RALS-LM Unit Root Test with Trend Breaks and Non-Normal Errors: Application to the Prebisch-Singer Hypothesis

Transcription

1 RALS-LM Unit Root Test with Trend Breaks and Non-Normal Errors: Application to the Prebisch-Singer Hypothesis Ming Meng Department of Economics, Finance and Legal Studies University of Alabama Box Tuscaloosa, AL Junsoo Lee* Department of Economics, Finance, and Legal Studies University of Alabama Box Tuscaloosa, AL James E. Payne Dean, and Professor of Economics J. Whitney Bunting College of Business Georgia College & State University Milledgeville, GA Revised and Resubmitted February 2015 *Corresponding Author. 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 Midwest Econometrics Meetings, for their helpful comments. 1

2 RALS-LM Unit Root Test with Trend Breaks and Non-Normal Errors: Application to the Prebisch-Singer Hypothesis 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. Moreover, the RALS procedure is more powerful than the usual LM test which does not incorporate information on non-normal errors, and is free of nuisance parameters that indicate the locations of structural break. 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 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 Key Words: Prebisch-Singer hypothesis; relative commodity prices; unit root; trend break; residual augmented least squares 2

3 RALS-LM Unit Root Test with Trend Breaks and Non-Normal Errors: Application to the Prebisch-Singer Hypothesis 1. Introduction In this study, we suggest a new Residual Augmented Least Squares Lagrange Multiplier (RALS-LM) unit root test that allows for multiple trend breaks and non-normal errors. The motivation of the new approach is to utilize all possible information to maximize the power of the test as well as to make the test free of nuisance parameters that indicate the locations of structural breaks. The new test is not subject to the spurious rejection problem which occurs by assuming that breaks are absent under the null hypothesis. The RALS methodology was initially suggested by Im and Schmidt (2008). Meng et al. (2014) adopts the RALS procedure for the LM test to show that the RALS-LM test gains improved power with non-normal errors and are fairly robust to some forms of non-linearity. However, as with the other unit root tests, the RALS-LM test also loses power when existing structural breaks are not taken into account. Thus, we modified the test in Meng et al. (2014) by allowing for trend shifts while properly handling the nuisance parameter problem. For this, we consider the transformed test to eliminate the dependency on the trend break locations while employing the RALS procedure in the presence of trend-shifts. Thus, we employ the most general and powerful test that utilize information on all major factors. As an application of the RALS-LM unit root test with trend breaks, we examine the Prebisch-Singer hypothesis (PSH) which postulates a secular decline in commodity prices relative to manufactured goods in the long-run (Prebisch, 1950; Singer 1950). Section 2 discusses the econometric methodology. The empirical application to the Prebisch-Singer hypothesis is presented in Section 3 with concluding remarks given in Section 4. 3

4 2. Econometric Methodology To begin, we consider the following data generating process (DGP) based on the unobserved component representation:, (1) where in the usual unit root test. The unit root null hypothesis is. Similar to the model C in Lee and Strazicich (2003), a more general model which allows for both levelshift and trend-shift can be described with, where and are the dummy variables denoting the positions of the th level and trend breaks, respectively. Specifically, we have for,, and zero otherwise; for and zero otherwise; is the maximum number of structural breaks where we set = 2; and is the location of the th structural break. For simplicity, we first assume the information about the structural breaks to be known a priori, and we will also adopt a procedure to estimate them. Following the LM (score) principle, the unit root test statistic is then obtained from the regression:, (2) where is the LM de-trended series, which is calculated using ; is the coefficient in the regression of on ; and and are the first difference of and, respectively. Let be the -statistic testing the null hypothesis from (2). 1 As shown in Lee and Strazicich (2003), in the presence of trend-breaks, will depend on the location 1 See Schmidt and Phillips (1992) for the derivation of the LM unit root test. Following the LM procedure, equation (2) amounts to the score vector of the maximum likelihood estimation. Testing for in (2) with (or with ) leads to testing for in (1). 4

