Factor models · Product inferenceEmpirical finding
ETF factor exposures, alpha, and multiple-testing control
Advertised exposures appeared; no tested alpha survived multiple-testing correction.
Factor regressions tested six style ETFs for their advertised exposures. A second stage asked what unexplained return survived multiple-testing correction.
Commit 4806df9Evidence ad24c49Data – · 2,892 rowsUniverse 15 instrumentsTests 239 / 239 passed
- 0.34MTUM momentum β
- 49MTUM momentum t-statistic (OLS)
- 0.31VLUE value (HML) β
- 29VLUE value t-statistic (OLS)
- +3.29%QQQ annualised alpha
- 0.024QQQ raw HAC p-value
The advertised exposures were recovered; no alpha survived multiple-testing correction.
MTUM loaded on momentum at β = 0.34 (t = 49). VLUE loaded on value at β = 0.31 (t = 29). QQQ's estimated +3.29% annualized alpha carried a raw HAC p-value of 0.024. Its Benjamini–Hochberg q-value was 0.147. Across the six ETFs, no alpha survived the 5% false-discovery threshold.
Factor loadings with 95% HAC intervals · six style ETFs
β · daily excess returns on FF5 + momentum · Newey–West HAC(5) · notebook 05
| factor | QQQ | QQQ · low | QQQ · high | USMV | USMV · low | USMV · high | MTUM | MTUM · low | MTUM · high | VLUE | VLUE · low | VLUE · high | QUAL | QUAL · low | QUAL · high | IWM | IWM · low | IWM · high |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mkt-RF | 1.10561 | 1.07625 | 1.13497 | 0.742938 | 0.698619 | 0.787258 | 1.04062 | 1.01464 | 1.06659 | 0.984251 | 0.964179 | 1.00432 | 0.976277 | 0.95761 | 0.994944 | 1.00352 | 0.988921 | 1.01812 |
| SMB | -0.121876 | -0.14973 | -0.0940219 | -0.124721 | -0.168068 | -0.0813742 | -0.096254 | -0.136873 | -0.0556354 | 0.107931 | 0.0642319 | 0.151629 | -0.0837018 | -0.10613 | -0.0612736 | 0.848244 | 0.810275 | 0.886213 |
| HML | -0.306236 | -0.342893 | -0.269579 | 0.0101894 | -0.0348243 | 0.0552031 | -0.0202303 | -0.0585054 | 0.0180449 | 0.310955 | 0.271237 | 0.350673 | -0.0435282 | -0.0631468 | -0.0239096 | 0.100439 | 0.078164 | 0.122714 |
| RMW | 0.0505008 | 0.0167816 | 0.08422 | 0.163288 | 0.122347 | 0.204229 | -0.156801 | -0.200634 | -0.112967 | 0.00422313 | -0.0446202 | 0.0530664 | 0.148468 | 0.129138 | 0.167798 | -0.110021 | -0.131989 | -0.0880523 |
| CMA | -0.185612 | -0.237528 | -0.133695 | 0.177752 | 0.0981771 | 0.257328 | -0.0148822 | -0.0787626 | 0.0489983 | 0.118833 | 0.0634586 | 0.174207 | 0.0250655 | -0.00747678 | 0.0576078 | -0.0176326 | -0.0478461 | 0.012581 |
| Mom | 0.0414621 | 0.0251993 | 0.0577248 | -0.0189788 | -0.0431255 | 0.00516793 | 0.339623 | 0.308575 | 0.370672 | -0.056308 | -0.0812792 | -0.0313368 | -0.0412127 | -0.0575118 | -0.0249137 | 0.0246866 | 0.0059725 | 0.0434007 |
Annualised alpha with HAC intervals · six style ETFs
% per year · 95% HAC(5) interval · notebook 05
| ETF | Alpha (% p.a.) | Raw p | Holm p | BH q | Survives 5% FDR |
|---|---|---|---|---|---|
| QQQ | 3.29498 | 0.0244544 | 0.146726 | 0.146726 | no |
| USMV | -1.23103 | 0.51895 | 1 | 0.778425 | no |
| MTUM | 0.020619 | 0.990158 | 1 | 0.990158 | no |
| VLUE | 0.487565 | 0.773771 | 1 | 0.928525 | no |
| QUAL | -0.739199 | 0.417677 | 1 | 0.778425 | no |
| IWM | -1.13975 | 0.110003 | 0.550015 | 0.330009 | no |
Rolling 252-day QQQ betas with HAC bands
β · trailing 252 observations, month-end · 95% HAC(5) band · notebook 05
| window end | Mkt-RF | Mkt-RF · low | Mkt-RF · high | HML | HML · low | HML · high | Mom | Mom · low | Mom · high |
