A Complete Introduction To Time Series Analysis (with R):: Prediction 1 → Best Predictors II
In the last article, we saw how we could obtain the best linear predictor (BLP) of X_{n+h} given a function of X_{n}. This week, we will see that the most appropriate model for this function has the form given above, by a series of manipulations. We will follow the same idea as before: minimize some objective function!
Best Linear Predictor of X_{n+h}
Why? Let’s see the derivation: we would like to find a model for
So we can consider a linear function, say
and then find a and b that minimizes the MSE, that is
Once again, we can take partials w.r.t a and b, obtaining the system
which we set to 0 for optimization purposes. Working this out a bit, this gives
so now we can solve for a, say. You can verify that the solution is given by
where we used “\mu” to represent the expectation as a function of the inner lag. Next, plugging back and solving for b, we obtain
so that
Does this look familiar? Recall that
So that the expression above becomes
Now, plugging this back once more into “a”, gives