In order to adjust our asset-pricing test results with other microstructure factors, especially other liquidity measures, measures of other liquidity dimensions are added in these asset-pricing tests. They are Lambda, dollar effective spread and proportional effective spread. These three liquidity measures could be affected by the same set of market/stock-specific factors as resiliency, which is discussed in section 5.
Unlike Brennan and Subrahmanyam (1996), we use the inverse value of Lambda to directly measure depth level. When Lambda is high, depth is lower and higher required rate of return is expected. Thus, a positive relation between this depth measure and stock’s realized return is assumed.
Dollar/proportional effective spread measures the direct transaction costs for the public traders and is a natural proxy for the first dimension of market liquidity. Amihud and Mendelson (1986, 1989) and Eleswarapu (1997) found that spreads, as a measure of market illiquidity, are priced by the market. We also expect that these two spread measures have significant positive relation with stock’s realized market returns.
From table 11, we find that dollar effective spread is positively correlated with stock’s realized return, while proportional effective spreads are negatively correlated with stock’s realized return. And the relation between Lambda and stock’s realized return is insignificant. Dollar effective spread is highly positively related with proportional effective spread in the time-series analysis, while the cross-section correlation is negative and very weak. This implies that cross-section variation of proportional effective spread is mainly driven by the price level. Multivariate regression test results including all these liquidity measures are presented at table 13 (cross-section test results), and table 14 (time-series test results).
First, let’s look at the asset-pricing test results on the depth measure, Lambda. At first, it is included in asset-pricing tests in place of resiliency and gets insignificant coefficients at both cross-section and time-series tests. When we combine both Lambda and resiliency in the regression, we find that coefficients of Lambda are still insignificant.
However, resiliency still keeps its results: significant and negative in both cross-section and time-series tests. These results show that, although these two factors are correlated (see table 11), their relationship is not strong enough to cover resiliency’s effect on the stock’s realized market return. And this depth measure is not related with stock’s realized return.
Second, proportional and dollar effective spreads get different results in these multi-variate asset-pricing tests. Proportional effective spread shows significant and negative coefficients in cross-section tests, and positive and insignificant coefficients in time-series tests. Thus, proportional effective spread shows cross-section negative relation with stock’s realized market return and is consistent with result of bivaraite correlation
analyses. Although it may be inconsistent with what is expected from illiquidity-compensation theories, this result is consistent with previous finding by Easley, Hvidkjaer and O’Hara (2002), which also reported negative relation between return and percentage spread using monthly return data. Dollar spreads get positive and significant coefficient at both cross-section and time-series test, which indicates that it may be compensated by higher realized market return. This result is consistent with previous findings by Amihud and Mendelson (1986, 1989) and Eleswarapu (1997). The significantly negative coefficients on resiliency at both cross-section and time-series asset-pricing tests still hold, when combining with any of these two spread measures.
This implies that, accounting for the effect of the relation between spread and resiliency, resiliency yields additional effect on stock’s realized return.
Finally, we include all the available liquidity measures in the asset-pricing tests. We find that resiliency still shows significant negative relation with market return. And dollar effective spreads still get positive and significant coefficients on realized market return.
In all, the negative relationship between stock’s realized return and resiliency seems to be robust with the inclusion of other different measures of market liquidity in both the cross-section and time-series asset-pricing tests. Our results indicate that resiliency does affect stock’s realized daily return.
7. Conclusions
This paper investigates, for the first time, the main features of resiliency as a dimension of liquidity, and its effect on stock’s realized returns. Specifically, we first address how we can formally define and measure resiliency. Second, we analyse the micro-structural and stock-specific factors that affect resiliency. Third, we examine the relationship between resiliency and the other two liquidity dimensions: spread and depth.
And finally, we test if resiliency affects stock’s realized market return. Our empirical investigation is based on 100 actively traded stocks on the New York Stock Exchange over 294 trading days.
