ccgarch2-package: ccgarch2: a package for simulation and estimation of...
In ccgarch2: Conditional Correlation GARCH Models

Description Details Note References

This package provides functions for simulation and estimation of conditional correlation (CC-) GARCH models. The CC-GARCH model includes the Constant Conditional Correlation (CCC-), Dynamic Conditional Correlation (DCC-) and corrected Dynamic Conditional Correlation (cDCC-) GARCH models.

The package ccgarch2 provides functions for simulation and estimation of conditional correlation (CC-) GARCH models. The CC-GARCH model includes the Constant Conditional Correlation (CCC-), Dynamic Conditional Correlation (DCC-) and corrected Dynamic Conditional Correlation (cDCC-) GARCH models.

The CC-GARCH model is defined as follows. Let y_t be an (N times 1) vector of returns

y_t = mu + e_t

where D_{t} = diag(h_1t, …, h_Nt) is a diagonal matrix whose diagonal entries are the conditional variances of y_t, H_{t} is the conditional covariance matrix of y_t and R_{t} is the conditional correlation matrix of y_t as well as the conditional covariance matrix of the standardized residuals, z_t = D_t^{-1/2}e_t.

One of the main properties of the CC-GARCH model is to specify the diagonal entries of H_{t} (that is, h_it) and R_{t} separately. While it is common to assume that h_it is described by a variant of univariate GARCH models, a number of formulations for R_{t} have been proposed.

In ccgarch2, the GARCH part h_t = (h_1t, …, h_Nt)' may be modelled as a vector GARCH (1,1) specification such that

h_t = a0 + A*e_{t-1}^{(2)} + B*h_{t-1}

where A and B are (N times N) matrices, and e_{t-1}^{(2)} denotes the elementwise squared. If A and B are not restricted to diagonal, one may estimate extended types of GARCH models that allow for interactions in conditional variances through off-diagonal elements of the matrices (see Nakatani and Ter\"asvirta (2009), for instance). However, estimation of the extended models has difficulty in finding estimates of A and B that satisfy the stationarity conditions of the vector GARCH process. Therefore, the codes for estimating CC-GARCH models in ccgarch2 are restricted to dealing with A and B being diagonal while for simulating data from a CC-GARCH process one may set non-zero off-diagonal elements.

For the correlation part, the following three types of specifications are incorporated in ccgarch2.

Constant Conditional Correlation; CCC-GARCH (Bollerslev, 1990)
The simplest way of specifying the conditional correlation part is to set the conditional correlation constant over time, namely,

R_t = R

For estimation of a CCC-GARCH model estimateCCC is available while simulateCCC simulates the data from a CCC-GARCH process.

Although the CCC-GARCH model has an advantage in reducing the number of parameters to be estimated, the assumption of constant conditional correlation is regarded too restrictive.

Dynamic Conditional Correlation; DCC-GARCH (Engle, 2002)
To relax the assumption of constant conditional correlation, Engle (2002) proposed the dynamic conditional correlation (DCC-) GARCH model. The correlation part of the model is specified as follows.

where Q_t^{*} is a diagonal matrix whose entries consist of diagonal elements of Q_t.

For Q_t to be positive definite, it is assumed that Q is positive definite, alpha >= 0, beta >= 0 and alpha + beta < 1.

For estimation of a DCC-GARCH model estimateDCC is available while simulateDCC simulates the data from a DCC-GARCH process.

Engle (2002) intended to formulate Q as the unconditional second moment of z_t. As Aielli (2013) shows, however, this turns out not to be the case and interpretation of Q is not immediate. Furthermore, Aielli (2013) discusses that Q != E[z_t z_t] results in inconsistency of the estimators in the DCC-GARCH model in general.

corrected Dynamic Conditional Correlation; cDCC-GARCH (Aielli, 2013)
To gain consistency of the estimators in the DCC-GARCH model, Aielli (2013) suggested the following modification of Q_t in the DCC-GARCH model:

where S is a unit-diagonal matrix. With this formulation, S = E[z_t z_t] holds, so that the consistency of the estimators is regained (see, Aielli (2013) for detail).

For estimation of a cDCC-GARCH model estimateCDCC is available while simulateCDCC simulates the data from a cDCC-GARCH process.

The function estimateAAA, where AAA is replaced with one of CCC, DCC and CDCC, returns an S3 class object "ccc", "dcc" or "cdcc". The output coisists of the outcomes from the first and second stage optimization, original data and matrices of standardized residuals, conditional variances and conditional correlations. All of them are combined as components of a list object.

Corresponding S3 methods summary and plot are available for each class object. summary summarizes the model estimation with coefficient table including estimates and inferencial statistics, information criteria such as AIC and diagnostic statistics of the standardized residuals. plot depicts multiple time series plot in a single plot window. plot has an option argument item that determines which item to be plotted. User may choose one of "correlation", "volatility", "std.residuals" and "level".

It is time consuming (usually takes five to tem minutes) to implement the summary method for ‘ccc’, ‘dcc’, ‘cdcc’ class objects. This is because obtaining numerical derivatives, which are used for computing the standard errors of the parameter estimates, is computationally demanding.

Aielli, G.P. (2013), “Dynamic Conditional Correlation: On Properties and Estimation.” Journal of Business and Economic Statistics 31, 282–299.

Bollerslev, T. (1990), “Modeling the Coherence in Short-Run Nominal Exchange Rates: A Multivariate Generalized ARCH Approach,” Review of Economics and Statistics, 72, 498–505.

Engle, R.F. and K. Sheppard (2001), “Theoretical and Empirical Properties of Dynamic Conditional Correlation Multivariate GARCH.” Stern Finance Working Paper Series FIN-01-027 (Revised in Dec. 2001), New York University Stern School of Business.

Engle, R.F. (2002), “Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models.” Journal of Business and Economic Statistics 20, 339–350.

Nakatani, T. and T. Ter\"asvirta (2009), “Testing for Volatility Interactions in the Constant Conditional Correlation GARCH Model”, Econometrics Journal, 12, 147–163.

ccgarch2 documentation built on May 2, 2019, 5:56 p.m.