Statistics and Probability Letters 79 (2009) 354–360

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Statistics and Probability Letters

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Unit root tests in the presence of an innovation variance break that haspower against the mean break stationary alternativeAmit Sen ∗Department of Economics, Xavier University, United States

a r t i c l e i n f o

Article history:Received 25 May 2008Received in revised form 4 August 2008Accepted 7 September 2008Available online 17 September 2008

a b s t r a c t

We show that Perron’s [Perron, P., 1990. Testing for a unit root in a time series with achanging mean. Journal of Business and Economic Statistics 8, 153–162] unit root test canbe oversized when there is a break in the innovation variance. We propose a modifiedPerron test that maintains its size, and has power against the mean-break stationaryalternative.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Owing to the works of Perron (1989, 1990) and Hendry and Neale (1991), it is now well recognised in the literature thatunit root tests should to be designed to have power against the alternative hypothesis that allows for a break in the mean.Consider, for example, the time plot of the exchange rate between the U.S. dollar and the Thai Baht shown in Fig. 1.1 TheBaht/$ rate moved from a fixed regime to a floating regime in July 1997, and the regime shift was immediately followedby a severe depreciation of the Thai Baht. Perron (1990) argued that a statistic designed to test for the presence of a unitroot in the Baht/$ exchange rate should, therefore, have power against a stationary alternative that allows for a break in themean. Conventional unit root tests that ignore the break under the alternative can spuriously fail to reject the unit root nullhypothesis. Therefore, Perron (1990) proposed a unit root test, denoted by tP , that is specifically designed to have poweragainst the alternative that allows for a one time break in the mean occurring at a known break-date.2A second feature of the time plot of the Baht/$ exchange rate is an increase in the volatility of the exchange rate in the

post June, 1997 sample. In this paper, we focus on the behaviour of Perron’s (1990) unit root test, tP , in the presence of aninnovation variance break, see Sen (2007) for a general discussion. We show that tP can suffer from serious size distortionsin the presence of an innovation variance break.3 Our results pertaining to tP are similar to that of the Dickey–Fuller unitroot test as demonstrated by Kim et al. (2002).4We use Kim et al.’s (2002) strategy to devise an extension of Perron’s (1990)statistic. Our modified Perron (1990) test, denoted by t∗P , is based on a modified GLS transformation using the estimated

∗ Corresponding address: Department of Economics, 3800 Victory Parkway, Xavier University, Cincinnati, OH 45207-3212, United States. Tel.: +1 513745 2931; fax: +1 513 745 3692.E-mail address: sen@xavier.edu.1 This series was originally considered by Busetti and Taylor (2003). Data on the Baht/$ exchange rate can be found at the Federal Reserve EconomicDatabase website maintained by the St. Louis Federal Reserve Bank, http://research.stlouisfed.org/fred2/.2 Several studies have proposed extension of the Perron’s (1990) test. For example, Perron and Vogelsang (1992) propose a test that is appropriate whenthe break-date is unknown, and Perron (1989, 1997)) propose tests that allow for a break in the drift under the unit root null.3 Brooks and Rew (2002) examine the effects of GARCH errors on a version of the Perron unit root tests, and find that the Perron test can be oversized inthe presence of GARCH errors.4 Kim et al. (2002) develop an extension of Dickey and Fuller’s (1979) unit root test that allows for a one time break in the innovation variance. The finitesample simulations presented in Cook (2002) demonstrate that the unit root test of Leybourne (1995) has greater power than the test suggested by Kimet al. (2002).

0167-7152/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2008.09.005

A. Sen / Statistics and Probability Letters 79 (2009) 354–360 355

Fig. 1. Thai Baht/US $ Exchange Rate, 1985:01–2007:12.

pre-break and post-break variance implied by the known break-date. Themodified Perron (1990) statistic maintains its sizein the presence of an innovation variance break, and also has power when there is a break in the mean under the alternativehypothesis. Our results, however, are based on the assumption that the break in the innovation variance coincides with thebreak in the mean, if it exists.The remainder of the paper is organized as follows. In Section 2, we discuss the asymptotic and finite sample behaviour

of tP under the unit root null hypothesis with a break in the innovation variance. We also propose a modified Perron (1990)test using the modified GLS transformation of Kim et al. (2002). An empirical application is discussed in Section 3, and someconcluding comments are presented in Section 4. All proofs are relegated to an Appendix.

