# RALS-LM unit root test with trend breaks and non-normal errors: application to the Prebisch-Singer hypothesis

Ming Meng , Junsoo Lee and James E. Payne

## Abstract

This study proposes a new unit root test that allows for structural breaks in both the intercept and the slope, and adopts the residual augmented least squares (RALS) procedure to gain improved power when the error term follows a non-normal distribution. The new test using the RALS procedure is more powerful than the usual LM test which does not incorporate information on non-normal errors. Our test is free of nuisance parameters that indicate the locations of structural break. It is also free of the spurious rejection problem. Thus, the rejection of the null hypothesis can be considered as more accurate evidence of stationarity. We apply the new test on the recently extended Grilli and Yang index of 24 commodity series from 1900 to 2007. Our empirical findings provide significant evidence that primary commodity prices are stationary with one or two trend breaks. However, compared with past studies, our findings provide even weaker evidence to support the Prebisch-Singer hypothesis.

JEL Classification: O13; C22

## Acknowledgments

The authors wish to thank Walt Enders, Matt Holt, M. Kejriwal, Robert Reed, Jun Ma, Kyung-so Im, Karl Boulware, and seminar participants at University of Alabama, and the 2nd International Workshop on “Financial Markets and Nonlinear Dynamics” (FMND) for their helpful comments. We are also grateful to two anonymous reviewers for their suggestions.

## Appendix

### Proof of the asymptotic distribution of the transformed LM tests

We first consider the case with R=1 and then extend the result to multiple breaks. We define: D1t=1 for tTB and 0 otherwise; and D2t=1 for tTB+1 and 0 otherwise. Similarly, we let DT1t*=t for tTB and 0 otherwise; and DT1t*=t–TB for tTB+1 and 0 otherwise. Then, the first step testing regression (3) can be alternatively written as:

(A.1)Δyt=δ1B1t+δ2B2t+δ3D1t+δ4D2t+ut.
Table A.1:

Results using ADF test and no-break LM unit root tests.

Aluminum–3.180*7–3.520**–3.593***0.5547
Banana–2.3902–1.547–1.5681.0152
Beef–2.0495–1.909–1.5030.5375
Cocoa–2.5572–2.250–5.317***0.6192
Coffee–3.315*0–3.353**–4.806***0.6480
Copper–1.5988–1.983–1.8050.7998
Cotton–2.9752–2.057–2.4220.8932
Hides–3.925**3–3.322**–3.527**0.8303
Jute–3.285*3–3.044*–3.125**0.9493
Lamb–3.411*4–3.442**–3.398**0.8214
Maize–5.667***0–1.817–1.5890.6954
Palm oil–4.829***0–4.082***–3.035**0.5770
Rice–4.024**7–3.546**–4.249***0.8407
Rubber–2.1140–2.212–2.883**0.5210
Silver–2.2422–2.336–4.434***0.4382
Sugar–3.923**2–3.920***–6.759***0.4492
Tea–2.0917–2.207–2.3380.8177
Timber–4.199***3–3.532**–3.470**0.9383
Tin–2.3550–2.321–2.0230.7590
Tobacco–1.4144–1.580–1.7210.9114
Wheat–4.042***4–1.503–2.912**0.6306
Wool–3.0934–1.554–1.7630.8194
Zinc–4.875***1–2.353–4.073***0.4292

Since our LM test and RALS-LM test share the same procedure when searching for the optimal lags, we only report one time to save space. k^ is the optimal number of lagged first-differenced terms. τADF, τLM and τRALS-LM denote the test statistics for the ADF test, LM test, and RALS-LM test, respectively. *, ** and *** denote the test statistic is significant at 10%, 5% and 1% levels, respectively.

Figure A.1:

Relative primary commodity prices.

Since Bjt are asymptotically negligible, we may drop these variables without a loss of generality:

Δyt=δ3D1t+δ4D2t+ut.

For tTB, we obtain

(A.2)δ˜3=1TB1t=2TBΔyt=1TB1t=2TB(δ3D1t+δ4D2t+ut)=δ3+1TB1t=2TBut, andT(δ˜3δ3)σW(λ)/λ.

Further, for rλ, by defining r*=r/λ, r*∈[0, 1], we have:

W(r)rW(λ)/λ=W(rλ)rλW(λ)/λ=λ[W(r)rW(1)],

where we define

(A.3)V1(r)W(r/λ)(r/λ)W(1)=W(r)rW(1).

