The role of centralization index in identifying momentum stage of stocks: empirical evidence from investor networks

Figures & data

Figure 1. Illustration of decentralized versus centralized firms. Empirical investor networks delineate various network structures. Each node represents the identification code for an investor. Six investors are arbitrarily selected from distinct company-specific networks. For each connected investor pair in the network (i, j ≠ i), it represents that they traded the same stock in the same direction (buy or sell) a minimum of three times within a time window of Δt during a month. The decentralized and centralized network graphs are illustrated in the left and right figures, respectively.

Table 1. Summary statistics.

Panel A. Summary Statistics
	Avg. # of firms	Mean	Std. dev	Min	25%	Median	75%	Max
DD	534	0.06	0.03	0.01	0.04	0.05	0.07	0.24
EIG	534	0.07	0.07	0.02	0.03	0.07	0.11	0.20
β	529	0.95	0.29	0.04	0.76	0.95	1.15	1.74
Size (in NT$ millions)	534	34,150	119,670	436	3,387	7,185	17,976	1,917,260
B/M	534	0.87	0.48	0.08	0.53	0.79	1.13	3.65
IVOL	529	0.02	0.01	0.01	0.02	0.02	0.02	0.07
Ret-1:-1	534	0.02	0.11	−0.29	−0.05	0.00	0.06	0.76
IO	533	0.40	0.22	0.01	0.22	0.38	0.57	0.97
NS (in thousands)	534	48	80	1	11	21	49	886
Illiquidity (×10^5)	534	3.21	22.36	0.00	0.19	0.58	1.69	451.73
Turnover	534	0.18	0.15	0.01	0.08	0.14	0.24	1.05
Dvolume (in NT $ millions)	534	3512	7891	31	454	1180	3262	91456

Table

Panel B. Correlations across Variables
	DD	EIG	β	Size	B/M	IVOL	ret-1:-1	IO	NS	Illiquidity	Turnover	Dvolume
DD	1
EIG	0.117	1
β	0.078	0.276	1
Size	0.052	0.44	0.189	1
B/M	−0.012	0.012	−0.024	−0.394	1
IVOL	0.17	−0.105	0.32	−0.317	−0.125	1
Ret-1:-1	0.05	−0.028	−0.058	0.013	−0.141	0.053	1
IO	0.01	0.051	0.036	0.6	−0.311	−0.171	0.003	1
NS	0.086	0.569	0.248	0.632	0.232	−0.269	−0.066	0.181	1
Illiquidity	−0.005	−0.54	−0.3	−0.882	0.403	0.251	0.055	−0.425	−0.571	1
Turnover	0.058	0.222	0.432	−0.184	−0.203	0.523	0.005	−0.312	−0.136	−0.194	1
Dvolume	0.066	0.55	0.435	0.826	−0.446	−0.004	−0.028	0.38	0.529	−0.942	0.384	1

This table presents descriptive statistics (Panel A) and correlations (Panel B) for monthly firm-level variables used in our study from January 2005 to December 2014. Network structure variables, directed-degree (DD) and eigenvector (EIG) centralization indices, are defined in the text. Firm-specific variables include β (the slope coefficient of the time-series regression of stock returns on the market returns), Size (closing price multiplied by outstanding shares), B/M (book-to-market ratio), IVOL (standard deviation of residual returns from a market model), and Ret_-1:-1 (returns from the previous month). Investor attention variables include IO (percentage of shares held by institutional investors) and NS (number of shareholders). Liquidity variables include Amihud (2002)’s illiquidity (the time-series average of absolute daily returns scaled by the daily dollar trading volume, multiplied by 10⁵), Turnover (average monthly traded shares relative to total outstanding shares), and Dvolume (average monthly dollar trading volume). These statistics and correlations are initially computed monthly, then averaged over the period. The t-statistics are estimated using Newey and West (1987) robust standard errors, with correlations significant at the 0.05 level highlighted in bold.

Table 2. Returns for the simple momentum strategy.

	Winner			Loser			MOM
	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted
K = 1	1.42**	1.13***	0.71***	1.08	0.78	0.33	0.33	0.35	0.38
	(2.14)	(2.22)	(2.32)	(1.50)	(1.45)	(1.67)	(1.02)	(1.11)	(1.22)
K = 3	1.32**	1.04***	0.62**	1.12	0.81	0.37*	0.20	0.23	0.25
	(2.02)	(2.99)	(2.16)	(1.55)	(1.59)	(1.97)	(0.68)	(0.77)	(0.91)
K = 6	1.27*	0.98***	0.56**	1.16	0.85*	0.41	0.10	0.13	0.16
	(1.95)	(2.89)	(2.17)	(1.61)	(1.77)	(1.42)	(0.37)	(0.49)	(0.66)
K = 9	1.18*	0.90***	0.49***	1.23*	0.92*	0.47**	−0.05	−0.02	0.02
	(1.85)	(2.60)	(2.54)	(1.70)	(1.83)	(2.00)	(-0.19)	(-0.07)	(0.08)
K = 12	1.13*	0.85***	0.44**	1.29*	0.98	0.52	−0.17	−0.13	−0.08
	(1.77)	(2.44)	(2.19)	(1.79)	(1.28)	(1.66)	(-0.69)	(-0.57)	(-0.38)

This table presents the average monthly returns, expressed as percentages, of the simple momentum strategy (MOM), over the sample period from January 2005 to December 2014. All stocks are ranked in ascending order at the end of each month based on their past 6-month cumulative returns. Stocks are then divided equally into three portfolios of equal weight and held for K months (1, 3, 6, 9, and 12). To mitigate microstructure bias, a 1-month interval was imposed between the end of the portfolio formation period and the start of the holding period. The stocks in the bottom one-third are assigned to the loser portfolio, and those in the top one-third are assigned to the winner portfolio. The momentum strategy (MOM) involves buying winner and selling loser portfolios. The monthly portfolio alphas were calculated by running a time-series regression of portfolio returns on risk factors, including the CAPM factor and the three factors of Fama and French (1993). Both raw and risk-adjusted returns were reported. The t statistics are adjusted for autocorrelation using the Newey and West (1987) method and are reported in parentheses, and *, **, and *** indicate significance at 0.1, 0.05, and 0.01 levels, respectively.

