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4. Empirical Design and Analysis

4.4 Small-cap Stock VAR Model

Lee (1991) first stated that investor sentiment is more of a size story: small firms are influenced more than large firms. Han and Li (2017) found that short-term momentum effect matter more for small sized firms. In this part, I’m going

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to repeat the steps in part 4.3 with the data of small-cap stock returns, which is, a small-cap stock VAR model.

4.4.1 Optimal Lag Order

Based on LogL, LR, FPE, AIC, SC and HQ, Table 4-10 shows the optimal lag order. Greater order indicates smaller degree of freedom.

Table 4-10 Optimal Lag Order

Lag LogL LR FPE AIC SC HQ

0 -457.251 NA 2.423 6.561 6.603 6.578

1 -385.224 140.966 0.917 5.589 5.715* 5.640*

2 -381.066 8.020 0.915 5.587 5.797 5.672

3 -373.465 14.442* 0.869* 5.535* 5.829 5.655

4 -370.966 4.678 0.888 5.557 5.935 5.710

Among the six standards, LR, FPE and AIC choose the third order, SC and HQ choose the first order. As a result, in this model I also choose the second order as compromise and get the VAR model below:

π‘†π‘šπ‘Žπ‘™π‘™π‘†π΄π‘… = 0.000167 πΌπ‘†πΌπ‘‘βˆ’1 - 0.00052 πΌπ‘†πΌπ‘‘βˆ’2 + 0.12715 π‘†π‘šπ‘Žπ‘™π‘™π‘†π΄π‘…π‘‘βˆ’1 -

0.06820π‘†π‘šπ‘Žπ‘™π‘™π‘†π΄π‘…π‘‘βˆ’1+ 0.03035 (4.3)

𝐼𝑆𝐼 = 0.60018 πΌπ‘†πΌπ‘‘βˆ’1 + 0.16560 πΌπ‘†πΌπ‘‘βˆ’2 + 60.29245 π‘†π‘šπ‘Žπ‘™π‘™π‘†π΄π‘…π‘‘βˆ’1

-13.44994πΏπ‘Žπ‘Ÿπ‘”π‘’π‘†π΄π‘…π‘‘βˆ’2+ 12.49488 (4.4)

Figure 4-11 Stability Test

31 -1.5

-1.0 -0.5 0.0 0.5 1.0 1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Inverse Roots of AR Characteristic Polynomial

From the model we can see that the return of small-cap stocks has a lag effect on investor sentiment index. To test the stability of this VAR model, I use an eigenvalue graph and get a result like Figure 4-11. The points represent reciprocal of AR eigenvalue. This model is also stable since all points fall within the unit circle.

4.4.2 Granger Causality Test

To further study the interrelation and compare that with the interrelation of large-cap stock return, I’m going to do Granger causality test on SmallSAR and ISI. The result is shown in Table 4-12.

Table 4-12 Granger Causality Test

H0 Chi-sq df Prob.

SMALLSAR does not Granger cause ISI 23.640 2 0.0000

ISI does not Granger cause SMALLSAR 1.229 2 0.5410

If p-value of the test is below 0.05, it means the null hypothesis can be rejected and X can Granger cause Y. From the table we can see that SmallSAR can Granger cause ISI but ISI cannot Granger cause SmallSAR. As is mentioned above, if only one of the two variables has the Granger causality relation, it means there exists unidirectional causality between them. As a result, there exists

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unidirectional causality between SmallSAR and ISI. SmallSAR is the Granger cause of ISI. At a significant level of 10%, the change in small-cap stock return can cause the change in investor sentiment index, however, the change in investor sentiment cannot cause the change in small-cap stock return. This result further reflects the rationality of model (4.4).

4.4.3 Impulse Response Function

To further test the interrelation and whole influence between ISI and SmallSAR, I’m going to do the impulse response function. The response process of a standard deviation innovation shock after orthogonalization to the future period can be clearly seen. Figure 4-13 and Figure 4-14 show the impact of the unit standard deviation of ISI and SmallSAR causes on ISI and SmallSAR.

