Measurement of Price Risk in Revenue Insurance: Implications of Distributional Assumptions

Transcription

1 Jurnal fagricultural and Resurce Ecnmics 25(1): Cpyright 2 Western Agricultural Ecnmics Assciatin Measurement f Price Risk in Revenue Insurance: Implicatins f Distributinal Assumptins Barry K. Gdwin, Matthew C. Rberts, and Keith H. Cble A variety f crp revenue insurance prgrams have recently been intrduced. A critical cmpnent f revenue insurance cntracts is quantifying the risk assciated with stchastic prices. Frward-lking, market-based measures f price risk which are ften available in the frm f ptins premia are preferable. Because such measures are nt available fr every crp, sme current revenue insurance prgrams alternatively utilize histrical price data t cnstruct measures f price risk. This study evaluates the distributinal implicatins f alternative methds fr estimating price risk and deriving insurance premium rates. A variety f specificatin tests are emplyed t evaluate distributinal assumptins. Cnditinal heterskedasticity mdels are used t determine the extent t which price distributins may be characterized by nncnstant variances. In additin, these mdels are used t identify variables which may be used fr cnditining distributins fr rating purpses. Discrete mixtures f nrmals prvide flexible parametric specificatins capable f recgnizing the skewness and kurtsis present in cmmdity prices. Key wrds: mixture distributins, price risk, revenue insurance Intrductin A variety f crp revenue insurance prgrams have recently been develped t supplement the standard Multiple Peril Crp Insurance that has existed in the U.S. since the 193s. In general, these prgrams guarantee prducer revenues by prtecting against any revenue-diminishing cmbinatin f lw prices and/r lw crp yields. The revenue insurance cntracts guarantee prducers a minimum level f revenues. If, because f any cmbinatin f pr yields and/r lw prices, revenues are beneath the guaranteed level, insured farmers receive an indemnity payment equal t the difference between realized and guaranteed revenues. Increased planting flexibility and recent farm prgram changes which included the eliminatin r reductin f direct price supprts have led many t anticipate increased price risk and uncertainty. Such cncerns have heightened interest in the revenue insurance prducts. Gdwin is prfessr and Rberts is graduate research assistant, bth in the Department f Agricultural and Resurce Ecnmics, Nrth Carlina State University; Cble is assistant prfessr, Department f Agricultural Ecnmics, Mississippi State University. We are grateful t tw annymus reviewers fr helpful cmments and suggestins. This research was supprted by the Ecnmic Research Service f the USDA. The pinins and cnclusins expressed herein are thse f the authrs and d nt necessarily reflect the views f the Ecnmic Research Service r the U.S. Department f Agriculture.

2 196 July 2 Jurnal fagricultural and Resurce Ecnmics Three alternative crp revenue insurance prducts currently exist: Crp Revenue Cverage (CRC), Incme Prtectin (IP), and Revenue Assurance (RA). 1 Cnventinal crp insurance prgrams have been hampered by actuarial prblems that have led t significant lsses. In particular, prgram utlays exceeded $8.9 billin between 199 and 1997 [U.S. General Accunting Office (U.S. GAO)]. These high lsses have been attributed t adverse selectin and mral hazard issues. Adverse selectin ccurs when prducers have mre infrmatin abut their risk than d insurers, such that premium rates are inaccurate. Mral hazard ccurs when insuring prducers alter their behavir in rder t increase the likelihd f cllecting indemnities. Inaccurate premium rates and perfrmance mnitring prblems underlie the actuarial shrtcmings f crp insurance prgrams. Cnventinal yield insurance prgrams need accurate measurements f an individual prducer's distributin f expected yields in rder t determine actuarially fair premium rates. In the case f revenue insurance, an additinal critical cmpnent f the prper insurance premium is setting a rate that accurately reflects the price dimensin f risk. A variety f methds fr measuring price risk have been prpsed. A reprt recently released by the General Accunting Office is critical f the actuarial methds underlying all three revenue insurance plans (U.S. GAO). It is imprtant t nte that yields and prices are likely t be negatively crrelated since lw yields are typically accmpanied by high prices. The extent f this crrelatin fr an individual prducer depends upn the degree f crrelatin between the prducer's yields and an aggregate yield, such as the natinal average. This, in turn, depends upn the spatial crrelatins f yields ver principal prductin regins. The three primary revenue insurance cntracts have different appraches t addressing this yield-price crrelatin issue. CRC, the largest f the three main revenue insurance prgrams, simply treats yield and price risk as thugh they are independent. Standard Multiple Peril Crp Insurance (MPCI) rates are added t a premium cmpnent that represents the price side f revenue risk. Negative crrelatin implies that the risks assciated with a revenue shrtfall are prbably less than thse assciated with price and yield shrtfalls when the latter are cnsidered in islatin. This is because lw yields wuld typically be expected t increase price, thus ffsetting a prtin f the revenue shrtfall. In this manner, CRC is smetimes said t be "cnservatively" rated-i.e., the CRC rate is higher than a rate which recgnized the negative yieldprice crrelatin wuld be. The emphasis f ur analysis is n evaluating methds fr rating the price side f risk in the largest revenue insurance prgram, CRC. As is the case with current CRC rating methds, we d nt cnsider the issue f yield and price crrelatin, thugh we d discuss belw hw ur analysis might be extended t cnsider such crrelatin. In cntrast t the CRC plan, the RA and IP plans d attempt t accunt fr yield and price crrelatin, thugh the adjustments made t accunt fr crrelatin have been questined (see U.S. GAO, pp , 71). 1 Additinal frms f revenue insurance, including insurance which utilizes Schedule F tax return infrmatin as a basis fr insurance and an areawide versin that utilizes cunty average yields as a basis fr insurance, are currently under develpment. Althugh the issues discussed in this study are pertinent t all three prducts, the specific prvisins f the cntract and examples are taken frm CRC.

