

bTu1c∑t=1

k∑i=1

(φi,s − φi,s)((1 − L4)Y4It+s−i −1T

T∑t=1

(1 − L4)Y4t+s−i)

+ op(1)

=1√

4T

0∑s=−3

bTu1c∑t=1

(ε(k)4It+s −

1T

T∑t=1

ε(k)4t+s) − BT (u1) −CT (u1) + op(1)

=1√

4T

b4Tu1c∑τ=1

(ε(k)iτ− Eε(k)

iτ) − BT (u1) −CT (u1) + op(1),

(C.3)

where BT (u1) and CT (u1) have obvious definitions.Now we show BT (u1)

p→ 0, and CT (u1)

p→ 0, uniformly in u1. For BT (u1), notice if π j,s , 0, then

Y j,4t+s is weakly stationary. Hence, by [35], πAj,s − π j,s = Op(T−1/2). It follows straightforwardly

that BT (u1)p→ 0 uniformly in u1. On the other hand, if π j,s = 0, then by Theorem 3.1, πA

j,s − π j,s =

Op(T−1). Let

QT (u1) =1√

4T

bTu1c∑t=1

(Y j,4It+s−1 −1T

T∑t=1

Y j,4t+s−1).

It suffices to show that sup0≤u1≤1 QT (u1) = op(T ). By continuous mapping theorem, it suffices toprove (4T )−1QT (·) ⇒ 0(·), where 0(·) ≡ 0. It is straightforward to show the weak convergence ofthe finite dimensional distributions of (4T )−1QT (·). Furthermore, for all r1 ≤ r ≤ r2,

E[(QT (r2)

T−

QT (r)T

)2(QT (r)

T−

QT (r1)T

)2]

=E[Var[QT (r2)

T−

QT (r)T

]Var[QT (r)

T−

QT (r1)T

]]→ 0.

By [49], pp. 146-147, (4T )−1QT (·) is tight. Hence (4T )−1QT (·)⇒ 0(·), and consequently BT (u1)p→

0 uniformly in u1. For CT (u1), in light of the derivation of Theorem 3.1, it can be shown that38

φi,s − φi,s = Op(T−1/2) holds not only under alternative hypotheses but also under the null. Hence,it follows that uniformly in u1, CT (u1)

p→ 0. Therefore, recalling (C.3), we have

sup0≤u1≤1

|1√

4T

b4Tu1c∑τ=1

ε?τ −1√

4T

b4Tu1c∑τ=1

(εiτ − Eεiτ)|p→ 0.

Further, it is straightforward to show E[B2T (1)]

p→ 0, and E[C2

T (1)]p→ 0. Using the same decom-

position as in (C.3), we have σ?1 − S td(ε(k)

iτ)

p→ 0. Hence we have proven (C.1).

Secondly we prove (C.2). Notice that the standard deviations in the definition of υ( j)T,τ are

bounded in probability. For example,

S td(ε(k)iτ

) = S td(ε(k)4It+s) = S td(ε(k)

4t+s) + op(1) = S td(ε4t+s) + op(1),

Further, conditional on Y4t+s, for fixed j = 1, . . . , 4, υ( j)T,1, υ

( j)T,2, . . . , υ

( j)T,T are row-wise independent

random variables. Finally, for all u ≥ 0,

Var[1√

4T

b4Tuc∑m=1

υ( j)T,m]

p→ u,

Cov(1√

4T

b4Tuc∑m=1

υ( j)T,m,

1√

4T

b4Tuc∑m=1

υ(i)T,m)

p→ 0 for i , j.

Hence, the conditions of Theorem 3.3 of [50] are satisfied. From [51], the convergence of R?T to

W? follows.

Appendix D. Proof of Proposition 4.1.

Proof. Without loss of generality, assume block size b is a multiple of four. Let im = I(m−1)b+1,where It is defined in Algorithm 4.1. Then the mth block of V∗t starts from Vim . Recall l = b4T/bcdenotes the number of blocks. Let υ( j)

l,m be the rescaled aggregation of the mth block, defined by

υ(1)l,m =

1√

b

b∑h=1

(Vim+h−1 − EVim+h−1)/S td(1√

b

b∑h=1

Vim+h−1)

υ(2)l,m =

1√

b

b∑h=1

(−1)h(Vim+h−1 − EVim+h−1)/S td(1√

b

b∑h=1

(−1)hVim+h−1)

υ(3)l,m =

1√

b

b∑h=1

√2 sin(

πh2

)(Vim+h−1 − EVim+h−1)/S td(1√

b

b∑h=1

√2 sin(

πh2

)Vim+h−1)

υ(4)l,m =

1√

b

b∑h=1

√2 cos(

πh2

)(Vim+h−1 − EVim+h−1)/S td(1√

b

b∑h=1

√2 cos(

πh2

)Vim+h−1)