5 parameter,, which denotes the fraction of the th sub-sample in each regime such that,,, and. In such cases, it will be difficult to combine the usual LM test with the RALS procedure which will induce a new parameter. Indeed, the dependency of tests on the nuisance parameter in the trend-shift models has been an issue in the literature. As such, we consider a simple transformation which can make the unit root test statistic free of the dependency on the break location as in Im, Lee and Tieslau (2014). The following transformation can remove the dependency on the nuisance parameter: (3) where is the untransformed series and is the transformed series. We then replace in the testing regression (2) with such that we have a new testing regression, and denote as the -statistic testing the null hypothesis. Then, the asymptotic distributions of the test statistic will be invariant to the nuisance parameter., (4) where is the projection of the process on the orthogonal complement of the space spanned by the trend function defined over the interval, where 5

6 , and is a Wiener process for. For a detailed proof, see Im, Lee and Tieslau (2004) and the Appendix. Note that this result is different from those in Lee and Strazicich (2003, equation A-13, p. 1089), who show that the distribution of the usual (untransformed) test depends on the projection. But, the transformed test depends on the projection which is free of the location parameter,. Following the transformation, the asymptotic distribution of the number of trend breaks, since the distribution is given as the sum of depends only on independent stochastic terms. In general, the distribution of with the structural breaks evenly distributed, or is the same as that of the untransformed test. Therefore, we do not need to simulate numerous critical values at all possible break point combinations. The critical values of are reported in Lee et al. (2012), but instead of using this test, we move on to the next step. To improve the power of the LM test, we adopt the procedure to utilize the information on non-normal errors. We adopt the residual augmented least squares (RALS) method as in Im et al. (2014). The RALS procedure augments the following term to testing regression (2)., (5) where is the OLS residual from regression (2), and. To capture the information of non-normal errors, we let second and third moments of. Then, letting, which involves the, the augmented term can be given as. (6) 6

7 The first term in is associated with the moment condition, which is the condition of no heteroskedasticity. The second term in improves efficiency unless, where. This condition improves the efficiency of the estimator of when the error terms are not symmetric. In general, knowledge of higher moments is uninformative if, known as the redundancy condition. The normal distribution is the only distribution that satisfies the redundancy condition. Thus, the above condition does not lead to efficiency gains when the error terms are normal and thus symmetric. However, if the distribution of the error term is not normal, the condition is not satisfied. In such cases, one may increase efficiency by augmenting the testing regression with. That is, the transformed RALS-LM test statistic with trend-breaks is obtained from the regression (7) We denote the corresponding -statistic for as. One may relate equation (2) to this regression with, where is uncorrelated with, as proved in Li and Lee (2015). Then, as shown in Im and Schmidt (2008), we have, since. Thus,, implying that the variance of the error term in (7) is smaller than that in (2). This result will yield to the asymptotic efficiency gain (thus, increase in power of the test) with non-normal errors. The asymptotic distribution of Meng et al. (2014). can be easily derived from the result in Meng et al. (2014). Specifically, it is given as, (8) 7

8 where reflects the relative ratio of the variances of two error terms such that. Meng et al. (2014) showed that the asymptotic distribution of the untransformed RALS-LM test not allowing for breaks ( ) is given as. The asymptotic distribution of the transformed RALS-LM test with trend-shifts ( ) is the same except that the first term is replaced with, which is given in equation (4). 2 The point is that the asymptotic distribution of the transformed RALS-LM test statistic with trend breaks ( ) no longer depends on the break location parameters. Thus, this result paves the way for us to employ the RALS procedure in the presence of trend-shifts. Since our newly suggested test with trend-shifts becomes free of all nuisance parameters, the critical values are tabulated and provided in Table 1 for 1 and 2, 50, 100, 300 and 1000, and 0 to [Insert Table 1 here] Next, we briefly examine the property of the new test using different combinations of non-normal errors and break locations. The DGP is given as the form of (1). The initial values is assumed to be generated from the N(0,1) distribution. We consider the following nonnormal errors: (i - iv) distribution with 1 to 4 and (v - vii) -distribution with 2 to 4. We also include the size and power for the case of (viii) standard normal. In addition, we 2 Note that this limiting distribution is similar to the distribution of the unit root test with the stationary covariates of Hansen (1995). If the usual Dickey-Fuller (DF) test were used instead of, the limiting distribution would be identical. However, this study considers a new test with trend breaks and non-normal errors, which were not considered in the DF and Hansen s tests. 3 All the critical values are simulated through Monte Carlo simulations with WinRATS 8.2 using 100,000 iterations. The codes are available upon request. 8

