What is the mathematical definition of "unbiased forecast"

Question

Assume we estimate a model from the data $(X, Y)$, with some estimator $W(X, Y)$, which is estimating parameters $\theta$ for the model we chose.

Then, we would like to perform a forecast for $Y_h$ from $X_h$ in the future. For this, we take the estimator $W(X, Y)$, and create another "estimator", or predictor, $T(W(X, Y), X_h)$ for forecasting the outcome $Y_h$ (this is how e.g. regression works).

However, the predictor itself is not a "traditional estimator", since it is guessing a random variable, not a parameter, like in standard inference. As such, there does not exist some "standard" measure of unbiasedness afaik.

For example, we could define unbiasedness for $W$ above as $E_{\theta}\big[W(X, Y)\big] = \theta$, where we simply take the expectation of the estimator over the data distribution given the parameter $\theta$ of the said distribution. But I do not see how something like this could be defined for $T(W)$ which "predicts" the random variable $Y_h$.

1. How can we define forecast unbiasedness, e.g. in terms of expectations? Is there some textbook reference?

2. If the estimated model is not linear regression (e.g. GLM, or random forest) is the forecast unbiasedness defined the same way as for linear regression?

If possible, use the notation I introduced.

Answer 1

The following reference: Economic forecasts and expectations describes the unbiased forecast well (on page 8):

"The forecast is unbiased if that point lies on $LPF$, that is if $E(P) = E(A)$. The difference $E(A)—E(P)=E(u)$ measures the size of bias. (...) Unbiasedness is a desirable characteristic of forecasting, but it does not, by itself, imply anything about forecast accuracy."

An illustrative figure is given on page 7 as well (which I find helpful):

I understand the forecast unbiasedness for GLMs as defined the same way as for linear regressions.