JOURNAL OF APPLIED ECONOMETRICS, VOL. 6, 239-254 (1991)

PRICE FORMATION IN COMMODITY MARKETS

M. J . LORD Development Policy Research Division, Inier-Atnericun Developtneni Bank, 20577 Washington, DC USA

SUMMARY

This paper develops a theory-consistent market model for storable commodities and illustrates its characterization of the data-generating process for a set of major traded commodities. The dynamics of the system incorporate recent advances in modelling techniques. Cointegrated variables in the demand functions are represented by the error correction mechanism (ECM), and expected prices in the stock demand relationship are generated by a rational expectations process. The outside-sample performance of the model is tested against the pure time-series model used to formulate expected prices, and is shown to have a smaller mean square error than that of the time-series model. Thus the model provides comparatively efficient forecasts and, unlike models constructed in their reduced form, permits consideration of key behavioural relationships in commodity markets.

1. INTRODUCTION

Despite widespread attempts to model the process of price formation in commodity markets, there has been a noticeable absence of theory-consistent econometric models that maintain a priori restrictions on parameters in their dynamic specification. Instead, commodity market models have typically used the reduced form of the system of equations, with the result that parameter estimates of the price equation are unrelated to the separate estimates of parameters in the original structural form of the system of the equations in the model. In a recent example of this approach, Hwa (1985) obtained a price equation from the equilibrium relationship between actual and desired stocks for a commodity, and used estimates of that relationship to evaluate the performance of the model. Parameter estimates of the price equation in that study were unrelated to the separate estimates also obtained for the parameters of the supply and demand functions, particularly since the price equation, which was nonlinear in the model, was conveniently hypothesized to be linear in its logarithmic form. Such direct estimation of price relationships in commodity markets have been commonly adopted by others in modelling commodity markets (for a survey of commodity market modelling methodologies, see Labys, 1987).

As a consequence of this common modelling procedure there has been little, if any, consideration given to dynamic stability conditions of the structural form of the system. Since production often has a greater lagged response to price changes than does consumption in commodity markets, it is unclear whether parameter estimates of the supply and demand functions in the system satisfy the dynamic stability conditions of the resulting cobweb model when the price of the commodity is obtained from the reduced form of the model.

The present modelling strategy departs from usual modelling practices by providing separate estimates of the supply and demand components in commodity markets, and by using those

0883-7252/91/030239-16$08.00 0 1991 by John Wiley & Sons, Ltd.

Received June 1988 Revised February 1991

240 M . J . LORD

parameter estimates to determine the steady-state dynamic equilibrium solution for market prices. The theory postulated provides a parsimonious interpretation of the supply and demand relationships in the market, and it suggests the equilibrium conditions that need to be imposed in order to formulate and estimate a complete market system. The dynamic specification of the relationships in the system adopts recent work on time-series models that explains observed disequilibria in the context of steady-state solutions of behavioural relationships in a market; as a consequence the dynamic specification of the system of equations is capable of reproducing the postulated theory of market price formation. Lord (1991) used this approach in the context of international commodity trade in the presence of imperfect competition. The implementation of the approach to commodity markets in what follows incorporates cointegration tests of postulated long-run relationships between variables in those markets.

Although a common framework is applied to the characterization of the underlying data- generating processes in commodity markets, the inherent features of each commodity market are retained. Commodity markets differ from one another in terms of the degree of competition that prevails among buyers and sellers, the extent of market fragmentation, and whether there are close substitutes. Since different market conditions give rise to different forms of competition, there are inherent difficulties in developing a single model to encompass those different forms of competition (see Lord, 1989). This study therefore seeks to represent only the key features of the processed that give rise to price formation in the particular markets being modelled.

The plan of the paper is as follows. Section 2 formulates the dynamic relationships that characterize the adjustment processes of consumers and producers in markets towards their steady-state equilibrium solutions. Section 3 applies the general specification of the model to seven major traded commodities, it presents estimates for each of the equations in the system, and it evaluates the performance of the system of equations as a whole for the individual commodities in terms of the pure time-series model used to characterize price expectations in the system. The final section presents the conclusions.

2 . THE MODEL

Commodity markets adjust with lagged responses by economic agents, even in time-series data with annual periodicity. Suppliers of coffee, cocoa, and copper, for example, take several years to react fully to price changes. Consumer delays arise from the derived demand for raw materials and for basic, pre-processed foods, from the filtering of price changes to the final demand for commodities. The lack of instantaneous adjustment in production and consumption gives rise to temporary states of disequilibria in commodity markets. In this section the dynamic relationships in commodity markets are formulated in terms of the adjustment processes of producers and consumers towards their steady-state equilibrium solutions.

The equilibrium relationship between consumption and income changes can be described by a cointegral process of the time-series of these variables when the individual series are integrated of order one, denoted Z(l), and a linear combination of these two series is integrated of order zero, denoted Z(0) (Granger, 1983; Granger and Weiss, 1983). The adjustment process of consumption to income changes can then be represented by the error correction mechanism (ECM), which corrects for the disequilibria that arise in the inter-temporal adjustment process between these cointegrated variables (Engle and Granger, 1987). The ECM representation of the relationship relating consumption, C, to economic activity, Y , and the commodity’s price,

COMMODITY MARKETS 24 1

P , relative to the general price index, D, is:

Acr = 010 + a lAy, + ~ Z ( C - y)r- 1 + a ~ y r - I + W A ( P - d)r + a s ( p - d) t - I + U l r (1)

where lower-case letters denote logarithms of corresponding capital letters, e.g. ( p - d ) = In(P/D), and the expected signs are a1 > 0 if the short-run income elasticity is to be po$itive, - 1 < 012 < 0 for convergence of the lag distribution, a3 > a2 if the long-run income elasticity is to be positive, and a4, a5 < 0 for the price elasticity to be negative. In ( I ) , Ac and A y are cointegrated i f a2 # 0, and the term a3y,- I accounts for any non-proportional response of demand for a commodity as a result of a change in the level of income. When a3 < 0, consumption is inelastic with respect to income; when a3 = 0, it has a unitary elasticity; and when a3 > 0, it is income elastic.