|---|---|---|---|---|---|---|---|---|---|
| 2016-01-29 | 1.0431 | 1.00906 | 1.07714 | -0.114465 | -0.210026 | -0.0189029 | 0.00933983 | -0.0362626 | 0.0549422 |
| 2016-02-29 | 1.04671 | 1.0119 | 1.08151 | -0.124627 | -0.232131 | -0.0171235 | 0.0139579 | -0.0395234 | 0.0674392 |
| 2016-03-31 | 1.05086 | 1.01554 | 1.08618 | -0.110771 | -0.21281 | -0.00873136 | 0.0212103 | -0.0274961 | 0.0699166 |
| 2016-04-29 | 1.04146 | 1.0044 | 1.07851 | -0.129357 | -0.224805 | -0.0339093 | 0.0143462 | -0.0373625 | 0.0660549 |
| 2016-05-31 | 1.0434 | 1.00705 | 1.07975 | -0.119298 | -0.21069 | -0.0279059 | 0.0177421 | -0.033232 | 0.0687163 |
| 2016-06-30 | 1.04597 | 1.00836 | 1.08358 | -0.115914 | -0.205009 | -0.0268188 | 0.022749 | -0.0277195 | 0.0732176 |
| 2016-07-29 | 1.06346 | 1.02606 | 1.10085 | -0.139649 | -0.231184 | -0.0481136 | 0.0275621 | -0.0204532 | 0.0755775 |
| 2016-08-31 | 1.06677 | 1.02093 | 1.11262 | -0.149418 | -0.24083 | -0.058006 | 0.0339202 | -0.0168405 | 0.0846809 |
| 2016-09-30 | 1.08866 | 1.0434 | 1.13392 | -0.190756 | -0.283014 | -0.0984969 | 0.0444552 | -0.00856915 | 0.0974796 |
| 2016-10-31 | 1.0858 | 1.04924 | 1.12235 | -0.170453 | -0.265651 | -0.0752536 | 0.0732324 | 0.0259694 | 0.120495 |
| 2016-11-30 | 1.10088 | 1.06106 | 1.14071 | -0.230305 | -0.360251 | -0.100359 | 0.0820427 | 0.035318 | 0.128767 |
| 2016-12-30 | 1.11748 | 1.07582 | 1.15914 | -0.235719 | -0.360282 | -0.111157 | 0.0883889 | 0.0406167 | 0.136161 |
| 2017-01-31 | 1.11532 | 1.06539 | 1.16525 | -0.279068 | -0.382223 | -0.175913 | 0.0746015 | 0.024889 | 0.124314 |
| 2017-02-28 | 1.13526 | 1.08621 | 1.18431 | -0.265473 | -0.360864 | -0.170081 | 0.0810392 | 0.0272357 | 0.134843 |
| 2017-03-31 | 1.12099 | 1.06905 | 1.17293 | -0.274333 | -0.367281 | -0.181386 | 0.0713002 | 0.00292866 | 0.139672 |
| 2017-04-28 | 1.11736 | 1.06198 | 1.17273 | -0.292987 | -0.396814 | -0.189161 | 0.0577965 | -0.0173293 | 0.132922 |
| 2017-05-31 | 1.12678 | 1.06724 | 1.18631 | -0.296072 | -0.404742 | -0.187401 | 0.0855098 | 0.000679407 | 0.17034 |
| 2017-06-30 | 1.1169 | 1.04628 | 1.18751 | -0.375655 | -0.481708 | -0.269602 | 0.142596 | 0.037649 | 0.247544 |
| 2017-07-31 | 1.09941 | 1.03836 | 1.16047 | -0.386373 | -0.482092 | -0.290653 | 0.214992 | 0.131119 | 0.298866 |
| 2017-08-31 | 1.10193 | 1.04631 | 1.15755 | -0.420923 | -0.500236 | -0.341611 | 0.260517 | 0.196204 | 0.324829 |
| 2017-09-29 | 1.14895 | 1.08884 | 1.20906 | -0.415359 | -0.494027 | -0.336691 | 0.233034 | 0.160253 | 0.305816 |
| 2017-10-31 | 1.15759 | 1.09318 | 1.22201 | -0.423138 | -0.495429 | -0.350847 | 0.235572 | 0.163783 | 0.30736 |
| 2017-11-30 | 1.15912 | 1.07636 | 1.24189 | -0.431684 | -0.500004 | -0.363364 | 0.174909 | 0.0883792 | 0.261439 |
| 2017-12-29 | 1.16251 | 1.07711 | 1.24791 | -0.428492 | -0.497951 | -0.359033 | 0.164748 | 0.085449 | 0.244047 |
| 2018-01-31 | 1.16022 | 1.07657 | 1.24386 | -0.434808 | -0.507361 | -0.362255 | 0.113174 | 0.0237557 | 0.202592 |
| 2018-02-28 | 1.10745 | 1.05197 | 1.16294 | -0.447878 | -0.523495 | -0.37226 | 0.113992 | 0.0282691 | 0.199714 |
| 2018-03-29 | 1.11829 | 1.06725 | 1.16933 | -0.453792 | -0.528911 | -0.378672 | 0.104598 | 0.0117192 | 0.197476 |
| 2018-04-30 | 1.12839 | 1.08023 | 1.17655 | -0.462323 | -0.535047 | -0.3896 | 0.0875431 | -0.0079499 | 0.183036 |
