Testing for Rational Bubbles under StronglyDependent Errors∗

Yiu Lim LuiSingapore Management University

November 12, 2019

Abstract

A heteroskedasticity and autocorrelation robust (HAR) test statistic is proposed todetect the presence of rational bubbles in financial assets when errors are strongly de-pendent. The asymptotic theory of the test statistic is developed. Unlike conventionaltest statistics that lead to a too large type I error under strongly dependent errors, thenew test does not suffer from the same size problem. In addition, it can consistentlytimestamp the origination and termination dates of a rational bubble. Monte Carlostudies are conducted to check the finite sample performance of the proposed test andestimators. An empirical application to the S&P 500 index highlights the usefulnessof the proposed test statistic and estimators.

JEL classification: C12, C22, G01Keywords: long memory, explosiveness, unit root test

1 Introduction

The standard no-arbitrage condition implies that

Pt =1

1 +REt (Pt+1 +Dt+1) , (1)

where R, Et, Pt, and Dt denote the discount rate, the expectation based on informa-tion at time t, asset price, and the fundamentals (such as the dividend for a stock orthe rent from a house) at time t, respectively. Solving (1) by forward substitution,

we can express Pt = PFt + Bt where PFt =∑∞

i=1

(1

1+R

)iEt (Dt+i) is the fundamen-

tal price and Bt = 11+REt(Bt+1) is the bubble component. Note that Bt is not related

to the fundamentals. If Pt is the price of a stock, PFt is determined by the sum of

∗I would like to thank Peter Phillips, Jun Yu, and Yichong Zhang for their useful discussions and com-ments. Yiu Lim Lui, School of Economics, Singapore Management University, 90 Stamford Rd, Singapore178903, Email: yl.lui.2015@phdecons.smu.edu.sg

1

the discounted dividends. Suppose that the transversality condition is satisfied, namely,limT→∞ (1 +R)−T EtPt+T = 0. This condition implies Bt = 0 and hence, Pt = PFt . Whenthe transversality condition is not satisfied, Bt 6= 0. In this case, Bt is an explosive processsince R > 0 and hence, 1 + R > 1. The explosiveness in Bt also makes Pt an explosiveprocess even when PFt is not explosive. This is how a rational bubble is related to explo-siveness in time series. In practice, empirical studies often verify the explosiveness of theprice-fundamental ratio for the purpose of bubble detection; see, for instance, Phillips etal. (2015a) (hereinafter PSY) and Pedersen and Schütte (2017). A natural approach tothe bubble detection is to employ a right-tailed unit root test, popularized by Diba andGrossman (1988), Phillips et al. (2011) (hereinafter PWY) and Phillips and Yu (2011).PSY (2015a, b) extend the work of PWY to detect multiple bubbles. Harvey et al. (2016,2019a, 2019b) extend it to account for heteroskedastic errors and Pedersen and Schütte(2017) for weakly dependent errors.

Considers the following simple first-order autoregressive (AR) model

yt = ρyt−1 + εt, y0 = Op(1), εtiid∼ (0, σ2), t = 1, ..., n. (2)

Under the null hypothesis, yt is a unit root process (i.e., ρ = 1). Under the alternativehypothesis, yt displays explosiveness (i.e., ρ > 1), suggesting why one would use the right-tailed unit root test.

This paper focuses on model (2) with a simple extension, where errors are assumed tofollow a strongly dependent process. The phenomenon of strong dependence is widespreadin economic and financial time series. Cheung (1993) and Baillie et al. (1996) find em-pirical evidence of strong dependence in exchange rates. Christensen and Nielsen (2007),Andersen et al. (2003) and Ohanissian et al. (2008) show empirical evidence of strongdependence in volatility of stock returns and exchange rate returns. In addition, empiricalstudies obtained by Gil-Alana et al. (2014) and Barros et al. (2014) suggest strong de-pendence in housing prices in US cities. More recently, Chevillon and Mavroeidis (2017)show statistical learning can generate strong dependence and find empirical evidence ofstrong dependence in the US monthly CPI inflation rates.

Consider the following model

yt = yt−1 + ut, t = 1, ..., n,

ut = (1− L)−dεt, d ∈ (0, 0.5), εtiid∼ (0, σ2),

(3)

where L is the lag operator with (1− L)−d defined as

(1− L)−d =∞∑j=0

Γ(j + d)

Γ(d)Γ(j + 1)Lj and Ljεt = εt−j .

Denote ψ(k) the kth order autocovariance function of ut, namely ψ(k) := E (utut−k). Forut to be strongly dependent, we have

∑∞k=−∞ |ψ(k)| = ∞. It can be shown that when

d ∈ (0, 0.5),∑∞

k=−∞ |ψ(k)| = ∞, suggesting ut is indeed strongly dependent. If d = 0

2

in (3), model (3) becomes model (2) with ρ = 1. In the time series literature, we oftensay ut ∼ I(d), fractionally integrated of order d. Since the first difference of yt is I(d),yt ∼ I(λ) with λ = 1 + d.

Although yt in (3) is a unit root process with strongly dependent errors, it is plausibleto see an explosive trajectory in yt. To see why this is the case, we can express yt =∑t

i=1 ui + y0. Since ut is strongly dependent, a positive realization of the error term islikely to generate a long stream of positive errors due to strong dependence. Since yt isthe cumulative sum of errors, a long stream of positive errors will generate an upwardtrend that looks like an explosive process.

Suppose that we use the least squares (LS) method to estimate the AR(1) coeffi cientwhen data come from in (3) and then construct the conventional t statistic. Sowell (1990)shows that the t statistic diverges with the sample size. Therefore, applying a traditionalright-tailed unit root test tends to reject the unit root null hypothesis when the samplesize is large. In the context of rational bubble detection, due to the diverging type I error,applying a unit root test that ignores the strong dependence in ut (i.e. assuming d = 0)tends to conclude explosiveness in yt, thereby incorrectly detecting a rational bubble inmodel (3) when there is no bubble.

To showcase the empirical relevance of this problem, Figure 1 plots the monthly price-dividend ratio of S&P 500.1 In particular, we consider three sampling periods: (a) May1948 to October 1955; (b) April 1977 to March 1987; and (c) November 1990 to April1998. It is noteworthy that each sampling period contains a trajectory in which themarket experiences exuberance with a rising price-dividend ratio. Under the assumptionthat the true model is (2), that is, the errors are not strongly dependent, we perform anLS regression with an intercept and calculate the Dickey-Fuller t statistic for each sample(denoted by DFn).

Table 1 Right-tailed unit root test for the price-dividend ratio

Sample Period n DFn cv95% d CI for d(a) May 1948 to October 1955 90 1.12 −0.08 0.25 0.02 0.48(b) April 1977 to March 1987 120 1.9 −0.08 0.21 0.001 0.42(c) January 1990 to April 1998 100 3.5 −0.08 0.24 0.02 0.46

Table 1 reports DFn and the 95% critical value (cv95%) for the explosive alternative.We always reject the unit root null hypothesis during these sampling periods. Therefore,we would conclude the “successful” detection of a rational bubble in each of the threesampling periods using the traditional right-tailed unit root test. In fact, adopting a formof recursive unit root test, PSY (2015a) also find bubbles during similar periods.

However, if we assume that the time series are fractionally integrated as in model(3), we can estimate λ and d. In particular, we apply the exact local Whittle methodof Shimotsu and Phillips (2005) to estimate d and construct the 95% confidence interval

1The price-dividend ratio was used in PSY (2015a).

3

Figure 1: Monthly price-dividend ratio of S&P 500

(CI) of d, both shown in Table 1.2 Table 1 shows that positive estimates d for all threesampling periods are found. Moreover, CIs always exclude 0, suggesting strong evidenceof strong dependence in ut. Therefore, it is possible that the true model is (3) and thatthe divergent t statistic leads to the rejection of the unit root null hypothesis in favor ofan explosive alternative. In other words, these rational bubbles can be spurious.

Motivated by the empirical evidence of strongly dependent errors and their implicationfor bubble detection, this paper proposes a method to address the spurious explosivenessproblem in detecting rational bubbles when a time series model has strongly dependenterrors. We construct a heteroskedasticity-autocorrelation robust (HAR) test statistic,which converges to a proper distribution under no bubble assumption but diverges whenthe underlying model has an explosive or a mildly explosive root. Therefore, we candistinguish between an explosive and unit root time series even when the error process isstrongly dependent. After a bubble is detected, a new estimator is proposed to consistentlytimestamp its origination and termination dates.

The remainder of this paper is organized as follows. Section 2 provides a brief review ofthe traditional right-tailed unit root test for bubble detection and the estimation methodto timestamp bubble origination and termination dates. Section 3 introduces our modelwith strongly dependent errors, proposes the new test, and derives the asymptotic theoryunder the null hypothesis of unit root. Section 4 examines the asymptotic properties of the

2We first estimate the fractionally integrated order (λ) of yt. Then we subtract 1 from the estimate ofλ to obtain d. The confidence interval obtained is based on the asymptotic of the ELW estimate where√m(d− d) d→ N(0, 1/4), with m = nδ. We set δ = 0.65 in all applications in this paper.

4

proposed test statistic under two explosive alternatives. Section 5 proposes new estimatorsof bubble origination and termination dates based on the new test statistic. Monte Carlosimulation studies are carried out in Section 6 to study the finite sample performance ofthe proposed test and estimators. Section 7 provides an empirical study using the S&P500 index. Finally, Section 8 concludes the paper. The proofs of the main results in thepaper are provided in the Appendix. We use the following notations throughout the pa-per:

p→, d→, as→,⇒, a∼ and iid∼ denote convergence in probability, convergence in distribution,almost sure convergence, weak convergence, asymptotic equivalence, and independent andidentically distributed, respectively.

2 A Brief Review of Literature

In this section, we briefly review the traditional right-tailed unit root test statistic forbubble detection and the estimation method to timestamp bubble origination and termi-nation dates. Consider model (2). Suppose that we perform an LS regression with anintercept from the full sample and obtain our LS estimator ρn of ρ. Denote the t statisticDFn = (ρn − 1) /se(ρn), where se(ρn) is the standard error of ρn. Under the assumptionthat ρ = 1, following Phillips (1987a), we have

DFn =⇒ DF∞ :=

∫ 10 W (s)dW (s)(∫ 10 W (s)2ds

)1/2, (as n→∞), (4)

where W (r) is the standard Brownian motion and W (r) = W (r) − 1r

∫ r0 W (s)ds is the

demeaned Brownian motion. To implement a right-tailed unit root test, we can obtainthe (1−β)% critical value as the (1− β) percentile of DF∞ and reject the null hypothesiswhen DFn is greater than the critical value.

In practice, a bubble usually starts not from the first observation of the full sample, butfrom the middle of the sample, say at τ e = bnrec where b·c denotes the integer part of itsargument and re ∈ (0, 1). Moreover, a bubble usually does not last forever, but collapseslater in the sample, say at τ f = bnrfc. The collapse of a bubble typically correspondsto a market correction. If a bubble emerges and collapses in the sample, Phillips and Yu(2009) show that DFn → −∞, suggesting that the traditional right-tailed unit root teststatistic, if calculated from the full sample, cannot reject the unit root null hypothesis.

To identify the bubble in the full sample, PWY (2011) propose a sup statistic basedon recursive regressions. The regression in the first recursion is

yt = µ+ ρτyt−1 + ut, (5)

where t = 1, ..., τ0(:= bnr0c), µ, ρτ , and ut are the intercept estimator, the AR coeffi cientestimator and the LS residuals, respectively, and 0 < r0 < re. Subsequent regressionsemploy this originating data set supplemented by successive observations giving a sampleof size τ = bnrc for r0 ≤ r ≤ 1. Let DFτ denote the DF t statistic based on the first

5

τ observations of the sample, that is,

DFτ :=ρτ − 1

sτ, (6)

where sτ =

(1τ

∑τt=1 u

2t∑τ

t=1 y2t−1−

1τ (∑τt=1 yt−1)

2

)1/2

is the standard error of ρτ . The test statistic

proposed by PWY is supτ∈[τ0,n]DFτ , Under the null hypothesis, PWY show

SDF := supτ∈[τ0,n]

DFτ =⇒ supr∈[r0,1]

∫ r0 W (s)dW (s)(∫ r0 W (s)2ds

)1/2, (as n→∞).

If SDF takes a value larger than the right-tailed critical value, the null unit root hypothesisis rejected in favor of the explosive alternative. In this case, the evidence of a bubble isfound.

After a bubble is detected, one may want to estimate bubble origination and conclusiondates, that is, re and rf . Assume the model under the alternative hypothesis is given by

yt = yt−11 t < τ e+ ρnyt−11 τ e ≤ t ≤ n

+

t∑k=τf+1

εk + y∗τf

1 t > τ f+ εt 1 t ≤ τ f , (7)

ρn = 1 +c

nα, c > 0, α ∈ (0, 1) , εt

iid∼ (0, σ2).

Model (7) has two structural breaks. Before the first break (i.e. t < τ e), yt follows a unitroot process. After the first break but before the second break (i.e. τ e ≤ t ≤ τ f ), it followsa mildly explosive process with a root above 1 taking the form ρn = 1+ c

nα . At τ f +1, thebubble terminates with a crash to y∗τf which is in the neighborbood of the fundamentalvalue prior to the emergence of the bubble. Here τ e and τ f are the true absolute bubbleorigination and termination dates. Since τ e = bnrec and τ f = bnrfc, re and rf are thetrue fractional bubble origination and termination dates.

PWY (2011) introduce the estimator of re as

rPWYe = inf

r≥r0r : DFτ > cvn. (8)

Conditional on finding some originating date rPWYe of a bubble, PWY introduce the

estimator of rf asrPWYf = inf

s≥re+ γ ln(n)n

s : DFs < cvn . (9)

In (8) and (9), cvn is a critical value function that increases with the sample size. Providedcvn goes to infinity at a slower rate than n1−α/2, Phillips and Yu (2009) showed thatrPWYe

p→ re and rPWYf

p→ rf under some general regularity conditions. In the empiricalapplications of PWY, cvn is set to be proportional to ln lnn.

6

3 Model, New Test and Asymptotic Null Distribution

Motivated by the empirical studies in section 1, we now consider the following model

yt = ρnyt−1 + ut, t = 1, ..., n,

ut = (1− L)−dεt, εtiid∼ (0, σ2), d ∈ [0, 0.5), (10)

y0 = op(n1/2+d).

