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

Annual data on market prices ($/tonne) for the six major crops in Manitoba (wheat, barley, canola, oats, rye, and flax) over 1950-2002 were obtained from the Manitoba Agriculture Yearbook (2002). Correlations between market prices and coefficients of variation for each crop are presented in table 1. The high correlations between prices indicate the importance of including price covariances in aggregation of price risk over crops. For simple autoregressive models of price, homoscedasticity is generally rejected at the 0.05 level for multiplicative heteroscedasticity (Harvey 1976, 1990), but homoscedasticity is not rejected for ARCH (by Engle Lagrange multiplier test). Nevertheless, we consider GARCH models of price risk as well as naive and multiplicative heteroscedastic models.

First, consider the common naive model of price risk (21)-(22) with annual data. In principle, this naive model is a poor forecast for the distribution of prices. Nevertheless, estimates of (e.g.) the univariate GARCH models for prices (discussed below) offer some support for this model for all crops excluding oats: coefficients for a one-year lag in price are very close to +1.0 and coefficients for other lags are smaller and approximately cancel out; and correlations between predicted prices from GARCH and a one-period lag in prices are high (ranging from 0.9729 for wheat to 0.9179 for oats).

The resulting estimates of time-varying conditional variances and covariances for all six crops are aggregated using the methodology developed above. In constructing indexes of price risk from (21)-(22), estimates of the covariance matrix Vp from (22) are used to calculate revenue uncertainty VR = [y.sup.T]Vpy. A chained Laspeyres index (4) [VP.sub.t]/[VP.sub.t-1] is calculated. This does not require the calculation of an aggregate output quantity index, in contrast to our other approaches based on (7). Since we do not have reliable time series data on average costs by crop, our Tornqvist and Fisher-type output quantity indexes (8) and (12) cannot be calculated. So instead we calculate standard chained Tornqvist, Fisher, and Laspeyres output quantity indexes [Y.sub.t]/[Y.sub.t-1] (in effect substituting [p.sub.it] for [AC.sub.it] in (8) and (12)). (10) Table 2 indicates that the Tornqvist and Fisher output quantity indexes are highly correlated with the Laspeyres index (r = 0.9912, 0.9992, respectively). Given these output quantity indexes, aggregate indexes for price risk are calculated from

(23) [VP.sub.t]/[VP.sub.t-1] = ([y.sup.T.sub.t][Vp.sub.t][y.sub.t]/ [y.sup.T.sub.t-l][Vp.sub.t-1][y.sub.t-1]) /[([Y.sub.t]/[Y.sub.t-1]).sup.2]

which is analogous to (7).

For comparison, we also construct a Tornqvist output price index [q.sub.t] [equivalent to] [([P.sub.t]/[P.sub.t-1]).sup.Torn] and then calculate its variance as [var.sub.t-1]([q.sub.t]) = 0.50 [([q.sub.t-1]- [q.sub.t-2]).sup.2] + 0.33 [([q.sub.t-2] - [q.sub.t-3]).sup.2] + 0.17 ([q.sub.t-3] - [q.sub.t-4]) similarly to (22). This provides an aggregate measure of price risk [VP.sup.B], which is in theory highly inferior to the other indexes of aggregate price risk.

Table 3 presents correlations between these price risk indexes based on naive expectations and annual data. [VP.sup.L] denotes the Laspeyres index (4). [VP.sup.Torn], [VP.sup.Fish,] [VP.sup.Lasp] denote the aggregate indexes of price risk constructed from (23) using Tornqvist, Fisher, and Laspeyres output quantity indexes, respectively. [VP.sup.B] denotes the price variance for the aggregate Tornqvist output price index. The first four indexes can be viewed as alternative index number approaches for aggregating price risk over commodities, but the last index VPB does not have a valid interpretation as an index number that preserves the contribution of commodity price variances and covariances to revenue risk.

The three indexes constructed from (23) are very highly correlated, with correlations r ranging between 0.9997 and 0.9846. This reflects very high correlations between the corresponding output quantity indexes (table 2). These price risk indexes are also highly correlated with the Laspeyres index (4), with r ranging between 0.9897 and 0.9678. However, all four of these indexes show much smaller correlations with the last index [VP.sup.B], with r ranging between 0.5129 and 0.4656. This last result illustrates that the distinctions in theory between the first four indexes and [VP.sup.B] are also important in practice. (11)

Although GARCH models are inappropriate for annual data, for comparison the two-step multivariate GARCH model under constant conditional correlations was also estimated with this data set. Current price Pi is specified as a function of a four-period lag in Pi (longer lags are insignificant) and a time trend. Univariate GARCH(1,1) models are estimated for each crop by maximum likelihood using Broyden--Fletcher--Goldfarb--Shanno (BFGS) algorithms as encoded in Shazam (White 1997). Second, assuming constant correlations for u's, pairwise regressions of standardized residuals are estimated and conditional covariances are calculated, as discussed above. Estimates of the covariance matrix are used to calculate measures of revenue risk [VR.sub.t] = [y.sub.t][Vp.sub.t][y.sub.t]. The correlation between this measure and the measure using Vp from naive models is 0.860. Then index numbers for price risk are calculated similarly to the naive case.

The three indexes constructed from (23) are very highly correlated, with correlations r ranging between 0.991 and 0.971. These indexes are also highly correlated with the Laspeyres index (4), with r ranging between 0.974 and 0.925. However, all four of these indexes are negatively correlated with the index [VP.sup.B] (based on a univariate GARCH model of a Divisia price index), with r ranging between -0.274 and -0.420.

A model of multiplicative heteroskedasticity (Harvey 1976, 1990) is more appropriate than GARCH with annual data. The variance of the disturbance for the price equation is specified as multiplicative in explanatory variables [z.sub.t] = (1, [p.sub.t-1], x , [p.sub.t-4], t), and price equations are estimated as in Harvey (1976) using a BFGS maximum likelihood algorithm in Shazam. As before, the three indexes constructed from (23) are very highly correlated, with correlations r ranging between 0.9998 and 0.9873, and are highly correlated with the Laspeyres index (4), with r between 0.9628 and 0.9147. These indexes are somewhat less highly correlated with the index [VP.sup.B] (based on a multiplicative heteroscedastic model of a Divisia price index), with r ranging between 0.8742 and 0.8191.

Since GARCH models are inappropriate with annual data, monthly data were also collected for Manitoba crop prices. Monthly price data for wheat, barley, canola, oats, and flax were obtained from January 1990 to July 2005 (Agriculture and Agri-Food Canada, Winnipeg). Correlations between market prices and coefficients of variation are presented in table 4. The high correlations between prices again indicate the importance of including price covariances in aggregation of price risk over crops.

For simple autoregressive models of price, homoscedasticity is rejected at the 0.01 level for ARCH (Engle Lagrange multiplier test) for all crops except canola. However, homoscedasticity is seldom rejected for multiplicative heteroscedasticity (Harvey 1976, 1990). So we focus on GARCH models with monthly data.

A multivariate GARCH model is estimated by the two-step process assuming constant conditional correlations as presented above. First, univariate GARCH models are specified for each of the five prices. Current price [p.sub.i] is initially specified as a function of a 24-month lag on [p.sub.i], monthly dummies and a time trend, and a GARCH(1,1) model is estimated by BFGS maximum likelihood in Shazam. Price lags beyond 13 months are statistically insignificant, so longer lags are omitted. Various monthly dummies are statistically significant, so first-step univariate GARCH models are estimated without and with dummies. Then conditional covariances are estimated as before, and aggregate indexes of price risk are constructed as before.

Table 5 presents correlations between these price risk indexes based on multivariate GARCH and monthly data, and omitting dummies. Notation is similar to table 2. [VP.sup.L] denotes the Laspeyres index (4). [VP.sup.Torn], [VP.sup.Fish], [VP.sup.Lasp] denote the aggregate indexes of price risk constructed from (23) using Tornqvist, Fisher, and Laspeyres output quantity indexes [Y.sub.t]/[Y.sub.0], respectively. [VP.sup.B] denotes the price variance for the aggregate Tornqvist output price index. The first four indexes can be viewed as alternative valid index number approaches for aggregating price risk over commodities, in contrast to the last index [VP.sup.B].

The three indexes constructed from (23) are very highly correlated, with correlations r ranging between 0.9994 and 0.9978. These indexes are also highly correlated with the Laspeyres index (4), with r ranging between 0.9568 and 0.9536. However, all four of these indexes show smaller correlations with the last index [VP.sup.B], with r ranging between 0.8043 and 0.7810. Results in this table illustrate again that the theoretical contributions of this article are important in practice.

Table 6 presents correlations between these price risk indexes based on multivariate GARCH including monthly dummies in the first-step univariate GARCH models. Results are similar to the previous table. Here, correlations of [VP.sup.B] with the first four indexes range from only 0.6819 to 0.6663.

Conclusion