Loss Functions in Time Series Forecasting

When a forecast ft,h of a variable Yt+h is made at time t for h periods ahead, the loss (or cost) will arise if a forecast turns to be different from the actual value. The loss function of the forecast error et+h = Yt+h − ft,h, is denoted as c(Yt+h, ft,h). The loss function can depend on the time of prediction and so it can be ct+h(Yt+h, ft,h). If the loss function is not changing with time and does not depend on the value of the variable Yt+h, the loss can be written simply as a function of the error only, ct+h(Yt+h, ft,h) = c(et+h).