Table 6 Relationship between changes in...

Table 6

Relationship between changes in regression coefficients and explanatory power (Adj. R²)

Base model	Compared model	Variable	Estimate	t-value	R²
Panel A. Effect of the sum of squared changes in regression coefficients
FF3	FF3_E1	$γ$	0.0207	1.94	0.0014
FF3	FF3_E3	$γ$	0.0350	5.6	0.0154
FF3	FF3_E5	$γ$	0.0572	9.83	0.047
Panel B. Effect of the difference in the sum of squared regression coefficients
FF3	FF3_E1	$δ$	0.00983	6.52	0.021
FF3	FF3_E3	$δ$	0.00557	4.05	0.0079
FF3	FF3_E5	$δ$	0.00577	4.35	0.0092

Base model	Compared model	Variable	Estimate	t-value	R²
Panel A. Effect of the sum of squared changes in regression coefficients
FF3	FF3_E1	$γ$	0.0207	1.94	0.0014
FF3	FF3_E3	$γ$	0.0350	5.6	0.0154
FF3	FF3_E5	$γ$	0.0572	9.83	0.047
Panel B. Effect of the difference in the sum of squared regression coefficients
FF3	FF3_E1	$δ$	0.00983	6.52	0.021
FF3	FF3_E3	$δ$	0.00557	4.05	0.0079
FF3	FF3_E5	$δ$	0.00577	4.35	0.0092

Note(s): This table presents regression analyses examining how the relative magnitude (vector sum) of changes in α and β coefficients, estimated from the original FF3 model and its alternatives, affects the model’s explanatory power (Adj. R²). Panel A shows how the sum of squared coefficient changes, $\sum_{j}^{3} {(∆ β_{j})}^{2}$ ⁠, relates to the increase in Adj. R². Panel B analyzes the impact of the difference in total squared coefficient magnitudes, $(\sum_{j}^{3} β_{j, F F 3_E s}^{2} - \sum_{j}^{3} β_{j, F F 3}^{2})$ on changes in Adj. R². Here, j refers to the betas on RM_RF, SMB and HML, respectively

Regression model

• Panel A: $∆ A d j . R^{2} = θ + γ \cdot \sum_{j}^{3} {(∆ β_{j})}^{2} + ϵ$

• Panel B: $∆ A d j . R^{2} = μ + δ \cdot (\sum_{j}^{3} β_{j, F F 3_E s}^{2} - \sum_{j}^{3} β_{j, F F 3}^{2}) + ϵ$

Source(s): Author’s own work

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