Trade co-occurrence, trade flow decomposition and conditional order imbalance in equity markets

Figures & data

Figure 1. Illustration of trade co-occurrence. This figure visualizes the idea of trade co-occurrence; given a user-defined neighbourhood size δ, trade $x_b$ arrives within the δ-neighbourhood of trade $x_a$, and thus they co-occur. In contrast, trade $x_d$ locates outside $x_a$'s neighbourhood, and thus the two trades do not co-occur. Both trades $x_b$ and $x_c$ co-occur with trade $x_a$, but they do not co-occur with each other.

Figure 2. Illustration of trade types, conditioning on co-occurrence. We showcase the distinct categorical labels of trade $x_a$. Colour indicates the stock corresponding to a trade. Thus $x_b$ is for the same stock as $x_a$, while $x_c$ is for a different stock. First line: $x_a$ is an isolated (iso) trade with empty δ-neighbourhood; second to fourth lines: $x_a$ is a non-isolated (nis) trade with nonempty δ-neighbourhood; second line: $x_a$ is a non-self-isolated (‘nis-s’) trade with only other trades for the same stock in its δ-neighbourhood; third line: $x_a$ is an non-cross-isolated (‘nis-c’) trade with only other trades for the different stocks in its δ-neighbourhood; last line: $x_a$ is a non-both-isolated (‘nis-b’) trade with both other trades for the same and different stocks in its δ-neighbourhood.

Table 1. Difference between null and empirical probability.

δ (ms)	0.05	0.0725	0.125	0.25	0.5	1	5	50
Average weighted distance	0.09	0.10	0.11	0.12	0.13	0.14	0.11	0.12

Note: This table reports the average weighted distance for different values δ indicated in the columns.

Figure 3. Co-occurrence probability: null versus empirical. The table plots the null and empirical probabilities of each type of trades, over 5 minute intervals, averaged over stocks and days, for selected values of δs.

Table 2. Null and empirical probability of each type of trade flows.

	Null (%)	Empirical (%)
iso	81.35	28.55
nis	18.65	71.45
nis-s	0.09	29.75
nis-c	18.53	17.27
nis-b	0.03	24.43

Note: This table shows the percentage, averaged over both days and stocks, of each type of trades under the null model and from the real data respectively, with $\delta =1\mathrm{ms}$ and T = 5 min.

Figure 4. Intraday distributions of the number of each type of trades. We calculate numbers of each types of trades in percentage of the total number of trades, for non-overlapping 5-min intervals during normal trading hours from 9:30 to 16:00 for all stock over the period from 2017-01-03 to 2020-12-31. This figure plots intraday 5-min counts of different types of trades, averaged over both time series and cross-section.

Table 3. Summary statistics for all groups of trades.

	Mean	Median	Std. dev.
Number of trades	3906.00	3010.85	3425.05
Percentage of iso trades	28.55	28.31	3.91
Percentage of nis trades	71.45	71.69	3.91
Percentage of nis-s trades	29.75	29.34	6.34
Percentage of nis-c trades	17.27	17.29	3.64
Percentage of nis-b trades	24.43	23.95	4.69

Note: This table documents the time-series average of the daily cross-sectional statistics of each type of trades. Our data include records of trades within normal trading hours of 457 stocks from 2017-01-03 to 2020-12-31.

Table 4. Summary statistics for all groups of trades and order imbalances.

Panel A: Statistics of daily order imbalances
	Mean	Median	Std. dev.
all	−0.0012	−0.0014	0.1144
iso	−0.0095	−0.0091	0.1561
nis	0.0030	0.0022	0.1196
nis-s	−0.0058	−0.0054	0.1445
nis-c	0.0161	0.0130	0.1622
nis-b	0.0119	0.0091	0.1724
Panel B: Average partial autocorrelation of order imbalances
lag	1	2	3
all	0.274	0.093	0.043
iso	0.243	0.097	0.048
nis	0.273	0.095	0.045
nis-s	0.226	0.092	0.049
nis-c	0.297	0.116	0.062
nis-b	0.254	0.099	0.056

Note: This table shows the summary statistics of COIs from 2017-01-03 to 2020-12-31 for the selected 457 stocks. Panel A documents the mean, median and standard deviations of COIs. Panel B presents the partial autocorrelations of COIs, averaged over all stocks.

Figure 5. Pearson correlation of order imbalances. For each type of order imbalance, we first consider the vector of daily values during 2017-01-03 to 2020-12-31, then compute the correlation matrix and finally average across all stocks.

Table 5. Contemporaneous regression against individual COIs.

	${\beta }_{\rho }$	t	adj.$R^2$ (%)
all	2.22${}^{\ast \ast \ast }$	34.22	5.66
iso	1.70${}^{\ast \ast \ast }$	37.39	5.91
nis	1.68${}^{\ast \ast \ast }$	24.78	4.48
nis-s	0.92${}^{\ast \ast \ast }$	16.62	3.44
nis-c	1.04${}^{\ast \ast \ast }$	22.76	3.94
nis-b	0.80${}^{\ast \ast \ast }$	18.41	3.50
control variables			2.63

Note: This table summarizes the coefficients of COIs by regressing again each type of COIs individually following equation (6(6) $\begin{array}{rl}{r}_{i,t}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t+{\beta }_c\mathrm{CM}{A}_t\\ & +{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t},\end{array}$(6) ). As a benchmark, the last row shows the result of regressing against control variables only. ‘${\beta }_{\rho }$’ denotes the regression coefficients and the superscript ${}^{\ast \ast \ast }$ indicates significant at $1\mathrm{\%}$ using a two-tailed t-test. ‘t’ denotes the t-value of each coefficient. ‘adj.$R^2$’ denotes the adjusted $R^2$ of regressions. Additional evaluation metrics are available in table A8.

Table 6. Contemporaneous regression against multiple COIs.

	iso	nis-c	nis-b	nis-s	nis	all	adj.$R^2$ (%)
iso + nis	1.42${}^{\ast \ast \ast }$				0.75${}^{\ast \ast \ast }$		6.19
	(28.35)				(10.79)
iso + nis-c	1.51${}^{\ast \ast \ast }$	0.50${}^{\ast \ast \ast }$					6.17
	(31.33)	(10.76)
iso + nis-c + nis-b	1.51${}^{\ast \ast \ast }$	0.30${}^{\ast \ast \ast }$	0.35${}^{\ast \ast \ast }$				6.29
	(31.19)	(6.35)	(8.85)
iso + nis-c + nis-b + nis-s	1.66${}^{\ast \ast \ast }$	0.21${}^{\ast \ast \ast }$	0.46${}^{\ast \ast \ast }$	−0.29${}^{\ast \ast \ast }$			6.33
	(32.06)	(5.09)	(12.75)	(−5.61)
iso + nis-c + nis-b + nis-s + nis	1.67${}^{\ast \ast \ast }$	−0.45${}^{\ast \ast \ast }$	−0.38${}^{\ast \ast \ast }$	−1.51${}^{\ast \ast \ast }$	2.74${}^{\ast \ast \ast }$		6.58
	(32.39)	(−7.47)	(−6.32)	(−14.24)	(13.35)
iso + nis-c + nis-b + nis-s + all	0.88${}^{\ast \ast \ast }$	−0.21${}^{\ast \ast \ast }$	−0.08	−1.06${}^{\ast \ast \ast }$		2.54${}^{\ast \ast \ast }$	6.51
	(8.82)	(−3.93)	(−1.43)	(−11.10)		(9.70)
iso + nis-c + nis-b + nis-s + nis + all	1.76${}^{\ast \ast \ast }$	−0.45${}^{\ast \ast \ast }$	−0.38${}^{\ast \ast \ast }$	−1.53${}^{\ast \ast \ast }$	2.95${}^{\ast \ast \ast }$	−0.27	6.59
	(10.97)	(−7.53)	(−6.55)	(−14.69)	(8.51)	(−0.61)

Note: We run regressions against multiple COIs following equation (6(6) $\begin{array}{rl}{r}_{i,t}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t+{\beta }_c\mathrm{CM}{A}_t\\ & +{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t},\end{array}$(6) ). The first column states the types of COIs input in each regression. Regression coefficients of different types of COIs are listed in the following columns as indicated by the column names. The superscripts ${}^{\ast }{,}^{\ast \ast }{.}^{\ast \ast \ast }$ indicate significant at $10\mathrm{\%},5\mathrm{\%}$ and $1\mathrm{\%}$ respectively using a two-tailed t-test and corresponding t-values are reported in the parentheses below. The last column‘adj.$R^2$’ presents the adjusted $R^2$ of regressions. Additional evaluation metrics are available in table A8.

Table 7. Predictive regression against individual COIs.

	${\beta }_{\rho }$	t	adj.$R^2$ (%)
all	0.00	0.07	0.1013
iso	0.08${}^{\ast \ast }$	2.30	0.1090
nis	−0.05	−0.85	0.1030
nis-s	0.04	0.94	0.1027
nis-c	−0.05	−1.44	0.1049
nis-b	−0.08${}^{\ast \ast }$	−1.97	0.1094
Control variables			0.1015

Note: This table summarizes the coefficients to COIs by regressing again each type of COIs individually following equation (7(7) $\begin{array}{rl}{r}_{i,t+1}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\tau }{r}_{i,t}+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t\\ & +{\beta }_c\mathrm{CM}{A}_t+{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t+1},\end{array}$(7) ). As a benchmark, the last row shows the result of regressing against control variables only. ‘${\beta }_{\rho }$’ denotes the regression coefficients and the superscript ${}^{\ast \ast \ast }$ indicates significant at $1\mathrm{\%}$ using a two-tailed t-test. ‘t’ denotes the t-value of each coefficient. ‘adj.$R^2$’ denotes the adjusted $R^2$ of regressions. Additional evaluation metrics are available in table A8.

Table 8. Predictive regression against multiple COIs.

	iso	nis-c	nis-b	nis-s	nis	all	adj.$R^2$ (%)
iso + nis	0.13${}^{\ast \ast \ast }$				−0.14${}^{\ast \ast }$		0.1182
	(4.32)				(−2.26)
iso + nis-c	0.12${}^{\ast \ast \ast }$	−0.10${}^{\ast \ast }$					0.1185
	(3.00)	(−2.31)
iso + nis-c + nis-b	0.12${}^{\ast \ast \ast }$	−0.05	−0.07${}^{\ast }$				0.1231
	(3.04)	(−1.22)	(−1.79)
iso + nis-c + nis-b + nis-s	0.11${}^{\ast \ast \ast }$	−0.05	−0.08${}^{\ast \ast }$	0.02			0.1230
	(3.18)	(−1.2)	(−2.05)	(0.56)
iso + nis-c + nis-b + nis-s + nis	0.11${}^{\ast \ast \ast }$	−0.03	−0.05	0.05	−0.07		0.1230
	(3.14)	(−0.6)	(−1.21)	(0.62)	(−0.45)
iso + nis-c + nis-b + nis-s + all	0.16${}^{\ast \ast }$	−0.02	−0.05	0.06		−0.15	0.1234
	(2.41)	(−0.48)	(−1.18)	(0.95)		(−0.77)
iso + nis-c + nis-b + nis-s + nis + all	0.21${}^{\ast \ast }$	−0.04	−0.06	0.04	0.16	−0.31	0.1234
	(2.26)	(−0.72)	(−1.37)	(0.49)	(0.77)	(−1.12)

Note: We run regressions against multiple COIs following equation (7(7) $\begin{array}{rl}{r}_{i,t+1}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\tau }{r}_{i,t}+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t\\ & +{\beta }_c\mathrm{CM}{A}_t+{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t+1},\end{array}$(7) ). The first column states the types of COIs input in each regression. Regression coefficients for different types of COIs are listed in the following columns as indicated by the column names. The superscripts ${}^{\ast }{,}^{\ast \ast }{.}^{\ast \ast \ast }$ indicate significant at $10\mathrm{\%},5\mathrm{\%}$ and $1\mathrm{\%}$ respectively using a two-tailed t-test and corresponding t-values are reported in the parentheses below. The last column‘adj.$R^2$’ presents the adjusted $R^2$ of regressions.

Table 9. Summary of single-sort portfolios.

Panel A: Annualized returns
	Low	2	3	4	High
all	−2.13	−2.70	−3.17	−5.12	−1.81
iso	−4.52	−4.72	−4.37	−3.57	2.24
nis	0.05	−4.44	−2.16	−4.04	−4.35
nis-s	−3.98	−4.20	−2.64	−4.50	0.39
nis-c	2.33	−3.63	−4.28	−5.69	−3.64
nis-b	1.55	−1.93	−4.73	−3.87	−5.98
Panel B: Average daily COIs
all	−0.16	−0.06	0.00	0.05	0.16
iso	−0.23	−0.09	−0.01	0.07	0.21
nis	−0.16	−0.06	0.00	0.06	0.17
nis-s	−0.21	−0.08	−0.01	0.07	0.19
nis-c	−0.18	−0.05	0.02	0.08	0.22
nisb	−0.21	−0.07	0.01	0.09	0.24

Note: This table shows the statistics of COI-sorted quintile portfolios. Each row contains five portfolios constructed by sorting all stocks every day by the type, indicated by its row index, of COI on the previous day from low to high and allocating each stock to the corresponding quintile portfolio indicated by the column names. The breakpoints are $20\mathrm{\%},40\mathrm{\%},60\mathrm{\%}$ and $80\mathrm{\%}$ of each type of COI calculated daily. Panel A presents the annualized return of each portfolio calculated by averaging its daily returns, from 2017-01-03 to 2020-12-31, and multiplying by 252. Panel B reports the average daily COIs of stocks included in portfolio over the sample period.

Table 10. Annualized returns of double-sort portfolios.

		Low	2	3	4	High		Low	2	3	4	High
		nis						nis-s
Low	iso	0.42	−8.85	−6.03	−6.51	−18.08	iso	−3.68	−7.31	−0.52	−7.20	−5.02
2		−2.53	−7.38	−2.45	−1.78	−15.45		−7.26	−6.62	−2.38	−1.49	−5.17
3		−2.11	−3.74	−2.15	−5.16	−10.44		−6.36	−4.11	−4.76	−5.08	−1.13
4		1.79	−3.65	−2.45	−5.63	−6.69		2.15	−1.67	−5.28	−10.61	1.07
High		5.25	6.14	5.06	−1.16	1.86		−4.12	4.00	2.94	0.29	2.51
	nis−c						nis−b
Low	iso	0.31	−1.93	−8.67	−10.36	−16.62	iso	1.05	−5.00	−6.75	−5.94	−13.04
2		4.73	−7.65	−5.02	−11.26	−10.33		0.00	−5.98	−2.92	−8.45	−10.93
3		−1.07	−5.19	−3.11	−6.27	−4.44		−0.77	−2.29	−5.66	−5.38	−8.37
4		2.67	−1.95	−4.98	−4.43	−7.48		−2.22	3.90	−5.93	−2.09	−9.44
High		18.20	3.22	1.95	−0.91	2.22		13.38	4.74	−1.03	1.92	1.31
	nis−s						nis−c
Low	nis	−1.43	3.15	−2.80	3.91	0.40	nis	2.55	−3.81	−3.46	6.11	3.79
2		−9.89	−6.34	−0.65	−5.19	13.00		−2.86	−5.19	−3.33	−5.28	−2.36
3		−2.04	−7.49	−1.60	−1.02	14.11		11.26	−2.75	−4.86	−5.74	−4.83
4		−10.97	−5.21	−4.43	−6.42	4.18		3.35	−4.77	−5.38	−6.71	−2.59
High		−7.43	1.55	−10.01	−8.20	−2.35		−14.04	−1.26	−4.37	−6.13	−4.25
	nis−b						nis−c
Low	nis	0.33	0.71	−3.88	3.43	5.35	nis−s	1.77	−7.17	−7.23	−5.56	−6.73
2		3.07	−7.56	−7.29	−9.52	−7.42		−0.29	−3.96	−4.18	−12.56	−0.36
3		1.93	4.12	−5.88	−2.19	−7.66		3.61	−2.44	−4.17	−3.78	−5.28
4		−1.78	7.22	−1.70	−6.60	−7.75		1.44	−7.25	−9.03	−6.97	−4.38
High		1.10	−1.40	−8.36	1.96	−5.63		4.67	3.7	3.44	−2.74	−2.76
	nis−b						nis−b
Low	nis−s	−1.30	−6.55	−8.65	−6.51	−1.94	nis−c	3.37	−1.65	0.10	7.58	−2.30
2		2.83	−4.92	−7.07	−5.63	−9.77		−2.70	−2.01	−1.37	−5.73	−8.39
3		2.07	0.47	−2.67	−5.66	−8.74		−0.92	−0.96	−8.62	−3.71	−7.54
4		1.27	−1.30	−5.71	−4.40	−8.50		3.75	−0.49	−7.94	−7.82	−7.70
High		0.61	16.53	−0.39	1.91	−3.37		6.87	−8.56	−3.42	−2.60	−5.45