For any steady-state growth path, Ac ,= gl and Ay,= gz are the growth rates of consumption and income respectively. Market prices would not be expected to have any long- run dynamic influence since, as pointed out by Currie (1981), relative price changes cannot continue indefinitely and the theoretical justification for their dynamic effects is absent. As such, their effect is constrained to zero so that Apt = 0 in the long run. Hence, following Currie (1981), the long-run dynamic relationship implicit in (1) is:

(2 )

where k l = exp( -a0/az + xlgZ), such that H I = (1 - a1)/a2 is the income growth elasticity. Thus, on a steady-state growth path, the level of consumption depends on the rate of growth of income, gz, as well as on the levels of income and of the price of the commodity.

The dynamics underlying the production function are often characterized by long lags, since the effects of price changes usually take a long time to work themselves through to supply. The lag structure of production can be represented by a stochastic difference equation whereby 'difference' formulations of the variables are nested in their level form. The advantage of this transformation is that it avoids multicollinearity between lagged values of the price variables when the dynamics are of a relatively high order, whereas reformulations with only difference variables meant to avoid serial correlation when the true relationship is in terms of levels will introduce an additional moving average term into the disturbance (Plosser and Schwert, 1977). The expression for production, Q, in terms of own market price, P, relative to the general price deflator, D , major disturbances, W, and a secular trend, T, measuring technological changes in the production process, is:

C = klY1 - U I / U L ( ~ / D ) - U I / O L L

11- I

Aqr = PO + P1qr-1 + C P2+kA(p - d1r-r + Y A ( P - d)i-n + ydT+ 7s wr + Uzr (3)

where, as before, lower-case letters denote logarithms of corresponding capital letters, and where the expected signs are - 1 < < 0; P Z + k , 73 > 0; 74, 7 5 > 0. Since production of a commodity has a transient response to the rate of change of its constant dollar price, the long-run equilibrium solution of (3) is:

k = O

where kz = exp [ - ( P o + Y ~ T + ys W)/Pl ] . When the production function takes on the constant elasticity of substitution (CES) form, the value of the price elasticity of supply, -y3/P1, is directly dependent on returns to scale: when there are decreasing returns to scale the supply function is strictly increasing; when there are constant returns to scale the supply function is constant; and when there are increasing returns to scale the supply function is strictly decreasing. Technological changes in the production process are reflected in the value of the

242 M. J. LORD

coefficient for T, and these changes bring about a long-term, or secular, shift in the supply function.

Equilibrium in the world market of a commodity is attained when the demand for stocks, K d , equals the supply of stocks, K':

K~ = K' ( 5 )

The supply of stocks is simply defined to be actual stocks on hand, and stock supply changes equal the difference between production and consumption:

A K : = Q t - Ci (6)

Demand for stocks arises from transaction, precautionary, and speculative motives. The transactions and precautionary demand for stocks originate from the desire of producers, consumers, and merchants to hold stocks to meet future consumption requirements, as well as to avoid delays and provide continuity in supplies. For these reasons consideration is usually given to the relationship between the demand for stocks and either consumption or production of the commodity (Labys, 1973: 61-83). In the first relationship agents adjust their stock levels to consumption; in the second they adjust their stocks to the level of output. In both of these stockpiling decision rules the control errors ulr = Kt - XICt and u2[ = Kt - X2Qt are expected to be Z(O), in which case production and consumption will be multi-cointegrated (Granger and Lee, 1989a).

The speculative demand for stocks is motivated by the desire of producers, consumers, and merchants to profit from changes in future market conditions. Thus, whereas production and consumption decisions are determined by backward-looking variables, the demand for stocks is related to forward-looking variables in so far as the speculative motive for holding stocks is related to expectations concerning future market conditions. ' These desires have typically been related to expected prices even though the formulation gives rise to an element of inconsistency, since the behavioural econometric model which solves for the market price uses a time-series model to generate price expectations.2 Thus the demand for stocks, Kd, depends on production, Q (or consumption, C), and the expected price, P':

(7)

where $1, $2 > 0. There are a number of costs of stocks primarily related to storage and interest costs and to depreciation (see Ghosh, Gilbert, and Hughes Hallett, 1987: 25-29); these have been included in the empirical estimates of the relationship, and have been made implicit in the specification to simplify the discussion that follows.

The expected price is assumed to be formed through a rational process. The rational expectations hypothesis of Muth (1961) uses conditional probability theory to argue that expectations about a variable, such as the price of a commodity, should be treated as a

kP = $0 + $14t + $2p: + V3t

' One could interpret estimates of relationships such as those of production and consumption as backward solutions of their implied forward-looking relationships. This interpretation would require that there be constancy in the parameters of the marginal models for the expectations processes whose solution values enter into the structural equation (see Hendry's (1988) critique of the marginal expectations models used in Cuthbertson's (1988) demand for money equation). 'In a recent study by Gilbert and Palaskas (1990), expectations have been formulated in terms of quantities rather than prices. This alternative formulation is legitimate in that producers and consumers anticipate the effects that lagged production responses will have on their future inventory levels. Price expectations are therefore a reflection of quantity expectations. Despite practical difficulties in the implementation of this approach therr are legitimate grounds for formulating expectations in terms of quantities, and this approach may eventually lead to greater power in modelling commodity markets.

