• **Robustness Test**

In this part, a zero-mean test for OLS and ridge regression model is adopted to check the robustness of models. Both OLS and ridge regression models are fixed-coefficient models, which means that they have an assumption that the covariance of errors between different units (stocks) are equal to zero. Thereafter, a residual covariance matrix between 1097 stocks' regression can be drawn by Python and calculate the mean and standard deviation of all numbers in covariance matrix. Then a *t*-test is conducted to test whether E(*ui*,*j*) is equal to zero.

Covariance matrix of residual

$$\begin{array}{ccccccccc} u\_{1,1} & u\_{1,2} & & \dots & u\_{1,1097} \\ & u\_{2,1} & & & \dots & \vdots \\ & & \vdots & & \ddots & \vdots \\ & & & & \dots & \dots & u\_{1097,1097} \\ \end{array} \tag{11}$$

*<sup>u</sup>*1,2 means the covariance between stock 1 and 2.

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$$\text{T-value} = \frac{\text{E}(u\_{i,j})}{\text{STDE}(u\_{i,j})} \tag{12}$$

3.3.2. Time-Series Analysis for Risk Factors
