# R language simulates arch process model to analyze the stationarity and volatility of time series

Time：2022-5-16

In the process of the development of things, it often shows complex fluctuations, real-time and slow fluctuations, and volatility clustering often occurs, which is often encountered in risk research. Engle proposed the autoregressive conditional heteroskedasticity model arch (autoregressive conditional heteroskedasticity model) to describe variance fluctuation in 1982. It developed from bollerslev (T., 1986) to generalized autoregressive conditional heteroscedasticity GARCH (generalized arch), and later developed into many special forms.

In the context of AR (1) process, we spent some time explaining whenWhat happens close to 1.

• IfThe process is smooth,
• IfThe process is random walk
• IfThis process will fluctuate greatly

Similarly, random walk is a very interesting process with puzzling characteristics. For example,

AsAnd the process will pass through infinite times_ x_ Axis

We carefully study the properties of arch (1) process, especially when, the results we get may be puzzling.

Consider some arch (1) processes, with Gaussian noise, i.e

among

Is an IID sequenceVariable. HereAndMust be positive.

Review # due to   . therefore

So the variance exists, and only if, in this case

In addition, if, you can get the fourth moment,

Now, if we go back to the attribute obtained when studying variance, if, or ?

If we look at the simulation, we can generate an arch (1) process, for example

``````> ea=rnorm
> eson=rnorm
> sga2=rep
> for(t in 2:n){

> plot``````

In order to understand what happened, we should remember that our good thing is,Must beCan be calculated betweenThe second moment. However, there may be a stationary process with infinite variation.

iteration

Iterating over and over again

among

Here, we have a sum of positive terms, and we can use the so-calledCauchy rule: definition

So, if，  Convergence. here,

It can also be written as

And according to the law of large numbers, because we have a sum of independent and identically distributed terms,

Therefore, if, thenThere will be restrictions whenTake infinity.

The above conditions can be written as

This is calledLyapunovCoefficient.

equation

yesOne condition

In this case, the value of this upper bound is 3.56.

``> 1/exp(mean(log(rnorm(1e7)^2)))``

In this case（）, the variance may be infinite, but the sequence is stationary. On the other hand, if, thenIt’s almost certain to go to infinity becauseTowards infinity.

But in order to observe this difference, we need a lot of observation. For example

And,

We can easily see the difference. I’m not saying it’s easy to see that the above distribution has infinite variance, but it’s still so. In fact, if we consider Hill’s picture in the above series, it’s on the front tailof

In fact, if we consider the Hill plot of the above series, in the positiveTail of

``> hil``

Or negativeTail of

``-epsilon``

We can see that the tail index (strictly speaking) is less than 2 (which means that the second-order moment does not exist).

Why is it puzzling? Maybe it’s because of hereNot weakly stationary (atIn a sense), but strong and stable. This is not the usual weak and strong relationship. This may be why we call it strict stationarity instead of strong stationarity.

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