imagine that when using the historical simulation approach for computing VaR, you want to update volatility using garch (1,1). you have 4 assets and therefore 4 time series.

then you link all 4 time series to 1 lambda value instead of 4 time series and 4 lambda values.

Whats wrong with having 1 lambda value per time series instead?

Also,

When is it a good time to have 4 lambda values if we are observing 4 time series?