Model (10) is different from model (2) in that ut can be strongly dependent. We firstconsider the asymptotic property of the traditional Dickey-Fuller t test when ρn = 1.

3.1 Asymptotic null distribution of DFτ

Lemma 3.1 Assume the true data generation process is model (10) with ρn = 1 andd ∈ (0, 0.5). Suppose that DFτ is constructed from the empirical regression (5) based onthe first τ observations. Then, for any r ∈ (0, 1], as n→∞,

DFτ= Op

(nd). (11)

Lemma 3.1 indicates a serious implication for bubble detection using the traditionalmethod. Namely, the t statistic DFτ diverges with the sample size. When r = 1,DFn = DFτ = Op

(nd). This divergence leads to excessive rejection of the null hypothesis

when the sample size is large and conclude with a spurious bubble.

Remark 3.1 To detect the presence of a bubble, PWY and PSY propose to use SDF andGSDF defined by

SDF = supτ∈[τ0,n]

DFτ ,

GSDF = supτ2∈[τ0,n],τ1∈[0,τ2−τ0]

DF τ2τ1 ,

where DF τ2τ1 is the t statistic based on the observations from τ1 = bnr1c to τ2 = bnr2c.As Lemma 3.1 holds uniformly for r ∈ (0, 1], under model (10) with ρn = 1 and

d ∈ (0, 0.5), we have

SDF = Op

(nd),

GSDF = Op

(nd).

Both statistics can lead to spurious rational bubble detection as they diverge to infinity.

Remark 3.2 Similar to PSY (2014), our model (10) can be generalized to have an as-ymptotically negligible intercept. In this case,

yt = µn + ρnyt−1 + ut, (12)

where µn = O(n−θ), with θ > 1/2 − d. It can be shown that, since µn is asymptoticallynegligible, the result in Lemma 3.1 remains valid.

7

Remark 3.3 Note that if the kth order augmented Dickey-Fuller test is used, the sameresult as in (11) can be obtained.

Remark 3.4 If the LS regression is carried out without intercept and r = 1, the resultof Lemma 3.1 coincides with that of Sowell (1990) (see Theorem 4 in Sowell, 1990) whend ∈ (0, 0.5).

3.2 New test statistic

The failure of standard t statistic stems from estimating the variance of ut by the averagesquared residuals 1

τ

∑τt=1 u

2t . As this estimator does not provide a proper normalization, it

results in the divergence of DFτ . In this paper, we use a properly self-normalized statisticthat converges to a proper distribution for d ∈ [0, 0.5). To design the new statistic, notingthat as ut is potentially strongly dependent, we propose to estimate the variance of ut byusing ΩHAR =

∑τj=−τ+1K

(jM

)γj , where K(·) is a kernel function with bandwidth M

and γj = 1τ

∑τt=j+1 utut−j is the j

th order sample autocovariance. Based on ΩHAR, wecan define the new t statistic as

DFτ ,HAR =ρτ − 1

sτ ,HAR, (13)

where

sτ ,HAR =

√√√√ ΩHAR∑τt=1 y

2t−1 − τ−1 (

∑τt=1 yt−1)2 .

In addition, we select the bandwidth by letting M = τ so that the bandwidth is thesame as the sample size τ in the regression window. This approach is popularized byKiefer and Vogelsang (2002a, 2002b, 2005), Bunzel et al. (2002) and Vogelsang (2003).The test statistic based on Ω is heteroskedasticity-autocorrelation robust (HAR). Our testalso shares the same spirit as the test proposed in Sun (2004), where the HAR test statisticis used to tackle the problem of spurious co-integration.

Theorem 3.1 Suppose M = τ , K(·) ≥ 0, K(x) = K(−x), and K(·) is twice differen-tiable. Under model (10), for r ∈ (0, 1] and d ∈ [0, 0.5), as n→∞,

DFτ ,HAR =⇒ Fr,d

:=

r3/2

∫ r0 W (s)dW (s)

(∫ r0 W (s)2ds

∫ r0

∫ r0 −K′′(

p−qr )Gr(0,p)Gr(0,q)dpdq)

1/2 if d = 0

r3/2

2 (BH(r))2−r1/2(

∫ r0 B

H(s)ds)BH(r)

((∫ r0 B

H(s)2ds)∫ r0

∫ r0 −K′′(

p−qr )Gr(d,p)Gr(d,q)dpdq)

1/2 if d ∈ (0, 0.5),(14)

where

Gr(d, p) = BH(p)− 1

rX(r, d)

∫ p

0BH(s)ds− Y (r, d)p,

8

X(r, d) =

r∫ r0 W (s)dW (s)∫ r0 W (s)2ds

if d = 0

r2(B

H(r))2−(

∫ r0 B

H(s)ds)BH(r)∫ r0 (BH(s))

2ds

if d ∈ (0, 0.5),

Y (r, d) =

(∫ r0 W (s)2ds)W (r)− 1

2 [(W (r))2−r]∫ r0 W (s)ds

r∫ r0 W (s)2ds−(

∫ r0 W (s)ds)

2 if d = 0

(∫ r0 B

H(s)2ds)BH(r)− 12

[(BH(r))

2] ∫ r

0 BH(s)ds

r∫ r0 B

H(s)2ds−(∫ r0 B

H(s)ds)2 if d ∈ (0, 0.5)

,

BH(t) is a fractional Brownian motion (fBm) with the Hurst parameter H = 1/2 +d, andBH(r) = BH(r)− 1

r

∫ r0 B

H(s)ds is the demeaned fBm.

Theorem 3.1 requires K(·) to be twice differentiable. This requirement rules out thepopular Bartlett kernel. The following theorem gives the limit distribution of DFBartlettτ ,HAR

which is DFτ ,HAR defined in (13) is calculated from the Bartlett kernel.

Theorem 3.2 Under the same set of assumptions as in Theorem 3.1, if K(x) = (1 −|x|)1 (|x| ≤ 1), for r ∈ (0, 1], as n→∞,

DFBartlettτ ,HAR =⇒ FBartlettr,d

:=

r∫ r0 W (s)dW (s)

[2∫ r0 W (s)2ds(

∫ r0 Gr(0,p)

2dp)]1/2 if d = 0[

r2(B

H(r))2−(

∫ r0 B

H(s)ds)BH(r)]

[2∫ r0 (BH(s))

2ds(

∫ r0 Gr(d,p)

2dp)]1/2 if d ∈ (0, 0.5)

. (15)

Theorems 3.1 and 3.2 imply that the test statistics DFn,HAR and DFBartlettn,HAR have theasymptotic distributions F1,d and FBartlett1,d , respectively. Unlike Lemma 3.1, Theorems3.1 and 3.2 show that the HAR test statistics converge to proper limit distributions forboth d = 0 and d ∈ (0, 0.5). Both limit distributions are functionals of fBm and can beobtained by simulations. To carry out right-tailed unit root tests based on DFn,HAR and

DFBartlettn,HAR , we use (1 − β) × 100% critical values cv(1−β)%HAR (d) and cv(1−β)%

HAR (d) which aredefined by

Pr(F1,d > cv

(1−β)%HAR (d)

)= β,

Pr(FBartlett1,d > cv

(1−β)%HAR (d)

)= β.

Moreover, the memory parameter (d) that appears in the critical values can be consistently

estimated (e.g. using the ELW method). Therefore, feasible critical values cv(1−β)%HAR

(d)

and cv(1−β)%HAR

(d)can be obtained.

Remark 3.5 Following PWY, a recursive version of DFτ ,HAR and its limit distributioncan be obtained. Let

SDFHAR = supτ∈[τ0,n]

DFτ ,HAR.

9

Based on the continuous mapping theorem, under the same set of assumptions as in The-orem 3.1, we can obtain

SDFHAR =⇒ supr∈[r0,1]

Fr,d.

As in PWY, this recursive formulation can identify a rational bubble when the time serieshas a mildly explosive root.

Remark 3.6 The limit distributions in Theorems 3.1 and 3.2 apply when the error term utfollows a stationary ARFIMA(p, d, q) process with d ∈ (0, 0.5). Indeed, n−1−2d

(∑τt=1 u

2t

)vanishes with d > 0 as n → ∞. It washes out the variance term which is dependent onthe specific form of the ARFIMA(p, d, q) process.

Remark 3.7 In principle, we can let ΩHAR =∑τ

j=−τ+1K(jM

)γj with M = b× τ , and

then obtain DF bτ ,HAR and DFb,Bartlettτ ,HAR with b fixed (say any b ∈ (0, 1]). A fixed-b asymptotic

distribution of DF bτ ,HAR and DFb,Bartlettτ ,HAR can be obtained. However, using simulated data

we find different values of b lead to very little change in the empirical size. Thus, in thispaper we only focus on b = 1.

3.3 Finite sample corrected critical value

It should be noted that the limit distributions F1,d and FBartlett1,d are discontinuous at d = 0.The normalized centered LS estimator has a component −n−1−2d

(∑τt=1 u

2t

). When d > 0,

this term is asymptotically negligible. However, when d = 0, this term cannot be ignored.If d is strictly positive and suffi ciently close to zero, −n−1−2d

(∑τt=1 u

2t

)converges in

probability to zero slowly. In this case, it is expected that the limit distribution may notwell approximate the finite sample distribution DFn,HAR unless the sample size is verylarge. In the simulations provided later, we show that the critical value based on the limitdistribution provides a test that is too conservative. To obtain a better size, we proposethe following 4-step procedure to obtain a finite sample corrected critical value.

1. Suppose d is known and d ∈ (0, 0.5). Let εt = (1 − L)1+dyt for t = 1, ...n. Letς2 = Γ(1−2d)

Γ(d)Γ(1−d)1n

∑nt=1 ε

2t with Γ(·) being the Gamma function.

2. Let σ2u = 1

n

∑nt=1 u

2t , where ut is defined in (5).

3. If a second-order differentiable kernel function is used to calculate DFn,HAR, wesimulate the distribution of

Fn1,d = F1,d −12

1n2d

σ2uς2((∫ 1

0 BH(s)2ds

) ∫ 10

∫ 10 −K ′′ (p− q)G1(d, p)G1(d, q)dpdq

)1/2.

10

If the Bartlett kernel function is adopted to calculate DFn,HAR, we simulate thedistribution of

Fn,Bartlett1,d = FBartlett1,d −2n2d

σ2uς2[∫ 1

0

(BH(s)

)2ds(∫ 1

0 G1(d, p)2dp)]1/2

.

4. Obtain the (1−β)% critical value cv(1−β)%n,HAR (d) for the right-tailed unit root test using

the following construction:

Pr(Fn1,d > cv

(1−β)%n,HAR (d)

)= β,

Pr(Fn,Bartlett1,d > cv

(1−β)%n,HAR (d)

)= β.

The idea of the finite sample corrected critical value is to reserve and approximate thesmaller order term

(−n−1−2d

(∑τt=1 u

2t

)). As d is unknown, we can replace d by d when

d ∈ (0, 0.5) to obtain a feasible critical value cv(1−β)%n,HAR

(d). If d = 0, we do not implement

the finite sample correction. In this case, the asymptotic critical value cv(1−β)%HAR (0) is used.

4 Alternative Hypothesis and Asymptotic Theory

To study the asymptotic behavior of the proposed test statistic under the alternativehypothesis, following the literature we consider two popular explosive models. The firstalternative adopts the local-to-unit-root framework of Phillips (1987b) to study the locallyexplosive time series; see Harvey et al. (2016, 2019a, and 2019b). The advantage of usingthe locally explosive model is that it facilitates the computation of local power.

The second alternative is the mildly explosive model of Phillips and Magdalinos (2007).It assumes that the AR parameter has a greater deviation from the unit root than thelocal-to-unit-root model; see PWY, PSY and Phillips and Yu (2011). Under this explosivealternative, a consistent test can be obtained.

4.1 Locally explosive model

We first consider the alternative hypothesis with the following locally explosive setting:

yt = (yt−1 + ut) 1t < τ e+ (ρnyt−1 + ut) 1τ e ≤ t ≤ n, t = 1, ..., n,

ut = 1− L)−dεt, εtiid∼ (0, σ2), d ∈ [0, 0.5),

ρn = 1 + c/n, c > 0,

y0 = op(n1/2+d).

(16)

In model (16), yt has a unit root before τ e. It becomes mildly explosive after τ e.That is, there is a structural break at τ e. During both periods, the error term in the ARmodel has strong dependence with the same memory parameter d. To study the limitdistributions of DFτ ,HAR and DFBartlettτ ,HAR , we first introduce the following lemma.

11

Lemma 4.1 Under the local alternative model (16), assuming 0 < re ≤ r ≤ 1, as n→∞,

1. 1n1/2+d

yτ ⇒ σd

(e(r−re)cBH(re) +

∫ rree(r−s)cdBH(s)

);

2. 1n3/2+d

∑τt=1 yt−1 ⇒ σdAr,d;

3. 1n2+2d

∑τt=1 y

2t−1 ⇒ σ2

dBr,d;

4. 1n1+2d

∑τt=1 yt−1ut ⇒

σ2d2 Cr,d; where

Ar,d =

∫ r

0

(e(x−re)cBH(re) +

∫ x

re

e(x−s)cdBH(s)

)dx,

Br,d =

∫ r

0

(e(x−re)cBH(re) +

∫ x

re

e(x−s)cdBH(s)

)2

dx,

Cr,d =

(e(r−re)cW (re) +

∫ rree(r−s)cdW (s)

)2− r − re, if d = 0(

e(r−re)cBH(re) +∫ rree(r−s)cdBH(s)

)2, if d ∈ (0, 0.5)

,

JHc (r) =

∫ r

0e(r−s)cdBH(s),

σ2d =

σ2, if d = 0,

ζ2 := Γ(1−2d)Γ(d)Γ(1−d)σ

2, if d ∈ (0, 0.5).

Lemma 4.2 Under model (16), as n→∞,

1. n(ρτ − ρc)⇒ Xc(r, d);

2. n(ρτ − 1)⇒ Xc(r, d) + c;

3. n1/2−dµ⇒ σdYc(r, d),

where Xc(r, d) :=12Cr,d− 1

rAr,dB

H(r)

Br,d− 1rA2r,d

and Yc(r, d) :=Br,dB

H(r)− 12Cr,dAr,d

r(Br,d−A2r,d).

We now consider the asymptotic behavior of the HAR test statistics.