| 2018-05-31 | 1.12417 | 1.07792 | 1.17042 | -0.485631 | -0.562555 | -0.408707 | 0.0368532 | -0.052823 | 0.126529 |
| 2018-06-29 | 1.11817 | 1.07494 | 1.1614 | -0.496972 | -0.576751 | -0.417194 | 0.0129372 | -0.0666241 | 0.0924984 |
| 2018-07-31 | 1.12107 | 1.07316 | 1.16898 | -0.447737 | -0.532289 | -0.363185 | 0.0030704 | -0.07932 | 0.0854608 |
| 2018-08-31 | 1.11204 | 1.06531 | 1.15877 | -0.439393 | -0.526378 | -0.352408 | 0.0102088 | -0.0712619 | 0.0916796 |
| 2018-09-28 | 1.10637 | 1.06049 | 1.15226 | -0.452825 | -0.544201 | -0.361449 | 0.00731301 | -0.0704495 | 0.0850755 |
| 2018-10-31 | 1.10382 | 1.05654 | 1.1511 | -0.43474 | -0.519507 | -0.349974 | 0.0307729 | -0.0470527 | 0.108599 |
| 2018-11-30 | 1.10183 | 1.06033 | 1.14334 | -0.368013 | -0.444587 | -0.291439 | 0.0638304 | -0.0119336 | 0.139594 |
| 2018-12-31 | 1.08664 | 1.05212 | 1.12117 | -0.34481 | -0.431694 | -0.257925 | 0.0556197 | -0.0254999 | 0.136739 |
| 2019-01-31 | 1.07736 | 1.04526 | 1.10946 | -0.309277 | -0.392767 | -0.225786 | 0.0586241 | -0.0129508 | 0.130199 |
| 2019-02-28 | 1.0821 | 1.04147 | 1.12273 | -0.300722 | -0.385408 | -0.216037 | 0.0655325 | -0.00610674 | 0.137172 |
| 2019-03-29 | 1.07224 | 1.02801 | 1.11647 | -0.292343 | -0.372653 | -0.212033 | 0.0535051 | -0.0143262 | 0.121336 |
| 2019-04-30 | 1.04498 | 1.00112 | 1.08884 | -0.246064 | -0.329761 | -0.162366 | 0.0346771 | -0.0348112 | 0.104165 |
| 2019-05-31 | 1.04518 | 1.0012 | 1.08916 | -0.234637 | -0.31826 | -0.151014 | 0.0274846 | -0.0423641 | 0.0973333 |
| 2019-06-28 | 1.02583 | 0.983552 | 1.0681 | -0.219332 | -0.300154 | -0.138509 | -0.0236288 | -0.0926943 | 0.0454367 |
| 2019-07-31 | 1.03599 | 0.992554 | 1.07943 | -0.225637 | -0.316259 | -0.135014 | -0.028592 | -0.0981336 | 0.0409497 |
| 2019-08-30 | 1.04585 | 1.00143 | 1.09027 | -0.252291 | -0.347878 | -0.156704 | -0.0629285 | -0.131418 | 0.00556116 |
| 2019-09-30 | 1.05818 | 1.01442 | 1.10195 | -0.249795 | -0.348687 | -0.150902 | -0.0864441 | -0.15247 | -0.0204179 |
| 2019-10-31 | 1.04364 | 1.00061 | 1.08667 | -0.27134 | -0.366945 | -0.175735 | -0.113345 | -0.174555 | -0.0521339 |
| 2019-11-29 | 1.03353 | 0.991895 | 1.07517 | -0.306962 | -0.403429 | -0.210496 | -0.128146 | -0.190546 | -0.0657456 |
| 2019-12-31 | 1.06075 | 1.00903 | 1.11247 | -0.28401 | -0.378862 | -0.189157 | -0.114432 | -0.180049 | -0.0488155 |
| 2020-01-31 | 1.09284 | 1.04156 | 1.14412 | -0.248375 | -0.343634 | -0.153117 | -0.0796321 | -0.144532 | -0.014732 |
| 2020-02-28 | 1.09093 | 1.04745 | 1.1344 | -0.301161 | -0.404396 | -0.197926 | -0.0987412 | -0.168026 | -0.0294567 |
| 2020-03-31 | 1.04631 | 1.01253 | 1.08008 | -0.458587 | -0.598099 | -0.319075 | -0.141841 | -0.186724 | -0.0969583 |
| 2020-04-30 | 1.03967 | 1.0056 | 1.07375 | -0.456399 | -0.58055 | -0.332249 | -0.107165 | -0.171907 | -0.0424221 |
| 2020-05-29 | 1.03339 | 0.99935 | 1.06742 | -0.418577 | -0.535958 | -0.301197 | -0.0936012 | -0.151274 | -0.0359282 |
| 2020-06-30 | 1.03161 | 0.998964 | 1.06425 | -0.407798 | -0.537439 | -0.278156 | -0.0589709 | -0.115394 | -0.00254791 |
| 2020-07-31 | 1.03163 | 0.998774 | 1.06449 | -0.392481 | -0.518375 | -0.266587 | -0.0456965 | -0.0987475 | 0.0073544 |
| 2020-08-31 | 1.03124 | 0.99751 | 1.06497 | -0.385616 | -0.512907 | -0.258326 | -0.0321885 | -0.0857079 | 0.021331 |
| 2020-09-30 | 1.04125 | 1.00624 | 1.07626 | -0.391554 | -0.522931 | -0.260177 | -0.019145 | -0.0776947 | 0.0394048 |
