Robustness checks on FM regressions
| Regressors | Panel A: formation periods | Panel B: number of portfolios | ||||
|---|---|---|---|---|---|---|
| 30 | 90 | 150 | 365 | 50 | 200 | |
| Constant | 0.21 (0.72) | 0.22 (0.79) | 0.22 (0.84) | 0.20 (0.63) | 0.19 (0.70) | 0.20 (0.69) |
| −0.70 (−5.66)* | −0.71 (−5.87)* | −0.67 (−5.15)* | −0.45 (−4.92)* | −0.33 (−4.08)* | −0.44 (−4.72)* | |
| −0.71 (−6.35)* | −0.96 (−7.12)* | −0.39 (−3.68)* | −0.45 (−3.76)* | −0.89 (−6.66)* | −0.91 (−6.55)* | |
| −0.39 (−3.44)* | −0.16 (−2.83)* | −0.13 (−2.81)* | −0.099 (−2.18)* | −0.09 (−1.93)*** | −0.16 (−2.84)* | |
| 3.50 (12.03)* | 2.78 (9.74)* | 3.93 (13.01)* | 4.77 (17.88)* | 2.16 (9.43)* | 3.88 (13.12)* | |
| −0.13 (−2.99)* | −0.12 (−2.45)* | −0.12 (−2.66)* | −0.11 (−2.01)** | −0.13 (−2.33)** | −0.27 (−3.05)* | |
| 0.015 (0.84) | 0.046 (2.45)* | 0.04 (2.14)** | 0.038 (1.66) | 0.028 (1.78)*** | 0.036 (1.77)*** | |
| −0.18 (−3.22)* | −0.26 (−3.33)* | −0.24 (−3.22)* | −0.24 (−3.44)* | −0.11 (−2.53)* | −0.16 (−2.78)* | |
| −0.0036 (−1.33) | −0.0012 (−0.41) | −0.0028 (−1.11) | −0.006 (−2.54)* | −0.0038 (−0.49) | −0.016 (−0.84) | |
| −0.12 (−3.99)* | −0.16 (−3.94)* | −0.19 (−3.93)* | −0.23 (−4.08)* | −0.11 (−2.36)* | −0.16 (−4.01)* | |
| −0.80 (−5.43)* | −0.92 (−5.99)* | −0.95 (−6.04)* | −0.96 (−5.88)* | −0.79 (−5.34)* | −0.95 (−5.96)* | |
| −0.060 (−2.91)* | −0.067 (−2.99)* | −0.081 (−3.86)* | −0.084 (−3.93)* | −0.054 (−2.52)* | −0.079 (−3.73)* | |
| −0.007 (−0.88) | −0.009 (−0.99) | −0.011 (−1.12) | −0.012 (−1.19) | −0.006 (−0.69) | −0.010 (−1.01) | |
| 962.15 [0.00] | 994.22 [0.00] | 1014.06 [0.00] | 1008.33 [0.00] | 977.22 [0.00] | 999.26 [0.00] | |
| Pricing error | 0.0036 | 0.0030 | 0.0026 | 0.0028 | 0.0034 | 0.0029 |
| Adj | 0.34 | 0.38 | 0.44 | 0.43 | 0.36 | 0.41 |
| Regressors | Panel A: formation periods | Panel B: number of portfolios | ||||
|---|---|---|---|---|---|---|
| 30 | 90 | 150 | 365 | 50 | 200 | |
| Constant | 0.21 (0.72) | 0.22 (0.79) | 0.22 (0.84) | 0.20 (0.63) | 0.19 (0.70) | 0.20 (0.69) |
| −0.70 (−5.66)* | −0.71 (−5.87)* | −0.67 (−5.15)* | −0.45 (−4.92)* | −0.33 (−4.08)* | −0.44 (−4.72)* | |
| −0.71 (−6.35)* | −0.96 (−7.12)* | −0.39 (−3.68)* | −0.45 (−3.76)* | −0.89 (−6.66)* | −0.91 (−6.55)* | |
| −0.39 (−3.44)* | −0.16 (−2.83)* | −0.13 (−2.81)* | −0.099 (−2.18)* | −0.09 (−1.93)*** | −0.16 (−2.84)* | |
| 3.50 (12.03)* | 2.78 (9.74)* | 3.93 (13.01)* | 4.77 (17.88)* | 2.16 (9.43)* | 3.88 (13.12)* | |
| −0.13 (−2.99)* | −0.12 (−2.45)* | −0.12 (−2.66)* | −0.11 (−2.01)** | −0.13 (−2.33)** | −0.27 (−3.05)* | |
| 0.015 (0.84) | 0.046 (2.45)* | 0.04 (2.14)** | 0.038 (1.66) | 0.028 (1.78)*** | 0.036 (1.77)*** | |
| −0.18 (−3.22)* | −0.26 (−3.33)* | −0.24 (−3.22)* | −0.24 (−3.44)* | −0.11 (−2.53)* | −0.16 (−2.78)* | |
| −0.0036 (−1.33) | −0.0012 (−0.41) | −0.0028 (−1.11) | −0.006 (−2.54)* | −0.0038 (−0.49) | −0.016 (−0.84) | |
| −0.12 (−3.99)* | −0.16 (−3.94)* | −0.19 (−3.93)* | −0.23 (−4.08)* | −0.11 (−2.36)* | −0.16 (−4.01)* | |
| −0.80 (−5.43)* | −0.92 (−5.99)* | −0.95 (−6.04)* | −0.96 (−5.88)* | −0.79 (−5.34)* | −0.95 (−5.96)* | |
| −0.060 (−2.91)* | −0.067 (−2.99)* | −0.081 (−3.86)* | −0.084 (−3.93)* | −0.054 (−2.52)* | −0.079 (−3.73)* | |
| −0.007 (−0.88) | −0.009 (−0.99) | −0.011 (−1.12) | −0.012 (−1.19) | −0.006 (−0.69) | −0.010 (−1.01) | |
| 962.15 [0.00] | 994.22 [0.00] | 1014.06 [0.00] | 1008.33 [0.00] | 977.22 [0.00] | 999.26 [0.00] | |
| Pricing error | 0.0036 | 0.0030 | 0.0026 | 0.0028 | 0.0034 | 0.0029 |
| Adj | 0.34 | 0.38 | 0.44 | 0.43 | 0.36 | 0.41 |
Note(s): This table presents the results of the Fama and MacBeth (1973) regressions and their mean coefficients using equation (13). It utilizes monthly sorted portfolios based on . In Panel A, columns 1 to 4 display the FM regression outcomes computed through measures , in accordance with equations (11) and (12), for the formation periods of 30, 90, 150, and 365 days, respectively. We formulated the expected IE measures at each month’s end from January 1988 to June 2019. The expected IE is estimated using the risk factors from models 7 to 10 in Table 2 for the cross-sectional regressions. Panel B’s columns 1 (2) showcase stocks sorted into 50 (200) portfolios for every month, based on expected entropy. These cross-sectional regressions are calculated monthly for our sample, from January 1988 to June 2019. For portfolio p, expected entropy and other risk factors are the value-weighted average of firm-level factors across all stocks in the portfolio. The measures , as detailed in equations (11) and (12), are computed using a 60-day formation period. Table 4 defines other risk factors. We present average coefficients and Newey and West (1987) t-statistics (in parentheses), alongside average adjusted-R2. Levels of significance are denoted as *, **, and *** for 1%, 5%, and 10%, respectively. The FM t-statistics are shown in parentheses, while the test results are in brackets
Source(s): Created by the author
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