3 Gdwin, Rberts, and Cble Measurement f Price Risk in Revenue Insurance 197 Within the academic cmmunity, the Risk Management Agency (RMA), and thrughut the insurance industry, there has been a cllabrative fcus n the develpment f prper actuarial methds fr rating revenue insurance cntracts. Stkes, Nayda, and English review research n revenue insurance pricing methds and discuss ptinsbased pricing methds fr rating revenue insurance. Cnsiderable disagreement exists regarding the prper apprach fr rating price risk. While it is widely recgnized that frward-lking, market-based measures f price risk are t be preferred, it is als the case that such market-based mechanisms d nt exist fr several f the crps cvered by revenue insurance. Fr example, apprpriate ptins markets d nt exist fr sft white wheat, which is cvered under new revenue insurance cntracts. In additin, because rates are set several mnths befre planting, ptins markets fr many crps have very lw vlumes and thus are inapprpriate fr rating. Recent discussins have addressed three alternative appraches t rating price risk. The current CRC prgram uses a histrical series ( ) f futures prices, quted at planting time (F t ) and harvest time (Pt) t derive a "frecast errr" (et = Pt - Ft), which is then assumed t be nrmally distributed. The prtin f premium assciated with price risk is then calculated using standard results fr a nrmal distributin. An apprach which utilizes prprtinal errrs (et/p t ) under the assumptin f nrmality has been recmmended as an alternative. This apprach assumes that errrs are prprtinally larger as prices are higher, and is thus smewhat analgus t assuming a lgnrmal distributin fr prices since lgnrmality suggests a prprtinal relatinship between the variance and the mean f the bserved data. A third apprach t rating price risk utilizes existing ptins markets t derive market-based measures f price risk. As nted, this apprach, while clearly preferable, is nt apprpriate fr all revenue insurance cntracts since the necessary ptins cntracts d nt exist fr all crps currently insured. The revenue assurance (RA) versin f revenue insurance utilizes crn and sybean ptins premia t rate revenue insurance cntracts. The assumptin f lgnrmality has cnsiderable precedent in the financial literature. Mdels f price variability and ptins price determinatin have typically assumed that prices are lgnrmally distributed. In particular, the Black-Schles ptin valuatin frmula, which is based n the assumptin f lgnrmally distributed prices, has gained widespread acceptance. Hwever, relatively little attentin has been given t evaluating the extent t which prices adhere t distributinal assumptins and the ptential implicatins f distributinal misspecificatin. Mre recent research (seare, e.g., Crnew, Twn, and Crwsn; Hudsn, Leuthld, and Sarassr; Hall, Brrsen, and Irwin; Hsieh) has dcumented leptkurtsis, skewness, and ther distributinal characteristics that may be incnsistent with nrmality and lgnrmality. Recgnitin f these prblems has led t the develpment f a variety f appraches t easing distributinal restrictins and prviding mdeling techniques that allw fr nnnrmal distributins. The distributin f market prices als may be sensitive t market cnditins. Distributinal shifts may ccur if market cnditins change. If the variance f prices is timedependent, and if this time dependence is nt explicitly mdeled, the distributin f prices bserved ver time may invlve a mixture f different variances and thus may exhibit characteristics incmpatible with nrmality. 2 The price series may als display ther distributinal characteristics such as skewness, kurtsis, and multiple mdes. 2 A mixture f tw zer mean nrmal prcesses with different variances will typically imply a distributin that exhibits kurtsis.

4 198 July 2 Jurnal fagricultural and Resurce Ecnmics Recent research has applied alternative techniques t derive price distributins that reflect characteristics nt cnsistent with nrmality (see, e.g., Hall, Brrsen, and Irwin; Hsieh). In ne apprach, finite mixtures f knwn distributins are used t represent distributinal characteristics that are nt cmpatible with nrmality. This apprach is ften mtivated by the assumptin that, althugh a standard distributin is apprpriate under a given set f market cnditins, changing market cnditins may result in different distributins. Thus, when the entire series f prices is bserved, the underlying prcess describing the aggregate distributin is a mixture f several distributins. In ther research, mixed-jump prcesses have been used t represent nnstandard distributins. Jump prcesses are apprpriate in situatins where randm shcks shift the entire distributin. In bth cases, the resulting distributins are capable f representing characteristics f a series that may nt be cnsistent with nrmality r lgnrmality. Fr example, a simple mixture f tw nrmals is capable f representing a standard, symmetric nrmal distributin as well as nnsymmetric distributins, skewness, bimdality, and leptkurtsis. Figure 1 (panels A and B) illustrates implied vlatilities fr crn and wheat, respectively. In bth cases, the vlatilities appear t imply tw general states f nature. 3 In the first and mst cmmn state (perhaps 75% f the time), vlatilities are relatively stable at arund 15%. In the secnd and less frequent state, vlatilities are much higher. Of curse, the patterns f vlatility als reflect seasnality in variance. While the implicatins f such a cursry examinatin f weekly intraseasn data fr the annual price data required fr revenue insurance prducts are unclear, the illustratin prvides at least anecdtal evidence cnsistent with a mixture f a lw variance and a high variance state. The bjective f this analysis is t explre the distributinal characteristics f crn and wheat prices, fcusing n the measurement f price risk fr determining premium rates fr crp revenue insurance prgrams. We utilize time-dependent cnditinal heterskedasticity mdels and mixture distributin mdels t evaluate price risk. The cnditinal heterskedasticity mdels evaluate the rle f time t maturity, cntract qute and expiratin dates, and annual fixed effects in mdeling wheat and crn price variability. Implicatins fr imprving actuarial methds utilized in the revenue insurance prgrams are als cnsidered. The article prceeds in the fllwing manner. We begin with a descriptin f crp revenue insurance prducts available in the U.S. The ecnmetric methds applied t the analysis f price risk are then develped. In the next sectin we analyze the time dependency f the variance f prices and prvide a discussin f cnditinal heterskedasticity mdels that relate price variatin t a number f explanatry factrs. Mdels f cnditinal crn and wheat price distributins btained under alternative distributinal assumptins, including nrmality, lgnrmality, and discrete mixtures, are als presented. The final sectin ffers a brief review f the analysis and sme cncluding remarks. 3 Of curse, ne culd argue in favr f mre than tw states. As we pint ut belw, identificatin f multiple states is an exercise generally cnstrained by the number f bservatins available fr empirical wrk.

5 Gdwin, Rberts, and Cble Measurement f Price Risk in Revenue Insurance > 4 -._ ac " 3.a) - Ė 2 1 n 8 vb t 3t a a' 9 9~~~~~~~~~~~~~~K A ( "I "I9 3 9 I ' 4 a> 3 a- 3 E 2 1 Figure 1. Implied vlatilities fr December crn and September wheat ptins cntracts