Table 3. Returns for the turnover-based momentum strategies.

	EARLY			LATE			EARLY ‒ MOM			LATE ‒ MOM
	Raw	CAPM- adjusted	FF3- adjusted	Raw	CAPM- adjusted	FF3- adjusted	Raw	CAPM- adjusted	FF3- adjusted	Raw	CAPM- adjusted	FF3- adjusted
K = 1	0.47*	0.49*	0.50*	−0.42	−0.45	−0.48*	0.14*	0.14*	0.13	−0.74	−0.80	−0.86*
	(1.83)	(1.82)	(1.81)	(−1.05)	(−1.16)	(−1.73)	(1.75)	(1.71)	(1.55)	(−1.61)	(−1.67)	(−1.77)
K = 3	0.26	0.28	0.28	−0.53	−0.56	−0.59*	0.07	0.06	0.03	−0.73	−0.79	−0.84*
	(1.65)	(1.66)	(1.51)	(−1.43)	(−1.54)	(−1.78)	(0.84)	(0.79)	(0.40)	(−1.58)	(−1.66)	(−1.86)
K = 6	0.15	0.17	0.20	−0.51	−0.53	−0.55*	0.05	0.05	0.04	−0.61	−0.66	−0.71*
	(0.54)	(0.65)	(0.85)	(−1.49)	(−1.56)	(−1.84)	(0.76)	(0.70)	(0.53)	(−1.50)	(−1.67)	(−1.88)
K = 9	0.04	0.07	0.11	−0.62	−0.63	−0.65	0.09	0.09	0.08	−0.56	−0.61	−0.67
	(0.14)	(0.26)	(0.51)	(−1.58)	(−1.61)	(−1.64)	(1.37)	(1.30)	(1.26)	(−1.66)	(−1.63)	(−1.61)
K = 12	−0.03	−0.01	0.06	−0.66	−0.68	−0.68*	0.14	0.13	0.14	−0.49	−0.54	−0.61*
	(−0.14)	(−0.03)	(0.30)	(−1.64)	(−1.67)	(−1.85)	(1.26)	(1.32)	(1.43)	(−1.62)	(−1.69)	(−1.78)

This table presents the average monthly returns, expressed as percentages, of early-stage and late-stage momentum strategies. All stocks are ranked in ascending order at the end of each month based on their past 6-month cumulative returns. Stocks are then divided equally into three portfolios of equal weight and held for K months (1, 3, 6, 9, and 12). To mitigate microstructure bias, a 1-month interval was imposed between the end of the portfolio formation period and the start of the holding period. The stocks in the bottom one-third are assigned to the loser portfolio, and those in the top one-third are assigned to the winner portfolio. The MOM represents a zero-investment portfolio that longs the winner portfolio and shorts the loser portfolio. The stocks in the winner and loser portfolios are further sorted into three subgroups based on their turnover ratios: high, middle, and low turnover portfolios. The EARLY represents a zero-investment portfolio that longs winners with low turnover ratios and shorts losers with high turnover ratios. The LATE represents a zero-investment portfolio that longs winners with low turnover ratios and shorts losers with high turnover ratios. The last two columns present the differences in returns for the two momentum strategies. The monthly portfolio alphas were calculated by running a time-series regression of portfolio returns on risk factors, including the CAPM factor and the three factors of Fama and French (1993). Both raw and risk-adjusted returns were reported. The t statistics are adjusted for autocorrelation using the Newey and West (1987) method and are reported in parentheses, and *, **, and *** indicate significance at 0.1, 0.05, and 0.01 levels, respectively.

Table 4. Returns for the early-stage momentum strategies.

	DD			EIG			DD ‒ MOM			EIG ‒ MOM			EIG ‒ DD
	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted
K = 1	0.98**	1.04**	1.04**	0.72*	0.82*	0.68*	0.65**	0.70***	0.66***	0.39	0.47*	0.30	−0.26	−0.23	−0.36
	(2.26)	(2.55)	(2.62)	(1.85)	(1.95)	(1.66)	(2.51)	(2.85)	(2.67)	(1.30)	(1.81)	(1.29)	(−0.79)	(−0.70)	(−1.16)
K = 3	0.78**	0.85**	0.85**	0.79*	0.89**	0.75**	0.58***	0.63***	0.60***	0.60**	0.67***	0.49**	0.01	0.04	−0.11
	(2.07)	(2.47)	(2.59)	(1.85)	(2.37)	(2.06)	(2.84)	(3.43)	(3.16)	(2.08)	(2.76)	(2.29)	(0.05)	(0.15)	(−0.44)
K = 6	0.57*	0.65**	0.65**	0.85**	0.95***	0.80**	0.48***	0.52***	0.50***	0.75***	0.82***	0.64***	0.28	0.30	0.15
	(1.67)	(2.15)	(2.36)	(2.13)	(2.77)	(2.44)	(2.75)	(3.61)	(3.32)	(2.81)	(3.60)	(3.24)	(1.29)	(1.42)	(0.78)
K = 9	0.40*	0.49*	0.51**	0.80**	0.90***	0.75**	0.46***	0.51***	0.49***	0.86***	0.92***	0.73***	0.39*	0.41	0.24
	(1.73)	(1.70)	(2.00)	(2.07)	(2.74)	(2.43)	(2.94)	(3.97)	(3.79)	(3.27)	(4.06)	(3.75)	(1.85)	(1.59)	(1.32)
K = 12	0.34	0.42**	0.46*	0.72**	0.82***	0.69**	0.51***	0.55***	0.53***	0.89***	0.96***	0.76***	0.38*	0.40*	0.23
	(1.52)	(2.21)	(1.96)	(2.01)	(2.71)	(2.39)	(3.55)	(4.64)	(4.57)	(3.43)	(4.20)	(3.86)	(1.80)	(1.96)	(1.28)

