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

Ming Meng; Junsoo Lee; James E. Payne

doi:10.1515/snde-2016-0050

Published by De Gruyter June 16, 2016

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

From the journal Studies in Nonlinear Dynamics & Econometrics

https://doi.org/10.1515/snde-2016-0050

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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.

Keywords: Prebisch-Singer hypothesis; relative commodity prices; residual augmented least squares; trend break; unit root

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: D_1t=1 for t≤T_B and 0 otherwise; and D_2t=1 for t≥T_B+1 and 0 otherwise. Similarly, we let DT_1t*=t for t≤T_B and 0 otherwise; and DT_1t*=t–T_B for t≥T_B+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.

	ADF		LM	RALS-LM		k^
	τ_ADF	k^	τ_LM	τ_RALS-LM	ρ²	k^
Aluminum	–3.180*	7	–3.520**	–3.593***	0.554	7
Banana	–2.390	2	–1.547	–1.568	1.015	2
Beef	–2.049	5	–1.909	–1.503	0.537	5
Cocoa	–2.557	2	–2.250	–5.317***	0.619	2
Coffee	–3.315*	0	–3.353**	–4.806***	0.648	0
Copper	–1.598	8	–1.983	–1.805	0.799	8
Cotton	–2.975	2	–2.057	–2.422	0.893	2
Hides	–3.925**	3	–3.322**	–3.527**	0.830	3
Jute	–3.285*	3	–3.044*	–3.125**	0.949	3
Lamb	–3.411*	4	–3.442**	–3.398**	0.821	4
Lead	–2.376	1	–1.858	–1.674	0.777	0
Maize	–5.667***	0	–1.817	–1.589	0.695	4
Palm oil	–4.829***	0	–4.082***	–3.035**	0.577	0
Rice	–4.024**	7	–3.546**	–4.249***	0.840	7
Rubber	–2.114	0	–2.212	–2.883**	0.521	0
Silver	–2.242	2	–2.336	–4.434***	0.438	2
Sugar	–3.923**	2	–3.920***	–6.759***	0.449	2
Tea	–2.091	7	–2.207	–2.338	0.817	7
Timber	–4.199***	3	–3.532**	–3.470**	0.938	3
Tin	–2.355	0	–2.321	–2.023	0.759	0
Tobacco	–1.414	4	–1.580	–1.721	0.911	4
Wheat	–4.042***	4	–1.503	–2.912**	0.630	6
Wool	–3.093	4	–1.554	–1.763	0.819	4
Zinc	–4.875***	1	–2.353	–4.073***	0.429	2

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. τ_ADF, τ_LM and τ_RALS-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 B_jt are asymptotically negligible, we may drop these variables without a loss of generality:

Δyt=δ3D1t+δ4D2t+ ut.

For t≤T_B, we obtain

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

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

W(r)–rW(λ)/λ=W(r∗λ) – r∗λW(λ)/λ=λ[W(r∗)– r∗W(1)],

where we define

(A.3)V1(r∗)≡W(r/λ)–(r/λ)W(1)=W(r∗)– r∗W(1).

Similarly, we can obtain

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

and T(δ˜4−δ4)→σW(1−λ)/(1−λ)=σ1−λW(1).

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

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

where we define

(A.4)V2(r+)≡W((r−λ)/(1−λ))–((r−λ)/(1−λ))W(1)=W(r+)–r+W(1).

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

V∗(r)=λV1(r/λ) for r≤λ,1−λV2((r−λ)(1−λ)) for r>λ.

Then, it is easy to see that

T−2∑t=2TS˜t2→σ2∫01V(r)2dr=σ2[λ∫0λV(r/λ)2dr+(1−λ)∫λ1V((r−λ)/(1−λ))2dr]=σ2[λ2∫01V1(r∗)2dr∗+(1−λ)2∫01V2(r+)2dr+].

