Why replace Pearson's correlation with DCC GARCH? (non-technical) How you would try convincing a non-technical audience that applying DCC GARCH for correlation estimation is better than Pearson's correlation?
The task becomes even more challenging since, as seen in the below image, the GARCH-based correlation follows quite closely the Pearson's correlation calculated with a rolling window.

I am considering the following:


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*In the above graph, in the beginning of 2016 the 'GARCH' correlation spikes under zero while the Pearson's rolling window doesn't react equally. It's true that there was a huge diversion between the two timeseries at this point so I could argue that, in this example, the 'GARCH' one is much faster to react.

*The 'GARCH' model includes the necessary mathematical framework to react to extreme conditions (the implementation I am using assumes a t-distribution for the posterior probabilities)

*Pearson's correlation was established around 1900 while Engle's paper regarding DCC-GARCH was published around 100 years later. There has been significant research activity in the meantime so the underlying maths are bound to be more advanced.

 A: Well, abusing a lot of technical details, DCC GARCH is a "weighted correlation" in some sense ... while your rolling window correlation is also a "weighted correlation" where weights follow a specific pattern ... then significantly simplifying the  only difference is in the weighting scheme for modelling association=correlation ...
Obviously DCC GARCH employs far more complicated weighting scheme than a simple rolling window (e.g. all observations in DCC are taken into consideration and etc.), so the question is, do you really needs this extra complexity? If you pursue "diagnostics" of association between variables goals then probably rolling window correlation is sufficient ... if refer to the fundamental summary work (at least in the context of volatility modelling)

Andersen, T. G., Bollerslev, T., Christoffersen, P. F., & Diebold, F. X. (2006). Volatility and correlation forecasting. Handbook of economic forecasting, 1, 777-878.

it all starts with a rolling window weighting scheme and gets more complicated as we move further on. Well, again to non technical audience the difference is how you assign weights to measure your dependence, rolling window is a very rough weighting scheme, only certain period equally weighted observations are taken into consideration. For example (the most simplified one I guess), observations that happened long before have no impact on your association measure while in DCC GARCH they are taken into account, though receive less attention=less weight (maybe even significantly less weight) but still influence your final association estimate. Hope it helps.
A: I assume that first you want to find out which of the two methods is better and only then try convincing the audience. Otherwise you might (inadvertently) be trying to deceive the audience.
So how do you find out which of the two methods is better? You need a performance criterion. You may be interested in how well a method fits the data in sample and/or how accurately it forecasts out of sample. Once you choose the performance criterion, you would measure the performance of the two methods and see which one is better according to the chosen criterion.
In-sample fit: For example, you may know the true conditional correlations, e.g. if you have simulated the data. Then you can measure the goodness of fit (e.g. $R^2$) of the fitted correlations due to rolling-windows and DCC. Whichever yields a better fit (a higher $R^2$) is better. Then it should be easy to convince the audience by showing the fitted vs. true and the errors. Better fit and smaller errors should be a convincing argument that one method works better than another.
Out-of-sample forecasting: Another example is when you do not know the true conditional correlations. Then you can compare the forecasting performance of the two methods. You would compare the forecast errors of the rolling-window and the DCC forecasts and see which ones are smaller. Again, it should be easy to convince the audience that a method that forecasts better is a better method for forecasting.
How do your three points fit into this framework?


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*
It's true that there was a huge diversion between the two timeseries at this point so I could argue that, in this example, the 'GARCH' one is much faster to react. 

This aligns well with my propositions. If you know what the true correlation was, you can measure the goodness of fit of the competing methods and argue for the one that has the better fit.  

*If you can demonstrate that 

The 'GARCH' model includes the necessary mathematical framework to react to extreme conditions (the implementation I am using assumes a t-distribution for the posterior probabilities)

analytically or by simulations and, morever, show that the rolling-window method does not have these benefits, then this is in line with my propositions. If not, it might not be very convincing.

*I find this more of an emotional rather than a rational argument. I suppose you could construct examples where one method beats another and vice versa, i.e. DCC is not always and everywhere superior to the rolling-window method. The interesting question is, which one tends to do better in practice. A more convincing argument could be to collect sufficiently many real-world out-of-sample forecasting examples that favour DCC over the rolling-window method.

