Aggregation of price risk over commodities: an economic index number approach.

--.1978. "Superlative Index Numbers and Consistency in Aggregation." Econometrica 46:883-900.

--. 1980a. "Capital and the Theory of Productivity Measurement." American Economic Review 70:260-67.

--. 1980b. "Aggregation Problems in the Measurement of Capital." In D. Usher, ed. The Measurement of Capital, Chicago: University of Chicago Press, pp. 433-528.

--. 1981. "The Economic Theory of Index Numbers: A Survey." In A. Deaton, ed. Essays in the Theory and Measurement of Consumer Behaviour. Cambridge: Cambridge University Press, pp. 163-208.

--. 1992. "Fisher Ideal Output, Input, and Productivity Indexes Revisited." Journal of Productivity Analysis 3:211-48.

--. 2002. "The Quadratic Approximation Lemma and Decompositions of Superlative Indexes." Journal of Economic and Social Measurement 28:63-88.

--. 2004a. "Index Number Theory and Measurement Economics." mimeo (Economics 580), Department of Economics, University of British Columbia.

--. 2004b. "Index Number Theory: Past Progress and Future Challenges." SSHRC Conference on Price Index Concepts and Measurement." Vancouver, Canada, June 30-July 2004.

Domberger, S. 1987. "Relative Price Variability and Inflation: A Disaggregated Analysis." Journal of Political Economy 95:547-67.

Eisgruber, L.M., and L.S. Schuman. 1963. "The Usefulness of Aggregate Data in the Analysis of Farm Income Variability and Resource Allocation." Journal of Farm Economics 45:587-92.

Engle, R.F. 2002. "Dynamic Conditional Correlation--A Simple Class of Multivariate GARCH Models." Journal of Business and Economic Statistics 20:339-50.

--. 2004. "Risk and Volatility: Econometric Models and Financial Practice." American Economic Review 94:405-20.

Engle, R.F., and K. Sheppard. 2001. "Theoretical and Empirical Properties of Dynamic Conditional Correlation Multivariate GARCH." Department of Economics, University of San Diego.

Feller, W. 1966. An Introduction to Probability Theory and Its Applications, Vol. II. New York: John Wiley.

Fischer, S. 1981. "Relative Shocks, Relative Price Variability, and Inflation." Brookings Papers on Economic Activity 2:381-431.

Fisher, I. 1911. The Purchasing Power of Money. London: Macmillan.

Frisch, R. 1930. "Necessary and Sufficient Conditions Regarding the Form of an Index Number Which Shall Meet Certain of Fisher's Tests." American Statistical Association Journal 25:397-406.

--. 1936. "Annual Survey of General Economic Theory: The Problem of Index Numbers." Econometrica 4:1-38.

Gorman, W.M. 1953. "Community Preference Fields." Econometrica 21:63-80.

Haas, M., S. Mittnik, M.S. Paolella. 2004. "Mixed Normal Conditional Heteroskedasticity." Journal of Financial Econometrics 2:211-50.

Hansen, B.E. 1994. "Autoregressive Conditional Density Estimation." International Economic Review 35:705-30.

Harvey, A.C. 1976. "Estimating Regression Models with Multiplicative Heteroscedasticity." Econometrica 44:461-65.

--. 1990. The Econometric Analysis of Time Series. Cambridge, MA: MIT Press.

Harvey, C.R., and A. Siddique. 1999. "Autoregressive Conditional Skewness." Journal of Financial and Quantitive Analysis 34:465-87.

Hercowicz, Z. 1982. "Money and the Dispersion of Relative Prices." Journal of Political Economy 89:328-56.

Hicks, J.R. 1936. Value and Capital. Oxford: Oxford University Press.

Kendall, M., and A. Stuart. 1977. The Advanced Theory of Statistics, Vol. I. New York: Macmillan.

Knight, F.H. 1921. Risk, Uncertainty and Profit. New York: Houghton Mifflin.

Lewbel, A. 1996. "Aggregation without Separability: A Generalized Composite Commodity Theorem." American Economic Review 86:524-43.

Moschini, G., and D.A. Hennessy. 2001. "Uncertainty, Risk Aversion and Risk Management for Agricultural Producers." In B. Gardner and G. Rausser, eds. Handbook of Agricultural Economics, Vol. I. Amsterdam: Elsevier Science, pp. 88-153.

Manitoba Agriculture Yearbook 2002. 2002. Manitoba Agriculture and Food.

Parks, R.W. 1978. "Inflation and Relative Price Variability." Journal of Political Economy 86:79-95.

Pope, R.D. 1988. "A New Parametric Test for the Structure of Risk Preferences." Economic Letters 27:117-21.

Pope, R.D., and R.G. Chambers. 1989. "Price Aggregation When Price-Taking Firms' Prices Vary." Review of Economic Studies 56:297-309.

Roberts, M.C. 2001. "Commodities, Options, and Volatility: Modelling Agricultural Futures Prices." Working Paper, The Ohio State University.

