Just got an idea. I guess that the allocation % in your factors are fixed. On way is to change this allocation over time with some filtering techniques. This is what they do in hedge fund replications. See Thierry Roncalli An Alternative Approach to Alternative Beta or Hedge Fund Replication and Alternative Beta. But good luck with the math.
I'm not sure I'm understanding exactly what you mean. When you mention the 'allocation % in your factors', are you speaking of the entries of the cointegration vector, which act as the weights of each underlying stock in the basket? If so, as of now, they would indeed be fixed. If they were allowed to vary, isn't the 'rolling window' one approach? Are you speaking of other methods for calculating the entries of a cointegration vector that is allowed to vary with time? The dialogue is appreciated, by the way.
No, I was not refering about the cointegration vector but about the weights. So I am talking about the % of each stock you use to "replicate" the ETF. The stocks that "explain" the movement of the ETF vary over time so the idea would be to change the weight of each stock over time so that the sum of themmore closely match the ETF. It should therefore mean revert better. It would indeed imply that you have some time varying weights for which you need to change your portfolio with the assiciated transaction costs and also the cointegration vector could also be a rolling window. The paper I was refering to is different from what you do for sure and does not use cointegration. It uses the Kalman filter to select the right allocations in each factor (stocks, gold, oil, bonds, ... for them) (stocks for you) so that they can replicate a hedge fund indice (ETF for you) with them. The goal of the time varying weights is to catch the time varying influence of each factor. So for hedge funds it means allocating more to stocks in a bull market and more to bonds in a bear market for example. You got to understand that the trends you see in you spreads is created that exactly that, the time varying effect of each stock depending on the conditions. So it would make sense to allow the weights in the stocks to change over time. And the Kalman filter is one technique. The are many other replicators that use simple linear regression techniques that are more easy to understand. See Andrew Lo for example.