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

Therefore, if we denote our B.L.P as

we see that

and this is precisely the formula you saw at the beginning! How is this useful? Suppose that you have, in addition, a stationary series. Then the BLP is given by

and if the ACF is between 0 and 1, we have a weighted average between the most recent observation and the overall trend. Isn’t that fascinating?! Note that we dropped the argument for the mean since a stationary series mean is constant, and further, the covariance and variance just become the autocovariance function of the series.

Next time

Next time, we will start studying Linear Processes. which will be extremely useful when dealing with ARMA , ARIMA and other models. Stay tuned, and happy learning!

Last time

Prediction 1 → Best Linear Predictor I

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