eeng


Total Posts: 23 
Joined: Dec 2014 


I am trying to approximate the returns of asset A by means of a linear combination of other assets A'=a*B0+b*B1+c*B2....
I have this quite figured out but I'm not sure what a good metric for goodness of fit would be, so far I am only considering relative error (e=(rArA')/rA), and I'm concerned with distortions when rA is close to 0.
What would a better metric could be? Ideally it would penalize sign errors more than absolue value errors (ie, it is worse that rA' is +ve when rA is ve). 
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nikol


Total Posts: 628 
Joined: Jun 2005 


it is difficult to understand specs of your model.
what is rA, rA' ? a, b  are model parameters? A and B_i  are assets? What is the meaning of ' (accent sign) ? 



ronin


Total Posts: 427 
Joined: May 2006 


This is just linear regression.
There is a decent coverage of linear regression in a general setting in The Elements of Statistical Learning by Hastie et al, or just google "lasso regression", "ridge regression" etc.
But I think you will need to modify your approach. You seem to want to regress prices, and that is a bit of a dead end. You are better off regressing returns or log returns, especially if you are worried about small price levels.
I don't know why you want an asymetric penalty function. It is not very difficult to do, but I don't think that will take you in the right direction.

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eeng


Total Posts: 23 
Joined: Dec 2014 


I hadn't noticed that the thread got this traction. Let me try to explain better: We have an asset A whose log returns we want to approximate by a linear combination of assets forming the synthetic asset A’ in this way
Coefficients may or may not be computed via linear regression or any other regression type, but my question is related to a good metric to measure the fitness of approximation.
At the moment I'm considering RMSE but I'd like something more fit that penalizes situations A going up +3% and synthetic asset going down say 1% more, such that when combined with a predictor synthetic asset may adequately replicate the PnL distribution of asset A. 
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jr


Total Posts: 3 
Joined: Apr 2017 


If you don’t want to fit the model wrt this loss, i.e. estimate the coefficients by its minimizing, can’t you just design whatever metric you want via indicator function on residual sign?
For instance similarly to quantile regression asymmetric loss. 


 