2. Behaviour of Perron’s (1990) unit root test and an extension

Perron (1990) considers the unit root process {yt}Tt=0 given by:

yt = γ D(T cb )t + yt−1 + εt (1)where T cb is the known break-date,D(T

cb )t = 1(t=T cb+1) is a dummy that takes the value one at time T

cb+1 and zero elsewhere,

and εt is the error term. In particular, Perron’s (1990) unit root test, denoted by tP , is the t-statistic corresponding to thecoefficient on the first lag of the dependent variable in the following regression:

yt = µ+ γ DU(T cb )t + d D(Tcb )t + ρ yt−1 +

k∗∑j=1

cj∆ yt−j + et (2)

where DU(T cb )t = 1(t>T cb ) is an intercept break dummy, and the lags of first differences {∆ yt−j}k∗j=1 are included in the

regression to account for any additional correlation in the series. The value of k∗ is typically unknown, and so it is chosenbased on a data-dependent selection process such as that proposed by Perron and Vogelsang (1992) or an informationcriterion such as the Modified AIC of Ng and Perron (2001). The unit root statistic tP is designed to have power when thereis a one time break in the mean under the alternative hypothesis at a known break-date. The limiting null distribution of tPis given in equation (8) of Perron (1990).We study the behaviour of tP in the presence of a break in the innovation variance at the known break-date T cb . The

limiting null distribution of tP is a function of the break-fraction given by T cb = [τcT ] where [.] is the smallest integer

function. Following Kim et al. (2002), we model the break in the innovation variance as:εt = σt ηt (3)

with σ 2t = σ 21 1[t≤τ cT ] + σ 22 1[t>τ cT ], and ηt is a martingale difference sequence with E(η2t |ηt−1, . . .) = 1 and

E(|ηt |4+γ |ηt−1, . . .) = κ < ∞ for some γ > 0. So, the innovation variance in the pre-break sample is σ 21 and that inthe post-break sample is σ 22 .

Theorem 1. Suppose the true data generating process for the time series {yt}Tt=0 is given by Eqs. (1) and (3). The limitingdistribution of the tP based on the OLS regression (2) is given by:

356 A. Sen / Statistics and Probability Letters 79 (2009) 354–360

Table 1Size of tP with sample size T = 100

τ c δ = σ2/σ1

4.00 2.50 1.67 1.25 1.00 0.80 0.60 0.40 0.25

0.1 0.046 0.045 0.044 0.048 0.051 0.055 0.067 0.112 0.2230.2 0.047 0.044 0.045 0.045 0.049 0.056 0.075 0.129 0.2460.3 0.060 0.055 0.050 0.048 0.048 0.057 0.073 0.124 0.2050.4 0.083 0.069 0.057 0.049 0.049 0.055 0.068 0.103 0.1520.5 0.114 0.086 0.060 0.054 0.050 0.052 0.063 0.083 0.1100.6 0.157 0.106 0.071 0.054 0.050 0.050 0.055 0.067 0.0790.7 0.210 0.125 0.076 0.058 0.051 0.050 0.050 0.054 0.0590.8 0.249 0.133 0.078 0.057 0.051 0.047 0.046 0.049 0.0480.9 0.218 0.110 0.069 0.057 0.051 0.050 0.046 0.046 0.047