Following Kyle (1985), we define resiliency as the rate at which pricing errors caused by temporary order-flow shocks are corrected in the market. In this context, the observed mean-reversion parameter in the pricing-error process is used to empirically measure a stock’s resiliency. An integrated estimation procedure based on the
Kalman-filter methodology, is used to estimate resiliency for each trading day for each stock from minute-by-minute high-frequency data of 100 NYSE stocks. We find that, for our sample of (heavily traded) stocks, the mean value of the estimated resiliency over a one-minute horizon is 0.60, which indicates that as much as 60% of the pricing error reverts to zero within one minute.
We analyse eight micro-structural time-series and stock-specific factors as determinants of resiliency. We find that transaction frequency, tick size relative to pricing level, average transaction size, realized spread adverse-selection ratio, informative-transaction ratio and stock’s unexpected intra-day volatility are significant determinants of resiliency, and in the expected direction.
We investigate the relationships among these three liquidity dimensions. Spread, measured by the dollar effective spread, is highly correlated with depth, measured by the inverse value of Kyle’s Lambda. However, the relationship between resiliency and the other two liquidity dimensions, while statistically significant, is much weaker. It appears that resiliency, as a dimension of liquidity, is relatively independent to the other two dimensions of liquidity, spread and depth, and hence can potentially provide significant new information.
Finally, we apply the Brennan and Subrahmanyam (1986) methodology, and the commonly used Fama and MacBeth (1972) and Fama and French(1993) asset-pricing test procedures to test if resiliency affects a stock’s realized market return, and find strong evidence that it does influence the stock’s realized market return, and in the expected direction.
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Figure 1. Simulation test results This figure shows the estimation errors of all the three estimation methods: Reduced-form, Kalman-filter, and Combined procedures. 100 sets of data are simulated with set resilience value of 0.8. Estimations are made using every set of data with different amount of data. The estimation error is measured by the average value of all the 100 pricing error ratios:
0 0 |
| ˆ α
α
= α −i
Ri , where
α0is the set resilience value of 0.8, is the estimated resilience value.
ˆ0
α
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
30.00%
35.00%
20000 10000 5000 1500 400 250
Number of data
Estimation error
Reduced-form Kalman-filter Combined
Table 1 Frequency distribution of estimated resiliency
Reported is the frequency distribution of the estimated results of market resiliency. Panel A reports frequency distribution of all the all estimation results. Panel B reports frequency distribution of significant estimation results.
Panel A Frequency Ratio Panel B Frequency Ratio Less than 0 71 0.24% Less than 0 49 0.17%
(0-0.2] 4376 14.90% (0-0.2] 2408 8.20%
(0.2-0.4] 3646 12.41% (0.2-0.4] 3208 10.92%
(0.4,0.6] 4207 14.32% (0.4,0.6] 3965 13.50%
(0.6,0.8] 8112 27.61% (0.6,0.8] 7899 26.89%
(0.8,1.0] 7167 24.40% (0.8,1.0] 7122 24.24%
(1.0,1.2] 1718 5.85% (1.0,1.2] 1718 5.85%
More than 1.2 79 0.27% More than 1.2 79 0.27%
Significant 26448 90.03%
Insignificant 2928 9.97%
All 29376 100.00% All 29376 100.00%
Table 2 Summary statistics of estimated resiliency
The table contains basic statistics of the estimated resiliency values. Panel A reports main statistics of all the estimated values:
mean (MEAN), standard deviation (STDEV), minimum value (MIN), maximum value (MAX), the first, second, and third quartiles. ALL column gives statistics of all resiliency values, while SIGNIFICANT column gives statistics of all significant resiliency values. Panel B reports t-values for the null hypothesis 1: mean of all estimated resiliency is zero (T-value 1), and null hypothesis 2: mean of all estimated resiliency is one (T-value 2).
