spreads of the residual densities vary noticeably across cities, indicating that weather risk is much greater in
some cities than in others. Fourth, all of the residual densities have only moderate negative skewness and
moderate excess kurtosis; the average residual skewness and kurtosis coefficients are -0.36 and 4.10.
In Figure 5 we display the correlograms of the squared residuals, taken to a maximum displacement
of 800 days. There is clear evidence of strong nonlinear residual dependence, driven by strong conditional
variance dynamics. This contrasts sharply with the correlograms of the residuals (not shown), which are
negligible and indicative of weak white noise.
In Figure 6 we plot the estimated residual conditional standard deviation from 1996 through 2001.
The basic pattern is one of strong seasonal volatility variation, with additional GARCH volatility effects,
the persistence of which varies across cities. For each city, seasonal volatility appears highest during the
winter months. Among other things, this indicates that correct pricing of weather derivatives may in
general be crucially dependent on the season covered by the contract. Some cities display a great deal of
seasonal volatility variation - the conditional standard deviation of Atlanta temperature shocks, for
example, roughly triples each winter - whereas temperature shock volatility in other cities such as Las
Vegas varies less across seasons.
It is interesting to note from Figure 6 that the GARCH volatility effects appear more pronounced in
winter, when the volatility seasonal component is high, which might suggest the desirability of a
multiplicative volatility specification. Nelson’s (1991) exponential GARCH(1,1) is one attractive such
specification, replacing the volatility specification for an in equation (1c) with an alternative but related
specification for lnσ,. Estimation of exponential GARCH models, however, produced fitted conditional
variance series nearly identical to those of the original GARCH models.
We also estimated the densities of the standardized residuals, (Tt-Tt)∕0t, where Tt is the fitted
value of daily average temperature. They still display negative skewness; the average across cities is -0.45.
Modeling the conditional heteroskedasticity does, however, reduce (but not completely eliminate) residual
excess kurtosis; the average across cities is now 3.74. Finally, we also computed the correlograms of
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