• **OLS Regression**

We conducted OLS regression on 47 quarters for each single stock, in order to examine the multi-factor model. According to OLS, to calculate the beta (coe fficient) of each independent variable, the matrix operation combines X and y. In this formula, X is a 47 × 7 matrix, in which 47 is the 47 time-series of data and 7 represents seven proposed risk factors. The y is a 47 × 1 matrix and 1 indicates the excess return of a single stock. Since there are 1097 stock, X matrix is fixed ye<sup>t</sup> y matrix is the data of di fferent stocks.

$$\boldsymbol{\beta} = (\boldsymbol{\lambda}^{\mathrm{T}} \boldsymbol{\lambda})^{-1} \boldsymbol{\lambda}^{\mathrm{T}} \mathbf{y} \tag{6}$$


We used computer simulation in Python, which can help us automatically run regression 1097 times. The python program would print out the result of regression and calculate the number of mean value of coe fficients and *t*-value, which were collected in a 7 × 1097 matrix. We also drew the frequency histogram for each factor to study the distribution of the coe fficients. Based on these assumptions, the majority of the coe fficients should be positive. If most of them come out negative, we may infer that the five-factor model is not suitable for the Chinese stock market.
