Learning report.

This text is based on the ‘Modelling Asymmetric Co-movement of Asset Returns’, authored by Kenneth F. Kroner and Victor K. Ng published on Review of Financial Studies, Vol.11, No.4. 1998.
The paper questioned the utilization of various time-varying co-variance models since these models have much too restrict formations in the pattern of how the stock performance in the history impacts the estimated, and thus forecasted, co-variance matrix. The paper examined four types of most widely adopted variations of GARCH model and exhibited how they could obtain very different results based on the same observations. This fact exhibited the substantial model risk when applying these GARCH models and it is naturally going to impact whatever application of the GARCH models, such as portfolio optimization where the forecasted co-variance matrix plays a very important role. Based on the finding, the author provided a general form of model which includes all four types of GARCH models. According to the report in the paper, the loosened constraint would make the estimation of the model more robust. An empirical test was implemented on the dynamic between the stock returns of the big size and small size companies to confirm the conclusion.

The four GARCH-variable type models include:
1. VECH
The VECH model has the following pre-defined form:

Perhaps the mostly mentioned edge of the VECH model is its simplicity which is virtually a ARMA(1,1) model for the error items. The VECH model estimates the variance the historical data with a geometrically falling weighting. The outstanding issues of VECH is highlighted by dimensional curse (too many parameters to be estimated) and its incapability to generate non-negative definite covariance matrix.
2. BEKK
The BEKK model has the following pre-defined format:

Clearly, the positive definite feature of the covariance matrix estimated and applied is promised by the quadratic form used in the model. Even the positive definite problem is solved by BEKK model, its application is largely restricted by the number of parameters imposed in the model.
3. F-ARCH
The F-ARCH model has the following pre-defined format:

The F-ARCH model brilliantly solves the problem of positive definitiveness and has less number of parameters than the BEKK model. The F-ARCH is driven by only one factor, contrary to N factors used in BEKK. The decrease of dimension in driven factor simplifies estimation while at the same time, impose more restrict constraint on the pre-defined pattern of covariance co-movement.
4. CCORR
The CCORR model has the following pre-defined format:

The CCORR model assumes the covariant parameter performs in a fixed ratio to the product of the standard deviation of the two related assets.

The four models briefly described above were applied to a weekly return sample from July 1962 to December 1988. Only two groups of corporate are considered, respectively the big size companies and small size companies. The paper adopted a two-step estimation strategy in which the residuals are firstly obtained through the calibration of the expectation equation. The conditional covariance matrix is thereby estimated with the maximum likelihood method with the calculated error generated in the first step. The estimations, and thereby forecast, from the four GARCH models are very much different from each other. The VECH and CCORR model were giving higher estimation for the volatility of the small sized companies than F-ARCH and BEKK. Interestingly, at the same time, VECH model and CCORR model were generating lower estimation of volatility than F-ARCH model and BEKK model on the large size companies. For the variance series generated from the four models, there was reported no big difference (correlation higher than 95%) in the large corporate. However, in the small corporate sample, the correlation of projected variance series from these models may vary a lot. The lowest correlation of the variance