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5. Conclusion
In our paper we suggest that investor sentiment positively predicts, because of different level of noise trader, excess returns between announcement days and non-announcement days. In the high sentiment period, investor sentiment plays an
extremely important roles in the excess returns difference between days. We use forty eight portfolios to provide robust check with these results. Moreover, we provide evidence that investor sentiment factors holds during high sentiment period, reconciling traditional risk factors with behavioral factors.
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Figure 1. Investor sentiment index from 1966 to 2010.
This figure shows investor sentiment index from 1966 to 2010. The sentiment index is component of six measures which includes closed-end fund discount, the NYSE share turnover, the number of and the average first-day returns on initial public offerings (IPOs), the equity share in new issues, and the dividend premium. High- and low- sentiment periods classified based on the median level of the index of Baker and Wurgler (2006). Using the index, we identify the late 1960s, mid-1980s, and late 1990s, and early 2000s as high sentiment periods, whereas others periods are identified as low sentiment periods.
-3 -2 -1 0 1 2 3
1966 1976 1986 1996 2006
INVESTOR SENTIMENT INDEX
YEAR
Investor sentiment index from 1966 to 2010
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Table 1 Summary statistics of daily stock market excess returns
This table reports the distribution of monthly stock market excess returns on
announcement days and non-announcement days. Announcement days are those days when macroeconomic numbers, which include CPI/PPI numbers, FOMC interest rate decisions, and employment numbers, are scheduled for release. Non-announcement days are others trading days. The sample covers the 1966-2010 periods (excluding July 1968 in which there is announcement scheduled for release). Monthly stock market excess returns are computed by month as 22 times average daily different between the CRSP value-weighted/equal-weighted market returns and risk-free rate, 22 is the approximate number of trading days in one month. The daily risk-free rate is computed from the 1-month risk-free rate provided by CRSP. All numbers are
expressed in percent, and those number in bold are of special interest.
Panel A: Value-Weighted
Announcement Non- Announcement Difference
Mean 1.458572 0.27655 1.182022
t-stat 2.73 1.25 2.10
1% percentile -35.7577 -16.0721 -19.6856
25% percentile -4.55127 -2.67379 -1.87748
Median 2.28905 0.568812 1.720238
75% percentile 7.925 3.606725 4.318275
99% percentile 29.62195 13.40227 16.21968
Std. Dev. 12.39269 5.135425
Skewness -0.31111 -0.58647
Kurtosis 5.533046 4.809443
N 539 539 539
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Panel B: Equal-Weighted
Announcement Non- Announcement Difference
Mean 2.65975 1.050178 1.160351
t-stat 5.72 3.86 3.66
1% percentile -33.3433 -17.9858 -15.3575
25% percentile -1.6456 -2.68226 1.03666
Median 3.9439 1.37145 2.57245
75% percentile 8.518467 4.849333 3.669134
99% percentile 28.8926 15.43837 13.45423
Std. Dev. 10.79151 6.30215
Skewness -0.69183 -0.43071
Kurtosis 6.798921 5.691633
N 539 539 539
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Table 1B. Correlation metrics of explanation variables
SENT RMTRF SMB HML UMD
SENT 1
RMTRF 0.3263 1
SMB 0.481 0.2792 1
HML -0.4298 -0.3574 -0.2512 1
UMD 0.0961 -0.1854 -0.0195 -0.0961 1
This table reports that investor sentiment index is not strongly related to other factors, indicating that sentiment index is not proxy of other factor and that the regression model we use will not have colinearity effect.
Table 1C. VIF test of explanation variables
Variable VIF 1/VIF
SENT 1.55 0.645385
HML 1.33 0.749696
SMB 1.33 0.750023
RMTRF 1.3 0.769814
UMD 1.09 0.921129
Mean VIF 1.32
This table reports the VIF of each variables and the mean VIF. This result suggests that the regression model we use will not have colinearity effect.
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Table 1D Correlation metrics of explanation variables
Evt Evt+1 SENT SENTL
Evt 1
Evt+1 0.5024 1
SENT 0.037 0.0683 1
SENTL 0.0574 0.0288 0.645 1
This table reports that investor sentiment index is not strongly related to expected variance, indicating that sentiment index is not proxy of expected variance and that the regression model we use will not have colinearity effect.
