Market capitalization nonlinearity
factor.formula
The calculation formula of market value nonlinear deviation factor is:
in:
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is the logarithmic market value of stock i at time t (i.e., the total market value converted to natural logarithm). This value reflects the relative size of the stock. Taking the logarithm can reduce the impact of extreme values, make the data more consistent with the assumption of normal distribution, and reduce heteroscedasticity.
At each time section t, a weighted least squares regression (WLS) is performed on all stocks. The dependent variable of the regression is the return of each stock, the independent variable is $LNCAP_{i,t}$, and the weight of the regression is the square root of the total market value of each stock. The use of weighted least squares regression is to reduce the interference of heteroskedasticity on the regression results, especially when the stock market value is unevenly distributed, stocks with larger market value tend to have smaller volatility. Using the square root of market value as a weight can make the regression results more robust.
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The intercept term of the regression equation at time section t. The intercept term itself has no clear economic interpretation, but it plays a corrective role in the regression model, allowing the regression line to better fit the data.
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The regression coefficient of the regression equation at time section t. This coefficient represents the linear relationship between the logarithmic market value $LNCAP_{i,t}$ and the rate of return. If the coefficient is positive, it means that the market value is positively correlated with the rate of return; otherwise, it means a negative correlation. It should be pointed out that in the traditional market value factor, we assume that the rate of return and the logarithmic market value are in a simple linear relationship, and the coefficient of $eta_t$ here is also a linear relationship.
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The residual term obtained from the regression represents the difference between the actual return of stock i at time t and the return predicted by the linear relationship of market value. This residual term reflects the degree of nonlinear deviation of the market value return relationship. The larger the residual value, the greater the degree to which the actual return of stock i deviates from the linear prediction. We will de-extreme and standardize this residual term to eliminate the influence of outliers and obtain standardized factor values.
factor.explanation
The market capitalization nonlinear deviation factor is designed to measure the nonlinear relationship between stock market capitalization and returns. The A-share market has long had a "small market capitalization effect", but the relationship between market capitalization and returns is not a simple linear relationship. As market capitalization increases, the impact of market capitalization on returns will weaken, causing the traditional linear market capitalization factor to overestimate the returns of medium-sized stocks. This factor captures this nonlinear effect by measuring the degree of deviation between the actual returns of a stock and the predicted value of its linear relationship with market capitalization. The higher the factor value, the greater the degree to which the stock deviates from the linear relationship between market capitalization and returns. After standardization, the factor value is expected to be negatively correlated with stock returns. Stocks with lower factor values (medium-sized market capitalization stocks with smaller deviations from the linear relationship) have relatively lower returns, and stocks with higher factor values (extremely large and small market capitalization stocks with larger deviations from the linear relationship) have relatively higher returns.