**4. Results**

*4.1. Multi-Factor Model Examination with Single Stock Regression Analysis*

4.1.1. Stability Test

> DV: Ri-Rf IV: Rm-Rf, SMB, RMW, HML, CMA, CRMHL, AMLH CV: Time, Timeˆ2, Season

According to the result of stability test (presented in Appendix E), the control variable including Time, Timeˆ2, Season did not pass the *t*-test, the significance level is far less than 0.9. Thus, they are removed from the model. It means that excess return is not affected by time.

4.1.2. Regression Models

• **OLS Regression**

**Model 1:** Five-factor model

> DV: Ri-Rf IV: Rm-Rf, SMB, RMW, HML, CMA

**Model 2:** Seven-factor model

> DV: Ri-Rf IV: Rm-Rf, SMB, RMW, HML, CMA, CRMHL, AMLH

**Model 3:** Modified optimal model

> DV: Ri-Rf IV: Rm-Rf, RMW, HML, CMA, CRMHL, AMLH

• **Ridge Regression**

**Model 4:** Ridge regression model

> DV: Ri-Rf IV: Rm-Rf, SMB, RMW, HML, CMA, CRMHL, AMLH

We conducted OLS regression onModel 1, 2, and 3, and ridge regression onModel 4. After conducting the regression analysis, we recorded each factors' coefficients and mean of R square. We created frequency histogramstoobservethedistributionofcoefficients.Thegraphsprovideageneralviewofthecoefficients.

 In Table 2, above each histogram, there are two numbers, mean of coe fficients and the proportion of coe fficients that passed the *t*-test (in the parenthesis).

To be specific, we divided the coefficients into uncorrelated, positively correlated, and uncorrected, based on 95% confidence interval. Table 3 shows the percentage of coefficient. SMB, CMA, Rm-Rf, CRMHL, and AMHL have more positive coefficients. Whereas RMW and HML have more negative coefficients.

In ridge regression, we obtain the λ by calculating the maximum of the objective function Q. When λ equals to 0.008, the result of ridge regression is maximized. In Figure 6, we can find out the largest Q value at 0.008.

**Figure 6.** Target function optimization.

Some factors have negative e ffects on the excess return, which deviates from our assumption. Moreover, regarding the factors that have an even distribution. We need to classify them in industry groups to discover a further pattern.

**Table 2.** R square mean, coefficients' mean and p-level for four models.


**Table 3.** Correlations between seven factors and excess return of stocks.

### 4.1.3. Robustness Test: Zero Mean Residual Testing

According to Table 4, since *t*-values in four models are less than 2.021, we have 95% confidence not to reject the null hypothesis, which means the residuals are randomly independent for all stocks, both the OLS and ridge regression model pass the zero mean residual test.


**Table 4.** Zero mean residual testing.
