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Since the seminal paper by Fama and French(1993) that uses size, market, and value factors to explain extra market returns on the equity market, people have conducted tons of research on equity factor investing. Many factors such as growth, quality, ROE, etc., were discovered and used in actual trading.

I want to test whether certain factors (for example, size, value, growth, etc.,) will have an impact on a trading strategy. For example, for size factor, I implemented a trading strategy, and I backtested the strategy with stocks in the top 30% by market valuation and holding everything else unchanged, I backtested the same trading strategy but traded with stocks in the lowest 30% by market valuation. Of course, I will get two different investment returns, sharpe ratios, etc., Looking at the two investment returns(for example, 37% and 23%), how can I conduct a statistical test so that I can argue that, statistically, with 95% or 99% confidence, the size factor is making a difference in the trading strategy and the difference in the two investment returns is not due to random noise?

Thank you in advance.

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    $\begingroup$ Hi: I don't know if it's the best way but one way be to run the strategy letting the top 30 percent be long the strategy and letting the bottom 30 percent be short the same strategy. If this long short strategy does significantly (statistically) better than the same long-short strategy not deciled for size, ( so each stock is long or short the strategy randomly ) , then that would be evidence that size is doing something good. $\endgroup$
    – mark leeds
    Sep 8 at 3:49
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The Carhart 1997 paper, "On Persistence in Mutual Fund Performance", is probably a good reference for doing exactly what you're looking to do. I think you would want to do a multivariate regression of your returns against the market's factor returns. Then the regression will give you both an intercept and the loadings (betas) to each risk factor. The statistical significance of the intercept is whether you earned excess returns. The loadings will tell you how much risk you took to get your total returns. Taken together, they'll tell you if you were really good (statistically significant intercept) or just lucky (high risk loadings.)

Remember, of course, this is just a back test. Markets change.

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