6 2 July 2 Jurnal fagricultural and Resurce Ecnmics Revenue Insurance Prgrams Standard Multiple Peril Crp Insurance (MPCI) has been in existence in varius frms since the 193s. This insurance pays indemnities at a predetermined price whenever realized yields are less than guaranteed yields. A shrtcming f standard MPCI can be bserved in the price (determined prir t planting) at which indemnities are paid. When yield lsses are widespread, market prices are likely t be higher. Farmers receiving indemnities fr lst yields may actually be reimbursed smewhat less (in bushel terms) than their guarantee since their indemnities likely reflect a price that is lwer than the market price at harvest time. Revenue insurance had its beginnings with an ptinal rider that paid indemnities at harvest-time market prices. This, in cnjunctin with a put ptin cntract, allwed prducers t guarantee a minimum level f crp revenues. This cverage was extended t frm the basis fr individual Crp Revenue Cverage (CRC). CRC is currently available in majr crn, sybeans, wheat, cttn, and grain srghum grwing regins. CRC has been quite successful, accunting fr ver 26% f crn crp insurance sales in 1997; the latest sales figures indicate that revenue insurance plans currently accunt fr abut 23% f the ttal acreage insured (RMA). Incme Prtectin (IP) was develped at Mntana State University under a directive f the Federal Crp Insurance Refrm Act t create a pilt insurance plan based n the actual csts f prductin. IP insurance is available fr crn, sybeans, grain srghum, cttn, and wheat in majr grwing regins. IP guarantees a minimum level f crp revenues, based n frecasted prices, individual farm yields, and area yields. If realized revenues fall beneath the revenue guarantee, prducers receive an indemnity payment fr the amunt f the shrtfall. Revenue Assurance (RA) was develped by the Iwa Farm Bureau as a pilt prgram fr crn and sybeans in Iwa. RA prvides the ptin fr "whle-farm" insurance in which prducers insuring bth crn and sybeans receive significant premium discunts. RA prvides a guaranteed minimum level f revenue which is determined by individual farm yields and futures prices (adjusted fr the lcal histrical basis). If realized revenues are beneath the guarantee because f either lw prices r lw yields, r bth, farmers receive an indemnity payment fr the amunt f the shrtfall. A unique characteristic f the RA prgram is the utilizatin f market-based measures f price risks that are available in ptins markets. In cntrast, the CRC and IP prgrams utilize histrical futures prices t develp measures f price risks. RA actuarial prcedures emply estimates f a beta distributin t mdel yield risks. Ecnmetric Methds Revenue insurance cntracts require a frecast f harvest-time prices, made cnditinal n infrmatin available prir t planting time. In additin, a measure f the uncertainty assciated with the price frecast is needed t cnstruct a premium rate reflecting the risk f adverse mvements in prices. In all three revenue insurance plans, futures prices are used t frecast harvest-time prices. In the case f RA, ptins markets are used t gauge the uncertainty assciated with prices. IP and CRC utilize histrical price mvements t evaluate price risks. The measurement f price risk in bth the RA and CRC prgrams is heavily dependent upn assumptins regarding the

7 Gdwin, Rberts, and Cble Measurement f Price Risk in Revenue Insurance 21 parametric distributins underlying price mvements. RA adpts standard Black- Schles results t cnstruct implied vlatilities frm bserved ptins prices. As nted abve, this apprach assumes lgnrmally distributed prices (r, t be mre precise, this mdel assumes a-gemetric Brwnian mtin prcess fr prices, which is the cntinuus time equivalent f a lgnrmal distributin). In cntrast, CRC assumes nrmally distributed prices in the cnstructin f the price cmpnent f the revenue insurance premium. IP utilizes a nnparametric "empirical distributin" apprach. 4 In this analysis we emply tw distinct appraches fr evaluating price risk. In the first, the interest lies in determining if the variance f histrical prices, which is used in rating revenue insurance prducts, is cnstant. Maximum-likelihd estimates f cnditinal heterskedasticity mdels are used t evaluate the exgenus determinants f price variability. If an empirical analysis cnfirms that variances are cnstant r, alternatively, identifies factrs which underlie a nncnstant variance, mdel estimates can prvide cnditinal variance frecasts t be used in revenue insurance cntract cnstructin. In the secnd segment f the analysis, a set f annual price data is utilized t estimate price distributins and t evaluate insurance premia under alternative distributinal assumptins. In the cnditinal heterskedasticity mdels, the variance f cnditinal prices (i.e., price differences) is assumed t be prprtinal t a functin f several exgenus factrs which are hypthesized t be related t price variability. In particular, it is assumed that the variance f prices fr an individual cntract i quted at time t is given by the fllwing: ~~~~(1) (~i2 = 2 f(zity). We assume that the cnditinal variance functin f(zty) is the square f a linear index functin-i.e., (Zity) 2. Such a mdel f multiplicative heterskedasticity is widely applied in the literature. (Fr a detailed discussin f this mdel and its many variants, see Harvey.) Our specificatin ensures nnnegative variances fr all bservatins. Under the assumptin f nrmality, the fllwing lg-likelihd functin is maximized t btain estimates f y and, if applicable, f parameters f a cnditinal mean equatin (pit = XiP): 5 (2) InL -n [ln( 2 c) + ln( 2 )] - ln((zit)2) 2 2 i=i 1 (Yit - Pit) 2 22 i= (ZitY) 2 4 Nnparametric density estimatin techniques ffer cmplete flexibility in representing characteristics f a distributin. Such flexibility, hwever, des nt cme withut a significant lss in efficiency. Thus, the nnparametric techniques may nt be apprpriate fr the small samples which are cmmnly available fr measuring price risk. In that prbability density functins are cmmnly used as kernel functins in nnparametric density estimatin, the nnparametric techniques are analgus t mixtures f a large number f cmpnents. Fr example, nnparametric estimatin with Gaussian kernels is analgus t a mixture f n nrmals with equal variance terms (i.e., as determined by the kernel bandwidth). 5 A cnditinal mean equatin represents mvements in expected prices, cnditined n bservable data (typically expressed as Yit = Xip). In ur applicatin, yi represents the price difference, and n cnditining variables are added t the mean equatin. As we explain belw, this equatin is mdified t allw fr first-rder autcrrelatin by replacing y, with yit - PYit-1

