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Autoregressive Conditional Heteroscedasticity in Commodity Spot PricesAuthor(s): Stacie BeckSource: Journal of Applied Econometrics, Vol. 16, No. 2 (Mar. - Apr., 2001), pp. 115-132Published by: John Wiley & SonsStable URL: http://www.jstor.org/stable/2678513Accessed: 01/03/2010 22:07

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Estimation techniques and data are described in Sections 3 and 4. In Sections 5 and 6, results andconclusions are discussed.

2. THEORY

By modifying the Muth model to allow demand and/or supply shocks to be drawn fromdistributions with zero means but non-constant (i.e. stochastic) variances, it can be shown thatprices follow an ARCH process of order one [ARCH(1)] if inventories are carried. It is notnecessary to assume that the ARCH process is caused by external shocks that follow an ARCHprocess (as assumed, e.g., in Engle and Bollerslev, 1986, or Diebold and Nerlove, 1989). Thisextension of Muth's model is:

Ct = -f8pt + ed (1)

Pt = ypt + es (2)

It = at+ (Pt+ - Pt) (3)

Ct +It = Pt +t- (4)

where Ct = consumption in period t

Pt = production in period t

It = inventory at the end of period t

Pt = (xt - -p)

p = equilibrium commodity price

xt = current price level

Pt = Et-1 (Pt)

at+1 = K/a2

K = the negative of the ratio of the marginal utility of expected profit to

the marginal disutility of profit variance, so K > 0.

If it is assumed thatsupply

and demandshocks, es,

ed, areindependently

distributed withzero means, and that their variances, u2d and ar2 are random variables drawn from a stationarydistribution, then ar2 and at are variable rather than constant. This implies that commodity pricesPtshould be modelled using a variant of the ARCH-M model in which the expected conditionalvariance appears in both the slope and intercept of the price level equation (see the Appendix fordetails):

Xt = ko(ht) + kl(ht)xt- + et (5)

ht = ao + ale_ (6)

where et = (ed - e);

ht = the expected variance of et conditional on et2.- LICCI~CLCUVLIILLICVI f LVLIILILICL t-1,_l

Copyright 2001 John Wiley & Sons, Ltd.

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ARCH PROCESSES IN COMMODITY SPOT PRICES

Theory indicates that ako/ah, > 0 and akl/aht < 0. When the commodity is not storable,equation (4) is

Ct = Pt (7)

hence the price equation is:xt = ko + klxt- + et (8)

where ko and kl are constants. In fact, kl equals zero.This extension of the Muth model implies that price variance follows an ARCH process if

supply and demand shocks are drawn from probability distributions whose variances change,even if those changes are random. This is because current price volatility transmits itself intofuture periods by creating volatility in inventory carryover, thus price variance is a function of itslagged value. When an ARCH process exists, market participants can forecast the variance, so theexpected variance (ht) should affect their inventory holding decisions if they are risk-averse. This

is because a large price variance, i.e. high price risk, reduces inventory speculation. Low inventorycarryover raises the current price level and reduces the effect of lagged price on current price.Thus, the expected variance affects price levels when expectations are rational. Without inventorycarryover, the expected price variance depends only on the expected variance of external shocks,which is constant here if the variances of the shocks are drawn from a stationary distribution.

To summarize, theory indicates that when the commodity is storable and the production lagis one period, there is an ARCH process of order one. Moreover, the intercept and lagged pricecoefficient are positive and negative functions, respectively, of the expected variance. When thecommodity is nonstorable there should not be an ARCH process nor should price levels be seriallycorrelated, assuming external shocks and their variances are uncorrelated.2

3. ESTIMATION

The first version of the model estimated here treats the parameters of the price level equation asconstants but allows a ARCH process to exist in the residuals. This price level equation is

Xt = co + cl time + C i+lXti + Cn2 dummy + C2+3(dummy-time) + et (10)i=l

Trend and dummy variables are included because productivity gains, technological improvementsand demand disturbances caused trends and structural breaks in some price series. Also, some

regressions include more than one lag of the price level because the decision/production period islonger than the observationtioninterval; similarly, higher-order ARCH processes or the generalizedARCH (GARCH) model (Bollerslev, 1986) may be appropriate for equation (6). Versions ofequation (6) investigated here are encompassed by

12 ln

ht = ao + aja2,_ + Ebkht-k (11)j=1 k=l

2 Models by Wright and Williams (1982,1991), Deaton and Laroque (1992) and Gilbert (1989) also rely on inventorycarryover to transmit volatility. However, in their models the source of heteroscedasticity is the non-negativity constrainton inventories which causes occasional stockouts and

corresponding spikesin

prices.These authors show that stockouts

are rare or non-existent, so this constraint was ignored in Beck (1993). Empirical tests in this paper are valid for eithermodel of autoregressive conditional heteroscedasticity.


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where et = ,-hAt and qt is a random variable independently and identically distributed with zeromean and finite variance. The procedure for determining the exact form of the model for eachcommodity is described in detail in Section 4.