Note: This table presents annualized returns of double-sort portfolios based on every pair of COIs. Each panel contains $5\times 5$ double-sort portfolios. To construct the portfolios, we sort all stocks every day by the two types, indicated by its row index and column name, of COIs on the previous day, from low to high, to five quintiles independently. Then intersections of the two sorts create 25 double-sort portfolios. The annualized return of each portfolio is calculated by averaging its daily returns, from 2017-01-03 to 2020-12-31, and multiplying by 252.

Table 11. Profitability of long-short portfolios.

Panel A: Annualized returns
	iso	nis	nis-s	nis-c	nis-b
iso	6.76	23.33	0.83	34.87	26.43
nis		4.40	0.91	6.90	5.99
nis-s			4.38	4.57	2.08
nis-c				6.00	8.89
nis-b					7.50
Panel B: Annualized Sharpe ratios
iso	1.20	1.08	−0.04	1.79	1.74
nis		0.48	−0.10	0.71	0.65
nis-s			0.47	0.37	0.07
nis-c				0.71	0.87
nis-b					0.97

Note: This table shows the annualized returns and Sharpe ratios of the long-short portfolios sorted on COIs indicated by the corresponding row indices and column names. The on- and off-diagonal values are for single- and double-sort portfolios respectively. Panel A presents the annualized return of portfolios calculated by averaging their daily returns, from 2017-01-03 to 2020-12-31, and multiplying by 252. Panel B reports the annualized Sharpe ratios over the sample period calculated by equation (8(8) $S{R}_p:=\frac{\mathit{mean}\mathit{(}{\mathit{R}}_{\mathit{p}\mathit{,}\mathit{t}}\mathit{)}\mathit{-}{\mathit{R}}_{\mathit{f}}}{\mathit{std}\mathit{(}{\mathit{R}}_{\mathit{p}\mathit{,}\mathit{t}}\mathit{)}}\times \sqrt{252},$(8) ).

Table 12. Abnormal returns of long-short portfolios.

Annualized alpha
	iso	nis	nis-s	nis-c	nis-b
iso	6.49${}^{\ast \ast \ast }$	20.36${}^{\ast \ast }$	1.42	29.72${}^{\ast \ast \ast }$	25.50${}^{\ast \ast \ast }$
	(3.12)	(1.96)	(0.15)	(3.25)	(3.86)
nis		2.83	−0.48	4.23	4.33
		(1.08)	(−0.17)	(1.17)	(1.41)
nis-s			2.53	1.51	0.79
			(1.05)	(0.40)	(0.25)
nis-c				2.35	6.46
				(0.77)	(1.56)
nis-b					5.98${}^{\ast \ast }$
					(2.14)

Note: This table documents the abnormal returns, α, of long-short portfolios after adjusting for factors. For each long-short portfolio, we run time series regressions on portfolio excess returns against factor returns $\begin{array}{rl}{R}_{p,t}-R_{f,t}& ={\alpha }_p+b_p\mathrm{MK}{T}_t+s_p\mathrm{SM}{B}_t+h_p\mathrm{HM}{L}_t+r_p\mathrm{RM}{W}_t\\ & +c_p\mathrm{CM}{A}_t+m_p\mathrm{MO}{N}_t+u_p\mathrm{UM}{D}_t+e_{p,t},\end{array}$ where ${\alpha }_p$ is the abnormal return of the portfolio, the explanatory variables are the market, size, value, profitability, investment and momentum factors, and $e_{p,t}$ is the idiosyncratic term. For inference, we apply the Newey–West estimator (Newey and West 1994) to correct for heteroscedasticity and autocorrelation in the residual terms. The on- and off-diagonal values are for single- and double-sort long-short portfolios respectively. The superscripts ${}^{\ast }$, ${}^{\ast \ast }$ and ${}^{\ast \ast \ast }$ indicate statistical significance at $10\mathrm{\%}$, $5\mathrm{\%}$ and $1\mathrm{\%}$, and the corresponding t-values are reported in the parentheses.

Figure 6. Cumulative returns of portfolios. This figure plots cumulative returns of five portfolios from 2017-01-03 to 2020-12-31. The portfolios include (1) ‘iso’: the long-short portfolio single-sorted on iso COI; (2) ‘iso$/$nis’: the long-short portfolio double-sorted on iso and nis COIs; (3) ‘iso$/$nis-c’: the long-short portfolio double-sorted on iso and nis-c COIs; (4) ‘all’: the long-short portfolio single-sorted on COI of undecomposed trade flows; (5) ‘return momentum’: the long-short portfolio single-sorted on previous day's returns; (6) ‘equal weight’: equally-weighted portfolio of the selected 457 stocks; (7) ‘SPY ETF (open to close)’: cumulative open to close returns of the SPDR S$\mathrm{\&}$P 500 ETF Trust which tracks the S$\mathrm{\&}$P 500 Index; (8) ‘SPY ETF’: cumulative close to close returns of the SPDR S$\mathrm{\&}$P 500 ETF.

Table 13. Abnormal returns of long-short portfolios.

	α	MKT	SMB	HML	RMW	CMA	MOM	UMD	$R^2$ (%)
iso	0.03${}^{\ast \ast \ast }$	−0.01	0.06${}^{\ast \ast }$	−0.01	0.03	−0.02	−0.02	0.11${}^{\ast \ast }$	6.24
	(3.12)	(−0.62)	(2.20)	(−0.35)	(1.16)	(−0.44)	(−0.83)	(2.15)
iso/nis	0.08${}^{\ast \ast }$	0.12${}^{\ast \ast }$	−0.11	−0.25${}^{\ast }$	0.04	0.45${}^{\ast \ast }$	−0.14	0.12	3.16
	(1.96)	(2.17)	(−0.85)	(−1.7)	(0.27)	(2.35)	(−0.94)	(0.56)
iso/nis-c	0.12${}^{\ast \ast \ast }$	0.02	0.26${}^{\ast \ast }$	−0.09	−0.03	0.15	0.04	−0.35${}^{\ast \ast }$	4.62
	(3.25)	(0.48)	(2.19)	(−0.73)	(−0.2)	(0.88)	(0.39)	(−2.42)
all	0.00	−0.04${}^{\ast \ast }$	0.15${}^{\ast \ast \ast }$	0.04${}^{\ast }$	0.09${}^{\ast \ast \ast }$	−0.11${}^{\ast \ast }$	0.01	0.00	13.35
	(−0.08)	(−2.24)	(3.17)	(1.65)	(3.16)	(−2.07)	(0.41)	(0.05)
return momentum	−0.01	−0.02*	−0.03	0.00	0.00	−0.03	0.01	1.71${}^{\ast \ast \ast }$	94.18
	(−0.96)	(−1.93)	(−1.57)	(−0.03)	(0.01)	(−1.25)	(0.40)	(68.38)
equal weight	−0.04${}^{\ast \ast }$	0.51${}^{\ast \ast \ast }$	0.14${}^{\ast \ast \ast }$	−0.07	−0.10	0.26${}^{\ast \ast \ast }$	−0.09${}^{\ast \ast }$	0.02	56.68
	(−2.19)	(11.21)	(3.01)	(−1.28)	(−1.52)	(2.96)	(−2.00)	(0.30)
SPY ETF (open to close)	−0.04${}^{\ast \ast }$	0.52${}^{\ast \ast \ast }$	−0.05	−0.21${}^{\ast \ast \ast }$	−0.13${}^{\ast \ast }$	0.21${}^{\ast \ast }$	−0.08	0.12	55.64
	(−2.04)	(10.93)	(−1.25)	(−3.26)	(−2.23)	(2.40)	(−1.51)	(1.62)
SPY ETF	−0.01${}^{\ast \ast \ast }$	0.98${}^{\ast \ast \ast }$	−0.1${}^{\ast \ast \ast }$	0.02${}^{\ast }$	0.04${}^{\ast \ast \ast }$	0.04${}^{\ast \ast \ast }$	−0.01${}^{\ast \ast }$	0.01	99.52
	(−3.81)	(136.87)	(−9.55)	(1.76)	(4.70)	(2.88)	(−2.16)	(0.57)

