misspecifying weather hedges that involve multiple underlying indexes. The current
work simplifies the analysis by investigating seasonal (June, July, and August)
temperature WDs.
Conceptual Model
Idiosyncratic effects may self-diversify when aggregated, leaving a greater proportion of
the total risk in the form of weather risk. Thus, WD hedging may be more effective for
aggregate rather than disaggregate yield exposures. The magnitude of the spatial
aggregation effect depends on the relative correlations of weather and idiosyncratic yield
effects across locations. To illustrate, assume yields can be decomposed into two effects,
weather effects, W, and all other effects,ε , which may be correlated. Consider a simple
model of crop yields which allows for non-linear terms
(1) Ytk=αk+fk(Wtk)+εtk
t,k k k t,k t,k
where t is the time index, k is the location index, Wt,k is a vector of weather variables,
fk(Wt,k) represents the systemic weather component of yields, εt,k represents the
idiosyncratic risk component, and E[εt,k]= 0. Summing across k locations, gives
(2) E[∑Yt,k]=∑αk+E[∑fk(Wt,k)]+E[∑εt,k]
kk k k
and
(3) Var[∑Yt,k]=Var[∑ fk(Wt,k)]+Var[∑εt,k]+Cov[∑ fk(Wt,k),∑εt,k].
kk kkk
If the εt,k's are relatively less positively correlated than the fk(Wt,k)'s across locations
then, as the individual yields are summed, more variation in yields may be able to be
attributed to the weather effects at larger levels of spatial aggregation. Thus, WD