Selvanathan, E.A., and D.S. Prasada Rao. 1994. Index Numbers: A Stochastic Approach. Ann Arbor: University of Michigan Press.

Shumway, C.R., and G.C. Davis. 2001. "Does Consistent Aggregation Really Matter?" Australian Journal of Agricultural and Resource Economics 45:161-94.

Theil, H. 1967. Economics and Information Theory. Chicago: Rand McNally.

Wang, K.L., C. Fawson, C.B. Barrett, and J.B. McDonald. 2001. "A Flexible Parametric GARCH Model with an Application to Exchange Rates." Journal of Applied Econometrics 16:521-36.

White, K.J. 1997. Shazam User's Reference Manual Version 8.0. New York: McGraw-Hill.

(1) Following Knight (1921), price risk is characterized by randomness that can be measured precisely, i.e., as probabilistic.

(2) Theil (1967, p. 155) referred to a variance of log-changes in individual prices as a measure of dispersion of price changes about the mean (change in the "price structure"), and others have used this or similar measures in studies of relations between inflation and relative price variability (Parks 1978; Fischer 1981; Hercowitz 1982; Domberger 1987). However, this "stochastic index number" approach (Frisch 1936; Selvanathan and Rao 1994) has not been intended as an aggregate measure of price risk: there has been no attempt to separate anticipated and unanticipated price changes in this measure (although some studies do attempt by other means to separate anticipated and unanticipated inflation). For example, standard errors of inflation estimates permit confidence intervals for the true rate of inflation rather than measuring unanticipated inflation or price risk (Clements and Izan 1987). Moreover, this approach would obviously provide a poor aggregate measure of price risk.

(3) Many indexes are discrete time approximations to a continuous time Divisia index, including Laspeyres and Paasche indexes (Diewert 1980b, p. 444; 2004a ch. 0.5).

(4) In general (under appropriate regularity conditions), moments of a probability distribution for p exist and uniquely determine the distribution (the problem of moments) (Feller 1966; Kendall and Stuart 1977). Given a k parameter distribution h(p,[[theta].sub.1],.,[[theta].sub.k]), assume existence of the first k moment functions [m.sub.p] = [m.sub.p]([theta]) and of [[nabla]m.sub.-1.sub.p]. Then by the implicit function theorem, d[theta] = [[nabla]m.sub.-1.sub.p] [dm.sub.p] and [theta] = [theta]([m.sub.p]). This implies that any change in the vector of first k moments, [dm.sub.p], can be supported by a particular change in the vector of parameters, d[theta]. Parameters [[theta].sub.1],.,[[theta].sub.k] can evolve differently over time. So in general the first k moments mp can evolve somewhat differently over time, i.e., evolution of (e.g.) the [k.sup.th] moment is not determined by evolution of the first k - 1 moments.

(5) It is well known that Laspeyres and Paasche consumer price indexes place lower and upper bounds on a true cost of living index assuming homotheticity (e.g., Diewert 2004a, chapter 4), but it is apparent that a similar result does not hold in general for aggregate indexes of output price risk under risk aversion.

(6) In the case of nonlinear mean-variance risk preferences, non-joint CRTS and a Translog cost function, the appropriate aggregate output index [Y.sub.1]/[Y.sub.0] is more complex: (14) with [[gamma].sub.it] [equivalent to] ([p.sub.it][y.sub.it] - [[alpha].sub.t] [VR.sub.t] = [partial derivative][[alpha].sub.t]/ [partial derivative][y.sub.i][y.sub.it]/2)[C.sub.t].

(7) In the Tornqvist case, this was accomplished by applying Diewert's quadratic lemma to a Translog cost function. A generalized quadratic lemma can be used to relate Fisher input price and quantity indexes to a unit cost function c(w) = [[summation].sub.i=1,.,m][[summation].sub.j=1,.,m] [b.sub.ij][w.sub.i][b.sub.j]].sup.1/2] (Diewert 2002), but this lemma does not apply to a multi-output cost function such as c(w, y) = [[summation].sub.i=1,.,m][[summation].sub.j=1,.,m] [b.sub.ij][w.sub.i][b.sub.j]].sup.1/2] [[summation].sub.i=1,.,m][[summation].sub.j=1,.,m] [a.sub.ij][y.sub.i][y.sub.j]].sup.1/2].

(8) Consistency of aggregation of price risk subindexes can be summarized as follows. It is well known that standard Laspeyres and Paasche subindexes for output quantity are consistent in aggregation to an index for all outputs, but Tornqvist and Fisher subindexes for output quantity are only approximately consistent in aggregation (Diewert 1978). This implies that price risk subindexes of the simple Laspeyres or Paasche form (4), but not the Fisher form (5), are consistent in aggregation to a price risk index over all commodities. Alternatively, we construct price risk subindexes similarly to (7), which depends on output quantity subindexes ([Y.sub.A],[Y.sub.B]). Consistent aggregation of these price risk subindexes requires Laspeyres or Paasche, rather than Tornqvist or Fisher, output quantity subindexes.

(9) Diebold and Nerlove (1989) present a tractable alternative to multivariate GARCH.