| 2020-10-30 | 1.04732 | 1.01158 | 1.08307 | -0.402577 | -0.536502 | -0.268653 | -0.0164551 | -0.0765556 | 0.0436454 |
| 2020-11-30 | 1.04765 | 1.0121 | 1.08321 | -0.391716 | -0.527639 | -0.255793 | -0.00633038 | -0.0624874 | 0.0498267 |
| 2020-12-31 | 1.04948 | 1.0138 | 1.08516 | -0.400935 | -0.535025 | -0.266844 | -0.0128069 | -0.0696912 | 0.0440773 |
| 2021-01-29 | 1.05154 | 1.01543 | 1.08765 | -0.424611 | -0.560897 | -0.288324 | -0.00865566 | -0.0676942 | 0.0503829 |
| 2021-02-26 | 1.05066 | 1.01303 | 1.08829 | -0.413148 | -0.548497 | -0.277799 | 0.0029656 | -0.0578825 | 0.0638137 |
| 2021-03-31 | 1.13675 | 1.07852 | 1.19498 | -0.307101 | -0.386926 | -0.227277 | 0.058528 | 0.0158935 | 0.101162 |
| 2021-04-30 | 1.1895 | 1.14689 | 1.2321 | -0.278328 | -0.346564 | -0.210092 | 0.0713819 | 0.0395991 | 0.103165 |
| 2021-05-28 | 1.18564 | 1.14053 | 1.23076 | -0.314764 | -0.393537 | -0.23599 | 0.0531057 | 0.0232791 | 0.0829323 |
| 2021-06-30 | 1.19624 | 1.14121 | 1.25128 | -0.328669 | -0.411191 | -0.246148 | 0.0405039 | 0.0155122 | 0.0654957 |
| 2021-07-30 | 1.17627 | 1.12105 | 1.2315 | -0.358211 | -0.432713 | -0.28371 | 0.0337458 | 0.00841678 | 0.0590748 |
| 2021-08-31 | 1.16793 | 1.11245 | 1.22342 | -0.364214 | -0.437008 | -0.29142 | 0.0327369 | 0.00774232 | 0.0577315 |
| 2021-09-30 | 1.13734 | 1.08484 | 1.18984 | -0.371664 | -0.44137 | -0.301959 | 0.0339202 | 0.00740771 | 0.0604327 |
| 2021-10-29 | 1.11582 | 1.05905 | 1.17259 | -0.372437 | -0.442786 | -0.302088 | 0.0346817 | 0.0070575 | 0.0623058 |
| 2021-11-30 | 1.06981 | 1.01551 | 1.12411 | -0.399565 | -0.460004 | -0.339127 | 0.0964211 | 0.0547874 | 0.138055 |
| 2021-12-31 | 1.05143 | 1.00404 | 1.09882 | -0.392238 | -0.450548 | -0.333928 | 0.125403 | 0.0874671 | 0.163338 |
| 2022-01-31 | 1.07577 | 1.0274 | 1.12414 | -0.331362 | -0.378219 | -0.284506 | 0.120152 | 0.0795949 | 0.160709 |
| 2022-02-28 | 1.08661 | 1.04106 | 1.13216 | -0.342628 | -0.397405 | -0.287851 | 0.0948675 | 0.0519421 | 0.137793 |
| 2022-03-31 | 1.09634 | 1.05704 | 1.13564 | -0.323833 | -0.37756 | -0.270106 | 0.0636217 | 0.0208169 | 0.106426 |
| 2022-04-29 | 1.11014 | 1.07069 | 1.14959 | -0.318951 | -0.377916 | -0.259986 | 0.0409495 | -0.00607734 | 0.0879763 |
| 2022-05-31 | 1.11357 | 1.0804 | 1.14674 | -0.306421 | -0.367035 | -0.245808 | 0.0259529 | -0.0233941 | 0.0753 |
| 2022-06-30 | 1.12091 | 1.09171 | 1.1501 | -0.289441 | -0.350766 | -0.228116 | 0.0248096 | -0.0207144 | 0.0703336 |
| 2022-07-29 | 1.11847 | 1.08726 | 1.14967 | -0.283235 | -0.342597 | -0.223873 | 0.0130849 | -0.0317274 | 0.0578971 |
| 2022-08-31 | 1.11789 | 1.08758 | 1.14819 | -0.282147 | -0.344012 | -0.220283 | 0.0107129 | -0.0332949 | 0.0547206 |
| 2022-09-30 | 1.12151 | 1.09242 | 1.15061 | -0.301661 | -0.364773 | -0.238548 | 0.0181648 | -0.0281991 | 0.0645288 |
| 2022-10-31 | 1.11902 | 1.09006 | 1.14799 | -0.311994 | -0.376876 | -0.247111 | 0.0179194 | -0.0291963 | 0.065035 |
| 2022-11-30 | 1.12126 | 1.08993 | 1.1526 | -0.302127 | -0.366966 | -0.237288 | -0.0118193 | -0.0594665 | 0.0358278 |
| 2022-12-30 | 1.11051 | 1.07913 | 1.14189 | -0.290284 | -0.358038 | -0.22253 | -0.0508557 | -0.101811 | 0.00010016 |
| 2023-01-31 | 1.10433 | 1.07346 | 1.1352 | -0.310163 | -0.390169 | -0.230157 | -0.0645213 | -0.12232 | -0.00672235 |
| 2023-02-28 | 1.1025 | 1.07023 | 1.13477 | -0.289609 | -0.380968 | -0.19825 | -0.0928476 | -0.164553 | -0.021142 |
