Table 1: Complete _Kepler_ O’Connell Effect Target List  Kepler | R.A. (J2000) | Dec. (J2000) | Period | LC | Morph. | BP – RP | Gaia | Gaia | Dist. | Lum. | Spectral Type | Ref.ff1 – Balona (2015); 2 – Gao et al. (2016); 3 – Davenport (2016); 4 – Tran et al. (2013); 5 – Balaji et al. (2015); 6 – Kouzuma (2018); 7 – Borkovits et al. (2016); 8 – Furlan et al. (2017); 9 – Smith et al. (2015); 10 – Pizzocaro et al. (2019); 11 – Gaulme et al. (2013); 12 – Liakos & Niarchos (2017); 13 – Kjurkchieva et al. (2017); 14 – Matson et al. (2017); 15 – Kobulnicky et al. (2022); 16 – Hartman et al. (2004); 17 – Hoffman et al. (2008) or Hoffman et al. (2009); 18 – Devor et al. (2008); 19 – Pigulski et al. (2009); 20 – Ramsay et al. (2014); 21 – Clark Cunningham et al. (2019); 22 – Debski et al. (2014); 23 – Yoldaş (2021); 24 – NegmEldin et al. (2019); 25 – Blättler & Diethelm (2000a) or Blättler & Diethelm (2000b); 26 – Kunt & Dal (2017); 27 – Molnar et al. (2017), Socia et al. (2018), Pavlenko et al. (2018), and Kovács et al. (2019); 28 – Almenara et al. (2012) and Almeida et al. (2019); 29 – Liakos (2017)gg* – paper discusses this system; † – signal produced by this system is included in the paper’s sample but is conflated with another source; ‡ – system is mentioned in the paper but is not included in the paper’s samplehhReferences 1-3 indicate flares; references 4-5 indicate spots; reference 6 indicates an ETV consistent with mass transfer | Samp.iiC – Core Sample; M – Marginal Sample | NotesjjTV – Temporal Variations; AM – Asymmetric Minima; WD – White Dwarf; CU – Concave-Up; SC indicates the system shows the previous three features only with its short-cadence data; F – literature indicates system shows flares; S – literature indicates system shows spots; M – literature indicates system shows mass transfer
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IDaaLink leads to system’s KEBC page | hh:mm:ss.s | +dd:mm:ss.ss | (days) | ClassbbAl – Algol; BL – $\beta$ Lyrae; WU – W Ursae Majoris | Param. | Color | BP Mag. | G Mag. | (pc) | (L⊙)ccIn _Gaia_ G band, assuming a G band absolute magnitude of $4.68\pm 0.02$ for the Sun (Čotar et al., 2019) | Lit.dd1 – Frasca et al. (2016); 2 – Ramsay et al. (2014); 3 – Almenara et al. (2012) | ColoreeCalculated using Dr. Mamajek’s table; a luminosity class of V is implied for each system |  |  |
2159783 | 19:25:26.2 | +37:33:11.90 | 0.3739 | WU | 0.87 | 0.938 | 15.365 | 14.982 | 1546 | 1.810 | F6 V1 | G9 | 4, 5, 6, 22* | C | AM, S, M
2161623 | 19:26:58.7 | +37:35:27.60 | 2.2835 | Al | 0.54 | 0.771 | 14.571 | 14.273 | 1901 | 5.262 |  $\cdots$ | F9.5 |  $\cdots$ | C |  $\cdots$
2302092 | 19:25:02.9 | +37:41:52.49 | 0.2947 | WU | 0.88 | 1.124 | 14.982 | 14.503 | 811 | 0.775 |  $\cdots$ | K2 | 5, 7 | C | S
2305372 | 19:27:57.7 | +37:40:21.83 | 1.4047 | Al | 0.58 | 0.929 | 14.214 | 13.846 | 1273 | 3.498 | F8 Vs1 | G9 | 2, 7, 13, 14*, 16, 19 | C | F
2437038 | 19:20:47.6 | +37:44:31.83 | 0.2677 | WU | 0.87 | 1.120 | 16.501 | 16.149 | 2784 | 2.004 |  $\cdots$ | K2 | 3, 5, 6 | C | F, S, M
2449084 | 19:30:46.9 | +37:46:19.13 | 0.7398 | BL | 0.70 | 0.842 | 15.319 | 14.983 | 2798 | 5.926 |  $\cdots$ | G4 | 3 | C | AM, F
2569494 | 19:20:42.9 | +37:50:56.26 | 1.5233 | Al | 0.58 | 1.209 | 17.799 | 17.288 | 4760 | 2.051 |  $\cdots$ | K3 | 8, 13 | C | TV, AM
2577756 | 19:27:26.9 | +37:49:47.77 | 0.8703 | Al | 0.63 | 0.911 | 15.860 | 15.491 | 2185 | 2.262 |  $\cdots$ | G8 | 2 | M | TV, AM, F
2694741 | 19:05:39.0 | +37:54:29.65 | 0.3266 | WU | 0.93 | 0.936 | 12.639 | 12.256 | 373 | 1.304 | G4 V1 | G9 | 3, 5 | C | F, S
2695740 | 19:06:55.4 | +37:56:39.53 | 3.6159 | WU | 0.85 | 1.305 | 13.704 | 13.126 | 1453 | 8.844 |  $\cdots$ | K4 | 3, 5, 19 | C | TV, AM, F, S
2854752 | 19:27:44.6 | +38:00:22.65 | 0.4704 | BL | 0.79 | 0.636 | 14.153 | 13.918 | 1594 | 5.132 |  $\cdots$ | F6 | 3, 6 | C | F, M
2858322 | 19:30:41.9 | +38:03:53.38 | 0.4364 | BL | 0.77 | 0.984 | 15.197 | 14.789 | 2133 | 4.118 |  $\cdots$ | K0 | 5, 6 | C | S, M
2972514 | 19:06:44.6 | +38:10:12.89 | 0.5035 | WU | 0.74 | 0.802 | 14.332 | 14.020 | 1361 | 3.405 |  $\cdots$ | G1 |  $\cdots$ | C |  $\cdots$
3124420 | 19:32:58.9 | +38:16:18.68 | 0.9490 | Al | 0.60 | 1.174 | 18.801 | 18.287 | 5120 | 0.946 |  $\cdots$ | K2.5 |  $\cdots$ | C | TV, AM
3218683 | 19:05:36.1 | +38:22:34.22 | 0.7717 | Al | 0.69 | 1.156 | 16.524 | 16.030 | 2046 | 1.208 |  $\cdots$ | K2.5 | 2, 5 | C | F, S
3338660 | 19:21:53.8 | +38:25:38.62 | 1.8734 | Al | 0.60 | 0.944 | 15.087 | 14.707 | 3515 | 12.053 |  $\cdots$ | G9 | 3, 7, 8 | C | F
3339563 | 19:22:59.0 | +38:26:35.33 | 0.8412 | BL | 0.72 | 1.030 | 14.624 | 14.193 | 1195 | 2.240 | G0 IIIp1 | K1 | 2, 3, 5 | C | TV, AM, F, S
3342425 | 19:26:03.9 | +38:27:16.06 | 0.3934 | WU | 0.94 | 0.898 | 15.489 | 15.137 | 1896 | 2.362 |  $\cdots$ | G8 | 3, 5 | C | TV, AM, F, S
3344427 | 19:28:00.6 | +38:26:06.69 | 0.6518 | Al | 0.57 | 1.448 | 16.423 | 15.776 | 866 | 0.274 |  $\cdots$ | K5 |  $\cdots$ | M |  $\cdots$
3351945 | 19:35:32.3 | +38:26:44.56 | 1.0805 | Al | 0.50 | 1.288 | 15.359 | 14.794 | 483 | 0.210 |  $\cdots$ | K3.5 |  $\cdots$ | M | TV
3557421 | 19:38:32.4 | +38:37:36.86 | 0.3937 | Al | 0.70 | 1.124 | 10.486 | 10.003 | 94 | 0.662 |  $\cdots$ | K2 | 5, 16, 19 | M | S
3656322 | 19:36:38.5 | +38:46:02.73 | 3.6637 | Al | 0.59 | 1.125 | 13.538 | 13.077 | 1036 | 4.706 |  $\cdots$ | K2 | 13 | M | TV
3659940 | 19:40:01.1 | +38:46:31.65 | 0.8963 | Al | 0.55 | 1.047 | 15.619 | 15.187 | 1197 | 0.899 |  $\cdots$ | K1.5 |  $\cdots$ | C | TV
3662635 | 19:42:16.6 | +38:43:33.81 | 0.9394 | Al | 0.57 | 0.988 | 15.273 | 14.965 | 6474 | 32.241 |  $\cdots$ | K0 | 3 | C | TV, F
3743834 | 19:22:11.1 | +38:52:21.35 | 0.2736 | WU | 0.76 | 1.218 | 14.775 | 14.241 | 517 | 0.402 |  $\cdots$ | K3 | 3, 5 | C | F, S
3832382 | 19:00:59.1 | +38:58:59.86 | 0.2726 | WU | 0.75 | 1.426 | 16.447 | 15.802 | 918 | 0.300 |  $\cdots$ | K5 | 5 | C | S
3839964 | 19:13:10.5 | +38:58:41.03 | 0.2561 | WU | 0.80 | 1.026 | 14.988 | 14.557 | 1065 | 1.272 |  $\cdots$ | K1 | 3, 7 | M | F
3848919 | 19:24:13.5 | +38:58:27.83 | 1.0473 | Al | 0.57 | 1.059 | 14.409 | 13.984 | 817 | 1.269 |  $\cdots$ | K1.5 | 21 | C |  $\cdots$
3954227 | 19:25:58.3 | +39:03:00.51 | 0.5943 | Al | 0.56 | 1.434 | 16.689 | 16.097 | 1151 | 0.359 |  $\cdots$ | K5 |  $\cdots$ | M | TV
3965242 | 19:37:17.1 | +39:02:20.00 | 0.9967 | Al | 0.53 | 1.074 | 14.536 | 14.084 | 478 | 0.396 |  $\cdots$ | K1.5 |  $\cdots$ | C | TV
3972629 | 19:43:43.9 | +39:01:29.58 | 0.2442 | BL | 0.85 | 1.140 | 15.448 | 14.957 | 720 | 0.403 |  $\cdots$ | K2.5 | 3, 5 | C | F, S
4037163 | 19:00:19.2 | +39:11:02.44 | 0.6354 | Al | 0.59 | 1.484 | 18.153 | 17.446 | 2139 | 0.358 |  $\cdots$ | K5.5 | 7, 8 | M | TV
4049124 | 19:17:15.4 | +39:07:52.90 | 4.8045 | Al | 0.17 | 1.007 | 15.063 | 14.647 | 934 | 0.900 |  $\cdots$ | K1 | 2, 13 | M | F
4077442 | 19:45:21.9 | +39:08:26.06 | 0.6928 | Al | 0.59 | 1.325 | 14.213 | 13.629 | 545 | 0.785 |  $\cdots$ | K4 | 21 | M | TV
4168013 | 19:36:10.5 | +39:15:59.55 | 0.3109 | WU | 0.85 | 1.092 | 15.527 | 15.079 | 1124 | 0.875 |  $\cdots$ | K2 |  $\cdots$ | M |  $\cdots$
4241946 | 18:59:36.6 | +39:20:50.52 | 0.2844 | WU | 0.75 | 1.329 | 16.472 | 16.073 | 5779 | 9.260 |  $\cdots$ | K4 | 3 | C | F
4249218 | 19:10:49.9 | +39:18:48.69 | 0.3162 | WU | 0.76 | 1.053 | 14.449 | 14.184 | 801 | 1.015 | G4 V1 | K1.5 | 3, 5, 19 | C | F, S
4349483 | 19:06:35.4 | +39:27:17.00 | 0.5091 | Al | 0.71 | 1.243 | 14.798 | 14.371 | 641 | 0.546 |  $\cdots$ | K3.5 |  $\cdots$ | C | TV
