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

Time:2022-5-16

Original link:http://tecdat.cn/?p=25007 

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 whenR language simulates arch process model to analyze the stationarity and volatility of time seriesWhat happens close to 1.  

  • IfR language simulates arch process model to analyze the stationarity and volatility of time seriesThe process is smooth,
  • IfR language simulates arch process model to analyze the stationarity and volatility of time seriesThe process is random walk
  • IfR language simulates arch process model to analyze the stationarity and volatility of time seriesThis process will fluctuate greatly

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

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

AsR language simulates arch process model to analyze the stationarity and volatility of time seriesAnd the process will pass through infinite times_ x_ Axis

We carefully study the properties of arch (1) process, especially whenR language simulates arch process model to analyze the stationarity and volatility of time series, the results we get may be puzzling.

Consider some arch (1) processesR language simulates arch process model to analyze the stationarity and volatility of time series, with Gaussian noise, i.e

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

among

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

R language simulates arch process model to analyze the stationarity and volatility of time seriesIs an IID sequenceR language simulates arch process model to analyze the stationarity and volatility of time seriesVariable. HereR language simulates arch process model to analyze the stationarity and volatility of time seriesAndR language simulates arch process model to analyze the stationarity and volatility of time seriesMust be positive.

Review # due toR language simulates arch process model to analyze the stationarity and volatility of time series  R language simulates arch process model to analyze the stationarity and volatility of time series . therefore

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

 R language simulates arch process model to analyze the stationarity and volatility of time seriesSo the variance exists, and only ifR language simulates arch process model to analyze the stationarity and volatility of time series, in this case

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

In addition, ifR language simulates arch process model to analyze the stationarity and volatility of time series, you can get the fourth moment,

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

R language simulates arch process model to analyze the stationarity and volatility of time seriesNow, if we go back to the attribute obtained when studying variance, ifR language simulates arch process model to analyze the stationarity and volatility of time series, orR language simulates arch process model to analyze the stationarity and volatility of time series ?

If we look at the simulation, we can generate an arch (1) process, for exampleR language simulates arch process model to analyze the stationarity and volatility of time series

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

> plot

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

In order to understand what happened, we should remember that our good thing is,R language simulates arch process model to analyze the stationarity and volatility of time seriesMust beR language simulates arch process model to analyze the stationarity and volatility of time seriesCan be calculated betweenR language simulates arch process model to analyze the stationarity and volatility of time seriesThe second moment. However, there may be a stationary process with infinite variation.

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

iteration

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

Iterating over and over again

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

among

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

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

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

So, ifR language simulates arch process model to analyze the stationarity and volatility of time series,  R language simulates arch process model to analyze the stationarity and volatility of time seriesConvergence. here,

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

It can also be written as

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

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

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

Therefore, ifR language simulates arch process model to analyze the stationarity and volatility of time series, thenR language simulates arch process model to analyze the stationarity and volatility of time seriesThere will be restrictions whenR language simulates arch process model to analyze the stationarity and volatility of time seriesTake infinity.

The above conditions can be written as

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

This is calledLyapunovCoefficient.

equation

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

yesR language simulates arch process model to analyze the stationarity and volatility of time seriesOne condition

In this caseR language simulates arch process model to analyze the stationarity and volatility of time series, the value of this upper bound is 3.56.

> 1/exp(mean(log(rnorm(1e7)^2)))

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

In this case(R language simulates arch process model to analyze the stationarity and volatility of time series), the variance may be infinite, but the sequence is stationary. On the other hand, ifR language simulates arch process model to analyze the stationarity and volatility of time series, thenR language simulates arch process model to analyze the stationarity and volatility of time seriesIt’s almost certain to go to infinity becauseR language simulates arch process model to analyze the stationarity and volatility of time seriesTowards infinity.

But in order to observe this difference, we need a lot of observation. For exampleR language simulates arch process model to analyze the stationarity and volatility of time series

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

AndR language simulates arch process model to analyze the stationarity and volatility of time series,

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

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 tailR language simulates arch process model to analyze the stationarity and volatility of time seriesof

In fact, if we consider the Hill plot of the above series, in the positiveR language simulates arch process model to analyze the stationarity and volatility of time seriesTail of

> hil

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

Or negativeR language simulates arch process model to analyze the stationarity and volatility of time seriesTail of

-epsilon

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

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 hereR language simulates arch process model to analyze the stationarity and volatility of time seriesNot weakly stationary (atR language simulates arch process model to analyze the stationarity and volatility of time seriesIn 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.


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

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