COMMODITY MARKETS 243

problem of optimal forecasting whereby the minimum mean square error forecast of that variable is produced by all available information at the time the forecast is made. Thus, at period I the expected price, Pe , of a commodity is the expected value of the price, conditional on all information, I , available at period t - rn to agents concerned with stocks in the commodity market:

where Et-rll is the expectation operator for expectations formed at t - m. The characterization of the data-generating process for expected prices in the stock demand

relationship in (7 ) is based on the work of Wallis (1980) (for a recent application to the market for the chicken broiler industry, see Goodwin and Sheffrin, 1982). It is assumed that the price of the commodity follows a general A R M A ( p , 4 ) process:

PF= Er-rn(PrlI) (8)

4(L)pr = O(L)ut (9)

where @ ( L ) and B ( L ) are polynomials in the lag operator, L , which are defined as

4 ( L ) = 1 - q5IL - 42LZ - ... - &LP

O ( L ) = 1 - OIL - 02L2 - '.. - 6,LQ and

Then Pr can be represented by an autoregressive process:

where

The minimum mean square error forecast based on information up to period t - n z is:

P : = u(L)Pt-rri (1 1)

which states that the forecast error is zero. Substitution of (1 1) into ( 7 ) gives:

k:' = $0 + $14r + $ ~ ( L ) p r - I + ~ 3 r

and multiplication of the expression through by O ( L ) gives:

where

The 8( L ) polynomial associated with the terms that contain K d and Q is of a relatively low order in annual time-series data. The ECM form of the stock demand relationship with a first- order polynomial in the lag operator is:

Ak:' = $6 + $lAqr + Ez(kd - 4)r- I + $2u(L)pr- I + ~ 4 , (13)

where t 2 = 8 - 1 and the expected signs are $1 > 0, - 1 < E2 < 0, and $2 > 0. Hence, the dynamic specification of the relationship for the demand for stocks yields a unitary elasticity of demand for stocks with respect to production (or consumption) of the commodity in its long-run dynamic equilibrium solution:

(14) ~d = k3Qpe-*','t'

244 M. J . LORD

where k3 = exp(- $;/tz + xZg3), such that H Z = ( 1 - $I)/& is the production or consumption growth elasticity. Thus, the demand for stocks of the commodity is related to both the level and rate of growth of production or consumption, as well as to the expected price level. The unitary elasticity of demand for stocks with respect to either production or consumption means that producers and consumers will maintain a fixed proportion of stocks relative to their levels of production and consumption. This proposition can be tested by the inclusion of a second term for lagged production in ( 1 3); a statistically significant coefficient in the estimated e~quation would imply that stockholdings have a non-proportional response to changes in the level of production in the market. As will be seen below, the results of our tests of the proportionality assumption in (13) show that the demand for stocks has a unitary elasticity with respect to either production or consumption.

Direct estimation of (13) in its ‘reduced form’ generates greater forecast error variance than does the separate estimation of the parameter value of $ 2 ( L ) for P I , and the use of those parameter values in the relationship for K d (Wallis, 1980). The reason is that rational producers and consumers know the data-generating process of prices, and use that information to minimize any errors in their stock requirement forecasts. Accordingly, the relationship for the commodity price, P , has first been estimated as an ARMA process; this relationship has then been used to estimate the stock demand relationship in (13). In this way forecasts of the commodity price, which are based on the past behaviour of the variable, are introduced into the dynamic specification used to generate forecasts of the demand for stocks.

3. EMPIRICAL ANALYSIS

The above system of equations has been used to construct market models for seven commodities: coffee, cocoa, copper, sugar, cotton, maize, and soybeans. These products are representative of the major internationally traded commodities. The Appendix describes the data and their sources. The equations were initially estimated by the method of two-stage least- squares; when consumption or production was found to have a non-contemporaneous response to price changes the equations were estimated by the method of ordinary least- squares, since a trade-off exists between consistency and efficiency in the method of instrumental variables (for details, see Judge et al., 1985: 167-79; Harvey, 1981: 77-80).

In order to examine the stability of the estimated coefficients a test of parameter constancy following the 1981 world-wide recession was performed with the Chow test (Maddala, 1977: 198-201). The empirical results indicate that the general dynamic specification provides a good representation of the data-generating process in the commodity markets. The test of parameter constancy showed the coefficients to be stable at the 5 per cent level of significance in all the estimated relationship^.^ 3 ’ Since there is a trade-off between consistency and efficiency in the method of instrumental variables (IV), it is appropriate to test the null hypothesis that the IV estimate is not significantly different from that of ordinary leajt- squares (OLS), using the test procedure derived by Durbin (1954); i f the difference is insignificant, then the hypothesis that OLS provides consistent estimates is not rejected, and the IV estimate is not used since it would involve a certain loss in efficiency. ‘Empirical estimates were performed with the GIVE computer software. In the case of sugar, separate considerations were given to consumption and production in the United States, which is one of the largest markets for sugar, and the European Economic Community (EEC), which has become the largest sugar-producing region. This distinction was motivated by the large amount of protection given to producers in the United States and the EEC which alterc the price to producers and consumers and, consequently, the behaviour of agents in those markets. Moreover, sugar consumption in the United States decelerated in the sample period as a result of the rapid substitution of high fructose corn syrup for sugar in the beverage industry. This occurrence was captured by the inclusion of a quadratic trend variable in the market.

COMMODITY MARKETS 245

Consumption

Empirical evidence provided by the coefficient estimates of the error-correcting term in the consumption equation points to the existence of a cointegral process between consumption and income in all the commodities. The coefficients were found to be close to unity in absolute terms in half of the commodities, which reflects the relatively quick response of consumers to income and price changes. In the consumption equation of the other commodities, Granger (1986) has noted that 'error-correcting' can be a somewhat too optimistic title for the term, since it takes a great deal of time for consumption to resume its long-run equilibrium growth path once a disequilibrium arises between consumption and income.

