Econ . 513 , Time Series Econometrics Fall 1999 Chris Sims

In other words, E [y | {x1, . . . , xn}] is the best linear predictor of y based on {x1, . . . , xn}. In order for this projection to be well defined, y must have finite variance. The definition obviously can be interpreted as applying to random vectors y and xj also. Here are some useful properties of E . i. E is linear, meaning E [aX + bY |Z] = aE [X |Z] + bE [Y |Z], where a and b are numbers and X, Y , and Z are vectors of random variables. ii. The prediction error Y − E [Y |X] is uncorrelated with any linear function of X. iii. There is a law of iterated projections, formally similar to the law of iterated expectations: