General Concepts

Statement of the Problem

To predict an unobserved variable is to find an approximation of it that is a function of the observations (see Chapter 2). Prediction problems are frequent in economic applications. They arise naturally in dynamic contexts when, for instance, one desires to know the level of unemployment in the next years, or when one's goal is to complete a time series of which some intermediate values are missing. Prediction problems, however, are also important in numerous other situations. For instance, a researcher wants to determine the expected change in consumption of a household whose income changes by 5%. Alternatively, one wants to find the price that would have prevailed in equilibrium on a market in desequilibrium (see Example 1.20).

We shall see in the following sections, that estimation problems and prediction problems are often closely related. As an example, consider estimation by ordinary least squares. This estimation will naturally appear when looking for an optimal prediction of a nonobserved endogenous variable by a linear function of the observed endogenous variables.

Thereafter, we let Y1,…,Yn denote the observations and W the variable to predict. To simplify, it is assumed that W takes its values in ℝ. A prediction or predictor of W is a function Ŵ(Y1,…,Yn) of the observations. The prediction error is the discrepancy between the predicted variable and the prediction.