Fixed-b Asymptotics for t-Statistics in the Presence of ......Fixed-b Asymptotics for t-Statistics...

Fixed-b Asymptotics for t -Statistics in the Presence ofTime-Varying Volatility∗

Matei Demetrescu,a Christoph Hanckb and Robinson Krusec

aChristian-Albrechts-University of Kiel† bUniversity of Duisburg-Essen‡

cCREATES, Aarhus University and Leibniz University Hannover§

Preliminary version: June 11, 2015

Abstract

The fixed-b asymptotic framework provides refinements in the use of heteroskedasticity andautocorrelation consistent variance estimators. The resulting limiting distributions of t-statistics are however not pivotal when the variance of the underlying data generating processchanges over time. To regain pivotal fixed-b inference under such time heteroskedasticity,we discuss three alternative approaches. We (1) employ the wild bootstrap (Cavaliere andTaylor, 2008, ET), (2) resort to time transformations (Cavaliere and Taylor, 2008, JTSA) and(3) suggest to pick suitable the asymptotics according to the outcome of a heteroskedasticitytest, since small-b asymptotics deliver standard limiting distributions irrespective of the so-called variance profile of the series. We quantify the degree of size distortions from using thestandard fixed-b approach and compare the effectiveness of the corrections via simulations.We also provide an empirical application to excess returns.

Keywords: Hypothesis tests, HAC estimation, HAR testing, Bandwidth, Robustness

JEL classification: C12 (Hypothesis Testing), C32 (Time-Series Models)

∗The first two authors gratefully acknowledge the support of the German Research Foundation (DFG) throughthe projects DE 1617/4-1 and HA 6766/2-1. Robinson Kruse gratefully acknowledges financial support fromCREATES funded by the Danish National Research Foundation.†Institute for Statistics and Econometrics, Christian-Albrechts-University of Kiel, Olshausenstr. 40-60, D-24118

Kiel, Germany, email: [email protected].‡Corresponding author: Faculty of Economics and Business Administration, University of Duisburg-Essen,

Universitätsstraße 12, D-45117 Essen, Germany, e-mail address: [email protected] .§CREATES, Aarhus University, School of Economics and Management, Fuglesangs Allé 4, DK-8210 Aarhus

V, Denmark, e-mail address: [email protected] . Leibniz University Hannover, School of Economics andManagement, Institute for Statistics, Königsworther Platz 1, D-30167 Hannover, Germany, e-mail address:[email protected] .

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1 Introduction

Most macroeconomic and financial variables are observed over time. When conducting statisticalinference using such data, it is therefore crucial to allow for serial dependence in the series. Forweakly stationary series, the seminal contribution of Newey and West (1987) (see also Andrews,1991) allows to asymptotically robustify the class of GMM (Hansen, 1982) hypothesis tests tothe potential presence of serial correlation. Relying on a heteroskedasticity- and autocorrelationconsistent [HAC] estimator of the variance of the estimators, this framework allows for normalor χ2 asymptotics. The asymptotic distributions turned out, however, to be fairly poor approx-imations to the actual finite-sample distributions, often leading to substantial size distortions inapplied work. As a consequence, test results are often sensitive to the choice of bandwidth B

and kernel k employed for estimating the variance. Also, the asymptotics require that a van-ishing fraction b := B/T → 0 of the number of time series observations T is used. In actualapplications, b must of course be positive.

To tackle these finite-sample issues with HAC variance estimation, a series of contributions, in-cluding Kiefer et al. (2000) and Kiefer and Vogelsang (2002a,b, 2005), proposes a new asymptoticframework, labelled fixed-b asymptotics, in which it is not required that b → 0. This leads tonew asymptotically heteroskedasticity- and autocorrelation robust [HAR] distributions (reviewedin more detail below) for the standard t and Wald-type test statistics in the GMM framework.Conveniently, the distributions reflect the choice of bandwidth and kernel even in the limit.The above-cited papers convincingly demonstrate that the asymptotic distributions may pro-vide substantially better approximations to actual finite-sample distributions. The usefulness ofsuch procedures has spawned a very active literature. An incomplete list of recent contributionsincludes Yang and Vogelsang (2011), Vogelsang and Wagner (2013) or Sun (2014a,b).

Fixed-b asymptotics, however, do not automatically lead to finite-sample improvements in allempirically relevant settings. In particular, variances—as a measure of volatility—varying intime (i.e. time heteroskedasticity) affect limiting distributions in the fixed-b framework and thuslead to a loss of asymptotic pivotality; see Müller (2014). Time-varying variances are present inmany financial (see among others Guidolin and Timmermann, 2006; Amado and Teräsvirta, 2014;Teräsvirta and Zhao, 2011; Amado and Teräsvirta, 2013) and macroeconomic (see e.g. Stock andWatson, 2002; Sensier and van Dijk, 2004; Clark, 2009, 2011; Justiniano and Primiceri, 2008)time series such as excess returns, economic growth or inflation. Time-varying variances include,but are not limited to, permanent breaks or trends in the variance properties of (the innovationsof) the series.1 Correspondingly, consequences of and remedies for time heteroskedasticity forinference with dependent data have received substantial attention in recent years.2 Yet, if onelets b → 0 as in Newey and West (1987) time-varying variance does not have an asymptotic

1In the macroeconomic literature, a particular such phase of declining volatility at the end of the millenniumis known as the “Great Moderation.”

2For stationary autoregressions see e.g. Phillips and Xu (2006) or Xu (2008); for unit root autoregressions, seeCavaliere and Taylor (2008b) or Cavaliere and Taylor (2009). The effects of time-varying volatility are amplifiedin panels of (nonstationary) series, making corrections all the more necessary; see e.g. Demetrescu and Hanck(2012) or Westerlund (2014).

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effect (cf. Cavaliere, 2004). Practitioners thus face a trade-off in the precision of the criticalvalues provided by the fixed-b approach, a trade-off which is determined by the strength of thetime heteroskedasticity in the data generating process [DGP].

Our first contribution is to quantify the extent of distortions due to time-varying volatility inSection 3. They are nontrivial and fixed-b asymptotics should only be used with additional care.The second contribution of the paper is to discuss methods for correctly sized fixed-b inferencein the presence of time-varying variances, thus making the trade-off irrelevant. To achieve this,we make use of suitably modified techniques from the unit root testing literature (Section 4).More specifically, we build on the work of Cavaliere and Taylor (2008a) that suggests to employ awild bootstrap scheme. Alternatively, we propose to time-transform heteroskedastic series as inCavaliere and Taylor (2008b) so as to recover homoskedasticity prior to conducting the test. SinceCavaliere and Taylor work with integrated series directly, while our setup assumes integration oforder zero, the transformation algorithm is adapted to suit our needs. Third, we demonstratethat a pretest for time-varying variance can also be used for robustification: depending on theoutcome of the test, one either uses small-b methods valid under heteroskedasticity or fixed-bmethods requiring homoskedasticity.

Simulation results presented in Section 5 support the asymptotic discussion: First and as iswell-known, the standard HAC tests are size distorted in finite samples when there is serial cor-relation, a distortion which can be remedied using fixed-b methods under homoskedasticity. Thelatter are however size distorted under heteroskedasticity. Second, the corrections suggested hereyield better finite-sample size under heteroskedasticity. They also show good performance underhomoskedasticity. The time-transformation is somewhat conservative, while the wild bootstrapis sometimes slightly upward size distorted. Third and as one would expect, the pre-test has anintermediate position. Fourth, the wild bootstrap turns out to be more powerful than the timetransformation.

Section 6 provides an empirical application to excess returns of US stocks and 10-year bonds over30-day US treasury bills, illustrating the potential empirical effect of using testing procedureswhich are and are not robust to time-varying variances. Section 7 concludes. The appendicescollect proofs and other derivations.

2 Fixed-b HAR testing

In this paper, we focus on the simple and prototypical case of tests for the mean of a series yt,E (yt) = µ. That is, we test H0: µ = µ0. The findings to be presented however generalize readilyto other testing problems, e.g. to the case of testing moment restrictions in a regression model,where both shocks or instruments could be subject to time-varying volatility. Our goal is toprovide tests which are robust to the potential presence of both (unconditional) heteroskedasticityand autocorrelation.

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The classical t-test for µ relies on the normalized sample mean,

√T

(y − µ0

ω

)with y = 1

T

∑Tt=1 yt the sample average of yt and ω2 = limT→∞Var

(√T (y − µ)

), the so-called

long-run variance of yt. We thus restrict ourselves to the case of√T -consistent sample averages,

which are given for independent, both identically and heterogeneously distributed random vari-ables, as well as serially correlated short memory series. The long-run variance—as opposed tothe variance of yt—captures the effect of possible serial correlation or heteroskedasticity on thesample average, hence the acronym HAC for its estimate.

Should yt be weakly stationary with absolutely summable autocovariances γj = Cov (yt, yt−j), itholds that ω2 =

∑∞j=−∞ γj . Regularity conditions assumed,3 a central limit theorem applies for

y and√T

(y − µ0

ω

)d→ N (0, 1)

under the null.

In practice, the long-run variance ω2 is unknown and has to be estimated, leading to the feasiblet-ratio

T =√T

(y − µ0

ω

). (1)

The most popular HAC estimators ω2 rely on suitably weighted sums of sample autocovariances;see Newey and West (1987) and Andrews (1991).4 Thus,

ω2 =

T−1∑j=−T+1

k

(j

B

)γj

where k is the employed kernel function, B denotes the so-called bandwidth and γj is the jth-order sample autocovariance, γj = 1

T

∑Tt=j+1 (yt − y) (yt−j − y). Under additional regularity

conditions (see e.g. Andrews, 1991), and in particular b = B/T → 0 at suitable rates, consistencyfollows, ω p→ ω, and

T d→ N (0, 1).

under H0. Although the asymptotics are the same for any suitable kernel and bandwidth choice,the finite-sample behavior of T does depend on the kernel and bandwidth chosen in the testsituation at hand. The quality of the asymptotic approximation thus depends on user input.To make this dependence explicit, Kiefer and Vogelsang (2005) let B/T = b ∈ (0, 1] for theasymptotic analysis. While the resulting limiting distribution is free of nuisance parameters, itis nonstandard. But, more interestingly, it depends directly on the kernel k and indirectly (viab) on the bandwidth B, thus offering second-order refinements to the usual, small-b asymptotics

3See e.g. Davidson (1994, Chapter 24) for sets of suitable assumptions.4Semiparametric estimates based on AR approximations (e.g. Berk, 1974) or on so-called steep origin kernels

(Phillips et al., 2006) are also available in the literature.

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where b→ 0; see Sun (2014b). Concretely,

T d→ B (k, b)

whereB (k, b) =

W (1)√Q (k, b)

(2)

with W (s) a standard Wiener process,

Q (k, b) = −∫ 1

0

∫ 1

0

1

b2k′′(r − sb

)(W (r)− rW (1)) (W (s)− sW (1)) drds

for kernels with smooth derivatives, and

Q (k, b) =2

b

∫ 1

0(W (r)− rW (1))2 dr − 2

b

∫ 1−b

0(W (r + b)− (r + b)W (1)) (W (r)− rW (1)) dr

for the Bartlett kernel.

The corresponding critical values for T are tabulated as a function of k and b in Kiefer andVogelsang (2005). For b→ 0, Q (k, b)

d→ 1, B (k, b)d→ N (0, 1) and small-b asymptotics are, in a

sense, a particular case of the fixed-b approach.

Note that the functional B (k, b) depends on the entire path of the Wiener process and not onlyon W (1), as is the case with the small-b approach. This has consequences when the volatility ofyt varies in time, as we shall see in the following section.

3 Failure of fixed-b HAR tests under time-varying variance

In order to analyze fixed-b asymptotics of T under time-varying variance, we assume a multi-plicative component structure of the series to be tested.

Assumption 1 Let the observed series yt be generated as

yt = htυt + µ, t = 1, . . . , T,

where the stochastic component υt is zero-mean stationary as specified by below, and the timeheteroskedasticity is induced by the function h, also specified below.

This multiplicative structure is common in the literature; see e.g. Cavaliere (2004). This makesyt a uniformly modulated process (Priestley, 1988, p. 165).5 To conduct the asymptotic analysis,we assume the stochastic component to have short memory in the following sense.

5Other contributions model υt explicitly as a linear process with modulated innovations; see e.g. Cavaliere andTaylor (2008a,b). Demetrescu and Sibbertsen (2014) argue that the two DGPs are essentially equivalent.

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Assumption 2 Let υt be a zero-mean strictly stationary series with unity long-run variance,L2+δ-bounded for some δ > 0, and strong mixing with coefficients α (j) for which∑

j≥0

α (j)1/p−1/(2+δ) <∞

for some 2 < p < 2 + δ.

Strong mixing with coefficients α satisfying some summability condition is a standard way ofimposing short memory onto υt (and thus yt); cf. e.g. Phillips and Durlauf (1986). Restrictingthe long-run variance to equal unity is not leading to any loss of generality, and allows one tointerpret h2

t as the localized long-run variance of the series yt. The assumption yields (see e.g.Davidson, 1994, Chapter 29) weak convergence of the partial sums of υt to a standard Wienerprocess,

1√T

[sT ]∑t=1

υt ⇒W (s) ,

so υt is integrated of order 0. While yt, being a modulated version of υt, is also strong mixing,its partial sums exhibit a limiting behavior depending on the modulating function ht.

Assumption 3 Let ht = h (t/T) with h (·) a deterministic, piecewise Lipschitz function, positiveat all s ∈ [0, 1].

This allows for general patterns of smoothly or abruptly changing variance, as long as the abruptchanges are not too frequent.6

Under the conditions spelled out by the above assumptions, we have (for details see Cavaliere,2004, Lemma 3)

1√T

[sT ]∑t=1

(yt − µ)⇒∫ s

0h (v) dW (v) ≡ Bh (s) .

The process Bh (s) is a Gaussian process, but not a Brownian motion: the covariance kernel ofBh (s) is given by

Cov (Bh (s) , Bh (r)) =

∫ min{s,r}

0h2 (v) dv

which is not proportional to min {s, r}, the covariance kernel of the standard Wiener process.

Clearly, the fact that the normalized partial sums of the centered yt do not converge weakly toa Brownian motion affects the fixed-b asymptotics of T . Under a time-heteroskedastic DGP, thelimiting distribution given by fixed-b asymptotics is stated in the following proposition.

6Seasonally varying variances are excluded, for instance. This is not critical, however, since the work ofBurridge and Taylor (2001) suggests that seasonally varying variances actually average out and do not affectconvergence to Wiener process.

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Proposition 1 Under H0 and Assumptions 1-3, it holds for b ∈ (0, 1] that

T d→ B (h, k, b) ≡ Bh (1)√Qh,k,b

as T →∞, where

Qh,k,b = −∫ 1

0

∫ 1

0

1

b2k′′(r − sb

)(Bh (r)− rBh (1)) (Bh (s)− sBh (1)) drds


Qh,k,b =2

b

∫ 1

0(Bh(r)− rBh(1))2 dr − 2

b

∫ 1−b

0(Bh(r + b)− (r + b)Bh(1)) (Bh(r)− rBh(1)) dr

for the Bartlett kernel.

Proof: See the Appendix.

Proposition 1 demonstrates that HAR testing is not robust to time heteroskedasticity for fixedb. Although Bh (1) is normal with mean zero and variance ω2 =

∫ 10 h

2 (s) ds, the distribution ofB (h, k, b) is different from that of B (k, b) whenever h is not constant almost everywhere. This isbecause Qh,k,b is essentially different from the denominator of (2) under time-varying volatility.

