Leads and Lags in Commodity Prices: An Application of Cross-Spectral Analysis Moshe Buchinsky Division Working Paper No. 1987-7 August 1987 International Commodity Markets Division international Economics Department The World Bank Division Working Papers report on work in progress and are circulated to stimulate discussion and comment. LEADS AND LAGS IN COMMODITY PRICES: AN APPLICATION OF CROSS-SPECTRAL ANALYSIS Moshe Buchinsky August 1987 The World Bank does not accept responsibility for the views expressed herein which are those of the author and should not be attributed to the World Bank or to its affiliated organizations. The findings, interpretations, and conclusions are the results of research supported by the Bank; they do not necessarily represent official policy of the Bank. The designations employed, the presentation of material, and any maps used in this document are solely for the convenience of the reader and do not imply the expression of any opinion whatsoever on the part of the World Bank or its affiliates concerning the legal status of any country, territory, city, area, or of its authorities, or concerning the delimitation of its boundaries, or national affiliation. i C (K Y-z TABLE OF CONTENTS SIUMMARY..........# * ......... ....e *0 eoae eaeoe gee a aa a.. . ................. * * .e.. v I.* INTRODUCTION c.................c.........e e o eeZaZ*e*o .. .. .. . .. . . o .*.. *.e ..lX ¢eee II. AN INTUITIVE EXPLANATION OF SPECTRAL ANALYSIS.e.................. e ...... 3 III. US CONSUMER PRICE INDEX-A/NALYSIS ........................... *...... .... 5 IV. APPLICATION OF SPECTRAL ANALYSIS TO GRAINS, SOYBEANS AND BEEF PRICES.-11,:), Gh- 1 . A ; 1: t^^ '- -' Beef Price ........... ...e.... * 6 a a .e.... * *. . *. a a a o o * * * . . . . * a a a a a a 12 Maize Price ......... a.. . . ... e*e.o..... .a...... ea*e............ 215 Soybean Price. . a a a a a a a. . . a a a a a a a a a a a a a a e a a a a a a a a e a a a a a a a a a a .21 Wheat Prices .... o.... e. . . . . . o .......e..... 00 vv eo o ............@e ee ee o.......... g 24 Riea Pres.........aaa.aaa .a.aaa ............3 Rice Pri ce . a a ... a a a a a a e a a a a a a a a a a a gao e. e.. . a a a a a a a a a a a ea.a 30 V. CROSS SPECTRAL ANALYSIS OF COMMODITY PRICESa_eaaaaaaaa ao aea.et....*.. 35 Relationships between Maize, Palm Oil and Soybean Prices.. ....... 40 Rice and Wheat Prices O....... a a a a a a a a a a a a a a. a a a a a a a . a a a a a a a a a a a a a a a o48 Maize and Wheat Prices ...........e ee............ ......... . 48 VI. SIMULATION AND FORECAST OF COMMODITY PRICE SERIES.L .. . . a .ee aaaaae 57 Maize Price Simulations and Forecasts...ao ..a........ a a a a e....a.a..a a 58 Beef Price Simulations and Forecasts .................................. e 60 Soybean Price Simulations and Forecasts, . .a....aa *...... . a,*..60 Wheat Price Simulations and Forecasts.. e ..a....... e e ee.....a.... ... -66 Rice Price Simulations and Forecasts.. a.a..a.aa. a. a. a, a. a-aaaa a,... .*66 READINGS . ....... . .. .. o e a a a a e a a a a a a a a a a ...... . a a a a a a a a a a a a .. ....... .. 71 Table 1 Major Frequencies and Parameter Estimates for USA CPI Growth Rate ......7 2 Major Frequencies and Parameter Estimates for Beef Prices a....ee .c .13 3 Major Frequencies and Parameter Estimates for Maize Prices ...... . 17 4 Major Frequencies and Parameter Estimates for Soybean Pricesaaaa..,.ee22 5 Major Frequencies and Paramieter Estimates for Wheat Prices ............ 26 6 Major Frequencies and Parameter Estimates for Rice Prices. ......... o.31 7a Beef, Maize Cross-Spectral Analysis Results........*..a...a..a..a.. P..a36 7b Beef, Maize Cross-Spectral Analysis Results ..........................a37 7c Beef, Maize Cross-Spectral Analysis Results...............ae38 8a Soybean, Maize Cross-Spectral Analysis Results......................e.41 8b Soybean, Maize Cross-Spectral Analysis Resultsa..................a...42 8c Soybean, Maize Cross-Spectral Analysis Results...o.....a..a.as.o.......a43 9a Soybean, Palm Oil Cross-Spectral Analysis Results*. ........a. a.a a45 9b Soybean, Palm Oil Cross-Spectral Analysis Results* .............* ....46 9c Soybean, Palm Oil Cross-Spectral Analysis Resultso .............g.....47 - iv - lOa Rice, Wheat Cross-Spectral Analysis Results ........... . .. .49 lOb Rice, Wheat Cross-Spectral Analysis Results ............... 50 lQc Rice, Wheat Cross-Spectral Analysis Results........................... 51 lla Maize, Wheat Cross-Spectral Analysis Results.......................... 53 llb Maize, Wheat Cross-Spectral Analysis Results .......................... 54 llc Maize, Wheat Cross-Spectral Analysis Results .......................... 55 Figure la USA CPI Growth Spectrum (Frequency Domain) ........... e............. 8 *e8 lb USA CPI Growth Spectrum (Time Domain) ...................e D.ee...8 2a Beef Spectrum (Frequency Domain) .. ................. .... .14 2b Beef Spectrum (Time Domain).0..................... 3a Beef Coherency with USA CPI (Frequency Domain)ain...................16 3b Beef Coherency with USA CPI (Time Domain)...................... e ...... 16 4a Maize Spectrum (Frequency Domain)..................... ....... .18 4b Maize Spectrum (Time Domain) ................ ................18 5a Maize Coherency with USA CPI (Frequency Domain) ......................20 5b Maize Coherency with USA CPI (Time Domain) .. ........... ..... 20 6a Soybean Spectrum (Frequency Domain).............................. 23 6b Soybean Spectrum (Time Domain) ...... ........................ .....0..23 7a Soybean Coherency with USA CPI (Frequency Domain) ..................... 25 7b Soybean Coherency with USA CPI (TimeDomain)..............0000...25 8a Wheat Spectrum (Frequency Domain) ......................... 27 8b Wheat Spectrum (Time Domain) ............................... .. . 0.27 9a Wheat Coherency with USA CPI (Frequency Domain).................29 9b Wheat Coherency with USA CPI (Time Domain) .. ..................... 29 lOa Rice Spectrum (Frequency Domain) ...... ............................ 32 lOb Rice Spectrum (Time Domain) ................................. 32 lla Rice Coherency with USA CPI (Frequency Domain) ........................ 34 llb Rice Coherency with USA CPI (Time Domain) ........................... 34 12 Maize Price Actual and Simulated (2 & 3 Frequencies).................59 13 Maize Price Actual and Simulated (6 & 12 Frequencies) ................ 59 14 Beef Price Actual and Simulated (2 & 3 Frequencies) ..........e. 61 15 Beef Price Actual and Simulated (6 & 9 Frequencies) ................... 61 16 Beef Price Linked to Maize (2 & 3 Frequencies) ........................ 26 17 Beef Price Linked to Maize (6 & 12 Frequencies) .........0........ 62 18 Soybean Price Actual and Simulated (2 & 5 Frequencies) ................64 19 Soybean Price Actual and Simulated (9 & 13 Frequencies) ............... 64 20 Soybean Price Linked to Maize AND Palm Oil (3 * 7 Frequencies)*.......65 21 Soybean Price Linked to Maize AND Palm Oil (10 & 15 Frequencies)......65 22 Wheat Price Actual and Simulated (2 & 5 Frequencies).............. 67 23 Wheat Price Actual and Simulated (8 & 11 Frequencies) , .......... 67 24 Rice Price Actual and Simulated (3 & 5 Frequencies)...................68 25 Rice Price Actual and Simulated (7 & 10 Frequencies)..................68 26 Rice Price Linked to Wheat (2 & 4 Frequencies) ................ 69 27 Rice Price Linked to Wheat (8 & 13 Frequencies)............. 69 This paper reports the results of an application of spectral analysis/ to the short- and medium-term forecastingm4f primary commodity prices.-4he study began with interest in the question of whether some primary commodity prices lead other commodity price series and, if they do, whether these rela- tionships can be used to improve forecasts of the lagging series. Interest in the identification of systematic behavior in one series which is reflected in another led to the use of spectral analysis, in particular to cross-spectral analysis, as a means of identifying frequencies common to two or more series and quantifying this interrelationship for forecasting purposes. The study identifies three categories of systematic behaviorti¸n the commodity price series: (i) behavior inherent to the commodity market itself; (ii) behavior which reflects the impact of systematic behavior in another commodity market, e.g., substitutes or complements; and (iii) systematic behavior reflecting general price movements--the aggregated result of all events in the economy. Spectral analysis is used to identify the major cycles in each of the commodity price series and the series chosen to indicate general price move- ments (US Consumer Price Index). The five commodity prices included in the analysis are beef, wheat, rice, maize and soybeans. The results from the spec- tral analysis can be summarized as follows. Beef is driven more than any other series by long-term cycles with the short-term cycles having little effect. The maize price is also driven by long-term cycles but with some influence from medium-term cycles. The effects of medium-term cycles are found to be of even greater importance in the wheat price, while the longer-term cycles are of less importance relative to the maize price. The relative importance of long- and medium-term cycles in the rice price fall in between the results for maize and wheat. Overall, these latter three series have very similar charac- teristics, which is not surprising considering the close relationships between \ the three markets. The soybean price is driven more by medium- and short-term cycles because of its dual relationships with maize and palm oil, products with markedly different characteristics. Movements in palm oil prices are mainly generated by short-term cycles. The US CPI analysis shows that, unlike commodity prices, the general price movements are driven equally by short-, medium- and long-term cycles. This result is not surprising since the CPI is a compound index of all prices and therefore reflects all events within the economy. The primary commodity prices appear to lead movements in the general price index and, therefore, while knowledge of their interrelationships is useful for understanding price behavior it provides no information for forecasting commodity prices. In the cross-spectral analysis several cross-commodity relationships were examined: (i) The impact of the maize price on the beef price. (ii) The impact of maize and palm oil prices on the soybean price. - vi (iii) The impact of the wheat price on the rice price. (iv) The relationship between wheat and maize. The choice of these relationships for analysis derived from knowledge about the interrelationships between commodity markets. The steps used in determining the lead-lag relationship between commodities were: (i) Identification of those frequencies common.to both series that have high "coherency" values (i.e., a notion akin to correlation in traditional statistical analysis). (ii) Identification of the important frequencies of the lagging series from the subset of frequencies identified in step Mi). (iii) Identification of the "phase-shift" and "time-shift," i.e., the parameters which imply the likely lag structure between the fre- quencies identified in step (ii). The results obtained from these steps were used to estimate equations that reflected the lagged relationships between commodities. The beef price was found to have a significant lagged relationship to the maize price. Short, medium and long-term cycles found in the maize price series are common to the beef price series, but with a lag. It is also obvious that the soybean price lags the maize price. The conclusions drawn about the relationship between soybean and palm oil prices depend on the period over which the analysis is done. In the 1960s it was soybeans (or, more specifi- cally, soyoil) that had a leading effect on palm oil prices. Conversely, palm oil is the leading series when the analysis is done over the 1970s and 1980s. This result is consistent with the fact that in the 1960s palm oil was a minor vegetable oil while in the 1970s its market share increased sharply. Rice was found to lag the wheat price with a very short-term response of a few months; an anticipated result because of the close competition between these food- stuffs. The pair of prices for which no firm conclusion can be drawn is maize and wheat. All that can be said from the cross-spectral analysis is that they appear to be mutually-determined. The results from the spectral and cross-spectral analysis were used to produce short- and medium-term projections for all commodity prices. The projections for price series that were found to lag other series were done in two ways: (i) based only on the important series from the single spectral analysis; (ii) based on their linkage to leading prices as well as the cycles found to be important only in the corumnodity market itself. The determined part was taken from the forecast of the leading series, and this was combined with the projection (using single spectral analysis) of the part not explained by the leading series. From the results of the analysis and the simulations of history, important leading relationships between commodity prices were identified and it is shown that these can be useful in simulating historical behavior. believed that 'these relationships can -also- improve forecasting performance. The forecast ,results, indicate that these commodity prices will increase substantially in the next two years.Q The prices have already bottomed out or are close to the bottom of the recent downswing. The 1985 US Food Security Act, which came into effect in 1986, had a strong depressing effect on all grains prices, particularly wheat and maize, which overflowed into the vegetable fats and oils markets. Any increase in wheat and maize prices is very likely to lead to increases in beef and soybeans. Forecasts based only on the inherent cycles of beef and soybeans would not take this change in wheat and maize prices into account and therefore are likely to be biased downward. The alternative forecasts for rice prices do not show major differences. This is due to the fact that wheat prices lead rice prices by only two to three months and therefore there is little new information in wheat price behavior to influence rice prices. It is worthwhile to highlight the importance of the stationarity assumption underlying this type of analysis and to give a warning about the possible results of detrending methods. It is assumed that the series are stationary after being detrended. Deviation from this basic assumption will bias the results--the longer the sample period, the greater the bias. The selection of the period for the analysis, January 1972 to May 1987, was made on the basis that stationarity held during this time. Deviation from stationarity, leading to biased results, can also be caused by the method of detrending employed. It is important to understand that the only reason for detrending a series is to decompose the stationary part of the series from the non-stationary part (the trend). Selection of an inappropriate detrending method may create cycles which do not exist in the series. A linear method was selected to detrend all series at all stages of this study, although no significant trend was observed for any of the series. bl-l -A I. INTRODUCTION 1. This paper reports the results of an application of sS3tra nays to the short- and medium-term forecasting of primary commodity prices. The use of spectral analysis arose from interest in the question o)f>i1ther some primary commodity prire series lead other series and,-- 'i they do, whether these relationships can be used to improve forecasts of the lagging series. Interest in the identification of systeinatic behavior in one series which is reflected in another led to the use of spectral analysis, in particular to cross-spectral analysis, as a means of identifying frequencies common to two series and quantifying this interrelationship. 2. 