The parameters of the consumption function are presented in Table I. On average the seven commodities were found to have a unitary income elasticity of demand, where income has been measured on the basis of total, rather than per capita, gross domestic product (GDP). However, individual elasticities vary, so that generalizations about elasticities on the basis of an average would be misleading. Among individual commodities, the estimated income elasticities are consistent with those calculated by others. In coffee, the 0 .6 long-term income elasticity compares with 0.45 calculated by Akiyama and Duncan (1982) and Hwa (1985). The elasticity estimate for cocoa is very similar to that calculated by Hwa (1985), though it is significantly higher than that estimated by Weymar (1968). The unitary income elasticities for cotton and copper are close to the estimates by Monke and Guisinger (1985) and Adams and Behrman (1976) for cotton, and by Hwa (1985) for copper.

In addition to the effects that changes in the level of economic activity have on consumption, there are generally strong dynamic effects on consumption arising from changes in the rate of growth of economic activity. These effects, which are measured by the income growth

Table I. Consumption functions

Income Price Income growth elasticitya elasticity elasticityb

Coffee Cocoa Maize Soybeans Cotton Sugar'

United States EEC Rest-of-World

Copper

0.62 0.62 1 a 0 0 1 .oo I *oo 0.89 1 *oo 0.11 1.00 1 .oo

-- 0.17 -0.19 -0.27 - 0.28 -0.75 - 0.07 -0.12

- 0.07 - 0.08

-

0.34 0.20 2.87

- 1.31 - 2.87 - 1.28

0.44 -0.09 - 1.63

7.28

a Measured with respect to a change in aggregate, rather than per capita, gross domestic product (GDP). bThe income growth elasticity is the percentage change in consumption brought about by a 1 per cent change in the rate of growth of economic activity, g, in equation (2). ' Trade-weighted average. Notes: ( I ) - Not significant at the 5 per cent level.

(2) Details of the estimates are found in Lord (1991, Annex C ) .

'Comparison of elasticity estimates with different sample periods is valid when the estimated parameters in the relationships are stable. As noted above, the tests for stability of coefficients did not reject the null hypothesis of parameter constancy in any of the estiniated relationship?.

246 M. J . LORD

elasticity, produce either a positive or negative impact on demand. When changes in the long- term rate of growth of income, denoted g2, bring about a less-than-proportional change in the rate of growth of consumption, gl, then the demand for the commodity will decrease in response to an acceleration in the rate of economic growth. On the other hand, when changes in g2 cause a more-than-proportional change in gl, then the demand for the commodity will increase as a result of an acceleration in the rate of economic growth.

As expected, the price elasticities of demand for all the commodities are significantly less than unity, the average elasticity for the commodities being -0.25. The very low price elasticity for sugar is consistent with that estimated by Hwa (1985), although it is somewhat lower than that estimated by Adams and Behrman (1976).6 The present calculation for copper is similar to that of Hwa (1985) but is lower than that calculated by Fisher, Cootner, and Baily (1972); the estimate for cotton lies between those of Monke and Guisinger (1985) and Adams and Behrman (1976); similarly, the present result for soybeans lies between the estimates by Houck and Mann (1968) and August0 and Pollak (1981). On the other hand, the range of elasticity calculations for coffee and cocoa by Weymar (1968), Adams and Behrman (1976), Akiyama and Duncan (1982), and Hwa (1985) were somewhat higher than those calculated in the present study.

Production

There are two fundamental questions concerning the supply of commodities: one dealing with price elasticities, the other dealing with lag structures. Production is inelastic with respect to price in all the commodities (see Table 11). Although this result is in line with other estimates,

Table 11. Production functions

Price elasticity Average lag No. of periods for

Impact a Total Mean Median 90% of response ~ ~~ ~~~~~ ~~

Coffee 0-14' 0.16' Brazil 0.28 (t - 4) 0.29 4.0 3.5 3.9

Cocoa 0.09 ( t - 8) 0.10 8.1 7.5 8 .0 Maize 0.09 (t - 2) 0.10 2.1 1.5 2.0

Rest of World 0.08 ( t - 3) 0.10 3.2 2.6 3.6

Soybeans 0.25 (t - I ) 0-33 1 . 3 0.7 1.8 Cotton 0.13 (f-3) 0.38 4.8 3.1 1.4 Sugar 0.04' 0.05'

USA 0.09 ( t - 2) 0.18 2.6 1.9 3.9 EEC 0-11 ( t - I ) 0.12 1 . 1 0-5 1.0 Rest of World 0.03 (f - 3) 0.03 3.0 2.5 2.9

Copper 0.13 ( t ) 0.61 17.4 13.8 40-0

"The impact elasticity measures the first-period response of production to a change in price. The notation in parentheses indicates the period in which production first responds to a change in price. bThe total, or long-run, elasticity measures the cumulative response of production to a change in pric'c. ' Trade-weighted average. Note: Details of the estimates are found in Lord (1991, Annex C ) .

6The reported elasticity estimates of Adams and Behrrnan for sugar, as well as those that follow for cotton by the same authors, and of Fisher, Cootner, and Baily (l??) for copper are trade-weighted averages of estimates for individual countries or regions.

COMMODITY MARKETS 247

the range of those estimates tends to be fairly wide. ' For instance, the long-run price elasticity of sugar production estimated in this study is similar to that estimated by Hwa (1985), but it is significantly below the estimate by Adams and Behrman (1976).' Although the calculated elasticities for cocoa and cotton are also below those estimated by Adams and Behrman, that of coffee is fairly close to that estimated by Adams and Behrman, and it is substantially below those estimated by Hwa (1985) and Akiyama and Duncan (1982). The calculated elasticity for copper is substantially higher than that obtained by Hwa, but it is lower than that calculated by Fisher, Cootner, and Baily (1972). In soybeans the present elasticity estimate is fairly similar to that calculated by Houck and Mann (1968), whereas Augusto and Pollak (1981) have obtained a much larger estimate. Accordingly, the present elasticity estimates are within the range of those calculated in other studies, but the range of those estimates tends to be wide.