8 22 July 2 Jurnal f Agricultural and Resurce Ecnmics The secnd part f the analysis evaluates the distributinal prperties f the price data cmmnly used t rate revenue insurance. Finite mixture distributins represent a flexible, parametric apprach t mdeling prbability distributin functins whse intrinsic characteristics are largely unknwn. A k-cmpnent mixture density functin is given by: (3) f(x) = [ifi(x)] i=l k where the prbability weights (Qi) satisfy the cnditins that E/= i = 1, and i > fr all i. In ur applicatin, we cnsider nly a mixture f tw distributins, such that there is a single mixing parameter X. Varius densities are cmmnly applied in representing the underlying cmpnents f the mixture. The mst cmmn apprach invlves utilizing nrmal densities: 1 (x-p) /-22 (4) fi(x) - e Mixtures f nrmals nest a cnventinal nrmal distributin (btained when, 1 = p2 =... = lk, and a = 2 =... = k). Asymmetric and bimdal distributins may result when the pi's are nt all equal. Kurtsis is implied when the pi's are nt all identical. Standard maximum-likelihd estimatin techniques are cmmnly used t estimate mixture distributins. There are, hwever, particular characteristics f mixture prblems that may cmplicate estimatin. Nnlinear estimatin techniques may have a tendency t cncentrate cmpnent densities n individual pints. In such a case, the ai assciated with that pint ges t zer and the likelihd functin becmes numerically unstable. T prevent such instabilities, the X and, terms must be cnstrained t be psitive. Estimatin must als recgnize that the mixing parameter X must be cnstrained t lie in the interval (, 1). Cnstrained maximum-likelihd estimatin techniques are used in this study t estimate the cmpnents f the mixture. We cnstrain ai t be greater than 1E-9, and X t lie in the clsed interval [O + e, 1 - e] fr E= 1E-9. The fact that the mixing parameter must be cnstrained and can lie n the bundary f the parameter space raises special cncerns fr hypthesis testing. In particular, test statistics may nt have cnventinal distributins when the true parameter value is n the bundary f the parameter space. Likewise, under the null hypthesis that the mixing parameter X is (r, equivalently, 1), the parameters f the cmpnent distributins may nt be fully identified. Prblems in tests where a subset f parameters may be unidentified under the null hypthesis are cmmn and can be addressed (see, e.g., the extensive literature underlying structural change tests with unknwn break pints, including the wrk f Andrews and f Hansen). In this applicatin, the parameters characterizing the cmpnents f the distributin (i.e., ai and pi) are unidentified if X = O.6 This precludes the applicatin f standard hypthesis testing techniques fr determining the number and nature f the cmpnent distributins. Btstrapping techniques are cmmnly used as an alternative t cnstruct empirical 6 It shuld be nted that cmpnent parameters f individual distributins can be estimated when X is cnstrained t be psitive, even when the estimate f X is very small.

9 Gdwin, Rberts, and Cble Measurement f Price Risk in Revenue Insurance 23 distributins frm which apprpriate critical values and assciated p-values can be btained. McLachlan, and Feng and McCullch (1994, 1996) discuss btstrapped likelihdrati tests fr evaluating hyptheses in finite mixture mdels. These tests are particularly apprpriate fr determining the number f cmpnents t include in a mixture density. The tests als may be used t evaluate a particular parametric distributinal specificatin. Fr example, an evaluatin f a nrmal versus a lgnrmal distributin can be cnsidered using a mixture f the frm: (5) f(x) = Xp(x) + (1 - A)(x), where (p(.) represents the lgnrmal prbability density functin (pdf), and 4(') represents the nrmal pdf. The estimate f A indicates whether the distributin is nrmal r, under the alternative, is a mixture f nrmal and lgnrmal densities. Because the likelihd-rati test statistic [given by -2(LR - LU), where LR and LU represent the restricted and unrestricted maximum lg-likelihd functin values, respectively] and parameters f the cmpnent distributins are defined even when X is at the bundary, this apprach prvides a straightfrward means fr evaluating the number f cmpnents. McLachlan recmmends a parametric btstrap, whereby the data are simulated using estimates btained under the null hypthesis. Fr each btstrap replicatin, the alternative mdel is fit and the likelihd-rati test statistic is cnstructed. The assciated p-values, which can be used t evaluate the significance f the likelihd-rati test statistic btained frm the estimatin sample, can be calculated using the replicated test statistics. We fllw this btstrapping prcedure t evaluate the number and nature f cmpnents in the price distributins. The randm variable x may als represent a cnditinal mean as in the standard linear regressin prblem. In this case, x may be replaced byy -Xf in equatin (3), and the parameters f the cnditinal mean equatin () may be estimated jintly with the parameters f the prbability distributin (a, pi, and X). We fllw this apprach in ur analysis. It shuld be nted that an additive intercept term is nt identified when the mixture des nt restrict estimates f pi. Estimates fr the intercept can be recvered by impsing restrictins n the means f the cmpnent distributins. In particular, the implicit assumptin f a zer mean fr the errrs prvides identificatin. Estimated Mdels and Results The prper treatment f nminal prices bserved ver a lng perid f time is an imprtant issue, especially in the secnd cmpnent f ur analysis which uses data cllected between 1899 and In particular, ne must cnsider whether the prices shuld be deflated. Indeed, this issue has arisen in actuarial debates ver the CRC prgram, where it was decided that nminal prices shuld be used. Of curse, inapprpriate deflatin causes heterskedasticity. Standard price deflatrs such as the CPI are nt apprpriate since they imply unreasnably high prices fr distant perids. This is because agricultural prices have nt fllwed the tendency f aggregate prices t rise ver time. Mdels utilizing lgarithmic transfrmatins f prices imply that the residuals (r price differences) are prprtinal t price levels, and thus that higher prices wuld be

10 24 July 2 Jurnal fagricultural and Resurce Ecnmics expected t crrespnd t larger price differences. Such an implicatin wuld suggest that the mean level f residuals in a lgarithmic mdel (r differences in lgarithmic prices) shuld be relatively stable ver time. In cntrast, mdels expressed in price levels suggest that the mean level f residuals (r price differences) des nt depend upn the price level. T the extent that prices (r residuals) are being driven by mvements in the verall price level, a plt f price differences shuld reveal increasing variability ver time. Such plts (nt presented here) were cnsidered fr bth price levels and lgarithmic transfrmatins f the prices. We did nt find evidence that price differences had trended upward in a manner cnsistent with aggregate price changes. 7 This was especially the case when lgarithmic prices were cnsidered. In light f these results and current rating practices used in the CRC prgram, bth segments f ur analysis emply nminal prices. The Bridge database f daily settlement prices is used t cnstruct mnthly average futures prices fr all cntracts in all mnths ver the perid Expiratin prices were the average in the mnth preceding the cntract's expiratin. This apprach is analgus t the treatment f futures prices in cnstructing CRC premium rates. These data are used t estimate the cnditinal heterskedasticity mdel [equatin (2)] t determine if prices are characterized by nncnstant variances and t prvide apprpriate cnditinal frecasts f price variances. Since the pled data set cnsists f many verlapping cntracts, a cmplex frm f mving-average errr crrelatin is inherent in the price differentials. T allw fr such crrelatin, we specify a first-rder autregressive crrelatin prcess amng the mnthly price differences. The crrelatin structure is restricted t prevent crrelatin crrectins acrss alternative cntracts. Maximum-likelihd estimates and summary statistics fr the cnditinal heterskedasticity mdels (Ziy) are presented in tables 1 and 2 fr crn and wheat, respectively. The mdels were expressed bth in price levels (crrespnding t a nrmal distributin), and in lgarithms f prices (crrespnding t a lgnrmal distributin). Fr crn, the default (mitted dummies) is a September cntract quted in the previus January. Fr wheat, the default is a July cntract quted in the previus January. Thugh the magnitudes f the estimates differ, the results fr the mdels expressed in levels and lgarithms are quite similar. The results strngly cnfirm that the variance f the price differentials is nt cnstant. They reveal that increased mnths t maturity decreases price vlatility. This is cnsistent with the "Samuelsn hypthesis" (Samuelsn) which maintains that prices will reflect mre infrmatin and thus be mre vlatile as cntract expiratin nears. In cntrast t ur findings, Hennessy and Wahl btained results that were nt cnsistent with the Samuelsn hypthesis. Our estimates als indicate that there are significant differences in price variability acrss alternative cntracts. Cntracts which expire in the mnths immediately preceding harvest (July fr crn and May fr wheat) appear t have the mst vlatile prices. Significant differences in the variability f prices ver the grwing seasn are als revealed in the estimates. Crn prices appear t be the mst variable in June, July, and August-the mst critical grwing perid. Likewise, wheat prices appear t be mre variable in April. Wheat prices als appear t be quite variable in June and August, 7 These plts are available frm the authrs n request. We shuld nte that, althugh n trend in price differences was apparent, the variability f price differences did appear t be larger after 197, thugh this pattern was nt reflected in the lgarithmic prices.