| 2023-03-31 | 1.09777 | 1.06615 | 1.12939 | -0.3121 | -0.400274 | -0.223925 | -0.0911897 | -0.159487 | -0.0228928 |
| 2023-04-28 | 1.0969 | 1.06352 | 1.13027 | -0.300046 | -0.388721 | -0.211371 | -0.0935594 | -0.160766 | -0.0263531 |
| 2023-05-31 | 1.07582 | 1.02793 | 1.12371 | -0.266587 | -0.353489 | -0.179686 | -0.0899443 | -0.159315 | -0.0205736 |
| 2023-06-30 | 1.02379 | 0.990413 | 1.05717 | -0.321598 | -0.383917 | -0.259279 | -0.0988675 | -0.148735 | -0.0489998 |
| 2023-07-31 | 1.03998 | 1.00203 | 1.07794 | -0.295769 | -0.364916 | -0.226622 | -0.0582612 | -0.11551 | -0.00101245 |
| 2023-08-31 | 1.06106 | 1.01707 | 1.10505 | -0.27949 | -0.346482 | -0.212498 | -0.0329903 | -0.0912435 | 0.025263 |
| 2023-09-29 | 1.06481 | 1.01698 | 1.11264 | -0.274877 | -0.345471 | -0.204283 | -0.0341547 | -0.0912686 | 0.0229593 |
| 2023-10-31 | 1.08451 | 1.03836 | 1.13066 | -0.271964 | -0.337385 | -0.206543 | -0.00633444 | -0.0602473 | 0.0475784 |
| 2023-11-30 | 1.10109 | 1.05944 | 1.14274 | -0.262229 | -0.330513 | -0.193945 | -0.014508 | -0.0695678 | 0.0405517 |
| 2023-12-29 | 1.12239 | 1.07919 | 1.1656 | -0.259675 | -0.332387 | -0.186963 | -0.0128974 | -0.0687584 | 0.0429635 |
| 2024-01-31 | 1.13007 | 1.08444 | 1.1757 | -0.263979 | -0.343498 | -0.184461 | -0.0039334 | -0.0630927 | 0.0552259 |
| 2024-02-29 | 1.11336 | 1.06344 | 1.16327 | -0.241794 | -0.329251 | -0.154336 | 0.0126195 | -0.043894 | 0.069133 |
| 2024-03-28 | 1.13663 | 1.08827 | 1.18499 | -0.160566 | -0.236994 | -0.0841376 | 0.0428995 | -0.00666269 | 0.0924616 |
| 2024-04-30 | 1.13461 | 1.08557 | 1.18365 | -0.158209 | -0.233101 | -0.0833174 | 0.042219 | -0.00923452 | 0.0936725 |
| 2024-05-31 | 1.12493 | 1.07517 | 1.1747 | -0.1463 | -0.219255 | -0.0733443 | 0.0291949 | -0.0313744 | 0.0897642 |
| 2024-06-28 | 1.14189 | 1.08524 | 1.19854 | -0.141285 | -0.215073 | -0.0674974 | 0.0329004 | -0.0294917 | 0.0952925 |
| 2024-07-31 | 1.22717 | 1.16521 | 1.28914 | -0.23774 | -0.316866 | -0.158614 | 0.0810147 | 0.0148239 | 0.147205 |
| 2024-08-30 | 1.21941 | 1.1636 | 1.27521 | -0.255099 | -0.327729 | -0.182469 | 0.0877779 | 0.0230842 | 0.152472 |
| 2024-09-30 | 1.22959 | 1.17382 | 1.28536 | -0.260679 | -0.330455 | -0.190902 | 0.0905871 | 0.0278212 | 0.153353 |
| 2024-10-31 | 1.2492 | 1.19305 | 1.30535 | -0.273564 | -0.346689 | -0.200438 | 0.0613323 | 0.000506838 | 0.122158 |
| 2024-11-29 | 1.24759 | 1.18898 | 1.3062 | -0.261909 | -0.327908 | -0.19591 | 0.0654329 | 0.0069105 | 0.123955 |
| 2024-12-31 | 1.23448 | 1.17688 | 1.29208 | -0.291381 | -0.36217 | -0.220592 | 0.0569003 | -0.00159957 | 0.1154 |
| 2025-01-31 | 1.23159 | 1.17478 | 1.28841 | -0.288139 | -0.35951 | -0.216769 | 0.0550124 | 0.00191451 | 0.10811 |
| 2025-02-28 | 1.24187 | 1.18722 | 1.29651 | -0.301315 | -0.372266 | -0.230365 | 0.0438658 | -0.00692054 | 0.0946522 |
| 2025-03-31 | 1.19711 | 1.13516 | 1.25906 | -0.322592 | -0.401056 | -0.244129 | 0.0591612 | 0.00622199 | 0.1121 |
| 2025-04-30 | 1.14849 | 1.10636 | 1.19062 | -0.331322 | -0.410875 | -0.251768 | 0.0852688 | 0.0414031 | 0.129134 |
| 2025-05-30 | 1.14491 | 1.10326 | 1.18657 | -0.330715 | -0.411542 | -0.249889 | 0.0845186 | 0.041491 | 0.127546 |
| 2025-06-30 | 1.14219 | 1.09941 | 1.18497 | -0.314242 | -0.397619 | -0.230866 | 0.0662861 | 0.0227936 | 0.109779 |
| 2025-07-31 | 1.13471 | 1.09408 | 1.17535 | -0.300634 | -0.378363 | -0.222905 | 0.044945 | 0.000903411 | 0.0889867 |