Theorem 4.1 Under model (16), for any r > re, as n→∞,

DFτ ,HAR =⇒ F cr,d :=Ar,dB

H(r)− r2Cr,d + cA2

r,d −Br,dcr1r

∫ r0

∫ r0 K

′′(p−q

r

)Gr,c(d, p)Gr,c(d, q)dpdq

, (17)

DFBartlettτ ,HAR =⇒ F c,Bartlettr,d :=r2Cr,d −Ar,dB

H(r) +Br,dcr − cA2r,d

2∫ r

0 Gc(d, r, p)2dp

, (18)

where Gr,c(d, p) = BH(p)−Xc(r, d)Ap,d − cAre ,d − Yc(r, d)p.

Both limit distributions in Theorem 4.1 depend on the non-centralized parameter c.This parameter departs the distribution of F cr,d and F

c,Bartlettr,d from Fr,d and FBartlettr,d ,

respectively. One can directly verify that if c = 0, F cr,d = Fr,d from (14) and F c,Bartlettr,d =

FBartlettr,d from (15). Since both F cr,d and Fc,Bartlettr,d are Op(1), one may use them to obtain

the local power of the proposed test.

12

4.2 Mildly explosive model

We then consider the alternative hypothesis with the following mildly explosive setting:

yt = (yt−1 + ut) 1t < τ e+ (ρnyt−1 + ut) 1τ e ≤ t ≤ n, y0 = op(n1/2+d1),

ut =

(1− L)−d1εt if t < τ e,

(1− L)−d2εt if τ e ≤ t ≤ n,and εt

iid∼ (0, σ2), d1, d2 ∈ [0, 0.5). (19)

where

ρn = 1 +c

nα, c > 0, α ∈

(0,min

1/2− d1

1/2− d2, 1

). (20)

In model (19), yt has a unit root before τ e. It becomes mildly explosive after τ e. That is,there is a structural break at τ e. During both periods, the error term in the AR modelhas strong dependence with memory parameters d1 prior to the break and d2 after thebreak. As 0 < α < 1, we obtain a higher degree of explosiveness than the local-to-unit-root explosiveness considered earlier. The upper bound on α (i.e. 1/2−d1

1/2−d2 ) is needed aswe have to ensure that the explosive observations dominate asymptotically the unit rootobservations with long memory errors.

Theorem 4.2 Under model (16) with (20), as n→∞,

DFn,HARp→∞ and DFBartlettn,HAR

p→∞.

Theorem 4.2 shows that the HAR test statistics diverge, and naturally, P (DFn,HAR >

cv|ρn = 1 + cnα ) → 1 for any given fixed critical value (cv) under model (16) with (20).

Therefore, we have a consistent test for this explosive alternative.

5 Estimation of Bubble Origination and Termination Dates

In this section, we discuss the estimation of the bubble origination and termination dates.Following PWY, we consider the following model:

yt = (yt−1 + ut) 1t < τ e+ (ρnyt−1 + ut) 1τ e ≤ t ≤ τ f+

t∑k=τf+1

uk + y∗τf

1t > τ f,

ρn = 1 +c

nα, c > 0, α ∈

(0,min

1/2− d1

1/2− d2, 1

),

ut = (1− L)−dtεt, εtiid∼ (0, σ2), (21)

dt = d1 for t ∈ [1, τ e) ∪ [τ f + 1, n], dt = d2 for t ∈ [τ e, τ f ], τ e = bnrec, τ f = bnrfc,y∗τf = yτe + y∗, and y∗ = Op(1).

Once again, we require α < 1/2−d11/2−d2 .

13

Theorem 5.1 Under model (21) with τ = bnrc , DFτ ,HAR has the following asymptoticbehaviour:

DFτ ,HARp→∞ if τ ∈ [τ e, τ f ],

DFτ ,HARp→ −∞ if τ ∈ [τ f + 1, n].

(22)

The estimators of re and rf are defined as

rHARe = infr≥r0r : DFτ ,HAR > cvn,HAR,

rHARf = infr>re+γ ln(n)/n

r : DFτ ,HAR < cvn,HAR.(23)

If re ≥ r0 and the critical value cvn,HAR satisfies the following condition

1

cvn,HAR+cvn,HAR

n(1−α)/2 → 0,

then, as n→∞, we have

rHARep→ re and rHARf

p→ rf .

Intuitively,⌊nrHARe

⌋represents the first observation when DFτ ,HAR > cvn,HAR and,

after a bubble is deemed to have emerged and lasts longer than γ ln(n)/n,⌊nrHARf

⌋represents the first observation when DFτ ,HAR < cvn,HAR. Note that we require a bubbleto have a minimum duration γ ln(n)/n where γ is a frequency dependent parameter.

Theorem 5.1 implies that DFτ ,HAR provides a consistent test when data do not containany observations after the bubble collapses. Otherwise, it diverges to negative infinity.Therefore, if the full sample is used to do the right-tailed unit root test, one should avoidincluding observations after a bubble collapses. Otherwise, the test cannot detect thepresence of a bubble. This is known as Evans’s critique (Evans, 1991). For the consistentestimation of the bubble origination and termination dates, we require that the criticalvalue cvn,HAR grows to infinity at a rate slower than n(1−α)/2, which is different from therate obtained in PWY.

PWY propose to estimate re using rPWYe in (6.3) with cvn increasing at the ln lnn

rate. However, in our model (21) with d1 ∈ (0, 0.5), we can easily obtain

rPWYe

p→ r0 ≤ re.

Indeed, for r < re, DFτ ,HAR diverges at the rate nd1 (as shown in Lemma 3.1), which isfaster than ln lnn. Therefore, we can expect rPWY

e with cvn increasing at the ln lnn rateto be inconsistent when d > 0 and r0 < re. In the Monte Carlo simulations presented insection 6, we show that rPWY

e with cvn increasing at the ln lnn rate tends to lead to a tooearly estimation of the bubble origination and termination dates. In contrast, rHARe is aconsistent estimator if ln lnn is the growth rate adopted for cvn,HAR.

14

Remark 5.1 If we use supDFτ to detect a rational bubble and use rPWYe with cvn increas-

ing at the ln lnn rate to estimate the bubble origination, under model (10) with d ∈ (0, 0.5),not only is it highly likely to have a false bubble detection, but also it generate a too earlyestimate of the bubble origination and termination dates.

6 Monte Carlo Studies

In this section, we design some Monte Carlo experiments to study the size and powerof our proposed test for bubble detection and our estimators of bubble origination andtermination dates in finite samples. The number of replications is always set to 2,500.

6.1 Empirical size

To investigate the empirical size of our test statistic, we perform a small-scale Monte Carlostudy. In particular, we consider the following DGP,

yt = yt−1 + ut, t = 1, ..., n,

ut = (1− L)−dεt, εtiid∼ N(0, 1),

(24)

with the following parameter settings: d ∈ 0, 0.05, 0.1, ..., 0.45, y0 = 0, and n ∈ 100, 500.

Table 2 Empirical size of the right-tailed unit root test

n = 100 d

test statistics, cv 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

DFn, cv95% 0.06 0.10 0.16 0.21 0.27 0.33 0.39 0.44 0.48 0.51

DFBartlettn,HAR , cv95%HAR(d) 0.06 0.01 0.01 0.02 0.02 0.03 0.03 0.03 0.04 0.04

DFBartlettn,HAR , cv95%n,HAR(d) 0.06 0.05 0.05 0.04 0.04 0.04 0.04 0.04 0.04 0.04

DFBartlettn,HAR , cv95%HAR

(d)

0.00 0.00 0.01 0.01 0.02 0.03 0.03 0.04 0.05 0.06

DFBartlettn,HAR , cv95%n,HAR

(d)

0.02 0.03 0.04 0.05 0.06 0.06 0.06 0.07 0.07 0.07

n = 500 d

DFn, cv95% 0.05 0.11 0.18 0.25 0.33 0.39 0.44 0.48 0.52 0.54

DFBartlettn,HAR , cv95%HAR(d) 0.05 0.01 0.01 0.02 0.03 0.04 0.04 0.04 0.05 0.05

DFBartlettn,HAR , cv95%n,HAR(d) 0.05 0.04 0.04 0.05 0.04 0.05 0.05 0.05 0.05 0.05

DFBartlettn,HAR , cv95%HAR

(d)

0.00 0.01 0.02 0.03 0.03 0.04 0.05 0.05 0.05 0.05

DFBartlettn,HAR , cv95%n,HAR

(d)

0.02 0.04 0.05 0.06 0.06 0.06 0.06 0.06 0.06 0.06

Under these parameter settings, we perform a right-tailed unit test and calculate DFnand DFBartlettn,HAR . For the standard right-tailed unit root test, we reject the null hypothesiswhen DFn is greater than the 95% asymptotic critical value cv95% = −0.08. For theHAR test, four critical values are used: two 95% asymptotic critical values (cv95%

HAR(d) and

cv95%HAR

(d)) and two finite sample corrected critical values (cv(1−β)%

n,HAR (d) and cv(1−β)%n,HAR

(d)).

15

Note that as d is unknown, cv95%HAR(d) and cv(1−β)%

n,HAR (d) are infeasible. We obtain the feasible

critical values by obtaining d using the ELW method proposed by Shimotsu and Phillips(2005).

Table 2 reports the empirical size of the 5% level right-tailed unit root test based onDFn and DFn,HAR and the corresponding critical values. Several observations can bemade from Table 2. First, the standard unit root test has a divergent size when d > 0.For example, it can be seen that when d = 0.3, the test rejects the null hypothesis about40% of the time, which is far above the nominal rate of 5%. These simulation results areconsistent with the prediction of the asymptotic theory in Sowell (1990) and Lemma 3.1.Second, when d is known to be 0, the standard unit root test and our HAR test have avery similar performance in size. When d is known but not equal to 0, cv95%

HAR(d) providesa very conservative test for small values of d. Table 2 shows that the rejection rates are0.01 for d = 0.05, 0.1 with n = 100, 500 and are far below 5%. However, when the finitesample correction is implemented, cv95%

HAR(d) provides a test with an empirical size veryclose to 5%. Third, when d is unknown, the test is extremely conservative when d is closeto 0, but not conservative when d > 0.4. Finally, when the finite sample correction is used,cv95%n,HAR

(d)has a good performance compared to cv95%

HAR

(d)when d is relatively small.

It is slightly outperformed by cv95%HAR

(d)when d > 0.3.

6.2 Power

To investigate the power of our test under finite sample, we design a Monte Carlo studybased on model (19) with the following parameter settings: n = 100, y0 = 100, re ∈0.3, 0.5, 0.7, d1 = d2 = d ∈ 0, 0.05, 0.1, ..., 0.45, ρn = 1 + c/nα, c ∈ 1, 2, andα = 0.75. Under these parameter settings, we perform a right-tailed unit root test andcalculate DFBartlettn,HAR . We use the same procedures as in section 6.1 and construct a finitesample corrected critical value, rejecting the unit root assumption if DFBartlettn,HAR is greater

than cv95%n,HAR

(d).

Table 3 Empirical power of the HAR test

d

re 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.3 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.97c = 1 0.5 1.00 1.00 1.00 1.00 0.99 0.97 0.94 0.90 0.86 0.82

0.7 0.95 0.92 0.87 0.82 0.82 0.77 0.72 0.65 0.61 0.59

0.3 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00c = 2 0.5 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.98

0.7 1.00 1.00 1.00 1.00 0.99 0.98 0.95 0.91 0.87 0.83

Table 3 reports the empirical rejection rates under different parameter settings. Severalobservations can be made from Table 3. First, the empirical power of our test decreasesas the value of d increases, because the random wandering behavior of the non-explosive

16

observations exhibits greater variation with a higher value of d. This can produce a randomwandering path to the negative side, which will yield a small test statistic DFBartlettn,HAR ,preventing it from exceeding the critical value in a finite sample. Second, higher powercomes with a higher c and a smaller re. This is expected as a higher value of c gives themodel a higher degree of explosiveness, while a lower value of re extends the duration of anexplosive path, both leading to an easier bubble detection. For a relatively short bubble(e.g re = 0.7), the successful detection of a rational bubble requires a higher degree ofexplosiveness.

6.3 Estimators of orgination and termination dates

To study the finite sample performance of the proposed estimators of bubble originationand termination dates, we design a Monte Carlo experiment based on model (21) withthe following parameter settings: n = 100, y0 = 100, c = 2, α = 0.75, d1 = d2 =

d ∈ 0, 0.1, 0.2, 0.3, 0.4, εtiid∼ N(0, 1), y∗τf = yτe , re = 0.6, rf = 0.8, r0 = 0.4, and

γ ln(n)/n = 0.1.To obtain rHARe and rHARf , we first calculate DFτ , DFτ ,HARnτ=bnr0c. Second, we use

bnr0c as the minimum window and obtain d using the ELW method. Third, we specifythe following critical value function cvn,HAR,

cvn,HAR =

ln(lnns)/100 if d = 0

cv90%n,HAR

(d)× ln(lnns)

1.3 if d ∈ (0, 0.5), (25)

where s ∈ (0, 1] is proportional to the sample size. Note that Theorem 5.1 implic-itly requires cvn,HAR > 0, so we specify cvn,HAR = ln(lnns)/100, when d = 0 (ascv90%n,HAR(0) < 0). This value is between the 95% and 99% critical value of cvn,HAR(0). For

d ∈ (0, 0.5), we let cvn,HAR = cv90%b

(d)× ln(lnns)

1.3 . This formulation always yields a posi-

tive number for cvn,HAR in our simulations and gives a critical value between cv90%n,HAR

(d)

and cv95%n,HAR

(d).

We also follow PWY and specify

cvn = ln(lnns)/100. (26)

By combining (8), (9), (23), (25), and (26), we can calculate rHARe , rHARf , rPWYe and rPWY

f .

Table 4 Finite sample performance of rHARe , rHARf , rPWYe and rPWY

f

d

0 0.1 0.2 0.3 0.4

rHARe 0.634 0.635 0.639 0.636 0.631rHARf 0.815 0.807 0.802 0.793 0.784

rPWYe 0.600 0.577 0.553 0.532 0.515rPWYf 0.792 0.767 0.743 0.727 0.721

17

Table 4 reports the average of rHARe , rHARf , re and rHARf across 2,500 replications. Withdifferent values of d, our estimators of re and rf are reasonably close to the true values of0.6 and 0.8, respectively. When d = 0, rPWY

e and rPWYf perform very well to estimate re

and rf . When d ∈ 0.1, 0.2, 0.3, 0.4, we can see a too early detection, as rPWYe is smaller

than re on average. Indeed, when d ∈ (0, 0.5) with the critical value increasing at the rateln lnn and r0 < re, rPWY

e is an inconsistent estimator, which converges in probability tor0. Interestingly, when d > 0, rPWY

f is also inaccurate, because when rPWYe is close to r0,

the region between r0 and re has a random wandering and non-standardized test statistic,as the test statistic occasionally falls below the too small critical value function (as ln lnn

is dominated by nd) in this region. In this experiment, the inconsistent estimator rPWYe

also induces an inaccurate estimate of rf .