Table 2Size of tP with sample size T = 200

τ c δ = σ2/σ1

4.00 2.50 1.67 1.25 1.00 0.80 0.60 0.40 0.25

0.1 0.046 0.045 0.047 0.048 0.049 0.055 0.070 0.112 0.2320.2 0.049 0.048 0.047 0.048 0.051 0.057 0.079 0.133 0.2510.3 0.059 0.055 0.051 0.045 0.049 0.056 0.076 0.123 0.2080.4 0.084 0.069 0.056 0.051 0.050 0.055 0.071 0.103 0.1580.5 0.118 0.086 0.065 0.054 0.050 0.053 0.063 0.085 0.1160.6 0.158 0.106 0.070 0.054 0.049 0.051 0.055 0.066 0.0820.7 0.205 0.121 0.075 0.055 0.050 0.047 0.048 0.053 0.0570.8 0.254 0.133 0.075 0.057 0.052 0.047 0.048 0.047 0.0490.9 0.229 0.112 0.069 0.055 0.050 0.048 0.048 0.048 0.045

tP ⇒1

{σ 21 τc + σ 22 (1− τ c)}

1τ c(1− τ c)

AVd

where

Vd = −(1− τ c)σ 21 B21 − τ

c(1− τ c)σ 22 B22 + τ

c(1− τ c)σ 21

∫ τ c

0W (r)2 dr + τ c(1− τ c) σ 22

∫ 1

τ cW (r)2 dr,

A = −(1− τ c)σ 21 B1W (τc)− τ cσ 22 B2{W (1)−W (τ

c)} +12τ c(1− τ c)σ 22 {W (τ

c)2 − τ c}

+ τ c(1− τ c)(σ 21 − σ22 )W (τ

c){W (1)−W (τ c)} +12τ c(1− τ c)(σ 21 − σ

22 ){W (τ

c)2 − 1},

B1 =∫ τ c

0W (r)dr, and B2 =

∫ 1

τ cW (r)dr.

The proof of Theorem 1 is outlined in the Appendix. In the presence of an innovation break, the limiting behaviour oftP depends on the pre-break variance (σ 21 ), the post-break variance (σ

22 ), and the break-fraction (τ

c). We use finite samplesimulations to evaluate the empirical size of tP . We used the same simulation design as Kim et al. (2002), that is, datais generated according to Eq. (1) with γ = 0, εt =

{1[t≤τ cT ] +

σ2σ11[t>τ cT ]

}ηt , and ηt ∼ i.i.d.N(0, 1). We used 10,000

replications for two different sample sizes, namely, T = 100 and T = 200. We set ρ = 1, µ = 0, and used all combinationsarising from τ c = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} and σ2

σ1= {4, 2.5, 1.67, 1.25, 1, 0.8, 0.6, 0.4, 0.25}. The

empirical size of tP is calculated using the critical values given in Table 4 on pp. 158 of Perron (1990). The empirical sizeof tP , shown in Tables 1 and 2, is much larger than the nominal size when there is a fall in the innovation variance occurringrelatively early in the sample, or if there is an increase in the innovation variance relatively late in the sample.We suggest using the modified GLS transformation proposed by Kim et al. (2002) to devise a unit root test that will

maintain its size in the presence of an innovation break. Specifically, we use the estimated residuals {et}Tt=1 from regression(2) to estimate the pre-break and post-break variances as:

σ 21 =1τ cT

τ cT∑t=1

e2t (4)

and

σ 22 =1

(T − τ cT )

T∑t=τ cT+1

e2t . (5)

A. Sen / Statistics and Probability Letters 79 (2009) 354–360 357

Table 3Size of t∗P with sample size T = 100

τ c δ = σ2/σ1

4.00 2.50 1.67 1.25 1.00 0.80 0.60 0.40 0.25

0.1 0.053 0.053 0.051 0.052 0.053 0.053 0.053 0.054 0.0570.2 0.050 0.049 0.051 0.050 0.052 0.052 0.052 0.052 0.0550.3 0.052 0.052 0.052 0.053 0.052 0.052 0.050 0.054 0.0540.4 0.051 0.052 0.052 0.051 0.052 0.053 0.052 0.053 0.0540.5 0.053 0.052 0.049 0.055 0.051 0.053 0.053 0.053 0.0530.6 0.052 0.054 0.052 0.052 0.053 0.053 0.052 0.052 0.0520.7 0.055 0.052 0.054 0.054 0.054 0.055 0.054 0.052 0.0530.8 0.059 0.056 0.056 0.054 0.054 0.054 0.053 0.055 0.0540.9 0.058 0.056 0.056 0.055 0.055 0.057 0.055 0.055 0.057

