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

Index number theory is commonly used to motivate aggregation of prices over commodities. In particular, three price indexes (Fisher, Tornqvist, Walsh) are favored by the most popular criteria (economic test and weighted stochastic approaches), and these three indexes closely approximate each other in practice (Diewert 2004a,b). Index number methods for aggregating prices and quantities over commodities are essential to empirical economics.

Other conceptual approaches are also used to aggregate prices over commodities or agents. Prices can be aggregated over commodities under distributional assumptions such as perfect contemporaneous covariance (Hicks' [1936] aggregation) or more generally mean scaling (Lewbel 1996; Davis, Lin, and Shumway 2000; Shumway and Davis 2001; Davis 2003). Different prices for the same commodity can be aggregated over firms assuming Gorman (Gorman 1953) polar form type relations in prices or more generally knowledge of second moments of cross-section distributions of price (Pope and Chambers 1989; Chambers and Pope 1991, 1994).

However, apparently an index number or related approach has not been presented for aggregation of price risk over commodities. (1,2) Such aggregation involves price covariances as well as variances, so this is a higher dimensional problem than aggregation over prices.

It is a stylized fact that agricultural producers are risk averse and decisions are influenced by risk and uncertainty (e.g., see references in Moschini and Hennessy 2001), and of course there are multiple agricultural commodities. So, it is important to develop summary measures of price risk in a multicommodity setting. The absence of an index number approach to this aggregation is a serious problem for policy and empirical research.

Common approaches to aggregating price risk are (a) to reduce the dimension of the problem by omitting many covariances and variances or (b) to calculate a variance of an aggregate price index. However, the first approach obviously ignores information about risk and the second approach has no basis in index number theory.

This article extends index number theory to aggregation of price risk over commodities in production, and includes an application to major Manitoba crops. The analysis is an extension of the economic approach to aggregation of prices under certainty due largely to Diewert.

We focus on price risk rather than quantity risk primarily because it is well known that output risk is underestimated by regional data on output (Eisgruber and Schuman 1963), and farm-level output data are less common than regional data. Since there is higher contemporaneous covariance of prices than output levels over farms, this problem is less serious for prices.

The study is organized as follows. The first section summarizes relevant standard index number theory. The next section presents preliminaries and initial results. The subsequent section presents the general economic theory approach to aggregation of price risk over commodities, and this is related to Tornqvist-type output quantity indexes. Then the analysis is extended to Fisher-type output quantity indexes, and the following section considers higher moments, subindexes, and output quantity risk. Finally the methodology is applied to price data for major Manitoba crops. Results illustrate the empirical importance of the index number approach to aggregation of price risk developed here.

Standard Index Number Theory

We first provide a brief summary of standard index number theory before extensions to price risk. Readers unfamiliar with standard index number theory should read Diewert (1976 or 2004a). Index number theory has focused on aggregation under certainty or risk neutrality. Consider aggregation of output prices and quantities, given output prices p = ([p.sub.1],., [p.sub.m]), output levels y = ([y.sub.1],., [y.sub.m] for m outputs and input prices w = ([p.sub.1],., [p.sub.n]), input levels x = ([x.sub.1],., [x.sub.n]) for n inputs, and two time periods 0, 1. A fundamental goal of index number theory (in both economic and test approaches) is to decompose the value (revenue) change between the two periods, [p.sub.1][y.sub.1]/[p.sub.0][y.sub.0] [equivalent to] [[summation].sub.i=1,.,m][p.sub.i1][y.sub.i1]/[[summation].sub.i=1,.,m] [p.sub.i0][y.sub.i0] into a price change part [P.sub.1][P.sub.0] and a quantity change part YflY0, i.e., [P.sub.1][P.sub.0] and [Y.sub.1]/[Y.sub.0] should satisfy

(1) ([P.sub.1][P.sub.0])([Y.sub.1]/[Y.sub.0]) = [p.sub.1][y.sub.1]/ [p.sub.0]/[y.sub.0].