11 Gdwin, Rberts, and Cble Measurement f Price Risk in Revenue Insurance 25 Table 1. Maximum-Likelihd Parameter Estimates and Summary Statistics fr Cnditinal Price Heterskedasticity Mdels: Crn Linear Lgarithmic Variables Estimate Std. Errr Estimate Std. Errr p Intercept Mnths t Maturity March Cntract May Cntract July Cntract September Cntract December Cntract February Qute March Qute April Qute May Qute June Qute July Qute August Qute September Qute Octber Qute Nvember Qute December Qute N Bera-Jarque Test (n annual effects) Bera-Jarque Test (annual effects) Test f Annual Effects *.3319*.21* * ,575 1,384.52* 47.65* 1,71.1* *.57*.656*.872*.124* *.469* *.14*.21* * * * * * , * 39.47* * Nte: An asterisk (*) dentes statistical significance at the a =.5 r smaller level. perhaps reflecting harvest realizatins r grwing cnditins fr substitute spring wheats. The strng seasnality in the prices cnfirms the findings f ther studies as well as cnventinal wisdm. The parameter estimates allw a frecast f the variance, cnditinal n cntract and mnth f qute. This is a frecast f the "average" variance fr the particular mnth f qute and cntract ver the years f available data. This frecast culd be used in cnjunctin with a price frecast t cnstruct premium rates fr the price cmpnent f revenue risk. This cnditinal variance may nt be cnstant acrss years. Changes in ther factrs that affect price vlatility (e.g., stcks, prductin, demand shcks, etc.) frm year t year wuld result in annual differences in the cnditinal variances. Annual dummy variables were added t estimate an expanded mdel (nt presented). Likelihd-rati tests (tables 1 and 2) strngly cnfirm the significance f annual effects, implying that the cnditinal variances are nt cnstant ver the years f the analysis.

12 26 July 2 Jurnal fagricultural and Resurce Ecnmics Table 2. Maximum-Likelihd Parameter Estimates and Summary Statistics fr Cnditinal Price Heterskedasticity Mdels: Wheat Linear Lgarithmic Variables Estimate Std. Errr Estimate Std. Errr p * * Intercept *.393.2* Mnths t Maturity * * March Cntract May Cntract July Cntract September Cntract December Cntract February Qute * March Qute * April Qute * * May Qute June Qute * * July Qute * August Qute * * September Qute Octber Qute * * Nvember Qute * December Qute * R N 2,8 2,8 Bera-Jarque Test (n annual effects) 1,949.81* * Bera-Jarque Test (annual effects) 9.72* 12.99* Test f Annual Effects 1,881.61* 5,955.15* Nte: An asterisk (*) dentes statistical significance at the a =.5 r smaller level. Hwever, the expanded mdel cannt be used fr frecasting variances ut f sample since nly infrmatin available at the time a frecast is made can be used t cnditin the frecast. The mdel which mits the annual dummy variables prvides an "average" variance frecast which culd be cnditined upn the mnths f cntract and qute and used t frecast the variance, and thus rate price risk. Such mdels may ffer advantages ver current prcedures which utilize nly a single cntract quted at a single perid f time by allwing use f a much larger sample (derived frm using many cntracts quted ver many different perids), thereby ptentially imprving the statistical efficiency f frecasts. Bera-Jarque cnditinal mment (chi-square) tests f nrmality were als applied t evaluate the extent t which the mdels were cnsistent with nrmality and, in the case f the lgarithmic mdels, lgnrmality. When the test is applied t the price-level mdels presented in tables 1 and 2, nrmality is strngly rejected in every

