in:

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  The return of asset i at time t, usually using monthly return data

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  The intercept term of asset i represents the portion of the return that cannot be explained by the model and can be regarded as the asset-specific return.

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  The sensitivity of asset i to the market risk premium RMRF measures the asset’s exposure to overall market risk, i.e., market beta

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  The market risk premium at time t, usually the market rate of return minus the risk-free rate

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  The sensitivity of asset i to the size premium SMB measures the exposure of the asset to size risk, i.e., the size factor beta value

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  The size premium at time t is usually the return on small-cap stocks minus the return on large-cap stocks.

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  The sensitivity of asset i to the value premium HML measures the exposure of the asset to value risk, i.e., the value factor beta

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  The value premium at time t is usually the return on high book-to-market stocks minus the return on low book-to-market stocks.

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  The residual of asset i at time t represents the part of the idiosyncratic return that cannot be explained by the Fama-French three-factor model and can be regarded as the unique return of the stock.

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  The mean of the residuals from T-12 to T-2 is used to calculate the standard deviation of the residual return.

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  The sum of the residual returns of asset i from T-12 to T-2, reflecting the cumulative momentum of the residual returns over the past 11 months

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  The standard deviation of the residual return of asset i from T-12 to T-2 is used to standardize the residual momentum to prevent the factor value from being affected by the volatility. The denominator is divided by 10 instead of 11 because the sample standard deviation is used here instead of the population standard deviation. The degree of freedom is n-1, and the sample size is 11, so it is divided by 10.