4350454 | 19:08:12.6 | +39:29:16.53 | 0.9657 | BL | 0.80 | 0.829 | 13.934 | 13.608 | 1861 | 9.305 |  $\cdots$ | G3 | 3 | C | F
4385109 | 19:45:02.2 | +39:25:17.15 | 0.2746 | WU | 0.90 | 1.181 | 15.580 | 15.071 | 875 | 0.535 |  $\cdots$ | K2.5 | 3 | C | F
4466691 | 19:29:07.0 | +39:30:18.22 | 0.4827 | BL | 0.78 | 0.688 | 14.576 | 14.308 | 2027 | 5.792 |  $\cdots$ | F8 |  $\cdots$ | C |  $\cdots$
4474193 | 19:36:57.5 | +39:33:30.45 | 0.3225 | WU | 0.89 | 1.067 | 15.628 | 15.173 | 1268 | 1.022 |  $\cdots$ | K1.5 | 3, 5 | C | TV, F, S
4474637 | 19:37:21.6 | +39:33:47.85 | 3.8867 | Al | 0.58 | 0.911 | 18.111 | 17.780 | 7742 | 3.450 |  $\cdots$ | G8 | 13 | C | TV, AM
4547308 | 19:08:14.6 | +39:39:54.54 | 0.5769 | WU | 0.88 | 0.552 | 12.617 | 12.457 | 1654 | 21.215 |  $\cdots$ | F4 | 3, 7 | C | F
4660997 | 19:34:03.3 | +39:42:41.00 | 0.5626 | Al | 0.62 | 0.885 | 12.606 | 12.250 | 431 | 1.742 | G1 V1 | G7 | 1, 3, 14, 16, 17, 19, 21 | M | F
4859432 | 19:50:13.1 | +39:59:00.27 | 0.3855 | WU | 0.77 | 1.064 | 15.651 | 15.214 | 1702 | 1.773 |  $\cdots$ | K1.5 | 3, 7 | C | F
4941060 | 19:44:39.7 | +40:04:47.95 | 0.3034 | WU | 0.75 | 1.160 | 15.892 | 15.403 | 1105 | 0.628 |  $\cdots$ | K2.5 | 5 | C | S
4991959 | 18:59:13.7 | +40:09:20.95 | 0.3609 | WU | 0.76 | 0.943 | 15.442 | 15.053 | 1353 | 1.300 |  $\cdots$ | G9 | 3, 5, 6 | C | F, S, M
5020034 | 19:37:12.9 | +40:11:33.18 | 2.1191 | Al | 0.57 | 0.884 | 16.110 | 15.761 | 4511 | 7.522 |  $\cdots$ | G7 |  $\cdots$ | C |  $\cdots$
5033682 | 19:48:57.3 | +40:10:15.60 | 0.3799 | WU | 0.94 | 0.892 | 13.640 | 13.278 | 741 | 2.000 |  $\cdots$ | G8 | 3, 4, 5 | C | F, S
5123176 | 19:50:33.6 | +40:14:09.90 | 0.7078 | WU | 0.82 | 0.622 | 11.713 | 11.481 | 765 | 11.153 | F5 IV1 | F6 | 3, 5, 6, 16, 17, 19 | C | F, S, M
5195137 | 19:36:25.2 | +40:18:05.89 | 0.3241 | WU | 0.92 | 1.007 | 14.426 | 14.010 | 770 | 1.099 |  $\cdots$ | K1 | 3, 5 | C | F, S
5215999 | 19:54:19.8 | +40:21:53.07 | 2.5312 | Al | 0.68 | 1.134 | 14.636 | 14.155 | 3216 | 16.783 |  $\cdots$ | K2 | 3, 5 | M | TV, F, S
5282464 | 19:35:47.0 | +40:26:58.34 | 0.4964 | BL | 0.73 | 0.491 | 12.708 | 12.524 | 918 | 6.144 |  $\cdots$ | F2 | 16, 19 | C |  $\cdots$
5283839 | 19:37:12.7 | +40:28:00.66 | 0.3152 | WU | 0.93 | 0.892 | 15.614 | 15.253 | 1587 | 1.486 |  $\cdots$ | G8 | 3, 4, 5 | M | AM, F, S
5294739 | 19:47:20.9 | +40:28:23.16 | 3.7360 | Al | 0.59 | 1.255 | 14.111 | 13.567 | 1688 | 7.949 |  $\cdots$ | K3.5 | 3, 13, 16, 19 | C | F
5300878 | 19:52:19.6 | +40:28:41.29 | 1.2794 | Al | 0.50 | 1.387 | 15.583 | 14.963 | 626 | 0.303 |  $\cdots$ | K4.5 | 5, 8 | C | CU, S
5357682 | 19:21:13.2 | +40:34:12.02 | 0.7177 | BL | 0.83 | 0.741 | 14.801 | 14.511 | 2118 | 5.242 |  $\cdots$ | F9 |  $\cdots$ | C |  $\cdots$
5425950 | 18:52:13.7 | +40:41:17.55 | 0.3782 | WU | 0.76 | 0.979 | 15.534 | 15.136 | 1418 | 1.323 |  $\cdots$ | K0 | 5 | C | S
5543482 | 19:38:34.2 | +40:43:47.36 | 0.2948 | WU | 0.80 | 1.154 | 16.171 | 15.680 | 1294 | 0.667 |  $\cdots$ | K2.5 |  $\cdots$ | C |  $\cdots$
5557368 | 19:50:56.4 | +40:46:04.24 | 0.2979 | WU | 0.93 | 1.086 | 13.220 | 12.756 | 360 | 0.765 |  $\cdots$ | K2 | 3, 5, 16, 19 | C | F, S
5563814 | 19:56:16.1 | +40:43:15.45 | 0.3082 | WU | 0.91 | 1.135 | 15.634 | 15.184 | 1259 | 0.997 |  $\cdots$ | K2 | 3, 5 | C | F, S
5615528 | 19:21:38.1 | +40:49:43.27 | 0.3392 | Al | 0.70 | 0.852 | 14.912 | 14.672 | 1304 | 1.714 |  $\cdots$ | G5 |  $\cdots$ | M | TV
5774375 | 19:03:42.2 | +41:05:26.93 | 1.5500 | Al | 0.60 | 1.092 | 14.375 | 13.915 | 1334 | 3.604 |  $\cdots$ | K2 | 2, 3 | C | TV, F
5820209 | 19:55:57.7 | +41:03:52.27 | 0.6561 | BL | 0.81 | 1.310 | 14.095 | 13.519 | 4153 | 50.251 |  $\cdots$ | K4 | 6, 11 | C | M
5956588 | 19:22:20.2 | +41:14:06.72 | 0.2729 | WU | 0.83 | 1.083 | 16.217 | 15.755 | 1296 | 0.625 |  $\cdots$ | K2 |  $\cdots$ | C |  $\cdots$
5984000 | 19:51:32.3 | +41:14:57.97 | 0.3231 | WU | 0.82 | 0.998 | 14.845 | 14.435 | 925 | 1.074 |  $\cdots$ | K0.5 | 3, 5 | C | F, S
6034812 | 19:17:20.8 | +41:19:06.82 | 0.3673 | WU | 0.93 | 0.765 | 13.143 | 12.844 | 696 | 2.633 | F51 | F9.5 | 3 | M | TV, F
6044064 | 19:29:24.8 | +41:19:27.26 | 5.0632 | Al | 0.54 | 1.070 | 15.527 | 15.081 | 2981 | 6.145 |  $\cdots$ | K1.5 | 2, 3, 13, 23* | C | CU, F
6050116 | 19:36:33.1 | +41:20:22.56 | 0.2399 | WU | 0.79 | 1.280 | 14.948 | 14.392 | 448 | 0.262 |  $\cdots$ | K3.5 | 3, 5, 19†, 24* | C | F, S
6103049 | 18:55:04.4 | +41:26:27.97 | 0.6432 | Al | 0.59 | 0.931 | 15.462 | 15.083 | 2358 | 3.839 |  $\cdots$ | G9 | 2, 7 | M | F
6106771 | 19:02:18.9 | +41:26:15.94 | 0.2638 | WU | 0.77 | 1.032 | 16.041 | 15.598 | 1136 | 0.554 |  $\cdots$ | K1.5 | 3 | C | F
6128248 | 19:33:07.4 | +41:26:08.39 | 0.2917 | WU | 0.76 | 1.238 | 15.808 | 15.267 | 850 | 0.421 |  $\cdots$ | K3 | 5 | C | S
6191574 | 19:08:42.5 | +41:31:59.23 | 1.0345 | Al | 0.50 | 1.560 | 15.013 | 14.310 | 271 | 0.103 |  $\cdots$ | K6 | 2 | M | TV, F
6197038 | 19:17:12.2 | +41:31:35.40 | 9.7517 | Al | 0.46 | 1.224 | 13.919 | 13.381 | 1468 | 7.136 |  $\cdots$ | K3 | 2 | C | TV, CU, F
6205460 | 19:27:55.3 | +41:33:33.90 | 3.7228 | Al | 0.58 | 1.065 | 12.957 | 12.558 | 778 | 4.282 |  $\cdots$ | K1.5 | 1, 2, 3, 13, 16, 19 | C | AM, F
6223646 | 19:47:32.0 | +41:31:07.90 | 0.3649 | WU | 0.84 | 0.474 | 13.655 | 13.482 | 1235 | 4.600 | B9 III1 | F2 | 19 | C |  $\cdots$
6264091 | 18:52:59.6 | +41:39:11.22 | 0.3250 | WU | 0.97 | 0.936 | 15.524 | 15.144 | 1607 | 1.685 |  $\cdots$ | G9 | 3, 5 | C | TV, F, S
6267702 | 19:00:00.8 | +41:38:29.04 | 0.2832 | BL | 0.90 | 1.128 | 16.021 | 15.783 | 1137 | 0.469 |  $\cdots$ | K2 | 3 | M | TV, F
6283224 | 19:24:46.0 | +41:40:01.94 | 0.7665 | Al | 0.57 | 1.128 | 16.021 | 15.541 | 1350 | 0.825 |  $\cdots$ | K2 | 2 | M | F
6421483 | 18:52:10.8 | +41:51:36.26 | 2.1584 | Al | 0.58 | 1.119 | 13.595 | 13.120 | 757 | 2.415 |  $\cdots$ | K2 | 3 | C | TV, AM, F
6424124 | 18:57:21.9 | +41:50:19.32 | 0.3855 | WU | 0.91 | 0.872 | 14.275 | 13.920 | 1021 | 2.102 | F7 III1 | G6 | 3, 5, 6, 19 | C | F, S, M
6431545 | 19:11:18.9 | +41:48:17.79 | 0.3168 | WU | 0.91 | 0.963 | 14.146 | 13.751 | 723 | 1.230 | G2 V1 | G9 | 3, 5*, 7*‡ | M | TV, F, S
6436038 | 19:18:10.1 | +41:53:25.24 | 0.3763 | WU | 0.96 | 0.917 | 15.588 | 15.219 | 1789 | 1.949 |  $\cdots$ | G8 | 3 | C | TV, F
6471048 | 19:56:30.8 | +41:49:41.89 | 0.2972 | WU | 0.77 | 1.159 | 13.287 | 12.784 | 309 | 0.549 | K0 V1 | K2.5 | 5, 19 | C | S
6615041 | 19:37:26.9 | +42:02:45.31 | 0.3401 | WU | 0.76 | 0.877 | 14.056 | 13.713 | 731 | 1.304 | Flat1 | G7 | 5, 7, 19 | C | S
6697716 | 19:34:22.7 | +42:08:35.19 | 1.4432 | Al | 0.44 | 1.172 | 14.918 | 14.417 | 806 | 0.828 |  $\cdots$ | K2.5 |  $\cdots$ | M | CU
6791604 | 19:42:42.1 | +42:14:26.14 | 0.5288 | BL | 0.70 | 0.555 | 14.098 | 13.896 | 1615 | 5.373 |  $\cdots$ | F4 | 6 | C | M
6836140 | 18:44:09.8 | +42:23:52.30 | 0.4877 | WU | 0.73 | 1.139 | 15.215 | 14.737 | 1000 | 0.950 |  $\cdots$ | K2.5 | 1, 3, 5 | C | TV, F, S
6962018 | 19:42:10.5 | +42:25:42.24 | 1.2699 | Al | 0.45 | 1.657 | 15.577 | 14.785 | 380 | 0.132 |  $\cdots$ | K6.5 |  $\cdots$ | C |  $\cdots$
7035139 | 19:29:08.1 | +42:35:40.46 | 0.3097 | WU | 0.80 | 1.011 | 15.950 | 15.543 | 1372 | 0.851 |  $\cdots$ | K1 | 5 | M | S
7037319 | 19:31:47.6 | +42:30:30.38 | 0.5781 | BL | 0.84 | 0.592 | 13.808 | 13.587 | 1850 | 9.375 |  $\cdots$ | F5 | 3 | C | F
7119757 | 19:29:35.6 | +42:37:47.63 | 0.7429 | Al | 0.64 | 1.134 | 16.118 | 15.636 | 2106 | 1.839 |  $\cdots$ | K2 | 2, 3, 5, 7 | M | TV, F, S
