Unit roots, flexible trends and the Prebisch-Singer hypothesis

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1 XXXII Encuentro de Economistas - Banco Central de Reserva del Perú Unit roots, flexible trends and the Prebisch-Singer hypothesis Mariella Marmanillo and Diego Winkelried winkelried_dm@up.edu.pe Universidad del Pacífico November 1

2 Outline 1. Motivation. Unit root econometrics 3. Testing for unit roots. Prebisch-Singer hypothesis 5. Closing remarks M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 / 8

3 Motivation Two theories (hypotheses) on the dynamics of primary commodity prices: 1 The Prebisch-Singer hypothesis (Cuddington, Ludema and Jayasuriya, 8): relative prices of primary commodities in terms of manufactures are driven by a secular downward trend. The Supercycle hypothesis (Cuddington and Jerrett, 8): long swings in primary commodity markets driven by the surge of industrial economies. Testing the validity of the Prebisch-Singer hypothesis has always been a subject of great empirical interest (Cuddington and Urzua, 1989; Cuddington, 199). A negative estimated time slope or drift of the relative price y t is taken as supportive evidence of the Prebisch-Singer hypothesis. To this end, it is necessary to decide first whether y t should be modeled as a trend stationary or as a difference stationary process. Conclusions are sensitive to the instabilities, such as structural breaks, that affect the performance of unit root tests. See Ghoshray (11) for a review. Refinements to unit root tests always find the Prebisch-Singer hypothesis to be an interesting application. This paper: Reexamination of the Prebisch-Singer hypothesis, using a new generation of unit root tests based on flexible trends. These trends may be interpreted as the long swings induced by Supercycles. M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 3 / 8

4 Outline 1. Motivation. Unit root econometrics 3. Testing for unit roots. Prebisch-Singer hypothesis 5. Closing remarks M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 / 8

5 Unit roots econometrics I Time series model for y t y t = τ(t) + u t, where Φ(L)u t = ε t. τ(t) is the trend function of y t. Define the slope as δ(t) = τ(t) τ(t 1). Objective: To model δ(t). Trend stationary model (TS): Φ(z) = contains no unit root, u t I(). In this case, E(y t ) = τ(t) and E( y t ) = δ(t). The trend function is explicitly estimated from the data in levels. Difference stationary model (DS): Φ(1) = so u t I(1). Here, E( y t ) = δ(t). The slope is directly modeled from the first differences of the data. Whether to use TS or DS depends on the results of unit root tests (H : DS model). M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 5 / 8

6 Unit roots econometrics II Usual choices τ(t) = α 1 + α t +Dummies for breaks in the intercept+dummies for breaks in the slope. Unit root tests, in general, are known to have low power (a high probability not to reject H, when it is false) if τ(t) is misspecified: 1 When τ(t) excludes important terms (critical). The leading example occurs when there are unmodeled structural breaks. In this case, the test confounds the breaks with a unit root. When τ(t) includes redundant terms (important but less critical). The leading example is the linear trend term t. Here, overfitting leads to a loss in power. Results are sensitive to whether the presence of structural breaks is considered in the alternative model, and how many breaks are modeled. This inconclusive picture is exactly what is found in the empirical literature of primary commodity prices. M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 6 / 8

7 Unit root tests under a flexible approach I Flexible trend approach (Becker, Enders and Hurn, ; Becker, Enders and Lee, 6; Enders and Lee, 1a,b): n τ(t) = α 1 + α t + k=1 ( ) πk β 1k cos T t n + k=1 ( ) πk β k sin T t. The cosine and sine terms form a Fourier expansion of an unknown and arbitrary function of t. The larger the n is, the better approximation. In practice, the instabilities brought by structural breaks are captured by the first few terms in the Fourier expansion. The unit root tests under the flexible approach use ( ) ( ) πk πk τ(t) = α 1 + α t + β 1 cos T t + β sin T t, where k is small (say, k = 1,). Controls for the effects of breaks of unknown form. Prevents overfitting. The (difficult) task of estimating the unknown break dates is changed by the (straightforward, linear) estimation of β 1 and β. M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 7 / 8

8 Flexible trends: Fourier approximations I 1. k = 1. k = k = 1. k = k = 1. k = k = 1. k = Notes: Fitted values of the function τ(t) = α 1 + α t + β 1 cos(θt) + β sin(θt), where θ = πk/t. The red (continuous) line shows the approximation for a given k = 1, whereas the blue (dotted) line gives the best single-frequency approximation by estimating k. M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 8 / 8

9 Flexible trends: Fourier approximations II k = 1. k = k = 1. k = k = 1. k = k = 1. k = Notes: Fitted values of the function τ(t) = α 1 + α t + β 1 cos(θt) + β sin(θt), where θ = πk/t. The red (continuous) line shows the approximation for a given k = 1, whereas the blue (dotted) line gives the best single-frequency approximation by estimating k. M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 9 / 8

10 Unit root tests under a flexible approach II Augmented Dickey-Fuller type test: ( ) ( ) πk πk y t = c 1 + c t + c 3 cos T t + c sin T t + ρy t 1 + The critical values depend on whether c =, and on k. p c +i y t i + error t. i= Interpretation: H : y t is DS, H 1 : y t is TS around a flexible trend, which can be rationalized by the Supercycle notion. Test for nonlinearities. F test for β 1 = β = under H. The critical values depend on whether α 1 =, and on k. These critical values are much larger (more than double) than those coming from an F distribution. The purpose is to enchance the power of unit root tests. Various simulation studies in Becker, Enders and Hurn (), Becker, Enders and Lee (6) and Enders and Lee (1a,b) support this conclusion. M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 1 / 8

11 Outline 1. Motivation. Unit root econometrics 3. Testing for unit roots. Prebisch-Singer hypothesis 5. Closing remarks M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 11 / 8

12 Data Annual data, over the period from 19 to 1 (T = 111 observations), for primary commodities (11 food, 7 nonfood and 6 metals). The dataset is the major extension of the popular Grilli and Yang (1988) dataset documented in Pfaffenzeller, Newbold and Rayner (7). The series of interest: Empirical strategy: y t = 1 log ( ) Prices of the primary commodity. Manufacturing unit value index 1 Begin with the most parameterized test (which is the least powerful). Rejections are conclusive. Upon non-rejection, use the F test to verify whether the nonlinearities are important. Rejections for the particular k selected by the F test are conclusive. 3 Upon non-rejection, evaluate whether the linear time trend term can be excluded and repeat the testing cycle. The commodity prices are classified into groups, according to the results of this procedure. M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 1 / 8

13 Group I: No unit root, weak instabilities 9 commodities. Strong UR rejection in the least powerful test (includes a linear trend) for all k (and, especially, for the relevant k pointed out by the F test). Evidence of slight nonlinearities. p k = k = 1 k = ρ t stat F stat ρ t stat F stat ρ t stat Hides Jute Maize Palmoil Rice Sugar Timber Wheat Zinc Critical values (5%) Critical values (1%) M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 13 / 8

14 Group II: No unit root, instabilities 9 commodities. UR rejection in the least powerful test for some k. Especially, strong rejection for the k supported by the F test. Evidence of nonlinearities in the underlying mean. p k = k = 1 k = ρ t stat F stat ρ t stat F stat ρ t stat Aluminum Lamb Cotton Wool Banana Tea Tobacco Beef Copper Critical values (5%) Critical values (1%) M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 1 / 8

15 Group III: No unit root, no time trend commodities. UR non-rejection in model with linear trend. This may be due to lack of power. Strong UR rejection in model without linear trend (no obvious drift in the data). p k = k = 1 k = ρ t stat F stat ρ t stat F stat ρ t stat Including a linear trend Coffee Cocoa Critical values (5%) Critical values (1%) Excluding a linear trend Coffee Cocoa Critical values (5%) Critical values (1%) M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 15 / 8

16 Group IV: Unit root commodities. Weak evidence against a unit root. However, if we were less conservative Rubber would belong to Group II, and Lead and Tin to Group III. p k = k = 1 k = ρ t stat F stat ρ t stat F stat ρ t stat Including a linear trend Rubber Lead Tin Silver Critical values (5%) Critical values (1%) Excluding a linear trend Rubber Lead Tin Silver Critical values (5%) Critical values (1%) M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 16 / 8

17 Taking stock UR rejection in (at least) out of cases (almost the probability of type I error). Previous literature is much more supportive of the unit root hypothesis. Nonlinearities. The critical value for the F test H : β 1 = β = is about 3 for I() series (it does not depend on k), and about twice as much for I(1) series (it does depend on k). Thus, the hypothesis of linearity is rejected, in order of appearance, for k = 1: Hides, Jute, Maize, Aluminum, Cotton, Wool, Banana, Tea, Tobacco, Coffee. k = : Palmoil, Rice, Sugar, Wheat, Zinc, Lamb, Beef, Copper, Cocoa, Rubber, Lead, Tin, Silver. It is not rejected only for Timber. Sharp structural breaks or supercycles? M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 17 / 8

18 Outline 1. Motivation. Unit root econometrics 3. Testing for unit roots. Prebisch-Singer hypothesis 5. Closing remarks M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 18 / 8

19 Evaluation of the Prebisch-Singer hypothesis (preliminary) After we decide whether to use a DS or a TS representation, we estimate the slope δ(t). Define PSH t = 1 if δ(t) < and PSH t = if δ(t). Standard errors by bootstrapping. The commodity prices are classified into 5 groups, according to the prevalence of the Prebisch-Singer hypothesis. M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 19 / 8

20 Group A: PSH does not hold ( prices) Lamb (PSH =, ) Silver (PSH = 1, ) Timber (PSH =, ) Copper (PSH = 53, ) Notes: Green line: Data (rescaled); Blue line: point estimate of δ(t); red (dotted) lines: confidence bounds of δ(t). In the title: PSH = (P 1,P ), where P 1 is the frequency of δ(t) < (dates marked by a hollow circle), and P is the frequency of a negative upper confidence bound (dates marked by a filled circle). M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 / 8

21 Group B: PSH barely holds ( prices) Zinc (PSH = 5, 1) Tin (PSH = 53, 19) Beef (PSH = 3, 17) Tobacco (PSH = 31, ) Notes: Green line: Data (rescaled); Blue line: point estimate of δ(t); red (dotted) lines: confidence bounds of δ(t). In the title: PSH = (P 1,P ), where P 1 is the frequency of δ(t) < (dates marked by a hollow circle), and P is the frequency of a negative upper confidence bound (dates marked by a filled circle). M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 1 / 8

22 Group C: PSH seldomly holds (7 prices) Banana (PSH =, 3) Coffee (PSH = 53, 33) Tea (PSH = 59, 3) Rubber (PSH = 1, 3) Other cases: Lead, Cocoa and Jute. Notes: Green line: Data (rescaled); Blue line: point estimate of δ(t); red (dotted) lines: confidence bounds of δ(t). In the title: PSH = (P 1,P ), where P 1 is the frequency of δ(t) < (dates marked by a hollow circle), and P is the frequency of a negative upper confidence bound (dates marked by a filled circle). M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 / 8

23 Group D: PSH holds sometimes (5 prices) Rice (PSH = 1, 51) Hides (PSH = 6, 5) Wheat (PSH = 1, 56) Wool (PSH = 66, 57) Other cases: Palmoil. Notes: Green line: Data (rescaled); Blue line: point estimate of δ(t); red (dotted) lines: confidence bounds of δ(t). In the title: PSH = (P 1,P ), where P 1 is the frequency of δ(t) < (dates marked by a hollow circle), and P is the frequency of a negative upper confidence bound (dates marked by a filled circle). M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 3 / 8

24 Group E: PSH often holds ( prices) Maize (PSH = 8, 59) Cotton (PSH = 76, 6) Sugar (PSH = 1, 63) Aluminum (PSH = 1, 69) Notes: Green line: Data (rescaled); Blue line: point estimate of δ(t); red (dotted) lines: confidence bounds of δ(t). In the title: PSH = (P 1,P ), where P 1 is the frequency of δ(t) < (dates marked by a hollow circle), and P is the frequency of a negative upper confidence bound (dates marked by a filled circle). M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 / 8

25 Outline 1. Motivation. Unit root econometrics 3. Testing for unit roots. Prebisch-Singer hypothesis 5. Closing remarks M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 5 / 8

26 Closing remarks The Supercycle story points out to the existence of very persistent cycles in primary commodity prices. Each phase of the cycle may last decades (as opposed to the common notion of business cycles ). Standard unit root tests will inevitably confound such a persistent component with the nonstationary behavior underlying a unit root. However, once we control for something that looks like a long-lasting cycle, the evidence against unit roots is much stronger. By construction, δ(t) is deterministic, and we may want to model the Supercycle as a stochastic process. This is part of our research agenda. The Prebisch-Singer hypothesis is weakened under the (complementary) notion of the Supercycle. M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 6 / 8

27 References I Becker, R., W. Enders and S. Hurn (), A general test for time dependence in parameters, Journal of Applied Econometrics, 19(7), Becker, R., W. Enders and J. Lee (6), A stationarity test in the presence of an unkwon number of smooth breaks, Journal of Time Series Analysis, 7(3), Cuddington, J. T. (199), Long-run trends in 6 primary commodity prices : A disaggregated look at the Prebisch-Singer hypothesis, Journal of Development Economics, 39(), 7-7. Cuddington, J. T. and D. Jerrett (8), Super cycles in real metals prices?, IMF Staff Papers, 55(), Cuddington, J. T., R. Ludema and S. A. Jayasuriya (7), Prebisch-Singer redux, in Lederman, D. and W. F. Maloney (eds.), Natural Resources: Neither Curse nor Destiny, Standford Univesity Press, ch. 5, Cuddington, J. T. and C. M. Urzua (1989), Trends and cycles in the net barter terms of trade: A new approach, Economic Journal, 99(396), 6-. Enders, W. y J. Lee (1a), A unit root test using a Fourier series to approximate smooth breaks, Oxford Bulletin of Economics and Statistics, 7(), Enders, W. y J. Lee (1b), The flexible Fourier form and Dickey Fuller type unit root tests, Economics Letters, 117(1), M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 7 / 8

28 References II Ghoshray, A. (11), A reexamination of trends in primary commodity prices, Journal of Development Economics, 95(), -51. Grilli, E. and M. C. Yang (1988), Primary commodity prices, manufactured goods prices, and the terms of trade of developing countries: What the long run shows, World Bank Economic Review, (1), 1-7. Pfaffenzeller, S., P. Newbold and A. Rayner (7), A short note on updating the Grilli and Yang commodity price index, World Bank Economic Review, 1(1), M. Marmanillo and D. Winkelried (UP) Prebisch-Singer hypothesis November 1 8 / 8

AJAE Appendix: The Commodity Terms of Trade, Unit Roots, and Nonlinear Alternatives

AJAE Appendix: The Commodity Terms of Trade, Unit Roots, and Nonlinear Alternatives Joseph V. Balagtas Department of Agricultural Economics Purdue University Matthew T. Holt Department of Agricultural