Panel A ALL SIGNIFICANT
MEAN 0.601 0.647
STDEV 0.305 0.277
MIN -0.032 -0.032
MAX 1.502 1.502
QUARTILE1 0.360 0.458
QUARTILE2 0.678 0.715
QUARTILE3 0.834 0.850
Panel B ALL SIGNIFICANT
T-value 1: 337.75 379.65
T-value 2: -223.98 -207.12
Table 3 Summary statistics of selected explanatory variables for resiliency
The table contains summary statistics of selected explanatory variables for resiliency: daily transaction number (TRADE), daily trading volume (TVOL), daily price level (PRICE) and daily average transaction size (SIZE). Summary statistics include mean (MEAN), standard deviation (STDEV), minimal value (MIN), maximum value (MAX), the first-, second- and third-quartiles.
.
Explanatory factors TRADE TVOL PRICE SIZE
MEAN 880.19 2180000 50.21 2150.20
STDEV 735.22 3314900 32.78 1436.00
MIN 49.00 35500 6.42 289.44
MAX 6029.00 119000000 332.87 36066.00
QUARTILE1 396.00 600800 30.84 1237.20
QUARTILE2 637.00 1130000 42.54 1785.40
QUARTILE3 1084.00 2310000 60.16 2630.30
Table 4.1 Summary statistics of daily average realized spread
The table contains basic statistics of daily average realized spreads: mean (MEAN), standard deviation (STDEV), minimum value (MIN), maximum value (MAX), the first, second, and third quartiles. NYSE(5), NYSE(15), NYSE(30) columns give statistics of 5-minute, 15-minute and 30-minute lag realized spreads, respectively. T-values (T-value) are provided for the null hypothesis 1:
mean of average realized spread is zero.
Realized-spread NYSE(5) NYSE(15) NYSE(30)
MEAN 1.02E-03 1.35E-03 1.63E-03
STDEV 2.98E-03 3.07E-03 3.24E-03
MIN -8.89E-03 -9.00E-03 -4.84E-02
MAX 1.14E-01 1.16E-01 1.17E-01
QUARTILE1 2.56E-04 4.91E-04 6.23E-04 QUARTILE2 6.70E-04 1.02E-03 1.29E-03 QUARTILE3 1.17E-03 1.64E-03 2.11E-03
T-value 58.737 75.198 86.218
Table 4.2 Frequency distribution of daily informative-transaction ratio
Reported is the frequency distribution of daily informative-transaction ratios. NYSE(5), NYSE(15), NYSE(30) columns give frequency distribution of 5-minute, 15-minute and 30-minute lag informative-transaction ratios, respectively.
NYSE(5) NYSE(15) NYSE(30) Informative-transaction
ratio Frequency Ratio Frequency Ratio Frequency Ratio Less than 0 16 0.05% 459 1.56% 2407 8.19%
(0-1] 29356 99.95% 28913 98.44% 26965 91.81%
More than 1 0 0.00% 0 0.00% 0 0.00%
All 29372 100.00% 29372 100.00% 29372 100.00%
Table 4.3 Summary statistics of daily informative-transaction ratio
The table contains basic statistics of daily average informative-transaction ratios: mean (MEAN), standard deviation (STDEV), minimum value (MIN), maximum value (MAX), the first, second, and third quartiles. NYSE(5), NYSE(15), NYSE(30) columns give statistics of 5-minute, 15-minute and 30-minute lag informative-transaction ratios, respectively. T-values (T-value) are provided for the null hypothesis 1: mean of average value-transaction ratios is zero.
Informative-transaction ratio NYSE(5) NYSE(15) NYSE(30)
MEAN 0.3927 0.23987 0.17215
STDEV 0.12981 0.12966 0.13698
MIN -1 -1 -1
MAX 1 1 1
QUARTILE1 0.30092 0.15117 0.081207
QUARTILE2 0.38697 0.22884 0.16179
QUARTILE3 0.47634 0.31618 0.25465
T-value 518.5 317.07 215.39
Table 4.4 Frequency distribution of daily average adverse-selection-ratio
Reported is the frequency distribution of daily average adverse-selection ratios. NYSE(5), NYSE(15), NYSE(30) columns give frequency distribution of 5-minute, 15-minute and 30-minute lag adverse-selection ratios, respectively.
