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In this lecture, we are going to discuss how to choose the final hyper perimeter of the Arima model,

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which is peh the order of the A.R. component.

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This is accomplished by looking at the pickoff, which stands for partial autocorrelation function.

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In my opinion, the partial autocorrelation function is a bit more difficult to understand than the

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AKF.

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However, the way that we apply it is the same as how we apply the akef for the moving average model.

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So in this lecture, first we will just simply discuss how to use the F, then we will discuss a tiny

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bit more about what the pickoff actually is.

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In case you are interested, you should not feel obligated to understand the second part in order to

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move on to the next lecture.

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So what is the perceive from a practical perspective, from a practical perspective, it's just a plot

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that looks very similar to the Akef and that it's some kind of auto correlation function.

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The vertical axis is still the value of the function and the horizontal axis still represents some like

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the way that we use it is exactly the same.

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As for the moving average, there will be some confidence interval on the Akef plot.

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If we see any statistically significant non-zero legs up to something like P, this would indicate that

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we have an auto regressive component of order.

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P, for example, if the highest non-zero value is five, then we would choose P equals five as before.

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It's usually the case that the values below five will also be non-zero.

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All right, so now that you know how the stuff works, what is the really we know that it results in

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a graph that looks like the AKF, but it is clearly not the same since it has a different name and a

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different purpose.

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There are two ways of thinking about the picture.

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The first way is to simply look at what it is in terms of its definition.

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The second way to look at the Sieff is to consider how it's calculated.

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Looking at both of these perspectives should help you understand the.

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So what is the partial autocorrelation function, the partial autocorrelation function at Leitao is

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defined as the correlation between a why a time T and Y a time T plus tau conditioned on all the ways

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in between a T plus tau.

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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.

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So one way to think about this is it's like a conditional auto correlation, whereas the regular auto

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correlation is unconditional.

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In other words, not conditioned on anything.

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The second way to think of the pickoff is how we calculate it, we usually define the coefficients using

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the Greek letter PHY and we use double indices, so we'll have five zero zero five one one and so on.

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In general, these will be represented by Tau Tau.

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Note that five zero zero will just be one as usual.

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This is because this is the ECF between Y a time T and Y a time T for any T and therefore there are

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no in between values and it's just the same as the.

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Furthermore, the same thing happens for five one one because again there are no values in between Y

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a time T and Y time T plus one.

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Therefore this just reduces to the autocorrelation at like one.

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What about for any other tau in this case we need to do a few auto regressions.

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Specifically, suppose that y half of time T plus tau is regressed on all of the ways in between a T

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plus 20.

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Note that this is just our usual auto regressive model.

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Next, let y had at time t be another regression again based on all the whys in between a T plus 20.

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This one is weird because it's like a backwards auto regression.

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We are using future values to predict a past value.

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So both Y had a time T plus tau and we had a time T or regressed on the same set of variables, the

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ones in between.

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Next, we have the big reveal via Tau Tau is just the correlation between why a time T plus tau minus

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Y hat at time T plus tau and why a time T minus Y hat at time T as defined on the previous slide.

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That is to say fire tau tau or the akef for like tau is the correlation between any noise after subtracting

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off the parts that could be predicted by the regressions.

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You may have to repeat this to yourself a few times, but it will make perfect sense.

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The partial autocorrelation function is just the correlation conditioned on the in between variables.

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Therefore, it's the correlation between whatever we can't explain using those in between variables.

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And of course that makes perfect sense for choosing the order p in an auto regressive model.

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If the pickoff is non-zero, that means there is a significant correlation that cannot be explained

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by the in between variables and therefore we should include this correlation in our final auto regressive

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model.

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So just to put some real numbers to this, suppose that we want to know if y one is helpful in predicting

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Y five.

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Obviously the same question applies to Y two and why six y three and Y seven and so forth.

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This is because we assume our signal is stationary and therefore any auto correlations only depend on

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the time difference.

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But let's stick to Y one in my five.

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The in between values are y two y three and Y for the pickoff will be large if there is a large correlation

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between Y one and Y five that cannot be predicted by Y to Y three and Y four.

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Therefore, we should use one in our auto regressive model for predicting Y five.

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That is to say, whenever we see large values in the pickoff, it indicates that these should be included

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in our auto regressive model.