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So with this library, obtaining the forecast is a bit complicated, there are three arguments we're

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going to consider in the forecast function.

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The first argument is simple, which is just the horizon.

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That is how many steps would we like to forecast?

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The second argument is called the index, which will be the main point of confusion.

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The third argument is start, which seems like it will be the start date of your forecast.

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However, as you'll see, this is not quite true.

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OK, so let's talk about this, a reenacts argument, in fact, the main reason we have to mention this

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is because of a new change in the art library.

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So if you do not set this argument explicitly, you're going to get a warning which says the default

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Tafari index is true after September twenty twenty one.

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This will change defaults.

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Since we don't want to get this warning, we're going to set it explicitly.

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Of course, in order to set this argument, we have to actually know what it means.

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So what does reenacts index do according to the documentation it states, whether to index the forecasts

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to have the same dimension as the series being forecast?

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Now, to me, this sounds very cryptic, so let's try to explain it in a simpler way.

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Basically, when you think of a forecast, you think of it like in stats models.

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You have your training time series and then you make a forecast which extends beyond the end of that

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Time series.

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Not so with this library.

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With this library, it's possible to make a forecast starting from any point within the Time series

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as well.

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For example, if your Time series goes from January twenty twenty one up to January twenty twenty two,

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it's possible to make a forecast starting at March 1st.

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Twenty twenty one.

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And note that this is not an example.

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One step prediction as you might expect, it's a legitimate multi-step forecast.

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So that's one thing you may not have expected.

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Of course, the latest forecast you can make corresponds to the final date in the training time series.

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OK, so here's another strange fact about how we can forecast with this library, not only can we start

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the forecast from any date within the time series, we can actually forecast from every date within

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the Time series at the same time.

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So if your data goes from twenty twenty one up to twenty, twenty two, you can potentially create a

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forecast for every day of the whole year.

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That brings us back to this reenacts index argument, basically reenacts equals true means you want

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the resulting data frame to have the same length as your training series.

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So if your training series went from twenty twenty one up to twenty twenty two, your forecast data

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frame will have one row for every day of the year.

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On the other hand, if you set index to false.

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This means that the resulting data frame will only contain the dates for which you wanted to make a

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forecast.

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Now this is a bit abstract, so it helps to simply look at examples.

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So suppose we said re indexed to true and we set the forecast horizon to five and we set the start to

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March six.

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Also, suppose that our data set only goes from March one up to March 10.

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In this case, the forecast data frame will have a row for every day from March one up to March 10,

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the entirety of the Input Time series, but only March six up to March 10 will be filled with actual

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numbers for the forecast.

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The other rows will be set to Enan.

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Now, let's consider what happens if you set reenacts to false in this case, you will only get the

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rose from March six up to March 10 and all of them will be filled.

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So as you can see in this case, the forecast goes from left to right.

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So suppose that we wanted to make a forecast for March six.

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The first column in this case contains the forecast for March seven.

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The second column contains the forecast for March 8th and so on.

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Also note the headings which are named as one H2 and so forth.

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Now, we've pretty much already explained the first argument, which is start essentially this argument

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controls the first date from which we want to generate a forecast.

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So this date and every date after it, we'll have a forecast as per the previous slide.

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The only thing left to mention is that if you do not pass in this argument, there will only be a single

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forecast for the final date of your training series.

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So if your data goes from March one to March 10, the forecasts will be assumed to start on March 10,

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which means that the first the actual forecast data point is a forecast for March 11.

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Now, again, as mentioned earlier, this so-called forecast is not just a single number when we call

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the forecast function, this actually returns an object of type Arche model forecast.

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From this, we can access certain attributes like the mean of variance and so forth.

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So when you call variance, this will return a data frame of variance forecasts.

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One point of confusion is that there are two attributes for variance.

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One is called residual variance and the other is simply called variance.

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These will only be different if you specify a mean model that can change like an error process.

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So for us, it doesn't matter which one we use since they will be the same.

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OK, so the final topic we have to discuss in this lecture is the difference between in sample and out

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of sample predictions.

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As you recall, we've seen that we can kind of do an end sample forecast, but this is not a true in

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sample prediction.

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In fact, the sample prediction is obtained by accessing that attribute, conditional volatility through

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the arch model result object.

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As you recall, this is the objects return by calling fit.

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Also recall that in finance we normally think of volatility as the square root of variance.

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Thus this is on a different scale than the actual forecast, which has variance.

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So when we do the forecast, we want to take the square root such that the sample predictions and the

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forecasts can be on the same scale.

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OK, so let's now summarize what we've learned about the code since that was probably more complex than

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you expected, as we saw, there are three main steps.

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The first step is to construct our model using the high level function arc model.

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This takes in several arguments, such as your training time series, the type of model, the model

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orders and the error distribution.

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The second step is to fit the model.

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After fitting the model, we can also call the summary function to check the result.

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The third step is to forecast.

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We learned about three arguments for the forecast function, including Horizon re index and start the

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most confusing of these were we index start, which allow us to quote unquote forecast starting not

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only from the end of the Time series, but from any point within the Time series.

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We also learn that what we are actually forecasting is not the Time series itself, but its variance.

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Furthermore, we learn that the sample predictions are the square root of the variance.

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Thus we should also take the square root of the forecast to put them both on the same scale.