Figure 1 quantifies this lack of pivotality, showing quantile-quantile (QQ) plots for the distribu-tions B (h, k, b) for b = {0.1, 0.5, 0.9} and four different h. We take T = 1000 and simulate theB (h, k, b) with 50,000 replications. The kernel k is taken to be the Quadratic spectral kernel.Under DGP1, volatility is constant over time. This case is reported as a benchmark, where wecompare the quantiles of B(k, b) with themselves. The first row of graphs in Figure 1 show theresults for a small, medium and large value of b. The negligible deviations are due to small MonteCarlo error. An early downward break in volatility is present in DGP2. Here, we compare thequantiles of B(h, k, b) on the y-axis with the corresponding quantiles of B(k, b) on the x-axis.The results shown in the second row of Figure 1 clearly demonstrate differences between the twodistributions. The larger b, the more pronounced is the discrepancy. For the third DGP (exhibit-ing a late upward break), differences are present but less visible, except for extreme quantiles.As indicated by the vertical bars representing the 5% critical values, this non-pivotality might beoverlooked by a test at that level. The results for the linear upward trend in volatility in DGP4nicely illustrates the difference between the distributions as well as the role of b.

For b = 0, however, robustness to time-varying volatility is recovered. In a nutshell, Bh (1) followsa normal distribution with mean zero and variance ω2, which can be interpreted as the averagelong-run variance of the series. Moreover, Cavaliere (2004) shows that, under mild conditions onthe rate at which b vanishes, the HAC variance estimator is consistent precisely for this variance,plim ω2 = ω2. Hence,

T d→ N (0, 1) for b→ 0.

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b = 0.1

DG

P1

b = 0.5 b = 0.9

DG

P2

DG

P3

DG

P4

Figure 1: Quantile-quantile plots to compare B(k, b) (x-axis) to the distributions B(h, k, b) undervarious variance profiles h and for different b. DGP1: constant volatility; DGP2: early down-ward break in volatility; DGP3: late upward break in volatility; DGP4: linear upward trend involatility. The quadratic spectral kernel is employed. The dashed vertical line is the 95% criticalvalue from the B(k, b) distribution.

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In other words, small-b methods asymptotically lead to pivotality under time-varying volatility asit does under weak stationarity. Recall however that the finite-sample quality of the asymptoticapproximation in the small-b framework is meager, so practitioners essentially have to choosebetween the devil and the deep blue sea when not knowing the properties of the DGP.

Section 5 quantifies the size distortions resulting from ignoring time-varying variance when usingfixed-b asymptotic approximations. It will also recall that, despite asymptotic robustness, thesmall-b approach will often exhibit fairly strong size distortions in small samples under bothhomo- and heteroskedasticity. Hence, corrections are required.

As a final comment, note that the behavior under local alternatives is affected as well.

Corollary 1 With E (yt) = c/√T , we have under the assumptions of Proposition 1 that

T d→ Bh (1) + c√Qh,k,b

.


4 Robust inference under time-varying volatility

The critical issue about the failing asymptotics is that the partial sums of yt do not converge toa Brownian motion anymore. In the following, three different corrections are discussed.

First, Cavaliere and Taylor (2008a) propose the wild bootstrap as a way to deal with the het-eroskedasticity issue in a unit root testing context. We therefore exploit the wild bootstrap toreproduce the actual null distribution of T under heteroskedasticity.7 The basic algorithm is asfollows.

1. Generate T iid standardized random variables r∗t .

2. Generate the wild bootstrap sample as y∗t = r∗t (yt − y).

3. Compute the bootstrap test statistic T ∗ based on the resampled series y∗t .

4. Repeat steps 1-3 to obtain a set of M resampled statistics T ∗m, m = 1, . . . ,M .

5. Use the 1− α-quantile of {T ∗m}m=1,...,M , say q∗1−α, as critical value for the test.

As choice for r∗t in Step 1 one usually picks the standard normal or the two-point Mammendistribution. To reduce the influence of the short-run dynamics in finite samples, we may alsouse the residuals of a parametric model fit, say ARMA, in Step 2; when doing so, we resort for

7Cavaliere and Taylor (2009) show the wild bootstrap to cope with stochastic volatility as well. The multivariatecase has been dealt with in a series of papers starting with Cavaliere et al. (2010).

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simplicity to an AR(1) fit such that ut = yt− c− ayt−1 are used instead of yt− y when generatingthe bootstrap sample.

It is not necessary to recolor the bootstrap shocks to obtain the correct limiting distribution,although it would of course be possible to do so. The following proposition shows that thebootstrapped residuals already replicate the variance profile and the wild bootstrap proceduregives size control in the limit.

Proposition 2 Under H0 and Assumptions 1-3, it holds as T →∞ and B/T = b ∈ (0, 1] that

Pr(T > q∗1−α

)→ α.


Alternatively to step 5, one could of course use bootstrap p-values for a test decision; it canbe seen from the proof of the proposition that the wild bootstrap p-values converge weakly toa uniform distribution U [0, 1]. Moreover, since the bootstrap procedure only generates criticalvalues which are invariant to the true mean µ of yt, the behavior of T under (local) alternativesremains as implied by Corollary 1.

But the wild bootstrap is computationally demanding, even in the simpler version withoutprewhitening. Our second correction therefore elaborates on the approach provided by Cava-liere and Taylor (2008b) which modifies the data in such a way that the series are in a sensetransformed back to homoskedasticity. Hence, it will be valid to apply fixed-b methods appliedto the transformed series. The time transformation approach of Cavaliere and Taylor (2008b)needs to be adapted to our setup, though, since they deal with I(1) processes under the nullwhen testing for a unit root, whereas we deal with I(0) processes. The procedure is as follows.

1. Subtract the mean of yt under the null and build the cumulated sums,

xt =t∑

j=1

(yj − µ0) .

2. Estimate the variance profile of xt, η (s) =∑[sT ]

t=1 (yj−y)2∑Tt=1(yj−y)2

, and build its inverse g (s).

3. Time-transform xt viaxt = x[Tg(t/T )].

4. Base the test on the differenced series, yt = ∆xt, i.e. compute

T =√T

¯y

ω,

where ω2 is an estimator of the long-run variance of yt using a bandwidth B = bT .

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The following proposition shows that fixed-b asymptotics are recovered.

Proposition 3 Under H0 and Assumptions 1-3, it holds as T →∞ and B/T → b ∈ (0, 1] that

T d→ B (k, b) .


In practice one often computes HAC estimators using some form of prewhitening; see Andrewsand Monahan (1992). Also, although serial correlation does not enter the asymptotic distribution,it may still impact the empirical size in small samples. The proposition also holds when the long-run variance estimator is computed on the basis of ARMA residuals and then adjusted for serialcorrelation.

Considering local alternatives, it turns out that the time transformation does have an asymptoticeffect, in contrast to the wild bootstrap (Corollary 1). The precise effect is given in the following

Corollary 2 With E (yt) = c/√T , we have under the assumptions of Proposition 3 that

T d→ W (1) + c/ω√Q

where

Q = −∫ 1

0

∫ 1

0

1

b2k′′(r − sb

)(W (r)− rW (1) + c/ω (g (r)− r)) (W (s)− sW (1) + c/ω (g (s)− s)) drds


Q =2

b

∫ 1

0(W (r)− rW (1) + c/ω (g (r)− r))2 dr

−2

b

∫ 1−b

0(W (r + b)− (r + b)W (1) + c/ω (g (r + b)− (r + b))) (W (r)− rW (1) + c/ω (g (r)− r)) dr

for the Bartlett kernel, and g is the inverse of the variance profile η (s) = ω−2∫ s

0 h2 (s) ds.