Note: This table documents the abnormal returns and relations of long-short portfolios with respect to factors. For each long-short portfolio, we run time series regressions on portfolio excess returns against factor returns $R_{p,t}-R_{f,t}={\alpha }_p+b_p\mathrm{MK}{T}_t+s_p\mathrm{SM}{B}_t+h_p\mathrm{HM}{L}_t+r_p\mathrm{RM}{W}_t+c_p\mathrm{CM}{A}_t+m_p\mathrm{MO}{N}_t+u_p\mathrm{UM}{D}_t+e_{p,t},$ where ${\alpha }_p$ is the abnormal return of the portfolio, the explanatory variables are the market, size, value, profitability, investment and momentum factors, the coefficients are the exposures to the corresponding factors and $e_{p,t}$ is the idiosyncratic term. For inference, we apply the Newey–West estimator (Newey and West 1994) to correct for heteroscedasticity and autocorrelation in the residual terms. The superscripts ${}^{\ast }$, ${}^{\ast \ast }$ and ${}^{\ast \ast \ast }$ indicate statistical significance at $10\mathrm{\%}$, $5\mathrm{\%}$ and $1\mathrm{\%}$, and the corresponding t-values are reported in the parentheses. The last column shows the $R^2$ (unadjusted) of each regression.

Table B1. Description of the sample universe.

	Number of stocks
Total	457
Sectors:
Communication services	24
Consumer discretionary	57
Consumer staples	32
Energy	24
Financials	62
Health care	55
Industrials	60
Information technology	64
Materials	24
Real estate	29
Utilities	26

Note: This table summarizes the total number of stocks as well as the number of stocks grouped by their sector membership.

Table C1. Yearly contemporaneous regression against individual COIs.

2017	${\beta }_{\rho }$	t	adj.$R^2$ (%)
all	2.43${}^{\ast \ast \ast }$	32.94	8.39
iso	1.96${}^{\ast \ast \ast }$	33.48	9.12
nis	1.72${}^{\ast \ast \ast }$	26.32	4.94
nis-s	1.10${}^{\ast \ast \ast }$	22.55	3.12
nis-c	0.89${}^{\ast \ast \ast }$	17.87	2.79
nis-b	0.66${}^{\ast \ast \ast }$	16.02	2.08
control variables			0.52
2018	${\beta }_{\rho }$	t	adj.$R^2$ (%)
all	2.49${}^{\ast \ast \ast }$	33.07	6.44
iso	1.73${}^{\ast \ast \ast }$	27.69	5.68
nis	2.06${}^{\ast \ast \ast }$	27.75	4.95
nis-s	0.91${}^{\ast \ast \ast }$	15.34	2.24
nis-c	1.48${}^{\ast \ast \ast }$	18.47	4.07
nis-b	1.17${}^{\ast \ast \ast }$	20.58	3.50
control variables			1.03
2019	${\beta }_{\rho }$	t	adj.$R^2$ (%)
all	2.53${}^{\ast \ast \ast }$	28.20	7.03
iso	1.76${}^{\ast \ast \ast }$	26.60	6.88
nis	2.05${}^{\ast \ast \ast }$	23.02	5.04
nis-s	1.38${}^{\ast \ast \ast }$	18.74	3.64
nis-c	0.98${}^{\ast \ast \ast }$	16.64	2.78
nis-b	1.02${}^{\ast \ast \ast }$	15.98	2.86
control variables			0.97
2020	${\beta }_{\rho }$	t	adj.$R^2$ (%)
all	1.47${}^{\ast \ast \ast }$	25.91	5.35
iso	1.38${}^{\ast \ast \ast }$	35.86	5.76
nis	0.97${}^{\ast \ast \ast }$	17.11	5.05
nis-s	0.21${}^{\ast \ast \ast }$	4.15	4.82
nis-c	1.11${}^{\ast \ast \ast }$	30.41	5.43
nis-b	0.54${}^{\ast \ast \ast }$	13.06	4.94
control variables			4.80

Note: This table summarizes the coefficients for COIs by regressing again each type of COIs individually following equation (6(6) $\begin{array}{rl}{r}_{i,t}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t+{\beta }_c\mathrm{CM}{A}_t\\ & +{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t},\end{array}$(6) ) on a yearly basis for 2017, 2018, 2019 and 2020 respectively. As a benchmark, the last row of each panel shows the result of regressing against control variables only. ‘${\beta }_{\rho }$’ denotes the regression coefficients and the superscript ${}^{\ast \ast \ast }$ indicates significant at $1\mathrm{\%}$ using a two-tailed t-test. ‘t’ denotes the t-value of each coefficient. ‘adj.$R^2$’ denotes the adjusted $R^2$ of regressions.

Table C2. Yearly contemporaneous regression against multiple COIs.