Remark 6.1 The Monte Carlo simulation exercise to estimate re and rf can also be seenas a pseudo real-time bubble detection exercise. Given the current data (ytbnr0ct=1 in oursimulations), we can obtain the estimator of d and thus cvn,HAR. As we can update theHAR test statistic DFτ ,HAR when new data become available, we can detect a rationalbubble in real time.

7 Empirical Studies

To highlight the usefulness of our HAR test statistic, we conduct an empirical study usingthe same time series as that was used to obtain Table 1. Because including observationsafter the collapse of a bubble asymptotically leads to a negatively divergent test statisticand jeopardizes the bubble detection procedure, we avoid any obvious market collapsewhen selecting our sampling periods. We implement our HAR test statistic with a Bartlettkernel function and use the finite sample corrected critical values when performing theright-tailed unit root test.

Table 5 Test for an explosive alternative in the S&P 500 index

Sampling Period DFBartlettn,HAR d cv90%n,HAR

(d)

cv95%n,HAR

(d)

(a) May 1948 to October 1955 0.98 0.25 2.54 3.86(b) April 1977 to March 1987 2.22 0.21 2.17 3.11(c) January 1990 to April 1998 11.35 0.24 2.48 3.72

Table 5 reports the HAR test statistic DFBartlettn,HAR and the 90% and 95% finite sample

corrected critical values (cv90%n,HAR

(d), cv95%

n,HAR

(d)) for the three sampling periods. In

Table 1, it is clear that the standard test statistic DFn is greater than its 95% criticalvalue, resulting in the rejection of the null hypothesis and confirming a rational bubble.In Table 5, our test statistic DFBartlettn,HAR fail to reject the null hypothesis for the samplingperiods (a) and (b) when using the 95% critical values in this test. Therefore, our proposedtest cannot find evidence of the presence of a rational bubble. For the sampling period(c), as DFBartlettn,HAR > cv95%

n,HAR

(d), our proposed test finds evidence of the presence of a

rational bubble.

18

Figure 2: Estimation of the bubble origination and termination dates

After a rational bubble is found, we proceed to estimate the bubble origination andtermination dates. Because we tried to avoid the stock market crash during the samplingperiod (c), to estimate the bubble termination date, we now use an extended samplingperiod where the stock market behaves more like model (21) with both exuberance andcollapse. Data from November 1990 to October 1998 with a sample size of 106 are includedin this analysis. Our estimators (rHARe , rHARf ) and the PWY estimators (rPWY

e , rPWYf )

are applied to evaluate the bubble origination and termination dates. We let the minimumbubble period be 6 months.

We let n∗ = 40 be the minimum data window and estimate d1 using the ELW method.We obtain d1 = 0.1965.3 Finally, we estimate rHARe , rHARf , rHARe and rHARf as in ourprevious Monte Carlo simulation in section 6.3.

In Figure 2, the blue, grey and red lines represent the DFBartlettτ ,HAR , the price dividendratio of the S&P 500 index, and cvn,HAR respectively. The left axis shows the price-dividend ratio and the right axis shows the values of DFBartlettτ ,HAR and cvn,HAR. The greenand brown vertical lines indicate the date DFBartlettτ ,HAR crosses cvn,HAR from below and fromabove, respectively. As the figure shows, our method detects the beginning of the bubblein November 1997 and the ending in August 1998. These estimates coincide with theAsian and Russian financial crises in the late 1990s. The PWY method obtains an earlier

3To check the robustness of this result, the memory parameter estimate proposed by Wang et al. (2019)

is applied, and we estimate d using the formula dWXY =12

[log2

(∑n∗i=1(yi+4−2yi+2+yi)

2∑n∗i=1(yi+2−2yi+1+yi )

2

)− 1

]. We obtain

dWXY = 0.19

19

estimate of the bubble origination date in July 1995. This empirical exercise confirms thatthe PWY method often provides an earlier estimate of the bubble origination when thetime series has a strongly dependent error.

8 Conclusion

This paper introduces a new test and dating algorithm for the purpose of bubble detection.We motivate our test by showing empirical evidence that an autoregressive model may havestrongly dependent errors. Because strongly dependent errors produce divergent Dickey-Fuller t statistics, the use of the traditional right-tailed unit root test statistics, such as thePWY statistic, spuriously detects a rational bubble. Not surprisingly, the PWY methodalso gives inaccurate estimators of the bubble origination and termination dates.

To avoid the spurious bubble detection, we propose a heteroskedasticity autocorrelationrobust (HAR) test statistic. The idea behind our test is to use a properly self-normalizedestimator of the standard error of the LS estimator of the AR(1) coeffi cient. We obtainthe limit distribution of the proposed test statistic.

Based on a sequence of proposed test statistic, we then introduce new estimators totimestamp the bubble origination and termination dates based on the first-time-crossingprinciple. We show that the proposed estimators consistently estimate the bubble origi-nation and termination dates when the true data generate process switches from a unitroot model to a mild explosive model with a crash at the end of the explosive period andthen switch back to a unit root model.

We have designed several Monte Carlo experiments to study the finite sample prop-erties of the proposed test and estimator. Via simulated data, we first show that thetraditional unit root test identifies too many bubbles when in an AR model with stronglydependent errors. We also show that the PWY estimator tends to have a too early esti-mation on the bubble origination and termination dates. Via the same simulated data, wethen show that the proposed HAR statistic, accompanied by the finite sample correctedcritical value, provides a test with well-controlled size and power in finite samples. Theproposed estimators also lead to much better finite sample performance than the PWYestimators for the bubble origination and termination dates.

Our proposed test and estimators are applied to the data of S&P 500 monthly price-dividend ratio. According to the new test, two of the rational bubbles ((a) from May1948 to October 1955 and (b) from April 1977 to March 1987) identified by the traditionalunit root test are spurious due to strongly dependent errors. However, a rational bubble isdetected by the proposed test during the 1990s, suggesting that a rational bubble originatesin November 1996 and collapses in August 1998.

While in this paper we have not addressed the issue of multiple bubbles, we shouldpoint out that our test statistic and the estimators can be extended to deal with themultiple bubbles in the same ways as in PSY (2015a, 2015b). The idea is to replace DFτwith DFτ ,HAR. Such an extension will be investigated in a future study.

20

APPENDIX

Proofs of Lemmas and Theorems

Before we prove Lemma 3.1, it is useful to list the following lemma.

Lemma 8.1 (Corollary 4.4.1 in Giraitis et al. (2012)) Suppose ut =∑∞

j=0 cjεt−j,

and εtiid∼ (0, σ2). Assume cj

a∼ γj−1+d with d ∈ (0, 0.5), γ being a constant, as j →∞,Then we have

n−( 12+d)bnrc∑t=1

ut ⇒ ςBH(r), (27)

in D[0, 1] with the uniform metric, where H = 12 + d, ς =

√σ2γ2 B(d,1−2d)

d(1+2d) with B(x, y) =Γ(x)Γ(y)Γ(x+y) , B

H(r) being an fBm with Hurst parameter H.

Note that Lemma 8.1 is general enough to include the fractional intergraded processwhere ut = (1−L)dεt, where εt

iid∼ (0, σ2). As one can show ut = (1−L)dεt =∑∞

j=0 cjεt−j ,

with cja∼ j−1+d

Γ(d) .

Lemma 8.2 Suppose that we have the following data generating process:

yt = yt−1 + ut, y0 = op(n1/2+d),

(1− L)dut = εt, εtiid∼ (0, σ2), d ∈ (0, 0.5)

Let τ = bnrc with r ∈ (0, 1], we have

1. 1n1+2d

∑τt=1 yt−1ut =⇒ ς2

2

[(BH(r)

)2]2. 1

n3/2+d

∑τt=1 yt−1 =⇒ ς

∫ r0 B

H(s)ds

3. 1n2+2d

∑τt=1 y

2t−1 =⇒ ς2

∫ r0 B

H(s)2ds

where ς =√σ2 Γ(1−2d)

Γ(d)Γ(1−d) .

Proof of Lemma 8.2

1.

n−1−2dτ∑t=1

yt−1ut = n−1−2d 1

2

(y2bnrc −

τ∑t=1

u2t

)+ op(1)

=1

2

[(n−1/2−dybnrc

)2− 1

n2d

(1

n

τ∑t=1

u2t

)]+ o(1)

=⇒ ς2

2

[(BH(r)

)2]. (28)

21

We obtain the last result since n−1/2−dybnrc ⇒ ςBH(r) (from Lemma 8.1), 1n

∑τt=1 u

2t =

τn

1τ

∑τt=1 u

2t ,

1τ

∑τt=1 u

2ta.s.→ E[u2

t ] (by ergodic theorem) andτn → r. These imply

1n2d

(1n

∑τt=1 u

2t

) p→ 0.

2.

1

n3/2+d

τ∑t=1

yt−1

=1

n3/2+d

τ∑t=1

(t−1∑i=1

ui +Op(1)

)

=1

n

τ∑t=1

(1

n1/2+d

t−1∑i=1

ui

)+ op(1)

=⇒ ς

∫ r

0BH(s)ds (by Lemma 8.1 and continuous mapping theorem (CMT)).

3.

1

n2+2d

τ∑t=1

y2t−1 =

1

n2+2d

τ∑t=1

(τ−1∑i=1

ui

)2

+ op(1)

=1

n

τ∑t=1

(1

n1/2+d

τ−1∑i=1

ui

)2

+ op(1)

=⇒ ς2

∫ r

0BH(s)2ds (by Lemma 8.1 and CMT).

Before we prove Lemma 3.1, we first introduce the following Lemma.

Lemma 8.3 Under model (10) with ρ = 1 and d ∈ (0, 0.5). Suppose an empirical re-gression (5) with observations indexed by t = 1, ..., τ = bnrc is applied. For r ∈ (0, 1], asn→∞,we have the following results:

τ(ρτ − 1) =⇒r2

(BH(r)

)2 − (∫ r0 BH(s)ds)BH(r)∫ r

0 (BH(s))2 ds− 1r

(∫ r0 B

H(s)ds)2

=r2

(BH(r)

)2 − (∫ r0 BH(s)ds)BH(r)∫ r

0

(BH(s)

)2ds

:= X(r, d). (29)

Proof of Lemma 8.3The normalized centered LS estimator can be written as

τ(ρτ − 1) =τ

nn

[∑τt=1 yt−1ut − 1

τ

∑τt=1 yt−1

∑τt=1 ut∑τ

t=1 y2t−1 − 1

τ (∑τ

t=1 yt−1)2

]

=τ

n

n−1−2d

n−2−2d

[∑τt=1 yt−1ut − 1

τ

∑τt=1 yt−1

∑τt=1 ut∑τ

t=1 y2t−1 − 1

τ (∑τ

t=1 yt−1)2

]

22

=τ

n

n−1−2d∑τ

t=1 yt−1ut − nτ n−3/2−d∑τ

t=1 yt−1n−1/2−d∑τ

t=1 ut

n−2−2d∑τ

t=1 y2t−1 − n

τ

(n−3/2−d∑τ

t=1 yt−1

)2=⇒ r

ς2

2

[(BH(r)

)2]− 1r ς

2∫ r

0 BH(s)dsBH(r)

ς2∫ r

0 BH(s)2ds− 1

r ς2(∫ r

0 BH(s)ds

)2 (by Lemma 8.1 and 8.2)

=

r2

[(BH(r)

)2]− ∫ r0 BH(s)dsBH(r)∫ r0 B

H(s)2ds− 1r

(∫ r0 B

H(s)ds)2 := X(r, d).

Proof of Lemma 3.1For the test statistics DFτ = ρτ−1

sτ, note that s2

τ =1τ

∑τt=1 u

2t∑τ

t=1 y2t−1−

1τ (∑τt=1 yt−1)

2 .

A ut = ut + (1− ρτ )yt−1− µ, where is µ the LS estimator for the intercept, so we have

1

τ

τ∑t=1

u2t

=1

τ

τ∑t=1

(ut + (1− ρτ )yt−1 − µ)2

=1

τ

τ∑t=1

u2t +

2(1− ρτ )

τ

τ∑t=1

yt−1ut +(1− ρτ )2

τ

τ∑t=1

y2t−1 − 2µ

1

τ

τ∑t=1

ut − 2(1− ρτ )µ

τ

τ∑t=1

yt−1 +µ2

τ

τ∑t=1

1

=1

τ

τ∑t=1

u2t +

2τ(1− ρτ )

τ2

τ∑t=1

yt−1ut +(τ(1− ρτ ))2

τ3

τ∑t=1

y2t−1 − 2τ(1− ρτ )µ

(1

τ2

τ∑t=1

yt−1

)+ op(1)

=1

τ

τ∑t=1

u2t + op(1)

p→ E[u2t ]

where the last equality follows from by Lemma 8.2.Note that

n−(2+2d)

τ∑t=1

y2t−1 −

1

τ

(τ∑t=1

yt−1

)2

= n−(2+2d)τ∑t=1

y2t−1 −

(n−3/2−d∑τ

t=1 yt−1

)2τn

=⇒ ς2

(∫ r

0

(BH(s)

)2ds− 1

r

(∫ r

0BH(s)ds

)2).

Therefore it is straightforward to show

n−dDFτ

=n(ρτ − 1)

(n2+2ds2τ )

1/2

23

=nτ τ(ρτ − 1)

(n2+2ds2τ )

1/2

=⇒ 1

r

r2

[(BH(r)

)2]− ∫ r0 BH(s)dsBH(r)∫ r0 B

H(s)2ds− 1r

(∫ r0 B

H(s)ds)2 ×

ς2(∫ r

0

(BH(s)

)2ds− 1

r

(∫ r0 B

H(s)ds)2)

E[u2t ]

1/2

.

This implies the test statistic DFτ is of Op(nd).

The following lemmas are useful to derive the limit distribution of DFτ ,HAR.

