Koutmos JBFA 1996 Multivariate EGARCH

[TomDoan]

There's nothing wrong with the model (though I'm not sure that cross product term in the z's is well-motivated). It's just their method of estimation is poorly described and a two-step estimator seems pointless. You should look at how Koutmos describes his model and estimator---the paper you're looking at has a somewhat different mean model and a very slightly different variance model, but is otherwise quite similar.

The model is quite doable with RATS. It's just if you want to apply it, you will need a better (more complete) technical explanation than was provided in the original paper.

[sultan[]]

Hm. Your observation was right that they generate z's by first implementing uEGARCH process. In fact, I have also found that this is how DCC GARCH model is implemented (by Engle and Kevin 2001). The z's are then transformed into correlation (in that case, covariance) matrix. By modelling this way, one can extract covariance matrix, which then need to be fed into conditional mean equation of stocks in order to prove/explain the hypothesis that stocks return are tied with market, through covariance. the reason to employ egarch process in this case is simply to capture asymmetric behavior of market shocks into stocks level through covariance. Therefore, the study is intra-market asymmetric behavior, where the Koutmos paper describes inter-market asymmetry (volatility spillover) through VAR(1) EGARCH.

That is, in the attached shot, it appears that equation 0 to 3 (mean and variance process) are already included in equation 4 to generate z's which are then jointly takes a EGARCH process (equation 5). This process produces conditional covariance which then need to be fed into conditional mean of stocks (equation 6). I would like to ask that if I follow these processes (particularly covariance into equation 6 as second step), whether the estimation is correct or makes sense?

I notice that in estimating DCC, the resulting conditional covariance matrix is three-dimensional (2*2*2345, that is, if we estimate 2 series and number of obs, 2345). In other words, it is pairwise covariance of each observation of the two series. However, conditional mean (equation 2) is univariate AR(1) process and the return series is two-dimensional. Since the covariance matrix is symmetric, we can extract only off-diagonal values of covariance matrix and stack them into two-dimensional series. In this way, we can obtain pairwise covariance of each observation (1 to 2345). Does this process sound ok?

Thanks.

[TomDoan]

Their conditional covariance (your (4) and (5), their (5) and (6)) doesn't really make much sense. They say that this is "based upon" EGARCH, but EGARCH takes the form it does because it's a log specification---the residuals are standardized because in a log additive model, they turn into multiplicative terms for the variance itself. The covariance isn't (and really can't be) modeled in log form since it could be negative, but what they're doing is taking a specification that makes sense in a log-additive model and applying it to an additive model. However, in an additive model, the z terms have the wrong scale (in fact, they're scale-free, unlike the lagged covariance term). With zero connection between the variances and the covariances and a rather poorly designed model for the covariances, the chances that this can actually work in practice, given that it has to produce a positive definite matrix at every single entry, is quite remote. That's the reason DCC gets used for time-varying correlations on large sets of series.

If you look at RePEc, there is just one citation to this paper, and even the authors don't cite it in future work. That's a rather strong sign that this was a dead end. RePEc citation statistics can sometimes be misleading ("Authors ABC did xxx, and we didn't" generates a citation, though not necessarily a positive one), but lack of citations on a paper more than ten years old is usually a good sign to look elsewhere.

[sultan[]]

Thanks TomDoan. Actually the theoretical base was appealing at me,but their modeling seems questionable to us. In fact, I was thinking whether they transformed Q_t process (of DCC model) as an E-GARCH form, however, you have confirmed that covariance won't take a log additive form since it could be negative. Therefore, I turned my attention to DCC model for covariance estimation, particularly E-GARCH ADCC in which the underlying conditional variances take E-GARCH forms and Q_t process takes GJR type (i guess, because the asymmetry is captured as dummy). I think this setup would be transparent (thus, useful) to obtain time-varying covariances since i look for asymmetry in both variances and covariances (see attached shot).

First, I estimate EGARCH ADCC on return series to get time-varying covariances (equation 1-5) and then I specify equation 1 (mean) as AR(1) process and in equation 2, I feed the time-varying covariance (using simply by regression or regression with arima errors, where arma lags are zero). The obtained residuals can again take the form of E-GARCH ADCC to produce covariances, which again can be used for regression (equation 2). This iterative process continues as many time as you want. How does this sound to you?