7173910 | 18:44:48.7 | +42:44:45.86 | 0.4022 | WU | 0.82 | 0.878 | 14.719 | 14.360 | 1298 | 2.263 |  $\cdots$ | G7 | 5 | M | S
7199183 | 19:25:43.0 | +42:47:10.56 | 0.3241 | WU | 0.97 | 1.003 | 16.086 | 15.667 | 1584 | 1.012 |  $\cdots$ | K0.5 | 3 | C | TV, F
7199353 | 19:25:56.4 | +42:44:15.93 | 0.2820 | WU | 0.78 | 1.054 | 14.953 | 14.530 | 736 | 0.623 |  $\cdots$ | K1.5 | 3, 5 | C | F, S
7259917 | 18:53:02.2 | +42:50:44.15 | 0.3847 | BL | 0.70 | 0.711 | 12.132 | 11.867 | 412 | 2.267 | F3 V1 | F9 | 5, 17, 19 | C | S
7272739 | 19:15:25.7 | +42:48:45.21 | 0.2812 | WU | 0.77 | 1.060 | 13.445 | 12.982 | 350 | 0.588 |  $\cdots$ | K1.5 | 3, 5, 7, 17, 19† | C | F, S
7284688 | 19:30:31.2 | +42:49:45.63 | 0.6460 | Al | 0.64 | 0.942 | 11.918 | 11.930 |  $\cdots$ |  $\cdots$ | G2 V1 | G9 | 2, 5, 9 | M | TV, F, S
7367833 | 19:33:20.4 | +42:55:18.87 | 0.2862 | Al | 0.77 | 0.940 | 15.793 | 15.412 | 1912 | 1.864 |  $\cdots$ | G9 |  $\cdots$ | M |  $\cdots$
7433513 | 19:12:27.9 | +43:05:39.41 | 0.7011 | Al | 0.70 | 0.966 | 15.872 | 15.482 | 1661 | 1.319 |  $\cdots$ | K0 |  $\cdots$ | C | TV, AM
7458285 | 19:42:10.3 | +43:02:56.97 | 0.6606 | BL | 0.71 | 0.813 | 13.848 | 13.527 | 1462 | 6.189 | F0 IV1 | G1 | 6 | C | M
7516345 | 19:12:15.0 | +43:07:39.28 | 0.4919 | BL | 0.75 | 0.506 | 12.499 | 12.311 | 873 | 6.761 |  $\cdots$ | F3 | 5, 19† | C | S
7542091 | 19:42:52.0 | +43:09:52.62 | 0.3905 | WU | 0.83 | 0.860 | 12.711 | 12.369 | 490 | 2.021 | G1 V1 | G6 | 3, 4, 5, 17, 19 | C | F, S
7546429 | 19:46:46.9 | +43:08:18.82 | 0.3521 | WU | 0.92 | 0.998 | 15.608 | 15.196 | 1580 | 1.553 |  $\cdots$ | K0.5 |  $\cdots$ | C |  $\cdots$
7584739 | 18:49:09.9 | +43:17:02.16 | 0.9116 | BL | 0.72 | 0.651 | 13.691 | 13.452 | 1860 | 10.721 | F3 V1 | F6 | 3, 6, 17†, 19† | C | F, M
7584826 | 18:49:23.1 | +43:17:04.20 | 0.6223 | Al | 0.62 | 0.992 | 15.654 | 15.255 | 1337 | 1.053 |  $\cdots$ | K0.5 | 5 | C | TV, S
7610486 | 19:30:55.6 | +43:14:41.50 | 0.4577 | WU | 0.80 | 0.940 | 16.107 | 15.788 | 2400 | 2.076 |  $\cdots$ | G9 | 5 | C | S
7627059 | 19:48:27.4 | +43:12:51.59 | 0.3237 | WU | 0.81 | 0.957 | 15.362 | 14.976 | 1310 | 1.308 |  $\cdots$ | G9 | 3 | C | F
7657914 | 18:39:52.3 | +43:23:08.43 | 0.4745 | BL | 0.72 | 0.581 | 13.178 | 12.979 | 951 | 4.336 | F5 Vaw1 | F5 | 17, 19 | C |  $\cdots$
7667885 | 19:03:30.3 | +43:23:22.47 | 0.3148 | Al |  $\cdots$ | 1.413 | 17.450 | 16.827 | 1374 | 0.262 | G2 | K4.5 | 3, 20* | C | F
7671594 | 19:10:32.6 | +43:22:14.95 | 1.4103 | Al | 0.34 | 2.369 | 16.980 | 15.719 | 209 | 0.017 |  $\cdots$ | M2.5 | 2, 13 | C | TV, CU, F
7685689 | 19:31:30.5 | +43:22:07.25 | 0.3252 | WU | 0.78 | 0.955 | 15.955 | 15.579 | 1587 | 1.101 |  $\cdots$ | G9 | 7 | C |  $\cdots$
7696778 | 19:45:09.1 | +43:23:55.39 | 0.3312 | WU | 0.79 | 1.209 | 15.931 | 15.409 | 1331 | 0.906 |  $\cdots$ | K3 |  $\cdots$ | C |  $\cdots$
7740566 | 19:07:08.0 | +43:25:50.82 | 0.2971 | WU | 0.81 | 0.951 | 16.030 | 15.696 | 1611 | 1.019 |  $\cdots$ | G9 | 3 | C | F
7773380 | 19:51:27.4 | +43:29:52.71 | 0.3076 | WU | 0.94 | 1.041 | 14.863 | 14.434 | 859 | 0.927 |  $\cdots$ | K1.5 | 3, 4, 5 | C | F, S
7871200 | 18:50:52.4 | +43:40:12.05 | 0.2429 | WU | 0.75 | 1.335 | 13.614 | 13.030 | 262 | 0.315 |  $\cdots$ | K4 | 3, 6, 17, 19, 25* | C | F, M
7877062 | 19:03:35.7 | +43:36:12.29 | 0.3037 | WU | 0.82 | 0.943 | 14.285 | 13.898 | 682 | 0.956 |  $\cdots$ | G9 | 3, 5, 7, 19 | C | F, S
7879399 | 19:08:29.9 | +43:36:00.64 | 0.3927 | WU | 0.74 | 0.900 | 12.980 | 12.599 | 506 | 1.744 |  $\cdots$ | G8 | 19† | C |  $\cdots$
7885570 | 19:19:53.7 | +43:39:13.77 | 1.7293 | Al | 0.58 | 0.890 | 12.046 | 11.686 | 510 | 4.115 | G0 IV1 | G7 | 1, 2, 3, 19, 26* | M | TV, AM:SC, F
7950962 | 19:13:38.5 | +43:42:26.77 | 0.8270 | WU | 0.94 | 0.524 | 10.794 | 10.621 | 644 | 17.477 |  $\cdots$ | F3 | 3, 19 | C | F
7959612 | 19:26:37.0 | +43:44:13.63 | 0.2432 | WU | 0.81 | 1.297 | 17.513 | 16.947 | 1444 | 0.258 |  $\cdots$ | K4 |  $\cdots$ | C |  $\cdots$
8029708 | 19:28:37.6 | +43:49:06.96 | 0.3136 | BL | 0.75 | 1.165 | 17.215 | 16.731 | 1816 | 0.499 |  $\cdots$ | K2.5 |  $\cdots$ | C |  $\cdots$
8112324 | 19:46:35.4 | +43:59:41.99 | 0.5759 | Al | 0.59 | 1.086 | 15.126 | 14.675 | 920 | 0.850 |  $\cdots$ | K2 |  $\cdots$ | C | TV
8190491 | 19:56:21.6 | +44:04:01.45 | 0.7779 | BL | 0.95 | 0.826 | 14.554 | 14.226 | 2705 | 11.116 |  $\cdots$ | G2 | 5, 7 | C | S
8211618 | 18:46:26.9 | +44:07:34.64 | 0.3374 | WU | 0.74 | 0.969 | 16.068 | 15.667 | 1668 | 1.121 |  $\cdots$ | K0 |  $\cdots$ | C |  $\cdots$
8211824 | 18:46:58.7 | +44:10:10.75 | 0.8411 | Al | 0.59 | 1.215 | 16.600 | 16.075 | 1315 | 0.479 |  $\cdots$ | K3 | 2 | C | TV, F
8230815 | 19:23:34.4 | +44:07:52.49 | 0.3452 | WU | 0.93 | 0.854 | 13.640 | 13.298 | 683 | 1.666 | G0.5 V1 | G5 | 3, 5 | C | F, S
8248967 | 19:47:46.6 | +44:09:59.05 | 0.5925 | WU | 0.83 | 0.799 | 15.439 | 15.112 | 3218 | 6.960 |  $\cdots$ | G1 |  $\cdots$ | C | AM
8285349 | 19:02:27.9 | +44:12:56.42 | 0.6671 | BL | 0.90 | 0.712 | 12.437 | 12.166 | 570 | 3.293 | F3/F5 V1 | F9 | 2, 3, 5, 18 | M | TV, F, S
8294484 | 19:19:45.7 | +44:13:32.55 | 1.0127 | Al | 0.58 | 0.832 | 15.152 | 14.827 | 1611 | 2.269 |  $\cdots$ | G3 |  $\cdots$ | C | TV, AM
8298344 | 19:25:08.6 | +44:12:57.21 | 0.3029 | WU | 0.93 | 1.120 | 15.780 | 15.305 | 1100 | 0.681 |  $\cdots$ | K2 | 3, 5 | C | TV, F, S
8386048 | 19:52:51.9 | +44:19:25.43 | 0.8174 | BL | 0.71 | 1.031 | 17.059 | 16.631 | 5100 | 4.314 |  $\cdots$ | K1.5 | 3 | C | F
8431389 | 19:28:12.2 | +44:28:25.71 | 0.3511 | WU | 0.84 | 0.938 | 15.058 | 14.689 | 1337 | 1.774 |  $\cdots$ | G9 | 6 | C | M
8479107 | 18:56:30.1 | +44:33:01.98 | 0.7676 | Al | 0.53 | 1.087 | 15.464 | 15.010 | 887 | 0.582 |  $\cdots$ | K2 | 2, 5 | M | TV, F, S
8608490 | 18:56:40.1 | +44:46:45.74 | 1.0828 | Al | 0.51 | 1.125 | 15.427 | 14.955 | 851 | 0.563 |  $\cdots$ | K2 | 1, 3, 5 | C | F, S
8620561 | 19:21:04.1 | +44:42:41.21 | 0.7820 | Al | 0.53 | 1.408 | 17.439 | 16.808 | 1259 | 0.223 |  $\cdots$ | K4.5 |  $\cdots$ | C |  $\cdots$
8690104 | 19:28:37.9 | +44:52:24.35 | 0.4088 | WU | 0.77 | 0.872 | 15.261 | 14.913 | 1572 | 1.994 |  $\cdots$ | G6 | 5, 6, 7 | C | S, M
8694926 | 19:36:08.7 | +44:49:46.34 | 0.3715 | WU | 0.80 | 0.995 | 16.050 | 15.693 | 4386 | 7.570 |  $\cdots$ | K0.5 |  $\cdots$ | M |  $\cdots$
8696327 | 19:38:02.7 | +44:48:24.22 | 0.8792 | BL | 0.75 | 0.708 | 14.850 | 14.575 | 2923 | 9.412 |  $\cdots$ | F9 |  $\cdots$ | C | AM
8703528 | 19:47:18.8 | +44:48:07.59 | 0.3999 | WU | 0.75 | 0.990 | 13.411 | 12.992 | 555 | 1.462 |  $\cdots$ | K0 | 3, 6, 17, 19† | C | F, M
8715667 | 20:00:06.5 | +44:49:05.02 | 0.4057 | WU | 0.85 | 1.034 | 13.873 | 13.516 | 1124 | 3.695 |  $\cdots$ | K1.5 | 3, 5, 19† | C | F, S
8822555 | 19:33:10.4 | +45:02:31.20 | 0.8528 | BL | 0.80 | 0.742 | 14.570 | 14.282 | 2501 | 9.029 |  $\cdots$ | F9 |  $\cdots$ | C | AM
8842170 | 19:57:14.5 | +45:03:48.88 | 0.3494 | WU | 0.90 | 1.025 | 15.200 | 14.771 | 1318 | 1.598 |  $\cdots$ | K1 | 10 | C | AM
8846978 | 20:02:43.3 | +45:00:07.76 | 1.3791 | Al | 0.77 | 1.098 | 13.716 | 13.254 | 887 | 2.929 |  $\cdots$ | K2 | 3, 5, 11 | C | TV, F, S
8904448 | 19:51:31.6 | +45:07:34.94 | 0.8660 | BL | 0.74 | 0.516 | 14.008 | 13.814 | 2734 | 16.603 | A2 V1 | F3 | 3, 7, 19 | C | F
8934111 | 18:53:23.9 | +45:17:50.46 | 0.3645 | WU | 0.92 | 0.943 | 15.553 | 15.169 | 1632 | 1.699 |  $\cdots$ | G9 | 3, 5 | C | F, S
8949316 | 19:22:04.6 | +45:14:07.36 | 0.6044 | Al | 0.51 | 2.247 | 17.317 | 16.148 | 423 | 0.046 |  $\cdots$ | M2 | 8 | M |  $\cdots$
