A Complete Introduction To Time Series Analysis (with R):: Best Linear Predictor (Part I)
In chapter 5 of this article series, we found that the best predictor of the n+h-th lag is given by the conditional expectation of X_{n+h} given X_{n}, that is
or extending to using all observations X_{1}, X_{2}, … , X_{n} , we can predict X_{n+h} by the conditional expectation of X_{n+h} given X_{1}, …, X_{n}. Further, we found that the actual form of this expectation when using only the last observation is given by
where, in the case that X_{n} were stationary this further simplified to
In this article, we will further explore a more precise and clear representation by using once more the principles of calculus optimization and linear algebra. Let’s jump into it!
The Projector Operator
Note that that the formula for the stationary case for predicting X_{n+h} given X_{n} is a (rather boring) linear combination. We would like to extend this to the case in which we use all observations X_{1}, X_{2}, …, X_{n}, say by finding some coefficients a_{0}, a_{1}, …, a_{n}, that is
, where P_{n} is called the projection operator. But how can we find these coefficients? Just as we did before, we will attempt to minimize the MSE of P_{n}X_{n+h} (prediction) and…
