The predictability of an event or quantity depends on:
A key step is knowing when something can be forecast accurately and when a forecast will be no better than a coin flip. Every environment is changing and a good forecasting model can capture the way in which it is changing.
There are many models to use for forecasting varying from simple to complex. The choice of method will depend on available data and the predictability of the object to be forecast.
Anything we try to forecast is a random variable; it is unknown in its essence. There are many possible futures to be imagined. A forecast is estimating the middle of the range of possible values of the variable. Forecasts are usually accompanied by a prediction interval giving a range of values the random variable could be with relatively high probability.This middle range or average is called the point forecast.
\(y_t\): an observation, \(y\), at time \(_t\)
Supposed all observed information is represented as: \(\mathcal{I}\).
\(y_t|\mathcal{I}\) would mean "the random variable \(y_t\) given what we know in \(\mathcal{I}\)
The set of values and relative probabilities of this random variable is the probability distribution. When forecasting, it is the forecast distribution.
\(\hat{y}_t\): Termed “y hat”, and is the point forecast of \(y_t\)
\(\hat{y}_{t|t-1}\): this would mean the forecast of \(y_t\) taking account of all previous observations
\(\hat{y}_{T+h|T}\): the forecast of \(y_{T+h}\) taking account of all observations up to time \(T\) via steps of \(h\) size