2017	iso	nis-c	nis-b	nis-s	nis	all	adj.$R^2$ (%)
iso + nis	1.68${}^{\ast \ast \ast }$				0.72${}^{\ast \ast \ast }$		9.70
	(28.47)				(11.71)
iso + nis-c	1.83${}^{\ast \ast \ast }$	0.39${}^{\ast \ast \ast }$					9.51
	(32.12)	(8.67)
iso + nis-c + nis-b	1.83${}^{\ast \ast \ast }$	0.18${}^{\ast \ast \ast }$	0.33${}^{\ast \ast \ast }$				9.78
	(32.17)	(4.77)	(10.17)
iso + nis-c + nis-b + nis-s	1.88${}^{\ast \ast \ast }$	0.16${}^{\ast \ast \ast }$	0.35${}^{\ast \ast \ast }$	−0.09${}^{\ast \ast }$			9.79
	(29.54)	(4.51)	(9.96)	(−2.03)
iso + nis-c + nis-b + nis-s + nis	1.89${}^{\ast \ast \ast }$	−0.35${}^{\ast \ast \ast }$	−0.23${}^{\ast \ast \ast }$	−1.19${}^{\ast \ast \ast }$	2.18${}^{\ast \ast \ast }$		10.21
	(29.84)	(−6.10)	(−4.34)	(−10.53)	(10.76)
iso + nis-c + nis-b + nis-s + all	1.04${}^{\ast \ast \ast }$	−0.19${}^{\ast \ast \ast }$	−0.04	−0.82${}^{\ast \ast \ast }$		2.32${}^{\ast \ast \ast }$	10.09
	(9.39)	(−3.59)	(−0.91)	(−8.04)		(8.56)
iso + nis-c + nis-b + nis-s + nis + all	2.04${}^{\ast \ast \ast }$	−0.35${}^{\ast \ast \ast }$	−0.23${}^{\ast \ast \ast }$	−1.20${}^{\ast \ast \ast }$	2.46${}^{\ast \ast \ast }$	−0.40	10.21
	(11.86)	(−6.33)	(−4.58)	(−11.18)	(7.85)	(−0.90)
2018	iso	nis-c	nis-b	nis-s	nis	all	adj.$R^2$ (%)
iso + nis	1.24${}^{\ast \ast \ast }$				1.22${}^{\ast \ast \ast }$		6.69
	(18.85)				(16.00)
iso + nis-c	1.41${}^{\ast \ast \ast }$	0.96${}^{\ast \ast \ast }$					6.79
	(22.41)	(13.16)
iso + nis-c + nis-b	1.37${}^{\ast \ast \ast }$	0.63${}^{\ast \ast \ast }$	0.52${}^{\ast \ast \ast }$				7.11
	(21.35)	(9.16)	(11.83)
iso + nis-c + nis-b + nis-s	1.63${}^{\ast \ast \ast }$	0.50${}^{\ast \ast \ast }$	0.70${}^{\ast \ast \ast }$	−0.48${}^{\ast \ast \ast }$			7.29
	(24.84)	(7.85)	(13.34)	(−7.16)
iso + nis-c + nis-b + nis-s + nis	1.63${}^{\ast \ast \ast }$	−0.45${}^{\ast \ast \ast }$	−0.41${}^{\ast \ast \ast }$	−1.87${}^{\ast \ast \ast }$	3.54${}^{\ast \ast \ast }$		7.86
	(25.22)	(−4.64)	(−4.02)	(−12.86)	(11.40)
iso + nis-c + nis-b + nis-s + all	0.53${}^{\ast \ast \ast }$	−0.14	−0.04	−1.38${}^{\ast \ast \ast }$		3.43${}^{\ast \ast \ast }$	7.70
	(3.80)	(−1.55)	(−0.49)	(−10.28)		(8.77)
iso + nis-c + nis-b + nis-s + nis + all	1.69${}^{\ast \ast \ast }$	−0.45${}^{\ast \ast \ast }$	−0.42${}^{\ast \ast \ast }$	−1.88${}^{\ast \ast \ast }$	3.67${}^{\ast \ast \ast }$	−0.17	7.86
	(8.10)	(−4.81)	(−4.16)	(−13.53)	(7.85)	(−0.28)
2019	iso	nis-c	nis-b	nis-s	nis	all	adj.$R^2$ (%)
iso + nis	1.37${}^{\ast \ast \ast }$				1.09${}^{\ast \ast \ast }$		7.73
	(21.22)				(13.93)
iso + nis-c	1.63${}^{\ast \ast \ast }$	0.33${}^{\ast \ast \ast }$					7.05
	(23.65)	(5.69)
iso + nis-c + nis-b	1.61${}^{\ast \ast \ast }$	0.02	0.63${}^{\ast \ast \ast }$				7.57
	(23.62)	(0.37)	(12.23)
iso + nis-c + nis-b + nis-s	1.53${}^{\ast \ast \ast }$	0.07	0.56${}^{\ast \ast \ast }$	0.17${}^{\ast \ast }$			7.59
	(21.81)	(1.28)	(10.32)	(2.49)
iso + nis-c + nis-b + nis-s + nis	1.54${}^{\ast \ast \ast }$	−0.61${}^{\ast \ast \ast }$	−0.44${}^{\ast \ast \ast }$	−1.13${}^{\ast \ast \ast }$	3.01${}^{\ast \ast \ast }$		7.97
	(21.38)	(−4.87)	(−3.41)	(−5.40)	(6.91)
iso + nis-c + nis-b + nis-s + all	0.74${}^{\ast \ast \ast }$	−0.38${}^{\ast \ast \ast }$	−0.11	−0.68${}^{\ast \ast \ast }$		2.78${}^{\ast \ast \ast }$	7.87
	(5.90)	(−3.84)	(−1.08)	(−4.10)		(6.01)
iso + nis-c + nis-b + nis-s + nis + all	1.52${}^{\ast \ast \ast }$	−0.61${}^{\ast \ast \ast }$	−0.44${}^{\ast \ast \ast }$	−1.13${}^{\ast \ast \ast }$	2.93${}^{\ast \ast \ast }$	0.09	7.97
	(8.96)	(−4.81)	(−3.36)	(−5.37)	(4.96)	(0.18)
2020	iso	nis-c	nis-b	nis-s	nis	all	adj.$R^2$ (%)
iso + nis	1.36${}^{\ast \ast \ast }$				0.05		5.76
	(11.75)				(0.25)
iso + nis-c	1.09${}^{\ast \ast \ast }$	0.59${}^{\ast \ast \ast }$					5.90
	(9.51)	(4.42)
iso + nis-c + nis-b	1.09${}^{\ast \ast \ast }$	0.62${}^{\ast \ast \ast }$	−0.06				5.90
	(9.51)	(4.33)	(−0.41)
iso + nis-c + nis-b + nis-s	1.55${}^{\ast \ast \ast }$	0.33${}^{\ast \ast \ast }$	0.43${}^{\ast \ast \ast }$	−1.04${}^{\ast \ast \ast }$			6.10
	(13.18)	(2.69)	(3.75)	(−6.16)
iso + nis-c + nis-b + nis-s + nis	1.57${}^{\ast \ast \ast }$	0.01	−0.03	−1.60${}^{\ast \ast \ast }$	1.34*		6.12
	(13.50)	(0.06)	(−0.11)	(−4.46)	(1.73)
iso + nis-c + nis-b + nis-s + all	1.44${}^{\ast \ast \ast }$	0.26	0.32	−1.17${}^{\ast \ast \ast }$		0.44	6.10
	(5.23)	(1.58)	(1.57)	(−3.85)		(0.53)
iso + nis-c + nis-b + nis-s + nis + all	2.30${}^{\ast \ast \ast }$	−0.04	−0.10	−1.71${}^{\ast \ast \ast }$	3.54${}^{\ast \ast \ast }$	−2.68*	6.14
	(5.47)	(−0.21)	(−0.42)	(−4.85)	(2.74)	(−1.95)

Note: We run regressions against multiple COIs following equation (6(6) $\begin{array}{rl}{r}_{i,t}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t+{\beta }_c\mathrm{CM}{A}_t\\ & +{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t},\end{array}$(6) ) on a yearly basis for 2017, 2018, 2019 and 2020 respectively. The first column states the types of COIs input in each regression. Regression coefficients to different types of COIs are listed in the following columns as indicated by the column names. The superscripts ${}^{\ast }{,}^{\ast \ast }{,}^{\ast \ast \ast }$ indicate significant at $10\mathrm{\%},5\mathrm{\%}$ and $1\mathrm{\%}$ respectively using a two-tailed t-test and corresponding t-values are reported in the parentheses below. The last column‘adj.$R^2$’ presents the adjusted $R^2$ of regressions.

Table C3. Yearly predictive regression against individual COIs.

2017	${\beta }_{\rho }$	t	adj.$R^2$ (%)
all	−0.01	−0.26	0.0273
iso	0.07${}^{\ast \ast }$	2.47	0.0360
nis	−0.06${}^{\ast }$	−1.69	0.0320
nis−s	0.05${}^{\ast \ast }$	2.18	0.0331
nis−c	−0.10${}^{\ast \ast \ast }$	−3.03	0.0569
nis−b	−0.07${}^{\ast \ast \ast }$	−2.75	0.0458
control variables			0.0281
2018	${\beta }_{\rho }$	t	adj.$R^2$ (%)
all	0.04	0.73	0.1248
iso	0.09${}^{\ast \ast }$	2.34	0.1361
nis	−0.02	−0.46	0.1242
nis-s	0.06	1.48	0.1285
nis-c	−0.06	−1.21	0.1292
nis-b	−0.05	−1.20	0.1275
control variables			0.1246
2019	${\beta }_{\rho }$	t	adj.$R^2$ (%)
all	0.04	0.91	0.0843
iso	0.09${}^{\ast \ast \ast }$	2.90	0.0979
nis	0.00	−0.10	0.0828
nis-s	0.11${}^{\ast \ast \ast }$	3.16	0.0993
nis-c	−0.08${}^{\ast \ast }$	−2.13	0.0953
nis-b	−0.07${}^{\ast }$	−1.79	0.0910
control variables			0.0837
2020	${\beta }_{\rho }$	t	adj.$R^2$ (%)
all	−0.07	−0.47	0.2804
iso	0.08	1.05	0.2820
nis	−0.15	−0.89	0.2850
nis-s	−0.06	−0.52	0.2806
nis-c	0.00	−0.04	0.2791
nis-b	−0.17	−1.25	0.2927
control variables			0.2799

