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Let's say you have a dependent variable and many independent variables. What are the preferred metrics for sorting and selecting variables based on explanatory power? Let's say you are not concerned with correlation among your inputs.

My thoughts are below. Let me know if I'm missing anything obvious, or if there is a metric that you feel dominates the others.

  1. Spearman correlation controlling for sector
  2. Granger causality
  3. Johansen test for cointegration
  4. Grinold's Information Ratio
  5. Economic performance of the factor during a factor backtest (i.e. drawdown, sharpe, etc.)
  6. Volatility of the factor premium
  7. Persistency of the factor premium
  8. Monotonic relation test
  9. Significance test on spread return (Q5-Q1) vs. Mean Return
  10. R^2 of factor after building some kind of model

My preference would be correlation coefficient.

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    $\begingroup$ You should be concerned with correlation among your inputs or at least used technics such as ridge regression when estimating your model. $\endgroup$ – Zarbouzou Jun 30 '11 at 9:03
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I'd add:

  1. Variance reduction
  2. Fraction same sign / Hit rate

Additionally, you might look at the relationship between the Q5-Q1 spread itself and the dependent (i.e. are larger/smaller spreads associated with some feature of the dependent).

Turnover may also be an issue as slippage and friction come into consideration. Measures such as percent turnover in the Q5-Q1 portfolio, and correlation coefficient decay over lagged periods can prove insightful in selecting factors with higher stability and persistence.

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  • $\begingroup$ Hi I-CJW, can you elaborate a bit more on variance reduction? $\endgroup$ – Ram Ahluwalia Jul 20 '11 at 19:09
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If you're using R, you might try:

http://cran.r-project.org/web/packages/relaimpo/index.html

https://stats.stackexchange.com/questions/8918/is-there-a-way-to-optimize-regression-according-to-a-specific-criterion/8932#8932

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I don't think any of the current answers, answer the question, there is one metric that dominates all of them and that is simply the t-stat of the factor. This is well shown across academic literature as well.

The rule of the thumb here is to use a t-stat greater than 2.

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There are a few other techniques to consider in addition to what has already been suggested.

Principal components regression (PCR). This would consist of doing a PCA on the independent variables and then regress the dependent variable on the factors. You can always translate the coefficients back into the original basis.

Partial Least Squares (PLS). Very similar to PCR, but while the PCR PCA is about choosing coefficients to maximize the explaned variance of the independent variables, in PLS you choose coefficients so that the independent variables explain the most variance of the dependent variable.

Information Criterion. If you can express the choice as between a simpler model versus a more complex model, the optimal choice under this approach is to choose the one with the most favorable information criterion. Most common is AIC, which increases in log likelihood of the model and decreases in number of paramters. Of course, this requires making some assumptions about the errors of the model.

Posterior predictive checks. Simulate data from the model. Evaluate if it makes sense.

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