[P.sub.1]/[P.sub.0] is an aggregate price index over commodities for period t = 1 relative to t = 0, and similarly [Y.sub.1]/[Y.sub.0] is an aggregate quantity index over commodities for t = 1 relative to t = 0. This criterion for index numbers was first proposed by Fisher (1911, p. 418) and was called the product test equation by Frisch (1930, p. 399).

Diewert (1976) notes that indexes aggregating input quantities or input prices should preserve the contributions to output quantity or marginal/average cost, respectively (the index should be exact), and he shows that Tornqvist indexes accomplish this under Translog flexible functional forms and common behavioral assumptions (the Tornqvist index is superlative). Analogous arguments apply to aggregation of output quantities and prices, as summarized next.

A common output quantity index is the following Tornqvist discrete time approximation [Y.sub.1]/[Y.sub.0] to a continuous Divisia aggregate index:

(2) log([Y.sub.1]/[Y.sub.0])

= [[summation].sub.i=1,.,m]{([S.sub.i1] + [S.sub.io]/2} log([y.sub.i1]/[y.sub.i0])

where [s.sub.it] [equivalent to] [p.sub.it][y.sub.it]/ [[summation].sub.J=1,.,n][W.sub.jt][X.sub.jt]. (3) [Y.sub.1]/[Y.sub.0] is a superlative output quantity index corresponding to a Translog joint cost function C(w, y) and static competitive profit maximization (Diewert 1980a, 1976). Given [Y.sub.1]/[Y.sub.0], a corresponding implicit index of output prices [P.sub.1]/[P.sub.0] is defined by the product test equation (1). This implicit index [P.sub.1]/[P.sub.0] can be interpreted as a superlative index for ratio of aggregate price (Diewert 1976).

A Fisher output quantity index is a prominent alternative:

(3) [Y.sub.1]/[Y.sub.0] = [[[p.sub.0][y.sub.1]/ [p.sub.0][y.sub.0]].sup.1/2] [[p.sub.1][y.sub.1]/[p.sub.1][y.sub.0]].sup.1/2].

This is also a superlative output quantity index, and it is superior to a Tornqvist index by the test approach (Diewert 1992).

Aggregation of Price Variances and Covariances: Preliminaries and Initial Laspeyres/Fisher Indexes

A common approach to approximating an index of price risk over commodities is simply to construct a variance from time series data for an aggregate price index such as the above Tornqvist (e.g., Coyle 1992, 1999). However, this ad hoc approach cannot be rationalized in terms of index number theory.

This weakness of the ad hoc approach can be explained as follows. A k parameter distribution for price implies that the first k moments generally characterize the distribution (higher moments are dependent on these moments); so these moments can evolve differently over time. (4) For example, assuming a normal distribution whose two parameters change over time, the mean and variance are independent constructs (determining higher moments) and hence can evolve separately over time. So, the relation between evolution of expected prices and of price variances/ covariances over time can be complex, i.e., price variances/covariances are unlikely to be determined by expected prices over time. In turn, the evolution of sample price, price mean, and price variance over time can be complex. Consequently, a simple transformation such as a variance of a correct price index is unlikely to provide a correct price risk index, which is an aggregation of price variances/covariances rather than of sample prices or expected prices. In sum, observed prices and risk can evolve differently over time (e.g., prices known with certainty can change over time, or consider a GARCH model of price). So, a variance of an index of observed prices does not provide a correct index number aggregate of risk.

A related approach is to specify a GARCH model for an aggregate price index and calculate the conditional variance of the disturbance. This approach is common in the literature on stock market price indexes (Bollerslev, Chou, and Kroner 1992). However, again the estimated aggregate of price risk is a (complex) transformation of data on aggregate prices rather than an aggregate of commodity level price variances and covariances, which evolve differently from observed prices or expected prices. A more appropriate procedure is to estimate a multivariate GARCH model over all prices and then aggregate the conditional price variances and covariances by a correct index number procedure.

In sum, common approaches for measuring aggregate price risk from a price index do not provide a correct index number approach to aggregating price risk over commodities. Moreover, as we shall demonstrate in the empirical section of this article, this distinction in theory between a variance of a price index and a correct price risk index is also important in practice.