13 Gdwin, Rberts, and Cble Measurement f Price Risk in Revenue Insurance 27 case. 8 Likewise, the tests reject nrmality in each f the lgarithmic versins f the mdel, suggesting that lgnrmality is als unsupprted. When the test is applied t the mdels cntaining annual dummy variables, nrmality is still rejected, thugh at a much lwer level f significance. This suggests that missin f fixed annual effects which are related t factrs that influence variability frm year t year results in a distributin that is much less cnsistent with nrmality than when such annual effects are accunted fr. The residual nnnrmality in the mdel withut dummy variables t accunt fr shifting annual variances may in part result frm the implied mixture f (pssibly nrmal r lgnrmal) distributins with different variances being assciated with each year's distributin. Such a cnclusin is smewhat tenuus, hwever, in light f the fact that the mdels cntaining annual effects still reject nrmality, albeit at a much lwer level f significance. Alternatively, these results may suggest that the distributin f futures prices is incnsistent with either nrmality r lgnrmality, and thus that mre flexible cnditinal heterskedasticity mdels perhaps shuld be cnsidered. 9 In summary, the results frm the first mdel shw that futures price variability may be cnditined upn a number f explanatry factrs, including mnths t maturity, mnth f cntract, and mnth f price qute. These findings shuld be useful fr cnstructing mre accurate premium rates fr the price-risk cmpnent f revenue insurance cntracts. The prpsed mdeling apprach allws a much larger sample t be used in cnstructing premium rates, ptentially imprving inferences and the accuracy f premium rates. Cnditinal mment tests reject nrmality, which may in part result frm the mixing f time-varying variance distributins. The secnd segment f the analysis utilizes a lng series f annual bservatins n planting- and harvest-time futures prices. Crn and wheat futures were cllected frm selected issues f the Chicag Bard f Trade's Annual Reprt f the Trade and Cmmerce f Chicag fr the perid cvering 1899 t 196. Data fr subsequent years were taken frm the Bridge financial database. Mnthly bservatins fr cntracts expiring at harvest (September fr crn and July fr wheat) were cnstructed by taking the midpint f the mnthly high and lw price qutes at planting times (January fr crn and December fr wheat). 1 The "harvest-time" price fr each cntract was that quted in the mnth preceding the cntract's expiratin. Maximum-likelihd techniques were emplyed t estimate alternative mdels f the annual price differentials in the secnd part f the analysis. A price relatinship f the frm Pt = a + pft was estimated, where Pt represents the harvest-time price, and Ft is the planting-time futures price. 11 In light f the prevailing assumptin f lgnrmality fr price distributins, five separate mdels differing in their distributinal assumptins 8 It shuld be nted that we d nt reprt rbust standard errrs, and thus ur estimated standard errrs may be incnsistent if remaining residual heterskedasticity is present. 9 Fr example, Ramirez presented a flexible autregressive cnditinal heterskedasticity (ARCH) mdel that accunts fr unimdal nnnrmality. Such mdels may have prmise in applicatins such as this ne. 1 This apprach was necessitated by the available data-daily prices were nt available befre An evaluatin f the difference in the mnthly price cnstructed in this manner and a mnthly average f daily clsing prices revealed n significant difference. In particular, the average differential between the alternative mnthly prices was nearly zer. Our use f these particular cntracts was als necessitated by the availability f data. 11 Similar results were btained when the mdels were cnstrained accrding t an "efficient-markets" type f relatinship, such that a =, and P = 1. We estimate the parameters rather than cnstraining them in rder t allw fr any biases r premia which may exist in the relatinship between prices. This is analgus t a linear mean frecast cnditined upn futures prices at planting.

14 28 July 2 Jurnal fagricultural and Resurce Ecnmics were estimated. These included nrmality, lgnrmality, a mixture f tw nrmals, a mixture f tw lgnrmals, and a mixture f a lgnrmal and a nrmal. The mixture mdels permit testing f standard distributinal assumptins using the btstrapping prcedures described abve. Tables 3 and 4 present the estimatin results fr the mdels f crn and wheat futures price relatinships, respectively. Nte that, with the exceptin f the variance estimate, maximum-likelihd estimates btained under nrmality are equivalent t rdinary least squares (OLS). Estimates labeled as "OLS" in tables 3 and 4 are actually the equivalent maximum-likelihd estimates btained under nrmality. Bera-Jarque nrmality tests are used t assess the extent t which the OLS residuals are cnsistent with nrmality and lgnrmality. As was the case abve fr the large sample f cntracts, the tests reject nrmality and lgnrmality fr bth crn and wheat. These results questin the validity f the assumptins f nrmality and lgnrmality used in the cnstructin f revenue insurance premia. They suggest that alternative, flexible distributinal specificatins may be preferred. The OLS estimates fr the nrmal and lgnrmal mdels have price cefficients which are slightly less than ne. The mixture f nrmals mdel fr crn has a price cefficient f.81-smewhat far frm the expected value f ne which wuld crrespnd t the futures price being an unbiased frecast f the future harvest-time price. The price cefficients fr the ther crn and wheat mdels are very similar, with values f abut Recall that the mixing parameter A characterizes the frequency f the alternative regimes. The estimated mixture f nrmals mdels pints t an envirnment characterized by a mixture f a frequent (75-89% f the time) lw-variance regime and a less frequent (11-25% f the time) high-variance regime. A similar pattern f variability is implied by the lgnrmal mixtures. 12 Such a finding is cnsistent with the pattern bserved fr ptins premia (figure 1). Our mixture apprach is smewhat analgus t mdeling heterskedasticity in that tw different variance estimates are used t characterize the aggregate distributin, thugh each distributin is permitted t have a unique mean. Analgusly, ur mixture mdels represent the nnnrmal distributin that results when distributins with different variances are cmbined (i.e., mixed). The btstrapped testing apprach described abve was used t calculate thep-values assciated with standard likelihd-rati test statistics fr the number f cmpnents (ne versus tw) t include in the mixture mdels. These are equivalent t testing H: X = O.13 The evaluatin f a mixture f nrmals versus a single nrmal results in a strng rejectin, implying that a standard nrmal distributin is nt suitable fr either crn r wheat. In the tests f a mixture f lgnrmals versus a single lgnrmal, the test statistic has a value f 9.17 fr crn (table 3) and 1.96 fr wheat (table 4). The btstrapped prbability values indicate that these test statistics d nt allw fr rejecting H: X =. Specifically, the crn and wheat test statistics havep-values f.29 and.25, respectively. This suggests that a single lgnrmal distributin is sufficient t mdel the price differentials when cmpared t a mixture f tw lgnrmals. The final mdel includes a mixture f a nrmal and a lgnrmal distributin. Estimatin f this mixture 12 A reviewer has crrectly nted that cnfidence intervals fr 2 cntain a, in several cases. Thugh this is nt a valid test f the significance f the differences in the tw alternative variance parameter estimates, it des suggest that ne shuld be cautius in cncluding that the variance terms are different. 13 In cases where the mixture cnsists f tw identical distributins (e.g., tw nrmals), this is als analgus t a test f X = 1, since the estimates are equivalent under either null. The simulatins used 3 replicatins.

15 Gdwin, Rberts, and Cble Measurement f Price Risk in Revenue Insurance 29 Table 3. Maximum-Likelihd Parameter Estimates and Summary Statistics: Crn Mixture Mixture Mixture f Nrmal Lgnrmal f Tw f Tw Lgnrmal/ Parameter OLS OLS Nrmals Lgnrmals Nrmal a (7.1627)* (.1693)* fi~~p ~ (.438)* (.353)* (.293)* (.334)* (.349)* a.263 a (2.5943)* (.156)* A b (.522)* (.282)* - pli (4.2833)* (.1616)* (1.991)* Oi (1.838)* (.164)* (4.6821)* A ( )* (.2197) (.1693)* ( )* (.162) (.156)* Lg-Likelihd Func Test f Mixing p-value P Pr {P<P} Rate Bera-Jarque Test 13, , Ntes: Numbers in parentheses are standard errrs. An asterisk (*) dentes statistical significance at the a =.5 r smaller level. a Maximum-likelihd estimate f standard deviatin. bvalue fixed by estimate n bundary f parameter space. mdel prduced estimates under which the density cllapsed int a single lgnrmal distributin (i.e., A was at its bundary value f 1E-9). The maximized lg-likelihd functin was nearly identical t that f the single lgnrmal. This bviates frmal hypthesis testing, thugh the implicatin is clear-lgnrmality has strng supprt ver nrmality. It shuld be nted that estimates f cmpnents f the mixture distributins are btained even when the mixing parameter estimate lies n its bundary. Such estimates are difficult t interpret since they apply nly t a very small fractin f the sample, as is implied by the restricted value f the mixing parameter. It is als desirable t test the lgnrmal specificatin against the mixture f nrmals. The btstrapping methd was extended t cnsider a cmpsite distributin cmprised f a lgnrmal distributin and a bivariate mixture f nrmals. In each case, the