NYSE(5) NYSE(15) NYSE(30) Adverse-selection-ratio
Frequency Ratio Frequency Ratio Frequency Ratio Less than 0 51 0.17% 957 3.26% 3529 12.01%
(0-1] 25874 88.08% 26042 88.65% 23484 79.94%
More than 1 3451 11.75% 2377 8.09% 2363 8.04%
All 29376 100.00% 29376 100.00% 29376 100.00%
Table 4.5 Summary statistics of daily average adverse-selection ratio The table contains basic statistics of daily average adverse-selection ratios: mean (MEAN), standard deviation (STDEV), minimum value (MIN), maximum value (MAX), the first, second, and third quartiles. NYSE(5), NYSE(15), NYSE(30) columns give statistics of 5-minute, 15-minute and 30-minute lag adverse-selection ratios, respectively. T-values are provided for the null hypothesis 1:
mean of average adverse-selection ratios is zero.(T-value1), and null hypothesis 2: mean of average adverse-selection ratios is one. (T-value2).
Adverse-selection-ratio NYSE(5) NYSE(15) NYSE(30)
MEAN 0.683 0.537 0.435
STDEV 0.264 0.325 0.403
MIN -0.270 -0.742 -1.754
MAX 2.276 2.408 3.815
QUARTILE1 0.499 0.318 0.172
QUARTILE2 0.668 0.513 0.409
QUARTILE3 0.852 0.733 0.670
T-value 1 442.5 283.35 184.97 T-value 2 -205.77 -244.58 -240.61
Table 5 Simple correlations between resiliency and its explanatory variables
The table contains bivariate correlation coefficients between resiliency and its explanatory factors. Panel A reports the time-series means of cross-section correlations at every trading day. Panel B reports the cross-section means of time-time-series correlations for every sample stock. RESI is the estimated daily resiliency value for every sample stock. TRADE is the logarithm value of daily transaction number. TVOL is the logarithm value of daily trading volume. REA is the daily average realized spread with 15-minute time lag. ITR is the daily informative-transaction ratio 15-minute time lag. RATIO is the daily average adverse-selection ratio with 15-minute time lag. PRICE is the inverse value of daily average transaction price.
SIZE is the logarithm value of daily average transaction size. CAP is the initial capitalization calculated at the end of year 1999 for every sample stock. VOLA is the unexpected standard deviation of 5-minute returns in every trading day. MARKET is the market resiliency, which is the weighted-average market resiliency across all other stocks. T-values for the time-series and cross-section means of correlations are provided in parentheses.
PANEL A TRADE TVOL REA ITR RATIO PRICE SIZE CAP VOLA RESI 0.102 0.068 -0.004 -0.085 -0.045 -0.065 -0.008 0.046 -0.060
(13.88) (9.61) (-0.59) (-11.72) (-6.26) (-9.80) (-1.28) (6.77) (-8.41) TRADE 0.857 -0.107 -0.509 -0.261 -0.282 0.269 0.525 -0.041
(590.66) (-13.29) (-90.00) (-42.74) (-58.10) (47.24) (149.17) (-4.19)
TVOL 0.055 -0.418 -0.274 -0.001 0.725 0.509 -0.028
(8.36) (-80.63) (-48.08) (-0.26) (239.44) (160.88) (-2.90) REA -0.339 -0.554 0.717 0.243 -0.054 0.068
(-21.86) (-63.01) (58.53) (37.17) (-20.59) (3.73)
ITR 0.789 0.175 -0.101 -0.267 0.054
(187.15) (17.98) (-17.18) (-68.46) (6.83)
RATIO -0.086 -0.164 -0.188 0.090
(-14.10) (-27.59) (-43.71) (12.35)
PRICE 0.374 -0.139 0.143