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

(A.5)Vi∗(r)={λ1∗V1(r/λ1)for r≤λ1λ2∗V2[(r−λ1)/(λ2−λ1)]for λ1<r≤λ2……λR+1∗VR+1[(r−λR)/(1−λR)]for λR<r≤1

Thus, using a common argument r we get:

T−2∑t=2TS˜t2→σ2∫01V(r)2dr=σ2[λ1∗∫0λ1∗V(r/λ1∗)2dr+λ2∗∫λ1∗λ2∗V((r−λ1∗)/(λ2∗−λ1∗)2dr+…+λR+1∗∫λR∗1V((r−λR∗)/(1−λR∗)2dr]=σ2∑i=1R+1λ1∗2∫01Vi(r)2dr.

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

(A.6)ϕ˜=(S˜1′MΔZS˜1)−1(S˜1′MΔZΔy),

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

It can be shown that:

(A.7)T−2S˜1′MΔZS˜1→σ2∑i=1R+1λι∗2∫01V_i(r)2dr.

Here, V_i(r) is the projection of the process V_i(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˜=argminδ∫01(Vi(r)−dz(λ, r)δ)2dr.

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

(A.8)T−1S˜1′MΔZΔy=T−1S˜1′MΔZε=T−1S˜1′ε_→−0.5σε2,

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

ρ˜=Tϕ˜→−12(σε2/σ2)[∑i=1R+1λι∗2∫01V_∗i(r)2dr]−1.

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

τ˜→−12ω[∑i=1R+1λι∗2∫01V_∗i(r)2dr]−1/2.

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

T−2∑t=2TS˜t2→σ2[(1/λ1∗)2λ1∗∫0λ1∗V(r/λ1∗)2dr+(1/λ2∗)2λ2∗∫λ1∗λ2∗V((r−λ1∗)/(λ2∗−λ1∗)2dr+…+(1/λR∗)2λR+1∗∫λR∗1V((r−λR∗)/(1−λR∗)2dr]=σ2∑i=1R+1∫01Vi(r)2dr,

where S˜1∗=(S˜1∗, .., S˜T−1∗)′ is used. Accordingly, we get:

(A.9)T−2S˜1∗′MΔZS˜1∗→σ2∑i=1R+1∫01V_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+1∫01V_i(r)2dr]−1/2.

References

Balagtas, J. V., and M. T. Holt. 2009. “The Commodity Terms of Trade, Unit Roots, and Nonlinear Alternatives: A Smooth Transition Approach.” American Journal of Agricultural Economics 91: 87–105.10.1111/j.1467-8276.2008.01179.xSearch in Google Scholar

Ghoshray, A. 2011. “A Reexamination of Trends in Primary Commodity Prices.” Journal of Development Economics 95: 242–251.10.1016/j.jdeveco.2010.04.001Search in Google Scholar

Grilli, E. R., and M. C. Yang. 1988. “Primary Commodity Prices, Manufactured Goods Prices, and Terms of Trade of Developing Countries: What the Long-Run Show.” World Bank Economic Review 2: 1–47.10.1093/wber/2.1.1Search in Google Scholar

Hansen, B. E. 1995. “Rethinking the Univariate Approach to Unit Root Testing: Using Covariates to Increase the Power.” Econometric Theory 11: 1148–1171.10.1017/S0266466600009993Search in Google Scholar

Harvey, D. I., N. M. Kellard, J. B. Madsen, and M. E. Wohar. 2010. “The Prebisch-Singer Hypothesis: Four Centuries of Evidence.” Review of Economics and Statistics 92: 367–377.10.1162/rest.2010.12184Search in Google Scholar

Im, K. S., and P. Schmidt. 2008. “More Efficient Estimation under Non-Normality when Higher Moments Do Not Depend on the Regressors, Using Residual-Augmented Least Squares.” Journal of Econometrics 144: 219–233.10.1016/j.jeconom.2008.01.003Search in Google Scholar

Im, K. S., J. Lee, and M. Tieslau. 2014a. “More Powerful Unit Root Tests with Nonnormal Errors.” The Festschrift in Honor of Peter Schmidt, edited by R. Sickles and W. Horrace, 315–342, Berlin, Germany: Springer Publishing Co.10.1007/978-1-4899-8008-3_10Search in Google Scholar