The shape of the lag distribution of the commodities is described by three statistics in Table 11. A comparison of the mean and median lags provides an indication of how rapidly the lag coefficients decline over time. As would be expected, an asymmetrical distribution which 'tails off' characterizes the supply response of all the commodities. The rate at which the lag coefficients converge to zero is also described by the difference between the median lag and the time it takes for 90 per cent of the total response to occur.

The lag distributions are closely associated with the gestation period between the planting decision and harvesting of agricultural commodities, and with the period between the investment decision and capacity initiation or expansion in minerals. For example, the gestation period for coffee is 3-4 years, and 8 years transpire between the planting response to a price change and when cocoa trees begin to bear fruit. The supply of these commodities can only be increased in the short run by the expansion of output through yield improvements. In contrast, over 90 per cent of the total response of the supply of maize and soybeans occurs within 2 years of a price change.

Although cotton is an annual crop in most areas of the world, there is a 3-year lag before production adjusts to changes in prices. Moreover, production takes several years to adjust completely to a price change. Sugar production adjustments to price changes are slow because the cane occupies the soil for several years, and there are large fixed costs of production which further slow down the response time. The response of producers, other than those in the United States and the EEC, begins to occur 3 years after a price change. In the United States and the EEC, producers have a faster response to price changes than do other producing countries, which are mainly composed of the developing countries.

In all commodities except cotton and copper, a trend variable, used to capture various forms of productivity improvements and institutional policies oriented towards influencing producer decisions, was found to be statistically significant. The signs of the coefficients were all positive. Those of soybeans and maize were found to be particularly high. In the relationship for EEC sugar production a quadratic form of the trend variable was included in order to capture a dampening effect on increased productivity changes that institutional policies have had on producer decisions in the EEC sugar industry.

Major disturbances tend to be more frequent in supply than in demand. Supply disturbances

'Numerous estimates exist for the production functions for these commodities (see the surveys by Labys and Hunkeler, 1974, and by Askari and Cuniniings, 1977). However, the present results have only been compared with estimates of other studie, in which supply and demand have been calculated in order to determine the market price of a commodity since the joint consideration of these parameters is central to the dynamic stability of equilibrium in commodity markets.

'See footnote 5.

248 M . J . LORD

related to weather and disease in the case of agricultural products, and labour disputes in minerals, occur with frequent and unpredictable regularity. Those that did occur in consumption in the sample period were concentrated around the mid-1970s and early 198Os, a period when sharp price movements in commodity markets caused unusually large, albeit temporary, shifts in consumption patterns.

Demand for Stocks

Prior to the estimation of the stock demand function, evidence of cointegration and multi- cointegration processes was sought that would provided theoretical justification for the use of the stockpiling rule being modelled. Following the procedure developed by Granger and Lee (1989a,b), we tested the following set of hypotheses: (a) each of the series for Q, C, and K is I(1); (b) the series for A K is Z(0); (c) the residuals in the relationship between K and both Q and C are Z(0); and (d) the ECM for Q and C provides evidence of multi-cointegration in each of the commodities.

Testing whether the series for Q , C, and K are I(1) Using the following autoregressive representation of each series:

n

AX; = ax,- I + C /3;AXt-; + V I I ; = I

where X represents any of the series for Q, C, and K , we tested the null hypothesis HO that the time-series are each Z(1) so that a = 0, which is an indication that the system has two unit roots. The alternative hypothesis H I is that each series is Z(0) so that OL is statistically significant and negative, which is an indication the system has one unit root. The augmented Dickey-Fuller (ADF) test provides an appropriate test statistic with which to assess the level of integration of these variables (see Engle and Granger, 1987). The ADF test uses the t-statistic, but its distribution is different from that of the normal t-statistic (see Fuller, 1976: Table 8.5.2). With the present sample size the ADF test must be less than - 3-00 i n order that HO be rejected at the 95 per cent level of confidence.

The estimated autocorrelations of all of the series decline fairly rapidly since the data are of annual periodicity and, for this reason, autoregressions of a low order were used in the estimates of equation (15). The results are shown in Table 111. The series for production, consumption, and stocks are clearly 1(1) in all the products.

Table 111. Augmented Dickey-Fuller statistics

Cl Kr A KI

Coffee - 0.79 - 1.26 - 1.79 -3.77 -2.49 ~ 2.54 Cocoa 1.28 - 0 . 5 5 - 1.05 - 2.75 - 1 *90 - 1.91 Maize - 0.57 -0 .51 0.67 - 5 . 8 1 - 3-79 -3.89 Soybeans - 1 . 1 1 - 1.08 - 0.01 ~ 4.02 - 2-84 -2.80 Cotton -0.51 1.14 -2.18 ~ 4.75 -3.38 - 3 3 1 Sugar - 1.45 -1 .39 -0.74 -4.19 . ~ 2.96 ~ 2.61 Copper .- 1.36 - 0.97 - 1.18 -3.19 - 2.75 -2.50

COMMODITY MARKETS 249

Tesiing whether the series for A K is I(0) In estimating the following autoregressive representation for the change in stocks:

111

A2K, = pKc- I + C 0jAKr-j + ~ 2 1 ; = I

we tested the null hypothesis that A K is I(1) so that p = 0, against the alternative hypothesis that the series is I (O), so that p is statistically significant and negative. The ADF test for the present sample size indicates that the negative value of the [-statistic must be less than - 3 . 0 for Ha to be rejected at the 95 per cent level of confidence.

