forecast of a time series model by taking into account over/under cost Let $T$ denote a univariate time series data, and $c_1$ represent the cost of one unit of over forecasting and $c_2$ represent the cost of one unit of under forecasting.
Suppose we can estimate $c_1$ and $c_2$ (i.e., cost of over forecasting and under forecasting), then, can we take them into account in our model such that the cost of forecast is minimized?
 A: I'll rename your costs to $c_u$ (cost of underforecasting) and $c_o$ (cost of overforecasting).
A cost optimal point forecast then is a quantile forecast for the $\tau$ quantile, where $\tau=\frac{c_u}{c_u+c_o}$ (e.g., Bruzda, 2018, Central European Journal of Operations Research). Such forecasts are important whenever goods or services are to be provided, e.g., in electricity, call center, or supply chain/inventory forecasting. You may want to look at the literature in these areas, depending on what your use case is. (Often, one can also learn from other areas.)
Gneiting (2011, IJF) gives a nice general overview on quantile forecasting. The recent M4 forecasting competition (IJF 36(1)) prominently featured quantile forecasts. IJF 35(1) was a special issue on supply chain forecasting, and IJF 35(4) contained a special section on energy forecasting. Energy forecasting competitions such as GEFCom2017 may also be interesting.
Generally speaking, there are different approaches to this problem. One is to optimize the point forecast directly using the appropriate linlin loss (e.g., the Bruzda paper above). The other is parametric: derive a full predictive distribution, then extract the $\tau$ quantile. The first is conceptually easier, but you will need to fit a new model for each new value of $\tau$, and you may get strange results with "crossing" quantile forecasts, where $\hat{y}_{\tau_1}<\hat{y}_{\tau_2}$ although $\tau_1>\tau_2$, which can also happen in quantile forecasting. I personally am a big fan of calculating predictive distributions (so the costs don't enter into the forecasting part at all) and then just taking the appropriate quantile.