This table presents the average monthly returns, expressed as percentages, of early stage momentum strategies for the directed-degree (DD) and eigenvector (EIG). All stocks are ranked in ascending order at the end of each month based on their past 6-month cumulative returns. Stocks are then divided equally into three portfolios of equal weight and held for K months (1, 3, 6, 9, and 12). To mitigate microstructure bias, a 1-month interval was imposed between the end of the portfolio formation period and the start of the holding period. The stocks in the bottom one-third are assigned to the loser portfolio, and those in the top one-third are assigned to the winner portfolio. The MOM represents a zero-investment portfolio that longs the winner portfolio and shorts the loser portfolio. The stocks in the winner and loser portfolios are further sorted into three subgroups based on their centralization index values: high, middle, and low centralization portfolios. DD represents a zero-investment portfolio that longs winners with low-directed-degree centralization and shorts losers with high-directed-degree centralization. EIG represents a zero-investment portfolio that longs winners with low eigenvector centralization, and shorts losers with high eigenvector centralization. The last three columns present the differences in returns for the two momentum strategies. The monthly portfolio alphas were calculated by running a time-series regression of portfolio returns on risk factors, including the CAPM factor and the three factors of Fama and French (1993). Both raw and risk-adjusted returns were reported. The t statistics are adjusted for autocorrelation using the Newey and West (1987) method and are reported in parentheses, and *, **, and *** indicate significance at 0.1, 0.05, and 0.01 levels, respectively.

Figure 2. Compound returns for the momentum and centralization momentum strategies. Time-series patterns of compound monthly returns for the momentum (MOM; dotted line), directed-degree momentum (DD; solid line), and eigenvector momentum (EIG; dashed line) strategies. All stocks are ranked in ascending order at the end of each month based on their past-6-month cumulative returns. The stocks in the bottom one-third are assigned to the loser portfolio and those in the top one-third are assigned to the winner portfolio. The MOM represents a zero-investment portfolio that longs the winner portfolio and shorts the loser portfolio. The stocks in the winner and loser portfolios are further sorted into three subgroups based on their centralization index values: high, middle, and low centralization portfolios. DD represents a zero-investment portfolio that longs winners with low-directed-degree centralization and shorts losers with high-directed-degree centralization. EIG represents a zero-investment portfolio that longs winners with low eigenvector centralization, and shorts losers with high eigenvector centralization. The sample period is from January 2005 to December 2014.

Table 5. Returns for the late-stage momentum strategies.

	LDD			LEIG			LDD ‒ MOM			LEIG ‒ MOM			LEIG ‒ LDD
	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted
K = 1	0.19	0.19	0.16	0.06	0.06	0.24	−0.13**	−0.16**	−0.22***	−0.27***	−0.28***	−0.14**	−0.14	−0.13	0.08
	(0.26)	(0.27)	(0.37)	(0.62)	(0.60)	(0.11)	(−2.32)	(−2.57)	(−3.07)	(−3.12)	(−3.29)	(−2.54)	(−0.55)	(−0.52)	(0.37)
K = 3	0.09	0.09	0.09	0.01	0.02	0.18	−0.11***	−0.13***	−0.17***	−0.19***	−0.20***	−0.07***	−0.08	−0.07	0.09
	(0.62)	(0.61)	(0.68)	(0.82)	(0.78)	(0.33)	(−3.42)	(−3.84)	(−4.20)	(−3.37)	(−3.54)	(−2.84)	(−0.45)	(−0.41)	(0.60)
K = 6	0.07	0.08	0.08	−0.05	−0.03	0.13	−0.02***	−0.04***	−0.08***	−0.15***	−0.15***	−0.03***	−0.12	−0.11	0.04
	(0.76)	(0.72)	(0.80)	(−1.13)	(−1.08)	(0.62)	(−3.00)	(−3.36)	(−3.71)	(−3.54)	(−3.69)	(−3.04)	(−0.76)	(−0.71)	(0.32)
K = 9	0.05	0.07	0.06	−0.10	−0.07	0.08	0.10*	0.09**	0.04**	−0.05***	−0.05***	0.05**	−0.15	−0.14	0.02
	(0.86)	(0.80)	(0.92)	(−1.37)	(−1.28)	(0.87)	(−1.76)	(−2.02)	(−2.49)	(−3.10)	(−3.19)	(2.47)	(−1.01)	(−0.96)	(0.13)
K = 12	−0.01	0.01	0.03	−0.15	−0.12	0.06	0.16	0.14	0.10	−0.15	−0.12	0.06	−0.13	−0.12	0.02
	(−1.20)	(1.12)	(1.18)	(−1.61)	(−1.52)	(1.06)	(−1.17)	(−1.39)	(−1.70)	(−1.61)	(−1.52)	(1.06)	(−0.95)	(−0.89)	(0.19)

This table presents the average monthly returns, expressed as percentages, of the late-stage momentum strategies for the directed degree (LDD) and eigenvector (LEIG). All stocks are ranked in ascending order at the end of each month based on their past 6-month cumulative returns. Stocks are then divided equally into three portfolios of equal weight and held for K months (1, 3, 6, 9, and 12). To mitigate microstructure bias, a 1-month interval was imposed between the end of the portfolio formation period and the start of the holding period. The stocks in the bottom one-third are assigned to the loser portfolio, and those in the top one-third are assigned to the winner portfolio. The MOM represents a zero-investment portfolio that longs the winner portfolio and shorts the loser portfolio. The stocks in the winner and loser portfolios are further sorted into three subgroups based on their centralization index values: high, middle, and low centralization portfolios. The LDD represents a zero-investment portfolio that longs winners with high directed-degree centralization and shorts losers with low directed-degree centralization. LEIG represents a zero-investment portfolio that longs winners with high eigenvector centralization, and shorts losers with low eigenvector centralization. The last three columns present the differences in returns for the two momentum strategies. The monthly portfolio alphas were calculated by running a time-series regression of portfolio returns on risk factors, including the CAPM factor and the three factors of Fama and French (1993). Both raw and risk-adjusted returns were reported. The t statistics are adjusted for autocorrelation using the Newey and West (1987) method and are reported in parentheses, and *, **, and *** indicate significance at 0.1, 0.05, and 0.01 levels, respectively.

