in:

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  It is the 25% quantile of the daily returns of all stocks in month t, representing the downside risk threshold of the overall market return for that month. Returns below this value can be considered to be in the tail.

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  It is the daily return rate of all stocks in the tth month that is less than $\mu_t$, indicating the kth specific return value of the market in the tail risk in that month.

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  is the total number of daily returns of all stocks in month t that are less than $\mu_t$, indicating how many trading days in the market that month are in a tail risk state.

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  It is a measure of the overall market tail risk in month t. Its value is equal to the average of all daily returns below the 25% percentile in that month divided by the 25% percentile, representing the relative size of the market tail risk.

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  is the daily return of stock i in month t.

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  is the intercept term of the regression of individual stock i, which represents the inherent return level of the stock when its return is not affected by the market tail risk.

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  is the exposure coefficient of stock i to the market tail risk factor (systematic tail risk exposure). This value measures the average change in the return of stock i when the market tail risk changes by one unit, and is the core output of this factor.

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  It is the residual term of the regression, indicating the part of the stock return volatility that the model cannot explain.

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  The time window for time series regression is usually 60 months (or adjusted according to actual conditions). In order to avoid forward-looking bias, the time of the market tail risk $\lambda_t$ should be lagged by one month, that is, $\lambda_{t-1}$ is used to regress $r_{i,t}$.