• **Chi-Square Test**

We use chi-square in two steps respectively. Firstly, it was used to examine the different effect of factors in different industries. In the test we divided the significance of factors based on industries and calculated the total value χ2.

Secondly, it was applied to find out the pattern in the inter-factor direction prediction analysis. We divided the pattern into four types with a combination of increase and decrease. Then we used chi-square to find out whether there is a pattern for factor change direction. With the increase and decrease pattern, we can predict the next quarter movement based on the current situation. As we can see in Figure 2, suppose "0" represents the increase of risk factor in one term (or one season) and "1" represents the decrease of risk factors.

$$R = \text{Max}(\text{F1, F2}) + \text{Max}(\text{F3, F4}) \tag{13}$$

$$\mathcal{W} = \text{Min(F1, F2)} + \text{Min(F3, F4)}$$

**Figure 2.** Pattern of factor's change.

In Figure 3, if investors spontaneously make investment decision, the frequencies of right and wrong decisions are equitable. Both are T/2. (T is number of total transaction time)

**H0**: *There is no relationship between right or wrong investment decisions and direction prediction rules.*

**H1**: *There is a relationship between right or wrong investment decisions and direction prediction rules.*



In Figure 4, the expected frequency of right decisions (*Re*) and wrong decisions (*We*) made by the rule can be calculated by Equations (15) and (16). Also, the expected frequency of right or wrong decision made by random selection ((*T*/2)*e*) can be calculated by Equation (17).

$$R\_c = (R + \mathcal{W}) \times (R + T/2) / (R + \mathcal{W} + T) \tag{15}$$

$$\mathcal{W}\_{\ell} = (R + \mathcal{W}) \times (\mathcal{W} + T/2) / (R + \mathcal{W} + T) \tag{16}$$

$$(T/2)\_{\varepsilon} = T \times \left(R + T/2\right) / \left(R + \mathcal{W} + T\right) \tag{17}$$


**Figure 4.** Expected frequencies: *fe.*

According to the significant table (Figure 5) of chi-square test, if total chi-square level is >2.706, we have 90% confidence to reject the null hypothesis (H0). If total chi-square level is >3.841, we have 95% confidence to reject the null hypothesis (H0). If total chi-square level is >6.635, we have 99% confidence to reject the null hypothesis (H0).

$$\text{Total chi-square level} = \left(\text{R} - \text{R}\_{\varepsilon}\right)^{2}/\text{R}\_{\varepsilon} + \left(\text{W} - \text{W}\_{\varepsilon}\right)^{2}/\text{W}\_{\varepsilon} + 2\left[\text{T}/2 - \left(\text{T}/2\right)\_{\varepsilon}^{2}\right]/\left(\text{T}/2\right)\_{\varepsilon} \tag{18}$$

 (14)

$$\text{The degree of freedom} = (\text{column} - 1)(\text{row} - 1) = (2 - 1)(2 - 1) = 1$$


**Figure 5.** Chi-square level: χ2 = (*fo* − *fe*) 2 *f*.

### • **Back-Test for Trading Strategy**

After factors' cyclical research and trend analysis for fluctuation of risk factors, the trading strategy based on moving average approach and correlations between factors and industries can be formed. By deciding the time to do the transactions and stocks which should be bought or sold, the float return from 2007S4 to 2018S3 and annual expected return can then be calculated.

### *3.4. Assumptions for Multi-Factor Examination*

The following assumptions are made in applying the multi-factor model:


The intuition and assumption behind the hypotheses are presented in the Appendix A with measurements of factors and graphs.

### *3.5. Models and Variable Definitions*

Model 1:

$$R\_i - R\_f = \mathbf{a} + \beta\_{\text{murkct}} \left( R\_{\text{m}} - R\_f \right) + \beta\_{\text{size}} \text{SMB} + \beta\_{\text{RM}} \text{HML} + \beta\_{\text{profitability}} \text{RMW} + \beta\_{\text{investment}} \text{CMA} + \varepsilon$$

Model 2:

 $R\_i - R\_f = \mathbf{a} + \beta\_{markt} (R\_m - R\_f) + \beta\_{size}$  $\text{SMB} + \beta\_{BM}\text{HML} + \beta\_{profiability}\text{RMW} + \beta\_{i,vestment}\text{CMA} + \beta\_{markt}$  $\text{CRMHL} + \beta\_{turnover}\text{AMHL} + \varepsilon$ 

The details of the factors (Rm-Rf, SMB, HML, RMW, CMA, CRMHL, AMHL) including explanation and graphs are presented in Appendix A.

Table 1 displays the dependent variable, independent variables and the corresponding labels and explanations.


**Table 1.** Dependent variable and independent variables.

 (19)


**Table 1.** *Cont.*