Table 1E VIF test of explanation variables
Variable VIF 1/VIF
LrNt 1.3 0.766374
LrAt 1.2 0.829936
Evt 1.15 0.868841
SENT 1.02 0.977873
Mean VIF 1.17
This table reports the VIF of each variables and the mean VIF. This result suggests that the regression model we use will not have colinearity effect.
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Table 2 Monthly excess returns difference against investor sentiment:
The table reports estimates of b in the following regressions A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENT𝑡+ cRMTRF𝑡+ ϵ𝑖,𝑡 (1) A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENT𝑡+ cRMTRF𝑡+ dSMB𝑡+ + ϵ𝑖,𝑡 (2)
A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENT𝑡+ cRMTRF𝑡+ dSMB𝑡+ eHML𝑡+ ϵ𝑖,𝑡 (3) A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENT𝑡+ cRMTRF𝑡+ dSMB𝑡+ eHML𝑡+ fUMD𝑡+ ϵ𝑖,𝑡 (4) Where A𝑖,𝑡 is the average daily excess returns on announcement days in month t on either value-weighted or equal-weighted market returns, N𝑖,𝑡 is the average daily excess returns on non-announcement days in month t that is pair with A𝑖,𝑡, SENT𝑡 is the changes of investor-sentiment index of Baker and Wurgler (2007). The sample periods is from 1966:1 to 2010:12 for all (excluding July 1968 in which there is announcement scheduled for release). Equation (1) estimates investor sentiment factor without control, and equation (2)(3)(4) estimate the factor by controlling possible factors including size (SMB), book-to-market (HML), and momentum (UMD) factors.
Panel A: Value-weighted
SENT 0.000906*** 0.000631** 0.000647** 0.000651**
(3.58) (2.26) (2.19) (2.19)
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Panel B: Equal-weighted
SENT 0.000620*** 0.000637*** 0.000596** 0.000591**
(3.09) (2.86) (2.54) (2.50)
RMTRF -0.00677*** -0.00675*** -0.00687*** -0.00684***
(-7.91) (-7.79) (-7.68) (-7.40)
SMB -0.000247 -0.000262 -0.000252
(-0.17) (-0.18) (-0.18)
HML -0.000851 -0.000820
(-0.55) (-0.53)
UMD 0.000139
(0.15)
N 539 539 539 539
R-sq 0.105 0.105 0.106 0.106
adj.R-sq 0.102 0.1 0.099 0.097
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Table 3 Monthly excess returns difference against the sign of investor sentiment This table presents estimate of b in the following equations
A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENTDUMMY𝑡+ cRMTRF𝑡+ ϵ𝑖,𝑡 (5) A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENTDUMMY𝑡+ cRMTRF𝑡+ dSMB𝑡+ ϵ𝑖,𝑡 (6)
A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENTDUMMY𝑡+ cRMTRF𝑡+ dSMB𝑡+ eHML𝑡+ ϵ𝑖,𝑡 (7) A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENTDUMMY𝑡+ cRMTRF𝑡+ dSMB𝑡+ eHML𝑡+ fUMD𝑡
+ ϵ𝑖,𝑡 (8)
Where SENTDUMMY is a dummy variable that equals 1 when changes in investor sentiment are positive. The sample periods is from 1966:1 to 2010:12 for all
(excluding July 1968 in which there is announcement scheduled for release). Equation (5) estimates investor sentiment dummy without control, and equation (6)(7)(8) estimate the factor by controlling possible factors including size (SMB), book-to-market (HML), and momentum (UMD) factors.
