Testing for a unit root against ESTAR stationarity

David I. Harvey; Stephen J. Leybourne; Emily J. Whitehouse

doi:10.1515/snde-2016-0076

Article

Testing for a unit root against ESTAR stationarity

David I. Harvey , Stephen J. Leybourne and Emily J. Whitehouse

Published/Copyright: June 16, 2017

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 22 Issue 1

Abstract

In this paper we examine the local power of unit root tests against globally stationary exponential smooth transition autoregressive [ESTAR] alternatives under two sources of uncertainty: the degree of nonlinearity in the ESTAR model, and the presence of a linear deterministic trend. First, we show that the KSS test (Kapetanios, G., Y. Shin, and A. Snell. 2003. “Testing for a Unit Root in the Nonlinear STAR Framework.” Journal of Econometrics 112: 359–379) for nonlinear stationarity has local asymptotic power gains over standard Dickey-Fuller [DF] tests for certain degrees of nonlinearity in the ESTAR model, but that for other degrees of nonlinearity, the linear DF test has superior power. Second, we derive limiting distributions of demeaned, and demeaned and detrended KSS and DF tests under a local ESTAR alternative when a local trend is present in the DGP. We show that the power of the demeaned tests outperforms that of the detrended tests when no trend is present in the DGP, but deteriorates as the magnitude of the trend increases. We propose a union of rejections testing procedure that combines all four individual tests and show that this captures most of the power available from the individual tests across different degrees of nonlinearity and trend magnitudes. We also show that incorporating a trend detection procedure into this union testing strategy can result in higher power when a large trend is present in the DGP.

Keywords: nonlinearity; trend uncertainty; union of rejections

A Appendix

Due to the invariance of all statistics to μ in (4), we set μ = 0 in what follows, without loss of generality.

Proof of Lemma 1

The DF_μ test statistic is given by

DFμ=∑t=2Tyμ,t−1Δytσ^2∑t=2Tyμ,t−12

where σ^2 denotes the error variance estimate from the OLS regression of Δy_t on yμ,t−1. Considering first the numerator of DF_μ, we can write

T−1∑t=2Tyμ,t−1Δyt=T−1∑t=2T(yt−1−y¯)(β−cTut−1G(θ,ut−1)+εt)=−cT−2∑t=2T(yt−1−y¯)ut−1G(θ,ut−1)+T−1∑t=2T(yt−1−y¯)εt+op(1).

Evaluating each term separately, and defining t¯=T−1∑t=1Tt , u¯=T−1∑t=1Tut,

cT−2∑t=2T(yt−1−y¯)ut−1G(θ,ut−1)=cT−2∑t=2T{β(t−1−t¯)+ut−1−u¯}ut−1G(θ,ut−1)=cT−1∑t=2T{κσT−1(t−t¯)+T−1/2(ut−1−u¯)}T−1/2ut−1G(θ,ut−1)+op(1)→dσ2c∫01(κ(r−12)+Xμ(r))X(r)G(g2X(r)2)dr

and

T−1∑t=2T(yt−1−y¯)εt=T−1/2∑t=2T{κσT−1(t−t¯)+T−1/2(ut−1−u¯)}εt+op(1)→dσ2∫01(κ(r−12)+Xμ(r))dW(r).

In the denominator of DF_μ, it is easily shown that σ^2→pσ2, and

T−2∑t=2T(yt−1−y¯)2=T−1∑t=2T(κσT−1(t−t¯)+T−1/2(ut−1−u¯))2→dσ2∫01(κ(r−12)+Xμ(r))2dr.

The DF_μ test statistic therefore has the limiting distribution

DFμ→d∫01(κ(r−12)+Xμ(r))dW(r)−c∫01(κ(r−12)+Xμ(r))X(r)G(g2X(r)2)dr∫01(κ(r−12)+Xμ(r))2dr.

The KSS_μ test statistic is given by

KSSμ=∑t=2Tyμ,t−13Δytσ^2∑t=2Tyμ,t−16

where σ^2 denotes the error variance estimate from the OLS regression of Δy_t on yμ,t−13. In the numerator,

T−2∑t=2Tyμ,t−13Δyt=T−2∑t=2T(yt−1−y¯)3(β−cTut−1G(θ,ut−1)+εt)=κσT−5/2∑t=2T(yt−1−y¯)3−cT−3∑t=2T(yt−1−y¯)3ut−1G(θ,ut−1)+T−2∑t=2T(yt−1−y¯)3εt→dκσ4∫01(κ(r−12)+Xμ(r))3dr−cσ4∫01(κ(r−12)+Xμ(r))3X(r)G(g2X(r)2)dr+σ4∫01(κ(r−12)+Xμ(r))3dW(r)

while in the denominator, σ^2→pσ2 and

T−4∑t=2Tyμ,t−16→dσ6∫01(κ(r−12)+Xμ(r))6dr

giving the limit for KSS_μ as

KSSμ→dκ∫01(κ(r−12)+Xμ(r))3dr+∫01(κ(r−12)+Xμ(r))3dW(r)−c∫01(κ(r−12)+Xμ(r))3X(r)G(g2X(r)2)dr∫01(κ(r−12)+Xμ(r))6dr.

The test statistics based on demeaned and detrended data are invariant to β, hence we set β = 0 without loss of generality in the remainder of this proof. The DF_τ test statistic is

DFτ=∑t=2Tyτ,t−1Δyτ,tσ^2∑t=2Tyτ,t−12

where σ^2 denotes the error variance estimate from the OLS regression of Δyτ,t on yτ,t−1. Here,

Δyτ,t=Δyt−β^=−cTut−1G(θ,ut−1)+εt−β^

and

T1/2β^=T−5/2∑t=1T(t−t¯)ytT−3∑t=1T(t−t¯)2→d12∫01(r−12)X(r)dr.

Also,

T−1/2yτ,⌊rT⌋=T−1/2y⌊rT⌋−T−1/2μ^−T1/2β^r=T−1/2y⌊rT⌋−(T−1/2y¯−T1/2β^T−1t¯)−T1/2β^r→dX(r)−∫01X(s)ds−12(r−12)∫01(s−12)X(s)ds=X(r)+(6r−4)∫01X(s)ds+(6−12r)∫01sX(s)ds≡Xτ(r).

The numerator of DF_τ is then given by

T−1∑t=2Tyτ,t−1Δyτ,t=T−1∑t=2Tyτ,t−1εt−cT−2∑t=2Tyτ,t−1ut−1G(θ,ut−1)−T1/2β^T−3/2∑t=2Tyτ,t−1→dσ2∫01Xτ(r)dW(r)−cσ2∫01Xτ(r)X(r)G(g2X(r)2)dr

and for the denominator we obtain

T−2∑t=2Tyτ,t−12→dσ2∫01Xτ(r)2dr

giving

DFτ→d∫01Xτ(r)dW(r)−c∫01Xτ(r)X(r)G(g2X(r)2)dr∫01Xτ(r)2dr.

Finally, the KSS_τ test statistic is

KSSτ=∑t=2Tyτ,t−13Δyτ,tσ^2∑t=2Tyτ,t−16

where σ^2 is the error variance estimate from the OLS regression of Δyτ,t on yτ,t−13. Now σ^2→pσ2 as before,

T−2∑t=2Tyτ,t−13Δyτ,t=T−2∑t=2Tyτ,t−13εt−cT−3∑t=2Tyτ,t−13ut−1G(θ,ut−1)−T1/2β^T−5/2∑t=2Tyτ,t−13→dσ2∫01Xτ(r)3dW(r)−cσ2∫01Xτ(r)3X(r)G(g2X(r)2)dr−12∫01(r−12)X(r)dr∫01Xτ(r)3dr

and

T−4∑t=2Tyτ,t−16→d∫01Xτ(r)6dr

KSSτ→d∫01Xτ(r)3dW(r)−c∫01Xτ(r)3X(r)G(g2X(r)2)dr−12∫01(r−12)X(r)dr∫01Xτ(r)3dr∫01Xτ(r)6dr.

Proof of Lemma 2

As before, the DF_μ test statistic is given by

DFμ=∑t=2Tyμ,t−1Δytσ^2∑t=2Tyμ,t−12

where σ^2 is the error variance estimate from the OLS regression of Δy_t on yμ,t−1. We find

T−3/2∑t=2Tyμ,t−1Δyt=T−3/2∑t=2T(σκ(t−t¯)+ut−1−u¯)(σκ+εt−cTut−1G(θ,ut−1))+op(1)=σκT−3/2∑t=2T(t−t¯)εt−σκcT−5/2∑t=2T(t−t¯)ut−1G(θ,ut−1)+op(1)→dσ2κ∫01(r−12)dW(r)−σ2κc∫01(r−12)X(r)G(g2X(r)2)dr

and

T−3∑t=2Tyμ,t−12=T−3∑t=2T(β(t−1−t¯)+ut−1−u¯)2=σ2κ2T−3∑t=2T(t−t¯)2+op(1)→pσ2κ2/12.

Also,

σ^2=T−1∑t=2T(Δyt−∑s=2T(ys−1−y¯)Δys∑s=2T(ys−1−y¯)2(yt−1−y¯))2=T−1∑t=2T(Δyt−T−3/2∑s=2T(ys−1−y¯)ΔysT−3∑s=2T(ys−1−y¯)2T−3/2(yt−1−y¯))2=T−1∑t=2T(Δyt)2+op(1)=σ2κ2+T−1∑t=2Tεt2+op(1)→pσ2(κ2+1).

Therefore the DF_μ test statistic has the limiting distribution

DFμ→d∫01(r−12)dW(s)−c∫01(r−12)X(r)G(g2X(r)2)dr(κ2+1)/12.

For KSS_μ we again have

KSSμ=∑t=2Tyμ,t−13Δytσ^2∑t=2Tyμ,t−16

where σ^2 is the error variance estimate from the OLS regression of Δy_t on yμ,t−13. We obtain

T−7/2∑t=2Tyμ,t−13Δyt=T−7/2∑t=2T(σκ(t−t¯)+ut−1−u¯)3(σκ−cTut−1G(θ,ut−1)+εt)+op(1)=3σ3κ3T−7/2∑t=2T(t−t¯)2(ut−1−u¯)+σ3κ3T−7/2∑t=2T(t−t¯)3εt−cσ3κ3T−9/2∑t=2T(t−t¯)3ut−1G(θ,ut−1)+op(1)→d3σ4κ3∫01(r−12)2Xμ(r)dr+σ4κ3∫01(r−12)3dW(r)−cσ4κ3∫01(r−12)3X(r)G(g2X(r)2)dr

and

T−7∑t=2Tyμ,t−16=σ6κ6T−7∑t=2T(t−t¯)6+op(1)→pσ6κ6/448

together with

σ^2=T−1∑t=2T(Δyt−T−7/2∑s=2T(ys−1−y¯)3ΔysT−7∑s=2T(ys−1−y¯)6T−7/2(yt−1−y¯)3)2=T−1∑t=2T(Δyt)2+op(1)→pσ2(κ2+1)

giving the KSS_μ limit distribution

KSSμ→d3∫01(r−12)2Xμ(r)dr+∫01(r−12)3dW(r)−c∫01(r−12)3X(r)G(g2X(r)2)dr(κ2+1)/448.

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Keywords for this article

nonlinearity; trend uncertainty; union of rejections

Testing for a unit root against ESTAR stationarity

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Abstract

A Appendix

Proof of Lemma 1

Proof of Lemma 2

References

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