Looking forward to your reply.

[TomDoan]

That generally seems like a reasonable approach, though I'm not sure why (1) would be different from (1')---the lagged value doesn't depend upon first stage estimates. It probably would only take one full iteration; it's unlikely you would see much difference in recomputing the DCC estimates of the covariance.

[sultan[]]

Instead of AR(1) term in equation (1)`, I was thinking to estimate it as egarch in-mean (or simply estimate first egarch (1,1) to get conditional variances and then regress the conditional variances on return (r_m) by fitting simple regression/regression with ARIMA errors, where arma lags are zeros). The idea is that variances are proportional to returns. How does it sound to you?

And yea, it is true that covariances do not change very in each iteration afterwards.

[TomDoan]

There's no reason to do a two-step estimator for an "M" effect on rm---unlike the M effects on the others, it depends only upon information within the series itself. And yes, it probably makes more sense to try to model any serial correlation in that using an M effect than to just shove an AR term into it.

[sultan[]]

Thank you Tom for your suggestion all the way to now. For hypothesis testing, I need to check how negative (positive) return shocks (both individually and jointly) affect conditional covariances and therefore, risk premium. The effects of any shocks can be identified as the changes in average covariance over the sample. I was thinking to implement Impulse Response for that. Could you have any suggestion for that? Since the study is intra-market, market-level (firm-level) shocks are defined as the average of the absolute standardized innovations from the estimation.

Looking forward to your reply.

[TomDoan]

A variance impulse response for a model like that doesn't exist. First, the variances themselves are governed by an EGARCH which doesn't admit an impulse response function. And DCC similarly doesn't have any simple way to convert shocks into changes in the covariances because the recursion only produces a matrix which has to go through a non-linear transformation to create correlations (and then another one to create covariances from those). You would have to do simulations and the results would depend upon the initial conditions.

[sultan[]]

Hi Tom, thanks. But what do we mean by simulations and is there any method to do that?. Could you please elaborate this point in details. There is a generalized impulse response function by Koop et al. (1996) and this model is used as correlation impulse function by Sevi and Pen (2009,6) in which they use boostrapped simulations. Also, their underlying covariance model is gjrGARCH ADCC. Unfortunately, it does not exist somewhere. I attach their paper.

[TomDoan]

Hi Tom, thanks. But what do we mean by simulations and is there any method to do that?. Could you please elaborate this point in details. There is a generalized impulse response function by Koop et al. (1996) and this model is used as correlation impulse function by Sevi and Pen (2009,6) in which they use boostrapped simulations. Also, their underlying covariance model is gjrGARCH ADCC. Unfortunately, it does not exist somewhere. I attach their paper.

Isn't the whole point of that paper that there really is no "variance impulse response function" with the DCC model and that all you can really do is differential simulations based upon specific histories? Bootstrapping is a type of simulation---they chose to do that rather than random-number simulations because the standardized residuals were too fat-tailed to be represented as Normals.

The ARCH/GARCH and Volatility e-course has two chapters on simulation methods, one for simulating the model given the parameters (which is what is needed for this) and one for inference on the model's underlying parameters.

[sultan[]]

If simulating the dcc model works in this case, then how to introduce shocks (positive and/or negative) to the model? Actually, I have got a function in R (since i am novice in RATS) for dcc model simulation (though i am not sure whether it fits for this case?) in which it simulates the estimated model given some starting values (eg., residuals,sigma,standardized residuals). Can you have a look at this please https://rdrr.io/rforge/rgarch/man/dccsim-methods.html
Furthermore, there are several inputs/arguments/instructions (I am confused what they actually indicate even though their explanations are given).

In my case, the standardized residuals are scalars (and positive) both for market and stocks (attach how they're computed). Therefore, in case of negative shocks what I have to do is to change sign either of them (or both)?? There's an argument preZ (that allows the starting standardized residuals to be provided by the user; I was thinking to provide those scalars here as vector form??)

Looking forward to your reply.