(72.03) (-37.59) (7.52)
SIZE 0.250 0.003
(32.53) (0.31)
CAP -0.025
(-4.97) PANEL B TRADE TVOL REA ITR RATIO PRICE SIZE MARKET VOLA RESI 0.031 0.009 0.051 -0.046 -0.030 -0.022 -0.017 0.032 -0.043
(3.58) (1.12) (5.22) (-5.78) (-3.36) (-2.81) (-2.09) (4.62) (-4.70) TRADE 0.707 -0.027 -0.180 -0.023 -0.016 0.182 0.014 0.497
(50.58) (-1.46) (-14.57) (-2.02) (-0.50) (7.84) (3.10) (32.73) TVOL 0.014 -0.106 -0.011 0.175 0.789 0.006 0.454
(0.76) (-8.75) (-0.90) (5.98) (63.69) (1.18) (36.67) REA -0.782 -0.803 0.246 0.051 0.021 0.043
(-46.68) (-101.66) (13.69) (3.34) (3.17) (2.33)
ITR 0.824 0.008 0.022 -0.013 -0.058
(102.67) (0.58) (1.94) (-2.10) (-5.64)
RATIO -0.044 0.003 -0.018 0.091
(-4.16) (0.28) (-2.94) (8.09)
PRICE 0.302 -0.007 0.203
(14.08) (-1.45) (9.88)
SIZE -0.010 0.228
(-1.72) (13.42)
MARKET -0.032
(-5.45)
Table 6 Regression test of resiliency explanatory variables
The table contains time-series/cross-section average of the coefficients in cross-section/time-series multivariate regression test of all explanatory variables to resiliency. Panel A reports time-series means of coefficients in cross-section regression tests. Panel B reports cross-section means of coefficients in time-series regression tests. Dependent variable is the daily resiliency for every sample stock. INTE is internal resiliency, which is also the intercept in every regression. TRADE is the logarithm value of daily transaction number. REA is the daily average realized spread with 15-minute time lag. ITR is the daily informative-transaction ratio with 15-minute time lag. RATIO is the daily average adverse-selection ratio with 15-minute time lag. PRICE is the inverse value of daily average transaction price. SIZE is the logarithm value of daily average transaction size. CAP is the initial capitalization calculated at the end of year 1999 for every sample stock. VOLA is the unexpected standard deviation of 5-minute returns in every trading day. MARKET is the market resiliency, which is the weighted-average market resiliency across all other stocks. All the independent variables are standardized. T-values of all the coefficient means are given in parentheses.
Panel A INTE TRADE REA ITR RATIO PRICE SIZE CAP VOLA
0.616 0.026 0.025 -0.025 -0.007 -0.002 -0.014
Set 1
(320.99) (10.70) (4.61) (-4.75) (-3.31) (-0.93) (-6.12)
0.616 0.023 -0.011 -0.004 -0.008 -0.002 -0.015
Set 2
(320.99) (8.73) (-5.14) (-1.54) (-4.10) (-0.98) (-6.85)
0.616 0.027 -0.006 -0.006 -0.009 -0.002 -0.014
Set 3
(320.99) (11.29) (-3.22) (-2.34) (-4.24) (-1.17) (-6.66)
0.616 0.026 0.033 0.003 -0.031 -0.007 -0.002 -0.014
Set 4
(320.99) (9.56) (3.80) (0.86) (-3.69) (-3.14) (-0.81) (-6.15)
0.616 0.026 0.038 0.011 -0.034 -0.005 -0.001 -0.014
Set 5
(320.99) (10.82) (5.69) (3.69) (-5.23) (-2.34) (-0.50) (-6.41)
0.616 0.025 0.040 -0.006 0.014 -0.034 -0.006 -0.001 -0.015
Set 6
(320.99) (8.50) (4.76) (-1.28) (4.07) (-4.08) (-2.60) (-0.37) (-6.90) Panel B INTE TRADE REA ITR RATIO PRICE SIZE MARKET VOLA
0.615 0.009 0.012 -0.002 -0.005 0.006 -0.016
Set 1
(167.78) (3.70) (4.33) (-0.81) (-2.40) (3.46) (-6.75)
0.615 0.006 -0.009 -0.001 -0.005 0.006 -0.017