| 2025-08-29 | 1.13659 | 1.09713 | 1.17605 | -0.296329 | -0.373822 | -0.218835 | 0.0457883 | 0.00443679 | 0.0871398 |
| 2025-09-30 | 1.12773 | 1.08994 | 1.16552 | -0.279356 | -0.357166 | -0.201546 | 0.0430977 | 0.000704528 | 0.0854908 |
| 2025-10-31 | 1.13383 | 1.09607 | 1.17158 | -0.266025 | -0.33718 | -0.194871 | 0.0480251 | 0.00943207 | 0.0866181 |
| 2025-11-28 | 1.12465 | 1.08872 | 1.16058 | -0.286419 | -0.356989 | -0.215849 | 0.0475805 | 0.00913988 | 0.0860211 |
| 2025-12-31 | 1.12664 | 1.08714 | 1.16614 | -0.265717 | -0.340534 | -0.1909 | 0.0493058 | 0.01061 | 0.0880017 |
| 2026-01-30 | 1.12386 | 1.08513 | 1.16258 | -0.253772 | -0.337594 | -0.169951 | 0.0407315 | 0.0021244 | 0.0793385 |
| 2026-02-27 | 1.13254 | 1.09201 | 1.17307 | -0.236664 | -0.326443 | -0.146884 | 0.0399499 | 0.00156402 | 0.0783357 |
| 2026-03-31 | 1.13571 | 1.0918 | 1.17962 | -0.172204 | -0.25802 | -0.0863874 | 0.0189084 | -0.00955843 | 0.0473752 |
| 2026-04-30 | 1.16255 | 1.11541 | 1.20969 | -0.0935731 | -0.164716 | -0.0224303 | 0.0194545 | -0.010896 | 0.0498051 |
| 2026-05-29 | 1.15858 | 1.10804 | 1.20912 | -0.046941 | -0.121659 | 0.0277765 | 0.0323238 | -0.00273804 | 0.0673857 |
In-sample versus holdout R² · chronological 50/50 split
R² (%) · coefficients frozen on the first half · notebook 05
| ETF | training R² | holdout R² |
|---|---|---|
| QQQ | 95.0542 | 92.933 |
| USMV | 89.5854 | 59.9519 |
| MTUM | 95.1999 | 86.3837 |
| VLUE | 94.5405 | 82.4719 |
| QUAL | 97.505 | 95.3112 |
| IWM | 98.453 | 98.0048 |
Factor exposures · OLS coefficients with HAC(5) intervals and plain-OLS t
| ETF | Factor | β | CI low | CI high | HAC t | OLS t |
|---|---|---|---|---|---|---|
| QQQ | Mkt-RF | 1.10561 | 1.07625 | 1.13497 | 73.8038 | 205.689 |
| QQQ | SMB | -0.121876 | -0.14973 | -0.0940219 | -8.5759 | -12.2298 |
| QQQ | HML | -0.306236 | -0.342893 | -0.269579 | -16.3735 | -33.7056 |
| QQQ | RMW | 0.0505008 | 0.0167816 | 0.08422 | 2.9354 | 4.07505 |
| QQQ | CMA | -0.185612 | -0.237528 | -0.133695 | -7.0073 | -12.2598 |
| QQQ | Mom | 0.0414621 | 0.0251993 | 0.0577248 | 4.997 | 7.18718 |
| USMV | Mkt-RF | 0.742938 | 0.698619 | 0.787258 | 32.8556 | 109.461 |
| USMV | SMB | -0.124721 | -0.168068 | -0.0813742 | -5.6394 | -9.9114 |
| USMV | HML | 0.0101894 | -0.0348243 | 0.0552031 | 0.4437 | 0.888155 |
| USMV | RMW | 0.163288 | 0.122347 | 0.204229 | 7.8171 | 10.4348 |
| USMV | CMA | 0.177752 | 0.0981771 | 0.257328 | 4.3781 | 9.29796 |
| USMV | Mom | -0.0189788 | -0.0431255 | 0.00516793 | -1.5405 | -2.60537 |
| MTUM | Mkt-RF | 1.04062 | 1.01464 | 1.06659 | 78.5147 | 161.416 |
| MTUM | SMB | -0.096254 | -0.136873 | -0.0556354 | -4.6445 | -8.05315 |
| MTUM | HML | -0.0202303 | -0.0585054 | 0.0180449 | -1.0359 | -1.85649 |
| MTUM | RMW | -0.156801 | -0.200634 | -0.112967 | -7.0111 | -10.5494 |
| MTUM | CMA | -0.0148822 | -0.0787626 | 0.0489983 | -0.4566 | -0.819576 |
| MTUM | Mom | 0.339623 | 0.308575 | 0.370672 | 21.4388 | 49.0852 |
| VLUE | Mkt-RF | 0.984251 | 0.964179 | 1.00432 | 96.1098 | 153.021 |
| VLUE | SMB | 0.107931 | 0.0642319 | 0.151629 | 4.8409 | 9.05063 |
| VLUE | HML | 0.310955 | 0.271237 | 0.350673 | 15.3447 | 28.6007 |
| VLUE | RMW | 0.00422313 | -0.0446202 | 0.0530664 | 0.1695 | 0.284774 |
| VLUE | CMA | 0.118833 | 0.0634586 | 0.174207 | 4.2061 | 6.55914 |
| VLUE | Mom | -0.056308 | -0.0812792 | -0.0313368 | -4.4196 | -8.15663 |
| QUAL | Mkt-RF | 0.976277 | 0.95761 | 0.994944 | 102.503 | 274.9 |