39

Let R∗T be the partial sum of the block aggregations above. Formally,

R∗T (u1, u2, u3, u4) = (1√

l

blu1c∑m=1

υ(1)l,m,

1√

l

blu2c∑m=1

υ(2)l,m,

1√

l

blu3c∑m=1

υ(3)l,m,

1√

l

blu4c∑m=1

υ(4)l,m)′

To prove theorem 4.1, it suffices to show

‖S ∗T − R∗T ‖p→ 0 uniformly in u1, u2, u3 and u4, (D.1)

and R∗T ⇒W∗ in probability, (D.2)

where ‖ · ‖ denotes the L2 norm.The proof of (D.1) is similar to that of (C.1). Here we only present the proof of (D.2). First we

assume Assumption 1.A and 2.A. In this scenario, it is sufficient to show that the following threeproperties hold:

bluc∑m=1

E[υ(i)2

l,m ]p→ u, ∀u ≥ 0, and ∀ i = 1, . . . , 4, (D.3)

bluc∑m=1

E[υ(i)2

l,m 1(|υl,m| > ε)]p→ 0, ∀u ≥ 0, ∀ i = 1, . . . , 4, (D.4)

bluc∑m=1

E[υ(i)l,mυ

( j)l,m]

p→ 0, ∀u ≥ 0, ∀ i, j ∈ 1, 2, 3, 4, i , j. (D.5)

Given (D.3), (D.4), and (D.5), [50] shows that if each row of υl,m is a martingale differencesequence, then

bluc∑m=1

υl,m ⇒W∗(u).

By Beveridge-Neilson Decomposition, e.g., Proposition 17.2, [44], p. 504, Helland’s result canbe generalized to the case when each row of υl,m is a convolution of a constant sequence anda martingale difference array. Further, Helland’s result can be generalized to the bootstrap worldwith [51]. Hence it suffices to show (D.3), (D.4), and (D.5).

To verify (D.3) and (D.4), notice that for all u ≥ 0, i = 1, . . . , 4,

bluc∑m=1

E[(υ(i)l,m)2] = bluc/l→ u,

and, by the dominated convergence theorem,

bluc∑m=1

E[(υ(i)l,m)21(|υl,m| > ε)]

p→ 0.

Hence, it remains to verify the (D.5), which indicates asymptotic independence between coordi-nates of R∗T . Notice that the (D.5) needs to be proved for all i, j ∈ 1, 2, 3, 4, i , j. Here we cite

40

as an example the case i = 1 and j = 3. The rest of cases can be shown by similar calculations.Notice,

bltc∑m=1

E[υ(1)l,mυ

(3)l,m] =

E[ 1√

b

∑bh=1 Vi1+h−1

1√

b

∑br=1

√2 sin(πr/2)Vi1+r−1]

S td[ 1√

b

∑bh=1 Vi1+h−1]S td[ 1

√b

∑br=1


−E[ 1

√b

∑bh=1 Vi1+h−1]E[ 1

√b

∑br=1


S td[ 1√

b

∑bh=1 Vi1+h−1]S td[ 1

√b

∑br=1


.

Since

E[1√

b

b∑h=1

Vi1+h−1]p→ 0, E[

1√

b

b∑r=1


p→ 0,

and both S td[ 1√

b

∑bh=1 Vi1+h−1] and S td[ 1

√b

∑br=1

√2 sin(πr/2)Vi1+r−1] converge in probability to

constants ([29]), we only need to show that

E[1√

b

b∑h=1

Vi1+h−11√

b

b∑r=1


p→ 0.

Notice,

E[1√

b

b∑h=1

Vi1+h−11√

b

b∑r=1


=

√2

b(T − b/4)

T−b/4∑i=1

b∑h=1

b∑r=1

sin(πr/2)V4i+h−4V4i+r−4

= −A + B + op(1),

where

A =

√2

b(T − b/4)

b/4∑h=1

T−b/4∑j=1

V4 j−3V4 j+4h−6,

B =

√2

b(T − b/4)

b/4∑h=1

T−b/4∑j=1

V4 j−3V4 j+4h−4.

By Assumptions 1.A and 2.A, it is straightforward to show

Ap→ 0, B

p→ 0. (D.6)

Thus, we complete the proof under Assumption 1.A and 2.A. Now we assume Assumption 1.Aand 2.B. Let υl,m = (υ(1)

l,m, υ(2)l,m, υ

(3)l,m, υ

(4)l,m)′. Let λl, j, j = 1, . . . , 4, be the eigenvalues of Var

∑lm=1 υl,m.

By Corollary 4.2 of [52], it is sufficient to show that the following two properties hold:

E(1√

l

bltc∑m=1

υ(i)l,m)(

1√

l

bltc∑m=1

υ( j)l,m)

p→ t1i = j, for all t ≥ 0, for all i, j = 1, . . . , 4, (D.7)

41

λ−1l, j = O(l−1), for all j = 1, . . . , 4. (D.8)

Notice, to show (D.7), it suffices to show (D.6), which follows from Assumptions 1.A and 2.B andLemma 3. In addition, (D.8) follows from the continuity of the eigenvalue function. Hence, wehave completed the proof under 1.A and 2.B.

Until now we assume that block size b is a multiple of four. When b is not a multiple of four,it is straightforward to show (D.1). For (D.2), let

R∗T,s = (1√

l/4

bblu1c/4c∑t=1

υ(1)l,4t+s,

1√

l/4

bblu2c/4c∑t=1

υ(2)l,4t+s,

1√

l/4

bblu3c/4c∑t=1

υ(3)l,4t+s,

1√

l/4

bblu4c/4c∑t=1

υ(4)l,4t+s)

′.

Since R∗T,s, s = −3, . . . , 0 are mutually independent with respect to P, and R∗T,s ⇒ W∗ in proba-bility for all s = −3, . . . , 0, we have R∗T = 1

2

∑0s=−3 R∗T,s + op(1)⇒W∗ in probability.

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