The possibility that variances respond asymmetrically to positive versus negative shocks was alsoinvestigated. This could occur as a result of the non-negativity constraint on inventories. Deatonand Laroque (1992) showed that, theoretically, the downward movement of price in response toan exogenous shock will be limited by the profitability of inventory holdings whereas stockoutsimply no limits to upward movements. If so, high volatility will be associated with positive shocksto price level and low volatility will be associated with negative shocks. The exponential GARCH(EGARCH) model allows for such asymmetries (Nelson, 1991). In this model, equation (11) isreplaced by

i) n inln(ht)=ao+Zai( et-)+ a2i

et- -E - )+ bln(ht_j) + ut (12)i1 ~t Z n(htt /h

The a2i parameters capture the ARCH process and the ali parameters capture asymmetry. Toillustrate, let n = 1. If all > 0, then a positive shock to the price level (et-_ > 0) increasesthe conditional variance whereas a negative shock reduces it; this would be consistent with pricebehavior during a stockout and a glut respectively (Trivedi, 1995). Although simulations by Deatonand Laroque indicated the impact of the non-negative inventory constraint is likely to be small,the exponential GARCH (EGARCH) model was estimated to determine if such asymmetries areprominent in these data, particularly for storable commodities.

Next, the (G)ARCH-M specification was estimated. The parameters of equation (6), ko and kl,are approximated by linear functions, thus equation (10) becomes

Xt = co + cltime + E Ci+lXt-i + c,+2 dummy + c,,+3(dummy.time)i=l

+ 01 t + 02(Xt- l Vfht) + et (13)

The hypotheses tested here are that 01 > 0 and 02 < 0, since a large expected price variance, i.e.high price risk, lowers inventory carryover thus raising price level and reducing dependence onprevious price.

4. DATAAnnual data were used because they conform most closely to the production time lags of Muth'smodel and they also avoid problems with seasonality. Twenty commodities were found for whichthere existed a consistent price series from the end of World War II or before. Both storable andnon-storable commodities are included. Annual data were obtained from the Commodity Yearbookfor cotton, coffee, copper, sugar, eggs, cocoa, hogs, cattle, soybeans, corn, lumber and wheat.Grape, pear and apple price series were obtained from the US Department of AgricultureFruit and Tree Nut Situation and Outlook. Lettuce, tomato, onion and potato data were obtainedfrom the Vegetables and Specialities Situation and Outlook Report.3

3 Series covered the following periods: cotton, 1875-1996, coffee, 1825-1996, copper, 1840-1996, sugar, 1845-1934,eggs, 1856-1995, cocoa, 1913-1983, hogs, 1930-1996, cattle, 1930-1993, soybeans, 1938-1995, corn, 1937-1996,


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All series were expressed in relative terms by dividing nominal prices by a price index. Theconsumer price index (CPI) and the producer price index (PPI, formerly the Wholesale PriceIndex) for commodities were obtained from the Bureau of Labor Statistics. The CPI was used todeflate all series except lumber which was deflated by the PPI.4

Graphs show that for storable commodities: cotton, coffee, copper, sugar, corn, cocoa, soy-beans, sugar, lumber and wheat, periods of high volatility are followed by high volatility,and low volatility with low volatility. However, for non-storable commodities, such as fruitand vegetables, price volatility shows no particular pattern.5 This suggests the presence ofan (G)ARCH process in the prices of storable commodities but not in the non-storable com-modities, which is consistent with the model in Section 2. The graphs also show that manynon-storable commodity prices have distributions as skewed or more skewed than storable com-modity distributions. If the non-negative inventory constraint is the principal explanation ofasymmetry in prices, storable commodities should appear more skewed than non-storable com-modities. This suggests that other factors are at work, supporting the conclusion by Deaton andLaroque (1992, 1996) that this constraint cannot adequately explain many features of commodityprice data.

The data series were tested for stationarity. Improved technology and productivity gains arelikely to lead to trends in relative prices, so the non-stationarity hypothesis was tested againstthe alternative of trend stationarity.6 Fourteen of the twenty series rejected non-stationarity basedon Dickey-Fuller statistics. Significant negative trends were detected in eight cases, which isconsistent with the effect that improved technology and productivity gains have on the relativeprices of commodities.

For the six cases that failed to reject non-stationarity: sugar, eggs, soybeans, lumber, milk andtomatoes, it is possible that structural breaks or segmented deterministic trends are responsible(Perron, 1989). These may be caused by major innovations or demand disturbances. The graphsshow structural breaks and/or trend changes for lumber and milk in 1949, sugar in 1863, soybeansin 1971, eggs in 1917 and tomatoes in 1978. In the case of lumber, sugar and milk, these aremost likely caused by postwar changes in demand conditions. Breaks occurred after World War IIfor milk and lumber, and after the Civil War for sugar. Soybeans experienced a rapid expansionin worldwide demand as a feedstock, and a series of harvest failures in the early 1970s. Supply

lumber, 1936-1987, wheat, 1930-1996, milk, 1936-1996, apples, pears and grapes, 1948-1996, lettuce and tomatoes,

1949- 1996, and onions, 1949-1995. The price series for raw sugar delivered in New York City is available for 1845 -1996,however, since 1934 the USA imposed an import quota system. Under this system price behaviour does not respond todemand and supply shocks, so the commodity market model is not applicable after this date. Gold and silver series werealso excluded from this study because of significant monetary demand which is not approximated by the assumptions ofthe commodity market model.4 The PPI may be considered to be a better choice for commodities that require a great deal of processing prior to consumeruse. However, in two cases, copper and cotton, the CPI rather than the PPI was used to deflate a series because the CPIwas available for the entire series whereas the PPI was not. In two other cases, hogs and cattle, series deflated by bothindices were used in many of the estimations described below. The results were so similar that only one set for eachcommodity, that deflated by the CPI, was retained.5 Graphs were not included here to conserve space. However, they are available from the author upon request.6 Theory suggests that the relative price of a commodity should converge to a level determined by cost and preferenceparameters. Although these parameters may shift over time, it is more theoretically plausible that they would shift in a

wayto

producedeterministic trends or trends with breaks, than stochastic trends in relative

prices.Thus, the

hypothesisof non-stationarity is tested against the null of trend stationarity, or trend stationarity with breaks (see below). The testingprocedure follows Mills (1993, pp. 56-57). The Dickey-Fuller statistics are available from the author upon request.