9 examine the untransformed RALS-LM test ( ), transformed LM test ( ), untransformed LM test ( ), and DF test for comparison. These are obtained as the usual t- statistic on = 0 in the corresponding regression. Specifically, and are obtained from (2) and and are given from (7), where ( ) is used for the untransformed (transformed) tests in these regressions. All simulation results are calculated using 10,000 simulations for sample size The size (frequency of rejections under the null when in the DGP) and power (frequency of rejections under the alternative when and in the DGP) of the tests are examined using their corresponding 5% critical values. The finite sample size and power property for the transformed RALS-LM test and comparing tests for 100 and 1 are presented in Table 2. [Insert Table 2 here] We consider DGP with two different break locations with 0.25 and 0.5, but we use only one set of critical values simulated using 0.5. For all five tests examined, we see little size distortion whether 0.25 or 0.5. For the power property, we observe a large power gain when the RALS procedure is used relative to non-rals type tests. When the non-normal error becomes normal (degrees of freedom become larger for the distribution and - distribution), the power gain is less. To the contrary, the transformation procedure suggests the test is a little less powerful than the un-transformed test due to the loss of degrees of freedom. 4 The maximum number of structural breaks ( ) is recommended to be 2 considering most time series available for testing for unit root are usually short. This is similar to the recommendations in the Lee and Strazicich (2003) and Lee et al. (2012). 9

10 However, since the LM type of unit root test statistics are dependent on the trend breaks, the transformation procedure is required for validity of the test. Fortunately, for all the non-normal distributions of the error term, the power gain from the RALS procedure is much higher than the power loss from the transformation procedure. Thus, we achieve improved power by using the information on non-normal errors, but without using any specific forms of nonlinear functions. 3. Empirical Application We apply the new test to examine the Prebisch-Singer hypothesis (PSH), which postulates a secular decline in commodity prices relative to manufactured goods in the long-run (Prebisch, 1950; Singer, 1950). The basis for the declining trend can be attributed to a low income elasticity of primary commodities, productivity differentials between industrial and commodity producing countries, and asymmetric market structures. This seemingly simple issue is more complicated than it appears due to the possibility of non-stationarity of the data. One additional complication is how to deal with structural breaks. Several recent studies introduce structural breaks in testing for the PSH with allowance for endogenously determined structural breaks; see Leon and Soto (1997), Zanias (2005), and Kellard and Wohar (2006). However, Ghoshray (2011) notes the spurious rejection problem of the endogenous unit root tests and adopts the LM endogenous test of Lee and Strazicich (2003) to provide different test results; see also Harvey et al. (2010) and Kejriwal et al. (2012). The new unit root test developed in this study differs from those used in the previous studies in several important aspects. First, we use a linear unit root test with structural breaks 5 Results for T = 300 and 1,000 are similar and available upon request from authors. 10

11 instead of non-linear unit root tests. The point is that non-normality can possibly mimic some unknown forms of non-linearity indirectly in a linear model framework. Second, the new test is free of the nuisance parameters that indicate the locations of trend-shifts and spurious rejection problems. Third, the new LM test selects the proper number of breaks determined from the data. Whether or not a structural break exists is an empirical issue that must be determined from the data. We use the recently extended annual data of Grilli and Yang (1988) on prices for twentyfour primary commodity prices spanning provided by Pfaffenzeller et al. (2007). The commodities examined are aluminum, banana, beef, cocoa, coffee, copper, cotton, hides, jute, lamb, lead, maize, palm oil, rice, rubber, silver, sugar, tea, timber, tin, tobacco, wheat, wool and zinc. For each primary commodity, we deflate the nominal prices with the United Nations Manufactures Unit Value Index (MUV) as consider a model with at most two level and trend breaks.. Throughout, we The test we developed in Section 2 assumes the information regarding the structural breaks are known a priori. However, the structural breaks may actually be unknown. The testing procedure for unknown structural breaks can be summarized as follows: In the first step, we set a maximum structural break number (in this study = 2) and identify the optimal number of breaks and locations along with the optimal lags. To do this, we first choose the optimal lag by using a general to specific approach with maximum lags equal to eight, 6 for each of the model 6 The general to specific method can be explained as follows: For each combination of breaks, we start with the first eight lags, and examine the significance of the eighth lag (or ). If it is significant at 10% level (the absolute value of the t-statistics is greater than 1.645), we select eight lags as the optimal lags; if not, we try the first seven 11