Table 4Size of t∗P with sample size T = 200

τ c δ = σ2/σ1

4.00 2.50 1.67 1.25 1.00 0.80 0.60 0.40 0.25

0.1 0.053 0.051 0.052 0.052 0.050 0.050 0.053 0.049 0.0520.2 0.052 0.051 0.052 0.052 0.052 0.051 0.054 0.051 0.0530.3 0.051 0.050 0.051 0.048 0.050 0.051 0.052 0.050 0.0510.4 0.051 0.051 0.052 0.052 0.051 0.051 0.051 0.049 0.0510.5 0.052 0.052 0.052 0.053 0.051 0.051 0.052 0.051 0.0530.6 0.052 0.052 0.051 0.051 0.050 0.052 0.051 0.050 0.0520.7 0.052 0.051 0.051 0.050 0.052 0.050 0.050 0.048 0.0490.8 0.054 0.051 0.052 0.051 0.054 0.052 0.052 0.051 0.0530.9 0.055 0.052 0.053 0.052 0.051 0.052 0.054 0.055 0.053

The modified Perron (1990) unit root statistic, denoted by t∗P , is based on the following modified GLS regression:

yt = α1 + α2 DU(T cb )t + α3 D(Tcb )t + ρ yt−1 +

k∗∑j=1

bj D(T cb + j)t +k∗∑j=1

cj∆yt + νt (6)

where yt = σ−11 yt 1(t≤τ cT ) + σ−12 yt 1(t>τ cT ), and D(T cb + j)t = 1(t=T cb+1+j) for j = 1, 2, . . . , k∗. The extra dummy regressors

D(T cb + j)t are included in the regression to remove the k∗+ 1 central observations. t∗

ρis the t-statistic for Ho : ρ = 1 based

on regression (6). The limiting distribution of t∗ρis given by equation (8) on pp. of Perron (1990). This result follows given

that the pre-break and post-break variance estimators given in (4) and (5) are consistent.The finite sample size of t∗P based on the simulation design discussed above are given in Tables 3 and 4. The empirical size

are calculated using the 5% critical values given in Table 4 of Perron (1990). As expected, the empirical size of t∗P are close tothe nominal size in all cases, and so use of t∗P is appropriate in the presence of a break in the innovation variance.

3. An empirical illustration

We test for the presence of a unit root in the exchange rate between the Thailand Baht and the U.S. $ using data overthe period 1985:01-2007:12, see Fig. 1. The break-date is specified as June 1997 at which point the Baht was allowed tofloat against the US $. First, we estimate Perron’s (1990) regression (2) with τ c = 0.56. Regression (2) allows for a breakin the mean under the alternative hypothesis, but does not allow for a break in the innovation variance. The lag-truncationparameter k∗ is determined using Perron and Vogelsang (1992) k(t − sig) procedure with kmax = 8.5 Perron’s (1990) unitroot statistic, tP , is equal to −4.0566. The critical value of tP , extrapolated from Table 4 of Perron (1990), implies that theBaht/$ exchange rate series is mean-break stationary. However, simulation evidence suggests that tP can be oversized in thepresence of an innovation break.We, therefore, consider the empirical evidence implied by the modified Perron (1990) test (t∗P ). Based on τ

c= 0.56,

the estimated pre-break variance (σ 21 ) is 0.0276 and the estimated post-break variance (σ22 ) is 1.6742. So, the post-break

standard deviation is more than seven times larger compared to the pre-break standard deviation. The modified Perronunit root statistic, t∗P , based on regression (6) with k

∗= 8 is equal to −2.7003.6 Given that the limiting distribution of t∗P

is given by expression equation (8) of Perron (1990), we can use the critical values in Table 4 of Perron (1990) to evaluate

5 Perron and Vogelsang (1992) suggest using the following data dependent procedure to determine the lag-truncation parameter. Pre-specify an upperbound for the order of the lagged first difference terms, denoted by kmax. The chosen lag-truncation parameter, k∗ , is such that the coefficient on the lastincluded lagged first difference term is significant and the coefficient on the higher order lagged first difference term is insignificant. The significance ofthe coefficient on the lagged first difference term is based on the critical values from a Normal distribution.6 We should note that the standard error of regression (6) is less than the standard error of regression (2).