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shocks, in particular the introduction of refrigeration around 1917 and a new cultivation methodduring the 1970s, most likely created the breaks in the egg and tomato series.7

Perron's (1989) approach was used first to determine whether unit roots existed in the presenceof structural breaks and/or segmented trends for these six series. In his approach, the point ofstructural break/trend reversal (break point) is determined from observation of the data, then thenull hypothesis is tested against the alternative of trend stationarity with that break point.8 Four ofthe six series rejected the null hypothesis of a unit root in favour of the altemative(s) of stationarityaround a trend that included a structural break and/or trend reversal.

A procedure proposed by Zivot and Andrews (1992) was used next. Zivot and Andrewsargued that, because Perron's approach relied on prior observation of the data to determinepoints of structural breaks and trend changes, it is subject to pre-testing problems. In theirprocedure, the break points are determined endogenously. Then Zivot and Andrews developedan asymptotic distribution of the test statistic that accounted for this endogeneity. With the

Zivot and Andrews procedure, 1855,1920,1945,1947,1972 and 1979 were identified for sugar,eggs, milk, lumber, soybeans and tomatoes respectively, which are close to those used forthe Perron procedure. Since theirs is a more objective procedure for determining the breakpoints caused by the demand and supply shocks described above, these are retained in theestimations that follow. Based on their asymptotic distribution, only the sugar and egg seriesrejected non-stationarity. This is not surprising since the sugar and egg series cover the longesttime span (89 and 139 years respectively) whereas the lumber, soybean, milk and tomato seriesare shorter (51, 57, 60 and 47 years respectively). The failure to reject the null hypothesisof non-stationarity in these latter cases is most likely due to the low power of the testas a consequence of their shorter time spans (Mills, 1993, p. 60). Hence, all series were

retained and used in the estimations below, with equation (10) modified to include lineartrends, structural break points and segmented trends for each commodity as indicated by thesetests.9

5. RESULTS

First, the ordinary least squares residuals of equation (10) were analysed, as in Engle (1982,pp. 1000-1003). The appropriate number of lags of the price level was determined by theSchwarz-Bayesian information criteria. Lagrange multiplier statistics were computed to checkfor residual serial correlation up to the fourth order, which is adequate for annual data. These

statistics indicated that one lag is appropriate in most cases (except corn, wheat and sugar) andthere is no significant residual serial correlation in any case. Skewness and excess kurtosis statisticswere also computed for the OLS residuals and reported in Table I.

7 This information concerning soybeans, eggs, milk, sugar and lumber is based on various issues of theCommodity Yearbook. I am indebted to Gary Lucier of the Economic Research Service, United States Department ofAgriculture, for the information on tomatoes.8 Perron tested the null against three different types of structural breaks and trend reversals. These are: a change in theintercept but not the slope of the trend line, second, a change in the slope but not the intercept of the trend, third, a changein both the intercept and the trend. All three were tested here.9 Examination of the graph plus the results of the Zivot and Andrews procedure suggested there may be two breakpoints inone case,

soybeans,in 1947 and 1972.

Equation (10)was re-estimated with two

breakpointsand with a

single breakpointat 1947. However, the fit of the equation and the time series properties of the residuals deteriorated relative to thespecification with one break point, 1972, hence the latter was retained in this paper.


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Table I. Skewness and excess kurtosis measures of ordinary least squares residuals3

Xt= co + c time + CiXt_i + C5 dummy + c6(dummy.time) + eti=l

Commodity No. of obs. Skewness Excess kurtosis Restrictions

Cotton 125 -1.04 9.81 Ci,C3-C6 = 0Coffee 169 1.53 10.05 cl,C3-C6 = 0Copper 154 4.86 6.38 C3-C6 = 0Sugar 84 0.74 8.19 cl = 0Eggs 137 1.48 0.19 C3 = C4 = 0Cocoa 68 7.70 14.35 C3-C6 =0Potatoes 70 0.81 0.01 3 -c6 = 0

Hogs 64 2.12 0.52 c3-C6= 0Cattle 61 0.54 0.41 Ci,C3-C6 = 0Soybeans 55 2.14 1.60 c3-c4 = 0

Corn 56 0.27 5.75 C4-c6 = 0Lumber 49 4.40 5.51 cl,C3-C4,C6 = 0Wheat 63 2.68 10.62 C4-c6 = 0Milk 58 -0.37 0.57 cl,C3-C4 = 0Lettuce 45 2.08 1.54 C3-C6 = 0Tomatoes 45 0.21 -0.61 C3-c4 = 0Onions 44 2.03 0.51 Cl,C3-C6 = 0Apples 46 1.13 -0.33 C3-c6 = 0

Grapes 46 3.37 7.21 cl,C3-C6 = 0Pears 46 2.21 0.17 c3-c6 = 0

Note: Kurtosis is measured relative to the normal distribution. Skewness and kurtosis statistics aredistributed N(0,1) under the assumption of normality (see TSP Version 4.4 Reference Manual, underthe MSD

procedure).