12 with different break locations. Then, we determine the optimal break locations for a given number of breaks where the test is maximized. 7 If the null of no trend break is not rejected or if the null of no trend break is rejected but one of the break dummy variables is not significant based on the standard -test, we return to the first step with the structural break number equal to. This procedure continues until the break number becomes zero or all the identified break dummy variables are significant. Thus, the optimal number of breaks, locations and the optimal lags are jointly determined. 8 Then in the second step, we run the transformed LM test, obtain the residuals, and construct the higher moment condition terms to augment the regression to obtain the RALS-LM statistic. While searching for the optimal number of breaks, we use the grid search within intervals of the whole sample period so that each subsample before and after the breaks will have enough observations to perform a valid test. 9 The results using our new transformed LM and RALS-LM unit root tests are shown in Table 3. We observe that the number of rejections of a unit root hypothesis is 21 (20) out of 24 from ( ). The null hypothesis is rejected much more often from these tests than from the preliminary results where the number of rejections of a unit root using the ADF, LM test and RALS-LM test is 12, 9, and 14, respectively; these results are shown in the Appendix lags, and repeat the previous check. The searching procedure ends when the coefficient of the last lag,, is significant in which case we select the optimal lags to be ; otherwise, we select the optimal lags to be zero. 7 The critical values for the maxf test are derived under the assumption of a unit root (see Lee et al. (2012) for more details about the maxf test). One may consider an alternative testing procedure which is robust to the unit root/stationary assumption; see Harvey et. al. (2010) and Kejriwal and Perron (2010). However, our suggested procedure performs fairly comparably. The Monte Carlo simulation results provided in Lee et al. (2012) show that the maxf test has decent size and power against trend-breaks under the unit root hypothesis, and the potential power loss is mild or negligible under the stationary alternative. Thus, we apply the maxf test in our first step on the premise that the estimates are consistent both under the unit root and stationary hypotheses. 8 One may determine the location of breaks first assuming no lags, and choose the optimal lags later. We believe that this sequential procedure can be sub-optimal. 12

13 Table 1. While it appears we obtain at least as many trend stationary series when we add more trend breaks to the model, it is not necessarily so. We examine the prevalence of trends in the primary commodity prices using the optimal breaks identified in Table 3 by estimating ARMA(p,q) or ARIMA(p,d,q) models given the level and trend breaks identified in the above stated transformed RALS-LM tests. As shown in Table 4, the results show that out of the 21 trend stationary price series, 12 relative commodity prices are found to exhibit a significant negative trend, though not necessarily for the entire sample. The results using the ARIMA models for three non-stationary relative commodity price series (copper, lamb, and palm oil) display a significant negative trend or a mixture of a significant negative trend and positive trend. We also constructed a measure of prevalence of different types of trends with (-), (+), and (.), which denotes the proportion of time periods for the prevalence of a negative trend, positive trend, and trendless behavior, respectively. Out of these 15 commodities that show the prevalence of a negative trend, only 7 commodities display a negative trend for more than 50% of the sample period. The prevalence of a positive trend is found in 10 commodities, while the prevalence of trendless behavior is found in 19 commodities, of which 8 commodities (aluminum, banana, copper, cotton, lead, palm oil, silver and tin) display trendless behavior for more than 90% of the sample period. Overall, the results suggest that the trend is variable, and there is no evidence of a single negative trend. Compared with past studies, our findings provide even weaker evidence to support the PSH When using more than one break, we set the minimum length between two breaks to be at least three observations. 10 Persson and Teräsvirta (2003) and Balagtas and Holt (2009) have adopted a flexible model and estimated a number of smooth transition autoregressions (STARs). They also find limited support for the PSH. Linear trends are then estimated using OLS to connect the break points as shown in the Appendix Figure 1. 13

14 4. Concluding Remarks This study employs a newly developed RALS-LM unit root test with trend-shifts to determine whether relative primary commodity prices contain stochastic trends. Unlike the endogenous break unit root test, our LM and RALS-LM tests always include the appropriate number of trend breaks in the model. Simulation results show a power gain from the RALS procedure when the error term follows a non-normal distribution. Given this feature, the null hypothesis of non-stationarity of relative primary commodity prices is rejected much more often from these tests than from the traditional tests. Also, compared with past studies, our findings provide even weaker evidence to support the PSH. Also, we believe that in light of the significantly improved power, the newly suggested RALS-LM test with trend-shifts can be useful in other time series applications in related areas. 14

