# On unbiasedness of an optimal forecast

Diebold "Forecasting in Economics, Business, Finance and Beyond" (v. 1 August 2017) section 10.1 lists absolute standards for point forecasts, with the first one being unbiasedness: Optimal forecasts are unbiased. There is a brief follow-up there, too: If the forecast is unbiased, then the forecast error has a zero mean.

I have a quibble with this. Let $$Y$$ be the random variable the realization of which we are trying to predict, and let $$z$$ be the forecast. If

1. we know at the time of making the forecast that it will be judged by a quantile loss function (say, a $$p$$-level quantile)

and

1. our goal is to minimize the expected loss,

we should target the relevant quantile of the distribution, $$z=F_Y^{-1}(p)$$, where $$F_{Y}$$ is the cumulative distribution function of $$Y$$. I would be willing to call $$z=z^*:=F_Y^{-1}(p)$$ an optimal forecast. However, unless the quantile coincides with the mean, $$z$$ will produce an error $$e:=Y-z$$ with $$\mathbb{E}(e)$$, breaking Diebold's optimality property.

Question: How could we reformulate the optimality property for it to make more sense? What if we defined unbiasedness w.r.t. the loss-minimizing target instead of the realization of $$Y$$ (so that in my example, an optimal forecast would have the expected value of $$F_Y^{-1}(p)$$ instead of $$\mathbb{E}(Y)$$ as per Diebold)? Or are there circumstances where such version of unbiasedness is not desirable?

Update: To be fair, Diebold writes the following in section 2.8 (on p. 38):

Quite generally under asymmetric $$L(e)$$ loss (e.g., linlin), optimal forecasts are biased, whereas the conditional mean forecast is unbiased. Bias is optimal under asymmetric loss because we can gain on average by pushing the forecasts in the direction such that we make relatively few errors of the more costly sign.

But he does not reiterate nor even hint at that in the beginning of chapter 10 (e.g. on p. 334-335) where he discusses forecast optimality – even though he does mention the loss function there:

This unforecastability principle is valid in great generality; it holds, for example, regardless of whether linear-projection optimality or conditional-mean optimality is of interest, regardless of whether the relevant loss function is quadratic, and regardless of whether the series being forecast is stationary.

(The first emphasis is the author's, the second is mine.)

• Maybe he discusses somewhere that the loss function is quadratic? Then, the mean would be the optimal forecast, which then leads to an unbiased forecast. Commented Feb 27 at 13:15
• @ChristophHanck, he does that in section 2.8 on p. 38: Quite generally under asymmetric L(e) loss (e.g., linlin), optimal forecasts are biased, whereas the conditional mean forecast is unbiased. Bias is optimal under asymmetric loss because we can gain on average by pushing the forecasts in the direction such that we make relatively few errors of the more costly sign. But he does not come back to it in chapter 10 (e.g. on p. 334-335) where he discusses forecast optimality – even though he does mention the loss function there once. Looks like an oversight to me. Commented Feb 27 at 13:20
• Yes, it seems like clarification may have been useful there! Commented Feb 27 at 13:23
• @ChristophHanck, while I have your attention, would you mind taking a look at this? stats.stackexchange.com/questions/639814 Commented Feb 27 at 13:28
• @StephanKolassa, thanks for your interesting perspective. "My" view is, e.g., influenced by Bayesian perspectives along the lines of "if you have a quadratic loss function then your Bayes rule is the posterior mean". Or my more down-to-earth classroom example: I have an asymmetric loss function when cycling to the train station in the morning in that I'd rather arrive 5 minutes early than two minutes late. Commented Feb 27 at 16:48