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.
Our approach to constructing index numbers for aggregate price risk
is in the spirit of the economic analysis of index numbers under
certainty. In our case, price risk for individual outputs contributes to
risk regarding total revenues: variances and covariances for output
prices contribute to variance of total revenue. Ignoring output
(quantity) risk or uncertainty, price risk and output levels jointly
contribute to revenue risk as [VR.sub.t] =
[y.sup.T.sub.t][Vp.sub.t][y.sub.t], where Vp is the price covariance
matrix and y is a vector of output levels. Our index number theory is
invariant to the time dimension of price risk, i.e., the price
covariance matrix Vp can be viewed as reflecting either (e.g.) annual,
monthly, or daily price risk. Of course, the particular time dimension
is important in empirical applications of the theory.
An appropriate aggregation procedure for price risk Vp will
preserve the contribution of Vp to revenue risk while controlling for
effects of output levels y. This is the fundamental criterion in
designing index number approaches to aggregation of price risk over
commodities, and it is similar in spirit to standard index number
theory. Standard index number problems are best addressed in terms of
value ratios rather than levels (Diewert 2004a), and we proceed in a
similar manner.
The following "Laspeyres" index is the most obvious
approach to aggregating price risk over commodities:
(4) [(VP.sub.1]/[VP.sub.0])L = [y.sup.T.sub.0][Vp.sub.1][y.sub.0]/
[y.sup.T.sub.0][Vp.sub.0][y.sub.0]
using base period weightings [y.sub.0] throughout the index. An
analogous "Paasche" index [(VP.sub.1][VP.sub.0]).sup.P] can be
defined using base period weightings [y.sub.1]. Such weightings would
obviously be appropriate if outputs were in fixed proportions. However,
in principle this approach misrepresents the contribution of price risk
Vp to revenue risk VR under general changes in output levels y, somewhat
as aggregation with standard Laspeyres indexes generally loses the
economic meaning of the subaggregates (Diewert 1981). (5) An analogous
"Fisher" index is
(5) [([VP.sub.1]/[VP.sub.0]).sup.F]
= [{[([VP.sub.1]/[VP.sub.0]).sup.L]([VP.sub.1]/
[VP.sub.0]).sup.P]}.sup.1/2].
Fisher indexes typically have better properties than do Laspeyres
or Paasche indexes. Nevertheless, I am unaware of any applications or
references even to these Laspeyres, Paasche, or Fisher indexes for
aggregation of price risk.
Aggregation of Price Variances and Covariances: A Tornqvist Index
Approach
Similar to the product test equation (1), a fundamental goal of
index number theory for aggregating output price risk should be to
decompose the change in revenue risk between two periods,
[y.sup.T.sub.1][Vp.sub.1][y.sub.1]/[y.sup.T.sub.0][Vp.sub.0][y.sub.0],
into a price risk change part [VP.sub.1]/[VP.sub.0] and a quantity
change part [Y.sub.1]/[Y.sub.0]. So in the spirit of standard index
number theory and elementary statistics, [VP.sub.1]/[VP.sub.0] and
[Y.sub.1]/[Y.sub.0] should satisfy the following equation analogous to
(1):
(6) [([VP.sub.1]/[VP.sub.0])([Y.sub.1]/[Y.sub.0]).sup.2] =
[y.sup.T.sub.1][Vp.sub.1][y.sub.1]/ [y.sup.T.sub.0][Vp.sub.0][y.sub.0].
Then a price risk index can be correctly calculated from (6) given
an appropriate aggregate output index. This connects aggregation of
price risk to the formulation of aggregate index numbers for multiple
outputs.
Suppose an output quantity index [Y.sub.1]/[Y.sub.0] is superlative
under appropriate behavioral assumptions. Then (6) implies an aggregate
price risk index [VP.sub.1]/[VP.sub.0]
(7) [VP.sub.1]/[VP.sub.0] = ([y.sup.T.sub.1][Vp.sub.1][y.sub.1]/
[y.sup.T.sub.0][Vp.sub.0][y.sub.0]/ [([Y.sub.1]/[Y.sub.0]).sup.2]
that is exact and superlative in terms of preserving the
contribution of price risk for commodities to revenue risk.