Note: This table summarizes the coefficients for COIs by regressing again each type of COIs individually following equation (7(7) $\begin{array}{rl}{r}_{i,t+1}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\tau }{r}_{i,t}+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t\\ & +{\beta }_c\mathrm{CM}{A}_t+{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t+1},\end{array}$(7) ) on a yearly basis for 2017, 2018, 2019 and 2020 respectively. As a benchmark, the last row of each panel shows the result of regressing against control variables only.‘${\beta }_{\rho }$’ denotes the regression coefficients and the superscript ${}^{\ast \ast \ast }$ indicates significant at $1\mathrm{\%}$ using a two-tailed t-test. ‘t’ denotes the t-value of each coefficient. ‘adj.$R^2$’ denotes the adjusted $R^2$ of regressions.

Table C4. Yearly predictive regression against multiple COIs.

2017	iso	nis-c	nis-b	nis-s	nis	all	adj.$R^2$ (%)
iso + nis	0.11${}^{\ast \ast \ast }$				−0.12${}^{\ast \ast \ast }$		0.0511
	(3.28)				(–2.75)
iso + nis-c	0.11${}^{\ast \ast \ast }$	−0.13${}^{\ast \ast \ast }$					0.0780
	(3.53)	(–3.49)
iso + nis-c + nis-b	0.11${}^{\ast \ast \ast }$	−0.11${}^{\ast \ast \ast }$	−0.03				0.0794
	(3.52)	(–3.19)	(–1.33)
iso + nis-c + nis-b + nis-s	0.09${}^{\ast \ast \ast }$	−0.11${}^{\ast \ast \ast }$	−0.04	0.02			0.0792
	(2.98)	(–3.12)	(–1.41)	(0.79)
iso + nis-c + nis-b + nis-s + nis	0.09${}^{\ast \ast \ast }$	−0.10${}^{\ast \ast }$	−0.03	0.04	−0.03		0.0784
	(3.00)	(–2.04)	(–0.57)	(0.41)	(–0.17)
iso + nis-c + nis-b + nis-s + all	0.14	−0.09${}^{\ast \ast }$	−0.01	0.06		−0.12	0.0792
	(1.52)	(–2.05)	(–0.36)	(0.83)		(–0.58)
iso + nis-c + nis-b + nis-s + nis + all	0.24${}^{\ast \ast }$	−0.10${}^{\ast \ast }$	−0.03	0.02	0.26	−0.41	0.0796
	(2.06)	(–2.16)	(–0.72)	(0.24)	(1.05)	(–1.37)
2018	iso	nis-c	nis-b	nis-s	nis	all	adj.$R^2$ (%)
iso + nis	0.13${}^{\ast \ast \ast }$				−0.11${}^{\ast }$		0.1432
	(2.67)				(−1.7)
iso + nis-c	0.12${}^{\ast \ast \ast }$	−0.11${}^{\ast }$					0.1488
	(2.63)	(−1.76)
iso + nis-c + nis-b	0.13${}^{\ast \ast \ast }$	−0.09	−0.03				0.1491
	(2.65)	(−1.47)	(−0.9)
iso + nis-c + nis-b + nis-s	0.12${}^{\ast \ast \ast }$	−0.08	−0.04	0.02			0.1485
	(2.64)	(−1.47)	(−1.0)	(0.46)
iso + nis-c + nis-b + nis-s + nis	0.12${}^{\ast \ast \ast }$	−0.06	−0.01	0.06	−0.10		0.1481
	(2.65)	(−0.84)	(−0.11)	(0.50)	(−0.40)
iso + nis-c + nis-b + nis-s + all	0.13	−0.07	−0.03	0.03		−0.05	0.1478
	(1.22)	(−1.14)	(−0.44)	(0.35)		(−0.18)
iso + nis-c + nis-b + nis-s + nis + all	0.06	−0.05	−0.01	0.06	−0.23	0.18	0.1475
	(0.39)	(−0.8)	(−0.06)	(0.56)	(−0.59)	(0.40)
2019	iso	nis-c	nis-b	nis-s	nis	all	adj.$R^2$ (%)
iso + nis	0.12${}^{\ast \ast \ast }$				−0.08		0.1022
	(3.10)				(−1.49)
iso + nis-c	0.14${}^{\ast \ast \ast }$	−0.14${}^{\ast \ast \ast }$					0.1265
	(3.77)	(−3.02)
iso + nis-c + nis-b	0.14${}^{\ast \ast \ast }$	−0.12${}^{\ast \ast \ast }$	−0.04				0.1275
	(3.77)	(−2.64)	(−1.00)
iso + nis-c + nis-b + nis-s	0.10${}^{\ast \ast }$	−0.09${}^{\ast \ast }$	−0.08${}^{\ast }$	0.10${}^{\ast \ast }$			0.1341
	(2.53)	(−2.08)	(−1.74)	(2.27)
iso + nis-c + nis-b + nis-s + nis	0.10${}^{\ast \ast }$	−0.13${}^{\ast }$	−0.13	0.03	0.16		0.1342
	(2.55)	(−1.75)	(−1.51)	(0.25)	(0.65)
iso + nis-c + nis-b + nis-s + all	0.11	−0.08	−0.07	0.11		−0.04	0.1332
	(1.38)	(−1.38)	(−0.97)	(1.20)		(−0.17)
iso + nis-c + nis-b + nis-s + nis + all	0.29${}^{\ast \ast }$	−0.14${}^{\ast }$	−0.14${}^{\ast }$	0.01	0.67${}^{\ast }$	−0.66${}^{\ast }$	0.1376
	(2.54)	(−1.91)	(−1.68)	(0.06)	(1.96)	(−1.85)
2020	iso	nis-c	nis-b	nis-s	nis	all	adj.$R^2$ (%)
iso + nis	0.17${}^{\ast \ast }$				−0.26		0.2952
	(2.00)				(−1.35)
iso + nis-c	0.10	−0.05					0.2821
	(0.92)	(−0.37)
iso + nis-c + nis-b	0.10	0.06	−0.22				0.2991
	(0.89)	(0.42)	(−1.50)
iso + nis-c + nis-b + nis-s	0.10	0.06	−0.22	−0.02			0.2983
	(1.08)	(0.42)	(−1.46)	(−0.17)
iso + nis-c + nis-b + nis-s + nis	0.10	0.11	−0.14	0.08	−0.23		0.2979
	(1.06)	(0.60)	(−0.63)	(0.26)	(−0.36)
iso + nis-c + nis-b + nis-s + all	0.14	0.08	−0.18	0.02		−0.13	0.2976
	(0.68)	(0.49)	(−0.98)	(0.10)		(−0.20)
iso + nis-c + nis-b + nis-s + nis + all	0.04	0.12	−0.13	0.09	−0.42	0.24	0.2971
	(0.14)	(0.63)	(−0.6)	(0.31)	(−0.49)	(0.27)

Note: We run regressions against multiple COIs following equation (7(7) $\begin{array}{rl}{r}_{i,t+1}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\tau }{r}_{i,t}+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t\\ & +{\beta }_c\mathrm{CM}{A}_t+{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t+1},\end{array}$(7) ) on a yearly basis for 2017, 2018, 2019 and 2020 respectively. The first column states the types of COIs input in each regression. Regression coefficients to different types of COIs are listed in the following columns as indicated by the column names. The superscripts ${}^{\ast }{,}^{\ast \ast }{,}^{\ast \ast \ast }$ indicate significant at $10\mathrm{\%},5\mathrm{\%}$ and $1\mathrm{\%}$ respectively using a two-tailed t-test and corresponding t-values are reported in the parentheses below. The last column‘adj.$R^2$’ presents the adjusted $R^2$ of regressions.