8977390 | 19:55:44.2 | +45:17:51.59 | 0.6533 | WU | 0.86 | 0.781 | 13.427 | 13.125 | 1399 | 8.199 | F0 IV1 | G0 | 3, 19 | C | F
9071104 | 18:51:56.6 | +45:24:28.73 | 0.3852 | WU | 0.82 | 0.820 | 13.958 | 13.627 | 923 | 2.249 |  $\cdots$ | G2 | 4, 5, 18, 19 | C | S
9071373 | 18:52:38.2 | +45:24:56.24 | 0.4218 | BL | 0.81 | 0.836 | 13.313 | 12.974 | 753 | 2.733 | G0 V1 | G3 | 3, 5, 18 | C | F, S
9091810 | 19:31:48.0 | +45:25:11.88 | 0.4797 | Al | 0.70 | 0.970 | 13.006 | 12.601 | 457 | 1.418 |  $\cdots$ | K0 | 1, 3, 5, 7, 11 | C | F, S
9101279 | 19:44:00.4 | +45:26:53.24 | 1.8115 | Al | 0.58 | 0.535 | 13.916 | 13.721 | 2116 | 10.832 |  $\cdots$ | F4 | 7*, 13 | C |  $\cdots$
9119652 | 20:03:49.7 | +45:26:57.13 | 2.2162 | Al | 0.43 | 1.156 | 14.821 | 14.327 | 790 | 0.864 |  $\cdots$ | K2.5 | 13 | C | CU
9137992 | 18:53:50.5 | +45:33:20.95 | 1.4700 | Al | 0.64 | 0.936 | 13.388 | 13.004 | 862 | 3.479 | G5.5 IIb1 | G9 | 3, 18 | C | TV, F
9145707 | 19:10:59.4 | +45:31:04.94 | 0.3208 | WU | 0.75 | 0.993 | 15.152 | 14.740 | 990 | 0.927 |  $\cdots$ | K0.5 | 3, 5, 6, 18 | C | F, S, M
9145846 | 19:11:19.8 | +45:35:24.98 | 0.2945 | WU | 0.96 | 1.246 | 15.818 | 15.276 | 559 | 0.181 |  $\cdots$ | K3.5 | 3, 5 | C | TV, F, S
9161428 | 19:37:42.0 | +45:30:21.51 | 0.3026 | WU | 0.92 | 0.967 | 15.515 | 15.123 | 1256 | 1.049 |  $\cdots$ | K0 | 3 | C | F
9164694 | 19:42:39.8 | +45:30:24.70 | 1.1114 | BL | 0.75 | 0.744 | 14.386 | 14.107 | 2674 | 12.125 |  $\cdots$ | F9.5 | 3, 15* | C | AM, F
9274472 | 19:15:58.7 | +45:46:44.84 | 0.3021 | WU | 0.90 | 1.103 | 14.420 | 13.948 | 595 | 0.697 |  $\cdots$ | K2 | 3, 5, 18 | C | F, S
9283826 | 19:32:10.9 | +45:44:08.92 | 0.3565 | WU | 0.84 | 0.854 | 13.460 | 13.124 | 645 | 1.749 |  $\cdots$ | G5 | 3, 5, 6, 7, 19 | C | AM, F, S, M
9290838 | 19:42:36.0 | +45:47:02.22 | 0.3630 | WU | 0.92 | 1.011 | 15.890 | 15.489 | 1718 | 1.402 |  $\cdots$ | K1 | 3, 5 | C | F, S
9307423 | 20:02:22.1 | +45:45:26.92 | 0.5586 | WU | 0.73 | 1.402 | 16.762 | 16.150 | 1933 | 0.965 |  $\cdots$ | K4.5 |  $\cdots$ | C |  $\cdots$
9328852 | 18:59:59.3 | +45:53:12.08 | 0.6459 | Al | 0.54 | 1.402 | 15.919 | 15.291 | 703 | 0.282 |  $\cdots$ | K4.5 | 1, 2, 3, 18† | C | TV, AM:SC, F
9345163 | 19:31:46.5 | +45:49:34.69 | 0.2451 | WU | 0.77 | 1.242 | 16.018 | 15.478 | 797 | 0.305 |  $\cdots$ | K3.5 | 3 | C | F
9345838 | 19:32:53.0 | +45:50:37.96 | 1.0459 | BL | 0.75 | 0.636 | 12.392 | 12.159 | 1279 | 16.703 | F6 III1 | F6 | 3, 5, 15*, 17, 19 | C | F, S
9388303 | 18:54:30.2 | +45:55:18.14 | 0.2667 | WU | 0.78 | 1.095 | 16.075 | 15.652 | 1028 | 0.432 |  $\cdots$ | K2 | 3 | C | F
9419603 | 19:49:51.1 | +45:58:04.19 | 1.0524 | BL | 0.77 | 0.944 | 15.635 | 15.255 | 3080 | 5.589 |  $\cdots$ | G9 | 3, 5 | M | TV, AM, F, S
9450883 | 18:58:46.3 | +46:04:46.12 | 0.3637 | WU | 0.94 | 0.897 | 15.543 | 15.188 | 1642 | 1.689 |  $\cdots$ | G8 | 3 | C | TV, F
9466316 | 19:29:42.9 | +46:02:01.08 | 0.3525 | BL | 0.84 | 1.078 | 16.155 | 15.697 | 1702 | 1.136 |  $\cdots$ | K2 | 5 | M | TV, S
9489411 | 19:59:27.2 | +46:04:26.58 | 6.6889 | Al | 0.66 | 1.666 | 14.559 | 13.748 | 2265 | 12.113 |  $\cdots$ | K6.5 | 3, 11, 19 | C | F
9527167 | 19:28:44.1 | +46:09:33.61 | 0.2809 | WU | 0.97 | 0.987 | 15.858 | 15.453 | 1383 | 0.939 |  $\cdots$ | K0 | 3 | C | TV, F
9532591 | 19:37:23.0 | +46:07:32.87 | 0.3584 | WU | 0.95 | 1.071 | 15.920 | 15.481 | 1578 | 1.192 |  $\cdots$ | K1.5 | 3, 5, 10 | C | F, S
9533706 | 19:39:02.6 | +46:11:05.76 | 0.2841 | WU | 0.78 | 0.971 | 15.797 | 15.407 | 1197 | 0.734 |  $\cdots$ | K0 |  $\cdots$ | C |  $\cdots$
9551200 | 20:00:28.6 | +46:08:41.53 | 0.7119 | Al | 0.56 | 1.185 | 15.807 | 15.299 | 658 | 0.245 |  $\cdots$ | K2.5 | 2 | M | TV, F
9654476 | 19:36:02.1 | +46:22:47.94 | 0.3963 | Al | 0.71 | 0.966 | 15.208 | 14.812 | 1093 | 1.060 |  $\cdots$ | K0 | 3, 5 | C | F, S
9662581 | 19:47:32.9 | +46:23:34.41 | 0.2580 | WU | 0.79 | 1.221 | 16.213 | 15.681 | 961 | 0.367 |  $\cdots$ | K3 | 3 | C | F
9664382 | 19:49:37.7 | +46:18:58.09 | 0.2908 | WU | 0.75 | 1.344 | 17.145 | 16.572 | 3885 | 2.643 |  $\cdots$ | K4 |  $\cdots$ | C |  $\cdots$
9717924 | 19:39:37.7 | +46:26:32.47 | 0.3956 | BL | 0.72 | 0.721 | 17.730 | 17.450 | 3825 | 1.141 |  $\cdots$ | F9 |  $\cdots$ | C | AM
9767437 | 19:21:58.6 | +46:32:11.09 | 0.7127 | Al | 0.67 | 1.131 | 13.296 | 12.814 | 355 | 0.705 |  $\cdots$ | K2 | 2 | C | TV, F
9772642 | 19:30:48.9 | +46:35:22.31 | 0.2754 | WU | 0.80 | 1.175 | 14.101 | 13.590 | 414 | 0.470 | K21 | K2.5 | 3, 5, 19 | C | F, S
9777984 | 19:39:10.8 | +46:30:23.74 | 0.2585 | WU | 0.76 | 1.592 | 18.471 | 17.733 | 1717 | 0.177 |  $\cdots$ | K6 | 7 | C |  $\cdots$
9786165 | 19:50:11.0 | +46:34:40.60 | 0.3326 | WU | 0.73 | 1.086 | 17.576 | 17.164 | 3238 | 1.064 | G2 | K2 | 3, 20* | C | AM:SC, F
9832227 | 19:29:16.0 | +46:37:19.86 | 0.4579 | WU | 0.95 | 0.794 | 12.599 | 12.284 | 626 | 3.562 |  $\cdots$ | G1 | 3, 4, 5, 19, 27* | C | F, S
9838047 | 19:38:38.5 | +46:41:26.77 | 0.4362 | WU | 0.85 | 0.905 | 13.828 | 13.462 | 1292 | 5.130 |  $\cdots$ | G8 | 5, 7 | C | S
9882280 | 19:10:53.5 | +46:47:46.30 | 0.2891 | WU | 0.77 | 1.010 | 14.988 | 14.560 | 766 | 0.656 |  $\cdots$ | K1 | 3 | C | F
9935245 | 18:52:51.1 | +46:50:05.97 | 0.7849 | Al | 0.63 | 1.065 | 16.632 | 16.190 | 2287 | 1.303 |  $\cdots$ | K1.5 |  $\cdots$ | C | TV, AM
9935311 | 18:52:59.0 | +46:51:03.85 | 0.2738 | BL | 0.82 | 1.355 | 15.048 | 14.454 | 486 | 0.291 |  $\cdots$ | K4.5 | 18 | C | TV
9954225 | 19:32:17.9 | +46:53:59.91 | 1.3407 | Al | 0.56 | 0.824 | 14.786 | 14.460 | 1540 | 2.905 |  $\cdots$ | G2 |  $\cdots$ | C |  $\cdots$
9957351 | 19:37:25.0 | +46:50:27.01 | 0.4956 | Al | 0.60 | 1.352 | 17.586 | 16.981 | 2583 | 0.801 |  $\cdots$ | K4.5 |  $\cdots$ | C |  $\cdots$
9991887 | 18:45:30.3 | +46:59:15.34 | 0.3224 | WU | 0.88 | 1.077 | 15.633 | 15.178 | 1225 | 0.949 |  $\cdots$ | K2 | 3, 5 | C | F, S
9994475 | 18:51:59.7 | +46:55:50.48 | 0.3184 | WU | 0.77 | 0.969 | 14.772 | 14.365 | 899 | 1.081 |  $\cdots$ | K0 | 3, 7, 18 | C | F
10020247 | 19:40:56.2 | +46:56:28.57 | 0.3456 | WU | 0.83 | 1.035 | 16.409 | 15.973 | 1691 | 0.870 |  $\cdots$ | K1.5 |  $\cdots$ | C |  $\cdots$
10123627 | 18:55:45.0 | +47:08:58.88 | 0.2949 | WU | 0.96 | 1.347 | 16.059 | 15.510 | 933 | 0.406 |  $\cdots$ | K4 | 3, 5 | C | F, S
10128961 | 19:06:19.1 | +47:07:52.99 | 0.3477 | WU | 0.86 | 1.007 | 15.945 | 15.553 | 2236 | 2.238 |  $\cdots$ | K1 | 5 | C | S
10189523 | 18:52:02.9 | +47:14:19.03 | 1.0139 | Al | 0.48 | 1.149 | 16.304 | 15.814 | 1150 | 0.466 |  $\cdots$ | K2.5 | 2 | M | TV, F
10226388 | 19:52:35.7 | +47:16:37.90 | 0.6607 | WU | 0.77 | 0.465 | 10.861 | 10.691 | 689 | 18.761 | F0 IV1 | F2 | 7, 19 | C |  $\cdots$
10259530 | 18:59:45.5 | +47:20:07.34 | 0.7072 | WU | 0.86 | 0.423 | 12.210 | 12.067 | 1125 | 14.057 |  $\cdots$ | F1 | 6, 17, 18 | C | M
10322582 | 18:54:09.9 | +47:29:06.18 | 0.2913 | WU | 0.88 | 0.912 | 15.092 | 14.730 | 1017 | 0.989 |  $\cdots$ | G8 | 5, 6 | C | TV, S, M
10346522 | 19:37:56.0 | +47:27:09.62 | 3.9891 | Al | 0.59 | 1.021 | 14.794 | 14.386 | 3677 | 17.732 |  $\cdots$ | K1 |  $\cdots$ | C |  $\cdots$
10350225 | 19:43:12.6 | +47:26:23.67 | 0.2821 | BL | 0.80 | 0.995 | 15.373 | 15.014 | 1023 | 0.771 |  $\cdots$ | K0.5 | 3, 5 | C | TV, AM, F, S
10351767 | 19:45:16.9 | +47:24:23.51 | 0.6351 | Al | 0.58 | 0.801 | 14.833 | 14.649 | 878 | 0.794 |  $\cdots$ | G1 | 2, 3, 5 | M | TV, F, S
10389982 | 18:58:40.6 | +47:34:39.33 | 0.4440 | BL | 0.74 | 0.539 | 11.934 | 11.765 | 573 | 4.815 |  $\cdots$ | F4 | 18 | C |  $\cdots$