Assuming risk aversion, index numbers for multiple outputs should
not be calculated from profit maximization, and in general index numbers
depend on knowledge of risk preferences or the corresponding dual
utility function (Chambers 1983). Nevertheless, we can show that a
Tornqvist-like aggregate output quantity index is appropriate assuming a
(static) Translog cost function and constant returns to scale (CRTS) in
nonjoint technologies, which are common assumptions in index number
theory. Later, we will relax these assumptions. This result is stated as
the following Proposition (see Appendix for proof).
PROPOSITION 1. Define the output quantity index
[Y.sub.1]/[Y.sub.0]:
(8) log([Y.sub.1]/[Y.sub.0])
= [[summation].sub.i=1,.,m]{([[theta].sub.i1] + [[theta].sub.i0]/2}
log([Y.sub.i1]/[Y.sub.i0])
where [[theta].sub.i0] [equivalent to] [C.sub.it]/[C.sub.t] is the
share of output i in total cost at time t. Assume CRTS non joint
technologies, (static) cost minimization, and a Translog cost function
C(w, y). Then
(9) [Y.sub.1]/[Y.sub.0] = ([C.sub.1]/[C.sub.0])/([AC.sub.1]/
[AC.sub.0])
where [C.sub.1]/[C.sub.0] is ratio of total cost and log
([AC.sub.1]/[AC.sub.0]) = [[summation].sub.i=1,.,n][S.sub.wj]
log([w.sub.j1]/[w.sub.j0]) [S.sub.wj] [equivalent to]
{[w.sub.j1][x.sub.j1]/[C.sub.1] + ([w.sub.j0][x.sub.j0]/[C.sub.0])}/2.
[AC.sub.1]/[AC.sub.0] is an index of aggregate average cost. Thus
[Y.sub.1]/[Y.sub.0] is an exact output quantity index, and it is
superlative.
In contrast, if technology is joint or not CRTS, then the above
result does not apply. Then an aggregate output index generally depends
on the properties of risk preferences as well as technology. For
example, assume constant absolute risk aversion (CARA) and the firm
solves the utility maximization problem
(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [alpha] is the constant coefficient of absolute risk
aversion. This leads to the following Proposition.
PROPOSITION 2. Define the output quantity index
[Y.sub.1]/[Y.sub.0]:
(11) log([Y.sub.1]/[Y.sub.0])
= [[summation].sub.i=1,.,m]{([[gamma].sub.i1] +
[[gamma].sub.i0]/2log([y.sub.i1]/[y.sub.i0])
where [[gamma].sub.it] [equivalent to] ([P.sub.it][y.sub.it] -
[alpha][VR.sub.t])/[C.sub.t] Assume CARA utility maximization (10) and a
Translog cost function C(w, y). Then [Y.sub.1]/[Y.sub.0] satisfies (9).
However, it is important to note that this index requires an
estimate of the coefficient of absolute risk aversion [alpha]. (6)
Extensions to a Fisher Index
A Fisher ideal index [([Y.sub.1]/[Y.sub.0]).sup.F] (3) is often
advocated for aggregation. The economic interpretation of this index is
typically based on revenue maximization: then an aggregator function of
the form f(y)= [[[summation].sub.i=1,.,m]
[[summation].sub.i=1,.,m][a.sub.ij][y.sub.i][y.sub.j]].sup.1/2] implies
that [([Y.sub.1]/[Y.sub.0]).sup.F] = f(y.sub.1])/f([y.sub.0]). Thus,
this index is superlative assuming revenue or profit maximization
(Diewert 1976, 2004a). However for our purposes, a Fisher output
quantity index should be defined from cost minimization and a cost
function, independently of profit maximization or risk preferences. (7)
A Fisher-like output quantity index can be developed from a cost
minimization model as follows. Assume nonjoint CRTS technologies, so
that average cost for output i is [AC.sub.i](w). Define a new
Fisher-like index by substituting [AC.sub.i](wt) for [p.sub.t] in (3):
(12) [([Y.sub.1]/[Y.sub.0]).sup.F]*
= [[[AC.sub.0]/[y.sub.1]/[AC.sub.0]/[y.sub.0].sup.1/2]
[[[AC.sub.1]/[y.sub.1]/[AC.sub.1]/[y.sub.0].sup.1/2]
where [AC.sub.t] [equivalent to] ([AC.sub.1]([W.sub.t]),.,
[AC.sub.m]([W.sub.t])). Here, e.g., [AC.sub.0][y.sub.1] [equivalent to]
[[summation].sub.i=1,.,m][AC.sub.i]([w.sub.0])[y.sub.i1] = C([w.sub.0],
[y.sub.1]). The following Proposition shows that (under standard
assumptions in the literature) this is a superlative output quantity
index.
