## Monday, June 25, 2018

### Peter Christoffersen and Forecast Evaluation

For obvious reasons Peter Christoffersen has been on my mind. Here's an example of how his influence extended in important ways. Hopefully it's also an entertaining and revealing story.

Everyone knows Peter's classic 1998 "Evaluating Interval Forecasts" paper, which was part of his Penn dissertation. The key insight was that correct conditional calibration requires not only that the 0-1 "hit sequence" of course have the right mean ((1-$$\alpha$$) for a nominal 1-$$\alpha$$ percent interval), but also that it be iid (assuming 1-step-ahead forecasts). More precisely, it must be iid Bernoulli(1-$$\alpha$$).

Around the same time I naturally became interested in going all the way to density forecasts and managed to get some more students interested (Todd Gunther and Anthony Tay). Initially it seemed hopeless, as correct density forecast conditional calibration requires correct conditional calibration of all possible intervals that could be constructed from the density, of which there are uncountably infinitely many.

Then it hit us. Peter had effectively found the right notion of an optimal forecast error for interval forecasts. And just as optimal point forecast errors generally must be independent, so too must optimal interval forecast errors (the Christoffersen hit sequence). Both the point and interval versions are manifestations of "the golden rule of forecast evaluation": Errors from optimal forecasts can't be forecastable. The key to moving to density forecasts, then, would be to uncover the right notion of forecast error for a density forecast. That is, to uncover the function of the density forecast and realization that must be independent under correct conditional calibration. The answer turns out to be the Probability Integral Transform, $$PIT_t=\int_{-\infty}^{y_t} p_t(y_t)$$, as discussed in Diebold, Gunther and Tay (1998), who show that correct density forecast conditional calibration implies $$PIT \sim iid U(0,1)$$.

The meta-result that emerges is coherent and beautiful: optimality of point, interval, and density forecasts implies, respectively, independence of forecast error, hit, and $$PIT$$ sequencesThe overarching point is that a large share of the last two-thirds of the three-part independence result -- not just the middle third -- is due to Peter. He not only cracked the interval forecast evaluation problem, but also supplied key ingredients for cracking the density forecast evaluation problem.

Wonderfully and appropriately, Peter's paper and ours were published together, indeed contiguously, in the International Economic Review. Each is one of the IER's ten most cited since its founding in 1960, but Peter's is clearly in the lead!