| QUAL | SMB | -0.0837018 | -0.10613 | -0.0612736 | -7.3146 | -12.7124 |
| QUAL | HML | -0.0435282 | -0.0631468 | -0.0239096 | -4.3486 | -7.25115 |
| QUAL | RMW | 0.148468 | 0.129138 | 0.167798 | 15.0541 | 18.1325 |
| QUAL | CMA | 0.0250655 | -0.00747678 | 0.0576078 | 1.5097 | 2.50579 |
| QUAL | Mom | -0.0412127 | -0.0575118 | -0.0249137 | -4.9558 | -10.8126 |
| IWM | Mkt-RF | 1.00352 | 0.988921 | 1.01812 | 134.728 | 318.814 |
| IWM | SMB | 0.848244 | 0.810275 | 0.886213 | 43.7865 | 145.352 |
| IWM | HML | 0.100439 | 0.078164 | 0.122714 | 8.8375 | 18.8777 |
| IWM | RMW | -0.110021 | -0.131989 | -0.0880523 | -9.8157 | -15.1604 |
| IWM | CMA | -0.0176326 | -0.0478461 | 0.012581 | -1.1438 | -1.98882 |
| IWM | Mom | 0.0246866 | 0.0059725 | 0.0434007 | 2.5855 | 7.30751 |
ETF alpha tests with family-wise and false-discovery corrections
| ETF | Alpha (annualised) | Alpha HAC SE (annualised) | Raw p | Holm adjusted p | BH q | Survives BH 5% | R² | n |
|---|---|---|---|---|---|---|---|---|
| QQQ | 0.0329498 | 0.0146449 | 0.0244544 | 0.146726 | 0.146726 | no | 0.949831 | 2,867 |
| USMV | -0.0123103 | 0.0190868 | 0.51895 | 1 | 0.778425 | no | 0.813279 | 2,867 |
| MTUM | 0.00020619 | 0.016715 | 0.990158 | 1 | 0.990158 | no | 0.917733 | 2,867 |
| VLUE | 0.00487565 | 0.016962 | 0.773771 | 1 | 0.928525 | no | 0.909099 | 2,867 |
| QUAL | -0.00739199 | 0.00912076 | 0.417677 | 1 | 0.778425 | no | 0.966977 | 2,867 |
| IWM | -0.0113975 | 0.00713155 | 0.110003 | 0.550015 | 0.330009 | no | 0.983704 | 2,867 |
Chronological holdout results; coefficients fitted only on the first half
| ETF | Training R² | Holdout R² | Change | Training window | Holdout window | Holdout RMSE (daily) |
|---|---|---|---|---|---|---|
| QQQ | 0.950542 | 0.92933 | -0.0212128 | 2015-01-05 – 2020-09-11 | 2020-09-14 – 2026-05-29 | 0.00374699 |
| USMV | 0.895854 | 0.599519 | -0.296335 | 2015-01-05 – 2020-09-11 | 2020-09-14 – 2026-05-29 | 0.00492912 |
| MTUM | 0.951999 | 0.863837 | -0.0881614 | 2015-01-05 – 2020-09-11 | 2020-09-14 – 2026-05-29 | 0.0048687 |
| VLUE | 0.945405 | 0.824719 | -0.120686 | 2015-01-05 – 2020-09-11 | 2020-09-14 – 2026-05-29 | 0.00470729 |
| QUAL | 0.97505 | 0.953112 | -0.0219379 | 2015-01-05 – 2020-09-11 | 2020-09-14 – 2026-05-29 | 0.00233515 |
| IWM | 0.98453 | 0.980048 | -0.00448234 | 2015-01-05 – 2020-09-11 | 2020-09-14 – 2026-05-29 | 0.00202335 |
Daily factor correlations, 2015–2026
| Factor | Mkt-RF | SMB | HML | RMW | CMA | Mom |
|---|---|---|---|---|---|---|
| Mkt-RF | 1 | 0.192215 | -0.0922692 | -0.223637 | -0.262154 | -0.110321 |
| SMB | 0.192215 | 1 | 0.273967 | -0.270396 | 0.0440945 | -0.30201 |
| HML | -0.0922692 | 0.273967 | 1 | 0.321309 | 0.559635 | -0.312252 |
| RMW | -0.223637 | -0.270396 | 0.321309 | 1 | 0.27769 | -0.0589468 |
| CMA | -0.262154 | 0.0440945 | 0.559635 | 0.27769 | 1 | -0.014175 |
| Mom | -0.110321 | -0.30201 | -0.312252 | -0.0589468 | -0.014175 | 1 |
Rolling QQQ beta ranges across windows
| Factor | Min | Max |
|---|---|---|
| Mkt-RF | 1.02379 | 1.2492 |
| HML | -0.496972 | -0.046941 |
| Mom | -0.141841 | 0.260517 |
Audience and decision
Research significance
ETF allocators, product researchers, and exposure owners
The result matters when a branded strategy is selected for a specific systematic exposure, or when a portfolio report attributes historical return to manager skill rather than priced factors.