Lemma 8.4 Let µ be the LS estimator of the intercept, under model (10), for r ∈ (0, 1],

as n→∞,we haven1/2−dµ =⇒ σdY (r, d), for d ∈ [0, 0.5)

where

Y (r, 0) : =

(∫ r0 W (s)2ds

)W (r)− 1

2

[(W (r))2 − r

] ∫ r0 W (s)ds

r∫ r

0 W (s)2ds−(∫ r

0 W (s)ds)2 ,

Y (r, d) : =

(∫ r0 B

H(s)2ds)BH(r)− 1

2

[(BH(r)

)2] ∫ r0 B

H(s)ds

r∫ r

0 BH(s)2ds−

(∫ r0 B

H(s)ds)2 ,

σd = σ if d = 0, σd = ζ if d ∈ (0, 0.5), ς :=√σ2 Γ(1−2d)

Γ(d)Γ(1−d) with Γ(·) being the Gammafunction.

Proof of Lemma 8.4

n1/2−dµ = n1/2−d

(∑τt=1 y

2t−1

∑τt=1 ut −

∑τt=1 yt−1ut

∑τt=1 yt−1

τ∑τ

t=1 y2t−1 − (

∑τt=1 yt−1)2

)

=1

n2+2d

∑τt=1 y

2t−1

1n1/2+d

∑τt=1 ut − 1

n1+2d

∑τt=1 yt−1ut

1n3/2+d

∑τt=1 yt−1

τn

1n2+2d

∑τt=1 y

2t−1 −

(1

n3/2+d

∑τt=1 yt−1

)2

=⇒

σ

(∫ r0 W (s)2ds)W (r)− 1

2 [(W (r))2−r]∫ r0 W (s)ds

r∫ r0 W (s)2ds−(

∫ r0 W (s)ds)

2 := σY (r, 0) if d = 0,

ς(∫ r0 B

H(s)2ds)BH(r)− 12

[(BH(r))

2] ∫ r

0 BH(s)ds

r∫ r0 B

H(s)2ds−(∫ r0 B

H(s)ds)2 := ςY (r, d) if d ∈ (0, 0.5).

where we utilize results in Lemma 8.2 to obtain the weak convergence.To prove Theorem 3.1 and Remark 3.7, we need the following lemma for the asymptotic

behaviour of s2τ ,HAR.

Lemma 8.5 Suppose that M = b × τ , K(.) ≥ 0 and K(x) = K(−x), and K(·) is twicedifferentiable, let ΩHAR =

∑τj=−τ+1K

(jM

)γj , with γj = 1

τ

∑τt=j+1 utut−j for j ≥ 0

24

and γj = γ−j for j < 0, and s2τ ,HAR = ΩHAR∑τ

t=1 y2t−1−τ−1(

∑τt=1 yt−1)

2 . Under model (10), for

r ∈ (0, 1], as n→∞, for d ∈ [0, 0.5), we have

1

n2dΩHAR =⇒ − σ2

d

b2r3

∫ r

0

∫ r

0K ′′(p− qbr

)G(d, r, p)G(d, r, q)dpdq (30)

τ2s2τ ,adj =⇒

∫ r0

∫ r0 −K

′′ (p−qbr

)G(d, r, p)G(d, r, q)dpdq

b2r∫ r

0 BH(s)2ds

(31)

Proof of Lemma 8.5, Theorem 3.1 and Remark 3.7

Let Ki,j = K(i−jbτ

), St =

∑ti=1 ui, ΩHAR =

∑τj=−τ+1K

(jbτ

)γj . We have

ΩHAR =

τ∑j=−τ+1

K

(j

bτ

)γj

=1

τ

τ∑i=1

τ∑i=1

uiKi,j uj

=1

τ

τ−1∑i=1

1

τ

τ−1∑j=1

τ2 [(Ki,j −Ki,j+1)− (Ki+1,j −Ki+1,j+1)]1√τSi

1√τSj

=1

τ

τ−1∑i=1

1

τ

τ−1∑j=1

τ2Dτ

(i− jbτ

)1√τSi

1√τSj (32)

where Dτ

(i−jbτ

)= (Ki,j−Ki,j+1)− (Ki+1,j−Ki+1,j+1). The last equality follows from

Kiefer and Vogelsang (2002a) equation A.1., and following the steps in proving theorem 4

in Sun (2004) we can show limn→∞

τ2Dτ

(i−jbτ

)= − 1

b2r2K ′′(p−qbr

), given (i/n, j/n)→ (p, q).

We first study the limit of partial sum of residual, under d = 0 and d ∈ (0, 0.5). Forp ∈ [0, 1], we have

1√τSbnpc =

(nτ

)1/2 1

n1/2

bnpc∑t=1

(yi − ρτyi−1 − µ)

=(nτ

)1/2 1

n1/2

n(1− ρτ )1

n

bnpc∑t=1

yi−1 +

bnpc∑t=1

ui − µ bnpc

For d = 0, we have

1√τSbnpc

25

=(nτ

)1/2

n(1− ρτ )1

n3/2

bnpc∑t=1

yi−1 +1

n1/2

bnpc∑t=1

ui −µ bnpcn1/2

=

(nτ

)1/2

1

n1/2

bnpc∑t=1

ui − n(ρτ − 1)1

n3/2

bnpc∑t=1

yi−1 − n1/2µbnpcn

=

σ

r1/2

[W (p)− 1

rX(r, 0)

∫ p

0W (s)ds− Y (r, 0)p

]:=

σ

r1/2Gr(0, p) (33)

For d ∈ (0, 0.5),we have

1

nd1√τSbnpc

=(nτ

)1/2

n(1− ρτ )1

n3/2+d

bnpc∑t=1

yi−1 +1

n1/2+d

bnpc∑t=1

ui −1

n1/2+dµ bnpc

=

(nτ

)1/2

1

n1/2+d

bnpc∑t=1

ui − n(ρτ − 1)1

n3/2+d

bnpc∑t=1

yi−1 − n1/2−dµbnpcn

=⇒ ς

r1/2

[BH(p)− 1

rX(r, d)

∫ p

0BH(s)ds− Y (r, d)p

]:=

ς

r1/2Gr(d, p) (34)

Combining (32), (33) and (34), we have

1

n2dΩHAR =

1

τ

τ−1∑i==1

1

τ

τ−1∑j=1

−Dτ

(i− jbτ

)1

nd1√τSi

1

nd1√τSj

=⇒− σ2

b2r3

∫ r0

∫ r0 K

′′ (p−qbr

)Gr(0, p)Gr(0, q)dpdq if d = 0,

− ς2

b2r3

∫ r0

∫ r0 K

′′ (p−qbr

)Gr(d, p)Gr(d, q)dpdq if d ∈ (0, 0.5).

(35)

And for the denominator in s2τ ,HAR, we have

1

n2+2d

τ∑t=1

y2t−1 − τ−1

(τ∑t=1

yt−1

)2 =⇒

σ2∫ r

0 W (s)2ds if d = 0,

ς2∫ r

0 BH(s)2ds if d ∈ (0, 0.5).

(36)

From (35) and (36), we therefore have

τ2s2τ ,HAR =

( τn

)2n2s2

τ ,HAR

=( τn

)2 1n2d

ΩHAR

1n2+2d

(∑τt=1 y

2t−1 − τ−1 (

∑τt=1 yt−1)2

)

26

=⇒

∫ r0

∫ r0 −K

′′( p−qbr )Gr(0,p)Gr(0,q)dpdqb2r

∫ r0 W (s)2ds

if d = 0,∫ r0

∫ r0 −K

′′( p−qbr )Gr(d,p)Gr(d,q)dpdqb2r

∫ r0 B

H(s)2dsif d ∈ (0, 0.5).

It implies the result in Lemma 8.5.For the HAR test statistic, suppose d = 0, we have

DFτ ,HAR =τ (ρτ − 1)(τ2s2

τ ,HAR

)1/2

=⇒br3/2

∫ r0 W (s)dW (s)∫ r

0 W (s)2ds

( ∫ r0 W (s)2ds∫ r

0

∫ r0 −K ′′

(p−qbr

)Gr(0, p)Gr(0, q)dpdq

)1/2

=br3/2

∫ r0 W (s)dW (s)(∫ r

0 W (s)2ds∫ r

0

∫ r0 −K ′′

(p−qbr

)Gr(0, p)Gr(0, q)dpdq

)1/2

Suppose d ∈ (0, 0.5).

DFτ ,HAR =ρτ − 1

sτ ,HAR=

τ (ρτ − 1)(τ2s2

τ ,HAR

)1/2

=⇒r2

(BH(r)

)2 − (∫ r0 BH(s)ds)BH(r)∫ r

0 BH(s)2ds

(b2r∫ r

0 BH(s)2ds∫ r

0

∫ r0 −K ′′

(p−qbr

)Gr(d, p)Gr(d, q)dpdq

)1/2

=br3/2

2

(BH(r)

)2 − br1/2(∫ r

0 BH(s)ds

)BH(r)((∫ r

0 BH(s)2ds

) ∫ r0

∫ r0 −K ′′

(p−qbr

)Gr(d, p)Gr(d, q)dpdq

)1/2

By letting b = 1, we obtain the results in Theorem 3.1.Proof of Theorem 3.2 and Remark 3.7

From straightforward calculation, we can show Dτ

(i−jbτ

)=

2bτ if |i− j| = 0,

− 1bτ if |i− j| = bbτc ,

0 otherwise.This implies

ΩHAC =

τ−1∑i=1

τ−1∑j=1

Dτ

(i− jbτ

)1√τSi

1√τSj

=2

bτ

τ−1∑i=1

(1√τSi

)2

− 2

bτ

τ−bbτc−1∑i=1

(1√τSi

)(1√τSi+bbτc/n

)

=2

b

n

bnrc1

n

τ−1∑i=1

(1√τSi

)2

− 2

b

n

bnrc1

n

τ−bbτc−1∑i=1

(1√τSi

)(1√τSi+bbτc/n

)

27

Thus, we have

1

n2dΩHAC

=2n

bτ

1

n

τ−1∑i=1

(1

nd1√τSbnpc

)2

− 2n

bτ

1

n

τ−bbτc−1∑i=1

(1

nd1√τSi

)(1

nd1√τSi+bbτc/n

)(37)

=⇒ 2

br

∫ r

0

( ς

r1/2Gr(d, p)

)2dp− 2

br

∫ (1−b)r

0

ς2

rGr(d, p)Gr(d, p+ br)dp

=2ς2

br2

(∫ r

0Gr(d, p)

2dp−∫ (1−b)r

0Gr(d, p)Gr(d, p+ br)dp

)(38)

Similar in proving Theorem 3.1, (38) and (36) imply

τ2s2τ ,HAR =

( τn

)2 1n2d

ΩHAR

1n2+2d

(∑τt=1 y

2t−1 − τ−1 (

∑τt=1 yt−1)2

)=⇒

2(∫ r

0 Gr(d, p)2dp−

∫ (1−b)r0 Gr(d, p)Gr(d, p+ br)dp

)b∫ r

0 BH(s)2ds

(39)

For d = 0, we have

DF b,Bartlettτ ,HAR

=τ (ρτ − 1)(τ2s2

τ ,HAR

)1/2

=⇒r∫ r

0 W (s)dW (s)∫ r0 W (s)2ds

b∫ r

0 W (s)2ds

2(∫ r

0 Gr(0, p)2dp−

∫ (1−b)r0 Gr(0, p)Gr(0, p+ br)dp

)1/2

=b1/2r

∫ r0 W (s)dW (s)[

2∫ r

0 W (s)2ds(∫ r

0 Gr(0, p)2dp−

∫ (1−b)r0 Gr(0, p)Gr(0, p+ br)dp

)]1/2

For d ∈ (0, 0.5), we have

DF b,Bartlettτ ,HAR

=τ (ρτ − 1)(τ2s2

τ ,HAR

)1/2

=⇒r2

(BH(r)

)2 − (∫ r0 BH(s)ds)BH(r)∫ r

0

(BH(s)

)2ds

b∫ r

0 BH(s)2ds

2(∫ r

0 Gr(d, p)2dp−

∫ (1−b)r0 Gr(d, p)Gr(d, p+ br)dp

)1/2

28

=rb1/2

2

(BH(r)

)2 − b1/2 (∫ r0 BH(s)ds)BH(r)[

2∫ r

0

(BH(s)

)2ds(∫ r

0 Gr(d, p)2dp−

∫ (1−b)r0 Gr(d, p)Gr(d, p+ br)dp

)]1/2

Proof of Lemma 4.1

1. The result for r < re is immediate after applying Lemma 8.1. For re ≤ r < 1, notethat by backward substitution, we can express ybnrc as

ybnrc = ρbnrc−(bnrec−1)t ybnrec−1 +

nr∑j=brec

(1 +

c

n

)bnrc−juj .

Note that ρbnrc−(bnrec−1)t = (1 + c/n)bnrc−(bnrec−1) ' exp(c (bnrc − (bnrec − 1)) /n) =

exp((r − re)c) + o(1). Therefore we have

1

n1/2+dybnrc = exp((r − re)c)

1

n1/2+dybnrec−1 +

1

n1/2+d

nr∑j=brec

(1 +

c

n

)bnrc−juj . (40)

Applying CMT and Lemma 8.1 with the similar argument as in Phillips (1987b), weobtain the second results.

2. 1n2/3+d

∑τt=1 yt−1 = 1

n

∑τt=1

(1

n1/2+dyt−1

)⇒ σd

∫ r0

(e(x−re)cBH(re) +

∫ xree(x−s)cdBH(s)

)dx

We obtain the result from CMT and the result in (40).

3. This is omitted due to its similarity to the proof of Lemma 4.1.2.

4. Note that∑τ

t=1 yt−1ut =∑bnrec−1

t=1 yt−1ut +∑τ

t=bnrec yt−1ut. Following the steps inPhillips (1987b) and applying Lemma 8.2, it is easy to obtain

bnrec−1∑t=1

yt−1ut ⇒σ2

2

(W 2(re)− re

)if d = 0,

ς2

2

[(BH(re))

2]

if d ∈ (0, 0.5).(41)

By squaring and summing yt over the observation from bnrec to τ , we haveτ∑

t=bnrecy2t = ρ2

c

τ∑t=bnrec

y2t−1 + 2ρc

τ∑t=bnrec

yt−1ut +

τ∑t=bnrec

u2t .