The third correction for time-varying volatility we propose is closely related to what practitionersoften do: only correct for a problem you have detected in the data. Essentially, we proposeto first test for heteroskedasticity, and then work with either fixed-b or small-b statistics andasymptotics, according to the outcome of the test. The intuition is that if a test for time-varyingvariance does not reject, then the departures from constant variances cannot be strong enoughto seriously distort the fixed-b asymptotics of T . If on the other hand the test rejects, then the

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small bandwidth-choice procedures may be preferable.8 The success of such a testing strategyobviously depends on the properties of the pre-test.

To this end, we resort to the test proposed by Deng and Perron (2008). In a recent simulationstudy, Bertram and Grote (2014) demonstrate that the test has good size and power properties,see also Xu (2013). It is based on the series zt = (yt − y)2. The test statistic is given by

Q = sup1≤t≤T

1√T

|Dt|ωz

where Dt =∑t

j=1 zj −tT

∑Tj=1 zj and ωz is a HAC estimator of the long-run variance of zt. The

test rejects for large values of Q.

Like for the wild bootstrap, the third solution just concerns getting suitable critical values andCorollary 1 applies.

5 Simulation evidence

This section studies the finite-sample behavior for the various statistics discussed above in dif-ferent settings.

We consider one-sided tests of H0 : µ = 0 against H1 : µ > 0. The DGP is given by

yt = µ+ vt (3)

(1− φL)vt = (1− θL)htεt (4)

with εt ∼ i.i.d.N(0, 1). We consider an AR(1)-process with (φ, θ) = (0.85, 0) as well as anARMA(1, 1)-process with (φ, θ) = (0.5,−0.45) and ht as before. The following deterministicvolatility DGPs for ht are studied

DGP1: Constant volatility (ht = 1);

DGP2: Downward break in volatility at t = [0.2T ] from σ0 = 5 to σ1 = 1;

DGP3: Upward break in volatility at t = [0.8T ] from σ0 = 1 to σ1 = 5;

DGP4: Linear upward trend in volatility: ht = σ0 +(σ1−σ0)(t/T ) with σ0 = 1 and σ1 = 5.

For power results, we take µT = c(V /T )1/2 with V being the average variance depending on theparticular DGP1-4. By doing so, we achieve comparable results among different DGPs. Underhomoskedasticity (DGP1), we have V = σ2, while V = T−1

∑Tt=1(σ2

0 + 1(t > [τT ])σ21) under

DGPs 2 and 3 for instance. The parameter c > 0 is a localizing constant and takes the values 5and 10.

8A more complex alternative would be to use either of the two robust versions discussed above when het-eroskedasticity is detected. We do not pursue this line of research here.

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Critical values for the fixed-b approach are taken from Kiefer and Vogelsang (2005), Table 1.The nominal significance level is 5%. We use the QS kernel and values of b ranging from 0.1 to1 in increments of 0.1. The number of wild bootstrap replications equals M = 399, while thenumber of Monte Carlo replications is 5,000. The sample sizes are T = 100 and T = 500. Forthe time-transformation, an AR(1) finite-sample correction is applied for all DGPs.

First, we present size results. Our leading case is the one with an AR(1) component; theARMA(1,1) results are discussed afterwards. Under the AR(1) DGP, the procedure appliesthe correct parametric pre-whitening correction to the series.

Under homoskedasticity (DGP1), the top-left entry of Table 2 reveals that, as is well-known,NW based on a Bartlett kernel with automatic bandwidth selection (see Andrews, 1991) facessubstantial size distortions for T as large as T = 100. Confirming the results of Kiefer and Vo-gelsang (2005), the remainder of the first row shows that fixed-b asymptotics provide a very goodapproximation to the finite-sample distribution of the t-ratio given in equation (1), all but elimi-nating the size distortions of NW. Unsurprisingly, the size of the pretest is intermediate betweenthat of NW and the fixed-b approaches. Similarly, the corrections based on time transformationsand the wild bootstrap provide accurate tests under homoskedasticity.

DGPs 2 and 4 confirm our analytical prediction from Section 3 that, in general, fixed-b asymp-totics are not pivotal under heteroskedasticity. In particular, the tests seem to be conservativeand increasingly so in b. That fixed-b asymptotics work relatively well for small b is not unex-pected, as they then operate very similarly to the standard NW approach which would be validasymptotically (see above and Cavaliere, 2004). Our suggested corrections are generally effectivein removing the size distortions, although the wild bootstrap has some upward size distortionsfor small b. Again, the pretest strikes a useful compromise between fixed-b and NW.

Table 3 highlights the finite-sample character of the size distortions of NW, which are largelyremoved for T = 500. Similarly, the upward size distortions of the wild bootstrap have all butdisappeared for T = 500. Table 3 moreover confirms that the distortions for fixed-b are not of afinite-sample nature.

The results for ARMA errors (Table 4) are qualitatively similar, although, unsurprisingly, theempirical sizes generally become somewhat less accurate when the prewhitening procedure doesnot soak up errors that follow the same parametric model as that applied in the procedure.

Tables 5 and 6 report power results. As expected, power increases in c. The particular type ofvariance break yields some variation in power, which may however be partly be explained by sizedistortions. Typically, power increases with decreasing values of b. This is plausible, as criticalvalues increase in b. Tests applied to the original series (i.e. Y) and the wild bootstrap analogues(YB) yield relatively high power. The pre-testing strategy performs similarly well. When thetime-transformation is applied, power is generally lower than for competing statistics.

This finding may be motivated as follows: under the alternative, the mean of the time-transformedseries yt is non-zero but not constant in general. Hence usual demeaning of the series yt to com-

13

pute a long run variance estimate does not eliminate its non-zero mean implying an inflatedvariance estimator due to the neglected deterministic component; see Corollary 2 for details.

In view of the size distortions observed for the wild bootstrap for small values of b as well asdecreasing power in b, the present simulations would therefore suggest to use the wild bootstrapwith an intermediate value of b (say, b = 0.4) in practice. The pretest would be a usefulalternative.

6 Excess returns

We illustrate the robustified procedures by an empirical application to financial excess returns zt.These are defined as the difference between speculative portfolio returns and risk-free returns.A primary research question in financial econometrics involves testing against positive excessreturns (on average) for speculative investments. Typically, monthly excess returns show amild degree of autocorrelation and strong heteroskedasticity. Thus, when testing the null thatE (yt) = 0 one needs to account for both features, such that the procedures discussed in thispaper may be valuable.

We use data from the Center for Research in Security Prices (CRSP) which have been analyzedin Guidolin and Timmermann (2006). The authors use multiple regime autoregressive Markovswitching models to study the joint dynamics of stock and bond returns. They find strongevidence for time-varying volatility during different regimes. See also Amado and Laakkonen(2014) for evidence on time-varying volatility in bond markets. Three time series of continuouslycompounded excess returns are analyzed here (cf. Figure 2): (1) returns for small caps (first andsecond size-sorted CRSP decile portfolios) (2) returns for large caps (deciles 9 and 10 size-sortedCRSP decile portfolios) and (3) returns of a portfolio consisting of 10-year bonds. The 30-dayT-bill rate is taken to be the risk-free rate. The sample period ranges from January 1954 toDecember 1999, resulting in T = 552 observations.

The sample averages z of the three series are (1) 0.822% for small caps, (2) 0.657% for largecaps and (3) 0.081% for the bond portfolio. The Deng and Perron (2008) pre-test against astructural change in volatility provides evidence against homoskedasticity. The test statistic issignificant at the nominal ten percent level for small and large caps, while its p-value is below onepercent for the bond portfolio series. Small and large caps show some degree of autocorrelation.Estimated first-order autocorrelation coefficients equal 0.190 and 0.078, respectively. Our testresults for the null hypothesis of zero excess returns against its positiveness, i.e. H0 : E (yt) = 0

against H1 : E (yt) > 0, are reported in Table 1. The finite-sample correction for autocorrelationis applied. For all three series, an AR(1) model is fitted. We use M = 2000 wild bootstrapreplications.