'The study posited and set aut to identify three categorief; of syste- matic behavior within the commodity price series: ti) behavior which is inherent to the commodity mark(fi. itself; (ii) behavior which reflects the impact of systematic behavior in another commodity market, e.g., substitutes or complements; and (iii) behavior which reflects movements which are common to all prices in the economy, as a result of, say, monetary policy changes. Because events in one commodity market take time to be reflected in other commodity markets, these cross-commodity effects should show up as lagged effects in the secondary market. Therefore, it is of interest to measure the size of this lag as well as the extent to which what happens in one market is reflected in another. These parameters can be utilized for forecasting purposes. ± e , . 3. The commodity prices included in the analysis are beef, wheat, rice, maize and soybeans. The pairs of price series analyzed reflect knowledge about the interrelationships between their markets. "For example, we know that wheat -2- and rice are competing foodstuffs. We know that maize and soybean meal are competing feedstuffs in beef production. And we know that soybean oil and palm oil are the major vegetable oils and that they compete in various end uses. 4. The format of the paper is as follows. The next section is a brief, non-technical description of spectral and cross-spectral analysis and its application in this study. (There is a full mathematical treatment in the Appendix.) Section III describes the results of the spectral analysis of the US Consumer Price Index, the series used to measure general price movements. Section IV present7 the results of spectral analysis of the individual primary commodity prices and for the cross-spectral analysis between the commodity price series and the US CPI. In Section V the results of the cross-spectral analysis of the interrelationships between the commodity price series are discussed. Finally, in Section VI the results are drawn together to derive simulations of the historical price series and forecasts of the series. -3- II. AN INTUITIVE EXPLANATION OF SPECTRAL ANALYSIS 5. Spectral analysis is an approach for examining time series behavior by investigating its underlying elements. Every stationary time series with cyclical components can be broken down into its generating elements which are called "frequencies." The frequencies are transformations of the time domain. Therefore, every frequency can be interpreted as a cycle of a certain length. 6. Every time series can be presented as a weighted sum of two series of trigonometric functions, cosine and sine. The squared sum of the weights. of each pair of these trigonometric functions gives the importance of a specific frequency relative to other frequencies. The series of summed coefficients gives the periodogram and a weighted (or non-weighted) transformation of the periodogram gives the spectral density function which therefore can be regarded as the distribution density function. The spectral density function is widely used in statistics to describe continuous variables--allowing identification of the major frequencies (or cycles) which are generating the observed time series. By estimating the coefficient of each pair of trigonometric functions, we can have an empirical estimator of the cycle length and its amplitude. 7. The basic approach to a single time series applies also to a complex of time series--or, irn its technical terminology, vector process. From the individual coefficients estimated for two time series there are a few other estimators that can be generated and which can give one an idea of the types of relationships between any pair of series. One important statistic that can be derived is the coherency, which, in terms of the relationship between two variables in traditional statistics, is the equivalent of the correlation - 4 - coefficient (p). The coherency is a measure of the degree to which two series are linked for each frequency. However, it does not provide information about the nature of this relationship. The Phase-Shift Statistic, the Gain Statistic, and the Cross Amplitude Statistic give information which defines, in turn, first, the leading series, second, the extent to which the lagging series reflects the cycle in the leading series and, lastly, the mutual amplitude of both series for any given frequency. 8. A mathematical derivation of spectral and cross-spectral analysis and its fundamentals is provided in the Appendix. -5- III. US CONSUMER PRICE INDEX ANALYSIS 9. Though the interrelationships involved are not well understood, it is generally agreed that most prices move, together in response to economy-wide real and monetary influences. Therefore, any individual price series should have some cyclical components similar to those in aggregate price series which reflect the aggregate effects of these monetary and real influences. Identifi- cation of these, what might be called, general price movements within the individual commodity price should make for an improved understanding of the movements of individual price series. However, their identification will not necessarily mean that they can be used in forecasting individual price series- -particularly as primary commodity prices may lead movements in the general price level. 10. The commodity prices which are analyzed in this exercise are US prices, or prices of commodities of which the United States is a very large trader. Therefore, in selecting a general price series with which to identify general price movements in the primary commodity prices it seemed most appro- priate to take price series which should largely reflect activity within the US economy. 11. Cross-spectral analysis was therefore carried out on several macro- economic prices (such as the US Consumer Price Index, the US Wholesale Price Index and the US Consumer Goods Index) to see which had most frequencies in conmron with the individual commodity price series. The US CPI was clearly superior in this respect to the other general price level series. Therefore, before going on to discuss the analysis of the commodity price series and the relationships between those prices--and between them and the general price level--I briefly describe the results of spectral analysis of the US CPI. -6- 12. The estimation of the spectral density function for the US CPI was carried out on monthly data over the period February 1957 to March 1987. The US CPI increased continuously throughout the sample period. While a trend can be observed in this series, the same behavior is not obvious in the commodity price series. This observation suggests that it is changes in the CPI growth rate that might be related to movements in commodity prices, rather than changes in the level of the CPI. Therefore, the spectral analysis was done on the growth rate rather than on the price levels to emphasize the cyclical behavior rather than the trend. The results, as can be seen from Table 1 and Figures la and lb provide some insights and conclusions. It can be observed that the first three frequencies and the fifth and sixth frequencies are of great importance. These high-level spectrum values correspond to the seasonal variation (periods that divide by 12). 13. In addition, there are peaks at the frequencies corresponding to periods of 90 months, 12 months, 51 month, 40 months, and 4 months, in that order of importance. An intuitive explanation for the 51-month and 90-month cycles that suggests itself is that it is related to the "presidential" or four-year cycle in the United States. The significant contribution of the 12- month cycle emphasizes the conclusion drawn about the first few cycles and the importance of the seasonality factor. These results suggest that the CPI is driven mainly by short- and medium-term cycles. This fact can be seen more clearly from the relatively higher importance of the 40-month and 32-month cycles, which correspond to the business cycle in different segments of the period over which this analysis has been done. -7- TABLE 1: MAJOR FREQUENCIES AND PARAMETER ESTIMATES FOR USA CPI GROWTH RATE -- --- ----- ------ ----- ------- -------- -------- ------------. --- --- ----- ----------------------- ------------------- CONTRIBUTION SPECTRUM COEFFICIENTS PHASE NO. FREQUENCY PERIOD PERIODOGRAM % CUMULATIVE % BASE WEIGHTED COSINE SINE SHIFT - --- ---- ------- --- ---- ------------- ---------- ----- ---------------------------------- --------- ---- ------------- 1 0.01740 361.00 7.275 19.81 19.81 0.58 0.23 -0.156 -0.127 -0.683 2 0.05221 120.33 2.472 6,73 26.55 0.20 0.19 -0.002 0,117 1.558 3 0.06962 90.25 2.093 5.70 32.25 0.17 0.17 0.043 0.099 -1.161 4 0.08702 72.20 2.011 5,48 37,73 0.16 015 0.085 -0.063 0.643 5 0.10443 60.17 1.971 5.37 43010 0,16 0.13 -0.102 0.023 0.223 6 0.03481 180,50 1.828 4.98 48.07 0.15 0.22 -0,072 0,070 0,771 7 0,52215 12.03 1,205 3,28 51.36 0.10 0.03 -0,032 0.075 1.164 8 0.12183 51.57 0.846 2.30 53.66 0.07 0,09 0.042 0.054 -0.914 9 0,19145 32,82 0.647 1.76 55,42 0,05 0.03 0.013 0.059 -1.356 10 0,15664 40.11 0.626 1,70 57.13 0,05 0,06 0,044 0.039 -0.719 11 1.58385 3,97 0.558 1,52 58.65 0,04 0,01 -0,027 -0.049 -1.063 12 0.13924 45,13 0,475 1,29 59e94 0.04 0.07 -0.030 0,042 0.955 13 0.62658 10.03 0,473 1,29 61.23 0.04 0,02 -0.046 0.022 0.436 14 0,66139 9.50 0,438 1,19 62.42 0.03 0.02 -0.007 0,049 1.424 15 0,38291 16.41 0,402 1.09 63.52 0.03 0,01 -0,001 -0,047 -1.544 16 1.30537 4.81 0.361 0,98 64,50 0,03 0.01 0,024 0.038 -1,017 17 1.13132 5.55 0.354 0.96 65.46 0.03 0,01 0.040 -0,019 0.439 18 2,85441 2.20 0,349 0.95 66.42 0.03 0,01 -0.043 0,007 0,163 19 0,71360 8,81 0.330 0.90 67,32 0.03 0,01 -0,040 0.016 0,389 20 0,90506 6.94 0.313 0.85 68,17 0.02 0.01 0.019 -0,037 1,106 21 0.50474 12.45 0.305 0.83 69,00 0,02 0,02 -0.038 0,016 0,386 22 1_60126 3,92 0.292 0.79 69,79 0.02 0.01 -0.040 -0,008 -0.191 23 2.15821 2,91 0,270 0,74 70.53 0.02 0.01 -0.026 0,029 0.844 24 0.60917 10.31 0,267 0.73 71,26 0,02 0.02 -0.027 -0,028 -0.796 25 2,52372 2,49 0,260 0.71 71,96 0,02 0.01 0.021 -0.032 0.997 26 1.20094 5,23 0,236 0.64 72.61 0,02 0,01 0.018 0.031 -1,044 27 2.45410 2.56 0.236 0.64 73,25 0,02 0.01 -0.023 -0.028 -0,890 28 2.10600 2.98 0.235 0.64 73.89 0,02 0.01 0,028 -0.023 0.674 29 1.54904 4,06 0.230 0.63 74.52 0.02 0.01 0.034 0,011 -0.319 30 0.93987 6,69 0,222 0.60 75.12 0,02 0.01 0.019 -0.030 1.005 31 118354 5,31 0,219 0.60 75.72 0.02 0.01 -0.035 -0,004 -0.113 32 0.73101 8.60 0.215 0,59 76.30 0.02 0.01 -0,021 0,027 0,902 33 2,59334 2,42 0,213 058 76.88 0,02 0.01 0,032 0.013 -0.400 34 0.45253 13,89 0,205 0.56 77,44 0.02 0,02 0.032 0,012 -0.361 35 0.83544 7.52 0,202 0.55 77.99 0.02 0.01 0.030 0,015 -0.456 36 2.00157 3.14 0,202 0,55 78.54 0.02 0,01 -0.030 0014 0.434 37 0,24367 25.79 0.200 0.54 79,09 0.02 0,01 -0.008 0,032 1.344 --------------------------------------------------------------------------O------------z------------------------ -8- FIGUE la. USA CPI GRWT SPCRU (IN FREQUENCY DOMAIN, 195701-196703) 0.22 - 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 01.02 0.0 0.2 0.4 0.6 0.7 0.9 1.1 1.3 1.4 1.6 1.8 1.9 2.1 2.3 FREQUENCY O BASE WEIGHTD FIGME lb: USA CP GRWT SPCRU (IN TIME DOMAIN 195701-198703) 0.22 0.20 0.18a 0.16a 0.14- 0.12- 0.10 0.08 0.06 0.04- 0.02- 0.00 - I 80.5 30.1 16.4 1 1.3 8.6 6.9 5.8 5.0 4.4 3.9 3.5 3.2 3.0 2.7 PERIOD 0 BASE - WEIGHTED -9- 4 14. As can be seen from Table 1 the contribution of each frequency (except for the first mentioned) to the variation in the growth rate of the US CPI is low. This result is also reflected in the flat shape of the spectral density function in Figures la and lb. As shown below, this result is in sharp contrast to the results for the commodity prices. While commodity prices appear to be mainly driven by behavior within each market, the CPI is an aggregate reflecting the contributions of many prices, with each dependent to a large degree on behavior in its own market. 15. It is important to point out that this analysis does not identify the sources for the various cycles, but rather provides a tool for forecasting the short-term future by transposing the past (or the regularities of past events) into the future. . I z I i I IV. APPLICATION OF SPECTRAL ANALYSIS TO GRAINS, SOYBEANS AND BEEF PRICES 16. Spectral analysis was carried out on the following primary commodity price series: (a) Wheat - Canadian, Western Red Spring, in store, Thunder Bay. (b) Maize - US No. 2 yellow fob Gulf ports. (c) Rice - Thai, 5% broken fob Bangkok. (d) Soybeans - US, cif Rotterdam. (e) Beef - US imported frozen boneless, fob port of entry. 17. Each of the commodity time series was initially analyzed individually regardless of any cause-and-effect relationship between the series or between them and any other series. This was done to be able eventually to answer three questions: 1. Is there behavior peculiar to a commodity market that deter- mines its own cyclical behavior? 2. Are there significant differences between the results obtained from spectral analysis of a single series as compared with the results from bivariate analysis of a particular series with other leading or lagging series? 3. How does the nature of the stochastic process of a series change when the leading or lagging effect of another series is introduced? 18. The analysis was carried out on average monthly data for the period January 1972 to March 1987. The period before 1972 can be characterized as stable with a generally smooth continuous increase of prices; from 1972 onward commodity prices were, in general, much more volatile and their variance much larger than in any other period in recent history--which raises the question of the stationarity of the underlying process. The judgement was made that the underlying process has changed since the early 1970s and that it is desirable, especially for forecasting purposes, to utilize only the data since 1972. - 12 - 19. As an extension of this point it is worthwhile to stress that the selection of the period length, as well as the selection of the period loca- tion in the time space, can be very critical to the analysis of time series and to the development of forecasts. The results obtained may be different for different time-length selections, and especially for different time-space locations. The importance of these factors for the robustness of the estimation procedure will not be discussed further in this paper. Beef Price 20. One of the most analyzed of all primary commodity price series is the beef price. The studies have resulted in numerous interpretations and conclusions about the behavior of the series. Attention has focused primarily on long-cycle behavior stemming from the nature of cattle herd dynamics. The results which are presented here coincide with the biological facts of the breeding process as well as with economic realities. 21, As can be seen from Table 2 and Figures 2a and 2b, there are twelve frequencies which explain about 95% of the variation of the beef price series. All of the major frequencies are medium- and long-term cycles (in terms of months). The two most important are of 15 and 7.5 years. The existence of these cycles can be explained in terms of the beef production cycle. When there is an increase in the market price there is a relatively long period of production adjustment before the initial change has any significant effect on the quantities supplied and therefore on the price. 