Forecasting extreme trajectories
Application
How to use it? You can either reproduce the results of the paper, or explore the semi-norm representation to predict the extreme evolutions of one of the many series available in the application.
Here is an example of use. Suppose we are interested in the Russell 2000 stock market index. First, you have to select the Financial data database, then the RUS2000 time series. Considering the 5% most extreme events, we see two large negative bubbles. Let’s try to predict the most recent one, whose 5% unconditional quantile is exceeded on 10/03/2020 according to the graph. By choosing this date in the application’s calendar, we thus define the end of the In-Sample.
By estimating various models, we find that the MAR(1,1) seems to be the most appropriate model with a causal coefficient of 0.8899 and a non-causal coefficient of 0.4154. The tail parameter is light (alpha = 1.8528) for a very low asymmetry. If you do not obtain these results, it is possible that you have converged on a local optimum and you must restart the estimation based on new initial parameters.
Now let’s move on to pattern recognition. As this is a negative bubble, select the « Flip » button and start the search for k0. The sequence is rather stable with three possible modalities for k. In the presence of a moderately extreme distribution, our simulations point us towards a rather large m. We therefore choose m = 10 and therefore k = 3. We also select h* = 10 as the prediction horizon.
As this is a MAR(1,1), one cannot obtain a degenerate probability on the future path of the series. But we can calculate crash probabilities and determine the most probable path. In the Crash probabilities tab, the calculation reveals a table of the probabilities of crashing before date h, surviving to date h and crashing at date h. As here k0 = 3, The first value calculated for this last column is therefore 0.992. As suggested above the table, we enter this value as the cutoff to determine the most probable path to use in predicting the trajectory. The Backtesting section allows us to verify that the procedure correctly identified 17/03 as the reversal date and that 10 periods later, on 2/06, the series should have risen to around -0.0959. The true Out-of-Sample value is -0.1077.
Abstract
For X(t) a two-sided alpha-stable moving average, this paper studies the conditional distribution of future paths given a piece of observed trajectory when the process is far from its central values. Under this framework, vectors of the form X(t)=(X(t-m),…,X(t),X(t+1),…,X(t+h))$, are multivariate alpha-stable and the dependence between the past and future components is encoded in their spectral measures. A new representation of stable random vectors on unit cylinders sets defined on an adequate semi-norm is proposed to describe the tail behaviour of vectors X(t) when only the first m+1 components are assumed to be observed and large in norm. Not all stable vectors admit such a representation and X(t) will have to be « anticipative enough » for X(t) to admit one.
The conditional distribution of future paths can then be explicitly derived using the regularly varying tails property of stable vectors and has a natural interpretation in terms of pattern identification. Through Monte Carlo simulations we develop procedures to forecast crash probabilities and crash dates and demonstrate their finite sample performances. As an empirical illustration, we estimate probabilities and reversal dates of El Niño and La Niña occurrences.