Im, K. S., J. Lee, and M. Tieslau. 2014b. “Panel LM Unit Root Tests with Trend Shifts.” Working Paper, University of Alabama.10.2139/ssrn.1619918Search in Google Scholar

Kejriwal, M., and P. Perron. 2010. “A Sequential Procedure to Determine the Number of Breaks in Trend with an Integrated or Stationary Noise Component.” Journal of Time Series Analysis 31: 305–328.10.1111/j.1467-9892.2010.00666.xSearch in Google Scholar

Kejriwal, M., A. Ghoshray, and M. Wohar. 2014. “Breaks, Trends and Unit Roots in Commodity Prices: A Robust Investigation.” Studies in Nonlinear Dynamics and Econometrics 18: 23–40.Search in Google Scholar

Kellard, N. M., and M. E. Wohar. 2006. “On the Prevalence of Trends in Commodity Prices.” Journal of Development Economics 79: 146–167.10.1016/j.jdeveco.2004.12.004Search in Google Scholar

Lee, J., and M. C. Strazicich. 2003. “Minimum Lagrange Multiplier Unit Root Test with Two Structural Breaks.” Review of Economics and Statistics 85: 1082–1089.10.1162/003465303772815961Search in Google Scholar

Lee, H., M. Meng, and J. Lee. 2011. “How Do Nonlinear Unit Root Tests Perform With Non-Normal Errors?” Communication in Statistics 40: 1182–1191.10.1080/03610918.2011.566972Search in Google Scholar

Lee, J., M. Meng, and M. C. Strazicich. 2012. “Two-Step LM Unit Root Tests with Trend-Breaks.” Journal of Statistical and Econometric Methods 1: 81–107.Search in Google Scholar

Leon, J., and R. Soto. 1997. “Structural Breaks and Long-Run Trends in Commodity Prices.” Journal of International Development 9: 347–366.10.1596/1813-9450-1406Search in Google Scholar

Li, J., and J. Lee. 2015. “Improved Autoregressive Forecasts in the Presence of Non-normal Errors.” Journal of Statistical Computation and Simulation 85: 2936–2952.10.1080/00949655.2014.945930Search in Google Scholar

Meng, M., K. Im, J. Lee, and M. Tieslau. 2014. “More Powerful LM Unit Root Tests with Non-Normal Errors.” The Festschrift in Honor of Peter Schmidt, edited by R. Sickles and W. Horrace, 343–357, Berlin, Germany: Springer Publishing Co.10.1007/978-1-4899-8008-3_11Search in Google Scholar

Persson, A., and T. Teräsvirta. 2003. “The Net Barter Terms of Trade: A Smooth Transition Approach.” International Journal of Finance and Economics 8: 81–97.10.1002/ijfe.198Search in Google Scholar

Pfaffenzeller, S., P. Newbold, and A. Rayner. 2007. “A Short Note on Updating the Grilli and Yang Commodity Price Index.” World Bank Economic Review 21: 151–163.10.1093/wber/lhl013Search in Google Scholar

Prebisch, R. 1950. The Economic Development of Latin America and Its Principle Problems. New York: United Nations Publications.Search in Google Scholar

Schmidt, P., and P. Phillips. 1992. “LM Tests for a Unit Root in the Presence of Deterministic Trends.” Oxford Bulletin of Economics and Statistics 54: 257–287.10.1111/j.1468-0084.1992.tb00002.xSearch in Google Scholar

Singer, H. W. 1950. “The Distribution of Gains between Investing and Borrowing Countries.” American Economic Review 40: 473–485.Search in Google Scholar

Zanias, G. P. 2005. “Testing for Trends in the Terms of Trade between Primary Commodities and Manufactured Goods.” Journal of Development Economics 78: 49–59.10.1016/j.jdeveco.2004.08.005Search in Google Scholar

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

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

Abstract

Acknowledgments

Appendix

Proof of the asymptotic distribution of the transformed LM tests

References

Supplemental Material:

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