Similarly, we can obtain

δ˜4=1TTBt=TB+1TΔyt=1TTBt=TB+1T(δ3D1t+δ4D2t+ut)=δ4+1TTBt=TB+1Tut,

and T(δ˜4δ4)σW(1λ)/(1λ)=σ1λW(1).

Further, for r>λ, by defining r+=(rλ)/(1–λ), r+∈[0,1], we have

W(r)(rλ)W(1λ)/(1λ)=W(r+(1λ))r+(1λ)W(1λ)/(1λ)=1λ[W(r+)r+W(1)],

where we define

(A.4)V2(r+)W((rλ)/(1λ))((rλ)/(1λ))W(1)=W(r+)r+W(1).

Combining (A.3) and (A.4), we obtain

V(r)=λV1(r/λ)for rλ,1λV2((rλ)(1λ))forr>λ.

Then, it is easy to see that

T2t=2TS˜t2σ201V(r)2dr=σ2[λ0λV(r/λ)2dr+(1λ)λ1V((rλ)/(1λ))2dr]=σ2[λ201V1(r)2dr+(1λ)201V2(r+)2dr+].

In the case of multiple breaks, we consider λι as defined in Proposition 1 and can easily show the expression for Vi(r) as:

(A.5)Vi(r)={λ1V1(r/λ1)forrλ1λ2V2[(rλ1)/(λ2λ1)]forλ1<rλ2λR+1VR+1[(rλR)/(1λR)]forλR<r1

Thus, using a common argument r we get:

T2t=2TS˜t2σ201V(r)2dr=σ2[λ10λ1V(r/λ1)2dr+λ2λ1λ2V((rλ1)/(λ2λ1)2dr++λR+1λR1V((rλR)/(1λR)2dr]=σ2i=1R+1λ1201Vi(r)2dr.

For the distribution of the test statistic, we examine regression (4) and obtain:

(A.6)ϕ˜=(S˜1MΔZS˜1)1(S˜1MΔZΔy),

where S˜1=(S˜1,..,S˜T1),ΔZ=(ΔZ2,..,ΔZT),Δy=(Δy2,..,ΔyT), and MΔZ=I–ΔZZ′ΔZ)−1ΔZ′.

It can be shown that:

(A.7)T2S˜1MΔZS˜1σ2i=1R+1λι201V_i(r)2dr.

Here, Vi(r) is the projection of the process Vi(r) on the orthogonal complement of the space spanned by the trend break function dz(λ*, r) as defined over the interval r∈[0, 1]. That is,

V_i(r)=Vi(r)dz(λ,r)δ˜,δ˜=argminδ01(Vi(r)dz(λ,r)δ)2dr.

We can show that for the second term in (A.6):

(A.8)T1S˜1MΔZΔy=T1S˜1MΔZε=T1S˜1ε_0.5σε2,

where ε=MΔZε. Combining this result with (A.7) we obtain

ρ˜=Tϕ˜12(σε2/σ2)[i=1R+1λι201V_i(r)2dr]1.

Accordingly, the limiting distribution of τ˜ is obtained as:

τ˜12ω[i=1R+1λι201V_i(r)2dr]1/2.

Now, when S˜t is divided by the fraction of each sub-sample, it is easy to see that:

T2t=2TS˜t2σ2[(1/λ1)2λ10λ1V(r/λ1)2dr+(1/λ2)2λ2λ1λ2V((rλ1)/(λ2λ1)2dr++(1/λR)2λR+1λR1V((rλR)/(1λR)2dr]=σ2i=1R+101Vi(r)2dr,

where S˜1=(S˜1,..,S˜T1) is used. Accordingly, we get:

(A.9)T2S˜1MΔZS˜1σ2i=1R+101V_i(r)2dr.

Then, it can be shown that the asymptotic distributions of τ˜ become invariant to the nuisance parameter λ as follows:

(A.10)τ˜12[i=1R+101V_i(r)2dr]1/2.

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## Supplemental Material:

The online version of this article (DOI: 10.1515/snde-2016-0050) offers supplementary material, available to authorized users.

Published Online: 2016-6-16
Published in Print: 2017-2-1

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