Next, we will sort it out according to the development process of GARCH family model.
Research object: the time series of volatility, that is, to study the relationship between current volatility and previous volatility. Often used in time series with violent fluctuations, the simplest relationship is a linear equation, namely:
ARCH and GARCH have the same meaning. The first equation is called the mean equation, which represents the relationship between the mean and the past mean. The second equation is the conditional variance equation, called ARCH equation, which represents the relationship between variance and past variance. GARCH can reduce the parameters to be estimated, that is, AR(q) is equivalent to GARCH( 1, 1), and GARCH( 1, 1) can generally meet the requirements in practical applications.
The two models require (1)xt to be static; (2)ut has arch effect.
(1) Only when xt is stable can we infer the future trend from the current state. If it is unstable, what is calculated according to the current data will be meaningless in the future, and the correlation between two variables may not be established (pseudo-regression problem). Stationary expansion can tell a lot (how to deeply understand stationarity in time series analysis? -Zhihu), let's talk about the mathematical derivation and software realization of stationarity.
The second equation requires not only stationarity, but also a coefficient greater than 0 (to ensure the nonnegativity of variance).
(2) There are two methods to test the ARCH effect: Ljung-Box Q test and arch-LM test.
The so-called arch effect is heteroscedasticity of residuals, which means that the square term (ut^2) of residuals has autocorrelation, the Q test directly looks at the autocorrelation coefficient (p=0 has arch effect), and the LM test looks at whether the coefficients of variance equation are all zero (p=0 has ARCH effect).
In Eviews, after xt autoregression, Qstatistic and sequence correlation LM test; Can be found in the remaining inspection view; In stata, the Q test should first calculate the residual sequence, and the command is: predict R, residual, and then do the autocorrelation Q test: corrgram r^2, and the LM test is to input the command: establish arch LM, lags (P) after completing the autoregression of xt.
(3) Some literatures will test the autocorrelation of residuals to see if the mean equation eliminates the autocorrelation of xt.
Software implementation of ARCH and GARCH
stata: (GARCH( 1， 1)): arch xt，arch( 1) garch( 1)
Personally, I prefer to use stata for ARCH and GARCH.
(1)TGARCH is called threshold ARCH model, which means that good news and bad news have different effects on conditional variance.
EGARCH, GJR-GARCH and APARCH are also GARCH derivative models considering leverage effect.
(2)ABSGARCH is called absolute ARCH model, which changes ut^2 into the absolute value of ut and reduces the amplitude of ut.
(3)I arch is called GARCH model with infinite variance, which combines the two parameters of GARCH into one and simplifies the calculation.
(4)GARCH-M is called mean GARCH model, and a variance variable is added to the mean equation, mainly because the greater the risk, the greater the return on investment.
……
Univariate GARCH is used to analyze the fluctuation clustering characteristics of sequences, and multivariate GARCH is used to analyze whether and how the fluctuations between different sequences are related.
The so-called pluralism. That is, the original sequence is expanded into a matrix including multiple sequences, so the variance sequence is also expanded into a covariance matrix (Ht). How to calculate the parameters of matrix equation is not a problem that we beginners can solve at present, but we are more concerned about how to use it.
According to the multivariate GARCH model, the time sequence is CCC( 1990), BEKK( 1995) and DCC(200 1). The estimation of DCC includes two steps:
In the results of DCC, the coefficient A+β < 1 shows that the model is stable, that is, the dynamic correlation is effective. A indicates the influence degree of residual on the variance correlation coefficient of different series, which is the influence degree of new information in economic language on the correlation of market fluctuation; β indicates the influence degree of the correlation of previous market volatility on the correlation of current market volatility, that is, the persistence of the correlation of market volatility.
The other is to look at the conditional dynamic correlation coefficient diagram and make an explanation in line with economic theory in combination with the actual situation.
After estimating the parameters of DCC model, hypothesis testing is needed to test whether there is significant difference between dynamic correlation coefficient and constant correlation coefficient. (stata's command is: test _ b [adjustment: lambda1] = _ b [adjustment: lambda 2]) You can refer to the paper: Calculation method of DCC-MVGARCH model and its application in financial markets.
GARCH is often used in combination with Copula function. Copula-GARCH and DCC-GARCH have similar functions, and both look at the correlation between different markets.
1.？ There are several DCC implementations of R package, which need to be studied.
2.Matlab can use Kevin Sheppard's Matlab? MFE toolbox;
3.Eviews is a menu-based operation, which can realize single-sequence garch. I don't know if I can do multiple arches; .
4. Stata