In this lecture, we are going to discuss how to choose the final hyper perimeter of the Arima model,
which is peh the order of the A.R. component.
This is accomplished by looking at the pickoff, which stands for partial autocorrelation function.
In my opinion, the partial autocorrelation function is a bit more difficult to understand than the
AKF.
However, the way that we apply it is the same as how we apply the akef for the moving average model.
So in this lecture, first we will just simply discuss how to use the F, then we will discuss a tiny
bit more about what the pickoff actually is.
In case you are interested, you should not feel obligated to understand the second part in order to
move on to the next lecture.
So what is the perceive from a practical perspective, from a practical perspective, it's just a plot
that looks very similar to the Akef and that it's some kind of auto correlation function.
The vertical axis is still the value of the function and the horizontal axis still represents some like
the way that we use it is exactly the same.
As for the moving average, there will be some confidence interval on the Akef plot.
If we see any statistically significant non-zero legs up to something like P, this would indicate that
we have an auto regressive component of order.
P, for example, if the highest non-zero value is five, then we would choose P equals five as before.
It's usually the case that the values below five will also be non-zero.
All right, so now that you know how the stuff works, what is the really we know that it results in
a graph that looks like the AKF, but it is clearly not the same since it has a different name and a
different purpose.
There are two ways of thinking about the picture.
The first way is to simply look at what it is in terms of its definition.
The second way to look at the Sieff is to consider how it's calculated.
Looking at both of these perspectives should help you understand the.
So what is the partial autocorrelation function, the partial autocorrelation function at Leitao is
defined as the correlation between a why a time T and Y a time T plus tau conditioned on all the ways
in between a T plus tau.
So that's why a T plus one, why a T plus two all the way up to Y of T plus tau minus one.
So one way to think about this is it's like a conditional auto correlation, whereas the regular auto
correlation is unconditional.
In other words, not conditioned on anything.
The second way to think of the pickoff is how we calculate it, we usually define the coefficients using
the Greek letter PHY and we use double indices, so we'll have five zero zero five one one and so on.
In general, these will be represented by Tau Tau.
Note that five zero zero will just be one as usual.
This is because this is the ECF between Y a time T and Y a time T for any T and therefore there are
no in between values and it's just the same as the.
Furthermore, the same thing happens for five one one because again there are no values in between Y
a time T and Y time T plus one.
Therefore this just reduces to the autocorrelation at like one.
What about for any other tau in this case we need to do a few auto regressions.
Specifically, suppose that y half of time T plus tau is regressed on all of the ways in between a T
plus 20.
Note that this is just our usual auto regressive model.
Next, let y had at time t be another regression again based on all the whys in between a T plus 20.
This one is weird because it's like a backwards auto regression.
We are using future values to predict a past value.
So both Y had a time T plus tau and we had a time T or regressed on the same set of variables, the
ones in between.
Next, we have the big reveal via Tau Tau is just the correlation between why a time T plus tau minus
Y hat at time T plus tau and why a time T minus Y hat at time T as defined on the previous slide.
That is to say fire tau tau or the akef for like tau is the correlation between any noise after subtracting
off the parts that could be predicted by the regressions.
You may have to repeat this to yourself a few times, but it will make perfect sense.
The partial autocorrelation function is just the correlation conditioned on the in between variables.
Therefore, it's the correlation between whatever we can't explain using those in between variables.
And of course that makes perfect sense for choosing the order p in an auto regressive model.
If the pickoff is non-zero, that means there is a significant correlation that cannot be explained
by the in between variables and therefore we should include this correlation in our final auto regressive
model.
So just to put some real numbers to this, suppose that we want to know if y one is helpful in predicting
Y five.
Obviously the same question applies to Y two and why six y three and Y seven and so forth.
This is because we assume our signal is stationary and therefore any auto correlations only depend on
the time difference.
But let's stick to Y one in my five.
The in between values are y two y three and Y for the pickoff will be large if there is a large correlation
between Y one and Y five that cannot be predicted by Y to Y three and Y four.
Therefore, we should use one in our auto regressive model for predicting Y five.
That is to say, whenever we see large values in the pickoff, it indicates that these should be included
in our auto regressive model.