15 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, Ghoshray, A. (2011), A Reexamination of Trends in Primary Commodity Prices, Journal of Development Economics, 95, 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, Hansen, B.E. (1995), Rethinking the Univariate Approach to Unit Root Testing: Using Covariates to Increase the Power, Econometric Theory, 11, 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, Im K.S., J. Lee and M. Tieslau (2014), More Powerful Unit Root Tests with Nonnormal Errors, The Festschrift in Honor of Peter Schmidt, Springer, Im, K., J. Lee, and M. Tieslau, 2014, Panel LM Unit Root Tests with Trend Shifts, mimeo. 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, Kejriwal, M. and Perron, P. (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, Kejriwal, M., A. Ghoshray, and M. Wohar (2012), Breaks, Trends and Unit Roots in Commodity Prices: A Robust Investigation, Studies in Nonlinear Dynamics and Econometrics, forthcoming. Kellard, N.M. and M.E. Wohar (2006), On the Prevalence of Trends in Commodity Prices, Journal of Development Economics, 79,

16 Lee, J. and M.C. Strazicich (2003), Minimum Lagrange Multiplier Unit Root Test with Two Structural Breaks, Review of Economics and Statistics, 85, 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(2), Leon, J. and R. Soto (1997), Structural Breaks and Long-Run Trends in Commodity Prices, Journal of International Development, 9, Li, J. and J. Lee (2015), Improved Autoregressive Forecasts in the Presence of Non-normal Errors, Journal of Statistical Computation and Simulation, forthcoming. 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, Springer, Persson, A. and T. Teräsvirta (2003), The Net Barter Terms of Trade: A Smooth Transition Approach, International Journal of Finance and Economics, 8, 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, Prebisch, R. (1950), The Economic Development of Latin America and Its Principle Problems, New York: United Nations Publications. 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, Singer, H.W. (1950), The Distribution of Gains between Investing and Borrowing Countries, American Economic Review, 40, Zanias, G.P. (2005), Testing for Trends in the Terms of Trade between Primary Commodities and Manufactured Goods, Journal of Development Economics, 78,

17 Table 1. Critical Values of Transformed RALS-LM Test with Trend Break % Notes: denotes the sample size; denotes the break number; denotes the coefficient in equation (8). The transformation does not influence the critical values, so this table can be used on non-transformed RALS-LM test when the structural breaks are evenly distributed. When = 0, the critical values are the same as those of the standard normal distribution; when = 1, the critical values are the same as transformed LM test or nontransformed LM test with structural breaks evenly distributed in the data. 17

18 Table 2. Size and Power Property ( 100, 1) (1) (2) (3) (4) (2) (3) (4) N (0,1) Size Property Power Property Notes: denotes the coefficient for the DGP; denotes the break location which defined as, where is the break location; denotes the test statistics.,, denote the test statistic for RALS-LM test, LM test and DF test, respectively; represents the transformed test. When 0.5, the size and power for the transformed tests and untransformed tests are the same, we report them together to save space. 18

19 Table 3. Results using LM and RALS-LM Tests LM RALS LM Aluminum 6.995*** 6.732*** Banana 3.680* 4.016** NA 6 Beef 6.604*** 5.109*** Cocoa 6.793*** *** Coffee 6.056*** 8.018*** Copper Cotton 8.361*** 8.641*** Hides 7.001*** 7.496*** Jute 5.494*** 5.487*** Lamb Lead 3.450* 4.525*** NA 1 Maize 7.776*** 6.226*** Palm oil Rice 6.500*** 6.294*** Rubber 8.242*** 9.396*** Silver *** *** Sugar 7.083*** *** Tea 5.105*** 4.414** Timber 7.518*** 7.799*** Tin 5.750*** 6.162*** Tobacco 4.766** 4.268** Wheat 7.989*** 8.748*** Wool 5.640*** 4.334** Zinc 5.904*** 6.332*** Notes: is the optimal number of lagged first-differenced terms. denotes the estimated break point. *, ** and *** denote the test statistic is significant at 10%, 5% and 1% levels, respectively. 19