22. Jarvis (1974) has described the cyclical behavior of beef production in the following way. If, for example, the price of beef rises, the first step on the part of producers is to accumulate breeding stock in order to produce more calves. Breeding stock are increased by withholding female cattle from TABLE 2: MAJOR FREQUENCIES AND PARAMETER ESTIMATES FOR BEEF PRICES CONTRIBUTION SPECTRUM COEFFICIENTS PHASE NO. FREQUENCY PERIOD PERIODOGRAM % CUMULATIVE % BASE WEIGHTED COSINE SINE SHIFT 1 0.03396 185.00 98835.6 33.27 33.27 7861.9 4320.1 -28.562 -15.896 -0.508 2 0.06793 92.50 90644.6 30.51 63.78 7210.4 3199.8 21.378 22.868 -0.819 3 0.10189 61.67 40307.8 13.57 77.34 3206.3 2819.1 -17.312 11.665 0.593 4 0.16982 37.00 15201.7 5,12 82,46 1209.2 1859.6 -12.818 i',.178 0.014 5 0.20378 30.83 10966.7 3.69 86.15 872.4 1340.9 -2.075 -10.689 -1.379 6 0.13585 46.25 8287.2 2.79 88.94 659.2 2322.5 5.623 7.614 -0.935 7 0.61134 10.28 5046.7 1.70 90.64 401.4 159.5 -3.690 6.398 1.048 8 0.57737 10.88 3744.9 1.26 91.90 297,9 154.9 3.709 -5.170 0.949 9 0.23774 26.43 3302.3 1.11 93.01 262.7 775.8 5.975 0.024 -0.004 10 0.27171 23413 1843.1 0.62 93.63 146.6 415.4 4.331 -1.081 0.245 11 0.71323 8.81 1785.0 0.60 94.23 142.0 105.3 2.760 -3,418 0.891 12 0.67926 9.25 1761.3 0.59 94.83 140.1 127.6 -3.255 2.906 0.729 13 0.30567 20.56 1583.1 0.53 95.36 125.9 247.8 2.654 3.173 -0.874 14 0.91701 6.85 1282.1 0.43 95.79 102.0 31.2 3.676 -0.588 0.159 14 - FIGtJRE 2a: BEEF SPECTRW (FREQUENCY DOMAIN) 7.0 - z 6.0 - 0 5.0 lb \ 4.0- tr 0 3.0 - 20 20- 0.0 0.03 0.20 0.37 0.54 0.71 0.88 1.05 1.22 FREQUENCY FIGIJRE 2b- BEEIF S;PEClRX (TIME tOMAIN) 8.0- 7.o - z 6.0 I.I 0 5.0 4.0 3.0 2.0 - 1.0 0.0 I o -4 185.0 30.8 1 6.8 1 1.6 8.8 7.1 6.0 5.1 PERIOD 15 sale, which means less meat coming to market despite the increase in price; in tur2n, this reduction in beef supplies will cause a further increase in the price above the initial change. It takes 2-3 years to generate a significant accumulation in breeding stock. It takes a further 1-1.5 years before the increase in breeding cattle is fully reflected in increased beef production. Altogether, a significant effect on the quantities of beef traded will not be seen before 3.5-4 years pass. It takes perhaps an additional 3 years until the change in the market is fully exploited, which means that it takes 7-7.5 years from the initial price change until the full impact is realized. 23. The next five frequencies in order of importance are of 60 months, 36 months, 30 months, 45 months, and 11 months. All of these frequencies have very close linkages to the global price movements as measured by the US CPI. The coherency graphs from the cross spectral analysis on the bivariate complex of the beef price and US CPI are presented in Figures 3a and 3b. These show that the peak points are located exactly in these frequencies. Maize Price 24. Like the beef price the maize price is dominated by medium- and long- term cycles. But it is also significantly affected by short-term cycles. As can be seen from Table 3 and Figures 4a and 4b, the most important cycles are the 90-month and 60-month cycles. The contribution of the longer, 15-year cycle is very limited compared to its contribution to the beef price. The other frequencies which have significant effects on the maize price range from 9 months to 4 years. 25. Significantly, the cycles which are simple multiples of 12 are of greater importance than the others. These frequencies are the 15-year cycle, the 5-year cycle, the 3-year cycle, the 2-year cycle, and the 12-month cycle itself. To explain the cause of these cycles we need to be able to explain the 12-month cycles, and we are presently unable to do so. - 16- FIM 3a- BEF COHEECY WITH USA OXP (FREQUENCY DOMAIN) 0.6 X 0.5 0.4 t z ix 0.3 o 0 0.2- 0.1 0.02 0.25 0.48 0.71 0.94 1.17 1 .40 1.63 1.86 2.09 2.32 2.56 2.79 3.02 FREQUENCY FIGME 3b. BEF COHERENCY WITH USA CP1 (TIME DOMAIN) 0.6 - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0.5 0.4 t 0.3- 0 0.2- 0.1 0 -1. .... ... 327.0 25.2 13.1 8.8 6.7 5.4 4.5 3.8 3.4 3.0 2.7 2.5 2.3 2.1 PERIOD TABLE 3: MAJOR FREQUENCIES AND PARAMETER ESTIMATES FOR MAIZE PRICES CONTRIBUTION SPECTRUM COEFFICIENTS PHASE NO. FREQUENCY PERIOD PERIODOGRAM % CUMUATIVE % SE WEIGHTED COSINE SINE SHIFT 1 0.06793 92.50 36633.0 35.59 35.59 2914.0 1093.2 -18.512 7.305 0.376 2 0.10i89 61.67 29597.0 28.75 64.34 2354.3 1042.6 -17.197 -4.921 -0.279 3 0.03396 185.00 15107.4 14.68 79.02 1201,7 1601.1 -12.513 -2,597 -0.205 4 0.20378 30.83 3453.0 3.35 82.38 274.7 517.4 -2.924 -5.365 -1.072 5 0.16982 37.00 2681.7 2.61 84.98 213.3 69307 3.346 -4.219 0.900 6 0.13585 46.25 2155.5 2.09 87.07 171.5 854e0 -3.357 3.469 0.802 7 0.23774 26.43 2038.0 1.98 89.05 162.1 333.4 -0.863 4.614 1.386 8 0.40756 15.42 1985.3 1.93 90.98 157.9 66.6 1.219 -0.266 9 0.33963 18.50 1438.6 1.40 92.38 114.4 85.3 3.871 -0.753 0.192 10 0.54341 11.56 951.2 0.92 93.31 75.7 46.5 1.924 -2.566 0.927 11 0.64530 9.74 880.7 0.86 94.16 70.1 31.8 -2.324 -2.030 -0.718 12 0.37359 16.82 857.7 0.83 94.99 68.2 74.4 -1.569 2.610 1.030 13 0.57737 10.88 450.1 0.44 95.43 35.8 40.6 1.303 -1.780 0.939 14 0.50945 12.33 444.5 0.43 95.86 35.4 51.4 1.958 0.985 -0.466 15 0.44152 14.23 408.3 0.40 96.26 32.5 58.5 -1.698 1.238 0.630 16 0.61134 10.28 387.7 0.38 96.64 30.8 34.7 -0.326 2.021 1.411 17 0.98493 6.38 349.4 0.34 96.98 27.8 11.6 1.519 1.212 -0.674 18 0.71323 8.81 305.9 0.30 97.27 24.3 21.3 1.723 0.583 -0.326 19 0.88304 7.12 291.9 0.28 97.56 23.2 11.3 0.474 -1.712 1.301 20 1.05286 5.97 189.9 0.18 97.74 15.1 10.3 0.030 1.433 -1.550 - 18 - FIGURE X AIZ SPECTRUM (FREQUENCY DOMAIN 5.0 - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 2.Z 2.A 2.4 a8 2.2 lu 2.0 z LIL 1.8 co 1.6 1.2 1.0 OAA 0.2 0.0-?' A) 0~.05 0.20 0.57 0.54 0.71 0.W 1 .05 1.2 FFEQUENJCY X EM - WEIGHrD FIGURE 4b LU ZE SPECTRUM C MiE DOMAIN) 2.8 - 2.6- 2.4- Z 2.2- 4)2.0- z IJ~ 1.8 1.4 icj 1.2 P910 X 1.0 0.6 0.4 0.2 0.0 - 1 8.0. 50.8 16.8 11.6 8.8 7.1 6.0 5.1 4.5 4.0 PERIOD X EASE - WYEIGKTED - 19 26. The flat shape of the spectral density function also suggests that the maize price is controlled to some degree by high frequencies (i.e. short- period cycles). The reason for that is obviously related to the fact that maize, like other crops, is planted at least once a year and sometimes more frequently. The resulting capability for making short-term adjustments in production shows up in the relative importance of the high frequencies. 27. The results from the bivariate cross-spectral analysis with the US CPI are presented in Figures 5a and 5b. The coherency series graph (see Figure 5b) peaks at six important frequencies which, in terms of period length, correspond to: 1. 3-year cycle with coherency of 0.57 2. 1-year cycle with coherency of 0.36 3. 6-month cycle with coherency of 0.35 4. 4.5-month cycle with coherency of 0.21 5. 3.5-month cycle with coherency of 0.51 6. 2.8-month cycle with coherency of 0.31 These results suggest two conclusions: a. The linkage between the maize price and the US CPI in the high fre- quency range suggests that the short-term movements of the maize price are the result of general price movements rather than being unique to the maize market. b. The relatively low coherency values for the frequencies that corres- pond to medium- and long-term cycles (cycles which are important to the determination of the maize price process) suggest that this part of the movement in the maize price may be determined more by what happens within the maize market than by what happens to general price movements. 28. There are some similarities in the cycles found to be important in the beef and maize prices. These similarities are not surprising because of the use of maize as feed for cattle. A closer look is taken at this relation- ship later in the paper. - 20 - FIGURE Sa: MAIZE COHERENCY WlE' USA CPI FRMUENY DOMIN) 0.6 - z S .. ... ... .. .. - - - ... . . 0.02 0-2 0o48 0.71 O."4 1 17 1.40 1.63 1.M 2.09 2.32 2.56 2.79 3. 2 FIRLQUENC FIGURE Sb: MAIZE COHERENCi' WITH USA CPI -nMSE DOMAINt 'A 0.3 OAA - .1 7.0 25.2 1 s.1 8 0. 47 5A 45 3 3A 3.0 2.7 25 2.3 2.1 PEIODl - 21 - Soybean Price 29. The soybean price is dominated by the same cycles as the maize price. As can be seen from Table 4 and Figures 6a and 6b, the cycles which are simple multinlications of 12 are of importance, as they are in the maize price. These are the 15-year cycle, the 5-year cycle the 2-year cycle and those cycles which are about 12 months in length. However, the contribution of the medium- term cycles is of major importance. The contribution of the three longest cycles to the total prediction power of all of the frequencies found in the maize price is 79%. If we take into account all of the cycles that are longer than 5 years, the contribution of the same number of cycles to the prediction power of the soybean price is only abouat 40%. In order to explain up to 80% of the variability in the soybean price, twice the number of frequencies have to be used as for the maize price, including shorter-term cycles which have very minor effects on the maize price. The spectral density function for the soybean price is very flat (see Figures 6a and 6b), which indicates the importance of the high frequencies (i.e. short-period cycles). The relative importance in the soybean price of cycles such as the 5-month cycle is suggestive that the soybean price is influenced by activities in other markets such as palm oil--commodities which are competitors. 30. The determination of soybean prices is complex. Soybeans can be used as soymeal or soyoil and it is the relationship between these and other pro- ducts that defines the soybean price. Soymeal may be complementary at some price ratios to maize and competitive at others whereas soyoil is competitive with other oils including palm oil. TABLE 4: MAJOR FREQUENCIES AND PARAMETER ESTIMATES FOR SOYBEAN PRICES CONTRIBUTION SPECTRUM COEFFICIENTS PHASE NO. FREQUENCY PERIOD PERIODOGRAM CUMULATIVE % BASE WEIGHTED COSINE SINE SHIFT 1 0.03396 185.00 120734.0 22.74 22.74 9603.8 4337.4 -36.045 -2.456 -0.068 2 0.10189 61.67 74339.5 14.00 36.73 5913.4 3737.8 -26.095 11.077 0.401 3 0.13585 46.25 40610.7 7.65 44.38 3230.4 3464.5 -17.738 11.154 0.561 4 0.16982 37.00 34824.1 6e56 50.94 2770.1 3099.9 1.995 -19.300 1.468 5 0.20378 30.83 33083.1 6.23 57.17 2631.6 2678.9 -0.738 -18.897 -1.532 6 0.27171 23.13 28571.3 5.38 62.55 2272.7 1957.2 -5.289 -16.760 -1.265 7 0.40756 15.42 26616.7 5.01 67.56 2117.2 990.9 14.166 9.331 -0.583 8 0.30567 20.56 20571.9 3.137 71.44 1636.4 1623.0 13.749 -5.776 0.398 9 0.06793 92.50 19396.1 3.655 75.09 1542.9 3959.9 -10.245 10.233 0785 10 0.33963 18.50 17038.8 3.21 78.30 1355.4 1367.4 10,094 -9.073 0,732 11 0.23774 26.43 16752.7 3.15 81.45 1332.6 2208.0 -12.615 4.689 0.356 12 0.64530 9.74 12714.8 2.39 83.85 1011.4 418.1 -5.900 -10.132 -1.044 13 0.44152 14.23 12280.6 2.31 86.16 976.9 809.7 -6.230 9.693 1.000 14 0.71323 8.81 719703 1.36 87.51 572.5 371.8 8.632 1.816 -0.207 15 0.54341 11.56 4931.2 0.93 88.44 392.3 486.9 -3.664 -6,315 -1.045 16 0.81512 7.71 4807.1 0.91 89.35 382.4 258.7 -4.829 5.353 0.837 17 0.61134 10.28 4320.7 0.81 90.16 343.7 412.2 -5.858 3.521 0.541 18 0.57737 10.88 4115.7 0.78 90.94 327.4 456.7 4.207 -5.177 0.888 19 0.88304 7.12 4013.6 0.76 91.69 319.3 206.6 1.986 -6.281 1.265 20 0.91701 6.85 2638.2 0.50 92.19 209.9 183.5 -3.761 -3.791 -0.789 - 23 - FIGUE X SOYBEAN SPECTRL (RE2QUENY IDOMAJN) 10.0 - 9.0 8s.0 z 7.0 z IL? 6.0 5.0 VI I -j - EI 4.0 I-^ IAJ 3.0 2.0 1.0 0Paon- .Cs 02 'a.7 0-54 0.71 0- I 0 IM 2 1.3 X BASE - WEIGWTE FIGURE 6b:(SOYEDOAIN)S E T 10.0 - nM DAI 9.0 8.0 z C P 7.0 z IL? 6.0 5.0 fc 4.0 - lu- 3.0 2.0- 1.0 10.0-WI1 I85.0 30.8 1 6.8 11.6 8.8 7.1 6i.0 5.1 4.5 PERIofl X( EASE - WEIGHTE - 24 - 31. It is obvious from the results from the maize price analysis that the longer-term cycles in the soybean price are closely related to the relation- ship between soymeal and maize. The shorter-term cycles can be related to the generally higher volatility of the oils. 32. The results of the cross-spectral analysis between the soybean price and the US CPI are presented in Figures 7a and 7b. The coherency series graph (Figure 7b) shows eight major peaks in frequencies. These correspond to: 1. 3-year cycle with coherency of 0.56 2. 11-month cycle with coherency of 0.32 3. 6,2-month cycle with coherency of 0.23 4. 4.4-month cycle with coherency of 0.29 5e 3.1-month cycle with coherency of 0.57 6. 2.7-month cycle with coherency of 0.29 7. 2.5-month cycle with coherency of 0.26 8. 2e3-month cycle with coherency of 0.39 33. The results coincide with the conclusion about the relative importance of the long-, medium- and short-term cycles of the soybean price. a. The soybean price has a stronger link to the US CPI than does the maize price. However, both linkages are fairly limited. This suggests that the general price movement can explain soy- bean price movements only partially. b. The importance of low frequencies indicates that they come from that part of the price which: (i) is not correlated with the maize price; (ii) is correlated with other prices which are more volatile, i.e. other oils; or (iii) mainly determined within the soybean market itself. Later in the paper there is a closer examination of some of these elements. Wheat Prices 34. The wheat price has a very simple cyclical structure. Very few fre- quencies explain most of its variation. Clearly, the wheat price is dominated by medium-term cycles. As can be seen from Table 5 and Figures 8a and 8b the three most important frequencies' are the 61-month cycle, the 92-month cycle and the 46-month cycle. The combined contribution of these three frequencies - 25 - FIGURE Ta: SOYBEAN COHER WITH USA CPI F REUENCY DMAIN) 0 -.. ... .. . . 0.02 0.2 0.o48 071 0.94 1-17 1.40 16 l .af 2.09 2.M 2.5 2.79 3.02 FREQUNCY FIGURE 7b: SOYBEAN COHER WITH USA CP 0.4 0.3 30.0 252 1 0.1 0.7 fi.7 5A 40 13 1A 2.0 2.7 2.5 2.3 2.1 P0R4D TABLE 5: MAJOR FREQUENCIES AND PARAMETER ESTIMATES FOR WHEAT PRICES CONTRIBUTION SPECTRUM COEFFICIENTS PHASE NO. FREQUENCY PERIOD PERIODOGRAM % CUMULATIVE % BASE WEIGHTED COSINE SINE SHIFT 1 0.10189 61.67 102780.8 42.80 42.80 8175.7 2572.3 -26.780 -19.849 -0.6378 2 0.06793 92.50 67364.1 28.05 70.85 5358.5 2603.9 -16.095 21.662 0.9313 3 0.13585 46.25 13226.5 5.51 76.35 1052.1 2196.8 -7.119 -9.608 -0.9331 4 0.03396 185.00 12077.3 5.03 8V.38 960.7 3766.3 -11.387 0.953 0.0835 5 0.16982 37.00 9919.3 4.13 85.51 789.0 1808.9 -2.981 -9.917 -1.2788 6 0.20378 30.83 8229.8 3.43 88.94 654.6 1383.1 -3.647 -8.699 -1.1738 o0 7 0.50945 12.33 4870.5 2.03 90.97 387.4 142.5 3.508 -6.352 1.0662 8 O.44152 14.23 2775.5 1.16 92.12 220.8 144.0 -4.166 -3.556 -0.7065 9 0.47548 13.21 2242.3 0.93 93.06 178.4 145.3 3.910 -2.992 0.6532 10 0.27171 23.13 2103.1 0.88 93.93 167.3 497.6 3.767 2.924 -0.6601 11 0.33963 18.50 1668.6 0.69 94.63 132.7 164.7 3.972 1.505 -0.3623 12 0.30567 20.56 1019.7 0.42 95.05 81.1 214.4 -1.276 3.065 1.1762 13 0.54341 11.56 883.7 0.37 95.42 70.3 121.1 3.077 -0.298 0.0966 14 0.78115 8.04 761.8 0.32 95.74 60.6 37.5 -2.148 -1.903 -0.7249 15 0.37359 16.82 742.8 0.31 96.05 59.1 138.8 -2.266 1.701 0.6439 16 0.74719 8.41 639.2 0.27 96.31 50.8 38.7 -2.440 0.979 0.3817 17 0.81512 7.71 637.5 0.27 96.58 50.7 35.1 -0.603 -2.555 -1.3391 18 0.23774 26.43 622.7 0.26 96.84 49.5 925.3 2.308 1.186 -0.4746 19 1.08682 5.78 578.7 0.24 97.08 46.0 18.6 0.362 -2.475 1.4257 20 0.67926 9.25 528.2 0.22 97.30 42.0 49.6 1.784 -1.590 0.7278 - 27 - FIGURE 8v WHEAT SPECTRUIW FRQUENCY DOMAIN) 7.0 j 8. 2.0 I-I i) .0 2.0 - - ' T 0.03 0.20 0.37 054 0..71 OJM 1 .c5 FFRQUENCY X BASE - WEIGXED FIGURE 8b WHEAT SPECTRUM C liME DOMAIN) 8-0 7.0 z I- 7" N 4.0 B.0 2.0 4.0 1Ri0 30.8 1 6.8 1 1...S 8. 7A 16.0 PER~IOD X E WEIGD - 28- accounts for 76%. of the predicting power of the whole set of frequencies. The significance of the 60-month cycle and the 46-month cycle can be explained by the importance of the 12-month cycle. The importance of the 12-month cycle is also emphasized by the relatively high-level spectrum value of the 37-month cycle and the 23-month cycle as well as those cycles which are of approxi- mately 12 months duration (the 12 month-cycle, the 13-month cycle and the 14- month cycle). 35. Longer-term cycles are not as important as the medium and short term cycles. The 185-month cycle contributes only 5% to the prediction power which is relatively low compared to other prices. Other important cycles are the 31- month cycle and the 10-month cycle. Altogether, the 11 most important frequencies presented in Table 5 explain about 95% of the total variation of the wheat price series. 36. Historically, the wheat price series is one of the least volatile of all agricultural products. This is reflected in the relatively high importance of the medium-term cycles. The only short-term cycles of any significance are the 12- and 18-month cycles. These can be explained by the fact that the production cycle is only 12 months and that production is important in both the northern and southern hemispheres with production in the southern hemisphere separated by a period of six months. The stability of the wheat price can be explained by the wide geographic dispersion of production, which leads to its being relatively stable and therefore price is stable even though demand is price and income inelastic. The results of the coherency analysis between the US CPI and the wheat price are presented in Figures 9a and 9b. The coherency series graph peaks at 9 points which in terms of cycle length correspond to: - 29 - FIGURE Se.- WHEAT COHRENCY WITH USA CPI F RQUENCY DDMCJN) 0.7 0-7 0.6 z I OA 02i 0 0.02 0.2 0.4a 0.71 0.4 t1.17 1 -41 16 i.8 2.09 2.32 2.5f 2.79 3.X FIGURE gb-- WHEAT COHERENCY WITH USA CPI MM DOMAaHIN 0.3 0.7 0.1 0- ig OA I )ZI 7.0 25 0 13.1 0.9 1.7 5A 4 1. 38 1A 3.0 2.7 2.5 29 2. FIGURE Gb: T COKlE EC P - 30 - 1. 41-month cycle with cohetency of 0.72 2. 13-month cycle with coherency of 0.27 3. 9.6-month cycle with coberen,cy of 0.19 4. 7.0-month cycle with coherency of 0.24 5. 4.3-month cycle with coherency of 0.33 6. 3.5-month cycle with coherency of 0.26 7. 2.8-month cycle with coherency of 0.26 8. 2.5-month cycle with coherency of 0.20 9. 2.3-month cycle with coherency of 0.20 37. However, all the coherencies except those that relate to the 41-month and the 4.3-month cycles are not significantly different from zero. It can be concluded that, unlike the maize price, the wheat price relates more to the medium-term movements of the general price level and is much l1z;ss correlated to the short-term movements. In fact, the wheat price leads the general price series by 4 to 5 months. 38. The structure of the coherency between the wheat price and the general price movements suggests that: a. The changes in the general price level are more frequent than those of wheat. b. The short term movements of the wheat price have much more to do with the wheat market itself than with general price movements, and therefore should be analyzed independently of the global price changes. Rice Price 39. Like the wheat price, variations in the rice price series can be traced by relatively few frequencies. The three most important frequencies are those of the 92-month cycle, the 85-month cycle and the 62-month cycle. These three cycles contribute about 60% of the total variability of the series. 40. From table 6 and Figures lCa and iOb the next two most important frequencies are those of the 46-month cycle and the 37-month cycle. They contribute 15% and 12% of the total variation of the rice prices respectively. Altogether these five frequencies explain more than 88% of the rice price variation. Thus, the break-down of the rice price into its generating cycles shows a balanced contribution of long-term and medium-term cycles. The other TABLE 6: MAJOR FREQUENCIES AND PARAMETER ESTIMATES FOR RICE PRICES CONTRIBUTION SPECTRUM COEFFICIENTS PHASE NO. FREQUENCY PERIOD PERIODOGRAM % CUMULATIVE % BASE WEIGHTED COSINE SINE SHIFT 1 0.06793 92.50 451679.0 20.77 20.77 35929.0 25792.2 -16.881 67.809 1.327 2 0.03396 185.00 434737.8 19.99 40.76 34581.4 30470.1 -68e544 1.267 0.019 3 0.10189 61.67 430557.8 19.80 60.56 34248.9 21988.0 -59.634 -33.143 -0.507 4 0.13585 46.25 333732.3 15.35 75.91 26546.9 20126.6 -47.218 -37.127 -0.666 5 0.16982 37.00 265048.4 12.19 88.10 21083.4 17323.6 36.396 -39.252 0.823 6 0.20378 30.83 76030.4 3.50 91.60 6047.9 13463.8 1.829 -28.611 1.507 7 0.23774 26.43 33644.8 1.55 93.14 2676.3 9325.0 19.070 0.236 -0.012 8 0.37359 16e82 29084.1 1.34 94.48 2313.5 1314.0 -7.882 -15.884 -1.110 9 0.27171 23.13 23417.3 1.08 95.56 1862.7 6004.6 11.576 10.916 -0.756 10 0.33963 18.50 15670.9 0.72 96.28 1246.5 2117.3 -5.135 11,960 1.165 11 0.47548 13.21 10160.6 0.47 96.75 808.2 601.7 0.380 -10.474 1.535 12 0.40756 15.42 10101.0 0.46 97.21 803.5 969.6 4.041 -9.637 1.174 13 0.30567 20.56 7485.0 0.34 97.56 595.4 3601.6 -7.615 -4.789 -0.561 14 0.54341 11.56 5670.2 0.26 97.82 451.0 336.7 5.180 -5.871 0.848 15 0.91701 6.85 3,922.6 0.18 98.00 312.0 123.9 4,117 -5.045 0.886 FIIGURE; 10a;!: RIClE SPlECTRUM C PREMUENCY CMIQbAN) 40.0- 35.0 z 20.0 I- hio 0.0- 0.05 0.20 0.37 0.54 0471 0o.M I . FFEQUENCY X RASE - WEIGWM FIGURE 1: CE SPECRUM C - .ME DOMAIN 40.0- z 30.0 F l ID) z 2. 22 0 20.0 ') 10D-\0 1i. .30.8 16.8 116 8.8 7.1 6.0 PERIOD X DSE - WEIGXM - 33 frequencies that have some effect on the rice price are those of the 31-month cycle, the 26-month cycle, the 17-month cycle and the 23-month cycle. Unlike the other prices analyzed here, except for wheat, the shorter-length cycles, such as the 12-month cycle, are of much lesser importance. Like wheat, rice is produced throughout the world and world production is relatively stable. 41. It seems that the "presidential" cycles affect the rice price as can be seen by the relatively high importance of the 92- and 46-month cycles. This is either because rice is a grain that is mainly traded by the United States and 'Thailand, or because rice is a commodity which is very heavily influenced by the intervention of the US government. 42. A comparison of the results of the cross-spectral analysis between the rice price and the US CPI and the results of cross-spectral analysis of other commodity prices with the general price index shows quite a difference. Unlike the other prices, the rice price has a close relationship with the US CPI--especially for the very low and very high frequencies. 43. An examination of the coherency series graph presented in Figures lla and lib shows that it peaks at 3 important points, which in terms of period length correspond to: 1. 13-14 year cycle with coherency of 0.74 2. 3.3-month cycle with coherency of 0.45 3. 2.3-month cycle with coherency of 0.40 44. The very high coherencies for the long-term cycles suggest that the long-term movements in the rice price are closely linked to the long-term movements of the general price level. The medium-term cycles are not very closely linked to the general price movements as can be seen from the low- level coherency values at these frequencies. The short-term cycles with high coherency values indicate some correlation with general price movements but it is not strong enough to explain the variation of the rice price due to the shorter term cycles. - 34 - FIGURE lla: RICE COHERENCY WIT CA CPI C FIEQUENCY DOMAIN) 0.7 0.6 0.5 1.)z 0.4 M .. .. ... ,, . - 0.0 m .2 0.48 0.71 0.94 1.17 1.40 1-.M 1.86 2.09 2.32 2.56 2-79 3.2 FUEQUE FIGURE llb: RICE COHIERENCY WITH USA CPI ,( ME MOMAIN 036 0.2 0.1 - lc 4V-- X ' OhA 0.6 0.5 0.1 0 3 327.0 25.2 13.1 8.8 6.7 5.4 4.5 3.8 3.4 3.0 2.7 2.5 2.3 2.1 PERIOD 35 - V. CROSS SPECTRAL ANALYSIS OF COMMODITY PRICES 45. In this section I examine the relationships between commodity prices. Cross-spectral analysis of the bivariate time series is used in order to answer three questions: (i) Whi ch major frequencies have a lagged effect in one series as compared to another series? (ii) What are the lagged effects in terms of: (a) the time over which the lagged series responds to changes in the leading series and (b) the size of the impact in terms of percentage changes? (iii) Given the size and nature of the relationship between series, is it possible to identify the type of relationship e.g., are they complementary or competitive. It can be noted at this point that it is not always possible to determine exactly what kinds of relationships exist without introducing additional evidence about the specific commodity markets. 46. Turning to the primary commodities being studies, the particular interrelationships of interest are the following: (i) Given that maize is an important input in beef production, what is the impact of changes in the maaize price on the beef price? What are the characteristics of the lag structure through which the. 1fe} yri - responds to changes in the price of maize? (ii) What is the nature of the effect, if any, of maize and palm oil prices on the price of soybeans? Is there anything special about the relationship between maize and soy meal (competing livestock feedstuffs) and between palm oil and soy oil (competing foodstuffs) that impact on the soybean price? (iii) What are the relationships between wheat and rice (in competition as food) and between wheat and maize (in competition as feedstuffs)? Are they simultaneously determined or is there a more complicated mechanism? Relationships Between Beef and Maize Prices 47. Tables 7a, 7b and 7c give a summary of the results obtained from the cross-spectral analysis of the beef and maize prices. While Table 7a lists the major frequencies according to the coherency values, Table 7b reorders the frequencies according to their relative importance in the series (beef). TABLE 7a: BEEF, MAIZE CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO COHERENCY VALUES) ------------ --- -------------.----------------------------------------------------------------~------------------------------------------------- PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT - GAIN N- LAGGED NO. FREQUENCY PERIOD BEEF MAIZE BEEF ZE BEEF MAIZE BEEF MAIZE COHERENCY PHASE TIME 1 ON 2 2 ON 2 VARIABLE ----- -----------------------------4------------------------------------------------------------------------------------------------------------- 1 2.2416 2.80 232e4 33.4 0.08 0.03 7e3 1.9 -0,711 -09771 0.6373 0.355 0.158 1.56 0.41 BEEF 2 2.2755 2.76 68.5 11.1 0.02 0.01 6,6 2.0 -0.347 -1.473 0.6292 0,453 0,199 1.43 0.44 BEEF 3 2.2076 2.85 69.6 8.2 0.02 0.01 7.0 1.8 -0.799 -0.735 0.5969 0,296 0.134 1.53 0.39 BEEF 4 2.3095 2,72 47,0 36.9 0.02 0.04 5.9 2.1 1,497 0.900 0.5897 0,524 0.227 1.28 0,46 BEEF 5 2,1736 2.89 119.9 36.2 0.04 0.04 6.8 1.7 0,428 0.284 0.5528 0,248 0,114 1.50 0.37 BEEF 6 2,3435 2,68 46.5 53.7 0,02 0.05 5,0 2.1 -0.807 1.350 0.5297 0,600 0.256 1.12 0.47 BEEF 7 2.1397 2,94 41.7 7.6 0,01 0.01 6,0 1.4 0.752 -0.722 0,4523 0.184 0,086 1.37 0.33 BEEF 8 2.3774 2.64 8.0 13.6 0,00 0,01 4.0 1,9 0.766 1,077 0,4406 0,692 0,291 0,95 0.46 BEEF 9 0,2377 26.43 3302,2 2038.0 1,11 1,98 775.8 333.4 -0,004 1,386 0.4323 1,309 5,504 1.00 0.43 BEEF 10 0,0340 185,00 98835,6 15107,4 33.27 14.68 4320,1 1601,1 -0.508 -0.205 0.3903 1.316 38,749 1.03 0.38 BEEF 11 0.2038 30,83 10966,7 3453.0 3.69 3.35 1340.9 517,4 -1,379 -1,072 0,3792 1.228 6.025 0.99 0.38 BEEF 12 0,1359 46.25 8287.2 2155,5 2.79 2.09 2322.5 854,0 -0,935 0,802 0,3734 1.197 8.810 1.01 0.37 BEEF 13 0,1698 37.00 15201,7 2681.7 5.12 2,61 1859.6 693,7 0,014 0,900 0,3723 1,223 7.202 1,00 0,37 BEEF 14 0,1019 61,67 40307,8 29597,0 13,57 28,75 2819.1 1042,6 0.593 -0.279 0,3572 1,136 11,150 0,98 0.36 BEEF 15 2,4114 2.61 110,5 12.2 0,04 0.01 3.4 1,8 -1.093 1.329 0,3522 0.787 0.326 0.81 0.43 BEEF 16 2,1057 2,98 17,9 22,8 0,01 0.02 5.2 1,3 -1.339 1,519 0.3384 0.081 0,038 1.15 0.29 BEEF 17 0.0679 92.50 90644,6 36633,0 30.51 35,59 3199.8 1093.2 -0.819 0,376 0.3330 1.210 17,809 0.99 0.34 BEEF 18 0,2717 23.13 1843,1 54,6 0.62 0,05 415,4 174.6 0.245 0,715 0,3188 0,975 3.589 0.87 0.37 BEEF ------------------------------------------------------------------------------------------------------------------------------------------------ TABLE 7b: BEEF, MAI ZE CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO BEEF FREQUENCIES CONTRIBUTION) PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT GAiNsN LAGGED NO. FREQUENCY PERIOD BEEF MAIZE BEEF MAIZE BEEF MAIZE BEEF MAIZE COHERENCY PHASE TIME 1 ON 2 2 ON 2 VARIABLE 1 0.0340 185.00 98835.6 15107.4 33.27 14.68 4320.1 1601.1 -0.508 -0,205 0.3903 1.316 38.749 1.03 0.38 BEEF 2 0,0679 92.50 90644.6 36633,0 30.51 35.59 3199.8 1093.2 -0.819 0,376 0.3330 1.210 17.809 0.99 0,34 BEEF 3 0.1019 61.67 40307.8 29597.0 13.57 28.75 2819.1 1042.6 0.593 -0.279 0.3572 1.136 11.150 0.98 0.36 BEEF 4 0.1698 37,00 15201.7 2681 .7 5.12 2.61 1859.6 693.7 0.014 0.900 0.3723 1.223 7.202 1.00 0.37 BEEF 5 0.2038 30.83 10966.7 3453.0 3.69 3.35 1340,9 517.4 -1.379 -1.072 0.3792 1.228 6.025 0.99 0,38 BEEF 6 0.1359 46,25 8287.2 2155.5 2.79 2.09 2322.5 854.0 -0.935 0,802 0.3734 1.197 8,810 1.01 0.37 BEEF 7 0,2377 26.43 3302.2 2038s0 1.11 1.98 775.8 333.4 -0.004 1.386 0.4323 1.309 5,504 1.00 0.43 BEEF 8 0.2717 23.13 1843,i 54,6 0.62 0.05 415.4 174.6 0.245 0.715 0.3188 0.975 3.589 0.87 0.37 BEEF 9 2.2416 2.80 232.4 33,4 0.08 0.03 7.3 1.9 -0.711 -0.771 0.6373 0.355 0.158 1,56 0.41 BEEF 10 2.1736 2.89 119.9 36,2 0.04 0.04 6.8 1.7 0.428 0.284 0.5528 0.248 0.114 1.50 0,37 BEEF 11 2,4114 2.61 110,5 12.2 0,04 0.01 3.4 1.8 -1.093 1,329 0.3522 0.787 0,326 0.81 0,43 BEEF 12 2.2076 2.85 69.6 8,2 0.02 0.01 7.0 1.8 -0.799 -0.735 0.5969 0.296 0.134 1.53 0.39 BEEF 13 2.2755 2.76 68.5 11,1 0.02 0.01 6.6 2,0 -0.347 -1.473 0.6292 0.453 0.199 1.43 0.44 BEEF 14 2.3095 2.72 47.0 36.9 0.02 0,04 5,9 2.1 1.497 0.900 0.5897 0.524 0.227 1.28 0.46 BEEF 15 2.3435 2.68 46,5 53,7 0,02 0.05 5.0 2,1 -0.807 1.350 0.5297 0.600 0.256 1.12 0.47 BEEF 16 2.1397 2.94 41.7 7,6 0,01 0.01 6,0 1.4 0.752 -0.722 0,4523 0.184 0.086 1.37 0.33 BEEF 17 2.1057 Z,98 17,9 22,8 0.01 0.02 5.2 1.3 -1,339 1,519 0.3384 0,081 0.038 1.15 0.29 BEEF 18 2.3774 2,64 8.0 13,6 0,00 0,01 4,0 1.9 0.766 1.077 0.4406 0.692 0.291 0.95 0.46 BEEF TABLE 7c: BEEF, MAIZE CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO MAIZE FREQUENCIES CONTRIBUTION) ------ ------- -- --- ---- -- -- ------ --- -- ------ --- - -- --- ------ - --- -------------- -------- --- --- ---------- --------- -------------- ---- ------ ---- --- PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT CAIN LAGGED NO. FREQUENCY PERIOD BEEF MAIZE BEEF AMIZE BEEF MAIZE BEEF MAIZE COHERENCY PHASE TIME 1 ON 2 2 ON 2 VARIABLE ------- ---- ----- ---- - ----- - ------ --- - ---- -- - ------ ---- ----- ---- - --- --------- ------------ -------- ---------- ---- --- ---- --- ----- -- - ------- - ---- 1 0.0679 92,50 90644,6 36633.0 30.51 35,59 3199,8 1093,2 -0.819 0.376 0.3330 1.210 17.809 0.99 0.34 BEEF 2 0.1019 61.67 40307.8 29597.0 13.57 28,75 2819.1 1042.6 0.593 -0.279 0,3572 1,136 11.150 098 0.36 BEEF 3 0,0340 185.00 98835.6 15107.4 33,27 14.68 4320.1 1601.1 -0.508 -0,205 0.3903 1.316 38,749 1,03 0,38 BEEF 4 0.2038 30.83 10966.7 3453.0 3.69 3.35 1346',9 517.4 -1.379 -1.072 0,3792 1.228 6,025 0,99 0,38 BEEF 5 0,1698 37.00 15201.7 2681.7 5.12 2.61 185!fe6 693.7 0,014 0.900 0,3723 1.223 7.202 1.00 0.37 BEEF 6 0,1359 46,25 8287.2 2155,5 2.79 2.09 2322.5 854,0 -0,935 0.802 0.3734 1.197 8,810 1,01 0,37 BEEF 7 0.2377 26,43 3302.2 2038.0 1,11 1,98 775.8 333.4 -0,004 1.386 0.4323 1.309 5e504 1,00 0,43 BEEF 8 0,2717 23.13 1843.1 54.6 0.62 0,05 415,4 174.6 0,245 0.715 0.3188 0.975 3.589 0.87 0,37 BEEF 9 2,3435 2,68 46.5 53,7 0,02 0.05 5.0 2.1 -0,807 1.350 0.5297 0.600 0,256 1,12 0.47 BEEF 10 2,3095 2,72 47.0 36.9 0,02 0.04 5.9 2,1 1,497 0.900 0.5897 0,524 0,227 1.28 0,46 BEEF 11 2,1736 2,89 119,9 36.2 0.04 0.04 6.8 1.7 0.428 0.284 0,5528 0.248 0,114 1.50 0.37 BEEF 12 2.2416 2.80 232.4 33.4 0.08 0.03 7,3 1.9 -0,711 -0.771 0,6373 0.355 0.158 1,56 0.41 BEEF 13 2,1057 2,98 17,9 22,8 0.01 0.02 5,2 1.3 -1,339 1.519 0,3384 0.081 0.038 1,15 0.29 BEEF 14 2,3774 2.64 8.0 13.6 0,00 0.01 4.0 1.9 0.766 1,077 0,4406 0.692 0,291 0.95 0b46 BEEF 15 2.4114 2,61 110.5 12,2 0,04 0.01 3.4 1,8 -1.093 1.329 0,3522 0,787 0.326 0.81 0.43 BEEF 16 2.2755 2,76 68.5 11,1 0.02 0.01 6,6 2.0 -0.347 -1.473 0,6292 0,453 0.199 1,43 0,44 BEEF 17 2,2076 2.85 69.6 8.2 0.02 0.01 7,0 1.8 -0.799 -0,735 0.5969 0.296 0.134 1.53 0.39 BEEF 18 2,1397 2.94 41.7 7.6 0,01 0.01 6.0 1.4 0,752 -0.722 0.4523 0.184 0.086 1.37 0.33 BEEF - ----- ----- -- - --- -- --- - --- - --------- - ---- --- ---------- ---------- - --- --- - ----- - --- --------- -- ---- - ----- --- - ----- --------- --- --- --- - --------------- - 39 - Table 7c sorts them according to their contribution to the variation in maize prices. (The frequencies for which the coherency was less than 0.3 have been dropped from the Table as well as from the discussion below as not being significant.) 48. On examination of the 18 major frequencies for which a "high" coherency was detected it is obvious that it is the beef price that lags the maize price. However, the structure of the lag is not clear-cut. Nevertheless, the results do give a hint of the structure. 49. The lag structure implied by the results of any bivariate analysis should be drawn from those frequencies which are important in the lagging series. If a frequency is not important for the lagging variable it will not have any explanatory power. 