Table 6. Fama–MacBeth cross-sectional regressions with centralization index.

	(1)	(2)	(3)	(4)	(5)	(6)
CRet	0.005	0.008	0.009*	0.002	0.004	0.006
	(0.63)	(1.44)	(1.87)	(0.31)	(0.82)	(1.31)
DD	0.073	0.074	0.115
	(0.80)	(0.94)	(1.59)
EIG				0.035	0.039	0.049
				(1.22)	(1.59)	(1.25)
$C\mathit{\text{Re}}t\times \widehat{\mathit{\text{DD}}}$	−0.118***	−0.113***	−0.102***
	(−5.18)	(−5.59)	(−5.22)
$C\mathit{\text{Re}}t\times \widehat{\mathit{\text{EIG}}}$				−0.045***	−0.043***	−0.013
				(−4.40)	(−4.23)	(−1.32)
β		−0.002	0.007		0.001	0.007
		(−0.29)	(1.51)		(0.15)	(1.47)
ln(Size)		−0.001	−0.001		−0.000	−0.001
		(−1.13)	(−0.15)		(−0.10)	(−0.27)
B/M		0.008***	0.007***		0.008***	0.007***
		(2.80)	(3.36)		(3.00)	(3.31)
Ret		0.002	0.003		0.004	0.004
		(0.25)	(0.41)		(0.42)	(0.52)
ln(Illiquidity)			−0.004			−0.004*
			(−1.50)			(−1.79)
ln(Turnover)			−0.002			−0.002
			(−0.47)			(−0.60)
ln(Dvolume)			−0.005			−0.005
			(−1.19)			(−1.23)
Attention variables	No	No	Yes	No	No	Yes
Avg Adj R^2	0.037	0.093	0.120	0.036	0.082	0.101

This table presents the estimated slope coefficients from the Fama–MacBeth regression. The dependent variable is the return on the next holding period. The independent variables are the past-6-month cumulative returns (excluding the most recent month), centralization indices, and the interaction between past cumulative return and centralization index, along with firm characteristics, liquidity variables, and investor attention metrics. The interaction terms ($C\mathit{\text{Re}}t\times \widehat{\mathit{\text{DD}}}$ and $C\mathit{\text{Re}}t\times \widehat{\mathit{\text{EIG}}}$) are calculated as the product of CRet and DD (or EIG) for stocks with cumulative returns above the median and as the product of CRet and -DD (or EIG) for those below the median. Firm characteristics include β (CAPM beta), ln(Size) (natural logarithm of market capitalization), B/M (book-to-market ratio), and Ret (current 1-month returns). The liquidity variables include ln(Illiquidity) (natural logarithm of Amihud (2002)’s illiquidity measure), ln(Turnover) (natural logarithm of turnover rate), and ln(Dvolume) (natural logarithm of dollar trading volume). Investor attention variables include institutional ownership and number of shareholders. The table reports the time-series average of these coefficient estimates and their associated t-statistics adjusted by Newey and West (1994). The t statistics are shown in parentheses, with *, **, and *** indicating significance at the 0.1, 0.05, and 0.01 levels, respectively.

Table 7. Post-holding-period returns for the various momentum strategies.

	MOM	DD	EIG	DD-MOM	EIG-MOM	EIG-DD
Year 2	2.28	−4.78	−5.49	−7.06	−7.77	−0.71
	(0.30)	(−0.48)	(−1.25)	(−0.74)	(−1.18)	(−0.08)
Year 3	−1.38*	−5.87	2.35	−4.49	3.72	8.21
	(−1.78)	(−1.00)	(0.35)	(−0.96)	(0.39)	(1.32)
Year 4	2.03	−2.39	4.89	−4.42	2.86	7.28
	(0.32)	(−0.72)	(0.61)	(−0.76)	(0.24)	(0.96)
Year 5	−4.32	−4.46*	−3.40*	−0.15	0.92	1.06
	(−0.82)	(−1.80)	(−1.83)	(−0.04)	(1.14)	(1.08)

This table presents the average annual returns in percentages for the three momentum strategies, namely MOM, DD, and EIG. At the end of each month, all stocks are ranked in ascending order based on their past-6-month cumulative returns. Stocks are then divided equally into three portfolios of equal weight and held over various holding horizons, ranging from 2 to 5 years subsequent to formation. The stocks in the bottom one-third are assigned to the loser portfolio, and those in the top one-third are assigned to the winner portfolio. The MOM represents a zero-investment portfolio that longs the winner portfolio and shorts the loser portfolio. The stocks in the winner and loser portfolios are further sorted into three subgroups based on their centralization index values: high, middle, and low centralization portfolios. DD represents a zero-investment portfolio that longs winners with low-directed-degree centralization and shorts losers with high-directed-degree centralization. EIG represents a zero-investment portfolio that longs winners with low eigenvector centralization, and shorts losers with high eigenvector centralization. The annual portfolio returns were calculated as the equal-weighted average of the annual returns of the stocks in the portfolio. The t statistics are adjusted for autocorrelation using the Newey and West (1987) method and are reported in parentheses, and *, **, and *** indicate significance at 0.1, 0.05, and 0.01 levels, respectively.

Table 8. Returns for the simple momentum strategy in the interval of 2015–2023.

	Winner			Loser			MOM
	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted
K = 1	1.54***	0.68**	0.32***	0.86	−0.10	−0.39	0.67	0.77	0.72
	(2.20)	(2.26)	(2.38)	(1.60)	(−0.31)	(−1.45)	(1.44)	(0.96)	(1.60)
K = 3	1.53***	0.64**	0.29***	0.83	−0.12	−0.40*	0.71	0.76	0.69*
	(2.12)	(2.06)	(2.65)	(1.59)	(−0.40)	(−1.88)	(0.98)	(1.39)	(1.84)
K = 6	1.48*	0.59**	0.24**	0.83	−0.11	−0.38	0.66	0.70	0.62
	(1.90)	(2.17)	(2.32)	(1.63)	(−0.38)	(−1.00)	(1.15)	(1.50)	(1.12)
K = 9	1.33*	0.44*	0.11	0.94	0.01	−0.28	0.39	0.43	0.39
	(1.75)	(1.69)	(1.23)	(1.58)	(0.05)	(−1.41)	(1.20)	(1.49)	(1.26)
K = 12	1.20*	0.31	−0.00	1.07	0.16	−0.15	0.13	0.15	0.15
	(1.90)	(1.21)	(−0.04)	(1.16)	(0.57)	(−1.47)	(0.86)	(1.01)	(0.98)

This table presents the average monthly returns, expressed as percentages, of the simple momentum strategy (MOM), over the sample period from January 2015 to December 2023. All stocks are ranked in ascending order at the end of each month based on their past 6-month cumulative returns. Stocks are then divided equally into three portfolios of equal weight and held for K months (1, 3, 6, 9, and 12). To mitigate microstructure bias, a 1-month interval was imposed between the end of the portfolio formation period and the start of the holding period. The stocks in the bottom one-third are assigned to the loser portfolio, and those in the top one-third are assigned to the winner portfolio. The momentum strategy (MOM) involves buying winner and selling loser portfolios. The monthly portfolio alphas were calculated by running a time-series regression of portfolio returns on risk factors, including the CAPM factor and the three factors of Fama and French (1993). Both raw and risk-adjusted returns were reported. The t statistics are in parentheses, and *, **, and *** indicate significance at the 0.1, 0.05, and 0.01 levels, respectively.

Table 9. Returns for the early-stage momentum strategies in split intervals of 2005–2009 and 2010–2014.

Panel A. 2005–2009
	DD			EIG			DD ‒ MOM			EIG ‒ MOM			EIG ‒ DD
	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted
K = 1	0.77**	0.85**	0.81**	0.45	0.59*	0.30	0.56*	0.62*	0.53*	0.24	0.36*	0.04	−0.32	−0.26	−0.51
	(2.04)	(2.23)	(2.01)	(1.60)	(1.89)	(1.31)	(1.78)	(1.82)	(1.72)	(1.46)	(1.82)	(1.04)	(−0.57)	(−0.47)	(−0.95)
K = 3	0.51*	0.61**	0.56*	0.51*	0.65**	0.36	0.56*	0.63*	0.56*	0.56*	0.67*	0.34	0.00	0.04	−0.21
	(1.90)	(2.17)	(1.90)	(1.74)	(2.10)	(1.44)	(1.83)	(1.98)	(1.77)	(1.78)	(1.82)	(1.34)	(0.00)	(0.10)	(−0.50)
K = 6	0.45	0.56*	0.55*	0.56*	0.70**	0.43	0.60*	0.68*	0.62*	0.71*	0.82**	0.48*	0.10	0.14	−0.13
	(1.68)	(2.00)	(1.70)	(1.88)	(2.31)	(1.64)	(1.85)	(1.99)	(1.85)	(1.93)	(2.07)	(1.95)	(0.82)	(0.93)	(0.23)
K = 9	0.22*	0.34**	0.36**	0.42**	0.57**	0.29**	0.64*	0.71**	0.65*	0.84**	0.94**	0.57**	0.20	0.23	−0.08
	(1.84)	(2.13)	(2.04)	(2.06)	(2.10)	(2.05)	(1.81)	(2.29)	(1.90)	(2.05)	(2.40)	(2.29)	(1.10)	(1.21)	(0.40)
K = 12	0.26	0.37*	0.38*	0.33*	0.48*	0.25*	0.61**	0.66***	0.60***	0.87*	0.97**	0.60*	0.07	0.11	−0.13
	(1.43)	(1.93)	(1.85)	(1.77)	(172)	(1.79)	(2.45)	(3.24)	(2.89)	(1.95)	(2.49)	(1.85)	(0.76)	(0.89)	(0.01)
Panel B. 2010–2014
	DD			EIG			DD ‒ MOM			EIG ‒ MOM			EIG ‒ DD
	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted
K = 1	1.18**	1.23**	1.27**	0.98*	1.04*	1.05**	0.74**	0.77***	0.77***	0.55*	0.58*	0.55*	−0.20	−0.19	−0.22
	(2.14)	(2.38)	(2.65)	(1.75)	(2.00)	(2.09)	(2.56)	(2.80)	(2.80)	(1.72)	(1.98)	(1.94)	(−0.57)	(−0.55)	(−0.63)
K = 3	1.04**	1.09**	1.13***	1.07**	1.13**	1.13**	0.61**	0.63***	0.63***	0.63**	0.67**	0.63**	0.03	0.04	0.00
	(2.08)	(2.37)	(2.84)	(2.04)	(2.39)	(2.59)	(2.60)	(2.93)	(2.95)	(2.22)	(2.61)	(2.60)	(0.10)	(0.13)	(0.00)
K = 6	0.68**	0.73**	0.76***	1.13**	1.19***	1.18***	0.34***	0.37***	0.37***	0.79***	0.83***	0.79***	0.45	0.46	0.42
	(2.01)	(2.32)	(2.94)	(2.34)	(2.74)	(3.04)	(3.10)	(3.86)	(3.93)	(2.86)	(3.42)	(3.44)	(1.12)	(1.18)	(1.10)
K = 9	0.58*	0.62**	0.65***	1.17**	1.23***	1.21***	0.28***	0.31***	0.31***	0.88***	0.91***	0.87***	0.59	0.60*	0.56*
	(1.94)	(2.20)	(3.03)	(2.61)	(2.99)	(3.42)	(3.05)	(3.79)	(3.99)	(3.21)	(3.73)	(3.78)	(1.67)	(1.75)	(1.73)
K = 12	0.42	0.46*	0.54***	1.11**	1.15***	1.13***	0.41***	0.45***	0.45***	0.91***	0.94***	0.90***	0.69*	0.70	0.59
	(1.62)	(1.88)	(2.67)	(2.61)	(2.99)	(3.51)	(2.74)	(3.52)	(3.71)	(3.33)	(3.83)	(3.92)	(1.80)	(1.57)	(1.65)

This table presents the average monthly returns, expressed as percentages, of early stage momentum strategies for the directed-degree (DD) and eigenvector (EIG). Panel A (Panel B) presents the average monthly returns from 2005 to 2009 (2010 to 2014). All stocks are ranked in ascending order at the end of each month based on their past 6-month cumulative returns. Stocks are then divided equally into three portfolios of equal weight and held for K months (1, 3, 6, 9, and 12). To mitigate microstructure bias, a 1-month interval was imposed between the end of the portfolio formation period and the start of the holding period. The monthly portfolio alphas were calculated by running a time-series regression of portfolio returns on risk factors, including the CAPM factor and the three factors of Fama and French (1993). Both raw and risk-adjusted returns were reported. The t statistics are adjusted for autocorrelation using the Newey and West (1987) method and are reported in parentheses, and *, **, and *** indicate significance at 0.1, 0.05, and 0.01 levels, respectively.

Table 10. Value-weighted returns for the various momentum strategies.

Panel A: Momentum returns
	MOM			DD			EIG
	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted
K = 1	0.60	0.62	0.68*	1.06**	1.09**	1.00**	0.91*	0.94**	0.82*
	(1.61)	(1.54)	(1.68)	(2.20)	(2.28)	(2.01)	(1.96)	(2.01)	(1.78)
K = 3	0.41	0.44	0.47	0.62*	0.66*	0.47*	0.62	0.67*	0.58
	(1.18)	(1.26)	(1.41)	(1.86)	(1.77)	(1.73)	(1.65)	(1.68)	(1.53)
K = 6	0.31	0.33	0.37	0.54	0.58*	0.43	0.66	0.71*	0.63*
	(0.93)	(1.02)	(1.24)	(1.43)	(1.76)	(1.23)	(1.62)	(1.81)	(1.80)
K = 9	0.24	0.27	0.29	0.55	0.60*	0.44	0.74*	0.79**	0.69**
	(0.73)	(0.84)	(1.01)	(1.53)	(1.71)	(1.37)	(1.87)	(2.08)	(2.04)
K = 12	0.08	0.11	0.13	0.44*	0.48*	0.32	0.62*	0.67*	0.56*
	(0.26)	(0.36)	(0.48)	(1.78)	(1.82)	(1.56)	(1.73)	(1.77)	(1.69)
Panel B: Difference in momentum returns
	DD ‒ MOM			EIG ‒ MOM			EIG ‒ DD
	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted
K = 1	0.46*	0.48*	0.33*	0.32*	0.33*	0.15	−0.14	−0.15	−0.18
	(1.85)	(1.79)	(1.70)	(1.93)	(1.85)	(1.44)	(−0.34)	(−0.35)	(−0.41)
K = 3	0.22*	0.23*	0.01	0.22*	0.24*	0.10	−0.00	0.01	0.11
	(1.71)	(1.75)	(1.03)	(1.71)	(1.80)	(1.36)	(−0.00)	(0.04)	(0.35)
K = 6	0.24	0.26	0.06	0.36	0.39*	0.27	0.11	0.13	0.21
	(1.27)	(1.33)	(1.37)	(1.32)	(1.74)	(1.29)	(0.39)	(0.44)	(0.75)
K = 9	0.32*	0.34*	0.15	0.51*	0.53**	0.40	0.19	0.20	0.25
	(1.71)	(1.80)	(0.89)	(1.90)	(2.01)	(1.63)	(0.71)	(0.73)	(0.95)
K = 12	0.37**	0.38**	0.19	0.55**	0.57**	0.42*	0.18	0.19	0.23
	(2.14)	(2.20)	(1.25)	(2.06)	(2.14)	(1.74)	(0.69)	(0.73)	(0.91)

This table presents the value-weighted monthly returns, expressed as percentages, of the simple momentum strategy (MOM) and early stage momentum strategies for directed-degree (DD) and eigenvector (EIG). All stocks are ranked in ascending order at the end of each month based on their past 6-month cumulative returns. Stocks are then divided equally into three portfolios of equal weight and held for K months (1, 3, 6, 9, and 12). To mitigate microstructure bias, a 1-month interval was imposed between the end of the portfolio formation period and the start of the holding period. The stocks in the bottom one-third are assigned to the loser portfolio, and those in the top one-third are assigned to the winner portfolio. The MOM represents a zero-investment portfolio that longs the winner portfolio and shorts the loser portfolio. The stocks in the winner and loser portfolios are further sorted into three subgroups based on their centralization index values: high, middle, and low centralization portfolios. DD represents a zero-investment portfolio that longs winners with low-directed-degree centralization and shorts losers with high-directed-degree centralization. EIG represents a zero-investment portfolio that longs winners with low eigenvector centralization, and shorts losers with high eigenvector centralization. Panel A reports the monthly value-weighted returns for momentum strategies. Panel B reports the difference in returns between the two momentum strategies. Monthly portfolio alphas were calculated by running a time-series regression of portfolio returns on risk factors, including the CAPM factor and the three factors of Fama and French (1993). Both raw and risk-adjusted returns are reported. The t statistics are adjusted for autocorrelation using the Newey and West (1987) method and are reported in parentheses, and *, **, and *** indicate significance at 0.1, 0.05, and 0.01 levels, respectively.

Table 11. Returns for an alternative momentum strategy.