PROPOSITION 3. Assume nonjoint CRTS technologies and a cost
function C(w, y) = c(w)h(y). Then
(13)
[([Y.sub.1]/[Y.sub.0]).sup.F*] = [[Y([w.sub.0],
[y.sub.1])/Y([w.sub.0], [y.sub.0])].sup.1/2]
x [[Y([w.sub.1], [y.sub.1])/Y([w.sub.1], [y.sub.0])].sup.1/2]
where Y(w, y) is an output aggregator function.
This implies that [([Y.sub.1]/[Y.sub.0]).sup.F*] is an output
aggregator, independently of profit maximization or risk preferences
(assuming cost minimization and nonjoint CRTS technologies). Since c(w)
can be a flexible functional form, [([Y.sub.1]/[Y.sub.0]).sup.F*] is a
superlative output quantity index.
Thus, we have developed two economic index numbers for aggregating
outputs based on cost minimization, the Tornqvist index
[([Y.sub.1]/[Y.sub.0]).sup.T] (8) and the Fisher index
[([Y.sub.1]/[Y.sub.0]).sup.F*] (12). In either case, the output quantity
index can be substituted into (7) to calculate a corresponding index of
price risk over commodities.
Extensions to Higher Moments, Subindexes, and Output Risk
The above analysis can easily be extended to aggregation over
commodities of higher moments of price risk. Given that y is
non-stochastic, the [k.sup.th] moment of the probability distribution
for revenue R = [[summation].sub.i=1,.,m][p.sub.i][y.sub.i] is defined
as [M.sup.k]R = E{[[[summation].sub.i][p.sub.i][y.sub.i] -
E([[summation].sub.i][p.sub.i][y.sub.i])].sup.k]} =
E{[[[[summation].sub.i]([p.sub.i] - E[p.sub.i])[y.sub.i]].sup.k]} (k
[greater than or equal to] 2) For example, the third moment of the
distribution for revenue with two commodities (m = 2) is [M.sup.3]R =
[m.sub.111][y.sup.3.sub.1] + 3 [m.sub.112][y.sup.2.sub.1][y.sub.2] + 3
[m.sub.122][y.sub.1][y.sup.2.sub.2] + [m.sub.222][y.sup.3.sub.2] where
(e.g.) [m.sub.112] = E[[([p.sub.1] - E[p.sub.1]).sup.2] ([p.sub.2] -
E[p.sub.2]) is a third moment of price. The Laspeyres and Fisher indexes
(4) and (5) extend in an obvious manner to higher moments.
Let [M.sup.k]p denote an aggregator of the kth moments of the joint
probability distribution for prices. In the spirit of index number
theory and elementary statistics, an aggregate output price risk index
[M.sup.k]P and an aggregate output quantity index Y should satisfy the
following identity in value ratios:
(14) [([M.sup.k][P.sub.1]/[M.sup.k][P.sub.0])([Y.sub.1]/
[Y.sub.0]).sup.k]
= [M.sup.k][R.sub.1]/[M.sup.k][R.sub.0] k [greater than or equal
to] 2.
This is a generalization of (6). Thus, an aggregate index of higher
moments of price risk can be calculated from (14) given an appropriate
aggregate output index, such as the Tornqvist and Fisher indexes (8,12)
of the previous section. Given such an output quantity index Y that is
superlative under our behavioral assumptions, then (14) implies an
aggregate price risk index
(15) ([M.sup.k][P.sub.1]/[M.sup.k][P.sub.0]
= [([M.sup.k][R.sub.1]/[M.sup.k][R.sub.0]/
([Y.sub.1]/[Y.sub.0]).sup.k]
that is exact and superlative in terms of preserving the
contribution of price risk for commodities to revenue risk.