50. It can be seen from Table 7b that the most likely lag structure will have within it lags of 39 months, 18 months, 12 months, a shorter-term effect of 5 - 8 months, as well as an immediate effect. In order to estimate parameters that represent this lag structure, one can estimate regressions on the lags implied by the results and choose the regression with the highest explanatory power. Taking the contribution of each frequency to the beef price variation to calculate the weighted average lag, we estimate that the average time lag is about 22 months. It is also clear from the results that the "gain" of the beef price will be close to 1 in terms of the difference between the maximum l.evel of the beef price and the maximum level of the maize price for a given frequency. (This does not necessarily mean that the elasticity of the beef price to the maize price is 1.) - 40 - Relationships between Maize, Palm Oil and Soybean Prices 51v Soybean is a somewhat unusual commodity in the ._nse that it is consumed in three different forms: as beans, meal and oil. Therefore, the soybean price interrelates with commodities that are in completely different markets. Because of the meal and oil content in soybean (80% and 17% respec- tively), one may think that the major impact on the soybean price would come from commodities that are either complementary or substitutes to the meal component. It would have been ideal if we could have examined the impact of maize on soy meal and of palm oil on soy oil and combined the effects from both the soy meal and soy oil prices to get the soybean price as a second stage result. But lack of monthly data on soy oil prices for most of the 1970s makes this strategy impossible. Therefore, a different approach was taken; the relationships between the soybean price and maize and between soybeans and palm oil were examined separately and the results from each analysis were combined systematically to obtain the lag structure influencing the soybean price. 52. The results from the cross-spectral analysis of soybean and maize are summarized in Tables 8a, 8b and 8c. As expected, the lagging series was found to be soybeans. The first six frequencies in Table 8b show a strong leading effect of maize on soybeans. Altogether, the six frequencies explain about 87% of the maize price variation and about 61% of the soybean price variation. This suggests very strong conclusions: (a) There is a strong leading effect of maize on soybeans, which is reflected both by the fact that soybeans is found to lag for the six major frequencies and by the fact that these freq,uencies explain most of the maize price variation. (b) The fact that these six frequencies explain up to 60% of the soybean price suggests that there are cycles in the soybean price that cannot be explained by the maize price movements. TABLE 8a: SOYBEAN, MAIZE CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO COHERENCY VALUES) ------------------------------------------------------------------------------------------------------------------------------ PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT- GAIN LAGGED NO, FREQUENCY PERIOD SOYBEAN MAIZE SOYBEAN MAIZE SOYBEAN MAIZE SOYBEAN MAIZE COHERENCY PHASE TIME 1 ON 2 2 ON 2 VARIABLE ------------------------------------------------------------------------------------------------------------------------------------------------- 1 0.0340 185.00 120734.0 15107.4 22.74 14.68 4337.4 1601.1 -0.068 -0,205 0.6643 0.396 11.660 1.34 0.50 SOYBEAN 2 0.0679 92.50 19396,1 36633.0 3.65 35.59 3959.9 1093,2 0.785 0,376 0,6204 0.308 4,530 1.50 0.41 SOYBEAN 3 0.1019 61.67 74339.5 29597.0 14,00 28,75 3737,8 1042.6 0.401 -0,279 0.5974 0,312 3.065 1.46 0,41 SOYBEAN 4 0.7132 8.81 7197.3 30509 1.36 0.30 371.8 21.3 -0.207 -0.326 0.5852 -0.066 -0.092 3.19 0.18 MAIZE 5 0,7472 8,41 2085,4 7.4 0e39 0.01 326.7 16.1 1.057 -0.877 0.5619 -0.040 -0.054 3.38 0.17 MAIZE 6 0,1359 46.25 40610.7 2155.5 7.65 2.09 3464,5 854.0 0,561 0.802 0.5507 0,320 2.355 1.49 0.37 SOYBEAN 7 2.2076 2.85 351.6 8,2 0.07 0.01 28.5 1.8 -1,294 -0.735 0.5379 -0.276 -0.125 2.93 0.18 MAIZE 8 0.1698 37,00 34824.1 2681,7 6.56 2.61 3099.9 693,7 1.468 0.900 0.5367 0.291 1.715 1.55 0.35 SOYBEAN 9 0.6793 9.25 2554.5 16.4 0.48 0.02 396.1 26.2 0.605 -1.364 0,5339 -0.130 -0.191 2.84 0.19 MAIZE 10 2.2416 2.80 566.2 33.4 0.11 0,03 27.0 1.9 -1,328 -0.771 0.5247 -0.232 -0.103 2,73 0.19 MAIZE 11 3.1246 2.01 644.7 39,8 0.12 0.04 20,0 1.3 -0.130 -0.198 0.5201 -0.568 -0.182 2.80 0.19 MAIZE 12 0.6453 9.74 12714,8 880.7 2.39 0.86 418.1 31.8 -1.044 -0.718 0.5186 -0,171 -0.265 2.61 0.20 MAIZE 13 0.8151 7.71 4807.1 144.9 0.91 0.14 258.7 11,5 0.837 0.047 0,5085 0.278 0,341 3.38 0.15 SOYBEAN 14 1.3925 4.51 2297,6 128,7 0.43 0.13 78,3 4.2 0.080 0,222 0,5051 -0.028 -0,020 3.07 0.16 MAIZE 15 2.1736 2.89 548,6 36.2 0.10 0.04 30.3 1.7 -0.008 0,284 0.5042 -0.316 -0.145 3,03 0.17 MAIZE 16 0,7812 8.04 1342.0 48.4 0.25 0.05 288.4 13.0 -1.303 -0.085 0.5026 0.103 0,132 3.33 0.15 SOYBEAN 17 0.8491 7.40 517.0 120,1 0.10 0,12 218.5 10.5 -0.023 0,016 0.4958 0.376 0.443 3.21 0.15 SOYBEAN 18 ('9849 6.38 2440.3 349,4 0,46 0.34 141,7 11.6 -0.287 -0.674 0.4860 0.510 0.518 2.44 0.20 SOYBEAN 19 0.8830 7.12 401306 291.9 0,76 0.28 206.6 11.3 1.265 1,301 0.4844 0.390 0.441 2.98 0.16 SOYBEAN 20 0.5774 10.88 4115.7 450.1 0,78 0.44 456.7 40.6 0.888 0.939 0.4725 -0,101 -0.174 2.30 0.21 MAIZE 21 0.2038 30.83 33083.1 3453.0 6.23 3.35 2678.9 517.4 -1,532 -1.072 0.4724 0.222 1.088 1.56 0,30 SOYBEAN 22 0.6113 10.28 4320.7 387.7 0.81 0.38 412.2 34.7 0.541 1.411 0,4693 -0.148 -0.242 2.36 0.20 MAIZE 23 3.0227 2.08 206,1 21.5 0.04 0.02 16,3 1.1 -0.459 1.289 0.4687 -0.909 -0.301 2.66 0,18 MAIZE 24 0,9510 6.61 236L 5 41.0 0.45 0.04 164.9 11.2 1.242 0,504 0,4647 0.492 0,518 2,62 0.18 SOYBEAN 25 1,4265 4.41 774,2 8.8 0.15 0.01 68.9 3,9 -0.115 0.216 0.4623 -0.012 -0.008 2.85 0.16 MAIZE 26 2,9888 2.10 10,4 3,3 0.00 0.00 15.0 1.2 1.002 0.886 0.4616 -0.852 -0.285 2.42 0,19 MAIZE 27 2.8189 2.23 514,2 65,2 0.10 0.06 23.4 1.3 -0.011 0,296 0.4611 -0,357 -0.127 2,89 0.16 MAIZE 28 2.2755 2,76 108,2 11.1 0.02 0.01 23,6 2.0 -0.272 -1.473 0,4593 -0.133 -0.058 2,31 0.20 MAIZE 29 2.8529 2,20 147,0 3.0 0.03 0.00 21.4 1.2 0.020 0,643 0.4515 -0,496 -0.174 2,87 0,16 MAIZE 30 0.4755 13.21 72,5 72,5 0.01 0.07 639.4 55.1 -0.892 -0.495 0.4486 0,031 0,064 2,28 0,20 SOYBEAN 31 2,9548 2.13 67,3 12,6 0.01 0.01 14.2 1,1 1.097 0,387 0.4455 -0.922 -0,312 2,41 0.18 MAIZE 32 2.8869 2.18 456,4 17e9 0,09 0.02 19,0 1,1 1.434 -0,810 0,4436 -0,672 -0.233 2.71 0.16 MAIZE 33 2.9208 2,15 12,6 2,0 0,00 0,00 15.5 1.1 0,397 -1.033 0.4426 -0.848 -0,290 2.53 0.18 MAIZE 34 1.0189 6,17 532.2 119,7 0,10 0,12 113.3 10,9 -0,499 -1,084 0.4406 0.619 0.607 2.14 0,21 SOYBEAN 35 0.9170 6.85 2638.2 15,2 0,50 0,01 183,5 10.9 -0.789 -0.843 0.4337 0.458 0,500 2,70 0.16 SOYBEAN 36 0,5095 12,33 2309.2 444.5 0,43 0.43 537,5 51,4 0.720 -0.466 0.4326 -0.026 -0,050 2.13 0.20 MAIZE ------------------------------------------------------------------------------------------------------------------------------------------------- TABLE 8b: SOYBEAN, MAIZE CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO CONTRIBUTION OF SOYBEAN FREQUENCIES) ---------- ----------- ------ ----- ---- ------ --- - --- --- --- --- - ---- --- ----- ----- --- ---- - ----- - ---- -- -- - ------ ----- ------------------ PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT GAIN LAGGED NO. FREQUENCY PERIOD SOYBEAN MAIZE SOYBEAN MAIZE SOYBEAN MAIZE SOYBEAN MAIZE COHERENCY PHASE TIME 1 ON 2 2 ON 2 VARIABLE 1 0,0340 185.00 120734.0 15107.4 22,74 14.68 4337.4 1601.1 -0.068 -0.205 0,6643 0.396 11.660 1.34 0.50 SOYBEAN 2 0.1019 61.67 74339.5 29597.0 14.00 28.75 3737.8 1042.6 0.401 -0.279 0.5974 0,312 3.065 1.46 0.41 SOYBEAN 3 0,1359 46.25 40610.7 2155.5 7.65 2.09 3464.5 854.0 0.561 0.802 0.5507 0.320 2.355 1.49 0.37 SOYBEAN 4 0.1698 37.00 34824.1 2681.7 6.56 2.61 3099.9 693.7 1.468 0.900 0.5367 0.291 1.715 1.55 0.35 SOYBEAN 5 02038 30.83 33083.1 3453.0 6.23 3.35 2678.9 517.4 -1.532 -1.072 0.4724 0.222 1.088 1.56 0.30 SOYBEAN 6 0.0679 92.50 19396.1 36633.0 3.65 35.59 3959.9 1093.2 OX785 0.376 0,6204 0.308 4.530 1.50 0.41 SOYBEAN 7 0.6453 9.74 12714.8 880.7 2.39 0.86 418.1 31.8 -1.044 -0.718 0.5186 -0.171 -0.265 2.61 0.20 MAIZE 8 0.4415 14.23 12280.6 408.3 2.31 0.40 809.7 58.5 1.000 0,630 0.4189 0.054 0.123 2.41 0.17 SOYBEAN 9 0,7132 8.81 7197.3 305.9 1.36 0.30 371.8 21.3 -0.207 -0.326 0.5852 -0.066 -0.092 3.19 0.18 MAIZE 10 0.5434 11,56 4931.2 951.2 0.93 0.92 486.9 46.5 -1.045 0.927 0.4289 -0.047 -0.087 2.12 0.20 MAIZE 11 0.8151 7.71 4807.1 144,9 0.91 0.14 258.7 11.5 0.837 0.047 0.5085 0.278 0.341 3.38 0.15 SOYBEAN !2 0,6113 10.28 4320.7 387,7 0.81 0.38 412.2 34.7 0.541 1.411 0.4693 -0.148 -0.242 2.36 0.20 MAIZE 13 0.5774 10,88 4115.7 450.1 0.78 0,44 456.7 40.6 0.888 0.939 0.4725 -0.101 -0,174 2.30 0.21 ',AiZE 14 0.8830 7.12 4013,6 291,9 0,76 0.28 206.6 11.3 1.265 1.301 0.4844 0.390 0.441 2.98 0.16 SOYBEAN 15 0,9170 6.85 2638.2 15,2 0.50 0,01 183.5 10.9 -0,789 -0.843 0.4337 0,458 0.500 2.70 0.16 SOYBEAN 16 0.6793 9,25 2554.5 16.4 0.48 0.02 396.1 26.2 0.605 -1.364 0.5339 -0.130 -0.191 2.84 0.19 MAIZE 17 0.9849 6.38 2440.3 349.4 0.46 0,34 141.7 11.6 -0.287 -0,674 0.4860 0.510 0.518 2.44 0.20 SOYBEAN 18 0.9510 6.61 2368.5 41,0 0.45 0.04 164.9 11.2 1.242 0.504 0.4647 0.492 0.518 2.62 0.18 SOYBEAN 19 0.5095 12.33 2309.2 444,5 0.43 0.43 537.5 51.4 0.720 -0.466 0.4326 -0.026 -0.050 2.13 0,20 MAIZE 20 1.3925 4,51 2297,6 128,7 0.43 0.13 78,3 4.2 0.080 0,222 0,5051 -0.028 -0.020 3.07 0,16 MAIZE 21 0.7472 8.41 2085,4 7.4 0.39 0.01 326.7 16.1 1.057 -0,877 0.5619 -0,040 -0.054 3.38 0.17 MAIZE 22 0.7812 8,04 1342,0 48,4 0.25 0.05 288.4 13.0 -1.303 -0.085 0.5026 0.103 0.132 3.33 0,15 SOYBEAN 23 1.0529 5.97 1119.0 189e9 0.21 0.18 96.6 10.3 0.105 -1.550 0,4249 0.796 0.756 2.00 0.21 SOYBEAN ---------------- ------------- ---- ------ --- ----- ----- ------------------- --- ---- -- --- ------ -------- ---- --- -- ----------------------------------- --- TABLE 8c: SOYBEAN, MAIZE CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO CONTRIBUTION OF MAIZE FREQUENCIES) ---------------------------------------------------------------------------------------------------------------------------------------------- PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT GAIN LAGGED NO. FREQUENCY PERIOD SOYBEAN MAIZE SOYBEAN MAIZE SOYBEAN MAIZE SOYBEAN MAIZE COHERENCY PHASE TIME 1 ON 2 2 ON 2 VARIABLE ----------------------------------------------------------------------------------------------------------- ------------------------------- 1 0.0679 92.50 19396.1 36633.0 3.65 35.59 3959.9 1093.2 0.785 0.376 0.6204 0,308 4.530 1.50 0.41 SOYBEAN 2 0.1019 61.67 74339,5 29597.0 14.00 28.75 3737.8 1042.6 0.401 -0.279 0.5974 0.312 3.065 1.46 0.41 SOYBEAN 3 0,0340 185.00 120734.0 15107,4 22.74 14.68 4337,4 1601.1 -0.068 -0.205 0.6643 0.396 11.660 1.34 0.50 SOYBEAN 4 0.2038 30.83 33083.1 3453.0 6,23 3.35 2678.9 517.4 -1.532 -1.072 0.4724 0.222 1.088 1.56 0.30 SOYBEAN 5 0.1698 37,00 34824.1 2681.7 6.56 2.61 3099,9 693.7 1.468 0.900 0.5367 0.291 1.715 1.55 0.35 SOYBEAN 6 0.1359 46.25 40610.7 2155.5 7.65 2.09 3464.5 854.0 0.561 0.802 0.5507 0.320 2.355 1.49 0.37 SOYBEAN 7 0,5434 11.56 4931.2 951.2 0.93 0,92 486.9 46.5 -1.045 0.927 0.4289 -0.047 -0.087 2.12 0.20 MAIZE 8 0.6453 9.74 12714.8 880.7 2.39 0.86 418.1 31,8 -1.044 -0.718 0.5186 -0.171 -0.265 2.61 0.20 MAIZE 9 0.5774 10.88 4115.7 450.1 0.78 0.44 456.7 40.6 0.888 0.939 0.4725 -0.101 -0.174 2.30 0.21 MAIZE 10 0.5095 12.33 2309.2 444.5 0.43 0.43 537.5 51.4 0.720 -0.466 0.4326 -0.026 -0.050 2.13 0,20 MAIZE 11 0.4415 14.23 12280.6 408.3 2.31 0.40 809.7 58.5 1.000 0.630 0.4189 0.054 0.123 2,41 0,17 SOYBEAN 12 0.6113 10.28 4320.7 387.7 0.81 0,38 412.2 34.7 0.541 1,411 0.4693 -0.148 -0.242 2.36 0.20 MAIZE 13 0.9849 6.38 2440.3 349.4 0.46 0.34 141.7 11.6 -0,287 -0.674 0.4860 0.510 0.518 2.44 0.20 SOYBEAN 14 0.7132 8.81 7197.3 305.9 1.36 0.30 371.8 21.3 -0.207 -0.326 0.5852 -0,066 -0.092 3.19 0.18 MAIZE 15 0.8830 7.12 4013,6 291.9 0.76 0.28 206.6 11,3 1.265 1.301 0.4844 0.390 0.441 2.98 0.16 SOYBEAN 16 1.0529 5.97 1119,0 189.9 0021 0.18 96,6 10.3 0.105 -1.550 0.4249 0.796 0,756 2.00 0.21 SOYBEAN 17 0.8151 7.71 4807.1 144.9 0.91 0,14 258.7 11.5 0.837 0.047 0.5085 0.278 0.341 3.38 0.15 SOYBEAN4 18 1,3925 4.51 2297.6 128.7 0.43 0,13 78.3 4,2 0,080 0.222 0.5051 -0.028 -0.020 3.07 0.16 MAIZE 19 0.8491 7.40 517,0 120,1 0.10 0,12 218,5 10.5 -0,023 0.016 0.4958 0.376 0.443 3.21 0.15 SOYBEAN 20 1.0189 6.17 532,2 119,7 0,10 0.12 113,3 10,9 -0.499 -1,084 0,4406 0.619 0.607 2.14 0,21 SOYBEAN 21 0.4755 13,21 72.5 72.5 0.01 0.07 639.4 55.1 -0.892 -0.495 0.4486 0.031 0.064 2.28 0.20 SOYBEAN 22 2.8189 2,23 514.2 65,2 0.10 0.06 23.4 1,3 -0.011 0.296 0,4611 -0.357 -0.127 2.89 0.16 MAIZE 23 2.3435 2.68 426,7 53.7 0,08 0.05 19,2 2.1 -1,300 1,350 0.4095 0.059 0.025 1,93 0.21 SOYBEAN 24 0,7812 8,04 1342.0 48,4 0.25 0.05 288,4 13.0 -1.303 -0.085 0.5026 0.103 0.132 3,33 0.15 SOYBEAN 25 1.1208 5.61 86.1 42.4 0.02 0.04 66e2 8.7 0,194 1.436 0.4134 0.925 0.825 1,77 0.23 SOYBEAN 26 0.9510 6.61 2368e5 41.0 0,45 0.04 164,9 11.2 1.242 0.504 0.4647 0,492 0,518 2,62 0,18 SOYBEAN ------------------------------------------------------------------------------------------------------------------------------------------ - 44 - 53. It is important to note at this point that since there have b',en major structural changes in the vegetable oils market, the selection of the period over which the analysis was performed is very crucial. Cross-spectral analysis for the period January 1960 to December 1971 shows a strong leading effect of soybean on palm oil, while cross-spectral analysis for the period January 1972 to May 1987 shows the reverse. These results coincide with the fact that during the 1960s it was the soy oil price that was the dominant oil and had the largest share of the trade market, while in the 1970's the palm oil share increased dramatically and it became if not the leading price then one of equal importance to soy oil. 54. Tables 9a, 9b and 9b present the results from the cross-spectral analysis of soybean and palm oil. The results emphasize the following points: (a) There is a strong short-term leading effect of palm oil on soybean oil which varies between 3 and 7 months. (b) The eight most important frequencies, in terms of their contribution to the soybean price, are relatively more important to the palm oil price (87%) then to the soybean price. (c) The coherency values for the major cycles (see Table 9b) are extren;cly high. This means that most of the variability of the soybean price is a reaction to changes in the palm oil price. 