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An infimum coefficient unit root test allowing for an unknown break in trend
Economics Letters 117 (2012) 298–302
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
An infimum coefficient unit root test allowing for an unknown break in trend David I. Harvey ∗ , Stephen J. Leybourne School of Economics, University of Nottingham, University Park, Nottingham NG7 2RD, UK
article
info
Article history: Received 10 February 2012 Received in revised form 23 April 2012 Accepted 22 May 2012 Available online 30 May 2012 JEL classification: C22 Keywords: Unit root test Trend break Minimum Dickey–Fuller test
abstract In this paper we consider testing for a unit root in the possible presence of a trend break at an unknown time. Zivot and Andrews (1992) [Journal of Business and Economic Statistics 10, 251–270] proposed using the infimum of t-ratio Dickey–Fuller statistics across all candidate break points in a trimmed range, however this procedure can have an asymptotic size of one when a break occurs under the unit root null. We show that if the same approach is used, but instead with coefficient Dickey–Fuller statistics in an additive outlier framework, the test is asymptotically conservative when a break is present under the null, provided the degree of trimming is appropriately controlled. The test is also shown to have superior local asymptotic power to the t-ratio version. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The issue of allowing for a break in linear trend when conducting a test for a unit root was first addressed by Perron (1989), who proposed implementing either an OLS-based t-ratio or coefficient Dickey–Fuller test, generalizing the deterministic component to include a trend break dummy defined using an assumed break date. Zivot and Andrews (1992) [ZA] extended this approach to the case where the break date is not assumed known, and considered using the infimum of the t-ratio version of Perron’s innovational outlier statistics, evaluated across all possible break dates in a trimmed range of the data. However, their analysis (and critical values) relied on an assumption that no break in trend occurs under the unit root null hypothesis. This assumption is crucial, since if a break is in fact present under the unit root null, both additive outlier and innovational outlier ZA-type tests can be severely oversized; see Vogelsang and Perron (1998) and Harvey et al. (2012). In this paper, we consider a test based on the infimum of Perron’s additive outlier coefficient statistic, and contrast its asymptotic behavior with that for the corresponding t-ratio version considered in Harvey et al. (2012). Using a local-to-zero trend break magnitude asymptotic framework, we show that the coefficient version of the statistic does not suffer from severe asymptotic oversize, regardless of the magnitude of the local break. Indeed, when the degree of trimming is appropriately controlled,
∗
Corresponding author. E-mail address: [email protected] (D.I. Harvey).
0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.05.023
comparison of this statistic with critical values derived in the no break case yields a test with near correct asymptotic size when a break occurs under the null. It also has similar local asymptotic power levels in the break and no break cases. In fact, in the latter case its power exceeds that of its t-ratio counterpart. In contrast, the t-ratio version of the test suffers from severe asymptotic oversize when a large local break occurs in the first two-thirds of the sample, a result that mirrors the Harvey et al. (2012) finding obtained assuming a fixed magnitude for the trend break. Therefore, while the t-ratio-based approach cannot be considered reliable for practical application since a trend break may be present under the unit root null, the coefficient-based variant offers a sizerobust alternative that does not compromise test power. 2. The model and test statistic Consider the following data generating process for a time series
{yt } yt = µ + β t + γT DTt (τ0 ) + ut , ut = ρT ut −1 + εt ,
t = 2, . . . , T
t = 1, . . . , T
(1) (2)
where DTt (τ0 ) := 1(t > ⌊τ0 T ⌋)(t − ⌊τ0 T ⌋), with ⌊τ0 T ⌋ the potential trend break point with associated break fraction τ0 , and break magnitude γT .1 A break in trend occurs in {yt } at time ⌊τ0 T ⌋ when γT ̸= 0. The true break fraction τ0 is treated as unknown, but
1 Here and throughout, 1(.) denotes the indicator function and ⌊.⌋ denotes the integer part of the argument.
D.I. Harvey, S.J. Leybourne / Economics Letters 117 (2012) 298–302
is assumed to satisfy τ0 ∈ Λ = [π , 1−π ] with 0 < π < 1−π < 1; the fraction π representing the degree of trimming, such that no break is deemed allowable to occur before π or after 1 − π . We assume u1 = op (T 1/2 ), while for purposes of tractability in what follows, we make the assumption that the innovation process {εt } satisfies εt ∼ IID(0, σε2 ); extension to cases where {εt } follows a dependent process is discussed in Section 4. Our focus is on testing the unit root null hypothesis H0 : ρT = 1, against a local alternative, H1 : ρT = 1 − c /T , 0 < c < ∞. We let the break magnitude be local-to-zero by setting γT = κσε T −1/2 , thereby adopting the appropriate Pitman drift for a trend break in a localto-unity process; cf. Vogelsang and Perron (1998).2 The test statistic we consider is an additive outlier Dickey–Fuller coefficient test variant of the ZA unit root statistic. This is obtained by minimizing the coefficient-based unit root statistic that allows for a break in trend at time ⌊τ T ⌋ over all possible break points, i.e. DF min = inf DF ρ (τ ) ρ τ ∈Λ
where T
ˆ DF ρ (τ ) = T φ,
φˆ =
∆uˆ t uˆ t −1
t =2 T
(3) u2t −1
ˆ
Table 1 Asymptotic critical values for DF min ρ .
π
Critical values
0.01 0.05 0.10 0.15
uˆ t = yt − µ ˆ − βˆ t − γˆ DTt (τ ) ˆ γˆ ]′ are obtained from an OLS regression of yt on where [µ, ˆ β, Xt ,τ = [1, t , DTt (τ )]′ . To implement the test, as in ZA, we will use
asymptotic critical values obtained under H0 in the case of no break in trend (κ = 0). 3. Asymptotic behavior The following theorem establishes the asymptotic behavior of DF min ρ under the null H0 and local alternative H1 , the proof of which is given in the Appendix. Here and throughout, we use the notation Iab = 1(a ≥ b). Theorem 1. Let yt be generated according to (1)–(2) with ρT = 1 − c /T , 0 ≤ c < ∞, where εt ∼ IID(0, σε2 ) with finite fourth order moment. Then, for γT = κσε T −1/2 , Kc (1, κ, τ0 , τ )2 − Kc (0, κ, τ0 , τ )2 − 1 2
1 0
Kc (r , κ, τ0 , τ )2 dr
(4)
where Kc (r , κ, τ0 , τ ) denotes the continuous time residual process from the projection of κ(r −τ0 )Irτ0 + Wc (r ) onto the space spanned by
{1, r , (r − τ )Irτ }, with Wc (r ) = Wiener process.
5%
1%
−30.77 −30.77 −30.73 −30.57
−35.22 −35.22 −35.20 −35.04
−44.76 −44.76 −44.69 −44.60
of magnitude κ = {0, 1, 2, . . . , 75} and break fractions τ0 = {0.3, 0.5, 0.7} at the nominal 0.10, 0.05 and 0.01 significance levels. Here we implement the tests using two representative degrees of trimming, π = {0.05, 0.15}. We find that at all three significance levels, its asymptotic size is virtually never above the nominal level. For very small values of κ , the test’s size can be very modestly in excess of the nominal value; however, for all other values of κ the test is a little undersized. To gain some (informal) insight into the relative size behavior of DF min for breaks of large magnitude, consider the following ρ heuristic. For large |κ| it can be shown that, by ignoring terms of order less than κ 2 ,
r 0
e−(r −s)c dW (s), W (r ) a standard
Table 1 presents critical values for DF min obtained from ρ simulating the limit distribution in Theorem 1 with c = 0 and κ = 0. Here and throughout the paper, we approximated the Wiener processes in the limiting functionals using NIID(0, 1) random variates, with the integrals approximated by normalized sums of 2000 steps, and used 20,000 Monte Carlo replications. Critical values at the 0.10, 0.05 and 0.01 nominal significance levels are provided for the degrees of trimming π ∈ {0.01, 0.05, 0.10, 0.15}, covering the breadth of trimming parameter choices considered for such problems in the literature. The critical values are seen to be quite insensitive to the degree of trimming, with very little change observed across the values of π considered. We next consider the asymptotic sizes of DF min under a local ρ break in trend. In Fig. 1 we show the sizes for local trend breaks
2 Scaling the trend break by σ is merely a convenience device allowing it to be ε factored out of the local limit distributions that arise in Section 3.
1
Kc (r , κ, τ0 , τ )2 dr 0 3 (1 + τ0 − 2τ ) (1 − 3τ0 + 2τ0 τ ) 2 (1 − τ0 ) (1 − τ ) (τ − 4τ0 + 3τ0 τ ) ≈ 3 (τ0 − 2τ ) (τ0 − 2τ + 2τ0 τ ) 2τ0 τ (τ0 − 4τ + 3τ0 τ ) 2
with
τ ∈Λ
10%
Kc (1, κ, τ0 , τ )2 − Kc (0, κ, τ0 , τ )2 − 1
t =2
DF min ⇒ inf ρ
299
for τ ≤ τ0
.
(5)
for τ > τ0
Hence, the limit of DF min in (4) for large |κ| approximates ρ a deterministic function that does not depend on κ . This approximation can be used to establish whether there are regions of the (τ0 , Λ) parameter space where DF min ρ will spuriously reject the null asymptotically when |κ| is large, i.e. whether regions exist where the infimum of the function (5) across τ ∈ Λ lies below the associated critical values at conventional significance levels. Evaluating (5) we find the infimum across τ ∈ Λ to be less than the 0.10-, 0.05- and 0.01-level critical values only when τ0 (∈ Λ) < 0.045, 0.040 and 0.033, respectively. Given that we assume τ0 lies within Λ should a break occur, asymptotic oversizing for large |κ| does not appear to be an issue for the DF min ρ test implemented using conventional significance levels, provided the degree of trimming is set to at least, say, π = 0.05. This result is illustrated in Fig. 2(a) and (b), where the asymptotic size of DF min is reported for π = ρ 0.01 with τ0 = 0.02 and τ0 = 0.06 respectively, obtained via simulation of (4) as in Fig. 1. In Fig. 2(a) we have τ0 (∈ Λ = [0.01, 0.99]) < 0.033, and in line with our heuristic we observe that asymptotic size increases rapidly above the nominal levels when κ is large. As expected, in Fig. 2(b) where τ0 (∈ Λ) > 0.045, we find no such problems, with DF min ρ undersized for large κ . For purposes of comparison, Fig. 1 also shows the asymptotic 0.05-level sizes of the corresponding t-ratio version of the DF min ρ test, that is: DF min = inf t
τ ∈Λ
φˆ ˆ s.e.(φ)
.
The limit distribution of DF min is given by (4) with the integral in t the denominator replaced by its square root. While the asymptotic size of DF min follows a very similar pattern to DF min t ρ for all values of κ when τ0 = 0.7, for τ0 = 0.3 and τ0 = 0.5 size is seen to follow that of DF min only up to a certain value of κ , after which ρ size increases rapidly in κ towards one. These values are around κ = 10 (for π = 0.05 and π = 0.15) when τ0 = 0.3, and about
300
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0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
0.00
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(a) π = 0.05, τ0 = 0.3.
0.00
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
10 15 20 25 30 35 40 45 50 55 60 65 70 75
0.00 0
5
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(c) π = 0.05, τ0 = 0.5.
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(e) π = 0.05, τ0 = 0.7.
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(d) π = 0.15, τ0 = 0.5.
0.30
0.00
5
(b) π = 0.15, τ0 = 0.3.
0.30
0.00
0
0.00
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(f) π = 0.15, τ0 = 0.7.
Fig. 1. Asymptotic size of nominal ξ -level tests: DFρmin , ξ = 0.10:___; DFρmin , ξ = 0.05: − −; DFρmin , ξ = 0.01: - - - ; DFtmin , ξ = 0.05: · · ·.
κ = 35 (κ = 50) for π = 0.05 (π = 0.15) when τ0 = 0.5. A heuristic analysis of DF min for large |κ| (similar to that discussed t
above for DF min ρ ) shows that its limit distribution approximates a deterministic function of the form |κ| infτ ∈Λ f (τ , τ0 ), where f (τ , τ0 ) is strictly positive for all τ ∈ Λ when τ0 (∈ Λ) ≥ 2/3 but is negative for some τ ∈ Λ for all τ0 (∈ Λ) < 2/3. Hence, for
large |κ|, this approximation suggests that DF min will spuriously t reject the null asymptotically whenever τ0 (∈ Λ) < 2/3. As we would expect, this finding is entirely consistent with the fixed magnitude break asymptotic analysis of DF min conducted in Harvey t et al. (2012), which shows that DF min has an asymptotic size of t unity for τ0 < 2/3.
D.I. Harvey, S.J. Leybourne / Economics Letters 117 (2012) 298–302
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(a) π = 0.01, τ0 = 0.02.
0.0
0
5
301
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(b) π = 0.01, τ0 = 0.06.
Fig. 2. Asymptotic size of nominal ξ -level tests: DFρmin , ξ = 0.10:___; DFρmin , ξ = 0.05: − −; DFρmin , ξ = 0.01: - - -.
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0
5
10
15
20
25
30
35
40
45
50
Fig. 3. Local asymptotic power of nominal 0.05-level tests with π = 0.15: DFρmin , κ = 0:___; DFρmin , κ = 5, τ0 = 0.5: − −; DFρmin , κ = 10, τ0 = 0.5: - -
- ; DFρmin , κ = 20, τ0 = 0.5: · · · ; DFtmin , κ = 0:_._.
To conclude this section, we examine the asymptotic local power of DF min both when a local break does and does not occur ρ in the DGP. We simulate the limit distribution in Theorem 1 for the no break case (κ = 0) and three local break cases (κ = {5, 10, 20}), for c = {0, 1, 2, . . . , 50}. We adopt the degree of trimming setting π = 0.15, and when a break in trend is present we set τ0 = 0.5. For purposes of comparison, we also simulate the asymptotic local power of DF min in the case of no break (which is t realistically the only situation where it can be considered to have reliable asymptotic size). The results for nominal 0.05 significance level tests are reported in Fig. 3. We find that when no break in trend occurs, the power of DFρmin is superior to that of DFtmin across almost all values of c. We also see that the power of DFρmin is hardly affected by the presence of a break, with local asymptotic power for κ = 5, 10 and 20 being very close to that when κ = 0, despite the fact that, as shown above, the procedure is undersized when κ ̸= 0.
the null hypothesis. In establishing this result, we have derived the limit distribution of the statistic under the null and local-to-unity alternative, modeling the putative break in trend as local-to-zero. Whereas the corresponding t-ratio version can have an asymptotic size of one whenever a trend break occurs in the first two thirds of the sample period, we find that the coefficient version is only subject to serious oversize when a break occurs in the first 5% of the data. Given that in practice the possibility of such very early breaks is typically discounted by a choice of trimming set to at least 5%, the coefficient test is, to all practical intents and purposes, asymptotically robust, while also sacrificing nothing in terms of local asymptotic power relative to the t-ratio version. Our focus here is on an OLS detrended procedure, and while more recent treatments of the unit root testing problem have largely focused on generalized least squares (GLS) detrending (e.g. Perron and Rodríguez, 2003), the results of, inter alia, Müller and Elliott (2003) show that the power of the OLS-based coefficient Dickey–Fuller test (in the no trend break context) is rather less adversely affected by the magnitude of the initial condition than the GLS-based t-ratio test. Consequently, if the goal is to achieve robust inference rather than maximize power under more restrictive assumptions governing the initial condition, the OLSbased coefficient test we consider here has clear appeal. Finally, note that while our analysis assumes serially uncorrelated innovations, the asymptotic results that we present for DF min ρ are also appropriate when the innovation process is serially correlated, provided the usual augmentation is made to the construction of the DF ρ (τ ) statistics to account for short run dynamics, cf. Chang and Park (2002). Appendix. Proof of Theorem 1 Standard asymptotic arguments give T
−1/2
uˆ ⌊rT ⌋ ⇒ σε Kc (r , κ, τ0 , τ ).
Then T −1 DF ρ (τ ) =
∆uˆ t uˆ t −1
t =2
T −2
4. Discussion We have demonstrated that in contrast to the additive outlier ZA-type t-ratio-based unit root test, a version that implements the same approach, but instead using the coefficient statistic, can yield robust asymptotic inference in the presence of a trend break under
T
T
uˆ 2t −1
t =2 T
T −1 uˆ 2T − T −1 uˆ 21 − T −1
(∆uˆ t )2
t =2
= 2T −2
T t =2
uˆ 2t −1
302
D.I. Harvey, S.J. Leybourne / Economics Letters 117 (2012) 298–302
Kc (1, κ, τ0 , τ ) − Kc (0, κ, τ0 , τ )2 − 1 2
⇒
2
1 0
Kc (r , κ, τ0 , τ )2 dr
2 p
ˆ t ) → σε2 . This result holds pointwise for all τ ∈ since T t =2 (∆u Λ. The result in Theorem 1 is then obtained via application of the continuous mapping theorem applied to the sequence of statistics across τ . The stated limiting representations for the infimum statistic follows uniformly from the pointwise τ representation, using a straightforward extension (accounting for the local trend break in the DGP) to the arguments in ZA or Perron (1997). −1
T
References Chang, Y., Park, J.Y., 2002. On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431–447.
Harvey, D.I., Leybourne, S.J., Taylor, A.M.R., 2012. On infimum Dickey–Fuller unit root tests allowing for a trend break under the null. Manuscript, School of Economics, University of Nottingham. Müller, U.K., Elliott, G., 2003. Tests for unit roots and the initial condition. Econometrica 71, 1269–1286. Perron, P., 1989. The Great Crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361–1401. Perron, P., 1997. Further evidence of breaking trend functions in macroeconomic variables. Journal of Econometrics 80, 355–385. Perron, P., Rodríguez, G., 2003. GLS detrending, efficient unit root tests and structural change. Journal of Econometrics 115, 1–27. Vogelsang, T.J., Perron, P., 1998. Additional tests for a unit root allowing for a break in the trend function at an unknown time. International Economic Review 39, 1073–1100. Zivot, E., Andrews, D.W.K., 1992. Further evidence on the Great Crash, the oil-price shock, and the unit-root hypothesis. Journal of Business and Economic Statistics 10, 251–270.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
An infimum coefficient unit root test allowing for an unknown break in trend David I. Harvey ∗ , Stephen J. Leybourne School of Economics, University of Nottingham, University Park, Nottingham NG7 2RD, UK
article
info
Article history: Received 10 February 2012 Received in revised form 23 April 2012 Accepted 22 May 2012 Available online 30 May 2012 JEL classification: C22 Keywords: Unit root test Trend break Minimum Dickey–Fuller test
abstract In this paper we consider testing for a unit root in the possible presence of a trend break at an unknown time. Zivot and Andrews (1992) [Journal of Business and Economic Statistics 10, 251–270] proposed using the infimum of t-ratio Dickey–Fuller statistics across all candidate break points in a trimmed range, however this procedure can have an asymptotic size of one when a break occurs under the unit root null. We show that if the same approach is used, but instead with coefficient Dickey–Fuller statistics in an additive outlier framework, the test is asymptotically conservative when a break is present under the null, provided the degree of trimming is appropriately controlled. The test is also shown to have superior local asymptotic power to the t-ratio version. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The issue of allowing for a break in linear trend when conducting a test for a unit root was first addressed by Perron (1989), who proposed implementing either an OLS-based t-ratio or coefficient Dickey–Fuller test, generalizing the deterministic component to include a trend break dummy defined using an assumed break date. Zivot and Andrews (1992) [ZA] extended this approach to the case where the break date is not assumed known, and considered using the infimum of the t-ratio version of Perron’s innovational outlier statistics, evaluated across all possible break dates in a trimmed range of the data. However, their analysis (and critical values) relied on an assumption that no break in trend occurs under the unit root null hypothesis. This assumption is crucial, since if a break is in fact present under the unit root null, both additive outlier and innovational outlier ZA-type tests can be severely oversized; see Vogelsang and Perron (1998) and Harvey et al. (2012). In this paper, we consider a test based on the infimum of Perron’s additive outlier coefficient statistic, and contrast its asymptotic behavior with that for the corresponding t-ratio version considered in Harvey et al. (2012). Using a local-to-zero trend break magnitude asymptotic framework, we show that the coefficient version of the statistic does not suffer from severe asymptotic oversize, regardless of the magnitude of the local break. Indeed, when the degree of trimming is appropriately controlled,
∗
Corresponding author. E-mail address: [email protected] (D.I. Harvey).
0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.05.023
comparison of this statistic with critical values derived in the no break case yields a test with near correct asymptotic size when a break occurs under the null. It also has similar local asymptotic power levels in the break and no break cases. In fact, in the latter case its power exceeds that of its t-ratio counterpart. In contrast, the t-ratio version of the test suffers from severe asymptotic oversize when a large local break occurs in the first two-thirds of the sample, a result that mirrors the Harvey et al. (2012) finding obtained assuming a fixed magnitude for the trend break. Therefore, while the t-ratio-based approach cannot be considered reliable for practical application since a trend break may be present under the unit root null, the coefficient-based variant offers a sizerobust alternative that does not compromise test power. 2. The model and test statistic Consider the following data generating process for a time series
{yt } yt = µ + β t + γT DTt (τ0 ) + ut , ut = ρT ut −1 + εt ,
t = 2, . . . , T
t = 1, . . . , T
(1) (2)
where DTt (τ0 ) := 1(t > ⌊τ0 T ⌋)(t − ⌊τ0 T ⌋), with ⌊τ0 T ⌋ the potential trend break point with associated break fraction τ0 , and break magnitude γT .1 A break in trend occurs in {yt } at time ⌊τ0 T ⌋ when γT ̸= 0. The true break fraction τ0 is treated as unknown, but
1 Here and throughout, 1(.) denotes the indicator function and ⌊.⌋ denotes the integer part of the argument.
D.I. Harvey, S.J. Leybourne / Economics Letters 117 (2012) 298–302
is assumed to satisfy τ0 ∈ Λ = [π , 1−π ] with 0 < π < 1−π < 1; the fraction π representing the degree of trimming, such that no break is deemed allowable to occur before π or after 1 − π . We assume u1 = op (T 1/2 ), while for purposes of tractability in what follows, we make the assumption that the innovation process {εt } satisfies εt ∼ IID(0, σε2 ); extension to cases where {εt } follows a dependent process is discussed in Section 4. Our focus is on testing the unit root null hypothesis H0 : ρT = 1, against a local alternative, H1 : ρT = 1 − c /T , 0 < c < ∞. We let the break magnitude be local-to-zero by setting γT = κσε T −1/2 , thereby adopting the appropriate Pitman drift for a trend break in a localto-unity process; cf. Vogelsang and Perron (1998).2 The test statistic we consider is an additive outlier Dickey–Fuller coefficient test variant of the ZA unit root statistic. This is obtained by minimizing the coefficient-based unit root statistic that allows for a break in trend at time ⌊τ T ⌋ over all possible break points, i.e. DF min = inf DF ρ (τ ) ρ τ ∈Λ
where T
ˆ DF ρ (τ ) = T φ,
φˆ =
∆uˆ t uˆ t −1
t =2 T
(3) u2t −1
ˆ
Table 1 Asymptotic critical values for DF min ρ .
π
Critical values
0.01 0.05 0.10 0.15
uˆ t = yt − µ ˆ − βˆ t − γˆ DTt (τ ) ˆ γˆ ]′ are obtained from an OLS regression of yt on where [µ, ˆ β, Xt ,τ = [1, t , DTt (τ )]′ . To implement the test, as in ZA, we will use
asymptotic critical values obtained under H0 in the case of no break in trend (κ = 0). 3. Asymptotic behavior The following theorem establishes the asymptotic behavior of DF min ρ under the null H0 and local alternative H1 , the proof of which is given in the Appendix. Here and throughout, we use the notation Iab = 1(a ≥ b). Theorem 1. Let yt be generated according to (1)–(2) with ρT = 1 − c /T , 0 ≤ c < ∞, where εt ∼ IID(0, σε2 ) with finite fourth order moment. Then, for γT = κσε T −1/2 , Kc (1, κ, τ0 , τ )2 − Kc (0, κ, τ0 , τ )2 − 1 2
1 0
Kc (r , κ, τ0 , τ )2 dr
(4)
where Kc (r , κ, τ0 , τ ) denotes the continuous time residual process from the projection of κ(r −τ0 )Irτ0 + Wc (r ) onto the space spanned by
{1, r , (r − τ )Irτ }, with Wc (r ) = Wiener process.
5%
1%
−30.77 −30.77 −30.73 −30.57
−35.22 −35.22 −35.20 −35.04
−44.76 −44.76 −44.69 −44.60
of magnitude κ = {0, 1, 2, . . . , 75} and break fractions τ0 = {0.3, 0.5, 0.7} at the nominal 0.10, 0.05 and 0.01 significance levels. Here we implement the tests using two representative degrees of trimming, π = {0.05, 0.15}. We find that at all three significance levels, its asymptotic size is virtually never above the nominal level. For very small values of κ , the test’s size can be very modestly in excess of the nominal value; however, for all other values of κ the test is a little undersized. To gain some (informal) insight into the relative size behavior of DF min for breaks of large magnitude, consider the following ρ heuristic. For large |κ| it can be shown that, by ignoring terms of order less than κ 2 ,
r 0
e−(r −s)c dW (s), W (r ) a standard
Table 1 presents critical values for DF min obtained from ρ simulating the limit distribution in Theorem 1 with c = 0 and κ = 0. Here and throughout the paper, we approximated the Wiener processes in the limiting functionals using NIID(0, 1) random variates, with the integrals approximated by normalized sums of 2000 steps, and used 20,000 Monte Carlo replications. Critical values at the 0.10, 0.05 and 0.01 nominal significance levels are provided for the degrees of trimming π ∈ {0.01, 0.05, 0.10, 0.15}, covering the breadth of trimming parameter choices considered for such problems in the literature. The critical values are seen to be quite insensitive to the degree of trimming, with very little change observed across the values of π considered. We next consider the asymptotic sizes of DF min under a local ρ break in trend. In Fig. 1 we show the sizes for local trend breaks
2 Scaling the trend break by σ is merely a convenience device allowing it to be ε factored out of the local limit distributions that arise in Section 3.
1
Kc (r , κ, τ0 , τ )2 dr 0 3 (1 + τ0 − 2τ ) (1 − 3τ0 + 2τ0 τ ) 2 (1 − τ0 ) (1 − τ ) (τ − 4τ0 + 3τ0 τ ) ≈ 3 (τ0 − 2τ ) (τ0 − 2τ + 2τ0 τ ) 2τ0 τ (τ0 − 4τ + 3τ0 τ ) 2
with
τ ∈Λ
10%
Kc (1, κ, τ0 , τ )2 − Kc (0, κ, τ0 , τ )2 − 1
t =2
DF min ⇒ inf ρ
299
for τ ≤ τ0
.
(5)
for τ > τ0
Hence, the limit of DF min in (4) for large |κ| approximates ρ a deterministic function that does not depend on κ . This approximation can be used to establish whether there are regions of the (τ0 , Λ) parameter space where DF min ρ will spuriously reject the null asymptotically when |κ| is large, i.e. whether regions exist where the infimum of the function (5) across τ ∈ Λ lies below the associated critical values at conventional significance levels. Evaluating (5) we find the infimum across τ ∈ Λ to be less than the 0.10-, 0.05- and 0.01-level critical values only when τ0 (∈ Λ) < 0.045, 0.040 and 0.033, respectively. Given that we assume τ0 lies within Λ should a break occur, asymptotic oversizing for large |κ| does not appear to be an issue for the DF min ρ test implemented using conventional significance levels, provided the degree of trimming is set to at least, say, π = 0.05. This result is illustrated in Fig. 2(a) and (b), where the asymptotic size of DF min is reported for π = ρ 0.01 with τ0 = 0.02 and τ0 = 0.06 respectively, obtained via simulation of (4) as in Fig. 1. In Fig. 2(a) we have τ0 (∈ Λ = [0.01, 0.99]) < 0.033, and in line with our heuristic we observe that asymptotic size increases rapidly above the nominal levels when κ is large. As expected, in Fig. 2(b) where τ0 (∈ Λ) > 0.045, we find no such problems, with DF min ρ undersized for large κ . For purposes of comparison, Fig. 1 also shows the asymptotic 0.05-level sizes of the corresponding t-ratio version of the DF min ρ test, that is: DF min = inf t
τ ∈Λ
φˆ ˆ s.e.(φ)
.
The limit distribution of DF min is given by (4) with the integral in t the denominator replaced by its square root. While the asymptotic size of DF min follows a very similar pattern to DF min t ρ for all values of κ when τ0 = 0.7, for τ0 = 0.3 and τ0 = 0.5 size is seen to follow that of DF min only up to a certain value of κ , after which ρ size increases rapidly in κ towards one. These values are around κ = 10 (for π = 0.05 and π = 0.15) when τ0 = 0.3, and about
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0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
0.00
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(a) π = 0.05, τ0 = 0.3.
0.00
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
10 15 20 25 30 35 40 45 50 55 60 65 70 75
0.00 0
5
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(c) π = 0.05, τ0 = 0.5.
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(e) π = 0.05, τ0 = 0.7.
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(d) π = 0.15, τ0 = 0.5.
0.30
0.00
5
(b) π = 0.15, τ0 = 0.3.
0.30
0.00
0
0.00
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(f) π = 0.15, τ0 = 0.7.
Fig. 1. Asymptotic size of nominal ξ -level tests: DFρmin , ξ = 0.10:___; DFρmin , ξ = 0.05: − −; DFρmin , ξ = 0.01: - - - ; DFtmin , ξ = 0.05: · · ·.
κ = 35 (κ = 50) for π = 0.05 (π = 0.15) when τ0 = 0.5. A heuristic analysis of DF min for large |κ| (similar to that discussed t
above for DF min ρ ) shows that its limit distribution approximates a deterministic function of the form |κ| infτ ∈Λ f (τ , τ0 ), where f (τ , τ0 ) is strictly positive for all τ ∈ Λ when τ0 (∈ Λ) ≥ 2/3 but is negative for some τ ∈ Λ for all τ0 (∈ Λ) < 2/3. Hence, for
large |κ|, this approximation suggests that DF min will spuriously t reject the null asymptotically whenever τ0 (∈ Λ) < 2/3. As we would expect, this finding is entirely consistent with the fixed magnitude break asymptotic analysis of DF min conducted in Harvey t et al. (2012), which shows that DF min has an asymptotic size of t unity for τ0 < 2/3.
D.I. Harvey, S.J. Leybourne / Economics Letters 117 (2012) 298–302
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(a) π = 0.01, τ0 = 0.02.
0.0
0
5
301
10 15 20 25 30 35 40 45 50 55 60 65 70 75
(b) π = 0.01, τ0 = 0.06.
Fig. 2. Asymptotic size of nominal ξ -level tests: DFρmin , ξ = 0.10:___; DFρmin , ξ = 0.05: − −; DFρmin , ξ = 0.01: - - -.
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0
5
10
15
20
25
30
35
40
45
50
Fig. 3. Local asymptotic power of nominal 0.05-level tests with π = 0.15: DFρmin , κ = 0:___; DFρmin , κ = 5, τ0 = 0.5: − −; DFρmin , κ = 10, τ0 = 0.5: - -
- ; DFρmin , κ = 20, τ0 = 0.5: · · · ; DFtmin , κ = 0:_._.
To conclude this section, we examine the asymptotic local power of DF min both when a local break does and does not occur ρ in the DGP. We simulate the limit distribution in Theorem 1 for the no break case (κ = 0) and three local break cases (κ = {5, 10, 20}), for c = {0, 1, 2, . . . , 50}. We adopt the degree of trimming setting π = 0.15, and when a break in trend is present we set τ0 = 0.5. For purposes of comparison, we also simulate the asymptotic local power of DF min in the case of no break (which is t realistically the only situation where it can be considered to have reliable asymptotic size). The results for nominal 0.05 significance level tests are reported in Fig. 3. We find that when no break in trend occurs, the power of DFρmin is superior to that of DFtmin across almost all values of c. We also see that the power of DFρmin is hardly affected by the presence of a break, with local asymptotic power for κ = 5, 10 and 20 being very close to that when κ = 0, despite the fact that, as shown above, the procedure is undersized when κ ̸= 0.
the null hypothesis. In establishing this result, we have derived the limit distribution of the statistic under the null and local-to-unity alternative, modeling the putative break in trend as local-to-zero. Whereas the corresponding t-ratio version can have an asymptotic size of one whenever a trend break occurs in the first two thirds of the sample period, we find that the coefficient version is only subject to serious oversize when a break occurs in the first 5% of the data. Given that in practice the possibility of such very early breaks is typically discounted by a choice of trimming set to at least 5%, the coefficient test is, to all practical intents and purposes, asymptotically robust, while also sacrificing nothing in terms of local asymptotic power relative to the t-ratio version. Our focus here is on an OLS detrended procedure, and while more recent treatments of the unit root testing problem have largely focused on generalized least squares (GLS) detrending (e.g. Perron and Rodríguez, 2003), the results of, inter alia, Müller and Elliott (2003) show that the power of the OLS-based coefficient Dickey–Fuller test (in the no trend break context) is rather less adversely affected by the magnitude of the initial condition than the GLS-based t-ratio test. Consequently, if the goal is to achieve robust inference rather than maximize power under more restrictive assumptions governing the initial condition, the OLSbased coefficient test we consider here has clear appeal. Finally, note that while our analysis assumes serially uncorrelated innovations, the asymptotic results that we present for DF min ρ are also appropriate when the innovation process is serially correlated, provided the usual augmentation is made to the construction of the DF ρ (τ ) statistics to account for short run dynamics, cf. Chang and Park (2002). Appendix. Proof of Theorem 1 Standard asymptotic arguments give T
−1/2
uˆ ⌊rT ⌋ ⇒ σε Kc (r , κ, τ0 , τ ).
Then T −1 DF ρ (τ ) =
∆uˆ t uˆ t −1
t =2
T −2
4. Discussion We have demonstrated that in contrast to the additive outlier ZA-type t-ratio-based unit root test, a version that implements the same approach, but instead using the coefficient statistic, can yield robust asymptotic inference in the presence of a trend break under
T
T
uˆ 2t −1
t =2 T
T −1 uˆ 2T − T −1 uˆ 21 − T −1
(∆uˆ t )2
t =2
= 2T −2
T t =2
uˆ 2t −1
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Kc (1, κ, τ0 , τ ) − Kc (0, κ, τ0 , τ )2 − 1 2
⇒
2
1 0
Kc (r , κ, τ0 , τ )2 dr
2 p
ˆ t ) → σε2 . This result holds pointwise for all τ ∈ since T t =2 (∆u Λ. The result in Theorem 1 is then obtained via application of the continuous mapping theorem applied to the sequence of statistics across τ . The stated limiting representations for the infimum statistic follows uniformly from the pointwise τ representation, using a straightforward extension (accounting for the local trend break in the DGP) to the arguments in ZA or Perron (1997). −1
T
References Chang, Y., Park, J.Y., 2002. On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431–447.
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