Table D1. Contemporaneous time series regressions.

	Average ${\beta }_{\rho }$	Standard deviation of ${\beta }_{\rho }$	Percentage positive	Percentage significant	Percentage positive and significant	Average adj.$R^2(\mathrm{\%})$
all	2.16	1.20	98.69	90.81	90.59	16.36
iso	1.72	0.97	98.91	94.09	94.09	16.72
nis	1.56	1.09	95.62	81.84	81.18	15.15
nis-s	0.95	1.05	86.21	63.46	61.71	14.33
nis-c	0.93	0.62	95.62	76.81	76.59	14.25
nis-b	0.77	0.59	91.90	71.12	70.46	14.03

Note: This table summarizes the results of 457 regressions, one for each stock, using equation (6(6) $\begin{array}{rl}{r}_{i,t}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t+{\beta }_c\mathrm{CM}{A}_t\\ & +{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t},\end{array}$(6) ), against each type of COI individually. ‘Average ${\beta }_{\rho }$’ denotes the mean of regressions coefficients over all stocks. ‘Percentage positive’ denotes proportion of stocks with positive ${\beta }_{\rho }$. ‘Significant’ denotes proportion of stocks with coefficients which are statistically significant at $5\mathrm{\%}$ significance level using a two-tailed t test. ‘Average adj. $R^2$’ denotes the adjusted $R^2$ averaged across all stocks.

Table D2. Predictive time series regressions.

	Average ${\beta }_{\rho }$	Standard deviation of ${\beta }_{\rho }$	Percentage positive	Percentage significant	Percentage positive and significant	Average adj.$R^2(\mathrm{\%})$
all	0.01	0.47	51.64	7.44	5.03	1.54
iso	0.09	0.38	65.86	9.41	7.88	1.57
nis	−0.05	0.42	44.20	5.03	2.41	1.54
nis-s	0.05	0.36	59.96	5.69	3.94	1.55
nis-c	–0.07	0.34	39.61	6.78	1.75	1.55
nis-b	–0.07	0.27	36.98	3.72	0.66	1.53

Note: This table summarizes the results of 457 regressions, one for each stock, using equation (7(7) $\begin{array}{rl}{r}_{i,t+1}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\tau }{r}_{i,t}+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t\\ & +{\beta }_c\mathrm{CM}{A}_t+{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t+1},\end{array}$(7) ), against each type of COI individually. ‘Average ${\beta }_{\rho }$’ denotes the mean of regressions coefficients over all stocks. ‘Percentage positive’ denotes proportion of stocks with positive ${\beta }_{\rho }$. ‘Significant’ denotes proportion of stocks with coefficients which are statistically significant at $5\mathrm{\%}$ significance level using a two-tailed t-test. ‘Average adj. $R^2$’ denotes the adjusted $R^2$ averaged across all stocks.

Figure A1. Distributions of coefficients of contemporaneous time series regressions. This figure displays the histogram and kernel density estimation of the coefficients of contemporaneous time series regressions. The orange line indicates the mean of the coefficients.

Figure A2. Distributions of adjusted $R^2$ of contemporaneous time series regressions. This figure displays the histogram and kernel density estimation of the adjusted $R^2$ of contemporaneous time series regressions. The orange line indicates the mean of the adjusted $R^2$.

Figure A3. Distributions of coefficients of predictive time series regressions. This figure shows the histogram and kernel density estimation of the coefficients of predictive time series regressions. The orange line indicates the mean of the coefficients.

Figure A4. Distributions of adjusted $R^2$ of predictive time series regressions. This figure displays the histogram and kernel density estimation of the adjusted $R^2$ of predictive time series regressions. The orange line indicates the mean of the adjusted $R^2$.

Table E1. Additional evaluation for regression analysis.

Panel A: Contemporaneous regression
	F-score	AIC	BIC	MSE	MAE
all	3014.97	1632679.36	1632783.54	2.1614	0.9661
iso	3160.07	1631448.98	1631553.16	2.1555	0.9665
nis	2356.34	1638306.63	1638410.81	2.1884	0.9750
nis-s	1789.82	1643203.51	1643307.69	2.2123	0.9839
nis-c	2062.61	1640838.93	1640943.11	2.2007	0.9803
nis-b	1824.98	1642898.05	1643002.23	2.2108	0.9827
iso + nis	2988.08	1630095.96	1630213.16	2.1491	0.9634
iso + nis-c	2977.28	1630197.28	1630314.48	2.1496	0.9639
iso + nis-c + nis-b	2760.67	1629639.74	1629769.96	2.1469	0.9627
iso + nis-c + nis-b + nis-s	2550.43	1629416.89	1629560.13	2.1459	0.9625
iso + nis-c + nis-b + nis-s + nis	2454.35	1628199.49	1628355.76	2.1401	0.9606
iso + nis-c + nis-b + nis-s + all	2423.37	1628575.80	1628732.07	2.1419	0.9609
iso + nis-c + nis-b + nis-s + nis + all	2279.23	1628196.95	1628366.24	2.1401	0.9606
Panel B: Predictive regression
all	46.96	1655230.52	1655347.72	2.2719	0.9950
iso	50.47	1655195.40	1655312.66	2.2717	0.9949
nis	47.77	1655222.42	1655339.62	2.2718	0.9950
nis-s	47.63	1655223.81	1655341.02	2.2718	0.9950
nis-c	48.60	1655214.09	1655331.29	2.2718	0.9949
nis-b	50.65	1655193.67	1655310.87	2.2717	0.9949
iso + nis	49.77	1655152.70	1655282.93	2.2715	0.9948
iso + nis-c	49.87	1655151.64	1655281.87	2.2715	0.9948
iso + nis-c + nis-b	47.55	1655129.59	1655272.83	2.2714	0.9948
iso + nis-c + nis-b + nis-s	43.94	1655128.98	1655285.25	2.2714	0.9948
iso + nis-c + nis-b + nis-s + nis	40.86	1655128.16	1655297.45	2.2713	0.9948
iso + nis-c + nis-b + nis-s + all	41.01	1655126.12	1655295.41	2.2713	0.9948
iso + nis-c + nis-b + nis-s + nis + all	38.34	1655125.05	1655307.36	2.2713	0.9948

Note: This table reports additional evaluation metrics, including F-score of regression, AIC, BIC, MSE and MAE. Panel A supplements the results of contemporaneous regressions in tables 5 and 6. Panel A supplements the results of predictive regressions in tables 7 and 8.

Figure A5. Average $R^2$ of regressing contemporaneous returns against COIs for different δs.

Figure A6. Sharpe ratios of COI-based long-short portfolios for different δs.

Table G1. Trades and COIs by universe of stocks.

Panel A: Average fractions
	S&P 500	S&P 100	Dow 30
iso	28.55	34.37	37.57
nis	71.45	65.63	62.43
nis-s	29.75	37.82	42.58
nis-c	17.27	11.45	8.24
nis-b	24.43	16.37	11.60
Panel B: Average correlations
	S&P 500 - S&P 100	S&P 500 - Dow 30	S&P 100 - Dow 30
iso	0.99	0.97	0.99
nis	0.99	0.99	1.00
nis-s	0.97	0.94	0.98
nis-c	0.92	0.85	0.95
nis-b	0.91	0.82	0.91
Panel C: Annualized Sharpe ratios
	S&P 500	S&P 100	Dow 30
iso	1.20	1.04	0.84
nis	0.48	0.59	−0.98
nis-s	0.47	0.26	0.02
nis-c	0.71	1.12	−1.59
nis-b	0.97	1.16	−1.65
iso/nis	1.08	1.22	0.76
iso/nis-c	1.79	1.34	1.03

Note: This table shows the results for three sets of stocks as the market index, including S&P 500, S&P 100 and Dow 30. Panel A documents the average fractions of each type of trade for each universe of stocks. Panel B reports the correlations of all types of COIs for each pair of universes. The columns indicate the pairs. Panel C presents the annualized Sharpe ratios, given by equation (8(8) $S{R}_p:=\frac{\mathit{mean}\mathit{(}{\mathit{R}}_{\mathit{p}\mathit{,}\mathit{t}}\mathit{)}\mathit{-}{\mathit{R}}_{\mathit{f}}}{\mathit{std}\mathit{(}{\mathit{R}}_{\mathit{p}\mathit{,}\mathit{t}}\mathit{)}}\times \sqrt{252},$(8) ), of single-sort and selected double-sort long-short portfolios based on COIs of the universes of stocks.