Our findings for the portfolio consisting of small firms suggest a clear and economically meaning-ful pattern: All statistics are significant at least at the nominal ten percent level, while most ofthem are significant at the five percent as well. When using small-b asymptotics, we find highly

14

(1) Small caps

Time

1960 1970 1980 1990 2000

−0.

3−

0.1

0.1

0.3

(2) Large caps

Time

1960 1970 1980 1990 2000

−0.

2−

0.1

0.0

0.1

(3) 10−year bonds

1960 1970 1980 1990 2000

−0.

050.

000.

050.

10

Figure 2: US Monthly excess returns (Jan 1954–Dec 1999, CRSP)

15

Table 1: Empirical test results for E(yt) = 0 against H1 : E(yt) > 0.

(1) Small caps fixed-b valuesNW 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Y 2.684a 3.260a 5.311a 7.385a 7.553a 7.812b 8.498b 9.414b 10.448b 11.542b 12.685b

X 2.349a 2.432b 2.239c 2.868b 3.460b 4.120b 4.948b 5.930b 7.027b 8.208b 9.467b

YB 2.684a 3.260a 5.311a 7.385a 7.553a 7.812a 8.498b 9.414b 10.448b 11.542b 12.685b

PT 3.260a 5.311a 7.385a 7.553a 7.812b 8.498b 9.414b 10.448b 11.542b 12.685b

(2) Large caps fixed-b valuesNW 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Y 3.695a 3.514a 2.933b 2.700c 2.779c 3.170c 3.751c 4.461c 5.262c 6.132c 7.071c

X 3.132a 3.095a 2.866b 2.838c 2.918c 3.311c 3.893c 4.595c 5.378c 6.215c 7.107c

YB 3.695a 3.514a 2.933b 2.700c 2.779c 3.170c 3.751c 4.461c 5.262c 6.132c 7.071c

PT 3.514a 2.933b 2.700c 2.779c 3.170c 3.751c 4.461c 5.262c 6.132c 7.071c

(3) 10Y bonds fixed-b valuesNW 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Y 0.824 0.872 0.861 0.811 0.815 0.860 0.931 1.018 1.114 1.216 1.324X 0.736 0.811 0.741 0.663 0.634 0.644 0.680 0.731 0.791 0.857 0.927YB 0.824 0.872 0.861 0.811 0.815 0.860 0.931 1.018 1.114 1.216 1.324PT 0.824 0.824 0.824 0.824 0.824 0.824 0.824 0.824 0.824 0.824

Note: NW stands for Newey and West (1987) standard errors; values ranging from b = 0.1 to b = 1 arefixed-bandwidth parameters. Y indicates that the test is carried for the untransformed series. X means that a

time-transformation is applied. YB stands for wild bootstrap versions. PT is a practitioner’s pre-testingstrategy (α = 5%). Superscripts ’a’, ’b’ and ’c’ refer to significance at the one-, five- and ten-percent level,

respectively. For further details see the text.

16

Tab

le2:

Size

resultsun

deran

AR(1)compo

nent,T

=10

0.NW

stan

dsforNew

eyan

dWest(198

7)stan

dard

errors;fi

xed-ba

ndwidth

parameter

btakesvalues

from

b=

0.1tob

=1.

Yindicatesthat

thetest

iscarriedfortheun

tran

sformed

series.X

means

that

atime-tran

sformationis

applied.

YB

stan

dsforwild

bootstrapversions.PT

isapractition

er’spre-testingstrategy

(α=

5%).

DGP1:

constant

volatility;

DGP2:

early

downw

ardbreakin

volatility;

DGP3:

late

upwardbreakin

volatility;

DGP4:

linearup

wardtren

din

volatility.

fixed

-bvalues

NW

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1DGP1

Y0.09

10.05

80.05

30.04

90.04

70.04

70.04

50.04

20.03

90.037

0.035

X0.05

20.04

70.04

90.05

20.05

20.05

20.05

20.05

20.05

20.051

0.048

YB

0.08

50.06

50.05

90.05

70.05

20.05

00.05

00.04

80.04

80.047

0.046

PT

0.05

80.05

30.05

10.04

90.04

80.04

60.04

30.04

10.038

0.037

DGP2

Y0.08

00.05

00.02

80.02

00.01

70.01

80.01

50.01

40.01

30.011

0.011

X0.03

40.02

10.03

10.03

40.03

70.03

80.03

80.03

90.03

80.037

0.037

YB

0.08

40.07

60.06

40.05

70.05

70.05

40.05

50.05

60.05

90.061

0.063

PT

0.06

10.04

90.04

30.04

20.04

30.04

10.04

10.04

00.040

0.039

DGP3

Y0.08

90.06

20.05

30.04

90.04

90.04

80.04

90.04

80.04

70.046

0.045

X0.04

40.04

10.04

40.04

30.04

40.04

30.04

20.04

10.04

30.041

0.040

YB

0.08

50.07

30.06

30.06

10.05

70.05

50.05

80.05

90.06

10.061

0.062

PT

0.06

30.05

40.05

10.05

10.05

00.05

10.05

00.04

90.048

0.046

DGP4

Y0.07

00.04

90.03

70.03

60.03

30.03

20.03

00.02

90.02

70.026

0.025

X0.04

10.04

00.04

50.04

90.05

20.05

10.04

70.04

70.04

60.044

0.041

YB

0.07

10.06

10.05

10.05

10.05

00.04

60.04

50.04

50.04

60.045

0.043

PT

0.05

10.04

10.03

90.03

70.03

60.03

40.03

30.03

20.031

0.029

17

Tab

le3:

Size

resultsun

deran

AR(1)compo

nent,T

=50

0.NW

stan

dsforNew

eyan

dWest(198

7)stan

dard

errors;fi

xed-ba

ndwidth

parameter

btakesvalues

from

b=

0.1tob

=1.

Yindicatesthat

thetest

iscarriedfortheun

tran

sformed

series.X

means

that

atime-tran

sformationis

applied.

YB

stan

dsforwild


isapractition

er’spre-testingstrategy

(α=

5%).

DGP1:

constant

volatility;

DGP2:

early

downw

ardbreakin

volatility;

DGP3:

late

upwardbreakin

volatility;

DGP4:

linearup

wardtren

din

volatility.

fixed

-bvalues

NW

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1DGP1

Y0.05

70.04

30.04

70.04

60.04

50.04

50.04

40.04

20.04

00.038

0.037

X0.05

00.04

50.04

60.05

00.04

80.04

60.04

70.04

80.04

60.046

0.042

YB

0.05

40.04

80.05

10.05

10.04

80.04

70.04

60.04

60.04

60.044

0.041

PT

0.04

40.04

70.04

70.04

60.04

60.04

50.04

30.04

10.039

0.038

DGP2

Y0.06

10.03

00.02

00.01

70.01

50.01

40.01

30.01

20.01

10.011

0.010

X0.04

50.04

70.05

00.05

10.05

00.05

00.05

10.05

30.05

20.053

0.052

YB

0.06

10.06

10.05

40.05

40.05

40.05

40.05

20.05

30.05

00.050

0.051

PT

0.05

80.05

70.05

70.05

60.05

60.05

60.05

60.05

60.056

0.056

DGP3

Y0.05

90.04

40.04

60.04

50.04

90.05

00.04

90.05

00.04

90.046

0.046

X0.04

90.04

60.04

70.04

70.04

70.04

90.04

90.04

70.04

70.047

0.047

YB

0.05

50.05

20.05

20.05

40.05

30.05

30.05

30.05

10.05

30.053

0.053

PT

0.04

80.05

00.05

00.05

20.05

30.05

20.05

30.05

20.051

0.050

DGP4

Y0.05

80.04

10.03

90.03

80.03

80.03

60.03

40.03

20.03

00.029

0.028

X0.05

30.05

00.05

10.05

20.05

10.05

30.05

20.05

10.05

10.048

0.047

YB

0.05

90.05

00.05

10.05

50.05

50.05

30.05

10.04

90.05

00.047

0.046

PT

0.05

60.05

60.05

60.05

60.05

60.05

60.05

50.05

50.055

0.054

18

Tab

le4:

Size

resultsun

deran

ARMA(1,1)compo

nent,T

=10

0.NW

stan

dsforNew

eyan

dWest(198

7)stan

dard

errors;fix

ed-ban

dwidth

parameterbtakesvalues

from

b=

0.1to

b=

1.Y

indicatesthat

thetest

iscarriedfortheun

tran

sformed

series.X

means

that

atime-

tran

sformationis

applied.