The results, reported in Table 111, are based on the use of low-order autoregressive estimates since the autocorrelations in the annual data for A K decline fairly quickly. There is clear evidence that production and consumption are cointegrated in all the products, except in cocoa. Nevertheless, the ADF statistic of the change in stocks of cocoa is close enough to the critical value of - 3 - 0 0 to suggest that production and consumption are cointegrated in this product.

Multi-cointegration tests In the following pair of equations:

K , = CY? + P2Cr + 1121 (18)

we tested whether the residuals, ul and u2, are /(0) by first estimating the equations and then performing a unit root test. The critical values of the ADF test differ from those cited in Fuller (1976), since they refer to the multivariate case in so far as the residuals calculated in (17) and (18) contain parameter estimates. For this type of higher-order system and for the present sample size, Engle and Yo0 (1987) found that the ADF test must be less than -3.29 for the null hypothesis to be rejected at the 95 per cent level of confidence. The Durbin-Watson test also provides a test of cointegration for the residuals in the equations. According to Bhargava (1 984), the Durbin-Watson will approach zero if the residuals are nonstationary. Engle and Yo0 (1987) have found that, in the multivariate case, the critical value must be greater than 1.03 for our present sample size in order that the null hypothesis be rejected at the 95 per cent level of confidence. However, they also found that this statistic is not as good a test o f cointegration as the ADF test.

Table I l l presents the ADF statistics for the residuals, and Table I V shows the

Table I V . Regression results of equations (17) and (18)

Equation (17) Equation ( 1 8)

81 R 2 DW Pz R 2 DW ~

Coffee 0.36 0.22 0.44 0.38 0.11 1 .oo Cocoa 0.34 0.26 0.34 0.25 0.09 0.41 Maize 0.16 0.36 0.83 0.15 0.27 0.98 Soybean5 0.24 0.82 0.65 0.24 0.80 0.87

Sugar 1.01 0.96 0.55 1.00 0-95 0.57 Copper 0.24 0.40 0.38 0.22 0.33 0.43

Cotton 0.29 0.33 1 .00 0.20 0.16 1 . 1 1

250 M. J . LORD

Durbin-Watson statistics. According to the ADF statistics there is clear evidence of multi- cointegration in maize and cotton, and there is some evidence of multi-cointegration in coffee, soybeans, sugar, and copper in so far as the ADF statistics are less than -2.5. The results for coffee were calculated with data beginning in 1970, since full sample calculations yielded poor estimates of the relationship between stocks and either production or consumption. The poor results obtained from full sample estimates may have been due to important differences between the 1962 and 1968 International Coffee Agreements and those which followed. In cocoa there is little, if any, evidence of multi-cointegration. The Durbin-Watson statistics in Table IV provide less support for multi-cointegration in the products, except in the case of cotton and, to a lesser extent, in coffee and maize. However, since this statistic has not been found to be as good a test of cointegration as the ADF statistic, less weight has been given to the Durbin-Watson statistics for the regression results of equations (17) and (18) in the present test of multi-cointegration.

These results generally support the empirical formulation of the stock demand function in term of the dynamic specification given in equation (13). The empirical results for the estimated stock demand functions are presented in Table V. Stocks have a proportional response to changes in the level of production or consumption in the long run. However, except in maize, the ratio of stocks to production or consumption is shown to vary with expected prices, the mean average elasticity being I .2.

The ARIMA estimates of expected market prices have low-order processes. All series, except one, are integrated of order one, in so far as stationarity is achieved from the first-difference of the series. The exception is sugar, which is integrated of order zero, i.e. sugar prices tend to fluctuate around their long-term average. Commodity prices have autoregressive (AR) processes of a low order, except coffee and cocoa which have autoregressive processes of a

Table V . Stock demand functions ~~ ~ -

Stock to Production production or

Elasticity with respect to Elasticity with respect to or consumption consumption output or consumption expected pricesa growth elasticity ratio at gb

Coffee 1.0 1.1 (El-2) - 3.0 0.76 (g = 3 . 3 % ) Cocoa 1.0 1.2 (El-2) 10.8 0 .35 ( g = 1.6%) Maize 1.0 - 1.6 0.16 ( g = 5 * 0 % ) Soybeans 1 .0 2.7 (Er-2) 1.3 0.16 (g = 5.9%)

Sugar 1.0 0.1 (El-1) - 1 . 2 0 .59 (g = 3.4%) Copper 1.0 3.1 (El-2) -2 .8 0. I6 (g = 2.9%)

'The expectations operator, E, next to the elasticity, shows whether it is conditional on information available at f - 1 or r - 2 . bThe ratio of stocks to production or consumption is given at the level when the production or consumption growth rate, g, equals the average rate over the period in which the equation was estimated. 'The current interest rate (3-month US Treasury Bill rate) was included in the relationship. Notes: (1) - Not significant at the 5 per cent level.

Cotton 1.0 0.3 (Et-z) - 0 . 5 0.38 (g = 2.8%)

(2) Derails of the estimates are found in Lord (1991, Annex C).

Interest rates were included in the relationship for copper since they motivate transactions and speculative demand for stocks. As a transactions motive, interest rates affect the cost of stock-holding; as a speculative motive, interest rates induce investors to move between financial and commodity markets. The elasticity of demand for copper stocks with respect to interest rates was found to be -2.1.

COMMODITY MARKETS 25 1

much higher order, viz. AR(11) for coffee and AR(12) for cocoa. The moving-average components indicate that some processes generating price movements have slow responses to innovations. Sugar and cotton prices are both affected by innovations that occurred six periods earlier. Nevertheless, other commodity prices are influenced only by current innovations and those of one period only.

In steady-state growth the demand for stocks of these commodities varies not only with the level, but with the rate of growth, g, of production or consumption. As Table V shows, the demand for stocks of three commodities increases when changes in consumption or production lead to a more-than-proportional change in stock demand, whereas the demand for stocks of four of the commodities decreases- when consumption or production causes a less-than- proportional change in stock demand.

Outside Sample Forecasting Performance lo

Although the commodity market models might be internally consistent with the sample period employed to estimate the system of equations, it is important to determine how well they perform outside the sample. Model adequacy has been assessed by comparing the predictive ability of the behavioural econometric models with that of time-series models. The rival time- series model is that which was incorporated into the behavioural econometric model in order to formulate the price expectations of rational producers and consumers determining their desired levels of stocks.

The one-step ahead post-sample forecast errors for observations in 1988-90 are reported in Table VI. Among the individual behavioural econometric models, those of cocoa, maize,

Table VI. Outside sample forecast performances for 1988-90

Projected percentage changes (annual average percentage

change) Root-mean-square error (070 )

Actual Behavioural Time-series Behavioural Time-series change (070) econometric model model econometric model model

Coffee Cocoa Maize Soybeans Cotton Sugar

World USA

Copper

- 5.9 - 13.3

14.5 7.0

23.4

-

24.9 2.3

16.5

6.7 10.9 16.1 14.8 23.3

16.6 0 -1

22.5

17-0 - 12-9

13.1 7 -3

19-2

16.8

12.7 -8.2

15-6 11.0 13.5 13.5 15.9

13-9 27.5 20-8

38.4 24-7 17.7 16.1 23.7

16.4 29.5 19.2

' O Labys and Lord (1988) presented a comparison of the forecasting performance of behavioural econometric models with that of time-series models at the Conference on International Commodity Market Modelling in Washington, DC, 24-26 October 1988, for four of the seven commodities covered by this study, and for a smaller outside sample period than the one used here.

"Actual, rather than simulated, values of past endogenous variables (as well as current and past exogenous variables) have been used in the one-step-ahead forecasts. The rationale for this approach is that agents always correct for their past errors, rather than accumulate errors over time.

252 M. J . LORD

soybeans, and sugar outperformed those of coffee, cotton, and copper. More importantly, the ARIMA forecasts were generally less accurate than those of the behavioural econometric models. The exception was copper, where the behavioural model overestimated the moderate rise that occurred in the 1989 price. Notwithstanding the model performance of this product, the predictive accuracy of the behavioural econometric models for all other commodities was clearly superior to that of the ARIMA forecasts.

These results provide support for the use of behavioural econometric models over time-series models in projecting commodity market prices. Deterministic models incorporating the fundamentals of supply and demand are, in general, more efficient in projecting commodity market price movements than time-series analyses which seek to identify regularities in movements of prices.

4. CONCLUSIONS

This paper developed a theory-consistent market model for storable commodities and illustrated its characterization of the data-generating process for a set of major traded commodities. The dynamics of the system incorporated recent advances in modelling techniques. Cointegrated variables in the demand functions were represented by an ECM, and expected prices in the stock demand relationship were generated by a rational expectations process. The outside-sample performance of the model was tested against the pure time-series model used to formulate expected prices, and was shown to have a smaller mean square error than that of the time-series model. Thus the model provides relatively efficient forecasts and, unlike models constructed in their reduced form, permits consideration of key behavioural relationships in commodity markets.

APPENDIX: DATA SOURCES AND DEFINITIONS

Sources for production, consumption, and stock data, and SITC (Rev. 2) description of commodities

Cocoa (SITC 072): Gill and Duffus Group, Ltd, Cocoa Market Report, various issues. Coffee (SITC 071 .l): United States Department of Agriculture (USDA), World Coffee

Copper (SITC 287.1 ; 682): World Bureau of Metal Statistics, Metal Statistics, various issues. Cotton (SITC 263.1): International Cotton Advisory Committee, Cotton: World Sfatisfics,

Maize (SITC 044): USDA, Grains, various issues. Soybeans (SITC 081.3 1 ; 222.2): USDA, Oilseeds and Products, various issues. Sugar (SITC 061.1; 062.2): International Sugar Organization, Sugar Yearbook, various issues.

Situation, various issues.

various issues.

Description and Sources of Market Prices

Cocoa: ICCO daily price, average, New York and London, nearest 3 future trading months. Coffee: ICO indicator price, other mild arabicas, average N.Y. Bremen/Hamburg markcts, ex-

dock; prior to January 1984, Guatemalan prime washed, ex-dock N.Y. for prompt shipment.

COMMODITY MARKETS 253

Copper: London Metal Exchange, Grade A cathodes, settlement price; prior to July 1986, high

Cotton: ‘A’ Index, Middling (I&’’), c.i.f. Europe; prior to January 1984, Mexican, middling

Maize: United States, No. 2, yellow, f.0.b. US Gulf ports. Soybeans: United States, c.i.f. Rotterdam. Sugar: ISA daily price, f.0.b. and stowed at greater Caribbean ports; prior to 1961, New York

All price data are from the World Bank, International Trade Division, International Bank for

grade.

(l<;’’), c.i.f. N. Europe.

World Contract No. 4, f.a.s. Cuba.

Reconstruction and Development, Washington, DC.

ACKNOWLEDGEMENTS

I am grateful to Walter Labys, Christopher Gilbert, and another referee of this journal for helpful comments. I am also indebted to Greta Boye for undertaking the calculations of the model described herein. The views expressed in this paper do not necessarily reflect those of the institution with which the author is affiliated.

REFERENCES

Adams, F. G. , and J . R. Behrman (1976), Econometric Models of World Agricultural Commodity Markets, Ballinger, Cambridge, Mass.

Akiyama, T., and R. C. Duncan (1982), Analysis of the World Coffee Market, World Bank Staff Commodity Working Paper No. 7. Commodities and Export Projections Division, International Bank for Reconstruction and Development.

Askari, H., and J . T. Cummings (1977), ‘Estimating agricultural supply response with the Nerlove model: a survey’, International Economic Review, 18, 257-292.