20 Table 4. Estimated Models with or Without Breaks, and Relative Measures of a Prevalence of a Trend Regime 1 Regime 2 Regime 3 ARMA Prevalence of a Trend Panel A. Estimated trend stationary models with breaks ( ) (+) (.) Aluminum 0.041(0.10) 0.164( 0.87) 0.011( 0.68) 2, Beef 0.004***(2.86) 0.042***(3.50) 0.02***( 3.34) 2, Cocoa 0.009**( 2.03) 0.053***(2.53) 0.005*( 1.68) 1, Coffee 0.005**(2.17) 0.018( 0.954) 0.011( 0.89) 1, Cotton 0.031( 0.28) 0.002(0.025) 0.032( 0.99) 2, Hides 0.068***(4.45) 0.076**(2.41) 0.004**( 2.09) 0, Jute 0.044( 0.42) 0.009(0.169) 0.040*( 1.86) 1, Maize 0.031(1.32) 0.012(10.29) 0.017***( 3.96) 0, Rice 0.066( 0.50) 0.008( 1.46) 0.030**( 2.14) 2, Rubber 0.037(0.24) 0.971***( 11.2) 0.027***( 4.02) 2, Silver 0.002( 1.07) 0.098***(4.73) 0.002( 0.26) 0, Sugar 0.035(1.30) 0.060( 1.06) 0.010**( 2.068) 0, Tea 0.018(0.28) 0.017(1.07) 0.024***( 2.65) 2, Timber 0.006***(3.84) 0.012(1.05) 0.002(0.36) 1, Tin 0.003(1.06) 0.010(0.59) 0.012( 1.031) 1, Tobacco 0.035***(2.71) 0.006***(4.39) 0.026***( 3.22) 2, Wheat 0.032**(2.25) 0.040***( 4.88) 0.168***( 7.40) 2, Wool 0.021(0.55) 0.014**( 2.30) 0.030***( 6.25) 0, Zinc 0.038**(2.00) 0.003(1.39) 0.027*(1.703) 0, Banana 0.002( 0.21) 0.008( 1.01) N.A. 1, Lead 0.012(0.22) ( 0.25) N.A. 2, Panel B. Estimated difference stationary models with breaks ( ) (+) (.) Copper 0.041(0.73) 0.136**( 2.01) 0.018(0.82) 0, Lamb 0.003(1.48) 0.018*( 1.82) 0.016***(3.13) 0, Palm oil 0.007( 0.42) 0.260***( 3.63) 0.006( 0.48) 2, Notes: The slope coefficients are reported for regime 1, 2 and 3. ***, ** and * denote significant at the 1%, 5% and 10% levels respectively. The final column represents the ARMA(p,q) specification. The numbers in parentheses denote the -ratios. 20

21 Appendix Appendix Table 1. Results using ADF Test and No-Break LM Unit Root Tests ADF LM RALS LM Aluminum 3.180* ** 3.593*** Banana Beef Cocoa *** Coffee 3.315* ** 4.806*** Copper Cotton Hides 3.925** ** 3.527** Jute 3.285* * 3.125** Lamb 3.411* ** 3.398** Lead Maize 5.667*** Palm oil 4.829*** *** 3.035** Rice 4.024** ** 4.249*** Rubber ** Silver *** Sugar 3.923** *** 6.759*** Tea Timber 4.199*** ** 3.470** Tin Tobacco Wheat 4.042*** ** Wool Zinc 4.875*** *** Notes: 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 the space. is the optimal number of lagged first-differenced terms., and 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. 21

22 Appendix Figure 1. Relative Primary Commodity Prices Aluminum - Banana - Beef Cocoa Coffee Copper Cotton Hides Jute 22

