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I am looking to get a better understanding of an output from a trading strategy. Basically I have a daily equity curve lets call it $Y_t$. I have defined a bunch of independent variables $X_{it}$ that I think can explain the movement in the daily PnL. The independent variables are not used in trading signal generation directly.

1) How can I go about deducing which independent variables explain my $Y_t$ , assuming that the relationship could be non-linear? I can start out with PCA but from what I understand it assumes a linear relationship.

2) Using a reduced independent variable set $X_{it}$ from 1) how do I go about defining a non-linear relationship with $Y_t$ . Neural Nets maybe?

I understand that doing 1) and 2) might result in overfitting but I just want to understand the equity curve better.

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The problem with Principle component analysis is that you moght lose information about what really drives your returns. Also as a rule of thumbs, before assuming that you have a non linear relationship make sure that a linear methods realy results in underfitting your problem. If you want to reduce the number of variables you could use shrinkage regression for example. –  Zarbouzou Jul 5 '12 at 8:22
    
I agree with @Zarbouzou, is there any particular reason you are so focused on non-linear relationships? What's wrong with PCA? –  Tal Fishman Jul 5 '12 at 17:04
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This is really just a generic attribution problem (see papers.ssrn.com/sol3/papers.cfm?abstract_id=1565134 for guidance on how to do that, but there's a large literature on other techniques). The biggest problem is that your holdings will not necessarily be constant over time. So in some periods you may be 100% long pork bellies and others you're 100% short the Sri Lankan rupee. The approach by Meucci I mention above can help with point in time attribution. –  John Jul 5 '12 at 21:44
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1 Answer

This is an pretty general question

You are essentially asking how to estimate the regression function

$$Y[t]-Y[t-1] = m(X_i [t], ..., X_p[t]) + \epsilon[t]$$

without any additional structure. Here is a basic list of questions to consider. Keep in mind that the more you can guide these procedures with domain knowledge the better your results will likely be.

  1. A couple of your questions suggest that you suspect many of your $X_i$ are irrelevant to $Y$. You suggest PCA. I think PCA is most useful when the $X_i$ are extremely correlated. If your $X_i$ are not correlated then selection techniques such as LASSO and stepwise regression may provide more insight.

  2. There are many methods for fitting non-linear forms for $m( )$. Before you launch into these it is useful if you can further restrict the function space. Remember, the set of all non-linear functions in $p$ variables can be vast and many of these functions will fit the data despite being nonsense. Common restrictions include additivity and smoothness.

  3. Some suggestions for fitting $m( )$ include: neural nets, regression trees, boosting, random forests, support vector regression, and multivariate adaptive regression splines. Some of these methods can automatically select variables as part of the fitting process. This avoids the problem with step (1) you point out, namely that those methods are linear and ignore potentially important non-linear relationships.

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