55. Taking into account both sets of cross-spectral results described above, the estimation of the structural form of the lag effect is straightfor- ward. The soybean price is a function of a very short-term effect from maize with a longer-effect from maize of about 12 months, combined with an effect from the palnii oil price of about 5 - 8 months lag, as well as a very short- term effect from palm oil with a one-period lag. TABLE 9a: SOYBEAN, PALM OIL CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO COHERENCY VALUES) PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT GAIN LAGGED PALM PALM PALM PALM NO. FREQUIENCY PERIOD SOYBEAN OIL SOYBEAN OIL SOYBEAN OIL SOYBEAN OIL COHERENCY PHASE TIME 1 ON 2 2 ON 2 VARIABLE -----------------------------------------------------------------------------------------------------~------------------------------------------- 1 0.0340 185.00 120734.0 1190291,0 22,74 25.17 4337.4 58308.3 -0.068 -0.060 0.8130 0,266 7,835 0.25 3.31 SOYBEAN 2 0.0679 92.50 19396.1 216285,5 3.65 4,57 3959.9 46383.1 0,785 -0,081 0.7515 0,325 4.788 0.25 2.97 SOYBEAN 3 0.1019 61.67 74339.5 1743950,5 14.00 36.87 3737.8 46638.4 0,401 0,208 0.7121 0,330 3.243 0,24 2,98 SOYBEAN 4 0,7472 8.41 2085.4 1742,3 0.39 0.04 326.7 873,1 1,057 0,480 0.7073 0.273 0.366 0,51 1.38 SOYBEAN 5 0.7812 8,04 1342,0 12214.2 0.25 0.26 288.4 808.0 -1.303 1,254 0.6885 0.284 0.363 0.50 1.39 SOYBEAN 6 0.7132 8.81 7197,3 12531,0 1,36 0.26 371,8 1101,8 -0.207 -0,611 0,6851 0,258 0,361 0,48 1.42 SOYBEAN 7 0.1359 46.25 40610.7 167311.9 7,65 3,54 3464.5 39132,9 0,561 0,418 0.6515 0.347 2,557 0.24 2.71 SOYBEAN 8 0.8151 7.71 4807.1 6245.0 0.91 0.13 258,7 740,4 0.837 0.620 0,6385 0,324 0,397 0.47 1,35 SOYBEAN 9 0.1698 37.00 34824.1 327209.5 6.56 6.92 3099.9 33757,0 1.468 0.230 0,6248 0,423 2.494 0.24 2,61 SOYBEAN 10 0.8491 7,40 517,0 14620.4 0.10 0.31 218.5 745.8 -0.023 -1.265 0,5809 0.372 0.438 0.41 1.41 SOYBEAN 11 0,6793 9,25 2554.5 10564.7 0,48 0.22 396.1 1369,6 0.605 0.535 0.5717 0.229 0,337 0.41 1.41 SOYBEAN 12 0.2038 30.83 33083,1 404578.0 6,23 8,55 2678.9 26971.7 -1.532 0.527 0.5561 0.492 2.416 0.24 2,37 SOYBEAN 13 0.8830 7.12 4013.6 11548.0 0.76 0.24 206,6 714.6 1.265 1,320 0.5491 0,360 0.407 0.40 1,38 SOYBEAN 14 2,2076 2.85 351.6 803.3 0.07 0.02 28.5 47.7 -1,294 -0.748 0,5470 -0,826 -0.374 0.57 0.96 PALM OIL 15 3.1246 2.01 644,7 1472.1 0.12 0.03 20.0 74.1 -0,130 0.203 0,5460 -0,544 -0.174 0.38 1.42 PALM OIL 16 2,2416 2.80 566.2 111.9 0,11 0,00 27.0 37,5 -1.328 -0.745 0,5330 -0,793 -0.354 0.62 0.86 PALM OIL 17 3.09'07 2.03 279.4 480.2 0.05 0.01 16.6 71,9 1,239 -1,425 0.5187 -0,465 -0.150 0.35 1,50 PALM OIL 18 2,1736 2,89 548.6 923.6 0,10 0.02 30.3 55.0 -0.008 0,799 0.4941 -0,864 -0,397 0.52 0.95 PALM OIL 19 0,6453 9.74 12714,8 12459,4 2.39 0.26 418.1 1621.0 -1,043 -1,001 0.4913 0,202 0,314 0.36 1.38 SOYBEAN 20 0,9170 6,85 2638.2 5576.3 0.50 0,12 183.5 666.4 -0,789 -0,798 0.4803 0,386 0.420 0.36 1,32 SOYBEAN 21 2.2755 2.76 108,2 10.9 0.02 0.00 23.6 29.3 -0.272 -0.069 0.4751 -0,783 -0,344 0.62 0.77 PALM OIL 22 0.2377 26.43 16752,7 47128.8 3.15 1.00 2208.0 18635.8 0,356 1.045 0,4528 0,554 2.332 0.23 1.95 SOYBEAN 23 2.6152 2,40 271.1 856,7 0,05 0,02 15.6 50.2 0,930 -1.549 0,4447 -0,492 -0,188 0.37 1,20 PALM OIL 24 3.0567 2.06 12402 1005,9 0.02 0.02 15.8 61,6 0,950 -0.354 (b.4390 -0,632 -0,207 0.34 1,31 PALM OIL 25 0.6113 10.28 4320.7 40492,7 0.81 0,86 412.2 1911.0 0.541 0.396 0.4380 0.209 0.341 0.31 1.42 SOYBEAN 26 2.5812 2.43 208.7 602.3 0,04 0.01 14.0 46.5 -1,235 -0.793 0.4315 -0.629 -0.244 0.36 1.20 PALM OIL 27 0,9510 6.61 2368.5 784.4 0.45 0.02 164.9 624.5 1.242 1.235 0.4143 0.452 0,476 0.33 1,25 SOYBEAN 28 2,1397 2.94 287.9 2274.6 0.05 0,05 30,1 62.9 0.763 -1.191 0,4017 -0,897 -0.419 0,44 0,92 PALM OIL ------------------------------------------ ------ ------ -- ---------------- -- ------- -- ----- ----- ----- --- ---------------.------------- .----------- --------- -- --- TABLE9b: SOYBEAN, PALM OIL CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO CONTRIBUTION OF SOYBEAN FREQUENCIES) ------------------------------------------------------------------------------------------------------------------------------------------- PERIODOGRAM % CONTRIBUTION - SPECTRUM PHASE-SH IFT SH IFT - GAIN LAGGED FRE- PALM PALM PALM F-ALM NO. QUENCY PERIOD SOYBEAN OIL SOYBEAN OIL SOYBEAN OIL SOYBEAN OIL COHERENCY PHASE TIME 1 ON 2 2 ON 2 VARIABLE ----------------------------------------------------------------------------------------------------------------------------------------------- 1 0.0340 185.00 120734.0 1190291.0 22.74 25.17 4337.4 58308.3 -0.068 -0.060 0.8130 0.266 7.835 0.25 3.31 SOYBEAN 2 O.lO9 61.67 74339.5 1743950.5 14.00 36,87 3737.8 46638.4 0.401 0.208 0.7121 0,330 3.243 0.24 2.98 SOYBEAN 3 0.1359 46.25 40610.7 167311,9 7.65 3,54 3464.5 39132.9 0,561 0,418 0.6515 0.347 2.557 0.24 2.71 SOYBEAN 4 0.1698 37.00 34824.1 3?7209.5 6.56 6.92 3099.9 33757.0 1.468 0.230 0.6248 0.423 2.494 0,24 2.61 SOYBEAN 5 0.2038 30.83 33083.1 404578.0 6.23 8.55 2678.9 26971.7 -1.532 0.527 0.5561 0.492 2.416 0.24 2.37 SOYBEAN 6 0.0679 92.50 19396.1 216285.5 3.65 4,57 3959.9 46383.1 0,785 -0.081 0.7515 0.325 4.788 0.25 2.97 SOYBEAN 7 0.2377 26.43 16752.7 47128.8 3.15 1.00 2208.0 18635.8 0.356 1.045 0.4528 0.554 2.332 0.23 1.95 SOYBEAN 8 0.6453 9.74 12714.8 12459.4 2.39 0.26 418.1 1621.0 -1.043 -1.001 0.4913 0.202 0.314 0.36 1.38 SOYBEAN 9 0.7132 8.81 7197.3 12531.0 1.36 0.26 371,8 1101.8 -0,207 -0.611 0.6851 0.258 0,361 0.48 1.42 SOYBEAN 10 0.8151 7.71 4807,1 6245.0 0.91 0,13 258.7 740.4 0.837 0.620 0.6385 0.324 0.397 0.47 1.35 SOYBEAN 11 0.6113 10.28 4320.7 40492.7 0.81 0.86 412.2 1911.0 0.541 0.396 0.4380 0.209 0.341 0.31 1.42 SOYBEAN 12 0.8830 7,12 4013.6 11548.0 0.76 0.24 206.6 714.6 1.265 1.320 0.5491 0.360 0,407 0.40 1.38 SOYBEAN 13 0.9170 6.85 2638.2 5576.3 0.50 0.12 183.5 666,4 -0.789 -0.798 0.4803 0.386 0.420 0.36 1.32 SOYBEAN 14 0.6793 9.25 2554,5 10564,7 0.48 0.22 396.1 1369.6 0.605 0.535 0.5717 0.229 0,337 0.41 1.41 PALMOIL 15 0.9510 6.61 2368.5 784.4 0.45 0.02 164.9 624.5 1,242 1,235 0.4143 0.452 0.476 0.33 1.25 PALMOIL 16 0.7472 8.41 2085.4 1742.3 0.39 0,04 326.7 873.1 1.057 0.480 0.7073 0.273 0.366 0.51 1.38 PALMOIL 17 0.7812 8.04 1342.0 12214.2 0.25 0.26 288,4 808.0 -1.303 1.254 0.6885 0,284 0,363 0,50 1.39 PALMOIL 18 3,1246 2.01 644.7 1472.1 0.12 0,03 20.0 74.1 -0.130 0.203 0.5460 -0,544 -0,174 0.38 1.42 PALMOIL 19 2.2416 2.80 566.2 111.9 0.11 0.00 27.0 37.5 -1.328 -0.745 0.5330 -0.793 -0,354 0.62 0.86 SOYBEAN 20 2,1736 2.89 548.6 923.6 0.10 0.02 30.3 55.0 -0.008 0.799 0.4941 -0.864 -0.397 0.52 0.95 SOYBEAN 21 0.8491 7.40 517,0 14620.4 0,10 0,31 218.5 745,8 -0.023 -1.265 0,5809 0.372 0.438 0.41 1.41 PALMOIL 22 2.2076 2.85 351.6 803.3 0.07 0.02 28.5 47.7 -1.294 -0,748 0.5470 -0,826 -0,374 0.57 0.96 SOYBEAN 23 2.1397 2.94 287,9 2274.6 0.05 0.05 30.1 62.9 0.763 -1.191 0,4017 -0.897 -0,419 0,44 0.92 PALMOIL 24 3.0907 2,03 279.4 480,2 0.05 0,01 16,6 71.9 1.239 -1,425 0.51W7 -0,465 -0,150 0.35 1.50 PALMOIL 25 2,6152 2,40 271.1 856.7 0.05 0.02 15,6 50.2 0.930 -1,549 O.4'447 -0.492 -0.188 0,37 1,20 SOYBEAN 26 2.5812 2,43 208.7 602.3 0.04 0.01 14.0 46,5 -1.235 -0.793 0.4315 -0.629 -0,244 0.36 1.20 PALMOIL 27 3,0567 2,06 124.2 1005.9 0.02 0.02 15.8 61,6 0,950 -0.354 0.4390 -0.632 -0,207 0,34 1.31 SOYBEAN 28 2.2755 2.76 108,2 10,9 0,02 0.00 23.6 29.3 -0.272 -0.069 0.47Sl -0,783 -0.344 0.62 0.77 PALMOIL -------------------------------------------------------------------------------------------------------------------------------------------- TABLE 9c: SOYBEAN, PALMOIL CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO CONTRIBUTION OF PALM OIL FREQUENCIES) PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT GAIN LAGGED FRE- PALM PALM PALM PALM NO. QUENCY PERIOD SOYBEAN - OIL SOYBEAN OIL SOYBEAN OIL SOYBEAN OIL COHERENCY PHASE TIME I ON 2 2 ON 2 VARIABLE ------------------------------------------------------------------------------------------------------------------------------------------------- 1 0.1019 61.67 74339.5 1743950.5 14.00 36.87 3737.8 46638.4 0.401 0.208 0.7121 0.330 3.243 0.24 2.98 SOYBEAN 2 0.0340 185.00 120734.0 1190291.0 22.74 25,17 4337,4 58308.3 -0.068 -0.060 0.8130 0.266 7.835 0.25 3.31 SOYBEAN 3 0.2038 30.83 33083.1 404578.0 6,23 8.55 2678.9 26971.7 -1.532 0.527 0,5561 0.492 2.416 0.24 2.37 SOYBEAN 4 0.1698 37.00 34824.1 327209.5 6.56 6.92 3099.9 33757.0 1.468 0.230 0.6248 0.423 2.494 0.24 2,61 SOYBEAN 5 0.0679 92.50 19396.1 216285.5 3.65 4.57 3959.9 46383.1 0.785 -0.081 0.7515 0.325 4.788 0.25 2.97 SOYBEAN 6 0,1359 46.25 40610.7 167311.9 7,65 3.54 3464,5 39132.9 0.561 0.418 0.6515 0.347 2.557 0,24 2.71 SOYBEAN 7 0.2377 26.43 16752.7 47128.8 3.15 1.00 2208.0 18635.8 0.356 1.045 0.4528 0.554 2.332 0.23 1.95 SOYBEAN 8 0.6113 10.28 4320.7 40492.7 0.81 0.86 412.2 1911,0 0.541 0.396 0.4380 0.209 0.341 0.31 1.42 SOYBEAN 9 0.8491 7.40 517.0 14620.4 0,10 0,31 218,5 745.8 -0.023 -1.265 0.5809 0.372 0.438 0.41 1.41 PALMOIL 10 0.7132 8.81 7197.3 12531.0 1.36 0.26 371.8 1101.8 -0.207 -0.611 0.6851 0.258 0.361 0.48 1.42 SOYBEAN 11 0.6453 9.74 12714.8 12459.4 2,39 0.26 418,1 1621.0 -1.043 -1.001 0.4913 0.202 0.314 0.36 1,38 SOYBEAN 12 0.7812 8.04 1342.0 12214,2 0.25 0.26 288,4 808.0 -1,303 1.254 0.6885 0,284 0.363 0.50 1.39 PALMOIL 13 0,8830 7.12 4013.6 11548.0 0.76 0.24 206.6 714.6 1.265 1.320 0.5491 0,360 0.407 0.40 1.38 SOYBEAN 14 0.6793 9.25 2554.5 10564.7 0,48 0.22 396.1 1369.6 0.605 0.535 0.5717 0.229 0.337 0.41 1.41 PALMOIL 15 0,8151 7.71 4807.1 6245.0 0.91 0.13 258.7 740.4 0.837 0.620 0.6385 0,324 0.397 0,47 1.35 SOYBEAN 16 0.9170 6.85 2638.2 5576.3 0.50 0.12 183,5 666.4 -0.789 -0.798 0.4803 0.386 0.420 0.36 1.32 SOYBEAN 17 2.1397 2.94 287,9 2274.6 0,05 0.05 30.1 62.9 0.763 -1,191 0.4017 -0.897 -0.419 0.44 0.92 PALMOIL 18 0.7472 8.41 2085.4 1742.3 0.39 0.04 326.7 873.1 1.057 0.480 0.7073 0.273 0.366 0.51 1.38 PALMOIL 19 3.1246 2.01 644,7 1472,1 0.12 0.03 20.0 74.1 -0.130 0.203 0,5460 -0.544 -0,174 0.38 1.42 PALMOIL 20 3,0367 2.06 124,2 1005.9 0.02 0.02 15.8 61.6 0.950 -0.354 0,4390 -0.632 -0.207 0.34 1,31 SOYBEAN 21 2.1736 2,89 548.6 923.6 0,10 0.02 30.3 55.0 -0.008 0.799 0.4941 -0,864 -0,397 0.52 0.95 SOYBEAN 22 2,6152 2.40 271.1 856.7 0,05 0.02 15.6 50.2 0.930 -1.549 0.4447 -0,492 -0.188 0.37 1.20 SOYBEAN 23 2.2076 2.85 351,6 803,3 0.07 0.02 28.5 47.7 -1,294 -0.748 0.5470 -0.826 -0,374 0.57 0,96 SOYBEAN 24 0,9510 6.61 2368.5 784.4 0.45 0,02 164.9 624,5 1.242 1.235 0,4143 0.452 0.476 0.33 1,25 PALMOIL 25 255812 2,43 208.7 602.3 0,04 0.01 14.0 46.5 -1,235 -0.793 0,4315 -0.629 -0,244 0.36 1.20 PALMOIL 26 3.0907 2,03 279.4 480,2 0,05 0.01 16.6 71.9 1,239 -1,425 0,5187 -0.465 -0,150 0,35 1,50 PALMOIL 27 2.2416 2.80 566.2 111.9 0,11 0.00 27.0 37,5 -1,328 -0,745 0,5330 -0,793 -0.354 0.62 0,86 SOYBEAN 28 2,2755 2,76 108.2 10.9 0.02 0.00 23,6 29.3 -0,272 -0.069 0,4751 -0,783 -0,344 0.62 0,77 PALMOIL ------------------------------------------------------------------------------------------------------------------------------------------------- - 48 - Rice and Wheat Prices 56. Tables lOa, lOb and 1Oc present the results from the cross-spectral analysis of rice and wheat prices. The results are at least as strong as in the previous cases. 57. The coherency level values--very high for all major frequencies--vary from 0.68 Fo 0.79. The total contribution of the first five frequencies to the explanation of the wheat price variation is about 85% and about 88% for the rice price. These results, along with the results of the time shift, explain the whole structural relationship between the rice price and the wheat price. 58. It is very obvious that it is the wheat price that drives the rice price and with a very short lag of about three months. This result is consistent with the fact that the two commodities are close substitutes with wheat having a much larger share of the market. Because of the amount of unexplained variation, it is also clear that there is some cyclical behavior in the wheat price which is not transmitted to the rice price, and that there are some characteristics of the rice price which cannot be explained by the movements of the wheat price. 59. The gain of the rice price is more or less constant for the major frequencies, at a level of around 2.5, which means that the difference (or aptitude change) between the maximum of each cycle is a movement of $2.5 in the rice price for every $1 change in wheat price. Maize and Wheat Prices 60. With respect to these two prices, the question of which series drives the other cannot be answered conclusively. There is some evidence for conclud- ing that it is the wheat price that follows the maize price, but the dominance is not as obvious as can be seen for the other prices. If there is a lagged TABLE 10a: RICE, WHEAT CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO COHERENCY VALUES) PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT GAIN LAGGED - - ----- --- - -- - ---- - ------ - - --- -- -- -- --- -- -- -- -- - ---- -- ------ - -- -- --- - -- -.I- - ----- NO. FREQUENCY PERIOD RICE WHEAT RICE WHEAT RICE WHEAT RICE PHASE TIME COHERENCY 1 ON 2 2ON2 VARIABLE 1 0.0340 185.00 434737.8 12077.3 19,99 5.03 30470.1 3766.3 0.019 0.084 0.7926 0,179 5,270 2.53 0,31 RICE 2 0.1019 61.67 430557.8 102780,8 19.80 42.80 21988.0 2572.3 -0.507 -0,638 0.7536 0.113 1,105 2.54 0.30 RICE 3 0,0679 92.50 451679.0 67364.1 20.77 28.05 25792.2 2603.9 1.327 0.932 0.7248 OJ111 1.639 2.68 0.27 RICE 4 0.1359 46.25 333732.3 13226.5 15,35 5.51 20126.6 2196.8 -0.666 -0.933 0.7228 0.090 0.660 2.57 0.28 RICE 5 0.1698 37.00 265048.4 9919.3 12.19 4.13 17323.6 1808.9 0.823 -1.279 0.6792 0.028 0.166 2,55 0l27 RICE 6 0.2038 30.83 76030.4 8229.8 3,50 3.43 13463.8 1383.1 1.507 -1.174 0.6605 -0,014 -0.066 2.54 0.26 WHEAT 7 0,2377 26.43 33644.8 622.7 1.55 0,26 9325.0 925.3 -0.012 -0.475 0.6184 -0.079 -0.333 2.50 0.25 WHEAT 8 0.2717 23.13 23417.3 2103.1 1.08 0.88 6004.6 497.6 -0.756 -0.660 0.5531 -0.270 -0.995 2.58 0.21 WHEAT 9 0.3057 20.56 7485.0 1019.7 0,34 0.42 3601,6 214.4 -0.561 1.176 0.4862 -0.597 -1.954 2.86 0.17 WHEAT < 10 0.3396 18.50 15670.9 1668.6 0,72 0.69 2117.3 164,7 1.165 -0,362 0.376 -0.858 -2.525 2.20 0.17 WHEAT TABLE lOb: RICE, WHEAT CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO CONTRIBUTION OF RICE FREQUENCIES) ------------------------------------------------------------------------------------------------------------------------------------------------------ PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT GAIN LAGGED NO. FREQUENCY PERIOD RICE WHEAT RICE WHEAT RICE WHEAT RiE E w'EAT PHASE TIME COHERENCY 1 ON 2 2 ON 2 VARIABLE ----------------------------------------------------------------------------------------------------------------------------------------------- -.---- 1 0.0679 92.50 451679.0 67364.1 20.77 28.05 25792.2 2603.9 1.327 0.932 0,7248 0.111 1.639 2.68 0.27 RICE 2 0.0340 185.00 434737.8 12077.3 19,99 5.03 30470.1 3766.3 0.019 0,084 0.7926 0.179 5.270 2.53 0.31 RICE 3 0.1019 61,67 430557.8 102780.8 19.80 42.80 21988.0 2572.3 -0.507 -0.638 0.7536 0.113 1.105 2.54 0.30 RICE 4 0.1359 46.25 333732.3 13226.5 15.35 5.51 20126.6 2196.8 -0.666 -0.933 0.7228 0,090 0.660 2,57 0.28 RICE 5 0,1698 37.00 265048.4 9919.3 12.19 4.13 17323.6 1808.9 0.823 -1.279 0.6792 0.028 0.166 2.55 0.27 RICE 6 0.2038 30.83 76030.4 8229.8 3.50 3.43 13463.8 1383.1 1.507 -1.174 0.6605 -0.014 -0.066 2.54 0,26 WHEAT 7 0.2377 26.43 33644.8 622.7 1.55 0.26 9325.0 925.3 -0.012 -0.475 0.6184 -0.079 -0.333 2.50 0.25 WHEAT o 8 0.2717 23.13 23417.3 2103.1 1,08 0.88 6004,6 497.6 -0.756 -0.660 0.5531 -0.270 -0.995 2.58 0.21 WHEAT 9 0.3396 18.50 15670.9 1668.6 0.72 0.69 2117.3 164.7 1.165 -0.362 0,376 -0.858 -2.525 2.20 0.17 WHEAT 10 0.3057 20.56 7485.0 1019.7 0.34 0.42 3601.6 214.4 -0.561 1.176 0.4862 -0.597 -1.954 2.86 0.17 WHEAT ----------------------------------------------------------------------------------------------------------------------------------------------- TABLE 1Oc: RICE, WHEAT CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO CONTRIBUTION OF WHEAT FREQUENCIES) PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT S-SHIFT -GAIN - LAGGED NO. FREQUENCY PERIOD RICE WHEAT RICE WHEAT RICE WHEAT RICE WHEAT PHASE TIME COHERENCY 1 ON 2 2 ON 2 VARIABLE 1 0.1019 61.67 430557.8 102780.8 19.80 42.80 21988.0 2572.3 -0.507 -0.638 0.7536 0.113 1.105 2.54 0.30 RICE 2 0.0679 92.50 451679.0 67364.1 20.77 28.05 25792.2 2603.9 1.327 0,932 0.7248 0.111 1.639 2.68 0.27 RICE 3 0.1359 46.25 333732.3 13226.5 15.35 5.51 20126.6 2196.8 -0.666 -0.933 0.7228 0.090 0.660 2.57 0.28 RICE 4 0.0340 185.00 434737,8 12077.3 19.99 5.03 30470.1 3766,3 0.019 0.084 0.7926 0,179 5.270 2.53 0.31 RICE 5 0.1698 37.00 265048.4 9919.3 12.19 4.13 17323.6 1808.9 0.823 -1e279 0.6792 0.028 09166 2.55 0.27 RICE 6 0.2038 30.83 76030.4 8229.8 3,50 3.43 13463.8 1383.1 1,507 -1.174 0.6605 -0.014 -0.066 2,54 0,26 WHEAT 7 0.2717 23.13 23417,3 2103.1 1.08 0,88 6004.6 497.6 -0.756 -0.660 0,5531 -0.270 -0.995 2.58 0.21 WHEAT 8 0.3396 18.50 15670.9 1668.6 0.72 0.69 2117.3 164.7 1.165 -0.362 0.376 -0.858 -2,525 2.20 0.17 WHEAT F 9 0.3057 20.56 7485,0 5019.7 0.34 0,42 3601.6 214.4 -0.561 1.176 0,4862 -0,597 -1.954 2.86 0.17 WHEAT 10 0.2377 26.43 33644.8 622.7 1.55 0.26 9325.0 925.3 -0.012 -0.475 0.6184 -0.079 -0.333 2.50 0,25 WHEAT -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ---- - ---- - - - - ---- - - - - - - - - - - - - - - - - - - --- - - - - - - - - - -- -------- ----- ---- ----- - -- -- --- ---- ---- ----- ---- - - ---- - - --------- ---- ----- ---- ---- ---- --------- ---- - -- - 52 - structure it appears to be a short-term response wherein the wheat price follows the maize price. 61. As can be seen from Tables lla, llb and llc it seems that for the most important frequencies of both commodity prices, wheat lags maize by about one month. However, there are other frequencies in which the maize price lags the wheat price, also by a period of one month. 