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 series are only 0.365, between F-ARCH model and CCORR model.
The author utilized ‘news impact surface’ implied by the four models to illustrate how the restrict format of pre-defined functional form would result in different estimation of the co-variance matrix, especially for the small companies. As of the observed stats from the models as well as the news impact surface, the author proposed thorough and careful selection of GARCH variant models to make the feature of the covariance co-movement of the data set as close as possible to the restriction imposed by the model.
In the paper, the author defined ‘generalized residual’ in order to measure the distance of the errors of the model to the news impact surface. The generalized residual should not be related to time t should the model under examine was correct. The author used the framework of robust conditional moment test from Wooldridge to finalize their testing process. The paper reported that even though Ljung-Box test were not rejecting any one of the four models, all of them were firmly rejected by the robust conditional moment test. The incapability of incorporation of asymmetric shock of the four models was clearly exhibited.
As per the above discussion, a general form with loosened pre-defined format of the model was necessary. The author gave in his paper a general form of the model which incorporates all of the four models tested. The format of the model is as,

The generalized dynamic covariance model is a composite form of CCORR model and BEKK model. Note that there were asymmetric extensions for all of the GARCH variant models discussed as well as for the generalized dynamic covariance model. The test revealed that generalized dynamic covariance model cannot be degenerated to any of the four models which illustrate the importance of loosen constraint. The significance of the asymmetric parameter in the model confirmed the well observed asymmetric impact on the stock return on past events. At last, the generalized dynamic covariance model passed the robust conditional moment test.
The paper was very enlightening in how should people compare, test and utilize econometric models. All of the parametric type models has pre-defined format that may or may not fit the observed data. Careful model selection is required for a successful modelling and people should check the validity of the models from various aspect rather than one certain test since the statistic test may not capture all of the perspective the model.

Reference:
[1] Kroner K E, Ng V K. Modeling asymmetric comovements of asset returns[J]. Review of Financial Studies, 1998, 11(4): 817-844.
[2] Wooldridge J M. A unified approach to robust, regression-based specification tests[J]. Econometric Theory, 1990, 6(01): 17-43.