358 A. Sen / Statistics and Probability Letters 79 (2009) 354–360

its significance. The extrapolated critical value for τ c = 0.56 implies that we cannot reject the unit root null hypothesisbased on t∗P . Therefore, allowing for the presence of a break in the innovation variance reverses the conclusion regarding theevidence of the presence of a unit root in the Baht/$ series.7

4. Conclusion

We derive the limiting distribution of the unit root test proposed by Perron (1990), denoted by tP , when there is a breakin the innovation variance. Perron’s unit root test is designed to have power against the stationary alternative with a breakin the mean at some known break-date. We assume that the location of the break in the mean, if it exists, coincides withthe location of the break in the innovation variance. Finite sample simulations show that tP can be oversized if the break inthe innovation variance is ignored. We, therefore, extend Perron’s unit root test by adapting a modified GLS strategy similarto that of Kim et al. (2002). The unit root test based on the GLS transformation has the same limiting distribution as that oftP described in Perron (1990), and so the practitioner can use the critical values tabulated in Perron (1990). Our simulationsshow that there are no size distortions with the new unit root statistic.

Acknowledgment

This research was partially supported by the D. J. O’Conor Professorship grant at Xavier University.

Appendix

Inwhat follows,we outline the proof of Theorem1. The location of the break-date is T cb = [τcT ]. All summations are taken

over the sample, that is, from t=1 to T unless otherwise specified. The results are based on the functional weak convergenceresult T−1/2

∑[r T ]t=1 ηt ⇒ W (r)∀r ∈ [0, 1]where isW (r) is theWiener Process defined on the unit interval, and ‘‘⇒’’ denotes

weak convergence. Based on the data generating process given in (1), and assuming without loss of generality that γ = 0and y0 = 0, we can show that:

T−1/2T cb∑t=1

εt ⇒ σ1W (τ c) (A.1)

T−1/2T∑

t=T cb+1

εt ⇒ σ2{W (1)−W (τ c)

}(A.2)

T−1T cb∑t=1

yt−1 εt ⇒12σ 21{W (τ c)2 − τ c

}(A.3)

T−1T∑

t=T cb+1

yt−1 εt ⇒12σ 22{W (1)2 − 1

}−12σ 22{W (τ c)2 − τ c

}+ σ2(σ1 − σ2)W (τ c)

{W (1)−W (τ c)

}(A.4)

T−3/2T cb∑t=1

yt−1 ⇒ σ1

∫ τ c

0W (r)dr (A.5)

T−3/2T∑

t=T cb+1

yt−1 ⇒ σ2

∫ 1

τ cW (r)dr + (σ1 − σ2)(1− τ c)W (τ c) (A.6)

T−2T cb∑t=1

y2t−1 ⇒ σ 21

∫ τ c

0W (r)2 dr (A.7)

T−2T∑

t=T cb+1

y2t−1 ⇒ σ 22

∫ 1

τ cW (r)2 dr + (σ1 − σ2)2(1− τ c)W (τ c)2 + 2(σ1 − σ2)σ2W (τ c)

∫ 1

τ cW (r)dr. (A.8)

7 Our findings are, for the most part, consistent with those of Busetti and Taylor (2003) who consider the Baht/$ exchange rate over a shorter period(1991:02-1999:12) using several different versions of their stationarity tests.