Panel A: Momentum returns
	MOM			DD			EIG
	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted
K = 1	0.31	0.33	0.32	1.23***	1.31***	1.31***	0.88*	0.98**	0.82**
	(1.00)	(1.11)	(1.05)	(3.11)	(3.66)	(3.58)	(1.92)	(2.41)	(2.07)
K = 3	0.24	0.27	0.29	0.88**	0.97***	0.98***	0.84**	0.94***	0.79**
	(0.86)	(0.99)	(1.08)	(2.48)	(3.07)	(3.07)	(2.05)	(2.63)	(2.27)
K = 6	0.19	0.21	0.23	0.70**	0.77***	0.78***	0.82**	0.90***	0.76***
	(0.88)	(0.99)	(1.19)	(2.43)	(3.01)	(3.16)	(2.38)	(3.01)	(2.63)
K = 9	0.08	0.10	0.14	0.55**	0.63***	0.66***	0.84**	0.93***	0.79***
	(0.41)	(0.56)	(0.82)	(2.06)	(2.80)	(3.25)	(2.55)	(3.32)	(2.97)
K = 12	−0.01	0.01	0.06	0.50*	0.57**	0.60***	0.82**	0.91***	0.78***
	(−0.07)	(0.07)	(0.36)	(1.91)	(2.57)	(3.02)	(2.61)	(3.44)	(3.05)
Panel B: Difference in momentum returns
	DD ‒ MOM			EIG ‒ MOM			EIG ‒ DD
	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted	Raw	CAPM-adjusted	FF3-adjusted
K = 1	0.92***	0.98***	0.99***	0.57*	0.65**	0.50**	−0.35	−0.33	−0.49
	(3.58)	(4.20)	(4.13)	(1.95)	(2.60)	(2.16)	(−1.12)	(−1.06)	(−1.60)
K = 3	0.64***	0.70***	0.68***	0.60**	0.67***	0.49**	−0.04	−0.03	−0.18
	(3.32)	(4.26)	(4.09)	(2.21)	(2.86)	(2.41)	(−0.17)	(−0.11)	(−0.77)
K = 6	0.52***	0.57***	0.55***	0.63**	0.70***	0.53***	0.12	0.13	−0.02
	(3.17)	(4.20)	(4.05)	(2.51)	(3.21)	(2.77)	(0.58)	(0.68)	(−0.12)
K = 9	0.48***	0.53***	0.52***	0.77***	0.83***	0.65***	0.29	0.30	0.13
	(3.16)	(4.37)	(4.43)	(3.13)	(3.86)	(3.53)	(1.46)	(1.55)	(0.79)
K = 12	0.51***	0.56***	0.54***	0.84***	0.90***	0.72***	0.33*	0.34*	0.17
	(3.53)	(4.71)	(4.63)	(3.47)	(4.23)	(3.92)	(1.68)	(1.79)	(1.09)

This table presents the average monthly returns, expressed as percentages, of the simple momentum strategy (MOM) and early stage momentum strategies for directed-degree (DD) and eigenvector (EIG). All stocks are ranked in ascending order at the end of each month, based on their past 3-month cumulative returns. Stocks are then divided equally into three portfolios of equal weight and held for K months (1, 3, 6, 9, and 12). To mitigate microstructure bias, a 1-month interval was imposed between the end of the portfolio formation period and the start of the holding period. The stocks in the bottom one-third are assigned to the loser portfolio, and those in the top one-third are assigned to the winner portfolio. The MOM represents a zero-investment portfolio that longs the winner portfolio and shorts the loser portfolio. The stocks in the winner and loser portfolios are further sorted into three subgroups based on their centralization index values: high, middle, and low centralization portfolios. DD represents a zero-investment portfolio that longs winners with low-directed-degree centralization and shorts losers with high-directed-degree centralization. EIG represents a zero-investment portfolio that longs winners with low eigenvector centralization, and shorts losers with high eigenvector centralization. Panel A reports the monthly value-weighted returns for momentum strategies. Panel B reports the difference in returns between the two momentum strategies. Monthly portfolio alphas were calculated by running a time-series regression of portfolio returns on risk factors, including the CAPM factor and the three factors of Fama and French (1993). Both raw and risk-adjusted returns are reported. The t statistics are adjusted for autocorrelation using the Newey and West (1987) method and are reported in parentheses, and *, **, and *** indicate significance at 0.1, 0.05, and 0.01 levels, respectively.

Table 12. Limits to arbitrage and momentum returns.

	High arbitrage friction		Low arbitrage friction		Difference
	DD	EIG	DD	EIG	DD	EIG
K = 1	0.93***	0.68***	0.17	0.36	0.76**	0.32*
	(5.48)	(3.01)	(0.40)	(0.77)	(2.09)	(1.77)
K = 3	0.78***	0.69***	0.37	0.30	0.40**	0.40*
	(4.84)	(3.15)	(1.08)	(0.70)	(1.97)	(1.75)
K = 6	0.59***	0.53**	0.11	0.28	0.49*	0.24*
	(4.13)	(2.35)	(0.33)	(0.70)	(1.75)	(1.78)
K = 9	0.48***	0.40*	0.11	0.33	0.38*	0.08
	(3.34)	(1.78)	(0.35)	(0.80)	(1.71)	(1.62)
K = 12	0.43***	0.32*	0.12	0.30	0.31	0.02
	(3.11)	(1.82)	(0.40)	(0.76)	(1.57)	(1.05)

This table presents the average monthly returns, expressed as percentages, of early stage momentum strategies for the directed-degree (DD) and eigenvector (EIG) under various arbitrage-constraint conditions. We developed an arbitrage-friction index through principal components analysis, incorporating six firm-specific characteristics: idiosyncratic volatility, bid-ask spread, Amihud’s illiquidity measure, institutional ownership, shareholder count, and trading volume. Following the creation of the arbitrage-friction index, stocks are equally divided into two groups: high and low arbitrage friction, with the index’s median acting as the separator. We divide the stocks within each arbitrage-friction group into two subportfolios based on their past-6-month cumulative returns. The portfolios are equally weighted and held for K (1, 3, 6, 9, and 12) months. To avoid microstructure bias, a 1-month lag is imposed between the end of the portfolio formation period and the beginning of the holding period. Stocks in the bottom half are categorized as loser portfolios, whereas those in the top half are included in the winner portfolio. We further classify stocks within the winner and loser portfolios into two subportfolios, based on their centralization index. DD represents a zero-investment portfolio that longs winners with low-directed-degree centralization and shorts losers with high-directed-degree centralization. EIG represents a zero-investment portfolio that longs winners with low eigenvector centralization, and shorts losers with high eigenvector centralization. The raw returns are reported. The t statistics are adjusted for autocorrelation using the Newey and West (1987) method and are reported in parentheses, and *, **, and *** indicate significance at 0.1, 0.05, and 0.01 levels, respectively.