High excess kurtosis in the OLS residuals is consistent with the presence of a GARCH process(Bera and Higgins, 1993, p. 315). To be consistent with our hypothesis, high excess kurtosis shouldbe more prominent in storable commodity series than in nonstorable commodities. Table I showsthat the excess kurtosis statistics follow the expected pattern in most cases; storable commodities

display higher kurtosis than nonstorable commodities with the exception of soybeans and grapes.Lagrange multiplier statistics for ARCH(1) and ARCH(2) processes were also computed usingOLS residuals, and are reported in Table II. These were not as supportive of the hypothesis aswere the excess kurtosis statistics. In particular, only the coffee, copper, corn and cattle statistics

indicate the presence of an ARCH process.To resolve the issue, (G)ARCH models were estimated for all commodities and likelihood ratio

tests were computed. To ensure proper fit, ARCH(2) and GARCH(1,1) processes were estimated aswell as ARCH(l). If likelihood ratio statistics showed that restrictions were significant, ARCH(2) orGARCH(l,1) was retained rather than ARCH(1).10 Estimation results are reported in Table III. The

10All estimations reported in Tables III-V were obtained using the maximum likelihood (ML) procedure in TSP version4.4. Only low-order processes of ARCH and GARCH were investigated since higher-order processes in financial time seriesare rare (Bera and Higgins, 1993, p.317). In this instance, where annual data are used, higher-order (G)ARCH processesare even less likely. When likelihood ratio tests of the restrictions for ARCH(1) relative to ARCH(2) or GARCH(1,1) weresignificant, either ARCH(2) or GARCH(1,1) was retained and reported in Table III. This version was then tested againstthe

hypothesisof no ARCH

process usingthe likelihood ratio statistic. Deb, Trivedi and

Varangis(1996,

pp.281-282)have observed that the hypothesis that there is no ARCH process is not independent of the hypothesis that there is noARCH(2) or GARCH(1,1) process. They present evidence that the degrees of freedom of the chi-square distribution of


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Table II. Lagrange multiplier statistics for ARCH

Commodity ARCH(1) ARCH(2)

Cotton 1.89 3.57Coffee 2.02 13.57*Copper 4.34* 5.55Sugar 0.33 5.43Eggs 0.27 1.08Cocoa 0.16 0.19Potatoes 2.49 2.14Hogs 0.26 1.33Cattle 1.55 7.10*Soybeans 1.71 1.61Corn 10.11* 9.95*Lumber 2.16 2.53Wheat 1.51 1.97

Milk 2.77 2.69Lettuce 0.43 0.84Tomatoes 0.43 0.42Onions 1.45 1.55Apples 0.16 3.15Grapes 0.18 0.30Pears 2.18 2.87

Note: Statistics are TR2 ~ X, where R2 is obtainedfrom the regression e = aO + ZE,7 aiei + ut andT is the number of observations (see Engle, 1982,p. 1000).

restriction was significant for every storable commodity except soybeans, and was not significantfor any non-storable commodity except cattle, thus generally supporting the hypothesis. Cattle,though non-storable, require more than one year to produce so an ARCH process in this case isreasonable. In two cases, lettuce and pears, convergence was not obtained. Their excess kurtosisstatistics are low and there is little serial dependence in price level (cl = 0). This evidence isinterpreted as supporting the hypothesis that an ARCH process does not exist, as the modelpredicts. Other non-storable commodities (eggs, potatoes, hogs, milk, tomatoes and apples) alsodo not exhibit an ARCH process, but they do have estimates of cl that are significantly differentfrom zero. This supports the premise of Deaton and Laroque's (1996) study in which they searchfor

seriallycorrelated external shocks to

explainserial correlation in commodity

prices.Estimates of ARCH parameters a,, a2 and bl indicate the process is stationary except for cocoa,lumber and wheat. In the case of lumber and wheat, the coefficient standard errors are too large toconclude that al is greater than one, hence the ARCH process for these two commodities may bestationary as well. Only for cocoa can it be concluded that the ARCH process is non-stationary.

Skewness statistics for residuals, ?t = etlf/ht, in Table III are significant for storable commodi-ties such as cotton, coffee, copper, cocoa, soybeans, and lumber, and for non-storable commoditiessuch as lettuce, onions, grapes and pears. To investigate asymmetry in the conditional varianceprocess, EGARCH models were estimated. The EGARCH estimates reported in Table IV assume

the likelihood ratio statistic are fewer thanusually assumed,

thusleading

tounder-rejection

of the nullhypothesis. Here,only for hogs might the hypothesis of no ARCH be accepted if one erroneously assumed too many degrees of freedom

for the chi-square distribution.


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Table III. Generalized autoregressive conditional heteroscedastic estimates3 2

Xt = Co + ctime + CiXt-i + C5 dummy + C6(dummy.time) + et with ht = ao + ajretj + blht-1i=l j=1

Cotton Coffee Copper Sugar Eggs Cocoa Potatoes

co 4.16* 7.72* 11.67* 8.76* 21.35* 1.48 2.48*(1.84) (1.76) (3.41) (1.32) (4.12) (0.97) (0.45)

cl -0.05*0.10.05 016 005* -0.02*(0.02) (0.05) (0.02) (0.01)

C2 0.80* 0.79* 0.83* 0.85* 0.64* 0.72* 0.26*(0.04) (0.04) (0.04) (0.08) (0.06) (0.02) (0.13)

C3 -0.64*(0.10)

C4 0.31*

(0.06)C5 5.69* -7.77

(1.09) (2.57)C6 -0.14* -0.41

(0.02) (0.10)ao 3.85 57.57* 35.88* 0.77* 45.47* 9.81* 0.39*

(2.68) (9.00) (6.75) (0.36) (7.02) (4.18) (0.10)a 1 0.20* 0.48* 0.64* 0.72* 0.02 2.97* 0.25