10395202 | 19:10:01.3 | +47:33:42.27 | 0.3059 | WU | 0.81 | 0.991 | 14.974 | 14.573 | 941 | 0.978 |  $\cdots$ | K0.5 | 3, 18 | C | F
10528299 | 19:15:57.7 | +47:44:17.71 | 0.3998 | WU | 0.81 | 0.775 | 14.146 | 13.837 | 1108 | 2.672 | G0 V1 | G0 | 3, 5, 18 | C | AM, F, S
10544976 | 19:42:37.2 | +47:45:48.39 | 0.3505 | Al | 0.55 | 1.302 | 19.292 | 18.845 | 520 | 0.006 | DA+M4 V3 | K4 | 3, 28* | M | WD, F
10676927 | 19:44:35.9 | +47:56:16.26 | 1.0311 | Al | 0.56 | 1.016 | 17.298 | 16.868 | 2625 | 0.919 |  $\cdots$ | K1 | 2, 8 | C | TV, AM, F
10680475 | 19:48:53.1 | +47:56:20.31 | 0.3498 | WU | 0.78 | 0.788 | 14.275 | 13.970 | 993 | 1.898 |  $\cdots$ | G0 | 3, 5, 19 | C | F, S
10711646 | 18:49:44.6 | +48:05:51.84 | 0.7377 | Al | 0.51 | 1.530 | 16.496 | 15.798 | 644 | 0.148 |  $\cdots$ | K5.5 |  $\cdots$ | C | TV
10723143 | 19:14:04.4 | +48:05:04.79 | 0.3076 | WU | 0.79 | 1.108 | 15.790 | 15.330 | 1097 | 0.662 |  $\cdots$ | K2 | 5 | C | S
10727655 | 19:21:43.9 | +48:03:57.17 | 0.3534 | WU | 0.74 | 0.872 | 13.760 | 13.412 | 723 | 1.684 |  $\cdots$ | G6 | 3, 7, 19, 25* | C | F
10737574 | 19:37:21.0 | +48:04:48.19 | 0.3656 | WU | 0.90 | 0.927 | 14.213 | 13.834 | 822 | 1.476 | G51 | G9 | 3 | C | F
10743600 | 19:45:32.0 | +48:02:38.42 | 0.8170 | Al | 0.58 | 0.886 | 18.562 | 18.194 | 5009 | 0.986 |  $\cdots$ | G7 |  $\cdots$ | C |  $\cdots$
10797126 | 19:27:11.0 | +48:08:35.09 | 0.4324 | BL | 0.77 | 0.711 | 14.148 | 13.871 | 1295 | 3.535 |  $\cdots$ | F9 | 5 | C | S
10802917 | 19:36:26.0 | +48:11:03.94 | 0.2725 | WU | 0.74 | 1.235 | 16.555 | 15.998 | 1090 | 0.353 |  $\cdots$ | K3 |  $\cdots$ | C |  $\cdots$
10815379 | 19:52:22.2 | +48:06:16.16 | 0.8905 | BL | 0.79 | 0.700 | 11.465 | 11.199 | 647 | 10.340 |  $\cdots$ | F8 | 3, 19 | C | F
10854621 | 19:15:07.4 | +48:15:08.01 | 0.3759 | WU | 0.81 | 0.861 | 15.615 | 15.271 | 1803 | 1.888 |  $\cdots$ | G6 | 5 | C | S
10857342 | 19:20:02.8 | +48:14:39.14 | 2.4159 | Al | 0.57 | 0.262 | 14.454 | 14.359 | 3956 | 21.036 |  $\cdots$ | A7 |  $\cdots$ | C |  $\cdots$
10861842 | 19:28:17.2 | +48:14:17.90 | 0.4719 | BL | 0.76 | 0.578 | 14.298 | 14.078 | 1717 | 5.138 | F1 V1 | F5 | 3, 19 | C | AM, F
10877703 | 19:51:08.4 | +48:14:55.43 | 0.4369 | WU | 0.80 | 0.909 | 14.759 | 14.392 | 1354 | 2.393 |  $\cdots$ | G8 | 5, 19 | C | S
10924462 | 19:35:23.3 | +48:19:18.86 | 0.3754 | Al | 0.67 | 1.201 | 14.121 | 13.603 | 405 | 0.443 |  $\cdots$ | K3 | 3, 5 | C | F, S
10979669 | 19:35:04.3 | +48:25:15.72 | 0.9258 | BL | 0.83 | 0.609 | 12.478 | 12.248 | 1376 | 17.801 | F7 IV1 | F5 | 3 | C | F
11017077 | 18:58:57.5 | +48:33:32.89 | 0.7086 | BL | 0.68 | 0.810 | 13.410 | 13.087 | 1009 | 4.421 | F5 V1 | G1 | 2, 5, 18, 19 | C | F, S
11036301 | 19:39:25.7 | +48:33:58.22 | 0.2852 | WU | 0.95 | 1.213 | 15.543 | 15.018 | 903 | 0.597 |  $\cdots$ | K3 | 3, 5 | C | F, S
11076176 | 19:10:18.4 | +48:40:45.75 | 0.7000 | Al | 0.58 | 1.123 | 16.154 | 15.671 | 1245 | 0.623 |  $\cdots$ | K2 | 2 | M | TV, AM, F
11084647 | 19:28:31.5 | +48:37:39.38 | 0.3117 | WU | 0.83 | 1.019 | 15.746 | 15.332 | 1350 | 1.000 |  $\cdots$ | K1 | 5 | C | S
11127048 | 19:04:13.7 | +48:44:20.12 | 0.2993 | WU | 0.94 | 0.985 | 15.636 | 15.227 | 1228 | 0.911 |  $\cdots$ | K0 | 3, 5 | C | TV, F, S
11154110 | 19:54:40.9 | +48:44:43.31 | 0.5297 | BL | 0.74 | 0.585 | 13.647 | 13.426 | 1402 | 6.240 |  $\cdots$ | F5 | 18†, 19† | C |  $\cdots$
11175495 | 18:50:11.3 | +48:51:23.06 | 2.1910 | BL | 0.66 | 0.291 | 12.998 | 12.896 | 1631 | 13.763 | A4 V1 | A7 | 12, 29* | M |  $\cdots$
11193447 | 19:33:31.6 | +48:49:29.33 | 0.6259 | Al | 0.64 | 0.768 | 13.439 | 13.618 |  $\cdots$ |  $\cdots$ |  $\cdots$ | F9.5 | 2, 5 | M | TV, F, S
11198068 | 19:41:44.0 | +48:48:58.23 | 0.4002 | Al | 0.71 | 1.168 | 16.019 | 15.513 | 1353 | 0.850 |  $\cdots$ | K2.5 | 5 | C | S
11244501 | 19:28:10.6 | +48:56:36.71 | 0.2967 | BL | 0.89 | 0.982 | 10.251 | 9.840 | 105 | 0.954 | G2 V1 | K0 | 3, 5, 19 | M | TV, F, S
11246163 | 19:31:29.9 | +48:59:02.50 | 0.2792 | WU | 0.79 | 1.024 | 15.032 | 14.603 | 762 | 0.623 |  $\cdots$ | K1 | 7 | C |  $\cdots$
11247386 | 19:33:54.8 | +48:57:26.05 | 0.3943 | BL | 0.76 | 0.471 | 14.442 | 14.269 | 2485 | 9.022 |  $\cdots$ | F2 | 7*‡ | C |  $\cdots$
11295026 | 19:23:18.9 | +49:02:08.05 | 0.2909 | WU | 0.90 | 1.053 | 13.188 | 12.749 | 362 | 0.780 |  $\cdots$ | K1.5 | 3, 5 | C | F, S
11303416 | 19:39:34.0 | +49:03:04.67 | 0.3208 | WU | 0.94 | 1.213 | 15.606 | 15.080 | 1020 | 0.720 |  $\cdots$ | K3 | 5 | C | S
11336707 | 18:55:12.5 | +49:07:43.68 | 0.2513 | WU | 0.91 | 1.327 | 15.801 | 15.476 | 793 | 0.302 |  $\cdots$ | K4 | 3, 5 | C | F, S
11341950 | 19:09:05.9 | +49:11:58.41 | 0.2884 | WU | 0.74 | 1.032 | 15.419 | 14.994 | 941 | 0.664 |  $\cdots$ | K1.5 | 3, 18† | C | F
11347875 | 19:21:52.1 | +49:11:55.10 | 3.4551 | Al | 0.86 | 1.491 | 14.806 | 14.124 | 884 | 1.306 |  $\cdots$ | K5.5 | 2*‡, 3, 19 | C | TV, AM, CU, F
11395645 | 19:11:37.0 | +49:17:13.26 | 0.4969 | BL | 0.75 | 0.674 | 15.235 | 14.968 | 2774 | 5.903 |  $\cdots$ | F7 | 18 | C | AM
11404698 | 19:31:07.9 | +49:13:49.45 | 0.5902 | Al | 0.58 | 1.569 | 15.569 | 14.863 | 444 | 0.167 |  $\cdots$ | K6 | 2 | C | TV, F
11494583 | 18:57:28.2 | +49:27:51.82 | 0.2483 | WU | 0.94 | 1.341 | 16.047 | 15.452 | 791 | 0.308 |  $\cdots$ | K4 | 3, 5, 6 | C | F, S, M
11498689 | 19:08:10.5 | +49:24:24.37 | 0.3063 | WU | 0.77 | 1.017 | 15.382 | 14.964 | 1050 | 0.850 |  $\cdots$ | K1 | 5, 18 | C | S
11566174 | 19:41:04.4 | +49:33:58.71 | 0.2769 | WU | 0.77 | 1.005 | 15.425 | 15.022 | 991 | 0.718 |  $\cdots$ | K0.5 | 3, 5 | C | F, S
11671660 | 19:46:10.7 | +49:45:36.77 | 8.7103 | WU | 0.76 | 1.237 | 13.836 | 13.294 | 1732 | 10.755 |  $\cdots$ | K3 | 11 | C | TV
11704155 | 19:07:55.7 | +49:53:01.87 | 0.2739 | WU | 0.94 | 0.995 | 13.048 | 12.639 | 398 | 1.041 | G8 V1 | K0.5 | 3, 5, 19 | C | TV, F, S
11717798 | 19:37:29.2 | +49:48:06.58 | 0.3747 | WU | 0.84 | 0.974 | 14.447 | 14.034 | 908 | 1.496 |  $\cdots$ | K0 | 3, 5, 6 | C | F, S, M
11819135 | 19:37:49.8 | +50:01:48.88 | 1.9016 | Al | 0.57 | 0.693 | 15.253 | 14.988 | 2715 | 5.553 |  $\cdots$ | F8 | 12‡ | C |  $\cdots$
11920266 | 19:39:31.3 | +50:16:00.89 | 0.4674 | WU | 0.91 | 0.698 | 11.789 | 12.083 | 489 | 2.619 | F7 IV1 | F8 | 3, 5 | M | TV, F, S
11924311 | 19:46:39.9 | +50:14:17.96 | 0.4451 | BL | 0.77 | 0.701 | 13.091 | 12.982 | 1047 | 5.239 | F3/F5 V1 | F8 | 6, 18, 19 | C | AM, M
12055517 | 19:11:25.2 | +50:30:57.27 | 0.6098 | BL | 0.77 | 1.250 | 16.409 | 15.866 | 1814 | 1.105 |  $\cdots$ | K3.5 | 2 | C | TV, F
12104285 | 19:10:09.5 | +50:41:48.92 | 0.2515 | WU | 0.87 | 1.323 | 16.440 | 15.845 | 931 | 0.297 |  $\cdots$ | K4 | 3, 5 | C | F, S
12109575 | 19:22:58.9 | +50:39:49.73 | 0.5317 | Al | 0.64 | 1.433 | 14.002 | 13.352 | 260 | 0.231 | K5 Ve1 | K5 | 2, 5 | C | TV, F, S
12164751 | 19:37:33.0 | +50:43:01.53 | 2.6301 | Al | 0.60 | 0.828 | 14.467 | 14.136 | 2743 | 12.426 |  $\cdots$ | G3 | 3, 13 | C | F
12218858 | 19:46:45.5 | +50:51:54.50 | 0.4267 | Al | 0.65 | 0.948 | 13.800 | 13.426 | 506 | 0.814 |  $\cdots$ | G9 | 18 | M | TV
12400729 | 19:09:39.3 | +51:12:36.43 | 0.9317 | Al | 0.52 | 1.211 | 15.883 | 15.355 | 749 | 0.301 |  $\cdots$ | K3 |  $\cdots$ | M | TV
12458797 | 19:22:07.4 | +51:18:30.85 | 0.2413 | WU | 0.94 | 1.268 | 14.434 | 13.876 | 388 | 0.317 |  $\cdots$ | K3.5 | 3 | C | TV, F
12602985 | 19:23:26.5 | +51:40:44.73 | 0.2509 | BL | 0.88 | 1.232 | 15.795 | 15.259 | 953 | 0.533 |  $\cdots$ | K3 | 3 | M | TV, F