Set 2
(167.78) (2.73) (-4.45) (-0.44) (-2.18) (3.64) (-6.77)
0.615 0.008 -0.004 -0.001 -0.006 0.006 -0.017
Set 3
(167.78) (3.53) (-1.98) (-0.33) (-2.54) (3.55) (-6.87)
0.615 0.012 0.027 0.017 -0.003 -0.005 0.006 -0.017
Set 4
(167.78) (4.68) (4.95) (3.58) (-1.32) (-2.44) (3.34) (-6.81)
0.615 0.010 0.030 0.022 -0.004 -0.005 0.006 -0.019
Set 5
(167.78) (4.31) (7.89) (6.62) (-1.81) (-2.39) (3.31) (-7.52)
0.615 0.011 0.036 0.005 0.022 -0.004 -0.005 0.006 -0.019
Set 6
(167.78) (4.23) (6.68) (0.95) (6.29) (-1.80) (-2.17) (3.32) (-7.87)
Table 7 Summary statistics of daily proportional and dollar spreads
The table contains basic statistics of daily proportional and dollar quoted/effective spreads. Panel A reports main statistics of all daily proportional and dollar spreads: mean (MEAN), standard deviation (STDEV), minimum value (MIN), maximum value (MAX), the first, second, and third quartiles. QUOTED column gives statistics of quoted spreads. EFFECTIVE column gives statistics of effective spreads. Panel B reports t-values for the null hypothesis 1: mean of the proportional/dollar spreads is zero (T-value)
Proportional Spread Dollar Spread Panel A
QUOTED EFFECTIVE QUOTED EFFECTIVE
MEAN 0.004 0.002 0.141 0.095
STDEV 0.004 0.003 0.071 0.058
MIN 0.001 0.000 0.014 0.011
MAX 0.136 0.113 0.800 1.229
QUARTILE1 0.002 0.001 0.100 0.064
QUARTILE2 0.003 0.002 0.125 0.081
QUARTILE3 0.004 0.003 0.164 0.109
Panel B
T-value 153.12 127.2 339.38 281.45
Table 8.1 Frequency distribution of estimated daily Lambda values
The table reports frequency distribution of all estimated Lambda values. Frequency columns give number of estimated Lambda values in every range. Ratio columns give percentage of estimated Lambda values in every range. Panel A reports frequency distribution of all estimated Lambdas. Panel B reports frequency distribution of all significant estimated Lambdas.
Panel A Frequency Ratio
Negative 235 0.80%
Zero1 10 0.03%
Positive 29131 99.17%
Total 29376 100.00%
Panel B Frequency Ratio
Negative 23 0.08%
Positive 24843 84.57%
Insignificant 4510 15.35%
Total 29376 100.00%
Table 8.2 Summary statistics of estimated daily Lambda values The table contains basic statistics of all the estimated Lambda values. Panel A reports main statistics of all estimated values:
mean (MEAN), standard deviation (STDEV), minimum value (MIN), maximum value (MAX), the first, second, and third quartiles. ALL column gives statistics of all Lambda values, while SIGNIFICANT column gives statistics of all significant Lambda values. T-values are provided for the null hypothesis 1: mean of estimated Lambda is zero (T-value).
Panel A ALL SIGNIFICANT
MEAN 5.80E-06 6.36E-06 STDEV 9.18E-06 9.41E-06
MIN -3.78E-05 -2.38E-05
MAX 2.83E-04 1.59E-04 QUARTILE1 1.10E-06 1.35E-06 QUARTILE2 2.81E-06 3.25E-06 QUARTILE3 6.71E-06 7.47E-06
Panel B
T-value: 108.29 106.58
1 This row shows the number and ratio of estimated results with zero value.
Table 9 Simple correlations between liquidity measures and its explanatory variables
Table 9 Simple correlations between liquidity measures and its explanatory variables