As ρc = 1 + o(1), we can express

τ∑t=bnrec

yt−1ut =1

2

[y2τ − y2

bnrec−1

]− 1

2

τ∑t=bnrec

u2t .

29

And with the normalization n−1−2d, note that 1τ−bnrec+1

∑τt=bnrec u

2tas→ E[u2

t ] (byergodic theorem), we have

1

n1+2d

τ∑t=bnrec

yt−1ut

=1

2

( 1

n1/2+dyτ

)2

−(

1

n1/2+dybnrec−1

)2

− 1

2

1

n2d

τ − bnrec+ 1

n

1

τ − bnrec+ 1

τ∑t=bnrec

u2t

⇒

σ2

2

[(e(r−re)cW (re) +

∫ rree(r−s)cdW (s)

)2−W 2(re)− (r − re)

]if d = 0,

ς2

2

[(e(r−re)cBH(re) +

∫ rree(r−s)c)dBH(s)

)2− (BH(re))

2

]if d ∈ (0, 0.5).

(42)

Finally, combining the results in (41) and (42), we have the result in Lemma 4.1.4.

Proof of Lemma 4.2

Let yt−1 = yt−1 − 1τ

∑τj=1 yj−1, we can express the normalized centered AR estimator

as:

(ρτ − ρc) =

∑τt=1 yt−1 (yt − ρcyt−1)∑τ

t=1 y2t−1

.

We now analyse the numerator and denominator,

τ∑t=1

yt−1 (yt − ρcyt−1) =

τe−1∑t=1

yt−1 (yt − ρcyt−1) +τ∑

t=τe

yt−1 (yt − ρcyt−1)

=

τe−1∑t=1

yt−1 (yt−1 + ut − ρcyt−1) +τ∑

t=τe

yt−1ut

=

τ∑t=1

yt−1ut −c

n

τe−1∑t=1

yt−1

where

1

n1+2d

τ∑t=1

yt−1ut

=1

n1+2d

(τ∑t=1

yt−1ut −1

τ

τ∑t=1

yt−1

τ∑t=1

ut

)

=1

n1+2d

τ∑t=1

yt−1ut −n

τ

1

n3/2+d

τ∑t=1

yt−11

n1/2+d

τ∑t=1

ut

⇒ 1

2σ2dCr,d −

1

rσ2dAr,dB

H(r) (Applying Lemma 8.1, 4.1.2 and 4.1.4)

30

and

c

n

τe−1∑t=1

yt−1 =c

n

τe−1∑t=1

yt−1 −c

n

1

τ

τe−1∑t=1

τ∑j=1

yj−1

=c

n

τe−1∑t=1

yt−1 −c

n

τ e − 1

τ

τ∑j=1

yj−1.

After normalization, we have

1

n1/2+d

c

n

τe−1∑t=1

yt−1 −c

n

τ e − 1

τ

τ∑j=1

yj−1

=

c

n3/2+d

τe−1∑t=1

yt−1 −τ e − 1

τ

c

n3/2+d

τ∑j=1

yj−1

⇒ cσd

∫ re

0BH(s)ds− creσdAr,d (Applying Lemma 8.2 and 4.1.2).

Therefore,∑τ

t=1 yt−1ut dominates cn

∑τe−1t=1 yt−1 as n→∞.

For the denominator, we have

1

n2+2d

τ∑t=1

y2t−1 =

1

n2+2d

τ∑t=1

y2t−1 −

1

τ

(τ∑t=1

yt−1

)2

⇒ σ2d

(Br,d −

1

rA2r,d

).

Eventually, we have

n(ρτ − ρc) =1

n1+2d

∑τt=1 yt−1 (yt − ρcyt−1)1

n2+2d

∑τt=1 y

2t−1

⇒ Xc(r, d) :=12Cr,d −

1rAr,dB

H(r)

Br,d − 1rA

2r,d

(by Lemma 4.1).

As n(ρτ − 1) = n(ρτ − ρc) + n(ρc − 1) = n(ρτ − ρc) + c, we have

n(ρτ − 1)⇒12Cr,d −

1rAr,dB

H(r)

Br,d − 1rA

2r,d

+ c. (43)

For the intercept estimator,

n1/2−dµ

31

=1

n2+2d

∑τt=1 y

2t−1

1n1/2+d

∑τt=1 ut − 1

n1+2d

∑τt=1 yt−1ut

1n3/2+d

∑τt=1 yt−1

τn

1n2+2d

∑τt=1 y

2t−1 −

(1

n3/2+d

∑τt=1 yt−1

)2

⇒ σdYc(r, d) :=σdBr,dB

H(r)− σd2 Cr,dAr,d

r(Br,d −A2

r,d

) (by Lemma 4.1).

Proof of Theorem 4.1

For the partial sum of residuals Sbnpc =∑bnpc

i=1 ui, we have

Sbnpc =

τe−1∑i=1

ui +

bnpc∑i=τe

ui

=

τe−1∑i=1

[(1− ρτ )yi−1 − µ+ ui] +

bnpc∑i=τe

[(ρc − ρτ )yi−1 − µ+ ui]

=

bnpc∑i=1

ui − (ρτ − ρc + ρc − 1)

τe−1∑i=1

yi−1 − (ρτ − ρc)bnpc∑i=τe

yi−1 − µ bnpc

=

bnpc∑i=1

ui − (ρτ − ρc)bnpc∑i=1

yi−1 − (ρc − 1)

τe−1∑i=1

yi−1 − µ bnpc .

Upon normalization, we have

1

n1/2+dSbnpc

=1

n1/2+d

bnpc∑i=1

ui − n(ρτ − ρc)1

n3/2+d

bnpc∑i=1

yi−1 − n (ρc − 1)1

n3/2+d

τe−1∑i=1

yi−1 − n1/2−dµbnpcn

⇒ σdBH(p)−Xc(r, d)σdAp,d − cσdAre ,d − σdYc(r, d)p = σdGr,c(d, p). (44)

Note that we can express the normalized long run variance estimator as

1

n2dΩHAR

=1

τ

τ−1∑i==1

1

τ

τ−1∑j=1

−Dτ

(i− jτ

)1

nd1√τSi

1

nd1√τSj

=⇒ −σ2d

r3

∫ r

0

∫ r

0K ′′(p− qr

)Gr,c(d, p)Gr,c(d, q)dpdq (From (44) and CMT). (45)

For the normalized HAR standard error, we have

τ2s2τ ,HAR

32

=( τn

)2 1n2d

ΩHAR

1n2+2d

(∑τt=1 y

2t−1 − τ−1 (

∑τt=1 yt−1)2

)=⇒

−1r

∫ r0

∫ r0 K

′′ (p−qr

)Gr,c(d, p)Gr,c(d, q)dpdq

Br,d − 1rA

2r,d

(From (45) and Lemma 4.1). (46)

Finally, the HAR test statistic has the following limit,

DFτ ,HAR =τ (ρτ − 1)(τ2s2

τ ,HAR

)1/2

=⇒

( 12Cr,d−1rAr,dB

H(r))rBr,d− 1

rA2r,d

+ cr

− 1b2r

∫ r0

∫ r0 K

′′( p−qr )Gr,c(0,p)Gr,c(0,q)dpdqBr,d− 1

rA2r,d

(From (46) and (43)).

=Ar,dB

H(r)− r2Cr,d + cA2

r,d −Br,dcr1r

∫ r0

∫ r0 K

′′(p−q

r

)Gr,c(d, p)Gr,c(d, q)dpdq

.

Under Bartlett kernel, from (37), (44) and CMT, we have

1

n2dΩHAC =

n

bnrc2

n

τ−1∑i=1

(1

nd1√τSbnpc

)2

=⇒ 2

r

∫ r

0

( σdr1/2

Gr,c(d, p))2dp

=2σ2

d

r2

(∫ r

0Gr,c(d, p)

2dp

).

Applying the results in Lemma 4.1 with CMT, we have

τ2s2τ ,HAR

=( τn

)2 1n2d

ΩHAR

1n2+2d

(∑τt=1 y

2t−1 − τ−1 (

∑τt=1 yt−1)2

)=

2∫ r

0 Gc(d, r, p)2dp

Br,d − rA2r,d

. (47)

It implies that

DFBartlettτ ,HAR =τ (ρτ − 1)(τ2s2

τ ,HAR

)1/2

=⇒

( 12Cr,d−1rAr,dB

H(r))rBr,d− 1

rA2r,d

+ cr

2(∫ r0 Gr,c(d,p)

2dp)Br,d− 1

rA2r,d

(From (47) and (43))

33

=r2Cr,d −Ar,dB

H(r) +Br,dcr − cA2r,d

2∫ r

0 Gr,c(d, p)2dp

.

Proof of Theorem 4.2

We skip the proof of this Theorem due to its similarity to Theorem 5.1.Proof of Theorem 5.1Before we prove Theorem 5.1, we need to utilize Lemma 8.6 to 8.13.

Lemma 8.6 Let B = [τ e, τ f ] be the bubble period, N0 ∈ [1, τ e) and N1 = [τ f + 1, n] arethe normal market period before and after the bubble period respectively. Under the dgp(21), with t = bnrc, we have the following asymptotic approximation :

1. For t ∈ N0,

yta∼ n1/2+d1σd1B

H1(re).

2. For t ∈ B,yt

a∼ ρ(t−τe)n n1/2+d1σd1B

H1(re).

3. For t ∈ N1,

ybnrca∼

n1/2+d1σd1B

H1(re) if d1 > d2,

n1/2+d2σd2(BH2(r)−BH2(rf )

)if d1 < d2,

n1/2+d1[σd2

(BH2(r)−BH2(rf )

)+ σd1B

H1(re)]

if d1 = d2

where σd1 =√σ2 Γ(1−2d1)

Γ(d1)Γ(1−d1) , σd2 =√σ2 Γ(1−2d2)

Γ(d2)Γ(1−d2) , BH1(r) and BH2(r) are frac-

tional Brownian motion with Hurst parameter H1 = 1/2 + d1 and H2 = 1/2 + d2,respectively.

Proof of Lemma 8.6

1. From Lemma 8.1, we have 1n1/2+d1

ybnrc =⇒ σd1BH1(re).

2. For t ∈ B, we have

yt = ρt−τe+1n yτe−1 +

t−τe∑j=0

ρjnut−j

Pre-multiplying the both term by ρ−(t−τe)n , we have

ρ−(t−τe)n yt = ρnyτe−1 +

t−τe∑j=0

ρ−jn ut−j (48)

34

From Lemma 3 in Magdalinos (2012), the second term in (48) is of orderOp(n(1/2+d2)α),

and the first term is of order Op(ρnn

1/2+d1). The order of ρ−(t−τe)

n yt depends on

the ratio n1/2+d1

n(1/2+d2)α.

As ρnyτe−1 is asymptotically dominant in (48), as ρn → 1, we have

ρ−(t−τe)n

1

n1/2+d1yt

a∼ ρnn1/2+d1

yτe−1a∼ σd1BH1(re).

3. For t ∈ N1, we have

ybnrc =

bnrc∑k=τf+1

uk + y∗τf =

bnrc∑k=τf+1

uk + yτe + y∗

ybnrc =

bnrc∑k=τf+1

uk + y∗τf =

bnrc∑k=1

uk −τf∑k=1

uk + yτe + y∗

Note that yτea∼ n1/2+d1σd1B

H1(re), we need to compare the order of∑bnrc

k=τf+1 ukand yτe .

Suppose that d1 = d2, we have

ybnrca∼ n1/2+d1

[σd2

(BH2(r)−BH2(rf )

)+ σd1B

H1(re)].

Suppose that d1 > d2, we have

ybnrca∼ n1/2+d1σd1B

H1(re).

Suppose that d1 < d2, we have

ybnrca∼ n1/2+d2σd2

(BH2(r)−BH2(rf )

).

Lemma 8.7 For the sample average,

1. For τ ∈ B,1

τ

τ∑j=1

yja∼ nα+d1−1/2ρτ−τen

1

rcσd1B

H1(re)

2. For τ ∈ N1,1

τ

τ∑j=1

yja∼ nα+d1−1/2ρ

τf−τen

1

rcσd1B

H1(re)

35

Proof of Lemma 8.7For τ ∈ B, we have

1

τ

τ∑j=1

yj =1

τ

τe−1∑j=1

yj +1

τ

τ∑j=τe

yj

The first term is

1

τ

τe−1∑j=1

yj = n1/2+d1 τ eτ

1

τ e

τe−1∑j=1

1

n1/2+d1yj

a∼ n1/2+d1 re

rσd1

∫ re

0BH1(s)ds (By Lemma 8.6.1 and CMT) (49)

For the second term,

1

τ

τ∑j=τe

yj

a∼ 1

τ

τ∑j=τe

ρ(j−τe)n n1/2+d1σd1B

H1(re)

= n1/2+d1σd1BH1(re)

1

τ

τ∑j=τe

ρj−τen

=n1/2+d1σd1B

H1(re)

τ

ρτ−τe+1n − 1

ρn − 1

=n1/2+d1σd1B

H1(re) [(ρτ−τen ρn)nα − nα]

bnrcca∼ nα+d1−1/2ρτ−τen

1

rcσd1B

H1(re) (as ρτ−τen nα asymptotically dominates nα) (50)

Comparing (50) with (49), as the second term is of higher order in all three cases,therefore, we have the results in Lemma 8.7.1.

For τ ∈ N1,

1

τ

τ∑j=1

yj =1

τ

τe−1∑j=1

yj +1

τ

τf∑j=τe

yj +1

τ

τ∑j=τf+1

yj .

For the first term, similar to (49), we have

1

τ

τe−1∑j=1

yja∼ n1/2+d1 re

rσd1

∫ re

0BH1(s)ds.

For the second term, similar to (50), we have

1

τ

τf∑j=τe

yja∼ nα+d1−1/2ρ

τf−τen

1

rcσd1B

H1(re).

36

For the last term, using Lemma 8.6.3, we have

1

τ

τ∑j=τf+1

yj =τ − τ fτ

n1/2+maxd1,d3Op(1)a∼ Op(n1/2+maxd1,d3).

Similar to the proof in Lemma 8.7, the second term has the highest order, thereforewe obtain the result in Lemma 8.7.2.

Lemma 8.8 Define the centered quantity yt = yt − 1τ

∑τj=1 yj−1.