62. Taking the 15 most important frequencies, sorted according to the contribution of the wheat frequencies to its price variation, gives the result that in frequencies where wheat lags the maize price the total contribution of those frequencies to the wheat price variation is about 85% and for maize about 84%. However, the average lag in the wheat price to changes in maize prices is 1.15 months; whereas the average lag for those frequencies in which maize lags wheat is about 0.8 months. Of course, one can argue that it is irrelevant to deal with frequencies which are of such little importance in terms of predictive power. Given the similarity in the behavior of these two price series, perhaps the best that one can say is that they are mutually determined. TABLE Hla: MAIZE, WHEAT CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO COHERENCY VALUES) ------------------------------------------------------------------------------------------------------------------------------------------------- PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT GAIN LAGGED ----------------------------------------------------------------------------------------------------------- NO. FREQUENCY PERIOD MAIZE WHEAT MAIZE WHEAT MAIZE WHEAT MAIZE WHEAT PHASE TIME COHERENCY 1 ON 2 2 ON 2 VARIABLE ----------------------------------------------------------------------------------------------------------------------------------------------------- 1 0.0340 185.00 15107.4 12077.3 14.68 5.03 1601,1 3766.3 -0.205 0.084 0.7113 -0.122 -3.603 0e55 1.29 WHEAT 2 0.1019 61.67 29597.0 102780.8 28,75 42.80 1042.6 2572.3 -0.279 -0.638 0.6855 -0.037 -0.359 0.53 1.30 WHEAT 3 0.1359 46.25 2155.5 13226,5 2.09 5.51 854.0 2196.8 0.802 -0.933 0.6442 -0.023 -0,167 0.50 1.29 WHEAT 4 0.0679 92.50 36633.0 67364.1 35.59 28.05 1093.2 2603.9 0.376 0.932 0.6427 -0.103 -1.514 0,52 1.24 WHEAT 5 0.1698 37.00 2681.7 9919.3 2.61 4,13 693.7 1808.9 0.900 -1,279 0.6313 -0.022 -0.127 0.49 1.28 WHEAT 6 0.2038 30.83 3453.0 8229.8 3.35 3,43 517.4 1383.1 -1.072 -1,174 0.6145 0.001 0.007 0,48 1,28 MAIZE 7 0,2377 26.43 2038.0 622.7 1.98 0.26 333.4 925.3 1,386 -0.475 0.5669 0.049 0,208 0045 1.25 MAIZE 8 2.2076 2.85 8,2 32.7 0,01 0.01 1.8 3.3 -0.735 -1.080 0.5281 -0.314 -0.142 0,54 0,99 WHEAT 9 2.1736 2,89 36.2 97.2 0.04 0.04 1.7 3.5 0.284 0.553 0.5272 -0.271 -0.125 0.51 1,04 WHEAT I 10 2.2416 2.80 33.4 83.3 0.03 0,03 1,9 3,3 -0.771 -0.656 0.4992 -0,411 -0.183 0.54 0.93 WHEAT Li 11 2.1397 2.94 7.6 4,4 0.01 0.00 1.4 3.3 -0.722 -1.329 0,4898 -0.199 -0,093 0.47 1,05 WHEAT 12 0,8830 7,12 291.9 406,3 0,28 0,17 11.3 27.1 1,301 1,271 0,4878 0.664 0,752 0,45 1408 MAIZE 13 0.8491 7.40 120,1 85.7 0,12 0,04 10,5 30.8 0,016 -1,242 0,4876 0,780 0,918 0,41 1.20 MAIZE 14 0.2717 23.13 54.6 2103.1 0.05 0.88 174,6 497.6 0.715 -0.660 0,4775 0.192 0.707 0.41 1.17 MAIZE 15 0,7132 8.81 305.9 315,7 0.30 0.13 21.3 39.3 -0.326 -1,306 0.4664 0.658 0,922 0,50 0.93 MAIZE 16 0.8151 7.71 144.9 637.5 0.14 0,27 11,5 35,1 0.047 -1.339 0.4502 0.795 0.976 0.38 1.17 MAIZE 17 0,9170 6.85 15.2 185.9 0,01 0,08 10.9 24.0 -0.843 -0,944 0.4490 0.637 0.694 0.45 0.99 MAIZE 18 2,2755 2.76 11,1 15,3 0,01 0,01 2,0 3.2 -1.473 0.690 0.4488 -0.539 -0.237 0,54 0.84 WHEAT 19 2.3095 2,72 36.9 0,2 0.04 0.00 2.1 3,1 0,900 0.140 0.4303 -0,621 -0.269 0.54 0.80 WHEAT 20 0.9510 6.61 41.0 103.8 0.04 0,04 11,2 21.1 0.504 -1,139 0.4294 0.641 0.674 0.48 0.90 MAIZE 21 2,1057 2,98 22,8 27,7 0.02 0,01 1,3 3.2 1,519 -0,758 0.4179 -0,209 -0,099 0.42 1.01 WHEAT 22 2,3435 2.68 53.7 61,5 0.05 0.03 2.1 3,3 1.350 -0,512 0,4163 -0.697 -0.297 0.52 0.80 WHEAT 23 0,7812 8,04 48,4 761.8 0.05 0.32 13,0 37.5 -0.085 -0.725 0.4149 0.764 0.978 0,38 1.09 MAIZE 24 0,7472 8.41 7.4 639,2 0.01 0,27 16,1 38.7 -0.877 0,382 0.4102 0,699 0.936 0.41 0.99 MAIZE ------------------------------------------------------------------------------------------------------------------------------------------------------ TABLE lib: MAIZE, WHEAT CROSS-SPECTRAL ANALYSIS RESULTS (FREQUENCIES SORTED ACCORDING TO CONTRIBUTION OF MAIZE FREQUENCIES) ------------------------------------------------------------------------------------------------------------------------------------------------------ PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT GAIN LAGGED NO. FREQUENCY PERIOD MAIZE WHEAT MAIZE WHEAT MAIZE WHEAT MAIZE WHEAT PHASE TIME COHERENCY 1 ON 2 2 ON 2 VARIABLE --------------------------------------------------------------------------------------------------------------------------------- ------------------- 1 0.0679 92.50 36633.0 67364.1 35.59 28,05 1093.2 2603.9 0.376 0.932 0,6427 -0.103 -1.514 0O52 1.24 WHEAT 2 0.1019 61.67 29597.0 102780.8 28.75 42.80 1042.6 2572.3 -0.279 -0,638 0.6855 -0.037 -0.359 0,53 1e30 WHEAT 3 0.0340 185.00 15107,4 12077.3 14.68 5.03 1601.1 3766.3 -0.205 0.084 0.7113 -0.122 -3.603 0.55 1.29 WHEAT 4 0.2038 30.83 3453.0 8229.8 3.35 3.43 517.4 1383.1 -1.072 -1.174 0.6145 0,001 0,007 0,48 1,28 MAIZE 5 0.1698 37.00 2681.7 9919.3 2.61 4.13 693.7 1808.9 0.900 -1.279 0.6313 -0.022 -0.127 0.49 1.28 WHEAT 6 0.1359 46.25 2155.5 13226.5 2.09 5.51 854.0 2196.8 0.802 -0.933 0.6442 -0.023 -0.167 0.50 1.29 WHEAT 7 0.2377 26.43 2038.0 622.7 1.98 0.26 333.4 925.3 1.386 -0.475 0.5669 0.049 0.208 0.45 1.25 MAIZE 8 0.7132 8.81 305.9 315.7 0.30 0.13 21.3 39.3 -0.326 -1.306 0.4664 0.658 0.922 0.50 0.93 MAIZE 9 0.8830 7.12 291.9 406.3 0.28 0.17 11,3 27,1 1.301 1.271 0.4878 0,664 0.752 0.45 1.08 MAIZE 10 0.8151 7.71 144.9 637.5 0.14 0.27 11.5 35.1 0.047 -1.339 0.4502 0.795 0,976 0.38 1,17 MAIZE - 11 0.8491 7.40 120.1 85.7 0.12 0.04 10.5 30.8 0.016 -1,242 0.4876 0.780 0.918 0.41 1.20 MAIZE 12 0.2717 23.13 54.6 2103.1 0605 0.88 174.6 497.6 0.715 -0.660 0.4775 0.192 0.707 0.41 1,17 MAIZE 13 2.3435 2.68 53.7 61.5 0.05 0.03 2.1 3.3 1.350 -0.512 0,4163 -0.697 -0297 0.52 0.80 WHEAT 14 0.7812 8.04 48.4 761.8 0.05 0.32 13,0 37e5 -0.085 -0.725 0.4149 0.764 0.978 0.38 1.09 MAIZE 15 0.9510 6.61 41.0 103,8 0,04 0,04 11.2 21.1 0.504 -1.139 0.4294 0.641 0.674 0,48 0.90 MAIZE 16 2.3095 2,72 36,9 0.2 0.04 0.00 2,1 3.1 0.900 0,140 0.4303 -0.621 -0.269 0,54 0,80 WHEAT 17 2.1736 2.89 36.2 97.2 0.04 0.04 1,7 3.5 0.284 0.553 0.5272 -0.271 -0.125 0,51 1.04 WHEAT 18 2.2416 2.80 33.4 83,3 0,03 0.03 1,9 3.3 -0.771 -0.6156 0,4992 -0,411 -0.183 0.54 0.93 WHEAT 19 2.1057 2.98 22.8 27.7 0.02 0,01 1,3 3.2 1,519 -0.758 0.4179 -0.209 -0.099 0,42 1,01 WHEAT 20 0.9170 6.85 15.2 185.9 0.01 0,08 10.9 24.0 -0.843 -0.944 0.4490 0.637 0.694 0.45 0.99 MAIZE 21 2.2755 2,76 11,1 15,3 0.01 0.01 2,0 3.2 -1.473 0.690 0.4488 -0,539 -0.237 0.54 0.84 WHEAT 22 2.2076 2,85 8.2 32.7 0,01 0.01 1.8 3.3 -0.735 -1.080 0.5281 -0.314 -0,142 0.54 0.99 WHEAT 23 2.1397 2.94 7,6 4.4 0.01 0.00 1.4 3.3 -0.722 -1.329 0.4898 -0.199 -0.093 0.47 1.05 WHEAT 24 0.7472 8.41 7.4 639.2 0.01 0.27 16.1 38,7 -0.877 0.382 0.4102 0.699 0.936 0,41 0.99 MAIZE ----------------------------------------------------------------------------------------------------------------------------- ------------------------ TABLE llc: MAIZE, WHEAT CROSS-SPECTRAL ANALYSIS RESULTS FREQEUENCIES SORTED ACCORDING TO MAIZE FREQUENCIES CONTRIBUTION PERIODOGRAM % CONTRIBUTION SPECTRUM PHASE SHIFT SHIFT GAIN LAGGED NO. FREQUENCY PERIOD MAIZE WHEAT MAIZE WHEAT MAIZE WHEAT MAIZE WHEAT PHASE TIME COHERENCY 1 ON 2 2 ON 2 VARIABLE ------------------------------------------------------------------------------------------------------------~------------------------------------------ 1 0.1019 61.67 29597.0 102780.8 28.75 42.80 1042.6 2572.3 -0.279 -0.638 0.6855 -0.037 -0.359 0,53 1.30 WHEAT 2 0.0679 92.50 36633,0 67364.1 35.59 28,05 1093.2 2603.9 0.376 0.932 0.6427 -0.103 -1,514 0.52 1.24 WHEAT 3 0.1359 46.25 2155.5 13226.5 2.09 5.51 854.0 2196.8 0.802 -0,933 0.6442 -0.023 -0.167 0.50 1.29 WHEAT 4 0.0340 185.00 15107,4 12077.3 14.68 5.03 1601.1 3766,3 -0.205 0.084 0.7113 -0.122 -3.603 0.55 1.29 WHEAT 5 0.1698 37,00 2681,7 9919.3 2.61 4.13 693.7 1808.9 0.900 -1,279 0,6313 -0.022 -0.127 0.49 1.28 WHEAT 6 0.2038 30.83 3453.0 8229.8 3.35 3.43 517.4 1383,1 -1,072 -1.174 0.6145 0.001 0,007 0.48 1,28 MAIZE 7 0.2717 23.13 54.6 2103.1 0.05 0,88 174,6 497.6 0.715 -0.660 0.4775 0.192 0,707 0.41 1,17 MAIZE 8 0,7812 8.04 48.4 761c8 0.05 0.32 13.0 37.5 -0,085 -0.725 0.4149 0.764 0.978 0.38 1.09 MAIZE 9 0.7472 8.41 7.4 639.2 0.01 0.27 16.1 38.7 -0,877 0.382 0,4102 0.699 0.936 0.41 0.99 MAIZE 10 0,8151 7,71 144.9 637.5 0.14 0.27 11.5 35.1 0,047 -1.339 0,4502 0,795 0.976 0.38 1,17 MAIZE 11 0.2377 26,43 2038.0 622,7 1,98 0,26 333.4 925.3 1.386 -0.475 0,5669 0.049 0.208 0,45 1,25 MAIZE 12 0.8830 7.12 291.9 406.3 0.28 0,17 11.3 27,1 1.301 1,271 0,4878 0.664 0,752 0,45 1,08 MAIZE .n 13 0.7132 8.81 305.9 315.7 0,30 0,13 21,3 39.3 -0,326 -1.306 0,4664 0.658 0.922 0.50 0,93 MAIZE 14 0.9170 6.85 15.2 185.9 0,01 0.08 10.9 24.0 -0.843 -0.944 0.4490 0.637 0.694 0,45 0.99 MAIZE 15 0.9510 6.61 41,0 103.8 0,04 0.04 11.2 21,1 0.504 -1,139 0.4294 0,641 0.674 0.48 0.90 MAIZE 16 2,1736 2.89 36.2 97.2 0.04 0.04 1.7 3,5 0.284 0.553 0.5272 -0.271 -0,125 0.51 1.04 WHEAT 17 0,8491 7.40 120.1 85,7 0.12 0.04 10,5 30.8 0,016 -1.242 0,4876 0.780 0.918 0,41 1.20 MAIZE 18 2.2416 2.80 33.4 83.3 0,03 0.03 1,9 3.3 -0,771 -0.656 0.4992 -0.411 -0,183 0.54 0.93 WHEAT 19 2,3435 2.68 53.7 61.5 0.05 0.03 2.1 3.3 1.350 -0,512 0.4163 -0,697 -0.297 0.52 0,80 WHEAT 20 2.2076 2,85 8.2 32.7 0.01 0.01 1,8 3.3 -0.735 -1,080 0.5281 -0,314 -0.142 0.54 0.99 WHEAT 21 2.1057 2.98 22,8 27.7 0,02 0.01 1.3 3.2 1.519 -0.758 0,4179 -0.209 -0,099 0,42 1.01 WHEAT 22 2.2755 2,76 11.1 15.3 0,01 0.01 2.0 3.2 -1.473 0.690 0,4488 -0.539 -0.237 0.54 0.84 WHEAT 23 2,1397 2,94 7.6 4e4 0.01 0.00 1.4 3.3 -0.722 -1,329 0.4898 -0.199 -0.093 0,47 1.05 WHEAT 24 2.3095 2.72 36.9 0.2 0.04 0.00 2.1 3.1 0.900 0.140 0.4303 -0.621 -0.269 0,54 0.80 WHEAT V- - 57 - VI SIMULATION AND FORECAST OF COMMODITY PRICE SERIES 63. Sections IV and V describe the main systematic behavior of each of the comnmodity price series as well as the cyclical interrelationships between pairs of prices. In this section, simulations based on the single spectral analysis are presented together with short- and medium-term forecasts of each series. For those series which were found to lag other commodity prices, additional simulations and forecasts are provided which take account of the leading effect. When the leading effect is included, only the part of the lagging series which is not explained by the leading series is simulated and forecast using the single spectral arnalysis in the same way as for the spectral analysis of individual series. The simulation and forecast values of the residual part are then combined with the part which is explained by the leading series. 64. The forecasts are presented in this way to show the differences between results based on single series spectral analysis and those based on the combination of single series spectral and cross-spectral analysis. It is clear that if major changes have occurred at the end of the historical period in the leading series they will not be reflected in the lagging series. Even if there has not been a major change in the leading series, the actual values of the leading provide information for projections of the lagging series. 65. There are four sets of results for each commodity price simulation and forecast. Simulation A uses those frequencies which are most important in explaining the series and which contribute more or less equally to the expla- nation of the price variation. Simulation B uses those frequencies used in simulation A together with frequencies of second-order importance. Simulation C uses the frequencies of simulation B and the frequencies of third-order - 58 - importance. Simulation D is created in the same manner with the addition of fourth-order importance frequencies. (The simulations are marked in each set of figures below. Also, the number of frequencies used in each trial is noted at the top.) 66. Caution should be exercised in using this technique for long-term projections. The forecasts are essentially based on the assumption that past behavior will continue into the future. The further into the future the projections are made the more likely it is that the actual series will develop different kinds of behavior than observed in the past. Maize Price Simulations and Forecasts 67. The results of the various simulations on the maize price are graphed in Figures 12 and 13. These are results only from single-series spectral analysis. Only frequencies that had a significant effect in explaining price variation were selected in these simulations. It can be seen that the more frequencies used, the more closely the simulated series tracks the actual series. However, the use of a large number of frequencies, particularly high frequencies, introduces an issue which must be considered when forecasting; high frequencies (short-term cycles) are more likely to change in a relatively short period of time. 68. The conclusion to be drawn from the simulations is that prices will soon increase from their low levels. According to simulations A and B, May 1987 was the low point. According to simulations C and D, August 1987 will be the low point in the present downswing. The level to which they will be raised is different under the different simulations. The maize price is forecast to increase over the next 2 years and to peak at the end of 1989 or the beginning of 1990. At that time prices are seen to be at least at the level observed in - 59 - FIGURE 12. IMIAE PRICE ACTUAL & SIM. ACTUAL, 2 & 3 FREQUENCIES 160*-.7 - .---....-...-.- ....-....- I3 Sim A 130 a 10 -4 L 1003 10 -' 710 -4 606 19S7201 1 97501 197801 198101 198401 198701 199001 FIG RE 13: MAIZE PRIC ACTUAL & SIM ACTUALs 6 & 12 REQUENCIES 70 T . . - . -.- ..- . -...... ... --- -. .. . -.... . . 18 ! s ill D ---./ 70i4 1SO -H 50 -7 197201 197501 197801 198101 198401 198701 199001 MONTHS - 60 - late-1983. The annual price average will be approximately $70-75Iton in 1988 and to increase to about $150 in 1990. After 1990 there is a moderate decline to about $135 in 1991 and $120 in 1992. Beef Price Simullations and Forecasts 69. The beef price simulations were carried out in two ways. The f irst ignores the linkage of the beef price to maize, while the second takes this linkage into account. A simple linear regression of the beef price on the maize price was estimated using the lagged structure explained in Section IV. The estimated equation for this part is: Beef price 9t) = 18.5 + 0.98 Maize price (t-39) - 0.51 Maize price (t-7) + 1.20 Maize price St) With adjusted R2 of 0.71 70. The results for the independent beef simulations are presented in and Figures 14 and 15. The results for the beef simulations, which were based on the linkage to the maize price, are presented in and Figures 16 and 17. Both exercises show similar patterns of price movement; a sharp increase in 1988-89 followed by a decrease in 1990. The swings are much stronger in the second exercise (see Figures 16 and 17), which is due to the projected steep increase and decline in the maize price. Soybean Price Simulations and Forecasts 71. Two sets of exercises were also carried out for the soybean price. The first is based on the results of the spectral analysis presented in Section IV. The second simulation follows the analysis reported in Section V, basing the simulated and predicted values on two elements: (a) the part which is explained by the maize and palm oil prices. (b) the residual which is not explained by those two prices. - 61 - FIGJE 14- BEf PR.ICE ACTUAL & SIM. ACTUAL. 2 & 3 FREQUENCIES 320 - 3:to 0 310 - 290- ACUL o P 280 - E 270 - Sim B 260 - 250 724 -0 230 - \ I 1320 - # Z 210 1200 - W1 D 190 170 - - 197201 1 97501 197801 1981 01 198401 198701 199001 MONIS ACTUAL, 6 & 9 FRE4UENCIES 320- 310 -0 :290 ACTUAL -- - E/P Simu C < 260t 120 I 97201 1 9750i 19o7801 19b810n1 198401 198701 1 99001 MONTHS \ 300I 220 260 0 2 10133 1800 1 40 -E 130- 120V 1 120 - I B~ 219701 19-1 170 911 180 97 190 wMONTH - 62 - FIGURE 16-- BEEF PRIM UWD TO MAIZE ACTUAL AND SIMULATED (2 & 3 FREQ.) 320 - 310 0 So ACTUAL- Sim A- 280 -n 270 - 260 250- ld40- 230 - y~220- Z 210 200 190 0- 190 Nim B & 180 - '173 - io 140 130 120 197201 1975301 197801 198101 1 98401 1 98701 199001 MONTHS ACTUAL AND SIMULATIEI (6 &12 FRE0.) 