(0.08) (0.17) (0.20) (0.24) (0.06) (0.70) (0.19)a2 0.61*

(0.21)bl 0.73*

(0.07)In L -422.89 -613.75 -533.67 -183.87 -457.61 -249.52 -76.45LR testa 15.86* 28.60* 22.06* 21.68* 0.16 28.86* 2.80SERb 1.003 1.000 1.000 1.003 1.003 0.907 1.007Skewnessb 2.01 3.66 3.19 1.08 1.38 3.80 0.92Kurtosisb 2.28 7.50 9.16 0.50 0.09 4.14 0.56Obs. no. 125 169 154 84 137 68 70

Hogs Cattle Soybeans Corn Lumber Wheat Milk

co 8.95* 2.03 304.73* 66.22* 29.63* 85.86* 1.31*(1.35) (1.62) (63.27) (16.11) (4.37) (11.90) (0.37)

cl -0.08* -4.08* -0.84* -1.06*(0.02) (1.09) (0.22) (0.16)

c2 0.69* 0.92* 0.30* 1.01* 0.73* 1.08* 0.78*(0.05) (0.06) (0.14) (0.11) (0.04) (0.15) (0.06)

C3 -0.32* -0.32*(0.09) (0.14)

C4

C5 126.86* 10.25* 0.33*(30.87) (3.79) (0.16)

C6 -6.07* -0.02*(1.82) (0.00)

ao 0.29 3.36* 1572.08* 187.74* 35.10 292.07* 0.04*

(0.27) (1.53) (2.07) (56.16) (21.83) (125.94) (0.01)

(continued overleaf)

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Table III. (continued)

Hogs Cattle Soybeans Corn Lumber Wheat Milk

al 0.28* 0.31 0.14 0.80* 1.98* 1.04* 0.12(0.10) (0.22) (0.13) (0.35) (0.89) (0.47) (0.14)

a2 0.47(0.24)

bl 0.70*(0.07)

In L -168.72 -155.13 -281.50 -245.96 -193.29 -294.22 3.86LR testa 4.84 11.96* 2.72 24.58* 19.66* 15.16* 1.30SERb 1.007 1.005 0.956 1.004 0.984 1.007 1.008Skewnessb 1.46 1.40 2.14 1.61 2.29 -0.38 -0.77Kurtosisb 0.22 -0.19 1.16 0.64 1.70 0.92 0.38Obs. no. 64 61 55 56 49 63 58

Lettuce Tomatoes Onions Apples Grapes Pears

co 5.40* 4.74* 3.03* 3.75* 20.78* 86.06*(1.42) (0.95) (0.65) (0.71) (8.27) (19.52)

Cl -0.04* 0.07* -0.03 -0.66*(0.01) (0.02) (0.91) (0.22)

c2 0.03 0.37* 0.15 0.32* 0.72* 0.22(0.04) (0.12) (0.17) (0.13) (0.10) (0.17)

C5 -2.10*(0.55)

C6 -0.12*

(0.05)ao 0.28* 0.59* 0.72* 0.36 165.12* 292.86*

(0.05) (0.14) (0.20) (0.21) (39.40) (6.74)

al -0.14 0.20 0.47 0.09(0.15) (0.20) (0.51) (0.11)

In L -35.23 -48.81 -59.47 -52.74 -184.84 -192.47

LR testa n.a 0.66 1.52 1.66 1.22 n.aSERb 1.011 1.011 1.011 1.008 0.989 0.938Skewnessb 2.08 0.23 2.55 1.21 3.40 2.24Kurtosisb 1.54 -0.83 0.65 -0.33 7.44 0.19Obs. no. 45 45 44 46 46 46

Note: Standard errors are in parentheses. Asterisks (*) indicates that the coefficient is significantly different fromzero at the 5% level.a Likelihood ratio tests the hypothesis that al (= a2 = bl) = 0.b These are the standard error, skewness and excess kurtosis of et/h/t. Kurtosis is measured relative to the normaldistribution. Both skewness and kurtosis statistics are distributed N(0,1).

a normal distribution. In many cases where Tables I-III indicated there was no ARCH process,convergence could not be obtained for the EGARCH model, i.e. potatoes, milk, lettuce, tomatoes,apples, grapes and pears. This supports the conclusion that an ARCH model is not appropriate forthe non-storable commodities. For commodities where estimates were obtained, they support theconclusions of Table III. Three non-storable commodities, eggs, hogs and onions, do not exhibit

ARCH processes, with hogs again a marginal case. Storable commodities (cotton, coffee, copper,sugar, cocoa, lumber, and wheat) exhibit ARCH processes. Cattle exhibit an ARCH process for the


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same reason as argued earlier, a long production cycle. The exceptions are soybeans and corn. Forsoybeans, the ARCH parameters are not significant in either Table III or IV, but the asymmetryparameter is significant in Table IV. For corn, the asymmetry parameter, al 1, is significant and the

ARCH, a21, is not; this is in contrast to Table III where an ARCH process was found.To provide indications of the goodness of fit between the GARCH and EGARCH models,

the values of the log likelihood function and residual standard deviations, skewness measures andexcess kurtosis measures are reported in both Tables III and IV. In most cases the EGARCH modelis not superior to the GARCH model (eggs, hogs, wheat, milk, and onions), and in some casesEGARCH is a worse fit (sugar, cocoa and corn). In two cases (coffee and lumber) the results are

Table IV. Exponential generalized autoregressive conditional heteroscedastic estimates assuming a normaldistribution

3

xt = co + cltime + cit_i + C5 dummy + c6(dummy-time) + et with

i=l2 2

log(ht) = ao + ali(et-i/h

it-) +Ea2i[et-Ji/Vhl - Elet-/l hi] + bl log(ht-l)i=1 i=l

Cotton Coffee Copper Sugar Eggs Cocoa

co 6.00*(1.68)

7.54*(1.73)

Cl

C2 0.82*

(0.04)

0.80*

(0.04)

11.70*(1.38)-0.05*(0.01)0.83*

(0.02)C3

C4

C5

C6

ao

all

3.92*(0.13)0.46*

(0.14)

4.48*(0.10)0.25*

(0.10)

4.14*(0.11)0.21

(0.11)a12

0.51*(0.20)

0.68*(0.15)

0.76*(0.18)

a22

In L

Obs. no.

SERa

Skewnessa

Kurtosisa

-418.78

125

0.987

1.91

1.70

-611.79

169

1.042

4.45

12.22

-532.54

154

0.999

2.57

3.68

7.70*(2.13)

0.87*

(0.11)-0.56*(0.12)0.27*(0.08)4.77*(1.41)-0.12*(0.03)1.77*(0.19)0.10

(0.08)0.33*(0.15)0.72*(0.24)0.60(0.37)-186.47

84

0.975

1.56

1.79

21.64*(4.72)-7.86*(2.25)0.64*

(0.07)

-0.42*(0.10)0.17*(0.05)3.84*(0.12)-0.05

(0.12)

-0.03(0.14)

-457.58

137

1.003

1.40

0.07

3.97(2.39)0.08(0.06)0.71*

(0.07)

4.65*(0.18)0.57*

(0.25)

0.58*(0.27)

-255.08

68

0.952

7.49

11.94

(continued overleaf)


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S. BECK

Table IV. (continued)

Hogs Cattle Soybeans Corn Lumber Wheat Onions

co 8.97* 2.85* 245.88* 72.54* 31.70* 85.89* 2.83*(2.46) (0.99) (28.95) (17.02) (5.59) (12.93) (0.66)

cl -0.08* -3.17* -0.92* -1.07*(0.02) (0.57) (0.23) (0.13)

C2 0.69* 0.89* 0.42* 0.81* 0.68* 1.07* 0.21(0.08) (0.03) (0.08) (0.09) (0.06) (0.07) (0.18)

C3 -0.12* -0.32*(0.05) (0.10)

c5 109.49* 19.58*(13.74) (6.67)

C6 -5.56*(1.77)

ao 0.70 2.16* 7.40* 5.99* 4.81* 6.72* -0.13(0.57) (0.19) (0.18) (0.20) (0.22) (0.22) (0.13)

all 0.03 0.68* 0.34* 0.71* -0.40 0.06 0.26(0.15) (0.16) (0.15) (0.16) (0.27) (0.25) (0.26)

al2 0.19 0.78*(0.24) (0.24)

a21 0.46 -0.02 0.12 0.35 1.28* 1.29* 0.30(0.32) (0.05) (0.13) (0.32) (0.37) (0.38) (0.38)

a22 1.11* -1.26*(0.33) (0.41)

bl 0.72*

(0.22)In L -168.75 -151.70 -281.07 -245.21 -188.39 -295.32 -58.86

No. obs. 64 61 55 56 49 63 44

SERa 0.999 1.045 1.010 1.041 1.876 0.915 1.004

Skewnessa 1.76 0.49 1.76 3.65 7.56 -0.46 2.79

Kurtosisa 0.78 0.05 1.16 2.72 12.73 0.61 0.85

Note: Standard errors are in parentheses. Asterisks (*) indicate significance (from zero) at the 5% level.a These are the standard error, skewness and excess kurtosis of et/h/t. Kurtosis is relative to the normal distribution.Skewness and kurtosis statistics are distributed N(0,1).

mixed since the likelihood function values are higher but residual kurtosis and skewness statisticsare larger. Only for cotton, cattle, soybeans, and possibly copper, does the EGARCH model appearto be superior to the GARCH model.

The EGARCH model was also estimated assuming the general error distribution for commoditiesin Table III with high excess kurtosis statistics, i.e. cotton, coffee, copper, cocoa and lumber. Thegeneral error distribution allows for greater kurtosis than the normal distribution (Nelson, 1991).1lTable V shows estimates for those cases in which convergence was obtained; convergence was notobtained for coffee and cocoa which, along with the results in Table IV, indicate that the EGARCHmodel is not appropriate for these two commodities. Table V shows that v is significantly less

l Another distribution that allows forhigher

kurtosis than the normal is the Student-t distribution(Bollerslev, 1987).Estimation of the model using the t distribution for these commodities was carried out, but reasonable estimates could

not be obtained.


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Table V. Exponential generalized autoregressive conditional heteroscedastic estimatesassuming a generalized error distribution

3

Xt = co + cltime +

ECiXt_i + C5 dummy + c6(dummy.time) + et with

i=1

2

log(h) = ao + ai(eti=l

2

-i/ hti) + a2i[eti=l

Cotton Copper Lumber

co 2.57 9.70* 29.30*(1.64) (0.23) (6.00)

cl -0.04*(0.00)

c2 0.91* 0.85* 0.77*(0.05) (0.01) (0.06)

C5 16.91*(5.64)

ao 0.77* 3.53* 4.84*(0.34) (0.24) (0.21)

all 0.36* 0.13* -0.47(0.13) (0.02) (0.29)

a12 0.78*(0.25)

a21 0.30* 0.79* 1.34*(0.15) (0.03) (0.43)

a22 -1.16*(0.25)

bl 0.79*(0.09)

v 1.20* 1.07* 1.63*(0.22) (0.06) (0.51)

