Weather Forecasting for Weather Derivatives

seasonal volatility component using a Fourier series, and we approximate the cyclical volatility component
using a GARCH process (Engle, 1982; Bollerslev, 1986). Assembling the various pieces, we estimate the
following daily average temperature model for each of our four cities:

η = Trend_t + Seasonal_t + ɪ^ɪ (_t)i^τt~ι ^{+ σ}t^εf                                      (1)

where

Trand_t = ∑%--₀V_mt^m                                                          (1a)

^(1b)onal_t -      ∣δ_c,_pcos∣2πp∣∣∣ + δ₄,sin(2π/^                                 ^(1b)

(¹ - ∑f=ι (Y^^cos(^2π⅛) ⁺ Wⁱⁿ(^2π⅛j) ⁺ ∏ι ɑ/ɑr-rɛr-ɔ^{2 +} H=ι β,^σL ^(1c)
ε_r~∏√(0,1),                                                                          (1d)

and d(t) is a repeating step funtion thatyles through 1, ..., 365 (as we drop Feruary 29 in all leap
years). In all that follows we set L=25, M=1, P=3, Q=3, R=1, and S=1, whihoth Akaike and Schwarz
informationriteria indiate are more than adequate for eahity. Maintaining the rather large value of
L=25osts little given the large numer of availale degrees of freedom, and it helps tocapture long-
memory dynamis, if present, as suggestedy results suh as those of Bloomfield (1992). Following
Bollerslev and Wooldridge (1992), weonsistently estimate this regression model with GARCH
disturanesy Gaussian quasi maximum likelihood.

Now let us disuss the estimation results. First, and perhaps surprisingly, mostcities display a
statistially signifiant trend in daily average temperature. In mostases, the trend is much larger than the
inrease in average gloal temperature over the same period. For example, the results indicate that the
daily average temperature in Atlanta has inreasedy three degrees in the last forty years. Such large trend
inreases are likely aonsequene of development and air pollution that inreased uran temperatures in
general, and uran airport temperatures in partiular, where most of the U.S. reording stations are located,
a phenomenon often dued the “heat island effet.” Seond, theonditional mean dynamics displayboth
statistially signifiant and eonomially important seasonality. Third,onditional mean dynamics also
display strongylial persistene. The estimated autoregressions display an interesting root pattern,

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