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OK, so the next step in our notebook, now that we know how the forecast function works, is to actually

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use the forecast function to see how our model behaves.

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We'll start by plotting the conditional volatility that our model has learned from some arbitrary date

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I chose beforehand.

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As you recall, I've stored this in a column called Arche one.

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Now, the reason I chose this date is because the conditional volatility on this day has a very high

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value compared to the others, as you can see.

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So what we want to observe is how does the forecast evolve over time starting from this value?

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OK, so just keep in your mind that this date is 2011, 08, 09.

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The next step is to create a forecast, well, sit index to true, although this isn't needed and will

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set start equal to August one, which includes the date of interest.

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The next step is to check the variance of our forecast for the dates surrounding our start date.

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This is just so we can observe once again that any date before the start date we passed in the forecast,

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we'll have not a number of values.

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OK, so we can see that any date before August one does not have a forecast.

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The next step as a sanity check is to get the shape of a single row of our variance forecast.

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So as you can see, we have five hundred values, which is what we set for the horizon.

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The next step is to get the index of the date we are interested in.

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The reason we want to do this is we would like to know what the date is, 500 steps ahead, since that

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will be the end of our forecast.

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OK, so the index of interest is four zero to.

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The next step is to check the index value at four zero two plus five hundred.

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Again, this represents the end of our forecast.

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OK, so the date at the end of our forecast is 12, 13, 08, 06.

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The next step is to take what we've just learned and store our variance forecast in our to data frame,

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so for the first index into the log function.

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Notice I've passed in 2011, 08, 10 up to 2013, 08, 06.

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For the second index, we're going to create a new column called Arche one forecast.

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On the right side, you can see that we're getting the variance forecasts for 2011, 08, 09.

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I've also converted this into an umpire, Ray, since we do not want this to be represented as a series

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or a data frame.

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If we did use a series or a data frame, it would not have the right indices to match up with the other

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two.

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So it's simpler to just convert it into an umpire.

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Furthermore, notice that we take the square root since we want the volatility.

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OK, so the next step is to plot our forecast, along with the predicted conditional volatility.

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Note that I've set the dates to surround at the beginning of our forecast, but to cut off around 2012,

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as you'll see, the forecast will converge much faster than the full five hundred steps.

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OK, so as expected, what we see is that the volatility forecast starts off at a very high value and

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then gets lower and lower until it converges.

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The next step is to get the index for another specially chosen date, which is 2011, 08 17.

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So the reason I chose this date is because this date happens to have an uncharacteristically low volatility.

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The goal of all this is to see for ourselves that no matter where the volatility starts from, they

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end up converging to the same value.

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OK, so the index of interest is four zero eight.

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The next step is to check the end date of our forecast, which is four zero eight plus five hundred.

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As you can see, it's 12, 08, 14.

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So the next step is to grab a forecast for the second date and stored in our data data frame.

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This code is pretty much the same as what we had before, except we have different dates and have stored

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this in a new column called Arche one for Carcillo.

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OK, so the next step is to plot both forecasts at the same time.

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So we can see that the result is what I've said you should expect, no matter what the starting volatility

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is, both forecasts end up converging to the same value.

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So the next step in this script will be to show that this value, which the forecasts converge to,

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is actually the theoretical, unconditional variance.

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So let's start by using the results objects from earlier and looking at the programs attribute.

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So we can see that this is a series that stores Mu, Omega and Alpha1.

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The next step is to use the formula we derived earlier for the unconditional variance.

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This is a mega divided by one minus Alpha one.

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OK, so we get about one point four.

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The next step is to check the final value of our first forecast, which started at a very high number,

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as you recall.

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As you can see, the final value is the same, which is about one point for.

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The next step is to check the final value of our second forecast, which started at a very low number.

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Again, you can see that the final value is still the same.

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About one point four.