Weather Forecasting for Weather Derivatives

common across all four cities, regardless of location. The dominant root is large and real, around 0.85, the
second and third roots are a conjugate pair with moderate modulus, around .3, and all subsequent roots are
much smaller in modulus. Fourth, the conditional variance dynamics, like the conditional mean dynamics
contain both seasonal (Fourier) and cyclical (GARCH) components. The Fourier terms appear to capture
adequately all volatility seasonality, and the GARCH terms capture the remaining “standard” volatility
persistence in daily average temperature. The Fourier volatility seasonality effect is relatively more
important; it is significant and sizeable for all cities. The GARCH volatility persistence effect is smaller,
although all cities have significant a, ranging across cities between 0.01 and 0.07. There is considerably
more range in the estimates of volatility persistence as summarized by the other GARCH parameter, β,
implying very different half-lives of volatility shocks across cities. For example, the half-life of a Las
Vegas volatility shock is approximately one day, whereas the half-life of a Chicago volatility shock is
approximately seven days.

In Figure 3 we show model residuals (σ_tε_r) over the last five years of the estimation sample. The
fit is typically very good, with Rb above 90%. Figure 3 also provides a first glimpse of an important
phenomenon: pronounced and persistent time-series heteroskedasticity in the residuals. In particular,
weather risk, as measured by its innovation variance, appears seasonal, as the amplitude of the residual
fluctuations widens and narrows over the course of a year. It seems that such seasonal heteroskedasticity in
temperature was first noted, informally, in an economic context by Roll (1984). Subsequently in this paper
we propose quantitative models and forecasts that explicitly incorporate the heteroskedasticity.

In Figure 4 we show kernel estimates or the residual densities for each of the models. Four features
stand out. First, as expected, average temperature residuals are much less variable than average
temperature itself; that is, weather surprises are much less variable than the weather itself, with residual
standard deviations only one third or so of the average temperature standard deviations. Second, again as
expected, all of the residual densities are uni- as opposed to bi-modal, in contrast to the unconditional
densities examined earlier, due to the model’s success in capturing seasonal highs and lows. Third, the

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