In L -406.40 -525.07 -188.25

Note: Standard errors are in parentheses. Asterisks (*) indicates significance at the 5% level. The loglikelihood function is

T

logL = -[log(v/) - 0.5(1et/(Xht)') - (1 + v-l)log(2) - log(r(l/v)) - 0.5log(ht)]t=l

where k = [2-2/ Fr(1/v)/r(3/v)]1/2

than two in two cases: cotton and copper. The EGARCH estimates for copper and lumber aresimilar to those obtained in Table IV; those for cotton are not, although all three still exhibit an

asymmetric ARCH process.To summarize the results on asymmetric price behaviour: cotton, copper, lumber and soybeans

show strong evidence of asymmetry in the conditional variance process but other storablecommodities do not. Moreover, skewness measures in Table III and the estimates for cattle inTable IV imply there is asymmetry in the price behaviour of some non-storable commodities.

These conflicting results are interpreted here as evidence against the hypothesis that the non-negative inventory constraint with stockouts is the primary explanation for skewness in price


127

i/ ht-l - Elet-i/ lht-i + bl log(ht-_l)

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128~~~~~~~~~S.BECK

Table VI. GARCH-in-parameters model estimates3

xt= co + cltime +ZEcixt- + C5 dummy +C6 (dummy-time) +Oi Vht + 02 (Xt 1.Vt +ei=1

2

and ht =ao +ZEajoJe,j +biht- Ij=1

Cotton Coffee Copper Sugar Cocoa

co -~~~9.12 3.93 21.19* 9.02* 4.16*(5.22) (4.29) (5.98) (1.21) (1.00)

Cl -~~~~~~~~~~~~~~0.05*(0.0 1)

C2 1.21* 0.99* 0.70* 0.76* 0.63*(0.1I1) (0.1I1) (0.08) (0.09) (0.03)

C3 -0.51*(0.1I0)

C4 0.27*(0.05)

C5 6.17*(0.92)

C6 -0. 15*(0.01)

01 ~~~2.07* 0.18 -1.31 0.23 0.24*(0.86) (0.47) (0.75) (0.29) (0.09)

02 -0.05* -0.01 0.01 -0.01 -0.01

(0.01) (0.01) (0.01) (0.01) (0.00)ao 5.78* 57.83* 32.86* 0.60 5.94

(2.41) (8.34) (5.93) (0.42) (3.61)al 0.20* 0.43* 0.67* 0.78* 3.28*

(0.07) (0.16) (0. 19) (0.29) (0.82)

a2 ~~~0.67 0.65*(4.95) (0.27)

In L -412.52 -611.82 -534.35 -182.25 -247.63

LR testa 20.74* 3.86 1.36 3.24 3.78co ~~~13.76* 311.57* 35.68 28.91* 77.31*

(2.91) (24.18) (21.03) (4.09) (12.97)Cl 0.07* -3.89* -0.81* -1.02*

(0.01) (0.99) (0.27) (0.17)C2 0.20 0.47* 1.26* 0.74* 1.04*

(0.18) (0.21) (0.14) (0.04) (0. 11)

C3 -0.43* -0.27*(0. 11) (0.12)

C5 ~~~~~~~117.24* 9.32*(27.03) (3.70)

C6 -5.42*(1.65)

01 -~~~1.44 -0.73 1.81* 0.09 0.23(0.88) (1.55) (0.93) (0.05) (0.14)

Copyright 2001 John Wiley & Sons, Ltd.J.Ap.En.1:1532(0)

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Table VI. (continued)

Hogs Soybeans Corn Lumber Wheat

02 0.13* -0.00 -0.01*(0.06) (0.01) (0.00)

ao 11.65* 1569.74* 180.61* 25.82 291.97*(1.16) (7.28) (59.10) (14.79) (3.76)

al 0.38 0.14 0.73* 2.26* 1.06*(0.20) (0.12) (0.32) (0.82) (0.38)

bl -0.39*(0.10)

In L -166.12 -280.91 -243.65 -192.35 -294.17

LR testa 5.20 1.18 4.62 1.88 0.10

Note: Standard errors are in parentheses. Asterisks (*) indicate significance at the 5% level.a The likelihood ratio statistic tests the restriction 01 = 02 = 0.

series. This reinforces Deaton and Laroque's (1992) conclusion that additional factors stronglyaffect commodity price time series behaviour. Nor do these results support a possible alternativeexplanation for price asymmetry based upon the existence of government support programmes(e.g. see Shonkwiler and Maddala, 1985). There is no discernable pattern in asymmetric pricemovement between supported commodities (e.g. corn, wheat, milk, cotton) and unsupportedcommodities (e.g. cattle, hogs, eggs, lettuce, tomatoes, onions and lumber) in Tables I andIV.12,13

The GARCH-in-parameters specification implied by the model in Section 2 was estimatedfor commodities where an GARCH process had been indicated in Table II.14 The signs of thecoefficients of x/ht and (xt-_l Vht) were expected to be positive and negative respectively. Thosefor cotton, cocoa and corn, are significant and have the correct signs; one coefficient for hogs issignificant with the wrong sign. However, the likelihood ratio test is not significant for cocoa,corn or hogs, but is significant only for cotton. In general, the evidence shows that agents do notrespond to predictable fluctuations in price variance in those markets where they exist. This isconsistent with Beck's (1993) results.