# Comparative analysis of Mixed-Data Sampling (MIDAS) model compared to Lag-
Llama model for inflation nowcasting

Adam Bahelka<EMAIL_ADDRESS>Harmen de Weerd
<EMAIL_ADDRESS>
###### Abstract

Inflation is one of the most important economic indicators closely watched by
both public institutions and private agents. This study compares the
performance of a traditional econometric model, Mixed Data Sampling
regression, with one of the newest developments from the field of Artificial
Intelligence, a foundational time series forecasting model based on a Long
short-term memory neural network called Lag-Llama, in their ability to nowcast
the Harmonized Index of Consumer Prices in the Euro area. Two models were
compared and assessed whether the Lag-Llama can outperform the MIDAS
regression, ensuring that the MIDAS regression is evaluated under the best-
case scenario using a dataset spanning from 2010 to 2022. The following
metrics were used to evaluate the models: Mean Absolute Error (MAE), Mean
Absolute Percentage Error (MAPE), Mean Squared Error (MSE), correlation with
the target, R-squared and adjusted R-squared. The results show better
performance of the pre-trained Lag-Llama across all metrics.

###### keywords:

Mixed-Data Sampling (MIDAS), Lag-Llama model, Economic nowcasting, Time series
forecasting, Harmonized Index of Consumer Prices (HICP), Euro area inflation,
Machine learning in econometrics

##  1 Introduction

Inflation nowcasting, the practice of predicting the current rate of
inflation, is crucial for both policymakers and market participants
(Monteforte and Moretti,, 2012). However, a significant challenge in inflation
nowcasting is the issue of mixed-frequency data. Inflation is typically
measured on a monthly basis, but many relevant indicators, such as financial
market data or commodity prices, are available on a daily basis. Traditional
econometric models, such as Autoregressive Integrated Moving Average (ARIMA)
or Vector Autoregression (VAR), are not equipped to handle data of varying
frequencies. A common workaround is to aggregate high-frequency data to match
the frequency of the target variable, but this approach can lead to the loss
of valuable information.