16 21 July 2 Jurnal f Agricultural and Resurce Ecnmics Table 4. Maximum-Likelihd Parameter Estimates and Summary Statistics: Wheat Mixture Mixture Mixture f Nrmal Lgnrmal f Tw f Tw Lgnrmal/ Parameter OLS OLS Nrmals Lgnrmals Nrmal a (6.5464) (.127)* P (.296)* (.235)* (.263)* (.234)* (.235)* a a.1376a (2.2665)* (.1)* A,~~~~X ~ (.86)* (.15)* - p.1i (4.1681)* (.1124)* (.1)* (1.8793)* (.375)* (.1)* U ( )* (.124) (.127)* (1.2585)* (.91) (.34)* Lg-Likelihd Func Test f Mixing p-value..25 P Pr {P<P} Rate Bera-Jarque Test 7, , Ntes: Numbers in parentheses are standard errrs. An asterisk (*) dentes statistical significance at the a =.5 r smaller level. a Maximum-likelihd estimate f standard deviatin. b Value fixed by estimate n bundary f parameter space. parameter defining the cmpsite mixture between the lgnrmal and the discrete mixture f nrmals was n the bundary crrespnding t an estimate f ne. This indicates that the distributin again cllapsed t a lgnrmal fr bth crn and wheat. Because the maximized lg-likelihd functin was again nearly identical t that f the single lgnrmal, frmal hypthesis testing is again precluded. This des, hwever, indicate strng supprt fr a single lgnrmal distributin when cmpared t a mixture f nrmals. Prices were frecast fr the last bservatin (1997) and insurance rates were based n a guarantee f 1% f this frecasted level. An insurance premium rate is given by expected lss ver ttal liability. Expected lss is given by the prduct f the prbability f a lss and the expected price given that a lss ccurs. Numerical integratin

17 Gdwin, Rberts, and Cble Measurement f Price Risk in Revenue Insurance 211 was used t estimate these prbabilities and expected lss levels. With the exceptin f the mixture f nrmals case fr crn, the predicted prices (given by P in tables 3 and 4) are very similar. As wuld be expected, rates based n lgnrmality are cnsiderably higher than thse based n nrmality. This reflects the psitive skewness inherent in the lgnrmal distributin and larger variance estimates. In cntrast, rates fr the mixture f nrmals are smewhat lwer than thse under nrmality r lgnrmality, particularly in the case f crn. This lwer rate in part reflects the lwer frecasted price, which implies a lwer price guarantee. The mixture f nrmals generates wheat premium rates that are smewhat smaller than thse under nrmality. The premium rate estimates using the lgnrmal mdels and the mdels invlving mixtures with lgnrmals are similar due t the similarity f the price distributins implied by the estimated mdels. The results suggest that rating prcedures assuming nrmality may underestimate the price cmpnent f risk. Lgnrmality appears t prvide mre accurate estimates. Differences in the premium rates and underlying distributins are revealed in plts f the densities implied by OLS and the mixturef nrmals cases. Figures 2(A) and 2(B) illustrate nnparametric kernel estimates f the densities assciated with the OLS residuals. 14 Strng psitive skewness is revealed in the estimates. In several cases, slight bimdality is revealed, suggesting that large, psitive errrs are smetimes bserved-i.e., that the distributins may be a mixture f an infrequent high-variance regime and a frequent lwer-variance regime. The distributins d nt resemble nrmal densities, and thus the assumptin f nrmality wuld again seem questinable. Figures 2(C) and 2(D) shw the nrmal distributins implied by the maximum-likelihd estimates btained under nrmality. Figures 2(E) and 2(F) illustrate the distributins under the mixture f nrmals case, which have a nticeably lwer variance. This lwer variance underlies the lw premium rates suggested by the mixture f nrmals mdels. Figures 2(G)-2(J) graphically prtray the distributins f the mixtures invlving lgnrmal densities. In all cases, the mdels were dminated by a single lgnrmal distributin, suggesting that the distributins are very similar t thse btained under a single lgnrmal distributin. The distributins are nearly identical in the case f the nrmal/lgnrmal mixture. In summary, this part f the analysis suggests that current premium rates which are based n nrmality are likely t be lwer than the underlying price risk estimate implied by lgnrmality. Rates calculated in this manner, hwever, are based slely n histrical infrmatin, and cnsequently may nt fully reflect the uncertainty underlying market participants' actins at the time cntracts are ffered. Cncluding Remarks This analysis evaluates distributinal implicatins f mdeling price uncertainty. The issue f price uncertainty has taken n increased imprtance with the intrductin f three revenue insurance prgrams. In additin, changes in the farm plicy envirnment that ccurred with the 1996 Farm Bill have led t increased cncerns regarding the stability f farm prices. 14 Nte that the nnparametric densities d nt assume nrmality. OLS is a nnparametric estimatin technique prviding unbiased parameter estimates regardless f the underlying distributin. It has been nted, hwever, that least-squares estimatin may make sample residuals mre symmetric than the actual errrs (see Huang and Blch).

