Linear ‘Prediction’ for AR Time Series

In the exercises for the MAT8181 course, there are two Exercises (16 and 17) about prediction and extrapolation based on MA(1) and AR(1) time series. But before discussing those exercises (I had some request for hints), I wanted to recall the definition of the linear prediction,

where

As discussed previously on this blog, we consider here on projection not on  (this would be the conditional expectancy) but on the linear subset.

The goal of Exercise 2 was to establish an important result, in the context of Gaussian random vectors. If  is a (multivariate) Gaussian vector, , then

where  is the vector .

Keep those results in mind, and let us look at Exercise 17, for instance. Here,  is an AR(1) process, with innovation ,

One observation (say ) is missing. We have here 3 questions:

  • what is the best linear prediction of  given  and 
  • what is the best linear prediction of  given  and 
  • what is the best linear prediction of  given  and 

Case 1. Here, we have to compute

Since we have an AR(1) process,  and . Thus, from the relationship above

which can be written

i.e. . Which makes sense actually: the AR(1) process is Markovian of order one, so

And we seen in class that for an AR(1) process

So far, so good.

Case 2. Here, we have to compute

Since we have an AR(1) process,  and . Thus, from the relationship above

i.e. .

Case 3. Finally, we have to compute

One more time, since we have an AR(1) process,  and . So here, the relationship above becomes

Here, we can write

i.e.

So finally, what we got here is

and

The mean squared errors for each of those estimates are obtained computing

I guess I should probably stop here… that’s a detailed hint actually.


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