Our analysis can also easily be extended to partition price risk
into aggregate subgroups. For example, variance of total revenue can be
partitioned in terms of two commodity groups A and B as
(16)
VR = [y.sup.T.sub.A][Vp.sub.A][y.sub.A] +
[y.sup.T.sub.B][Vp.sub.B[y.sub.B] + [2y.sup.T.sub.A][Vp.sub.AB[y.sub.B].
Matrices [Vp.sub.A] and [Vp.sub.B] can be aggregated similarly to
(7) using (e.g.) Tornqvist subindexes [Y.sub.A] and [Y.sub.B]. A
Laspeyres index analogous to (4) could be defined for [Vp.sub.AB].
However, a more appropriate approach is to define a
"Tornqvist" or "Fisher" index for [Vp.sub.AB] using
the following simple generalization of (7):
(17) [VP.sub.AB1]/[VP.sub.AB0] = ([y.sup.T.sub.A1][Vp.sub.AB1]
[y.sub.B1]/[y.sup.T.sub.A0][Vp.sub.AB0][y.sub.B0])
/{(Y.sub.A1]/[Y.sub.A0])([Y.sub.B1]/[Y.sub.B0])}. (8)
Extensions of the analysis to include risk for both output price
and output quantity are more problematic. Assuming both p and y are
stochastic but coy(p, y) = 0, then variance of revenue is
(18) VR = [Ey.sup.T]VpEy + [Ep.sup.T]VyEp +
[[summation].sub.i][[summation.sub.j][Vp.sub.ij][Vy.sub.ij]
where Vy is the covariance matrix for output risk, and Ey is the
vector of expected outputs. The more realistic assumption that p and y
covary typically leads to a more complex expression for variance of
revenue (see Bohrnstedt and Goldberger [1969] for an exception). An
aggregate index [VP.sub.1]/[VP.sub.0] can be defined as in (7), and an
aggregate index for Vy can be defined similarly as
(19) [VY.sub.1]/[VY.sub.0] = ([Ep.sup.T.sub.1][Vy.sub.1]/
[Ep.sup.T.sub.0][Vy.sub.0][Ep.sub.0])/[([EP.sub.1]/[EP.sub.0]).sup.2]
where [EP.sub.1]/[EP.sub.0] is an appropriate Tornqvist or Fisher
(expected) output price index. Since the last term in (18) depends
solely on Vp and Vy (independently of Ey and Ep), it directly defines an
aggregate Z or [Z.sub.1]/[Z.sub.0]. These indexes [VP.sub.1]/[VP.sub.0],
[VY.sub.1]/[VY.sub.0] (and [Z.sub.1]/[Z.sub.0]) can be viewed as
aggregate measures in ratio form of the components of revenue risk VR.
For certain purposes, this may be satisfactory.
On the other hand, note that (18) implies that VR is additive
rather than multiplicative in these components. Therefore (in contrast
to the product test equation (1) and its generalization (6)) the value
ratio [VR.sub.1]/[VR.sub.0] for revenue risk is not a simple
transformation of these indexes, e.g., [VR.sub.1]/[VR.sub.0] [not equal
to] ([VP.sub.1]/[VP.sub.0])[([EY.sub.1]/[EY.sub.0]).sup.2] +
([VY.sub.1]/[VY.sub.0]) [([EP.sub.1]/[EP.sub.0]).sup.2] +
[Z.sub.1]/[Z.sub.0]. Thus these indexes may provide a poor approximation
to [VR.sub.1]/[VR.sub.0].
Alternatively, a Laspeyres-type index for revenue risk under both
price risk and quantity risk can be defined as
(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
However, this index does not facilitate decomposition into price
and quantity aggregates. Paasche and Fisher indexes can be defined
similarly.
Empirical Applications
This article has shown that price risk should be aggregated over
commodities by applying index number theory to aggregate price variances
and covariances rather than by calculating a variance of an aggregate
price index, and several alternative measures have been suggested for
such aggregation. This section illustrates the practical importance of
this methodology in the case of major Manitoba crops.