Augusto, S., and P. Pollak (1981), ‘Structure and prospects of the world fats and oils economy’, in Work/ Bank Commodity Models, Vol. I . Commodity and Export Projections Division, International Bank for Reconstruction and Development.

Bhargava, A. (1984), ‘On the theory of testing for unit roots in observed time series’, Manuscript, ICERD, London School of Economics.

Currie, D. (1981), ‘Some long-run features of dynamic time-series models’, Economic Journal, 91,

Cuthbertson, K . (1988), ‘The demand for M I : a forward-looking buffer stock model’, Oxford Economic

Durbin, J . (1954), ‘Error in variables’, Revue de I’lnstitut International de Statistique, 22, 23-32. Engle, R. F., and C. W. J . Granger (1987), ‘Co-integration and error correction: representation,

estimation, and testing’, Econometrica, 55, 251-276. Engle, R . F., and B. S. Yo0 (1987), ‘Forecasting and testing in cointegrated systems’, Journal of

Econometrics, 35, 143-159. Fisher, F. M. , P. H. Cootner and M. Bailey (1972), ‘An econometric model of the world copper

industry’, Bell Journal of Economics and Management Science, 3, 568-609. Fuller, W. A. (1976), Introduction to Statistical Time Series, John Wiley and Sons, New York. Ghosh, S., C. L. Gilbert and A. J. Hughes Hallett (1987), Stabilizing Speculative Commodify Markers,

Clarendon Press, Oxford. Gilbert, C. L., and T. B. Palaskas (1990), ‘Modelling expectations formation in primary commodity

markets’, in D. Sapsford and L. A. Winters (eds), Prirnary Commodify Prices: Econoinic Models and Policy, Cambridge University Press, Cambridge.

Goodwin, T. H., and S. M. Sheffrin (1982), ‘Testing the rational expectations hypothesis in an agricultural market’, Review of Economics and Sfatistics, 64, 658-667.

Granger, C . W. J. (1983), ‘Co-integrated variables and error-correcting models’, University of California at San Diego Discussion Paper.

704-7 15.

Paper.s, 40, 110-131.

254 M. J. LORD

Granger, C. W . J . (1986), ‘Developments in the study of cointegrated economic variables’, Oxford Bulletin of Economics and Statistics, 48, 213-228.

Granger, C - W. J., and T-H. Lee (1989a), ‘Multicointegration’, Journal of Applied Econometrics, 4, s 145-SI59.

Granger, C. W. J., and T.-H. Lee (1989b), ‘Investigation of production, sales and inventory relationships using multicointegration and nonsymmetric error correction models’, in G. F. Rhodes, Jr and T. B. Fomby (eds), Advances in Econometrics: Cointegration, Spurious Regressions, and Unit Roots, JAI Press, New York.

Granger, C. W. J . , and A. A. Weiss (1983), ‘Time series analysis of error-correcting models’, in S. Karlin, T. Amemiya, and L. A. Goodman (eds), Studies in Econometrics, Time Series, and Multivariate Statistics, Academic Press, New York.

Harvey, A. C. (1981), The Econometric Analysis of Time Series, Phillip Allen, London. Hendry, D. (1988), ‘The encompassing implications of feedback versus feedforward mechanisms in

econometrics’, Oxford Economic Papers, 40, 132- 149. Houck, J . P., and J . S. Mann (1968), An Analysis of Domestic and Foreign Denrand for U.S. Soybeans

and Soybean Products, Technical Bulletin 256. Agricultural Experiment Station, University of Minnesota.

Hwa, E. C. (1985), ‘A model of price and quantity adjustments in primary commodity markets’, Journal of Policy Modeling, 7, 305-338.

Judge, G. G., W. E. Griffiths, R. C. Hill, T-C. Lee (1985), The Theory and Prdctice of Econometrics, John Wiley and Sons, New York.

Labys, W. C. (1973), Dynamic Commodity Models: Specification, Estimation and Simulation, Lexington Books, Lexington, Mass.

Labys, W. C. (1980), Market Structure, Bargaining Power, and Resource Price Formation, Lexington Books, Lexington, Mass.

Labys, W . C. (1987), Primary Commodity Markets and Models, Cower, Hampshire. Labys, W. C., and J . Hunkeler (1974), Survey of Commodity Demand and Supply Elasticities, Research

Memorandum No. 48. UNCTAD Research Division, UNCTAD/RD/70. Labys, W . C., and M. J. Lord (1988), ‘A comparison of time-series and behavioral econometric models

of commodity markets’, Paper presented at the Conference on International Commodity Market Modelling, Washington, DC, 24-26 October.

Lord, M. J. (1989), ‘Product differentiation in international commodity trade’, Oxford Bulletin of Economics and Statistics, 51, 35-53.

Lord, M. J. ( I 991), lmperfect Competition and International Commodity Trade: Theory, Dynamics, and Policy Modelling, Clarendon Press, Oxford.

Maddala, G. S. (1977), Econometrics, McGraw-Hill, New York. Maizels, A. (1984), ‘A conceptual framework for analysis of primary commodity markets’, World

Monke, E. A., and S. E. Guisinger (1985), ‘International trade constraints and commodity market

Muth, J . F. (1961), ‘Rational expectations and the theory of price movements’, Econornetrica, 29,

Plosser, C. I . , and G. W. Schwert (1977), ‘Estimation of a non-invertible moving average process: the

Wallis, K. F. (1980), ‘Econometric implications of the rational expectations hypothesis’, Econometrim,

Weymar, H. F. (1968), The Dvnamics of the World Cocoa Market, MIT Press, Cambridge, Mass.

Development, 12, 25-41.

models: an application to the cotton market’, Review of Economics and Statistics, 67, 98-107.

315-335.

case of overdifferencing’, Journal of Econometrics, 6, 199-224.

48, 49-73.