1. For τ ∈ B,if t ∈ N0

yta∼ −nα+d1−1/2ρτ−τen

1

rcσd1B

H1(re), (51)

if t ∈ B,yt

a∼(ρ(t−τe)n − nα

nrcρτ−τen

)n1/2+d1σd1B

H1(re). (52)

2. For τ ∈ N1,

if t ∈ N0,

yta∼ −nα+d1−1/2ρ

τf−τen

1

rcσd1B

H1(re), (53)

if t ∈ B

yta∼(ρ(t−τe)n − nα

nrcρτ−τen

)n1/2+d1σd1B

H1(re), (54)

if t ∈ N1

yta∼ −nα+d1−1/2ρ

τf−τen

1

rcσd1B

H1(re). (55)

Proof of Lemma 8.8

1. Suppose τ ∈ B.

yt = yt −1

τ

τ∑j=1

yj−1

If t ∈ N0, from Lemma 8.6.1, yt = Op(n1/2+d1), for the second term, 1

τ

∑τj=1 yj−1,

following Lemma 8.7.1, we can obtain 1τ

∑τj=1 yj−1

a∼ nα+d1−1/2ρτ−τen1rcσd1B

H1(re).

Therefore if t ∈ N0, the second term has a higher order and we obtain

yta∼ −nα+d1−1/2ρτ−τen

1

rcσd1B

H1(re).

If t ∈ B, from Lemma 8.6.2,

yta∼ ρ(t−τe)

n n1/2+d1σdBHu (re)− nα+d1−1/2ρτ−τen

1

rcσd1B

H1(re).

37

2. Suppose that τ ∈ N1.

If t ∈ N0,

yt = yt −1

τ

τ∑j=1

yj−1

Similar to the proof in Lemma 8.8.1, as yt is asymptotically dominated by the latterterm, we can express

yta∼ −nα+d1−1/2ρ

τf−τen

1

rcσd1B

H1(re).

If t ∈ B,

yta∼(ρ(t−τe)n − ρτf−τen

nα

nrc

)n1/2+d1σd1B

H1(re).

If t ∈ N1, components in yt will be dominated by the components in 1τ

∑τj=1 yj−1,

eventually, following the proof of Lemma we have

yta∼ −nα+d1−1/2ρ

τf−τen

1

rcσd1B

H1(re).

Lemma 8.9 The sample variance terms involving yt behave as follows.

1. If τ ∈ B,τ∑j=1

y2j−1

a∼ n1+2d1+α ρ2(τ−τe)n

2cσ2d1B

H1(re)2.

2. if τ ∈ N1,

τ∑j=1

y2j−1

a∼ n1+α+2d1 ρ2(τf−τe)n

2cσ2d1B

H1(re)2.

Proof of Lemma 8.9

1. For τ ∈ Bτ∑j=1

y2j−1 =

τe−1∑j=1

y2j−1 +

τ∑j=τe

y2j−1. (56)

For the first term in (56),

τe−1∑j=1

y2j−1

a∼τe−1∑j=1

(−nα+d1−1/2ρτ−τen

1

rcσd1B

H1(re)

)2

=(τ e − 1)

nn2(α+d1)ρ2(τ−τe)

n

1

r2c2σ2d1B

H1(re)2

38

a∼ rer2c2

n2(α+d1)ρ2(τ−τe)n σ2

d1BH1(re)

2.

For the second term in (56),

τ∑j=τe

y2j−1

a∼τ∑

j=τe

[(ρ(j−τe)n − nα

nrcρτ−τen

)n1/2+d1σd1B

H1(re)

]2

= n1+2d1σ2d1B

H1(re)2

τ∑j=τe

(ρ(j−τe)n − nα

nrcρτ−τen

)2

= n1+2d1σ2d1B

H1(re)2

τ∑j=τe

(ρ2(j−τe)n − 2ρ(j−τe)

n

nα

nrcρτ−τen +

n2α

n2r2c2ρ2(τ−τe)n

)

= n1+2d1σ2d1B

H1(re)2

[nαρ

2(τ−τe)n

2c− 2

n2α−1ρ2(τ−τe)n

nrc+r − re + 1/n

r2c2n2α−1ρ2(τ−τe)

n

]a∼ n1+2d1+ασ2

d1BH1(re)

2 ρ2(τ−τe)n

2c(since α > 2α− 1).

Since 1 + 2d1 + α > 2(α + d1),∑τ

j=τey2j−1 dominates

∑τe−1j=1 y2

j−1 asymptotically,and we have

τ∑j=1

y2j−1

a∼ n1+2d1+ασ2d1B

H1(re)2 ρ

2(τ−τe)n

2c.

2. For τ1 ∈ N0 and τ2 ∈ N1

τ2∑j=τ1

y2j−1 =

τe−1∑j=τ1

y2j−1 +

τf∑j=τe

y2j−1 +

τ∑j=τf+1

y2j−1. (57)

For the first term in (57), we have

τe−1∑j=1

y2j−1

a∼ rer2c2

n2(α+d1)ρ2(τf−τe)n σ2

d1BH1(re)

2.

For the second term, we have

τf∑j=τe

y2j−1

a∼ n1+α+2d1σ2d1B

H1(re)2 ρ

2(τf−τe)n

2c.

For the third term,

τ∑j=τf+1

y2j−1

a∼τ∑

j=τf+1

(−nα+d1−1/2ρ

τf−τen

1

rcσd1B

H1(re)

)2

39

=τ − τ fn

n2(α+d1)ρ2(τf−τe)n

1

r2c2σ2d1B

H1(re)2

a∼ (r − rf )

r2c2n2(α+d1)ρ

2(τf−τe)n σ2

d1BH1(re)

2.

As 1 + α+ 2d1 > 2 (α+ d1) , the middle term dominates, and we have

τ∑j=1

y2j−1

a∼ n1+α+2d1σ2d1B

H1(re)2 ρ

2(τf−τe)n

2c.

Lemma 8.10 The sample variances of yt and ut behave as follows:

1. For τ ∈ B,τ∑j=1

yj−1uja∼ ρτ−τen n

12

(1+α)+d1+d2α (r − re)(1/2+d2)α σd1BH1(re)Φ. (58)

2. For τ ∈ N1,

τ∑j=1

yj−1uja∼ ρτ−τen n

12

(1+α)+d1+d2α (rf − re)(1/2+d2)α σd1BH1(re)Φ. (59)

Proof of Lemma 8.10

1. For τ ∈ B,τ2∑j=1

yj−1uj =

τe−1∑j=1

yj−1uj +

τ∑j=τe

yj−1uj . (60)

The first term in (60) is

τe−1∑j=1

yj−1uja∼

τe−1∑j=1

(−nα+d1−1/2ρτ−τen

1

rcσd1B

H1(re)

)uj

=

(−nα+d1−1/2ρτ−τen

1

rcσd1B

H1(re)

) τe−1∑j=1

uj

=−nα+d1−1/2+d1+1/2

rcρτ−τen σd1B

H1(re)1

n1/2+d1

τe−1∑j=1

uj

a∼ −nα+2d1

rcρτ−τen σ2

d1BH1(re)

2. (61)

For the second term in (60),

40

τ∑j=τe

yj−1uj

a∼τ∑

j=τe

[(ρ(j−τe)n − ρτ−τen

nα

nrc

)n1/2+d1σdB

H1 (re)

]uj

= n1/2+d1σd1BH1(re)

ρτ−τen

τ∑j=τe

ρ−(τ−j+1)n uj − ρτ−τen

nα

nrc

τ∑j=τe

uj

= n1/2+d1σd1B

H1(re)

ρτ−τen(τ−τe+1)(1/2+d2)α

(τ−τe+1)(1/2+d2)α

∑τj=τe

ρ−(τ−j+1)n uj

−ρτ−τennα−1/2+d2

rc1

n1/2+d2

∑τj=τe

uj

a∼ n1/2+d1σd1B

H1(re)

[ρτ−τen [n (r − re)](1/2+d2)α Φ− ρτ−τen

nα−1/2+d2

rcσd2

(BH2(r)−BH2(re)

)]a∼ n1/2+d1σd1B

H1(re)ρτ−τen [n (r − re)](1/2+d2)α Φ.

= ρτ−τen n12

(1+α)+d1+d2α (r − re)(1/2+d2)α σd1BH1(re)Φ. (62)

As (1/2+d2)α−(α− 1/2 + d2) = (1/2−d2)(1−α) > 0, we have the last asymptoticequivalence.

The assumption that 1/2−d11/2−d2 > α implies 1

2(1 + α) + d1 + d2α > α + 2d1, so the∑τj=τe

yj−1uj asymptotically dominates∑τe−1

j=1 yj−1uj , eventually, we have

τ∑j=1

yj−1uja∼ ρτ−τen n

12

(1+α)+d1+d2α (r − re)(1/2+d2)α σd1BH1 (re)Φ.

2. For τ1 ∈ N0 and τ2 ∈ N1,

τ2∑j=τ1

yj−1uj =

τe−1∑j=τ1

yj−1uj +

τf∑j=τd

yj−1uj +

τ2∑j=τf+1

yj−1uj .

As in (61), the first term is

τe−1∑j=τ1

yj−1uja∼ −n

α+2d1

rcρτ−τen σ2

d1BH1(re)

2.

As in (62), the second term is

τf∑j=τe

yj−1uja∼ ρτf−τen n

12

(1+α)+d1+d2α (r − re)(1/2+d2)α σd1BH1(re)Φ.

41

The third term isτ∑

j=τf+1

yj−1uja∼

τ∑j=τf+1

(−nα+d1−1/2ρτ−τen

1

rcσd1B

H1(re)

)uj

= −nα+d1−1/2ρτ−τen

1

rcσd1B

H1(re)

τ∑j=τf+1

uj

= −nα+d1−1/2ρτ−τen

1

rcσdB

H1(re)n1/2+d1

(BH1(r)−BH1(rf )

)a∼ −nα+2d1ρτ−τen

1

rcσdB

H1(re)(BH1(r)−BH1(rf )

).

Note that 1/2−d11/2−d2 > α implies 1

2 + 12α + d1 + d2α > α + 2d1. So the second term

dominates the first and third term and we haveτ2∑

j=τf+1

yj−1uja∼ ρτ−τen n

12

(1+α)+d1+d2α (r − re)(1/2+d2)α σd1BH1(re)Φ.

Lemma 8.11 The sample covariances of yj−1 and yj − ρnyj−1 behave as follows:

1. For τ ∈ B,τ∑j=1

yj−1(yj − ρnyj−1)a∼ ρτ−τen n

12

(1+α)+d1+d2α (r − re)(1/2+d2)α σd1BH1(re)Φ.

2. For τ ∈ N1,

τ∑j=1

yj−1(yj − ρnyj−1)a∼ −ρ2(τf−τe)

n n1+2d1σ2d

(BH1(re)

)2.

Proof of Lemma 8.11

1. Note that we can separate∑τ

j=1 yj−1(yj − ρnyj−1) into two parts such that

τ∑j=1

yj−1(yj − ρnyj−1) =

τe−1∑j=1

yj−1(yj − ρnyj−1) +

τ∑j=τe

yj−1(yj − ρnyj−1)

=

τe−1∑j=1

yj−1 [(1− ρn) yj−1 + ut] +τ∑

j=τe

yj−1ut

=τ∑j=1

yj−1ut −c

nα

τe−1∑j=1

yj−1yj−1.

From (58),∑τ

j=1 yj−1uta∼ ρτ−τen n

12

(1+α)+d1+d2α (r − re)(1/2+d2)α σd1BH1(re)Φ, for

the second term

42

c

nα

τe−1∑j=1

yj−1yj−1a∼ c

nα

τe−1∑j=1

(−nα+d1−1/2ρτ−τen

1

rcσd1B

H1(re)

)yj−1 (From (51))

=

(−nd1−1/2ρτ−τen

1

rσd1B

H1(re)

)τ e

1

τ e

τe−1∑j=1

yj−1

a∼(−nd1−1/2 1

rρτ−τen σd1B

H1(re)

)τ e

(n1/2+d1σd1

∫ re

0BH1(s)ds

)= −n2d1+1ρτ−τen

rerσ2d1B

H1(re)

∫ re

0BH1(s)ds. (63)

Under the assumption α < 1/2+d11/2+d2

, we can show 12(1 + α) + d1 + d2α > 2d1 + 1 thus∑τ

j=1 yj−1ut asymptotically domininates − cnα∑τe−1

j=1 yj−1yj−1, and we have

τ∑j=1

yj−1(yj − ρnyj−1)a∼ ρτ−τen n

12

(1+α)+d1+d2α (r − re)(1/2+d2)α σd1BH1(re)Φ.

2. For τ ∈ N1, we can express

τ∑j=1

yj−1(yj − ρnyj−1)

=

τe−1∑j=1

yj−1(yj − ρnyj−1) +

τf∑j=τe

yj−1(yj − ρnyj−1) + yτf (yτf − ρnyτf ) +

τ∑j=τf+2

yj−1(yj − ρnyj−1)

=

τe−1∑j=1

yj−1

[− c

nαyj−1 + ut

]+

τf∑j=τe

yj−1ut + yτf (yτf+1 − ρnyτf ) +τ∑

j=τf+2

yj−1

[− c

nαyj−1 + ut

]

=

τ∑j=1

yj−1ut −c

nα

τe−1∑j=1

yj−1yj−1 −c

nα

τ∑j=τf+2

yj−1yj−1 − ρnyτf yτf .

For the first term, similar to (58), we have

τ∑j=1

yj−1uta∼ ρτ−τen n

12

(1+α)+d1+d2α (rf − re)(1/2+d2)α σd1BH1(re)Φ,

for the second term, following the step to obtain (63), we have

c

nα

τe−1∑j=1

yj−1yj−1a∼ −n2d1+1ρ

τf−τen

rerσ2d1B

H1

∫ re

0BH1(s)ds,

for the third term,

c

nα

τ∑j=τf+2

yj−1yj−1a∼ c

nα

τ∑j=τf+2

(−nα+d1−1/2ρ

τf−τen

1

rcσd1B

H1(re)

)yj−1

43

= −nd1−1/2ρτf−τen

1

rcσd1B

H1(re)

τ∑j=τf+2

yj−1

= −nd1−1/2ρτf−τen

1

rcσd1B

H1(re) (τ − τ f − 1)n1/2+d1

τ − τ f − 1

τ∑j=τf+2

1

n1/2+d1yj−1

a∼ −n2d1+1ρτf−τen

1

rcσ2d1B

H1(re)2 (r − rf )

∫ r

rf

BH1(s)ds.