Panel A: Value-weighted
SENTDUMMY 0.00259*** 0.00191** 0.00192** 0.00194**
(3.30) (2.35) (2.31) (2.32) RMTRF -0.00981*** -0.0105*** -0.0104*** -0.0106***
(-6.14) (-6.57) (-5.89) (-5.85)
SMB 0.00675*** 0.00682*** 0.00683***
(2.88) (2.78) (2.78)
HML 0.000273 -0.0000591
(0.09) (-0.02)
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Panel B: Equal-weighted
SENTDUMMY 0.00154*** 0.00134** 0.00120* 0.00120*
(2.60) (2.15) (1.89) (1.89)
RMTRF -0.00678*** -0.00697*** -0.00757*** -0.00760***
(-5.62) (-5.72) (-5.62) (-5.49)
SMB 0.00203 0.00148 0.00148
(1.13) (0.79) (0.79)
HML -0.00228 -0.00233
(-1.04) (-1.03)
UMD -0.000114
(-0.09)
N 269 269 269 269
R-sq 0.108 0.113 0.116 0.116
adj.R-sq 0.102 0.103 0.103 0.1
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Table 4 Monthly excess returns difference against investor sentiment during periods of high and low investor sentiment.
This table shows difference during each sentiment periods and the estimate of investor sentiment. We regress equations (5)(6)(7)(8) during each high- and low- sentiment periods. High- and low- sentiment periods classified based on the median level of the index of Baker and Wurgler (2006). All data periods is from 1966:01 to 2010:12 (excluding July 1968 in which there is announcement scheduled for release).
Panel A: Value-weighted
high low high low high low high low
SENT 0.00130*** 0.000309 0.000852** 0.000244 0.000953** 0.000266 0.000996** 0.000238
(3.59) (0.85) (2.07) (0.62) (2.10) (0.68) (2.17) (0.60)
SENT 0.000919*** 0.000166 0.000879*** 0.000268 0.000834** 0.000274 0.000849** 0.000255
(3.37) (0.54) (2.82) (0.81) (2.42) (0.82) (2.44) (0.76) RMTRF -0.00743*** -0.00619*** -0.00745*** -0.00590*** -0.00760*** -0.00587*** -0.00771*** -0.00569***
(-6.02) (-5.21) (-6.01) (-4.78) (-5.69) (-4.70) (-5.60) (-4.40)
SMB 0.000512 -0.00182 0.000433 -0.00185 0.000409 -0.00178
(0.26) (-0.86) (0.22) (-0.87) (0.21) (-0.83)
HML -0.000741 0.000384 -0.000866 0.000451
(-0.31) (0.17) (-0.36) (0.20)
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Table 5 Investor sentiment and monthly excess returns difference: predictive regressions for investor sentiment on long-short strategy returns.
This table presents estimate of b in the following regressions:
A𝑖,𝑡− N𝑖,𝑡 = a𝑖+ bSENT𝑡−1+ cRMTRF𝑡+ ϵ𝑖,𝑡 (9) A𝑖,𝑡− N𝑖,𝑡 = a𝑖+ bSENT𝑡−1+ cRMTRF𝑡+ dSMB𝑡+ ϵ𝑖,𝑡 (10) A𝑖,𝑡− N𝑖,𝑡 = a𝑖+ bSENT𝑡−1+ cRMTRF𝑡+ dSMB𝑡+ eHML𝑡+ ϵ𝑖,𝑡 (11)
A𝑖,𝑡− N𝑖,𝑡 = a𝑖+ bSENT𝑡−1+ cRMTRF𝑡+ dSMB𝑡+ eHML𝑡+ fUMD𝑡+ ϵ𝑖,𝑡 (12) Where SENT𝑡−1 is the lag 1 periods of the changes of investor sentiment index.
Long-short strategy is that we long the market portfolios in the announcement days and short the market portfolios in other trading days. The sample periods is from 1966:2 to 2010:12. Equation (9) estimates the predictability of investor sentiment without control, and equation (10)(11)(12) estimate the predictability of investor sentiment by controlling possible factors including size (SMB), book-to-market (HML), and momentum (UMD) factors.
Panel A: Value-weighted
SENT𝑡−1 0.000672*** 0.000651*** 0.000649*** 0.000652***
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Panel B: Equal-weighted
SENT𝑡−1 0.000414** 0.000409** 0.000403** 0.000408**
(2.17) (2.15) (2.12) (2.14) RMTRF -0.00590*** -0.00618*** -0.00654*** -0.00641***
(-7.25) (-7.30) (-7.36) (-7.01)
SMB 0.00151 0.00121 0.00121
(1.17) (0.93) (0.93)
HML -0.00196 -0.00180
(-1.34) (-1.21)
UMD 0.000539
(0.60)
N 537 537 537 537
R-sq 0.097 0.099 0.102 0.103
adj.R-sq 0.093 0.094 0.095 0.094
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Table 6 Monthly excess returns against conditional variance and investor sentiment during each high- and low- sentiment periods
The table reports OLS estimates of the investor sentiment and of the conditional variance using quarterly data from 1966 to 2010.