Table H1. COIs by time period.

Panel A: Contemporaneous regression $R^2(\mathrm{\%})$
	9:30–10:00	10:00–15:30	15:30–16:00	9:00–16:00
all	3.64	5.16	3.75	5.66
iso	4.70	5.32	3.18	5.91
nis	2.93	4.17	3.56	4.48
nis-s	2.89	3.27	2.84	3.44
nis-c	3.04	3.69	3.00	3.94
nis-b	2.65	3.36	3.09	3.50
Panel B: Annualized Sharpe ratios
	9:30–10:00	10:00–15:30	15:30–16:00	9:00–16:00
all	−1.43	−0.67	0.43	−0.25
iso	−0.94	1.33	1.61	1.20
nis	0.82	0.40	−0.56	0.48
nis-s	−0.85	0.18	1.06	0.47
nis-c	1.03	0.64	−0.53	0.71
nis-b	0.74	0.83	0.20	0.97

Note: We calculate COIs of 9:30 − 10:00, 10:00 − 15:30 and 15:30 − 16:00 each day from 2017-01-03 to 2020-12-31 for the selected 457 stocks. Panel A summarizes the adjusted $R^2$ of regressions on contemporaneous open-to-close market excess returns against each type of COIs, using equation (6(6) $\begin{array}{rl}{r}_{i,t}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t+{\beta }_c\mathrm{CM}{A}_t\\ & +{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t},\end{array}$(6) ). Panel B presents the annualized Sharpe Ratios, given by equation (8(8) $S{R}_p:=\frac{\mathit{mean}\mathit{(}{\mathit{R}}_{\mathit{p}\mathit{,}\mathit{t}}\mathit{)}\mathit{-}{\mathit{R}}_{\mathit{f}}}{\mathit{std}\mathit{(}{\mathit{R}}_{\mathit{p}\mathit{,}\mathit{t}}\mathit{)}}\times \sqrt{252},$(8) ), of single-sort long-short portfolios based on COIs of different intraday time periods. The last column of both panels reports the daily COIs as a benchmark.

Table I1. COIs measured by volume.

Panel A: Contemporaneous regression
	${\beta }_{\rho }$	t	adj.$R^2$ (%)
all	1.58${}^{\ast \ast \ast }$	24.72	4.70
iso	1.30${}^{\ast \ast \ast }$	30.85	5.02
nis	1.19${}^{\ast \ast \ast }$	19.21	3.90
nis-s	0.75${}^{\ast \ast \ast }$	15.52	3.37
nis-c	0.58${}^{\ast \ast \ast }$	16.24	3.14
nis-b	0.61${}^{\ast \ast \ast }$	16.46	3.27
Panel B: Annualized Sharpe ratios
	iso	nis	nis-s	nis-c	nis-b
iso	0.88	1.23	0.15	1.49	0.93
nis		0.59	−0.29	0.76	0.72
nis-s			0.24	−0.30	−0.19
nis-c				0.91	0.86
nis-b					1.01

Note: We calculate COIs measured by volumes, as equation (A1(A1) $\mathrm{CO}{I}_{i,t}^{\mathrm{type}}=\frac{V_{i,t}^{\mathrm{type},\mathrm{buy}}-V_{i,t}^{\mathrm{type},\mathrm{sell}}}{V_{i,t}^{\mathrm{type},\mathrm{buy}}+V_{i,t}^{\mathrm{type},\mathrm{sell}}},$(A1) ), from 2017-01-03 to 2020-12-31 for the selected 457 stocks. Panel A summarizes the coefficients to COIs by regressing again each type of COIs individually following equation (6(6) $\begin{array}{rl}{r}_{i,t}& =\alpha +\sum _{\rho \in \mathrm{Types}}{\beta }_{\rho }\mathrm{CO}{I}_{i,t}^{\rho }+{\beta }_{\sigma }{\sigma }_{i,t}+{\beta }_v\mathrm{vo}{l}_{i,t}\\ & +{\beta }_b\mathrm{MK}{T}_t+{\beta }_s\mathrm{SM}{B}_t+{\beta }_h\mathrm{HM}{L}_t+{\beta }_r\mathrm{RM}{W}_t+{\beta }_c\mathrm{CM}{A}_t\\ & +{\beta }_m\mathrm{MO}{N}_t+{ϵ}_{i,t},\end{array}$(6) ). ‘${\beta }_{\rho }$’ denotes the regression coefficients and the superscript ${}^{\ast \ast \ast }$ indicates significant at $1\mathrm{\%}$ using two-tailed t-test. ‘t’ denotes the t-value of each coefficient. ‘adj.$R^2$’ denotes the adjusted $R^2$ of regressions. Panel B shows the annualized Sharpe ratios of the long-short portfolios sorted on COIs indicated by the corresponding row indices and column names. The on- and off-diagonal values are for single- and double-sort portfolios respectively. The annualized Sharpe ratios over the sample period are given in equation (8(8) $S{R}_p:=\frac{\mathit{mean}\mathit{(}{\mathit{R}}_{\mathit{p}\mathit{,}\mathit{t}}\mathit{)}\mathit{-}{\mathit{R}}_{\mathit{f}}}{\mathit{std}\mathit{(}{\mathit{R}}_{\mathit{p}\mathit{,}\mathit{t}}\mathit{)}}\times \sqrt{252},$(8) ).

Table J1. Annualized returns and Sharpe ratios of selected and benchmark portfolios.

Panel A: Annualized returns
portfolio / cost (bps)	0	1	2	3	4	5
iso	6.76	4.24	1.72	−0.80	−3.32	−5.84
iso/nis	23.33	20.81	18.29	15.77	13.25	10.73
iso/nis−c	34.87	32.35	29.83	27.31	24.79	22.27
all	0.31	−2.21	−4.73	−7.25	−9.77	−12.29
return momentum	−14.81	−17.33	−19.85	−22.37	−24.89	−27.41
equal weight	−4.00
SPY ETF (open to close)	−1.00	−3.52	−6.04	−8.56	−11.08	−13.6
SPY ETF	10.04
Panel B: Annualized Sharpe ratios
iso	1.20	0.62	0.03	−0.55	−1.14	−1.72
iso/nis	1.08	0.95	0.83	0.70	0.58	0.45
iso/nis−c	1.79	1.66	1.52	1.39	1.25	1.11
all	−0.25	−0.74	−1.23	−1.71	−2.20	−2.69
return momentum	−1.23	−1.41	−1.60	−1.79	−1.98	−2.17
equal weight	−0.39
SPY ETF (open to close)	−0.19	−0.38	−0.57	−0.76	−0.95	−1.14
SPY ETF	0.42

Note: This table exhibits the results of selected and benchmark portfolios in figure 6. The column names indicate different levels of transaction costs in basis points (bps). Panel A presents the annualized return of portfolios calculated by averaging their daily returns, from 2017-01-03 to 2020-12-31, and multiplying by 252. Panel B reports the annualized Sharpe ratios over the sample period calculated in equation (8(8) $S{R}_p:=\frac{\mathit{mean}\mathit{(}{\mathit{R}}_{\mathit{p}\mathit{,}\mathit{t}}\mathit{)}\mathit{-}{\mathit{R}}_{\mathit{f}}}{\mathit{std}\mathit{(}{\mathit{R}}_{\mathit{p}\mathit{,}\mathit{t}}\mathit{)}}\times \sqrt{252},$(8) ).

Newey, W.K. and West, K.D., Automatic lag selection in covariance matrix estimation. Rev. Econ. Stud., 1994, 61, 631–653.

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