18 I "I &v sv,,.w V _1 I I &_,, I-.,, -. _ -,._ July 2 Jurnal fagricultural and Resurce Ecnmics e A. CORN: OLS Residuals B. WHEAT: OLS Residuals CN f- I - I _ , I... I....,....,,,.,,,, rrr,,,..,,.,,,, : d - q.r.... _ xi b u. I Price A 4. _ In 21n ' W- Price II I C. CORN: Nrmal d D. WHEAT: Nrmal - I IX]I l- ---, I,ZA 6I I (: R ~ Price C ren I7^ 27 I' I U 7^ 533U U 3/U JYU 41U I rnf% 434DU3 IA A- _I 4 7n u O--- Price n d E. CORN: Mixture f Nrmals r= t-... F. WHEAT: Mixture f Nrmals I-I rr.i I- d d w I,- U,. Price U U-U---U U -- V 3. U JI J~ n V 't 't n U 27/U ZY 31U 33U 3U A /U JYU 4lV 4JU 43u 4 Price G. CORN: Mixture f Lgnrmals Ml I H. WHEAT: Mixture f Lgnrmals l d cz I......; F; q, Price d ci n IV , Price a- a: d (N 6E I. CORN: Mixture f Nrmal and Lgnrmal... I I- d O J. WHEAT: Mixture f Nrmal and Lgnrmal In ;:! II I Q... n -.. Al,. -,.-2.1 ICA n7n n.nn in T,'n MAn- Ain q./n Ynn d'i. ri"n i n nn n l' dn ltn 'in 'ii n ''./" 7n n is n Ain AIrn A7n 17 uu5 19U 21 23U 25 2/ 29U u 41 du U 2 Li2 2 u 31 J 3J3 3/U JUU 41i 4JU 4U 4iv Price Price Figure 2. Estimatins f price densities btained under five alternative mdels fr crn and wheat

19 Gdwin, Rberts, and Cble Measurement f Price Risk in Revenue Insurance 213 An analysis f the cnditinal variance f crn and wheat prices revealed that variance decreases as time t maturity rises, and is highest during imprtant grwing perids. The findings f this analysis als imply that a nncnstant variance may cntribute t significant departures frm nrmality when data are aggregated ver time. The results als indicate that cnventinal appraches t measuring price variability and rating revenue insurance may be misspecified. Our empirical results strngly reject nrmality. Althugh cnditinal mment tests als reject lgnrmality, testing results btained frm flexible mixtures f nrmals and lgnrmals prvide reasnably strng supprt fr a lgnrmal distributin. Insurance rates based n lgnrmality are cnsiderably higher than thse implied by nrmality. Althugh ur research fidings have imprtant implicatins fr rating revenue insurance cntracts, many imprtant research issues remain. Mst fundamentally, we have fllwed current CRC revenue insurance ncerating prcedures and ignred yield-price crrelatin. Our methds culd be extended t cnsider bivariate density estimatin using mixture distributins that explicitly mdel such crrelatin. Hwever, such an extensin f ur methds faces the same hurdle as nearly all insurance prgrams-a general lack f available yield data. In particular, nearly all crp insurance prgrams have been hampered by the fact that individual prducer yield data are almst always scarce. Extensin f the methds described here t yield mdels wuld als raise a number f ther issues, including representatin f reginal differences in yield patterns and the apprpriate gegraphic area fr which t cnsider cmmn yield distributin mdels. Avenues fr making use f the limited data that are available within the cntext f mixture distributin estimatin methds remain an imprtant tpic fr further research. Future research will cnsider additinal explanatry factrs (such as ptins premia, stcks, demand shcks, and grwing cnditins) which may be used t cnditin variance frecasts. Additinal attentin will als be given t mdeling the cmplex crrelatin structure underlying ur analysis f verlapping cntracts. [Received January 1999; final revisin received September 1999.] References Andrews, D. W. K. "Tests fr Parameter Instability and Structural Change with Unknwn Change Pint." Ecnmetrica 61(1993): Bera, A., and C. Jarque. "Efficient Tests fr Nrmality, Heterscedasticity, and Serial Independence f Regressin Residuals: Mnte Carl Evidence." Ecn. Letters 6(198): Black, F., and M. Schles. "The Pricing f Optins fr Alternative Stchastic Prcesses." J. Plit. Ecn. 81(1973): Bridge Infrmatin Systems, Inc. "Histrical Financial and Cmmdity Market Database." Chicag IL, January Chicag Bard f Trade. Annual Reprt f the Trade and Cmmerce f Chicag. Selected issues, Crnew, R. W., D. E. Twn, and L. D. Crwsn. "Stable Distributins, Futures Prices, and the Measurement f Trading Perfrmance." J. Futures Mkts. 4(Fall 1984): Feng, Z. D., and C. E. McCullch. "On the Likelihd Rati Test Statistic fr the Number f Cmpnents in a Nrmal Mixture with Unequal Variances." Bimetrics 5(1994):

20 214 July 2 Jurnal fagricultural and Resurce Ecnmics. "Using Btstrap Likelihd Ratis in Finite Mixture Mdels." J. Ryal Statis. Sc. B,58(1996): Hall, J. A., B. W. Brrsen, and S. H. Irwin. "The Distributin f Futures Prices: A Test f the Stable Paretian and Mixture f Nrmals Hyptheses." J. Finan. and Quant. Anal. 24(1989): Hansen, B. E. "Inference in TAR Mdels." Studies in Nnlinear Dynamics and Ecnmetrics 2(1997). Online. Available at [Nte: Access t subscribers nly.] Harvey, A. "Estimating Regressin Mdels with Multiplicative Heterscedasticity." Ecnmetrica 44(1976): Hennessy, D. A., and T. I. Wahl. "The Effects f Decisin Making n Futures Price Vlatility." Amer. J. Agr. Ecn. 78(August 1996): Hsieh, D. A. "Mdeling Heterscedasticity in Daily Freign-Exchange Rates." J. Bus. and Ecn. Statis. 7(July 1989): Huang, C. J., and B. W. Blch. "On the Testing f Regressin Disturbances fr Nrmality." J. Amer. Statis. Assc. 69(June 1974): Hudsn, M. A., R. M. Leuthld, and G. F. Sarassr. "Cmmdity Futures Price Changes: Recent Evidence fr Wheat, Sybeans, and Live Cattle." J. Futures Mkts. 7(June 1987): McLachlan, G. J. "On Btstrapping the Likelihd Rati Test Statistic fr the Number f Cmpnents in a Nrmal Mixture." Appl. Statis. 36(1987): Ramirez,. A. "Estimatin and Use f a Multivariate Parametric Mdel fr Simulating Heterskedastic, Crrelated, Nnnrmal Randm Variables: The Case f Crn Belt Crn, Sybean, and Wheat Yields." Amer. J. Agr. Ecn. 79(February 1997): Risk Management Agency (RMA). Unpublished crp revenue insurance prduct participatin statistics. RMA, Washingtn DC, Samuelsn, P. A. "Is Real-Wrld Price a Tale Tld by the Idit f Chance?" Rev. Ecn. and Statis. 58(February 1976): Stkes, J. R., W. I. Nayda, and B. C. English. "The Pricing f Revenue Insurance." Amer. J. Agr. Ecn. 79(May 1997): U.S. General Accunting ing"crp Office. Revenue Insurance: Prblems with New Plans Need t Be Addressed." Pub. N. GAO/RCED , Washingtn DC, April 1998.