In this Eviews session we briefly investigate the dynamic, time series properties of the correlations between pairs of

monthly international stock returns, for a 1975:01 – 2016:12 sample, a total of 504 observations per series. Because so

far EViews has not implemented DCC models (but in Excel we have seen how to implement them), we simply focus on

multivariate GARCH models. The number of series is n = 5.

A simple kernel density-based analysis reveals that all the returns series display critical deviations (of the leptokurtic

type) from a Gaussian benchmark, and in particular fat tails – that may be induced by the presence of volatility

clustering.

With only a minor exception for Singapore (that may require some AR(1) modelling), there is evidence or little or no

structure (i.e., serial correlation) in the raw return data. Below, I just copy the estimated ACFs for three of the five

countries, but these all tend to be similar. Therefore in what follows, we shall disregard the conditional mean function

that will be set to a constant (note: in the case of monthly data, it is implausible that the monthly mean be zero).

Also cross-serial correlations (i.e., correlations that indicate whether lags of some variables predict the future of

another variable) tend to be small, not significant, and in fact much less than 5% of them is significant. This further

corroborates our decision to disregard the conditional mean function that will be set to a constant.

These are just standard, instantaneous correlation coefficients

However, the correlogram for squared returns of the majority of the stock returns series gives evidence of significant

serial correlation in squares, which is normally taken as indication of ARCH-type effects.

France Germany Singapore

Moreover, there is some evidence that for some pairs of countries, past products of stock returns (e.g., France x Germany

2 months ago) forecasts future products of stock returns. This means that when in the past stock returns have been

simultaneously large (small) and with the same (different) sign, these are expected to carry a large (small) magnitude and

the same (different) sign also in subsequent periods, which is equivalent to say that covariance is persistent.

Therefore we have sufficient justification to try and estimate conditional covariance models, to forecast correlations.

In EViews a multivariate GARCH-type (including BEKK) model is built by collecting the series of interest in a SYSTEM and

applying “ARCH-Conditional Heteroskedasticity” methods. The C(1)-C(5) coefficients in the SYSTEM panel just

correspond to the need to estimate one constant for each of the national stock market return series. Three types of

multivariate models are available.

In this case, even a simple diagonal VECH(1,1) model implies the estimation of 50 parameters! Because the saturation

ratio remains 50.4 (2,520/50), most of them are precisely estimated and EViews is rather fast (4-5 seconds on my

laptop). However, interpreting the meaning of the coefficients is not a small task.

Possibly, the wisest thing to do is to look at the transformed coefficients as they get to be arranged inside the matrices

M, A, B (in class M has been called C). It is then fast in EViews to derive the times series of filtered volatilities from the

model.

These are the recursive (filtered) forecasts of correlations for pairs of markets obtained by the VECH(1,1) model. Many

pairs of markets are characterized by a substantial increase of correlations that takes place starting in the late 1990s.

Average correlations between developed stock markets have come to exceed 0.7 for all pairs by the end of the sample,

that illustrates a fading diversification power; however, pairwise correlations with Singapore remain around 0.5.

One can also estimate a BEKK(1,1), which in this case implies some parsimony – only 30 parameters because of the

special, triangular based sandwich form. As far as one can see, the model implies roughly identical filtered volatilities

over time.

Here are the forecasts of the correlations during our sample derived from a BEKK(1,1). The general comments

concerning the time variation of the predicted correlations are roughly the same that we have reported above.

One wonder whether selecting VECH(1,1) or BEKK(1,1) can make any difference. Let’s see: try to play spot the

difference? Occasionally a few differences are visible…

Gaussian BEKK (1,1) Gaussian VECH (1,1)

Let’s examine a few close-up pictures… Some small differences remain visible, but whether or not one model

outperforms the other, remains a testable empirical issue – i.e., as always we would like to see whether and which

model outperforms the other in terms of forecasting future, realized correlations.

Gaussian BEKK (1,1) Gaussian VECH (1,1)

It is also possible to capture asymmetries in correlations, i.e.. the fact that potentially past negative Ri,t-kRj,t-k move

future, predicted covariance more/less than positive past Ri,t-kRj,t-k do. If you work in Eviews, you will note that

estimation get very slow in a BEKK model with asymmetries. In this case, the estimated parameters become 35.

It is also possible to estimate both more complex models and to specify multivariate distributions for the shocks that

are not simply multivariate Gaussian, for instance some multivariate t-Student in which, however, there is only one

degree of freedom parameter that is common to all equity markets (here it is not reported, but it came up to be 6.62

with a standard error of 1.03 – so these data do need very thick tails even after accommodating a complex BEKK(2,2,2).

Finally, E-Views also carries some relative of DCC, called CCC – constant conditional correlation model. What is it? It is a

DCC model in which the matrix “Gamma” is constant and set to the average, historical sample correlation matrix. This

model implies constant correlations and this is highly counter-factual. The estimated volatilities are now a bit different!

Constant correlations!