The index number theory developed above is formally independent of
the time dimension of risk, e.g., the price covariance matrix Vp above
can be defined as an annual, monthly, or daily measure of price risk.
The appropriate time dimension may vary with the application.
Consequently, this section considers variation in crop prices at both
the annual and monthly level (daily data were not as readily available).
Different measures of price risk have been used in empirical
studies. The most common alternatives are ad hoc naive models and GARCH
models. This section considers both models of price risk.
In all cases, the aggregation measures proposed here are highly
correlated with each other but are less correlated with variances of
aggregate price indexes. So the common approach to aggregation, a
variance of an aggregate price index, is not a close proxy in practice
to approaches that are superior in theory. Thus, our theoretical
arguments have considerable practical importance.
A simple measure of price risk based on naive expectations
approximates expected prices by a one-period lag and price variances and
covariances using a simple three period weighted average of prediction
errors:
(21) [E.sub.t-1][p.sub.t] = [P.sub.t-1]
(22)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Such approximations are common with annual data (we follow the
weightings of Chavas and Holt [1990]). Price covariances and higher
moments are easily calculated in this manner, in contrast to GARCH
models.
Univariate ARCH and GARCH models are commonly used to estimate the
second moment of price risk, but extensions to price covariances and
higher moments are problematic. A multivariate GARCH model with
time-varying conditional covariances was developed by Bollerslev, Engle,
and Wooldridge (1988), but this model has been intractable for empirical
research. Alternatively, Bollerslev (1990) simplifies the model by
assuming constant conditional correlations and then jointly estimates
variances and covariances. More recently, Engle (2002) advocates a
two-step multivariate GARCH process: (a) first univariate GARCH models
are estimated for each separate price or return, and then (b)
standardized residuals from the GARCH models are used to estimate
parameters of correlation. Engle and Sheppard (2001) extend this
analysis. The two-step procedure is particularly simple under constant
conditional correlations. ARCH/GARCH models have recently been extended
to higher moments (Hansen 1994; Harvey and Siddique 1999; Wang et al.
2001; Roberts 2001; Haas, Mittnik, and Paolella 2004; Alexander and
Lazar 2006). (9)
This article adopts the simple two-step process for estimating
price risk (price variances and covariances): univariate GARCH models
are first estimated, then standardized residuals are used to estimate
constant conditional correlations (as assumed by Bollerslev), and in
turn time varying conditional covariances are calculated. First,
univariate GARCH models are specified for each of the six prices
assuming a GARCH(1,1) conditional disturbance [u.sub.it] =
[[epsilon].sub.it][h.sup.1/2.sub.it]. Second, assuming constant
correlations for u's, pairwise regressions of standardized
residuals are estimated: [e.sub.it] = [[beta].sub.ij0] +
[[beta].sub.ij][e.sub.jt] + [v.sub.ijt] where e are estimates of
[[epsilon].sub.it] = [u.sub.it]/[h.sup.l/2.sub.it], and coefficient
[[beta].sub.ij] estimates a constant
cov([[epsilon].sub.it][[epsilon].sub.jt]) over t assuming
var([[epsilon].sub.jt]) - 1. Then estimates of regression coefficients
[[beta].sub.ij] and conditional variances hit are used to calculate
time--varying conditional covariances, cov([u.sub.it][.sub.jt]) =
[[beta].sub.ij][([h.sub.it][h.sub.jt]).sup.1/2]. More complex approaches
can be used to estimate covariances (Engle 2002), but research on this
is still preliminary (Engle 2004).
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
This article develops index number methods for aggregating price
risk over commodities in production. The standard economic theory
approach to construction of index numbers is extended here to price
risk, and the resulting index numbers are closely related to modified
Tornqvist and Fisher output quantity indexes. Given the multicommodity
nature of price risk, these index number methods should have important
empirical applications.
The methodology is applied to annual market price data for the six
major Manitoba crops over 1950-2002 and to monthly price data for five
crops over 1990-2005. Four index number formulas suggested by this
article are compared with a variance of a Tornqvist price index, a
common but in principle inferior measure of aggregate price risk.