YB

stan

dsforwild


isapractition

er’s

pre-testingstrategy

(α=

5%).

DGP1:

constant

volatility;

DGP2:

earlydo

wnw

ardbreakin

volatility;

DGP3:

late

upwardbreakin

volatility;

DGP4:

linearup

wardtren

din

volatility.

fixed

-bvalues

NW

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1DGP1

Y0.15

90.09

10.06

70.06

50.06

00.05

50.05

40.05

20.04

90.047

0.045

X0.15

40.10

10.07

80.07

10.06

90.06

60.06

30.06

20.06

00.056

0.054

YB

0.15

40.10

10.07

60.07

30.06

80.06

30.06

30.06

30.06

20.061

0.062

PT

0.09

80.07

70.07

70.07

30.06

80.06

70.06

50.06

20.060

0.058

DGP2

Y0.16

70.07

70.03

40.02

80.02

30.02

00.02

00.01

80.01

60.014

0.013

X0.13

10.10

40.09

40.08

90.08

40.07

90.07

70.07

50.07

10.068

0.067

YB

0.17

30.11

90.08

40.07

30.06

90.06

80.06

70.07

00.07

10.074

0.076

PT

0.14

40.13

50.13

30.13

30.13

30.13

20.13

20.13

20.132

0.132

DGP3

Y0.16

90.09

80.06

70.06

40.06

00.05

90.05

70.05

60.05

50.053

0.051

X0.15

80.11

50.08

50.07

50.06

90.06

50.06

20.06

00.05

80.056

0.054

YB

0.16

50.10

70.08

00.07

50.07

00.06

80.06

80.06

70.06

80.070

0.072

PT

0.11

50.09

20.09

00.08

50.08

70.08

40.08

50.08

40.083

0.082

DGP4

Y0.14

40.07

80.05

00.04

20.03

90.03

90.03

70.03

70.03

50.035

0.034

X0.14

30.10

60.08

10.07

40.07

10.06

60.06

40.06

20.06

00.058

0.055

YB

0.14

50.09

50.07

30.06

50.06

00.05

70.05

50.05

50.05

60.056

0.055

PT

0.11

80.11

20.11

00.10

70.10

80.10

70.10

70.10

50.105

0.105

19

Tab

le5:

Pow

erresultsforc

=5un

deran

AR(1)compo

nent,T

=10

0.NW

stan

dsforNew

eyan

dWest(1987)

stan

dard

errors;fix

ed-

band

width


from

b=

0.1

tob

=1.

Yindicatesthat

thetest

iscarried

fortheun

tran

sformed

series.

Xmeans

that

atime-tran

sformationis

applied.

YB

stan

dsforwild


isapractition

er’s

pre-testingstrategy

(α=

5%).

DGP1:

constant

volatility;

DGP2:

earlydo

wnw

ardbreakin

volatility;

DGP3:

late

upwardbreakin

volatility;

DGP4:

linearup

wardtren

din

volatility.

fixed

-bvalues

NW

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1DGP1

Y0.50

80.41

20.36

70.33

70.31

30.28

50.28

10.27

00.25

40.239

0.229

X0.24

30.24

70.27

30.27

60.26

60.25

90.24

50.24

20.22

80.215

0.210

YB

0.49

00.43

50.38

80.36

40.34

50.32

40.30

40.30

10.29

00.284

0.279

PT

0.41

50.37

30.34

40.32

00.29

30.28

90.27

80.26

20.247

0.237

DGP2

Y0.56

20.49

50.44

40.37

30.30

40.26

00.24

00.21

60.19

50.177

0.164

X0.26

40.08

40.06

20.06

80.07

10.06

70.05

90.05

70.05

80.058

0.057

YB

0.55

20.54

50.58

30.57

30.55

40.54

30.52

70.51

60.51

30.512

0.505

PT

0.50

90.46

60.40

80.35

30.31

80.30

20.28

50.26

60.252

0.241

DGP3

Y0.49

90.40

10.37

20.35

60.33

80.33

00.31

60.30

60.30

10.286

0.270

X0.21

20.20

60.22

20.21

10.20

80.19

70.18

70.18

30.17

70.176

0.167

YB

0.47

90.43

40.40

50.38

00.36

80.35

30.34

60.33

90.34

00.331

0.324

PT

0.40

10.37

50.35

80.34

00.33

30.31

90.30

90.30

40.289

0.274

DGP4

Y0.54

50.46

30.41

30.38

20.34

20.31

70.30

30.29

60.28

40.271

0.263

X0.18

90.16

60.20

40.19

70.18

70.17

20.15

70.14

90.13

70.129

0.117

YB

0.53

20.49

30.46

50.44

30.42

20.40

70.39

40.38

80.38

00.367

0.353

PT

0.46

60.41

80.39

00.35

10.32

50.31

20.30

40.29

60.283

0.275

20

Tab

le6:

Pow

erresultsforc

=10un

deran

AR(1)compo

nent,T

=10

0.NW

stan

dsforNew

eyan

dWest(1987)

stan

dard

errors;fix

ed-

band

width


from

b=

0.1

tob

=1.

Yindicatesthat

thetest

iscarried

fortheun

tran

sformed

series.

Xmeans

that

atime-tran

sformationis

applied.

YB

stan

dsforwild


isapractition

er’s

pre-testingstrategy

(α=

5%).

DGP1:

constant

volatility;

DGP2:

earlydo

wnw

ardbreakin

volatility;

DGP3:

late

upwardbreakin

volatility;

DGP4:

linearup

wardtren

din

volatility.

fixed

-bvalues

NW

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1DGP1

Y0.88

80.82

70.77

20.72

10.65

80.61

50.58

20.56

10.52

90.498

0.467

X0.40

30.45

60.52

50.52

00.48

10.44

10.41

90.39

70.37

50.357

0.340

YB

0.87

40.84

30.79

70.74

60.68

40.64

80.61

90.60

20.58

30.570

0.554

PT

0.82

70.77

40.72

80.66

70.62

60.59

30.57

20.54

00.509

0.479

DGP2

Y0.87

00.82

50.80

00.75

70.69

00.62

90.57

10.53

00.50

30.467

0.436

X0.29

90.04

00.03

90.04

00.04

40.04

00.03

50.03

40.03

40.033

0.033

YB

0.85

20.85

00.85

90.84

60.82

80.81

30.80

40.79

50.79

40.789

0.784

PT

0.82

80.80

40.76

40.69

70.64

00.58

80.55

00.52

60.495

0.466

DGP3

Y0.87

80.82

90.77

90.73

30.69

90.67

80.65

40.63

30.61

40.596

0.575

X0.29

30.32

60.39

10.39

30.37

40.33

90.30

50.28

80.27

80.261

0.248

YB

0.86

70.83

40.79

60.75

80.72

20.69

70.68

50.66

80.66

10.654

0.646

PT

0.82

90.78

00.73

60.70

60.68

50.66

10.64

00.62

20.604

0.583

DGP4

Y0.92

40.88

20.82

60.78

80.74

10.70

20.68

00.65

70.62

70.599

0.567

X0.20

30.22

90.28

10.25

60.22

60.19

80.17

80.15

80.14

00.131

0.115

YB

0.91

60.89

20.86

80.83

20.80

00.77

60.75

50.73

30.71

10.694

0.681

PT

0.88

20.82

70.79

30.74

80.71

30.69

10.66

90.63

90.611

0.579

21

significant statistics pointing towards positive excess returns. For the fixed-b approach, whilewe also find evidence against the null, the results are in line with those of the simulations: forlarger values of b (i.e. b > 0.4), we find weaker evidence against H0. Another finding is thatthe time-transformation procedure leads to smaller statistics at least for some values of b, whichmay be related to its lower power. As the pre-test does not reject at the five percent level, apractitioner would follow the decisions of the fixed-b approach.