Decision context
Evidentiary threshold for attributing alpha
First verify the intended exposure, then test residual return with robust uncertainty, multiplicity correction, and a chronological holdout. A raw p-value from one of several regressions is not the conclusion.
The regression found product identity more clearly than unexplained return
The study aligned 2,867 daily observations from 5 January 2015 through 29 May 2026. It ran six-factor time-series regressions for MTUM, VLUE, QUAL, IWM, USMV, and QQQ. The factors were the Fama–French five plus momentum, estimated with Newey–West HAC(5) intervals.
The named style loadings came out economically meaningful and precisely estimated. The alpha tests went the other way. Holm and Benjamini–Hochberg corrections ran across six simultaneous hypotheses, and none met a 5% false-discovery threshold. That does not mean the funds delivered nothing. It means the measured return read mainly as exposure under this factor model.
A chronological 50/50 holdout froze the first-half coefficients and applied them to the second half. Holdout R² fell by 0.4 to 29.6 percentage points across the ETFs. USMV decayed most. Exposure estimates can stay useful and still depend on the state.
Separate exposure, inference, and temporal stability
- Check the intended sign and an economically meaningful loading on a branded ETF's named factor.
- Correct alpha inference for the number of products or strategies searched.
- Use heteroskedasticity- and autocorrelation-consistent intervals for daily return regressions.
- Freeze coefficients into a chronological holdout instead of reporting full-sample fit alone.
- Monitor rolling betas. A full-sample coefficient is a time average, not a permanent product identity.
Inspect every loading, interval, holdout result, and adjusted test
Numbers
| Statement | Value | As stated | Note |
|---|---|---|---|
| MTUM momentum β | 0.34 | 0.34 | |
| MTUM momentum t-statistic (OLS) | 49 | 49 | Plain-OLS t as printed in notebook 05; HAC(5) t = 21.4. |
| VLUE value (HML) β | 0.31 | 0.31 | |
| VLUE value t-statistic (OLS) | 29 | 29 | Plain-OLS t as printed in notebook 05; HAC(5) t = 15.3. |
| QQQ annualised alpha | +3.29% | +3.29% | |
| QQQ raw HAC p-value | 0.024 | 0.024 | |
| QQQ Benjamini–Hochberg q-value | 0.147 | 0.147 | |
| alphas surviving the 5% false-discovery threshold | 0 | 0 | |
| aligned daily observations | 2,867 | 2,867 | |
| First aligned observation | 2015-01-05 | 5 January 2015 | |
| Last aligned observation | 2026-05-29 | 29 May 2026 | |
| Smallest holdout R² decay (percentage points) | 0.4 | 0.4 | |
| Largest holdout R² decay (percentage points) | 29.6 | 29.6 | |
| ETF with the largest holdout decay | USMV | USMV | |
| IWM size (SMB) β | 0.85 | 0.85 | |
| IWM R² | 0.98 | 0.98 | |
| USMV market β | 0.74 | 0.74 | |
| QQQ market β | 1.11 | 1.11 | |
| QQQ value (HML) β | −0.31 | -0.31 | |
| QUAL profitability (RMW) β | 0.15 | 0.15 |
Limitations
- The chosen factor menu defines the leftover alpha. Omitted technology, industry, or intangible-capital effects can matter.
- Fama–French factor portfolios are paper long–short portfolios, not costless tradable products.
- Eleven years remains one historical sample, including an unusual growth-over-value period.
- The chronological 50/50 split is one holdout path rather than repeated rolling-origin evaluation.
- HAC lag choice and vendor adjustments to ETF prices can move inference.
Sources
- 1Fama, E. F., and K. R. French (2015), “A Five-Factor Asset Pricing Model,” Journal of Financial Economics 116(1), 1–22. doi:10.1016/j.jfineco.2014.10.010
- 2Benjamini, Y., and Y. Hochberg (1995), “Controlling the False Discovery Rate,” Journal of the Royal Statistical Society: Series B 57(1), 289–300. doi:10.1111/j.2517-6161.1995.tb02031.x
- 3Harvey, C. R., Y. Liu, and H. Zhu (2016), “...and the Cross-Section of Expected Returns,” Review of Financial Studies 29(1), 5–68. doi:10.1093/rfs/hhv059
Cite this
Wisniewski, K. (2026, August 8). ETF factor exposures, alpha, and multiple-testing control. Quantitative Markets & Institutions Lab. https://www.kylewisniewski.com/lab/factor-etfs
@misc{wisniewski2026factor,
author = {Wisniewski, Kyle},
title = {ETF factor exposures, alpha, and multiple-testing control},
year = {2026},
month = {aug},
howpublished = {\url{https://www.kylewisniewski.com/lab/factor-etfs}},
note = {Empirical finding · Quantitative Markets & Institutions Lab · commit 4806df9}
}