Forecasting with Time Series Models

Abstract

Having selected a model and fitted its parameters to a given times series, the model can then be used to estimate new data of the time series. If such data are estimated for a time period following the final data value X T of the given time series, we speak of a prediction or forecast. The estimation of data lying between given data points is called interpolation. The question now arises as to how a model such as those given in Equations 33.6 or 33.13 could be used to obtain an "optimal" estimate. To answer this question the forecasting error $${X_{T + k}} - {\widehat X_{T + k}},\quad k \in \mathbb{N}$$ between the estimated values \({\widehat X_{T + k}}\) and the actual observed time series values XT+k can be used if the last value used in the calibration of the model was X T . The best forecast is that which minimizes the mean square error (MSE for short). The MSE is defined as the expectation of the squared forecasting error $$MSE: = E\left[ {{{\left( {{X_{T + k}} - {{\widehat X}_{T + k}}} \right)}^2}} \right]$$ ((34.1)) This expression is the mathematical formulation of the intuitive concept of the "distance" between the estimated and the actual values which is to be minimized "on average" (more cannot be expected when dealing with random variables). Minimizing this mean square error yields the result that the best forecasting estimate (called the optimal forecast) is given by the conditional expectation $${\widehat X_{T + k}} = E\left[ {\left. {{X_{T + k}}} \right|{X_T}, \ldots ,{X_2},{X_1}} \right]$$ ((34.2)) This is the expectation of XT+k, conditional on all available information about the time series up to and including T.KeywordsTime SeriesForecast ErrorConditional ExpectationConditional VarianceTime Series ModelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.