The results for large firms are quite similar, although the evidence is weaker for choices of b > 0.2.The test statistics are highly significant only for a relatively small value of b = 0.1 and with Neweyand West (1987) standard errors. Test decisions are very similar across methods and differ withb. Finally, no single statistic is significant for the 10-years bond portfolio, yielding no evidenceagainst the null hypothesis. This is not too surprising given the relatively small sample averageof 0.081% resulting from the numerous instances of inverted yield curves.

7 Concluding remarks

Fixed-b asymptotics are a tremendously useful device to enable more accurate finite-sampleinference when dealing with serially correlated data. Serial correlation is however not the onlynuisance practitioners need to pay attention to when aiming to conduct reliable hypothesis tests:many important macroeconomic and financial time series are subject to time-varying variancessuch as variance breaks. We show that the standard fixed-b approach no longer yields pivotaltests under such heteroskedasticity. Based on wild bootstrap schemes (Cavaliere and Taylor,2008a), on time transformations (Cavaliere and Taylor, 2008b) or on a pre-test procedure, weprovide corrections that restore size control of fixed-b methods even under heteroskedasticity.Simulations illustrate the useful size and power properties of the corrections, in particular of thewild bootstrap approach. An empirical application to excess returns reveals the importance ofproperly accounting for time-varying variances in practice.

Appendix

A Proofs

Proof of Proposition 1

Note that the arguments in the proof of Theorem 2 in Kiefer and Vogelsang (2005) can be usedwithout further modification to conclude that

T =

1√T

∑Tt=1 (yt − µ0)√

− 1T 2

∑T−1i=1

∑T−1j=1

T 2

B2k′′(i−jB

)1√T

∑it=1 (yt − y) 1√

T

∑jt=1 (yt − y)

+ op (1) .

22

for kernels with smooth derivative or

T =

1√T

∑Tt=1 (yt − µ0)√

2bT

∑Ti=1

(1√T

∑it=1 (yt − y)

)2− 2

bT

∑[(1−b)T ]i=1

(1√T

∑it=1 (yt − y)

)(1√T

∑i+[bT ]t=1 (yt − y)

)+op (1) .

for the Bartlett kernel. The weak convergence

1√T

[sT ]∑t=1

(yt − µ0)⇒ Bh (s)

and the continuous mapping theorem [CMT] then establish the desired limiting null distribution.

Proof of Corollary 1

Under a local alternative we have the weak convergence

1√T

[sT ]∑t=1

(yt − µ0)⇒ Bh (s) + cs

and the result follows with the same arguments as in the proof of Proposition 1.


We begin with the case of Gaussian bootstrap variables r∗t and no prewhitening. Let S∗T (s)

denote the normalized partial sums of the bootstrapped centered sample,

S∗T (s) =1√T

[sT ]∑t=1

(yt − y) r∗t

To guarantee size control in the limit, it suffices to show that the bootstrap partial sums convergeweakly in probability to

√Var(vt)Bh (s), since Var (vt) would cancel out in the bootstrapped t-

ratio. Note that, conditional on the sample yt, t = 1, . . . , T , S∗T (s) is a Gaussian process withindependent increments. Its covariance kernel is given by

Cov (S∗T (s) , S∗T (r)) =1

T

[min{s,r}T ]∑t=1

(yt − y)2E(

(r∗t )2)

=1

T

[min{s,r}T ]∑t=1

(yt − y)2 .

Then, following the proof of Lemma A.5 in Cavaliere et al. (2010), it suffices to establish theweak convergence

1

T

[sT ]∑j=1

(yj − y)2 ⇒ Var (vt)

∫ s

0h2 (r) dr

23

i.e. that the wild bootstrap correctly replicates the variance profile of the sample yt in thelimit. Assumption 2 guarantees pointwise convergence of 1

T

∑[sT ]j=1 (yj − y)2 via a Law of Large

Numbers for strong mixing processes (see Davidson, 1994, Section 20.6), and the monotonicityof the quadratic variation function leads to uniformity of the convergence, as required for theresult.

We then examine the case of prewhitening. Let a and c be the pseudo-true value of the AR(1) co-efficient and of the intercept in the autoregression9 yt = c+ ayt−1 +ut and let ut = ht (vt − avt−1)

such that ut satisfies indeed the same assumptions as yt up to the (for this step irrelevant) unitylong-run variance. Then,

S∗T (s) =1√T

[sT ]∑t=1

utr∗t +

1√T

[sT ]∑t=1

(ut − ut) r∗t

and the result follows along the lines of the case without prewhitening if the second summand onthe r.h.s. vanishes uniformly in s ∈ [0, 1]. Given that r∗t are serially independent and independentof yt, this is implied by

sups∈[0,1]

1

T

[sT ]∑t=1

(ut − ut)2 p→ 0

which is in turn implied by sups∈[0,1]\D∣∣u[sT ] − u[sT ]

∣∣ p→ 0 where D denotes the set of disconti-nuities of h (·), which are negligible given that there is a finite number of jump discontinuities.Now, at all continuity points of h,

u[sT ] − u[sT ] = µ+ h[sT ]v[sT ] − c− a(µ+ h[sT ]−1v[sT ]−1

)− h[sT ]

(v[sT ] − av[sT ]−1

)= µ (1− a)− c− v[sT ]−1

((a− a)h[sT ]−1 − a

(h[sT ] − h[sT ]−1

)),

where a and c are√T consistent (Phillips and Xu, 2006), supt∈{1,...,T} |vt| = op

(√T)

thanksto the uniform L2+δ boundedness of vt, and h[sT ] − h[sT ]−1 = O

(T−1

)uniformly in s thanks to

the piecewise Lipschitz condition on h. Hence sups∈[0,1]\S∣∣u[sT ] − u[sT ]

∣∣ p→ 0 as required for theresult.

Finally, should r∗t follow the Rademacher distribution, say, S∗T (s) is not Gaussian, but weakconvergence to a Gaussian process (conditional on the sample) holds and the proof follows alongthe same lines.


Cavaliere and Taylor (2008b, proof of Theorem 1) show that

1√T

[sT ]∑t=1

yt ⇒ ωW (s) .

9See Phillips and Xu (2006) for the precise details.

24

The result then follows like in the proof of Proposition 1 when no prewhitening is used.

To discuss prewhitening for computing the HAC estimator, we focus for simplicity on the AR(1)case; the extension to ARMA is not difficult but just time-consuming.

Use the Phillips-Solo device to write, with y0 = 0 for convenience,

ut = yt − ayt−1 − c = yt (1− a) + a∆yt−1 − c,

such that1√T

[sT ]∑t=1

ut = (1− a)1√T

[sT ]∑t=1

yt + ay[sT ] −√T c.