23 Lamb Lead Maize Palm oil - Rice - Rubber Silver 0 - Sugar Tea 23

24 Timber - Tin - Tobacco Wheat - Wool - Zinc 24

25 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 t and 0 otherwise; and D2t = 1 for t +1 and 0 otherwise. Similarly, we let DT1t* = t for t and 0 otherwise; and DT1t* = t-tb for t +1 and 0 otherwise. Then, the first step testing regression (3) can be alternatively written as: yt = B1t + B2t + D1t + D2t + ut. (A.1) Since Bjt are asymptotically negligible, we may drop these variables without a loss of generality: yt = D1t + D2t + ut. For t, we obtain = T 1 B TB-1 yt = t=2 T 1 B TB-1 ( D1t + D2t + ut) = + t=2 T B 1 TB-1 ut, and t=2 T ( - 3) W( ) /. (A.2) Further, for r, by defining r * = r/, r* [0, 1], we have: W(r) r W( ) / = W(r * ) r * W( ) / = [W(r * ) r * W(1)], where we define V1(r * ) W(r/ ) (r/ )W(1) = W(r * ) r * W(1). (A.3) Similarly, we can obtain 4 = 1 T-TB T yt = t=tb+1 1 T-TB T t=tb+1 ( D1t + D2t + ut) = + T 1 ut, T-TB t=tb+1 25

26 and T ( 4-4) W(1- ) /(1- = 1- W(1). Further, for r, by defining r + = (r- ), r + [0,1], we have W(r) (r- )W(1- )/(1- = W(r + (1- ) r + (1- W(1- )/(1- = 1- [W(r + ) r + W(1)], where we define V2(r + ) W((r- )) ((r- ))W(1) = W(r + ) r + W(1). (A.4) Combining (A.3) and (A.4), we obtain V * (r) = V1(r/ ) for r, 1- V2((r- ) for r > Then, it is easy to see that T -2 T t=2 1 V(r) 2 dr = [ 0 V(r/ ) 2 dr + (1-0 V((r- )) 2 dr] [ 1 V1(r * ) 2 dr * + (1- ) V2(r + ) 2 dr + ]. 0 In the case of multiple breaks, we consider as defined in Proposition 1 and can easily show the expression for Vi(r) as: V V ( r/ ) for r * * * 2 V2[( r 1) /( 2 1)] i () r for 1 r 2... V [( r ) /(1 )] for r 1 * R 1 R 1 R R R. (A.5) Thus, using a common argument r we get: T -2 T t=2 1 V(r) 2 dr 0 26

27 = * [ V(r/ * ) 2 * dr + 0 V((r- * )/( * - * ) 2 dr +. + R * V((r- R * )/( - R * ) 2 dr] R = R+1 1 Vi(r) 2 dr. 0 i=1 For the distribution of the test statistic, we examine regression (4) and obtain: = (S1 Z S 1) -1 (S 1 Z y), (A.6) where S 1 S 1.. S T-1, Z=( Z2,.., ZT), y=( y2,.., yt), and Z = I - Z( Z Z) -1 Z. It can be shown that: T -2 S 1 Z S 1 R+1 1 V_ i (r) 2 dr. (A.7) 0 i=1 Here, V_ i(r) is the projection of the process Vi(r) on the orthogonal complement of the space spanned by the trend break function dz(, r) as defined over the interval r [0,1]. That is, V_ i(r) = Vi (r) dz(, r), with = argmin 1 (Vi(r) dz(, r) ) 2 dr. 0 We can show that for the second term in (A.6): T -1 S 1 Z y = T -1 S 1 Z = T -1 S 1 _ 0.5 2, (A.8) where _ = Z. Combining this result with (A.7) we obtain = T - 0.5( 2 / 2 R+1 ) [ 1 V_ * 0 i(r) 2 dr] 1. i=1 Accordingly, the limiting distribution of is obtained as: 1-2 [ R+1 1 V_ * i(r) 2 dr] 1/2. 0 i=1 Now, when T -2 T t=2 is divided by the fraction of each sub-sample, it is easy to see that: [ (1/ * ) 2 * 0 V(r/ * ) 2 dr 27

28 + (1/ * ) 2 * V((r- * )/( * - * ) 2 dr +. + (1/ R * ) 2 R * V((r- R * )/( - R * ) 2 dr] R R+1 = 1 Vi(r) 2 dr, 0 i=1 where S 1 * S 1 *.. S T-1 * is used. Accordingly, we get: R+1 T -2 S 1 * Z S 1 * 1 V_i(r) 2 dr. 0 i=1 (A.9) Then, it can be shown that the asymptotic distributions of * become invariant to the nuisance parameter as follows: * 1-2 [ R+1 1 V_i(r) 2 dr] -1/2. 0 i=1 (A.10) 28