Commodity-level price risk is measured both from a naive expectations
model and a two-step multivariate GARCH model assuming constant
conditional correlations. Although we advocate modified Tornqvist and
Fisher output quantity indexes in constructing our measures of aggregate
price risk, data limitations required use of standard Tornqvist and
Fisher output quantity indexes. Even so, results indicated that our four
measures of aggregate price risk are highly correlated with each other
and less highly correlated with the variance of the Tornqvist price
index. These results illustrate the potential empirical importance of
the index number methods for aggregating price risk over commodities.
Moreover, the economic index number approach to aggregation of
price risk over commodities developed here has empirical importance in
cases where the standard index number theory does not. Although the
standard index number theory for aggregation of prices or quantities
over commodities is in principle superior to Laspeyres or Paasche
indexes, in practice correlations can be quite high (e.g., Diewert
1976). Similarly in this study, Tornqvist and Fisher output quantity
indexes are very highly correlated with a Laspeyres index (table 2).
Nevertheless, the aggregate indexes of price risk proposed here are not
highly correlated with the standard index of price risk, i.e., a
variance of an aggregate price index. In this sense, the extensions of
index number theory to aggregation of price risk should be particularly
important in practice.
Appendix
Proof of Proposition 1: Nonjoint CRTS technologies imply that the
total cost function over all outputs can be expressed as
(A1) C(w,y) = [[summation].sub.i=1.,.m] [C.sup.i](w,[y.sub.i])
and marginal cost equals average cost
(A2) [partial derivative][C.sup.i](w, [y.sub.i])/ [partial
derivative][y.sub.i] = [C.sup.i] (w, [y.sub.i])/[y.sub.i] i = 1,., m.
Define the following Tornqvist-like output quantity index
[Y.sub.1]/[Y.sub.0]:
(A3)
log[Y.sub.1]/[Y.sub.0]) = [[summation].sub.i=1,.,m]
{([theta].sub.i1] + [[theta].sub.i0])/2} log([y.sub.i1]/[y.sub.i0])
where [[theta].sub.it] [equivalent to] [C.sub.it]/[C.sub.t] is the
share of output i in total cost at time t. This is similar to a standard
Tornqvist input quantity index or output quantity index.
Proceeding similarly to Diewert (1976), we can show that this is a
superlative output quantity index corresponding to a Translog cost
function C(w, y) and nonjoint CRTS technologies, irrespective of risk
aversion and output price risk or uncertainty. Since a Translog cost
function C(w, y) is quadratic in logarithms, the quadratic lemma of
Diewert (1976) implies
(A4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
by Shephard's lemma (assuming all inputs are in static
equilibrium) and (A1)-(A2)
= log([W.sub.1]/[W.sub.0]) + log([Y.sub.1]/[Y.sub.0])
where [S.sub.wj] [equivalent to] {([w.sub.j1][x.sub.j1]/[C.sub.1])
+ ([w.sub.j0][x.sub.j0]/[C.sub.0])}/2 and [S.sub.yi] [equivalent to]
([[theta].sub.i1] + [[theta].sub.i0])/2 is defined as for (A3). The two
sums in the second equality of (A4) are designated as
log([W.sub.1][W.sub.0]) and log([Y.sub.1]/[Y.sub.0]), respectively. The
index [W.sub.1]/[W.sub.0] can be interpreted as an index for the ratio
of an aggregate average cost [AC.sub.1]/[AC.sub.0] under our
assumptions, by arguments similar to the standard case (Diewert 1976,
1980a). Then (A4) implies
(A5) [Y.sub.1]/[Y.sub.0] = ([C.sub.1]/[C.sub.0])/
([AC.sub.1]/[AC.sub.0])
i.e., [Y.sub.1]/[Y.sub.0] is equal to ratio of total cost divided
by an average cost index. This implies that [Y.sub.1]/[Y.sub.0] is an
exact output quantity index, and it is superlative.
Proof of Proposition 2: Assuming a Translog multioutput cost
function C(w, y) and proceeding similarly to (A4),
(A6) log([C.sub.1]/[C.sub.0]) = log([W.sub.1]/[W.sub.0])
+log([Y.sub.1]/ [Y.sub.0])
by Shep