A𝑡+1= A𝑡+ N𝑡+ Ev𝑡+ SENT𝑡 (13)(ℎ𝑖𝑔ℎ 𝑠𝑒𝑛𝑡𝑖𝑚𝑒𝑛𝑡)
Where A𝑡 is quarterly aggregate announcement-day market excess returns, N𝑡 is quarterly aggregate non-announcement-day market excess returns, and Ev𝑡 is expected variance of the market returns, which computed using French et al. (1987) methodology that shows in equation (17). In equation (17), 𝑟𝑡−𝑑 is daily market returns in month t, 𝑁𝑡 is the number of trading days in month t, and 22 is approximate number of trading days in one month.
(13) (14) (15) (16)
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Table 7 Monthly excess returns difference against investor sentiment: using VIX as proxy of investor sentiment
This table presents the results of OLS regressions of monthly excess returns difference on VIX that regards as proxy of investor sentiment and other controls including size (SMB), book-to-market (HML), and momentum (UMD) factors.
A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bVIX𝑡+ cRMTRF𝑡+ ϵ𝑖,𝑡 (18)
A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bVIX𝑡+ cRMTRF𝑡+ dSMB𝑡+ eHML𝑡+ fUMD𝑡+ ϵ𝑖,𝑡 (19) Where VIX is Chicago Board Options Exchange Volatility Index, and term VIX𝑡 is average of each day VIX in month t. VIX(S&P500) is the volatility of S&P500 index, and VIX(S&P100) is the volatility of S&P100 index. Data periods is from 1990:01 to 2010:12.
Panel A: Value-weighted
VIX(S&P500) -0.000780*** -0.000791***
(-7.03) (-6.96)
VIX(S&P100) -0.000702*** -0.000714***
(-6.66) (-6.59)
RMTRF -0.0170*** -0.0166*** -0.0172*** -0.0168***
(-9.89) (-9.65) (-9.30) (-9.04)
SMB 0.000747 0.000692 0.000134 0.0000965
(0.33) (0.30) (0.06) (0.04)
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Panel B: Equal-weighted
VIX(S&P500) -0.000538*** -0.000561***
(-6.04) (-6.18)
VIX(S&P100) -0.000472*** -0.000493***
(-5.56) (-5.68)
RMTRF -0.0119*** -0.0116*** -0.0125*** -0.0121***
(-8.65) (-8.38) (-8.43) (-8.14)
SMB -0.00157 -0.00154 -0.00281 -0.00275
(-0.85) (-0.83) (-1.44) (-1.40)
HML -0.00379* -0.00359*
(-1.80) (-1.69)
UMD 0.000596 0.000682
(0.49) (0.55)
N 251 251 251 251
R-square 0.256 0.241 0.268 0.252
Adj.R 0.247 0.232 0.253 0.237
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Table 8 Monthly excess returns difference against investor sentiment: using the closed-end fund discount as a proxy of investor sentiment
This table presents the results of OLS regressions of monthly excess returns
difference on closed-end fund discount that regards as proxy of investor sentiment and other controls including size (SMB), book-to-market (HML), and momentum (UMD) factors.
A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + b𝑐𝑒𝑓𝑑𝑡+ cRMTRF𝑡+ ϵ𝑖,𝑡 (20) A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + b𝑐𝑒𝑓𝑑𝑡+ cRMTRF𝑡+ dSMB𝑡+ ϵ𝑖,𝑡 (21) A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + b𝑐𝑒𝑓𝑑𝑡+ cRMTRF𝑡+ dSMB𝑡+ eHML𝑡+ ϵ𝑖,𝑡 (22)
A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + b𝑐𝑒𝑓𝑑𝑡+ cRMTRF𝑡+ dSMB𝑡+ eHML𝑡+ fUMD𝑡+ ϵ𝑖,𝑡 (23) Where the close-end funds discount is the difference between the net asset value of a fund’s actual security holding and the fund’s market price, and 𝑐𝑒𝑓𝑑𝑡 is the close-end funds discount at month t. Lee et al. (1991) argued that discount increasing when retail investors are bearish, suggesting the close-end funds discount be a proxy of investor sentiment. Data periods is from 1966:01 to 2010:12 (excluding July 1968 in which there is announcement scheduled for release).