Table 13. Sentiment variation and momentum returns.

	HIGH		LOW		Difference
	DD	EIG	DD	EIG	DD	EIG
K = 1	1.40**	1.36**	0.57	0.09	0.83**	1.27**
	(2.44)	(2.32)	(0.86)	(0.12)	(2.08)	(2.02)
K = 3	1.23**	1.22**	0.61	0.32	0.62**	0.90**
	(2.16)	(2.06)	(1.06)	(0.45)	(1.97)	(1.97)
K = 6	1.07*	1.10*	0.15	0.26	0.92*	0.83*
	(1.89)	(1.76)	(0.25)	(0.35)	(1.83)	(1.76)
K = 9	1.11*	1.25*	−0.03	0.01	1.14*	1.25
	(1.86)	(1.91)	(-0.38)	(0.01)	(1.72)	(1.52)
K = 12	1.16*	1.33*	0.00	0.10	1.16	1.23
	(1.81)	(1.90)	(0.20)	(0.13)	(1.36)	(1.40)

This table presents the average monthly returns, expressed as percentages, of early stage momentum strategies for the directed-degree (DD) and eigenvector (EIG) under various sentiment conditions. All stocks are ranked in ascending order at the end of each month based on their past 6-month cumulative returns. Stocks are then divided equally into three portfolios of equal weight and held for K months (1, 3, 6, 9, and 12). To mitigate microstructure bias, a 1-month interval was imposed between the end of the portfolio formation period and the start of the holding period. The stocks in the bottom one-third are assigned to the loser portfolio, and those in the top one-third are assigned to the winner portfolio. The MOM represents a zero-investment portfolio that longs the winner portfolio and shorts the loser portfolio. The stocks in the winner and loser portfolios are further sorted into three subgroups based on their centralization index values: high, middle, and low centralization portfolios. DD represents a zero-investment portfolio that longs winners with low-directed-degree centralization and shorts losers with high-directed-degree centralization. EIG represents a zero-investment portfolio that longs winners with low eigenvector centralization, and shorts losers with high eigenvector centralization. The Taiwan Consumer Confidence Index is used to partition the sample period into two distinct periods: high- and low-sentiment periods. To determine whether each holding month belongs to a high- or low-sentiment period, we count the number of K formation months that correspond to the respective periods. If the K formation months are classified as high- or low-sentiment months for all portfolios, the holding month is classified as a high- or low-sentiment month. However, if some portfolios exhibit K formation months that belong to high-sentiment periods, while others exhibit K formation months belonging to low-sentiment periods, then the holding month is deemed as a mild period. To test whether the average returns are equal to zero in each period, we regress the time series of average monthly returns on the HIGH, LOW, and MILD dummy variables with no intercept. To test whether the average returns in the high- and low-sentiment periods are equal, we regress the time series of average monthly returns on the MILD and HIGH dummy variables and an intercept. The t statistics are adjusted for autocorrelation using the Newey and West (1987) method and are reported in parentheses, and *, **, and *** indicate significance at 0.1, 0.05, and 0.01 levels, respectively.

Table 14. Market states and momentum returns.

	UP		DOWN		Difference
	DD	EIG	DD	EIG	DD	EIG
K = 1	1.17**	1.03**	0.35	−0.30	0.82*	1.33*
	(2.49)	(2.03)	(0.34)	(−0.28)	(1.82)	(1.72)
K = 3	1.15***	1.32***	−0.42	−0.92	1.56***	2.23***
	(4.84)	(4.99)	(−0.87)	(−1.61)	(2.91)	(3.56)
K = 6	0.86***	1.21***	−0.36	−0.33	1.22***	1.53***
	(5.78)	(6.91)	(−1.10)	(−0.88)	(3.37)	(3.73)
K = 9	0.54***	1.01***	−0.04	0.10	0.58**	0.91***
	(4.54)	(7.20)	(−0.16)	(0.36)	(2.08)	(2.80)
K = 12	0.36***	0.79***	0.30*	0.50**	0.06	0.30
	(3.53)	(6.71)	(1.67)	(2.30)	(1.30)	(1.21)

This table presents the average monthly returns, expressed as percentages, of early stage momentum strategies for the directed-degree (DD) and eigenvector (EIG) under various market conditions. All stocks are ranked in ascending order at the end of each month based on their past 6-month cumulative returns. Stocks are then divided equally into three portfolios of equal weight and held for K months (1, 3, 6, 9, and 12). To mitigate microstructure bias, a 1-month interval was imposed between the end of the portfolio formation period and the start of the holding period. The stocks in the bottom one-third are assigned to the loser portfolio, and those in the top one-third are assigned to the winner portfolio. The MOM represents a zero-investment portfolio that longs the winner portfolio and shorts the loser portfolio. The stocks in the winner and loser portfolios are further sorted into three subgroups based on their centralization index values: high, middle, and low centralization portfolios. DD represents a zero-investment portfolio that longs winners with low-directed-degree centralization and shorts losers with high-directed-degree centralization. EIG represents a zero-investment portfolio that longs winners with low eigenvector centralization, and shorts losers with high eigenvector centralization. We adopt Cooper, Gutierrez, and Hameed’s (2004) method for classifying market conditions. If the 3-year value-weighted market return prior to the beginning of the strategy’s holding period is positive or negative, the market state is classified as UP or DOWN. To test whether the average returns are equal to zero in each period, we regress the time series of the average monthly returns on UP, DOWN, and MILD dummy variables with no intercept. To test whether the average returns in UP-market and DOWN market periods are equal, we regress the time series of the average monthly returns on the MILD and UP dummy variables and an intercept. The t statistics are adjusted for autocorrelation using the Newey and West (1987) method and are reported in parentheses, and *, **, and *** indicate significance at 0.1, 0.05, and 0.01 levels, respectively.

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Data availability statement

The data that support the findings of this study are available on request from the corresponding author, W.R. Liu, upon reasonable request. The data are not publicly available due to the presence of information that may compromise privacy.