Weather Forecasting for Weather Derivatives

CumHDD, which we then use to estimate the density, as follows. First, we simulate 250 151-day
realizations of the temperature shock ε_r by drawing with replacement from the empirical distribution of
estimated temperature shocks (ε_f). Second, we run the 250 151-day realizations of temperature shocks
through the estimated model (1) to obtain 250 simulated 151-day realizations of daily average temperature.
Third, we convert the 250 simulated 151-day realizations of daily average temperature into 250 simulated
151-day realizations of HDD, which we cumulate over the November-March heating season,
CumHDD_s = 5~^₁¹ HDD_ty, s = 1, 2, ..., 250. Finally, we form the empirical distribution function of
(JumelDIp, based upon Cu=HDD_s, s = 1, ..., 250.

After passing through the entire sample, we have 41 assessed distribution functions, F_y(∙∖
y o I960,..., 2000, one governing each of CumHDD_y, y = I960,..., 2000. We assess the conditional
calibration of those distributional forecasts via the probability integral transform, as suggested and
extended by Rosenblatt (1952) and extended by Dawid (1984), Diebold, Gunther and Tay (1998), and
Diebold, Hahn and Tay (1999). In particular, if the estimated distribution and true distribution coincide
year-by-year, then the series of assessed distribution functions F_y(∙) evaluated at the corresponding series
of realized values of CumHDD_y should be approximately iid and uniformly distributed on the unit interval.

.            Hd

Formally, z ≡ F_j,(C∞wffi)Zζ,) ~ t7(0, 1). For each city, we check uniformity by examining histograms
of z, and we check independence by examining correlograms of the first four powers of z. The sample of
size 41 is of course small, but the framework has previously been applied successfully in small samples, as
for example by Diebold, Tay and Wallis (1999).

First consider assessing uniformity. We estimate the density of z using simple four-bin histograms,
which we present in the leftmost column of Figure 9, accompanied by 95% pointwise error bands under the
iid U(0, 1) null hypothesis. Interestingly, the z series differ rather noticeably from uniformity, and
moreover, they display a common pattern: too many large CumHDD realizations occur relative to the
assessed distributions, as evidenced by the increase in the histograms when moving from left to right. The
common nature of uniformity violations may indicate a neglected common temperature component, due for

-14-

<<   14   15   16   17   18   19   20   >>

More intriguing information