What is an "unbiased forecast"? 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.

*

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


*If the estimated model is not linear regression, but 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.

We can write your predictor for $Y_h$ as
$$
\widehat{Y_h}=T\left(W(x,y),X_h\right)
$$
Then $\widehat{Y_h}$ is unbiased as a predictor for $Y_h$ if
$$ \DeclareMathOperator{\E}{\mathbb{E}}
  \E \{\widehat{Y_h} - Y_h \} =0 
$$
where the expectation is calculated according to the joint distribution of all involved random variables.