Some OLS algebra indicates that c = (1− a) y + Op(T−1

), and, thanks to the uniform L2+δ-

boundedness of yt, it holds thatsups∈[0,1]

∣∣y[sT ]

∣∣ = op

(√T), hence

1√T

[sT ]∑t=1

ut = (1− a)1√T

[sT ]∑t=1

(yt − y) + op (1)

with the op (1) term uniform in s. Since the adjustment of the HAC estimator is done preciselyby division with (1− a), this term cancels out and we have the same behavior of Qb,κ as underno prewhitening.

Proof of Corollary 2

Begin by writing

xt =t∑

j=1

(yj − µ0) =t∑

j=1

hjvj + ct√T

and note that the variance profile estimate is invariant to µ. Now, the transformation is

xt = x[Tg(t/T )]

with g the inverse of η, implying that

t∑j=1

yj = xt =

[Tg(t/T )]∑j=1

hjvj + c√Tg

(t

T

).

Since η converges weakly to the variance profile η, its inverse converges weakly to the inverse gof the variance profile, which is, like η, monotonic and continuous. Hence, using the CMT andCavaliere and Taylor (2008b, proof of Theorem 1) again, we obtain the required weak convergence

1√T

[sT ]∑t=1

yt ⇒ ωW (s) + cg (s) .

25

References

Amado, C. and H. Laakkonen (2014). Modeling time-varying volatility in financial returns:Evidence from the bond markets. In N. Haldrup, M. Meitz, and P. Saikkonen (Eds.), Essaysin Nonlinear Time Series Econometrics, pp. 139–160. Oxford University Press.

Amado, C. and T. Teräsvirta (2013). Modelling volatility by variance decomposition. Journalof Econometrics 175 (2), 142–153.

Amado, C. and T. Teräsvirta (2014). Modelling changes in the unconditional variance of longstock return series. Journal of Empirical Finance 25 (1), 15–35.

Andrews, D. W. (1991). Heteroskedasticity and autocorrelation consistent covariance matrixestimation. Econometrica: Journal of the Econometric Society 59 (3), 817–858.

Andrews, D. W. and J. C. Monahan (1992). An improved heteroskedasticity and autocorrelationconsistent covariance matrix estimator. Econometrica 60 (4), 953–966.

Berk, K. N. (1974). Consistent autoregressive spectral estimates. The Annals of Statistics 2 (3),489–502.

Bertram, P. and C. Grote (2014). A comparative study of volatility break tests. mimeo.Burridge, P. and A. M. R. Taylor (2001). On regression-based tests for seasonal unit roots in thepresence of periodic heteroscedasticity. Journal of Econometrics 104 (1), 91–117.

Cavaliere, G. (2004). Unit root tests under time-varying variances. Econometric Reviews 23 (3),259–292.

Cavaliere, G., A. Rahbek, and A. M. R. Taylor (2010). Testing for co-integration in vectorautoregressions with non-stationary volatility. Journal of Econometrics 158 (1), 7–24.

Cavaliere, G. and A. M. R. Taylor (2008a). Bootstrap unit root tests for time series withnonstationary volatility. Econometric Theory 24 (1), 43–71.

Cavaliere, G. and A. M. R. Taylor (2008b). Time-transformed unit root tests for models withnon-stationary volatility. Journal of Time Series Analysis 29 (2), 300–330.

Cavaliere, G. and A. M. R. Taylor (2009). Heteroskedastic time series with a unit root. Econo-metric Theory 25 (5), 1228–1276.

Clark, T. E. (2009). Is the Great Moderation over? An empirical analysis. Economic Review 4,5–42.

Clark, T. E. (2011). Real-time density forecasts from bvars with stochastic volatility. Journal ofBusiness and Economic Statistics 29, 327–341.

Davidson, J. (1994). Stochastic Limit Theory. Oxford university press.Demetrescu, M. and C. Hanck (2012). Unit root testing in heteroskedastic panels using theCauchy estimator. Journal of Business & Economic Statistics 30, 256–264.

Demetrescu, M. and P. Sibbertsen (2014). Inference on the long-memory properties of timeseries with non-stationary volatility. Technical Report 531, Leibniz Universität Hannover,Wirtschaftswissenschaftliche Fakultät.

Deng, A. and P. Perron (2008). The limit distribution of the CUSUM of squares test undergeneral mixing conditions. Econometric Theory 24 (3), 809–822.

Guidolin, M. and A. Timmermann (2006). An econometric model of nonlinear dynamics in the

26

joint distribution of stock and bond returns. Journal of Applied Econometrics 21, 1–22.Hansen, L. P. (1982). Large sample properties of generalized methods of moments estimators.Econometrica 50 (2), 1029–1054.

Justiniano, A. and G. Primiceri (2008). The time-varying volatility of macroeconomic fluctua-tions. American Economic Review 98 (3), 604–641.

Kiefer, N. M. and T. J. Vogelsang (2002a). Heteroskedasticity-autocorrelation robust standarderrors using the Bartlett kernel without truncation. Econometrica 70 (5), 2093–2095.

Kiefer, N. M. and T. J. Vogelsang (2002b). Heteroskedasticity-autocorrelation robust testingusing bandwidth equal to sample size. Econometric Theory 18, 1350–1366.

Kiefer, N. M. and T. J. Vogelsang (2005). A new asymptotic theory for heteroskedasticity-autocorrelation robust tests. Econometric Theory 21 (6), 1130–1164.

Kiefer, N. M., T. J. Vogelsang, and H. Bunzel (2000). Simple robust testing of regressionhypotheses. Econometrica 68 (3), 695–714.

Müller, U. K. (2014). HAC corrections for strongly autocorrelated time series. Journal of Business& Economic Statistics 32 (3), 311–322.

Newey, W. K. and K. D. West (1987). A simple, positive semi-definite, heteroskedasticity andautocorrelation consistent covariance matrix. Econometrica: Journal of the Econometric So-ciety 55 (3), 703–708.

Phillips, P. C. and S. N. Durlauf (1986). Multiple time series regression with integrated processes.The Review of Economic Studies 53 (4), 473–495.

Phillips, P. C. B., Y. Sun, and S. Jin (2006). Spectral density estimation and robust hypothesistesting using steep origin kernels without truncation. International Economic Review 47 (3),837–894.

Phillips, P. C. B. and K. L. Xu (2006). Inference in autoregression under heteroskedasticity.Journal of Time Series Analysis 27 (2), 289–308.

Priestley, M. B. (1988). Non-linear and non-stationary time series analysis. Academic PressLondon.

Sensier, M. and D. van Dijk (2004). Testing for volatility changes in U.S. macroeconomic timeseries. The Review of Economics and Statistics 86 (3), 833–839.

Stock, J. H. and M. W. Watson (2002). Has the business cycle changed and why? NBERMacroeconomics Annual 17 (1), 159–218.

Sun, Y. (2014a). Fixed-smoothing asymptotics in a two-step generalized method of momentsframework. Econometrica 82 (6), 2327–2370.

Sun, Y. (2014b). Let’s fix it: Fixed-b asymptotics versus small-b asymptotics in heteroskedas-ticity and autocorrelation robust inference. Journal of Econometrics 178 (3), 659–677.

Teräsvirta, T. and Z. Zhao (2011). Stylized facts of return series, robust estimates and threepopular models of volatility. Applied Financial Economics 21 (1-2), 67–94.

Vogelsang, T. J. and M. Wagner (2013). A fixed-b perspective on the Phillips-Perron unit roottests. Econometric Theory 29, 609–628.

Westerlund, J. (2014). Heteroscedasticity robust panel unit root tests. Journal of Business &

27

Economic Statistics 32 (1), 112–135.Xu, K.-L. (2008). Bootstrapping autoregression under non-stationary volatility. EconometricsJournal 11, 1–26.

Xu, K.-L. (2013). Powerful tests for structural changes in volatility. Journal of Economet-rics 173 (1), 126–142.

Yang, J. and T. J. Vogelsang (2011). Fixed-b analysis of LM-type tests for a shift in mean. TheEconometrics Journal 14, 438–456.

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