A. Sen / Statistics and Probability Letters 79 (2009) 354–360 359

Let θ = (µ, γ , d, ρ)′ denote the parameter vector corresponding to regression (2). Without loss of generality, we canassume that k∗ = 0. Let X denote the corresponding matrix of explanatory variables, and so regression (2) can be written asY = X θ + ε, where X = [1,DUt ,Dt , yt−1], Y = [yt ], and ε = [εt ]. Therefore, θ = (X ′X)−1 X ′Y . Let X1 = [1,DUt , yt−1] andX2 = [Dt ]. It follows that regression (2) will yield numerically equivalent results as: Y ∗ = X∗1 θ1+ε

∗, where θ1 = (µ, γ , ρ)′,and Y ∗, X∗1 , and ε

∗ are respectively the projections of Y , X1, and ε on the space spanned by the columns of X2. That is,

θ1 =(X∗′

1 X∗

1

)−1X∗1′Y ∗. Therefore, we can write the OLS estimator of θ1 as follows:

DT (θ1 − θ1) =[D−1T (X

∗′

1 X∗

1 )D−1T

]−1 [D−1T X

∗′

1 ε∗

](A.9)

where DT = diag(T 1/2, T−1/2, T ). Based on the limiting behaviour of the moments in (A.1)–(A.8), we can show that:

VT = D−1T X∗′

1 X∗

1D−1T ⇒ V , and (A.10)

MT = D−1T X∗′

1 ε∗⇒ M (A.11)

where

VT [1, 1] = 1−1T→ 1, VT [1, 2] = VT [2, 2] = (1− τ c)−

1T→ (1− τ c)

VT [1, 3] = T−3/2τ cT∑t=1

yt−1 + T−3/2T∑

t=τ cT+2

yt−1

⇒ σ1

∫ τ c

0W (r)dr + σ2

∫ 1

τ cW (r)dr + (σ1 − σ2)(1− τ c)W (τ c)

VT [2, 3] = T−3/2T∑

t=τ cT+2

yt−1 ⇒ σ2

∫ 1

τ cW (r)dr + (σ1 − σ2)(1− τ c)W (τ c)

VT [3, 3] = T−2τ cT∑t=1

y2t−1 + T−2

T∑t=τ cT+2

y2t−1

⇒ σ 21

∫ τ c

0W (r)2 dr + σ 22

∫ 1

τ cW (r)2 dr + (σ1 − σ2)2(1− τ c)W (τ c)2 + 2(σ1 − σ2)σ2W (τ c)

∫ 1

τ cW (r)dr

MT [1, 1] = T−1/2τ cT∑t=1

εt + T−1/2T∑

t=τ cT+2

εt ⇒ σ1W (τ c)+ σ2{W (1)−W (τ c)

}MT [2, 1] = T−1/2

T∑t=τ cT+2

εt ⇒ σ2{W (1)−W (τ c)

}MT [3, 1] = T−1

τ cT∑t=1

yt−1 εt + T−1T∑

t=τ cT+2

yt−1 εt

⇒12σ 21{W (τ c)2 − 1

}+12σ 22{W (1)2 − 1

}−12σ 22{W (τ c)2 − τ c

}+ σ2(σ1 − σ2)W (τ c)

{W (1)−W (τ c)

}.

Combining Eq. (A.9), (A.10) and (A.11), we get:

DT (θ1 − θ1)⇒ V−1M (A.12)

and, the error variance from regression (2) is given by:

σ ∗2 =1

(T − 4)

[Y ∗ − X∗1 θ1

]′ [Y ∗ − X∗1 θ1

]⇒ τ cσ 21 + (1− τ

c)σ 22 . (A.13)

Therefore, the limiting distribution of the unit root statistic:

tP =(ρ − 1)√ˆvar(ρ)

=T (ρ − 1)√

σ ∗2(T 2(X∗′1 X

∗

1 )−1)[3,3]

⇒

(V−1M

)[3,1]√[

σ 21 τc + σ 22 (1− τ c)

] (V−1

)[3,3]

follows from (A.10) and (A.12), the limiting behaviour of σ ∗2, and some tedious calculations.

360 A. Sen / Statistics and Probability Letters 79 (2009) 354–360

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