Panel A: Value-weighted
cefd 0.0000709** 0.0000702** 0.0000693** 0.0000693**
(2.19) (2.19) (2.16) (2.15)
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Panel B: Equal-weighted
cefd 0.0000695*** 0.0000694*** 0.0000672*** 0.0000672***
(2.73) (2.72) (2.63) (2.63)
RMTRF -0.00591*** -0.00618*** -0.00653*** -0.00643***
(-7.30) (-7.33) (-7.36) (-7.05)
SMB 0.00148 0.00121 0.00121
(1.16) (0.93) (0.93)
HML -0.00183 -0.00171
(-1.25) (-1.15)
UMD 0.000420
(0.47)
N 539 539 539 539
R-square 0.101 0.104 0.106 0.107
Adj.R 0.098 0.099 0.1 0.098
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Table 9 Monthly excess returns difference against investor sentiment during periods of high investor sentiment
This table shows difference during low sentiment periods and the estimate of investor sentiment. We regress equations (5)(6)(7)(8) during high sentiment periods. In panel A, we test size-formed portfolios including trisections, which is showed in first row, quintile, which is showed in second row, and ten equal parts, which is showed in third row. In panel B, we test book-to-market-ratio-formed portfolios including trisections, which is showed in first row, quintile, which is showed in second row, and ten equal parts, which is showed in third row. In panel C, we test mixed-form portfolios, including 6 fama-french portfolios formed by size and book-to-market ratio, which is showed in the left three column, and 6 fama-french portfolios formed by size and momentum, which is showed in the right three column. This table reports that investor sentiment holds on most portfolios, especially on those portfolios of growth firms and of small firms. Data periods is from 1966:01 to 2010:12 (excluding July 1968 in which there is announcement scheduled for release).
Panel A: Size-formed portfolios
First Small Big
0.00114*** 0.00118*** 0.000929**
(2.80) (2.72) (1.97)
Second Small Big
0.00112*** 0.00129*** 0.00111** 0.00121*** 0.000903*
(2.84) (2.94) (2.56) (2.69) (1.90)
Third Small Big
0.000972*** 0.00124*** 0.00117*** 0.00140*** 0.00115** 0.00109** 0.00114** 0.00124*** 0.00110** 0.000870*
(2.69) (2.85) (2.62) (3.18) (2.50) (2.57) (2.58) (2.70) (2.32) (1.80)
Panel B: Value-formed portfolios
First Negative low high
0.00156*** 0.00118** 0.000635 0.000390
(2.68) (2.37) (1.52) (0.98)
Second Low High
0.00120** 0.000847* 0.000614 0.000582 0.000383
(2.35) (1.86) (1.44) (1.46) (0.95)
Third Low High
0.00120** 0.00111** 0.000992** 0.000660 0.000681 0.000522 0.000686* 0.000501 0.000351 0.000457
(2.23) (2.28) (2.07) (1.45) (1.50) (1.25) (1.69) (1.21) (0.84) (1.07)
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41 Panel C: Mixed-formed portfolios
Low High LOSS WIN
Small 0.00169*** 0.00100*** 0.000790** Small 0.00132*** 0.00108*** 0.00137***
(3.32) (2.60) (2.15) (2.65) (2.89) (2.93)
Big 0.00115** 0.000588 0.000283 Big 0.000928 0.000726 0.000964*
(2.27) (1.36) (0.68) (1.63) (1.60) (1.92)
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Table 10A Monthly excess returns difference against investor sentiment (orthogonal):
The table reports estimates of b in the following regressions A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENT𝑡+ cRMTRF𝑡+ ϵ𝑖,𝑡 (1) A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENT𝑡+ cRMTRF𝑡+ dSMB𝑡+ + ϵ𝑖,𝑡 (2)
A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENT𝑡+ cRMTRF𝑡+ dSMB𝑡+ eHML𝑡+ ϵ𝑖,𝑡 (3) A𝑖,𝑡− N𝑖,𝑡 = a𝑖 + bSENT𝑡+ cRMTRF𝑡+ dSMB𝑡+ eHML𝑡+ fUMD𝑡+ ϵ𝑖,𝑡 (4) Where A𝑖,𝑡 is the average daily excess returns on announcement days in month t on either value-weighted or equal-weighted market returns, N𝑖,𝑡 is the average daily excess returns on non-announcement days in month t that is pair with A𝑖,𝑡, SENT𝑡 is the changes of investor-sentiment index (orthogonal) of Baker and Wurgler (2007).
