# How to improve the consistency of explained variance statistics in a linear equity model?

I have an intraday equity returns linear model that, overall, shows good values in terms of $R^2$, p-value and other explained variance statistics. Around 70% of the stocks show consistent R-squared (in the range of 5% explained) and significance parameters, but the remaining 30% have degraded $R^2$ and p-values.

Would it help to include beta, liquidity and sector returns in the model to have a more consistent $R^2$ across all stocks?

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Since you mention beta, I assume you're familiar with the capital asset pricing model (CAPM). The concept is that an asset's expected returns are linearly correlated with the market's returns. Of course, there are other ways "normalize" returns, as you put it. We can extend CAPM with Fama-French, which adds market cap and relative value to the equation.

Within the realm of statistical arbitrage, sector-neutrality is very common. Stocks are compared with their sector or industry peers. An added dimension is region or country affiliation, if you were to trade globally.

Then there's arbitrage pricing theory (APT), which defines an asset's price in terms of numerous possible factors. I wouldn't call it "state of the art", but it takes a more realistic view of fair value. You'll need to define your own factors, which you can cull through either fundamental analysis or principal component analysis. You can also buy commercial risk models from numerous vendors. In this case, you may find yourself balancing stocks within the portfolio according to their exposure to things like interest rates or the price of oil.

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About CAPM, usually it's applied to longer-term models, EOD at least from what I read. Can Fama/French factors or APT be applied to a 1-minute time-series model? Are there factors that more relevant for high-frequency? –  Robert Kubrick Dec 31 '11 at 14:51
@RobertKubrick Any computation of beta has an implicit time frame. If you're using EOD data to derive beta, then you'll get an EOD beta. The same can be same for using PCA to derive factors and exposures. Just use intra-day data consistently for your calculations, and you'll get an intrad-day model. –  chrisaycock Dec 31 '11 at 16:29
I've found this thread about Beta. gappy argues against using daily Beta values to higher-frequency series. –  Robert Kubrick Dec 31 '11 at 18:43
@RobertKubrick Yeah, gappy correctly points to the Epps effect, which essentially says that correlation becomes harder to measure for higher-frequency data samples because of asynchronous trading. Note, though, that Epps wrote his paper in 1979. Most S&P 500 stocks today trade so often that one-minute discretization should be wide enough. (From experience, there are a few of the high-dollar S&P stocks that don't actually trade very often, but those are very rare and shouldn't impact PCA.) –  chrisaycock Dec 31 '11 at 19:02
@RobertKubrick You could use daily beta, but that could be a very different number from some phenomenon you're trying to capture. Just compute your own one-minute factors/betas and see what happens. –  chrisaycock Dec 31 '11 at 22:35

When trying to predict returns, I think you should never look at in-sample statistics like R-squared. Only look at out-of-sample prediction results. Cross validation is a useful tool in at least the initial phase of modelling.

I suggest you look for common characteristics amongst the 70% of stocks with decent $R^2$ and the 30% with a degraded in-sample fit. If you find that all the stocks that fit your model best were in one industry and all those that didn't fit were in another industry, then it may be that your model is actually picking up an industry effect. Controlling for sector/industry explicitly as part of your model, whether by imposing sector neutrality in the weights for a hypothetical backtest portfolio or by including it as an additional factor in the linear model, may reveal what is really going on.