The left side of Figure 4-13 reflects the impact ISI gives on itself. After a positive impact in the current period, ISI itself shows the largest positive impulse in the first period and gradually declines. It declined to 0 around the tenth period.

The right side of Figure 4-13 reflects the impact SmallSAR gives on ISI. After a positive impact in the current period, ISI didn’t show an immediate fluctuation in the first period. There’s an increasing positive impulse between the first and second period, which declines gradually after the third period and converges to 0 after the tenth period. It means the impact of SmallSAR would cause positive fluctuation on ISI in short term, and the single impulse response tend to converge to zero in the long term. The Hypothesis 2 is verified in terms of small-cap stocks, since there is also a lag of one period.

Figure 4-13

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0 4 8 12

1 2 3 4 5 6 7 8 9 10

Response of ISI to ISI

0 4 8 12

1 2 3 4 5 6 7 8 9 10

Response of ISI to SMALLSAR Response to Cholesky One S.D. (d.f. adjusted) Innovations ?2 S.E.

The left side of Figure 4-14 reflects the impact ISI gives on SmallSAR. After a positive impact in the current period, SmallSAR shows the largest positive impulse immediately in the first period and gradually declines between the first and third period. There’s even a slight negative impact in the fourth period and then declines, converging to 0 in the tenth period. It means the impact of ISI would cause positive fluctuation on SmallSAR in short term, and the single impulse response tend to converge to zero in the long term. The right side of Figure 4-14 reflects the impact SmallSAR gives on itself. After a positive impact in the current period, SmallSAR shows an immediate fluctuation in the first period and declines quickly to zero around the sixth period.

Figure 4-14

-.02 .00 .02 .04 .06 .08

1 2 3 4 5 6 7 8 9 10

Response of SMALLSAR to ISI

-.02 .00 .02 .04 .06 .08

1 2 3 4 5 6 7 8 9 10

Response of SMALLSAR to SMALLSAR Response to Cholesky One S.D. (d.f. adjusted) Innovations ?2 S.E.

4.4.4 Variance Decomposition

The variance decomposition process is always applied with impulse response function to show the contribution rate of each impact. Table 4-15 and Table 4-16

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show the result of variance decomposition of ISI and SmallSAR.

As is shown in Table 4-15, the contribution rate of SmallSAR to ISI is 0 in the first period and gradually increases. At around the tenth period, ISI can explain 87.057% of its variance variation, and SmallSAR can explain 12.943% of its variance variation. It means in the long run, the change of ISI is mainly affected by itself.

Table 4-15 Variance Decomposition of ISI

Period S.E. ISI SMALLSAR

1 11.546 100.000 0.000

2 14.975 88.708 11.292

3 16.057 86.044 13.956

4 17.132 86.982 13.018

5 17.850 87.126 12.874

6 18.255 86.986 13.014

7 18.596 87.059 12.941

8 18.866 87.077 12.923

9 19.049 87.052 12.948

10 19.186 87.057 12.943

As is shown in Table 4-16, the contribution rate of ISI to SmallSAR is 12.080%

in the first period and gradually increases. At around the tenth period, ISI can explain 13.288% of its variance variation, and SmallSAR can explain 86.712% of its variance variation. It means in the long run, the change of SmallSAR is mainly affected by itself.

Table 4-16 Variance Decomposition of SMALLSAR

Period S.E. ISI SMALLSAR

1 0.08190 12.080 87.920

2 0.08262 12.235 87.767

3 0.08291 12.195 87.805

4 0.08360 12.919 87.081

5 0.08375 13.221 86.780

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6 0.08378 13.230 86.770

7 0.08379 13.241 86.760

8 0.08380 13.263 86.737

9 0.08381 13.276 86.724

10 0.08382 13.288 86.712