The basic idea of ??ARCH model is that under the previous information set, the occurrence of noise at a certain time obeys the normal distribution. The mean of this normal distribution is zero, and the variance is a quantity that changes with time (that is, conditional heteroskedasticity). And this time-varying variance is a linear combination of the squares of the past finite term noise values ??(that is, autoregressive). This forms an autoregressive conditional heteroskedasticity model.

Due to the need to use conditional variance, we do not use Engel's more rigorous and complex mathematical expressions here, but adopt the following expression so that we can grasp the essence of the model. See the following mathematical expression:

Yt = βXt+εt (1) Among them,

* Yt is the explained variable,

* Xt is the explanatory variable,

* εt is the error term.

If the square of the error term obeys the AR(q) process, that is, εt2 =aa1εt-12 + a2εt-22 + …… + aqεt-q2 + ηt t =1,2,3…… (2) Among them,

If etat is independently and identically distributed and satisfies E(etat) = 0, D(etat) = λ2, then the above model is said to be an autoregressive conditional heteroskedasticity model. Abbreviated as ARCH model. The process that the sequence εt obeys the ARCH of order q is called εt - ARCH(q). In order to ensure that εt2 is positive, it is required that a0 >0, ai ≥0 i=2,3,4….

The model composed of equations (1) and (2) above is called a regression-ARCH model. ARCH models usually model and analyze the random disturbance terms of the main model. In order to fully extract the information in the residual, the final model residual etat becomes a white noise sequence.

It can be seen from the above model that since the variance of the noise at the current moment is the regression of the square of the noise value of the finite term in the past, that is to say, the fluctuation of the noise has a certain degree of memory. Therefore, if at the previous moment If the variance of the noise becomes larger, the variance of the noise at this moment will often also become larger; if the variance of the noise became smaller at the previous moment, then the variance of the noise at this moment will also become smaller. Reflected in the futures market, that is, if the price fluctuation of the futures contract became larger in the previous stage, then the market price fluctuation will also tend to be larger at this moment, and vice versa. This is the characteristic of the ARCH model that describes the clustering of fluctuations, which also determines that its unconditional distribution is a peaked and fat-tailed distribution.