320 - 310 o 300 ATA r S irti C 290 C 280 270 260 -C 250 - o 240- 230 - E cO 220- Z 210 200- M 190 170 1 1S0 1S0 140 130 S im D I110 .. ..... i 97201 1 97501 107801 1981301 198401 198701 199001 MONTHS - 63 - The estimated equation which takes into account the lagged effect of maize and palm oil prices is: Soybean price (t) = 11.02 - 0.14 Maize price (t-li) + 1.87 Maize price (t) - 0.04 Palm oil price (t-7) + 0.13 Palm oil price Ct-1) With adjusted R2 of 0.82 The results of the first simulation are presented in and Figures 18 and 19 while the results of the second are presented in and Figures 20 and 21. 72. The general pattern of all four simulations in each exercise is similar and shows an increase in price in early 1988 which tapers out by mid 1989. The difference between the simulations is in the short-term movement and in the magnitude of the changes. As mentioned earlier, maize is mainly driven by long-term cycles, while the palm oil price is driven more by short-term cycles. The combined effect of these two price series underlies movements in the soybean price. 73. Using the first set of simulations the soybean price is forecast to peak twice during the next 5 1/2 years with the peak in annual average prices occurring in 1989 at a level of about $280/ton and bottom out in 1991 at about $230/ton. Prices would increase in the next year, 1992, to about $290. 74. The forecasts under the second set of simulations are not very different in terms of the annual averages although they are quite different in terms of the month-to-month movements. The 1988 average is about $260/ton followed by an increase in 1989 to a level of about $280. There will be almost no change in the 1990 average price, with a sharp decrease in 1991 to $220 and a subsequent increase in 1992 to about $300. - 64 - FIGUE 18- SOYBEAN PRICE ACTUAL & SIM. ACTUAL. 2 & 8 FREQUENCIES 0 450 ACTUAL 400 - 0 oo 300 ° Sim A DW 13 4 9 So- 260 200- 150 w 100 - 197201 197501 197801 198101 198401 198701 199001 MONTHS FIGURE 19: SOMEA4 PRICE ACTUAL & SIM. ACTUAL, 9 & 13 FREQUENCIES 0 450 ACTUAL 400 Sim D 35M0 - /3 S Po o3 q 00 350 0 1 200 Sim C 100 - 197201 197501 197801 198101 198401 198701 199001 MONTHS - 65 - FIGURE 20: SO'EN PRICE LWD TO MAIZE AND PALM OIL, ACT. 3 & 7 FR. B00 - 00 450 400 ACTUAL 300- 250 - 200-Si 160 100 - 97201 197501 197801 198101 198401 198701 199001 MONTHS FIGURE 21- SOYBEAN PRICE LIWD TO MAIZE AND PALM OIL, ACT, 10 A 16 FR. 0o 4600 13 400 - ACTUAL o 0 Sim D 350 300 250 - 100 - 1e7201 197601 197801 198101 198401 198701 199001 MONTHS 66 - Wheat Price Simulations and Forecasts 75. The wheat price simulation and forecast were done in the same manner as for maize. The results are presented in Figures 22 and 23. The wheat price is mainly driven by long-term cycles as discussed earlier, and therefore should fluctuate vigorously in the next 5 1/2 years. Because short-term cycles do not influence the wheat price greatly there is not a large difference in the patterns observed in the four simulations reported. The turnabout in wheat prices is forecast for the latter half of 1987. A sharp increase in the wheat price is expected in the next 3 years, tapering out in mid-1990. After that there will be a moderate decrease. By the end of 1992 the wheat price is forecast to be approximately $160 per ton. Rice Price Simulations and Forecasts 76. As observed in Section V, the rice price lags the wheat price with a very short duration. It was also clear that the relationship between these two prices is more complicated than can be learned from cross-spectral analysis. A simple type of relationship was assumed, consistent with the results presented in Section V. The simulation values given in Figures 26 and 27 take into account the leading effect of wheat on rice, Preceding these results are the rice price projections and the simulated values based on the spectral analysis of the rice price only (see Figures 24 and 25). 77. The estimated equation that represents the relationship between the rice price and the wheat price is: Rice price = 40.77 + 1.74 Wheat price (t-3) With adjusted R2 of 0.66 - 67 - FIGURE 22-- WHEAT PRICE ACTUAL & SA.' ACTUAL, 2 & 5 FREQUENCIES 250 - 240 230 ACTUAL Sim A 220-I 210 - 200 190 - z I80- 170- 1502 Sim B 1401 130 120- I 110 -i 100 90 80 - 50 197201 197501 197801 198101 19841 198701 199001 MONTHS FIGURE23- WHEAT PRICE ACTUAL &SM ACTUAL. 8 & I11 FREQUENCIES 250 - 240 HACTUAL S imD--- 230 - 210 - 200 - 190 180-13i im C o 170 * 150- 1401 130- 120 - 90 197201 197501 197801 198101 196401 196701 199001 MONTHS - 68 - FIGlRE 2+ RICE PRICE ACTUAL & SKA ACTUAL, 3 & 5 FREQUENCIES 700 --. .-.-.......-.........1 e ACTUAL 500 - * I a ,/ g 400] 300Slii A 50 - a 200< V 100l 197201 197501 197801 198101 198401 198701 199001 MONTHS FIGURE 255a RICE PRICE ACTUAL & SIM. ACTUAL. 7 & 10 FREQUENCIES 700T 70oT- eACTUAL 500 Slim D- o 400- S C 100-J 0 197201 197501 197801 l98101 198401 198701 199001 MONTHS - 69 - FIGRE 26.- RICE PRICE LWED TO WHEAT ACTUAL AND SIMULATED (2 & 4 FREQ.) 700 600 ACTUAL | v S im B 0 400- Sool 200 Sim A 100 0 -. 197201 197501 197801 198101 198401 198701 199001 MONTHS FIGURE 27: RICE PRICE LINKED TO WHEAI ACTUAL AND SIMULATED (8 & 13 FREQ.) 700 -- - - - 600 ACTUAL 9~ Sim C 500 -H O 400-SnD Y) 300- 2004 0 197201 197501 197801 198101 198401 1 98701 199001 MONTHS - 70 - This equation implies a very short adjustment by the rice price to changes in the wheat price series. As can be seen, there are only small differences between the results of the two sets of simulations in regard to the pattern of the price movement, although the average annual prices for the years 1987-89 are different. Using the first method, the average price of rice is forecast to be about $115/ton for 1987 and using the second method about $120. Using the second method the price will be about $110 in 1988 and almost triple by the end of 1989 to average about $340 for the year. For the same years the single series analysis gives an average price of about $120 for 1988 and $390 for 1989. For the next 3 years, 1990-92, the projections are very close; $360 in 1990, $197 in 1991 and $175 in 1992 under the first method and $350 in 1990, $173 in 1991 and $179 in 1992 under the second. 78. In terms of these forecasts, therefore, the rice price is expected to increase in the short-term in a similar way to wheat and maize. It is of interest to indicate that this forecast pattern of movement in these prices was the US government's goal under the 1985 Food Security Act. The seeds of the recovery in prices are already evident in the changes that have occurred recently in the grain markets in terms of the global reductions in production and increases in consumption. - 71 - READINGS Anderson, T.W., The Statistical Analysis of Time Series, 1971, New York; John Wiley and Sons. Barnett, R.C., D.A. Bessler and R.L. Thompson, "The Money Supply and Nominal Agricultural Prices," American Journal of Agricultural Economics, 65(2) 1983: 303-7. Barras R., and D. Ferguson, "A Spectral Analysis of Building Cycles in Britain," Environment and Planning, 17 (1985): 1369-91. Barth, J.R. and J.T. Bennett, "Cyclical Behavior, Seasonality, and Trend in Economic Time Series," Nebraska Journal of Economics and Business, 13(1) 1974: 48-69. Bartlett, M.S., An Introduction to Stochastic Processes, 1966, Second Edition, Cambridge: Cambridge Press. Bennett, R.J., Spatial Time Series, Analysis-Forecast-Control, 1979, Piorn Limited, London. Browne, F.X., "Departure from Interest Rate Parity: Further Evidence," Journal of Banking and Finance, 7(1983): 253-72. Callen, J.L., C.C.Y4 Rivan and P.C.Y. Yip, "Foreign-Exchange Rate Dynamics: An Enmpirical Study Using Maximum Entropy Spectral Analysis," Journal of Business and Economic Statistics, 3(2) 1985: 149-55. Chan, W.S. and H. Tang,, "On Tests for Non-Linearly in Time Series Analysis," Journal of Forecasting, 5(1986): 217-28. Dahlhaus, R., "Spectral Analysis with Tapered Data," Journal of Time Series Analysis, 4(3) 1983: 163-75. Eberts, R.W., B.M. Streece, "A Test for Granger-Causality in a Multivariate ARMA Model," Journal of Empirical Economics, 9(1984):51-8. Eichenbaum, M., "Vector Autoregressions for Causal Inference: Comment," Carnegie-Rochester Conference Series on Public Policy i2(1985): 305-18. Fuller, W.A., Introduction to Statistical Time Series, 1976, New York: John Wiley and Sons. Granger, C.W.J., "Investigating Causal Relations by Econometric Models and Cross Spectral Methods," Econometrica, 37(3) 1969. - 72 - Griffith, G.R., "A Cross-Spectral Approach to Measuring Efficiency in the New South Wales Pigmeat Market," Review of Marketing and Agricultural Economics, 43(4) 1975: 163-83. HIavemer, A. and P.A.V. Swamy, "A Random Coefficient Approach to :Jeasonal Adjustment of Economic Time Series," Journal of Econometrics, 15(1981): 177-209. Jarvis, L.S., "Cattle as Capital Goods and Ranchers as Portfolio Managers: An Application to the Argentine Cattle Sector," Journal of Political Economy, 82(1974): 489-520. Nelson, C.R. and S. Beveridge, "A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the Business Cycle,' Journal of Monetary Economics 7(1981): 151-74. Nelson C.R. and C.I. Plasser, "Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications," Journal of Monetary Economics, 10(1982): 139-62. Pierce D.A. and L.D. Haugh, "Causality in Temporal Systems; Charaterizations and a Survey," Journal of Econometrics, 5 (1977): 265-93. Pindyck, R.S. and D.L. Rubinfeld, Econometric Models and Economic Forecasts, 1981, Second Edition, McGraw-Hill, Inc. New York. Sims, C.A., "Money, Income and Causality," American Economic Review, 62(4) 1972: 540-52. van Dijk G. and C. Mackel, "Fundamental Changes in Price Relationships: An Investigation of the UK Feed Grain Market, 1971/72 to 1978/79, Using Spectral Analysis," European Review of Agricultural Economics, 10(1) 1983: 15-31. Van Eaijk C., "A Spectral Analysis of the Kondratieff Cycle," Kyklos, 35(1982): 468-99. - 73 - APPENDIX THE THEORETICAL BACKCROUND OF SPECTRAL AND CROSS-SPECTRAL AALYSIS In many areas of applied mathematics it is convenient to use an approximating function consisting of a linear combination of elementary function, as was stated very clearly in Weierstrers theorem. The general implication of this theorem is that any continuous function on a compact set may be approximated by a polynomial. It is sometimes very useful to construct a set of vectors as a basis so that all other vectors in the space can be expressed as a linear combination of the vectors' elements of the basis. Let us assume that we have a function defined on a finite number of points N and let us further assume the't we have a sat of criteria m,r where tz, r = 0, l, ...y L(N) and L(N) is the largest integer: less than equal to N It follows that N-i 21Im 211rN 1) I cos N-t cos N t = 14 if m = r O or 2 N N -t if m = r Oor 2 -O if m * r N-i. 211m 211r (2) , sin -t cos -N t = 0 V m,r t=O NN N-1 2I l (3) sin 21mt co2s N if m Nr or t=I N N 2 2 -o if m r - 74 It follows from (1), (2) and (3) that the N functions form an orthogonal basis, and th1-.efore any function f(*) defined in N integers can be represented by L[N] (4) f(t) = x (am cos wmt + bm sin umt), for t=O, 1, ..., N-1 m=Om where W -= 2IIm m-Op 19 .s.e L[N] m N N-1 2 1 f(t) cos w t a N , for m = 1, 29 *.,v L[-N] N-1 X f(t) cos wmt t=0 __ _ _ _ _ _N , for m 0, and m - if N is even N-1 I f(t) sin t b - tO N - for m - 1, 2, ..., L[N-1] am and bm at e called Fourier coefficients. One way to obtain representation (4) is to find am and bm such that we minimize N-1 LINI 2 (5) 1 {f(t) - I (am cosamt ibm n m} The underlying model assumed for this type of approach is - 75 - (6) yt = f(t) + ut, t=1, ..., N where E (Ut) = 0; E(U 2) = a and E (U ) 0 ° V t *s t t S The normal equations for ao, am and bm ( m=l, *.. L[N]) are: (7) N 0 0 ...........0 a0 t 0 ½N 0 .,..*..0 a1 N 2Q11 L=N 0 0 N ........... .0 b = sin 211.1 N 2I1L[N] 00 0 N bL[N] yt sin- N Because the trigonometric functions are orthogonal the matrix of coefficients of the estimates is diagonal. Thus the solution to these normal equations are ^ 1 N (8) a0o x t=Y t=1 2 N 211k a 2 Cos -t- k=l, ...s L[N] 2 ! xY N 211 fit=l b N s in 211k- k=1, ..., L[N] If N is even and period 2 is desired then equation (4) can be modified to include the additional term, i.e. (4) will be L N] (4W) f(t) = a0 + I1(a cos w t + b sin w t) + ml m m m m - 76 - Lf N3 () f(t) =a + ](ar cos wmt + b 8sin wmt) + aIN(-lI) m=l and therefore an additional term will be estimated to get NNt- yy (-4)t %½N N t t=1~ For an infinite function y(t) it can be shown that the analogous procedure for obtaining estimates for am and bm gives T (9) am J cos I xf(x), dx k=O l 2, ..., L[N] T bm = sin JT f(x)dx k=1, 2, . Q. L[N] where T is the length of the interval on which y(t) is defined, and x T1 k=O, 1, ..., L[N] In this case equation (4) can be regarded as the approximation sum of the infinite function Yte N ao .L N] ,2 ' Note that 2 -2 L (a + bk)) is completely analogous to the sum of 2 2 k= k k squares due to regression on LIN] orthogor;al independent variables of finite least square theory. n Let us define S (x) I (ak cos XkX + bk sin XkX) k0O as the partial sum of an infinite function y(t), where Xk is defined as in (9). Then we can state the theorem. - 77 - Theorem 1: Let f(x) be a continuous periodic function of period 211 with derivative fV(X) that is square intergrable. The fourier series of f(x) converges to f(x) absolutely and uniformly. 1/ It follows from the proof of the theorem that the only required restriction is that 1 (a hi +l bh ) converges. It also follows from Theorem 1 that: Theorem 2: If p(h), the correlation function of a stationary time series is absolutely summable (i.e. f p(h) dh converges). Then there exists a continuous function f(w) such that: (i) p(h) = , f(w) cos whd (ii) f(W) > 0 Vw (iii) f f(w) dw =1 -11 (iv) f (w) is an even function. The results of Theorem 1 and 2 can be stated in the compact form of if See Fuller 1976 pp. 107-109. - 78 - (10) f(x) f=w ) cos kw cos kx dw k=l +I J f(w) sin kw sin kx dw 1 ikxr( ikwd 211 e f1 7 e dw p(k) can be written therefore (by theorem 2) as: (1) p(k) = , ei I y p(h) e1kdw -h=-m The functions in (10) and (11) of f(x) and p(h) is called the transform pair. We can associate the constant and the negative exponential with one transform which is called the Fourier transform or spectral density, and thus f (w) defined by I -iwh (12) f(w) I p(h) e 2Hh=-c is the Fourier transform of p(h). The positive exponent is called the inverse transform or (in its more common name in mathematics) the characteristic function and thus the correlation function is defined by (13) p(h) J' f(w) elhod is the inverse transform of f(w) or the characteristic function of f(w). 79 - The form of (13) is known in the more general form p(h) =eWhdG(M) where CGM) is the statistical distribution function of the process and the integral is Lebesque-Stieltjes. Vector Process The spectral representation of a vector time series comes straightforward from that of scalar time series. Lets denote the cross variance of two zero mean stationary time series xit and xmt by yjm(h) where yjm(h) = {r(h))jm = E {x. x } then jt jm jmjt xm,t+hl under the same assumption of the cross variance as of the single covariance -iw (14) f. (w) jm(h) e = f. (w) jm h-_. jm is a continuous periodic function of w. It is called the cross spectral function of xit and xmt. fjm(w) is, in general, a complex valued function and as such it can be written as (15) fjm(w) cjm (w) - iqj. (w) where din (w), and gjm (w) are both real valued function of w. cjm is called coincident spectral density (or cospectrum) and qjm (w) is called quadrature spectral density cjm(w) is the cosine transform and is even function whereas qim(w) is the sine position and therefore an odd function of w. Thus these two functions can be more specifically be defined as: (16) cm (w) = 1 2 jm(h) + Yjm(h)]eih iq m(w21 [Yjm h) - jm( j jL2 2 h h=LY. - 80 - For a stationary vector time series of dimension k satisfying co ly'. (h)l 00 Then, for 11 (d /n)