The Mixed Data Sampling (MIDAS) model, developed by Ghysels et al., (2004,
2005, 2007), offers a solution by allowing the use of high-frequency data
without the need for aggregation (Knotek and Zaman,, 2017; Kronenberg et al.,,
2023). This model has been widely adopted by public institutions and private
entities due to its ability to enhance forecasting accuracy by incorporating
real-time data that reflects the latest market conditions or economic
indicators (Knotek and Zaman,, 2017; Kronenberg et al.,, 2023). Recently,
advancements in machine learning have introduced new models for time series
forecasting. One such model is Lag-Llama, a foundational time series
forecasting model based on Long Short-Term Memory (LSTM) neural networks.
Developed by Rasul et al., (2024), Lag-Llama is trained on a large number of
time-series datasets and can perform zero-shot forecasting, making accurate
predictions on datasets it has never been explicitly trained on. However, Lag-
Llama does not support mixed-frequency data, which can limit its effectiveness
in applications where the target variable and predictors are measured at
different frequencies. In this study, we compare the performance of the MIDAS
model and the Lag-Llama model in nowcasting the Harmonized Index of Consumer
Prices (HICP) in the Euro area. Additionally, we compare these models to a
simple benchmark: an autoregressive model that uses one previous value, known
as AR(1). By evaluating these models performance, with the MIDAS regression
under optimal conditions, our study aims to introduce foundational time-series
forecasting models into the field of econometrics.

###  1.1 Problem Description

Inflation is a crucial economic metric for both institutional and private
operators. For institutions, timely economic data and projections are
essential for macroeconomic policy (Monteforte and Moretti,, 2012). Central
banks, for instance, increasingly base their monetary policy on variations of
the Taylor rule (Castro,, 2011), which describes the relationship between
inflation and other economic variables, namely interest rates and the output
gap (Taylor,, 1993). Taylor, (1993) argues that central banks can manage
inflation through changes in interest rates. As for the private agents,
reliable inflation data is important for adjusting investment strategies
(Monteforte and Moretti,, 2012) and for successful price setting (Ang et al.,,
2007). Typically, as inflation is measured on a monthly basis, a common
approach to forecasting inflation is to construct models with monthly
frequency data. However, these models lack very recent information that can be
found in daily data (Monteforte and Moretti,, 2012). Furthermore, data on
inflation is often published with a time lag (Monteforte and Moretti,, 2012).
For example, the U.S. Bureau of Labor Statistics publishes its consumer price
index for a given month only in the middle of the following month111These
dates and times can be found at:
https://www.bls.gov/schedule/news_release/cpi.htm. Its EU counterpart Eurostat
follows a very similar timeline.222Dates for releasing the HICP can be found
here: https://ec.europa.eu/eurostat/news/euro-indicators/release-
calendar?start=1714514400000&type=listMonth&indicator=Inflation Due to this
time lag, there is a space for inflation nowcasting, which involves
forecasting inflation for the near future, present, or recent past.

###  1.2 Background

Over the years, research on improving inflation predictions has advanced.
Numerous studies have examined different models suitable for this task. In
this research, we focus on MIDAS regression and the Lag-Llama LSTM model.

#### MIDAS regression background

Several econometric models are suitable for time series forecasting, such as
Autoregressive Integrated Moving Average (ARIMA) developed by Box and Jenkins,
(1970) and Vector Autoregression (VAR) developed by Sims, (1980). However,
none of these can handle mixed-frequency data. When it comes to nowcasting,
this leads to a loss of valuable information present in the higher frequency
data. A common approach is data aggregation, but this approach results in the
loss of some information (Ghysels et al.,, 2004). Several models in
econometrics can combine mixed-frequency data. We focus on a model known as
Mixed Data Sampling (MIDAS) regression, developed by Ghysels et al., (2004,
2005, 2007), to integrate data of different frequencies. The primary advantage
of MIDAS regression is that it allows for the use of higher-frequency data
such as daily or weekly measurements without the need to aggregate this
information to the frequency of the target variable, typically observed at
lower frequencies like monthly or quarterly intervals.

One of the primary advantages of MIDAS regressions is their ability to enhance
forecasting models by incsorporating real-time data that reflects the latest
market conditions or economic indicators (Ghysels et al.,, 2007). This
inclusion not only improves the timeliness of the forecasts but can also
increase their accuracy, which is crucial for both macroeconomic planning and
financial investment strategies. In practice, MIDAS models have been
implemented by several research institutes and public institutions, including
the Federal Reserve Bank of St. Louis (Knotek and Zaman,, 2017) and the ETH
Nowcasting Lab (Kronenberg et al.,, 2023). They use these models to enhance
the precision of economic forecasts that inform monetary policy decisions. The
ability of MIDAS models to include high-frequency financial indicators, such
as stock prices or interest rate spreads, provides a more nuanced view of
economic trends, allowing policymakers to react more swiftly and appropriately
to economic shifts (Ghysels et al.,, 2007).

In conclusion, the MIDAS regression represents a traditional econometric model
specifically designed to incorporate mixed-frequency data, thereby improving
prediction accuracy in tasks where this is necessary. As economic and
financial data continue to grow in volume and velocity, the relevance and
utility of MIDAS models may increase, given their ability to adjust forecasts
in real-time.

#### Machine Learning Background

In the past three decades, the rapid increase in computational power and
resources in the field of machine learning has constantly developed new ways
of predicting tasks, including time-series predictions. One of the first
significant developments was the introduction of Long Short-term Memory neural
networks (LSTM) by Hochreiter and Schmidhuber, (1997). Over the years,
improvements in architecture, such as the introduction of the attention
mechanism (Bahdanau et al.,, 2016) and transformers (Vaswani et al.,, 2017),
have improved their efficiency and capabilities. Although neural network
models are capable of achieving great results, they are typically trained for
specialized tasks. These specialized models have already been the subject of
studies and comparisons with different econometric models, such as ARIMA or
VAR (Siami-Namini et al.,, 2018; Parmezan et al.,, 2019). These studies have
found that machine learning models can often outperform traditional models
like ARIMA and VAR.

There are a few challenges in the deployment of machine learning models.
First, substantial computational power and data are necessary to train a
machine learning model, resources often unavailable to many institutions and
private entities. In recent years, machine learning has experienced a paradigm
shift due to the development of foundation models (Bommasani et al.,, 2022).
These are large-scale general-purpose networks trained on large datasets and
designed to be adaptable to a wide range of tasks without requiring task-
specific training. This approach gained significant traction following the
introduction of large-scale language models such as ChatGPT models (OpenAI et
al.,, 2024) or the open-source model LLaMA (Touvron et al.,, 2023). The
architectural principles of LLaMA have also been applied in the development of
Lag-Llama, one of the pioneering foundational models for time series
forecasting. Lag-Llama is distinguished by its ability to perform zero-shot
forecasting, enabling it to make accurate predictions on datasets it has never
been explicitly trained on (Rasul et al.,, 2024). This pre-trained model
combines the robustness of specific models with lower resource demands on
computing power or data.

###  1.3 Research Question

The aim of the study is to compare the MIDAS model with a pre-trained Lag-
Llama model in their ability to capture changes in monthly inflation based on
various daily data. The reason for choosing these two models is that the MIDAS
model symbolizes the state-of-the-art model often used by public institutions
and private organizations. Lag-Llama, on the other hand, is the result of
recent developments in the machine learning field and its relative ease of use
has the potential to spread among public institutions and private agents. The
aim of the study is therefore to determine which of these two models works
better in the task of predicting inflation. This gives us the following
research question:

Can a pre-trained neural network Lag-Llama model outperform the MIDAS
regression across all the performance metrics in inflation nowcasting?

##  2 Methods

In this section, we will describe the data used and provide a detailed theory
and implementation of the two previously mentioned models.

###  2.1 Data

####  2.1.1 Data choice and description

For this study, a dataset was created based on previous research on MIDAS
modelling, spanning from June 2010 to June 2022. From the literature,
specifically daily data that are available in real time were chosen. The
advantage of this is that it makes it suitable for nowcasting by being able to
adjust the forecast with the introduction of new data every day. Therefore,
predictions can dynamically change as soon as the new data are published. Our
dependent variable is the Harmonized Index of Consumer Prices (HICP)333HICP
retrieved from https://ec.europa.eu/eurostat/web/hicp/overview with a monthly
frequency. This index captures the measurement of inflation in the Euro area
and ensures the same method is used for calculating inflation in all Eurozone
countries. HICP represents the average change in household spending for
various goods and services (European Central Bank,, 2023).
Our independent variables are daily prices of Brent crude oil444Brent crude
oil prices retrieved from https://fred.stlouisfed.org/series/DCOILBRENTEU.
This represents potential supply shocks in the economy as oil prices have an
effect on inflation (Blanchard,, 2021), and this data was also used by the
Federal Reserve Bank of St. Louis (Knotek and Zaman,, 2017). Furthermore, we
are using daily data on the interest rate spread calculated as the difference
between the price of the 10-year German Bund555Prices of 10-year German Bunds
retrieved from: https://www.investing.com/rates-bonds/germany-10-year-bond-
yield-historical-data and the 3-month German interbank rate66690-day German
interbank rate retrieved from:
https://fred.stlouisfed.org/series/IR3TIB01DEM156N, short-term interest rates
represented by the 3-month Libor777LIBOR retrieved from:
http://iborate.com/eur-libor/ and long-term interest rates represented by the
price of the 10-year German Bund. All these data points are supposed to reveal
the latest market and inflation expectations and were used by (Monteforte and
Moretti,, 2012). Moreover, we are using USD/EUR888USD/EUR exchange rate data
retrieved from: https://www.investing.com/currencies/usd-eur-historical-data
exchange rates as well as CNY/EUR999CNY/EUR exchange rate data retrieved from:
https://www.investing.com/currencies/cny-eur-historical-data exchange rates as
they can capture the changes in import and export prices, as used by
Monteforte and Moretti, (2012). Lastly, the Eurostoxx50101010Eurostoxx50 index
data retrieved from Eurostoxx50 index was chosen, as stock market indices
contain information about inflation expectations (Stock and Watson,, 1999). In
Table 7 in the Appendix, we can see the descriptive statistics of our
variables. The statistics include measures such as the count, mean, standard
deviation, minimum, maximum, and various percentiles.

####  2.1.2 Data pre-processing and post-processing

First, we will focus on our dependent variable. A common way of analyzing a
time series is autocorrelation. Autocorrelation measures the correlation
between a time series’ own past values, indicating how past data points
influence future data points. It helps identify patterns such as trends or
seasonal effects within the data. The blue shaded area in the autocorrelation
plot represents the 5% confidence interval for the autocorrelation
coefficients at each lag. Our autocorrelation results are shown in Figure 1,
which shows significant positive correlation values for the initial 8 lags,
gradually declining towards zero. This indicates that the past three quarters
of a year might have an influence on our forecast.

Figure 1: Autocorrelation of HICP, measuring the correlation coefficients
between the time series and HICP’s past values

A common practice in time series analysis is seasonal decomposition. Seasonal
decomposition is a method used to analyze time series data by breaking it down
into three main components: trend, seasonality, and residual (noise). The
trend component represents the long-term progression of the series, showing
the general direction in which the data is moving. This is done by applying a
centered moving average to the time series with a window of 12 months. This
helps to smooth out short-term fluctuations and highlight longer-term trends.
The seasonality component captures the repeating yearly short-term cycle in
the data. This is done by averaging the data points at corresponding yearly
periods across multiple cycles. Lastly, the residual component represents the
irregular fluctuations that are not explained by the trend or seasonality. The
results of our decomposition are shown in Figure 2. The top plot shows the
original HICP series, highlighting the overall trends and fluctuations from
2010 to 2021. The second plot depicts the trend component, which captures the
long-term movement in the data, showing a general decline until around 2016,
followed by a slight increase and subsequent stabilization. The third plot
illustrates the seasonal component, which represents the regular, repeating
patterns observed within each year. Although there was a weak seasonality
found between $0.05-0.05$ of inflation change, the data from the European
Central Bank are already seasonally adjusted. Finally, the residual component
in the bottom plot shows the irregular fluctuations or noise remaining after
removing both the trend and seasonal effects.