12 Some commodities were covered by a type of support programme, market orders, during part of the time span coveredby the data series, e.g. apples and potatoes. Others were covered by international agreements that had varying degrees of

success over the period, e.g. copper, cocoa, sugar and coffee. It is difficult to determine how responsible these programmesare for creating the skewness observable in the data. US government programs changed every few years or less, and therewas often doubt about their final form and support level. Moreover, the political process accompanying these changesprobably affected producers' expectations and behaviour. Credibility was often a problem, especially for internationalagreements. See, for example, Cochrane and Runge (1992, Chapters 3 and 4) and Gilbert (1987).13Another explanation for asymmetric price behaviour that is not pursued here is that there is an absence of transactionswhen the price drops below average variable cost. Suppose, for example, a plentiful harvest develops or demandunexpectedly decreases. Initially, price may start above average variable cost, but then as price drops, harvest or productionwill cease, thus limiting supply. This creates a floor in the price series, making the series appear skewed. This effectmay exist for either non-storable or storable commodities. Table III shows considerable skewness in the price equationresiduals of non-storable commodities such as lettuce, onions, grapes and pears that could be due to this effect.14A GARCH process, rather than an EGARCH process, was assumed in the GARCH-in-parameters estimations. Asdiscussed earlier, the evidence suggests that the GARCH is a superior specification in most cases. Even in the fewcases where EGARCH

appearedto be a better

fit,the GARCH

specificationwas retained since reasonable

parameterestimates were obtained whereas convergence to reasonable parameter estimates could not be obtained with the EGARCHspecification.


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S. BECK

6. CONCLUSION

A model of an autoregressive conditional heteroscedastic price variance process in commod-

itymarkets has been

developedfrom a

generalizationof Muth's rational

expectationsmodel.

This model indicates that a serially correlated variance process will exist for storable com-modities but not for non-storable commodities. Tests conducted here on both storable andnon-storable commodity price series indicate that a serially correlated price variance processexists only for storable commodities, as expected. There is little evidence that storable com-modities have a more asymmetric price distribution than non-storable commodities, as thepresence of a non-negative constraint on inventories in the model might imply. The ratio-nal expectations model indicates that expected price variability should influence price levelif agents are risk-averse. However, there is no evidence that the expected variance has asignificant effect on price behaviour, yielding the counter-intuitive conclusion that expectedprice risk does not play an important role in commodity markets. These results supportthose obtained in Beck (1993) who found that an ARCH process exists in the futuresprices of storable commodities but that expected price variance had no effect on futuresprices.

APPENDIX

Muth's (1961) model is modified here to include demand and supply shocks with stochasticvariances. The model results in an ARCH(1) process if the commodity is storable. In addition,parameters in the price level equation depend on expected variances. Testable hypotheses are

obtained concerning the effects of the expected price variance on the coefficients of the price levelequation. See Beck (1993) for a more detailed derivation.Substituting equations (1)-(3) into the equilibrium condition (4) yields

-rpt + at+l cpt -Pt) = yte +at (Pttu - )-l +et (A 1)

where ed, es are independently distributed supply and demand shocks

et = ed - es is distributed with a zero mean and a stochastic variance, a2

a2,= a random variable with a constant mean, c2, and a constant (finite) variance.

Because parameter at is time-dependent, the solution was found by specifying a terminal conditionin the future (t = T) and solving the model backward. The general solution is

Pt = tPt-l (A2)et

and Pt = tt-l + et (A3)t+(l - t+l)+ +

where At =+ y + at + at+i(l -t+l)

When the commodity is not storable, At = 0 so pe = 0 and pt = -etl//3.The conditional variance of Pt is derived from equation (A2). It can also be found by using the

terminal condition and iteratively substituting backward. Because a2 appears in the parameters, AtCopyightU 2001ILV LJohnLWileyU& Sons OLtd.J.Api. co.6:115-13

(2001)Y~UCV y

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and at, the conditional variance expression must be linearized and solved for a,2 at each iteration.15Recalling that the expected value of the variance of et is constant, the general solution is:

a2 = 0+ 10a2 (A4)Pt Pt-1

Equation (A4) indicates that price volatility creates price volatility in future periods by creatingvolatility in inventory carryover. Here production and decision lags are one period long, sothere is an ARCH process of order one. Without inventory carryover, At = 0, therefore pricevariance depends only on the expected variance of external shocks, which is assumed to beconstant here.

Equations (A3) and (A4) imply a variation of the ARCH-M model of Engle, Lilien and Robins(1987). To determine the effect that changes in the expected variance have on the parameters ofthe price level equation, first rewrite equation (A3) as

etXt = Xt -++ [tx t_1tat+I(l - )t+li)+/5where pt = (xt - p)

xt = (1 - Xt)

dX*t _ 8t dotThe inequality - < 0

da2 aaot da2,

dao -Kholds because d = < 0da2 (o2 )2P,t P

and -d >0da, (a, + y + f)2

The price coefficient of the price level equation is inversely related to the expected variancebecause higher price ris-kiness, reflected in a large price variance, reduces inventory carryoverwhich reduces the influence of lagged price on current price. The inequality

dXt dXt aXt dlt

da2 a>.t aat de2,

holds because the second and third terms are positive and negative respectively and

dXtd - p < 0

Large price variance reduces inventory carryover which reduces current market supply, henceraising the current price level. From these results it is inferred that ko is a positive function andkl is a negative function of the expected variance.

15 The perfect foresight assumption was used for expectations of the price variance whereas rational expectations wereassumed for the price level. This was done to keep the model tractable. A rational expectations

assumptionfor the variance

results in moments of the a2 distribution appearing in these equations. Since that distribution is assumed to be stationary,its moments are constants so they are ignored here.


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