For the last term, from Lemma 8.6.2 and (54)

ρnyτf yτfa∼ yτf yτf (as ρn → 1)

a∼(ρτf−τen − ρτf−τen

nα

nrc

)n1/2+d1σd1B

H1(ρτf−τen n1/2+d1σd1B

H1(re))

= ρ2(τf−τe)n n1+2d1σ2

d1

(BH1(re)

)2.

Note that the last component dominates the previous terms as ρ2(τf−τe)n is over-

whelming. Finally, we have

τ∑j=1

yj−1(yj − ρnyj−1)a∼ −ρ2(τf−τe)

n n1+2d1σ2d1

(BH1(re)

)2.

Lemma 8.12 For the LS estimator ρτ , we have the following asymptotic results:

1. When τ ∈ B,n (ρτ − 1)

p→∞.

2. When τ ∈ N1,

n (ρτ − 1)p→ −∞.

Proof of Lemma 8.12

We first consider the centered statistics ρτ − ρn =∑τj=1 yj−1(yj−ρnyj−1)∑τ

j=1 y2j−1

.

1. When τ ∈ B,∑τj=1 yj−1(yj − ρnyj−1)∑τ

j=1 y2j−1

a∼ ρτ−τen n12

(1+α)+d1+d2α (r − re)(1/2+d2)α σd1BH1(re)Φ

n1+2d1+α ρ2(τ−τe)n

2c σ2d1BH1(re)2

(By Lemma 8.9.1 and 8.11.1)

=2c (r − re)(1/2+d2)α Φ

ρτ−τen n1/2+d1+(1/2−d2)ασd1BH1(re)

44

= Op(n−αρ−(τ−τe)

n ).

As n(ρτ − 1) = n(ρn − 1) + n(ρτ − ρn),

n(ρn − 1) + n(ρτ − ρn) = n1−αc+Op(n1−αρ−(τ−τe)

n )

= n1−αc+ op(1)→∞. (64)

2. When τ ∈ N1,∑τj=1 yj−1(yj − ρnyj−1)∑τ

j=1 y2j−1

a∼−ρ2(τf−τe)

n n1+2d1σ2d1

(BH1(re)

)2n1+α+2d1 ρ

2(τf−τe)n

2c σ2d1BH1(re)2

= −n−α2c.

As n(ρτ − 1) = n(ρn − 1) + n(ρτ − ρn),

n(ρn − 1) + n(ρτ − ρn) = n1−αc− n(n−α2c

)= −n1−αc→∞. (65)

Lemma 8.13 Under model (21), we have the following asymptotic result.

ΩHAR = Op(n2d1ρ2(τ−τe)

n ).

Proof of Lemma 8.13

From (32), we have

ΩHAR =1

τ

τ−1∑i=1

1

τ

τ−1∑j=1

τ2Dτ

(i− jτ

)1√τSi

1√τSj . (66)

To study the order of ΩHAR, we only need to study the limit of 1√τSi.

Suppose τ ∈ B,

1√nSτ =

1√n

τ∑i=1

(yi − ρτ yi−1)

=1√n

[τe−1∑i=1

(yi − ρτ yi−1) +

τ∑i=τe

(yi − ρτ yi−1)

]

=1√n

τe−1∑i=1

(ui − (ρτ − 1)yi−1) +1√n

τ∑i=τe

(ui − (ρτ − ρn)yi−1)

=1√n

τ∑i=1

ui − (ρτ − 1)1√n

τe−1∑i=1

yi−1 − (ρτ − ρn)1√n

τ∑i=τe

yj−1. (67)

45

We now compare the order of the three terms. It is clear that 1√n

∑τi=1 ui = Op(n

maxd1,d2).

For the second term (ρτ − 1) 1√n

∑τe−1i=1 yi−1, note that (ρτ − 1)

a∼ cnα and

√n

1

n

τe−1∑i=1

yi−1 =√nbnrcn

1

τ

τe−1∑i=1

yi−1

=√nbnrcn

(1

τ

τe−1∑i=1

yi−1 −τ e − 1

τ

1

τ

τ∑i=1

yi

)= O(

√n)(Op(n

1/2+d1)−Op(nα+d1−1/2ρτ−τen

))= Op

(nα+d1ρτ−τen

)This makes (ρτ − 1) 1√

n

∑τe−1i=1 yi−1 = c

nαOp(nα+d1ρτ−τen

)= Op

(nd1ρτ−τen

). For the

last term, note that from (64), we have (ρτ − ρn) = Op

(1

nαρτ−τen

), and

1√n

τ∑i=τe

yj−1 = r√nbnrcnr

1

τ

τ∑i=τe

yj−1

= r√nbnrcnr

1

τ

τ∑i=τe

(yj−1 −

1

τ

τ∑i=1

yj

)

= r√nbnrcnr

(1

τ

τ∑i=τe

yj−1 −τ − τ e + 1

τ

1

τ

τ∑i=1

yj

).

Note that from Lemma 8.7.1 and (50) we have

1

τ

τ∑j=τe

yj−1 = Op

(nα+d1−1/2ρτ−τen

),

and

1

τ

τ∑j=1

yj = Op

(nα+d1−1/2ρτ−τen

).

So we have

1√n

τ∑i=τe

yj−1 = O(n1/2

)Op

(nα+d1−1/2ρτ−τen

)= Op

(nα+d1ρτ−τen

).

This implies

(ρτ − ρn)1√n

τ∑i=τe

yj−1 = Op

(1

nαρτ−τen

)Op

(nα+d1ρτ−τen

)= Op

(nd1). (68)

Comparing the order of three terms in (67), we obtain

46

1√nSτ

a∼ −(ρτ − 1)1√n

τe−1∑i=1

yi−1 = Op

(nd1ρτ−τen

). (69)

Then (66) implies ΩHAR = Op(n2d1ρ

2(τ−τe)n ).

Suppose τ ∈ N1,

1√nSτ

=1√n

τ∑i=1

(yi − ρτ yi−1)

=1√n

τe−1∑i=1

(yi − ρτ yi−1) +

τf∑i=τe

(yi − ρτ yi−1) +

τ∑i=τf+1

(yi − ρτ yi−1)

=

1√n

τe−1∑i=1

(ui − (ρτ − 1)yi−1) +1√n

τf∑i=τe

(ui − (ρτ − ρn)yi−1) +1√n

τ∑i=τf+1

(ui − (ρτ − 1)yi−1)

=1√n

τ∑i=1

ui + (ρτ − 1)1√n

τe−1∑i=1

yi−1 − (ρτ − ρn)1√n

τ∑i=τe

yj−1 + (ρτ − 1)1√n

τ∑i=τf+1

yi−1. (70)

Note that the order of the first three terms in (70) are Op(nd1), Op(nd1ρτ−τen

), and

Op(nd1), respectively. For the last term, we have

(ρτ − 1)1√n

τ∑i=τf+1

yi−1a∼ c

nα1√n

τ∑i=τf+1

(−nα+d1−1/2ρ

τf−τen

1

rcσd1B

H1(re)

)(From (55) and (65))

= Op(nd1ρ

τf−τen ).

As τ ∈ N1, τ f < τ, eventually we have the same expression as (69) and it implies

ΩHAR = Op(n2d1ρ

2(τ−τe)n ).

Proof of Theorem 5.1

Note that DFτ ,HAR =(∑τ

i=1 y2i−1

ΩHAR

)1/2

(ρτ − 1).

Suppose that τ ∈ B, applying the results in Lemma 8.9.1, (64) and Lemma 8.13 , weobtain

(∑τi=1 y

2i−1

ΩHAR

)1/2

(ρτ − 1) = Op

(n1+α+2d1ρ

2(τ−τe)n

n2d1ρ2(τ−τe)n

)1/2c

nα

= Op

(n1−α2

)→∞.

Suppose that τ ∈ N1,applying the results in Lemma 8.9.1, (65) and Lemma 8.13, wehave

47

(∑τi=1 y

2i−1

ΩHAR

)1/2

(ρτ − 1) = Op

(n1+α+2d1ρ

2(τf−τe)n

n2d1ρ2(τ−τe)n

)1/2 (− c

nα

)= −Op

(n1−α2

)→ −∞.

To show rHARep→ re and rHARf

p→ rf , note that if τ ∈ N0,

limn→∞

Pr(DFτ ,HAR > cvn,HAR) = Pr (Fr,d >∞) = 0.

If τ ∈ B, limn→∞ Pr(DFτ ,HAR > cvn,HAR) = 1, given that cvn,HARn(1−α)/2

→ 0. If τ ∈ N1,

limn→∞ Pr(DFτ ,HAR > cvn,HAR) = 0, as DFτ ,HAR = −Op(n1−α2

).

It follows that for any η, ϑ > 0,

Pr(rHARe > re + η)→ 0, and Pr(rHARf < rf + ϑ)→ 0,

due to the fact that Pr(DF(τe+αη/n),HAR > re + η) → 1 for all 0 < αη < η andPr(DF(τf−αϑ/n),HAR > cvn,HAR) → 1 for all 0 < αϑ < ϑ. As η and ϑ are arbitrary and

Pr(rHARe < re) → 0 and Pr(rHARf > rf ) → 0, we can deduce Pr(∣∣rHARe − re

∣∣ > η) → 0

and Pr(∣∣∣rHARf − rf

∣∣∣ > ϑ)→ 0 as n→∞, provided that

1

cvn,HAR+cvn,HAR

n(1−α)/2 → 0.

References

Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2003). Modeling andForecasting Realized Volatility. Econometrica, 71(2), 579—625.

Baillie, R. T., Bollerslev, T. & Mikkelsen, H. O. (1996). Fractionally integrated general-ized autoregressive conditional heteroskedasticity. Journal of Econometrics, 74(1),3-30.

Barros, C., Gil-Alana, L., & Payne, J. (2013). Tests of convergence and long memorybehavior in US housing prices by state. Journal of Housing Research, 23(1), 73-87.

Cheung, Y W. (1993). Long memory in foreign-exchange rates. Journal of Business andEconomic Statistics, 11(1): 93-101.

Chevillon, G., & Mavroeidis, S. (2017). Learning can generate long memory. Journal ofeconometrics, 198(1), 1-9.

Christensen, B. J., & Nielsen, M. Ø. (2007). The effect of long memory in volatility onstock market fluctuations. The Review of Economics and Statistics, 89(4), 684-700.

48

Diba, B. & Grossman, H., (1988). Explosive Rational Bubbles in Stock Prices? AmericanEconomic Review, 78(3), 520-30.

Evans, G.W. (1991). Pitfalls in Testing for Explosive Bubbles in Asset Prices. TheAmerican Economic Review, 81, 922—30.

Gil-Alana, L. A., Barros, C., & Peypoch, N. (2014). Long memory and fractional inte-gration in the housing price series of London and Paris. Applied Economics, 46(27),3377-3388.

Harvey, D. I., Leybourne, S. J., Sollis, R., & Taylor, A. R. (2016). Tests for explosivefinancial bubbles in the presence of non-stationary volatility. Journal of EmpiricalFinance, 38, 548-574.

Harvey, D. I., Leybourne, S. J., & Zu, Y. (2019a). Testing explosive bubbles with time-varying volatility. Econometric Reviews, 38(10), 1131-1151.

Harvey, D. I., Leybourne, S. J., & Zu, Y. (2019b). Sign-based Unit Root Tests for Ex-plosive Financial Bubbles in the Presence of Nonstationary Volatility. EconometricTheory, 1-48.

Kiefer, N. M., & Vogelsang, T. J. (2002a). Heteroskedasticity-autocorrelation robusttesting using bandwidth equal to sample size. Econometric Theory, 18(6), 1350-1366.

Kiefer, N. M., & Vogelsang, T. J. (2002b). Heteroskedasticity-autocorrelation robuststandard errors using the Bartlett kernel without truncation. Econometrica, 70(5),2093-2095.

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

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

Kruse, R., & Wegener, C. (2019). Explosive behaviour and long memory with an ap-plication to European bond yield spreads. Scottish Journal of Political Economy,66(1), 139-153.

Magdalinos, T. (2012). Mildly explosive autoregression under weak and strong depen-dence. Journal of Econometrics, 169(2), 179-187.

Ohanissian, A., Russell, J. R., & Tsay, R. S. (2008). True or spurious long memory? Anew test. Journal of Business & Economic Statistics, 26(2), 161-175.

Pedersen, T. Q., & Schütte, E. C. M. (2017). Testing for explosive bubbles in the presenceof autocorrelated innovations. Available at SSRN 2916616.

49

Phillips, P. C. B. (1987a). Time Series Regression with a Unit Root. Econometrica,55(2), 277-301.

Phillips, P. C. B. (1987b). Towards a unified asymptotic theory for autoregression. Bio-metrika, 74(3), 535-547.

Phillips, P. C. B., & Magdalinos, T. (2007). Limit theory for moderate deviations froma unit root. Journal of Econometrics, 136(1), 115-130.

Phillips, P. C. B., Shi, S., Yu, J., (2014). Specfication Sensitivity in Right-Tailed UnitRoot Testing for Explosive Behaviour. Oxford Bulletin of Economics and Statistics,76(3), 315-333.

Phillips, P. C. B., Shi, S., Yu, J. (2015). Testing for multiple bubbles: Historical episodesof exuberance and collapse in the S&P 500. International Economic Review, 56(4),1043—1078.

Phillips, P. C. B., Wu, Y., Yu, J. (2011). Explosive behavior in the 1990s NASDAQ:When did exuberance escalate asset values? International Economic Review 52 (1),201—226.

Phillips, P. C. B. & Yu, J. (2011). Dating the Timeline of Financial Bubbles During theSubprime Crisis. Quantitative Economics 2, 455-491.

Shimotsu, K., & Phillips, P. C. B. (2005). Exact local Whittle estimation of fractionalintegration. The Annals of Statistics, 33(4), 1890-1933.

Sowell, F. (1990). The fractional unit root distribution. Econometrica, 58(2), 495-505.

Sun, Y. (2004). A convergent t statistic in spurious regressions. Econometric Theory,20(5), 943-962.

Vogelsang, T. J. (2003). Testing in GMM models without truncation. In MaximumLikelihood Estimation of Misspecified Models: Twenty Years Later (pp. 199-233).Emerald Group Publishing Limited.

Wang, X., Xiao, W., & Yu, J., (2019). Estimation and Inference of Fractional VasicekProcess with Discrete-Sampled Data. Working paper.

50