The sample periods is from 1966:1 to 2010:12 for all (excluding July 1968 in which there is announcement scheduled for release). Equation (1) estimates investor sentiment factor without control, and equation (2)(3)(4) estimate the factor by controlling possible factors including size (SMB), book-to-market (HML), and momentum (UMD) factors.
Panel A: Value-weighted
SENT 0.000432* 0.000401* 0.000409* 0.000408*
(1.79) (1.68) (1.71) (1.70)
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43
Panel B: Equal-weighted
SENT 0.000179 0.000171 0.000186 0.000183
(0.94) (0.90) (0.97) (0.96)
RMTRF -0.00589*** -0.00616*** -0.00656*** -0.00647***
(-7.22) (-7.25) (-7.36) (-7.06)
SMB 0.00146 0.00113 0.00113
(1.14) (0.87) (0.87)
HML -0.00216 -0.00205
(-1.47) (-1.37)
UMD 0.000389
(0.43)
N 539 539 539 539
R-sq 0.09 0.093 0.096 0.097
adj.R-sq 0.087 0.088 0.09 0.088
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Table 11B Monthly excess returns difference against investor sentiment during periods of high and low investor sentiment (orthogonal).
This table shows difference during each sentiment periods and the estimate of investor sentiment. We regress equations (5)(6)(7)(8) during each high- and low- sentiment periods. High- and low- sentiment periods classified based on the median level of the index of Baker and Wurgler (2006). All data periods is from 1966:01 to 2010:12 (excluding July 1968 in which there is announcement scheduled for release).
Panel A: Value-weighted
high low high low high low high low
SENT 0.00126*** -0.000724** 0.00131*** -0.000769** 0.00134*** -0.000766** 0.00135*** -0.000776**
(-3.65) (-2.19) (-3.89) (-2.31) (-3.95) (-2.30) (-3.98) (-2.32)
SENT 0.000702*** -0.000558** 0.000721*** -0.000544* 0.000771*** -0.000544* 0.000774*** -0.000551*
(-2.68) (-2.00) (-2.76) (-1.93) (-2.95) (-1.93) (-2.95) (-1.95) RMTRF
-0.00583*** -0.00615*** -0.00637*** -0.00600*** -0.00748*** -0.00599*** -0.00754*** -0.00577***
(-5.09) (-5.38) (-5.43) (-4.90) (-5.65) (-4.83) (-5.54) (-4.51)
SMB 0.00334* -0.000713 0.00223 -0.000719 0.00224 -0.000672
(-1.96) (-0.36) (-1.24) (-0.36) (-1.24) (-0.34)
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Table 12C Monthly excess returns against conditional variance and investor sentiment (orthogonal) during each high- and low- sentiment periods
The table reports OLS estimates of the investor sentiment and of the conditional variance using quarterly data from 1966 to 2010.
A𝑡+1= A𝑡+ N𝑡+ Ev𝑡+ SENT𝑡 (13)(ℎ𝑖𝑔ℎ 𝑠𝑒𝑛𝑡𝑖𝑚𝑒𝑛𝑡)
Where A𝑡 is quarterly aggregate announcement-day market excess returns, N𝑡 is quarterly aggregate non-announcement-day market excess returns, and Ev𝑡 is expected variance of the market returns, which computed using French et al. (1987) methodology that shows in equation (17). In equation (17), 𝑟𝑡−𝑑 is daily market returns in month t, 𝑁𝑡 is the number of trading days in month t, and 22 is approximate number of trading days in one month.
(13) (14)