virtue of our approach is its immediate and simple generalization to provide entire density forecasts. The
key feature of daily average temperature conditional density dynamics, apart from the seasonal conditional
mean dynamics, is the highly seasonal conditional variance dynamics, which we have modeled
parsimoniously and successfully. This facilitates simple modeling of time-varying scale of the conditional
density, and it is as relevant for long horizons as for short. All of this adds up to a simple yet powerful
framework for producing density forecasts of weather variables, to which we now turn. It is telling to
observe that in what follows we must evaluate the performance of our density forecasts in absolute terms,
rather than relative to EarthSat density forecasts, because EarthSat, like almost all forecasters, does not
even produce density forecasts.
Density Forecasting
In this section, we shift our focus to long-horizon density forecasting, and to cumulative heating
degree days, all of which is of crucial relevance for weather derivatives. Heating degree days for day t is
simply HDDt = max(0, 65-7,f). We use our model of daily average temperature to produce density
forecasts of cumulative HDDs from November 1 through March 31, for each city and for each year
between 1960 and 2000, defined as CumHDD6i = ɪ^ɪɪ ffi¾,1∙> for У = I960, ∙∙∙, 2000, i = 1, ∙∙∙, 4.
Because we remove February 29 from each leap year, each sum contains exactly 151 days. We use full-
sample as opposed to recursive parameter estimates, as required by the very small number of CumHDD
observations. To avoid unnecessarily burdensome notation, we will often drop the У and i subscripts when
the meaning is clear from context.
We focus on CumHDD for two important reasons. First, weather derivative contracts are often
written on the cumulative sum of a weather related outcome over a fixed horizon, as with the cumulative
HDD and CDD contracts traded on the CME. Second, and related, the November-March HDD contract is
one of the most actively traded weather-related contracts and hence is of substantial direct interest.
On October 31 of each year, and for each city, we use the estimated daily model to produce a
density forecast of CumHDD for the following winter’s heating season. We simulate 250 realizations of
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