Figure 2: Deseasoning of HICP

Although we found a trend that could represent a form of economic cycles in
the economy, this trend cannot be fitted easily into a simple linear or
polynomial equation. Therefore, after separating the trend from the target
data, another forecasting model would be necessary for the trend of our data,
which likely would not increase the performance of our predictions. For this
reason, we have decided not to de-trend our data as it most likely would not
improve our forecasts.

For the independent variables, the daily values of fractional changes in the
past 7 days were calculated. For example, the calculated data for the variable
$OIL$ is shown in Equation 1. This normalizes the data as different variables
are using different scales.

$\Delta\text{OIL}_{t}=\frac{\text{OIL}_{t}-\text{OIL}_{t-7}}{\text{OIL}_{t-7}}$
(1)

For the Lag-Llama model, there was additional pre-processing of the data. This
consisted of downsampling the target variable to a daily frequency by filling
in the values for the respective month. This was necessary to make it suitable
for the Lag-Llama model as the model doesn’t support mixed frequency input
data.

###  2.2 Multicollinearity and Variance Inflation Factor (VIF)

In the context of econometric modeling, multicollinearity refers to the
scenario where predictor variables are highly correlated, which can undermine
the statistical significance of individual predictors. To assess
multicollinearity in our dataset, we computed the Variance Inflation Factor
(VIF) for each predictor variable. The VIF quantifies how much the variance of
a regression coefficient is inflated due to multicollinearity in the model. A
VIF value above 5 generally indicates high multicollinearity, which could be
problematic for model estimation Kim, (2019).

In our analysis, the VIF values for each predictor converge to values close to
1, except for the USD/EUR exchange rates and CNY/EUR exchange rates with VIF
values of 1.4 and 1.37 respectively. This suggests that multicollinearity is
not expected to be an issue in our dataset. Table 1 presents the VIF values
for all predictors used in our models.

Table 1: VIF Factors for Predictor Variables Features | VIF Factor
---|---
const | 1.053
OIL | 1.104
LIBOR_3M | 1.018
INTEREST_SPREAD | 1.001
USD/EUR exchange rate | 1.400
Eurostoxx50_index | 1.089
CNY/EUR exchange rate | 1.366
German_bund | 1.001

To further understand the relationships between the predictors, a correlation
matrix was computed, as shown in Figure 3. This matrix helps to identify the
degree of the linear relationship between each pair of variables. The
correlation matrix reveals that most predictors have low to moderate
correlations with each other, with the highest correlation being 0.5 between
the USD/EUR exchange rate and the CNY/EUR exchange rate. This further confirms
that multicollinearity is not expected to be an issue in our dataset as
multicollinearity typically starts to be present at correlation coefficients
of around 0.8 among the independent variables (Shrestha,, 2020).

Figure 3: Correlation matrix of independent variables

###  2.3 Theory behind MIDAS

MIDAS is a type of regression that is able to accommodate for predicting the
target sampled with a lower frequency while its features can be sampled at a
higher frequency. First, we will introduce the theory behind the model,
followed by a polynomial specification for the model.

#### Theory

Based on the theory of distributed lag models, Ghysels et al., (2004) has
introduced a slightly different approach in order to accommodate data sampled
at different frequencies. A simple MIDAS regression with one endogenous
variable $y_{t}$ and one exogenous variable $X$ is shown in Equation 2.
Suppose that $y$ is sampled at some fixed frequency, in our case monthly. Now
let $X^{(m)}$ be sampled $m$ times faster. In our case, this would be daily,
which gives us approximately $m=30$. With this notation, the simple MIDAS
regression model would have the following form:

$y_{t}=\beta_{0}+\beta_{1}B(L^{1/m};\theta)X_{t}^{(m)}+\epsilon_{t}^{(m)}$ (2)

where:

$B(L^{1/m};\theta)=\sum_{j=0}^{J}B(j;\theta)L^{j/m}$ (3)

Equation 3 shows a polynomial of length $J$ where $B(j;\theta)$ is the
weighting function, $\theta$ is the vector of hyperparameters defining the
weighting function. This describes how each lagged value of $X^{(m)}_{t}$ is
weighted in the regression. The lagged polynomial notation $(L^{j/m})$ shows
the period of our endogenous variable as shown in Equation 4.

$L^{j/m}X_{t}^{(m)}=X^{(m)}_{t-j/m}$ (4)

To put it in another way, the $L^{j/m}$ operator produces the value of
$X^{(m)}_{t}$ lagged by $j/m$ periods. In our example, this would mean that
the above equation is a prediction of monthly inflation ($y_{t}$) on the basis
of daily data ($X^{(m)}_{t}$) using up to $J$ daily lags (our context
horizon).

Based on this theory, we can reach an immediate conclusion. The MIDAS
regression model is optimized over the vector of hyperparameters in $\theta$
which results in $J$ parameters to estimate for every exogenous variable,
namely $B(L^{1/m};\theta)$. When adding more exogenous variables, we might
face a problem of parameter proliferation, or as it is called in the machine
learning community, the ”curse of dimensionality”. This is the trade-off when
constructing a MIDAS regression. There is a benefit in exploiting data of
higher frequencies compared to aggregating all the data to the same low
frequency. However, when adding too many variables with high frequency, there
is a risk of overfitting the model.

#### Polynomial specification

One of the important parts of the MIDAS architecture is the choice of the
parametrization of $B(j;\theta)$. This is essentially the function that will
describe how different lags will have different weights in our regression. One
popular function is the exponential Almon lag, which originated from the
literature about distributed lag models (Almon,, 1965). Another popular
polynomial is the Beta lag introduced by Ghysels et al., (2005). Figure 4
shows the difference between Almon and Beta polynomials. Ghysels et al.,
(2007) argues that variables with Beta lag have better statistical
significance, which is also why we used them in our research.

Figure 4: Almon and Beta functions with 22 lags

###  2.4 Lag-Llama

Following the new paradigm shift created by foundation models trained on large
datasets (Bommasani et al.,, 2022), Lag-Llama is an attempt to bring this
success to the field of time-series forecasting. Lag-Llama is an LSTM neural
network with a decoder-only transformer-based architecture that was inspired
by Llama (Touvron et al.,, 2023). Rasul et al., (2024) took the well-known
architecture from the Llama model and changed tokenization and prediction in
order to accommodate for time series forecasting. Instead of tokenizing text
into individual words, as is standard practice in LLMs, tokenization
constructs lagged features from the prior values of the time series also with
a time series indices (this can be second, hour, day, month…). Besides the
lagged features, Rasul et al., (2024) added date-time features to provide
additional information such as the second of a minute, hour of the day, and up
to the quarter of the year. This inclusion allows the model to implicitly
understand the frequency of the series and provide additional temporal context
that may influence predictions.

When it comes to the last layer of the architecture that is making
predictions, Lag-Llama utilizes a so-called distribution head, which projects
the features to the parameters of a probability distribution. A Student’s
t-distribution is used (Student,, 1908) in the distribution head that outputs
the degrees of freedom, mean, and scale of the distribution. This is a
difference compared to MIDAS regression that isn’t able to perform
probabilistic forecasting as it produces point estimates based on ordinary
least squares (OLS) method of lagged values.

###  2.5 Implementation

The Python code with implementation of both models can be found on
GitHub.111111Link to the GitHub repository:
https://github.com/AdamBahelka/MIDAS_LagLama_inflation_nowcasting.git

#### Implementation of the MIDAS model

Python was used for the implementation of both models. The implementation of
the MIDAS model was made possible by the work by Yoseph Zuskin121212Class for
MIDAS model can be found at https://github.com/Yoseph-
Zuskin/midaspy?tab=readme-ov-file. For the model, the horizon was chosen at 22
lags for the exogenous variable and 1 lag for the endogenous variable, as the
same values were used used by Ghysels et al., (2007).

#### Implementation of Lag-Llama

The implementation of the Lag-Llama model was made according to Rasul et al.,
(2024). This model can be found on Hugging Face131313You can access the model
on Hugging Face at https://huggingface.co/time-series-foundation-models/Lag-
Llama and on GitHub.141414You can access the model on GitHub at
https://github.com/time-series-foundation-models/lag-llama For the model, the
context length that demonstrates the best performance was 2400 points of daily
data, around seven and a half years. A rolling window was constructed to
predict the next month on the basis of previous inflation data and the chosen
daily features for the above mentioned seven and a half year period.

###  2.6 Performance measures

In order to compare the two models, we have looked at multiple measures. Our
null hypothesis was that there is no difference between the performance of the
two models. In order to answer our research question, we have evaluated
whether the Lag-Llama model can outperform MIDAS regression. As mentioned,
MIDAS is given the best case scenario, evaluating it on in-sample predictions
and on the specialized task of mixed frequency data that Lag-Llama is not
optimized for. We have defined Lag-Llama as outperforming the MIDAS model only
if all the below mentioned performance metrics were lower than 95% (or higher
than 105% where it is appropriate) compared to the MIDAS regression.

#### Mean absolute error

The mean absolute error (MAE) is one of the simplest ways of evaluating a
model. The equation calculates the average of the difference between the
predicted value and the target value. The formula for calculating MAE is:

$\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}\left|y_{i}-\hat{y}_{i}\right|$ (5)

where $y_{i}$ is the true value of inflation and $\hat{y}_{i}$ is the
prediction of our model. In our case, $n=24$ as we are evaluating the model on
two years of data.

#### Mean absolute percentage error

Mean absolute percentage error (MAPE) shows a similar statistic as Equation 5,
but as a percentage difference between the target value and predicted value.
The equation is as follows:

$\text{MAPE}=\frac{100}{n}\sum_{i=1}^{n}\left|\frac{y_{i}-\hat{y}_{i}}{y_{i}}\right|$
(6)

#### Mean squared error

The mean squared error (MSE) is a measure that evaluates the accuracy of the
predictions. The equation is used to calculate the average of the difference
between the predicted value and the target value. Because of its squared term,
this method imposes proportionally greater penalty the further the prediction
is from the target. The formula for calculating MSE is:

$\text{MSE}=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}$ (7)

#### Correlation

Another measure suitable for the evaluation of the quality of predictions is
the correlation. We will use the formula for the Pearson correlation
coefficient between the predicted time-series and the actual time-series. The
formula for calculating the correlation is:

$r=\frac{\sum_{i=1}^{n}(y_{i}-\bar{y})(\hat{y}_{i}-\bar{\hat{y}})}{\sqrt{\sum_{i=1}^{n}(y_{i}-\bar{y})^{2}}\sqrt{\sum_{i=1}^{n}(\hat{y}_{i}-\bar{\hat{y}})^{2}}}$
(8)

where $y_{i}$ represents the actual values of the data, $\hat{y}_{i}$
represents our predicted values, $\bar{y}$ is the mean of the actual values,
and $\bar{\hat{y}}$ is the mean of the predicted values. The numerator of the
formula, $\sum_{i=1}^{n}(y_{i}-\bar{y})(\hat{y}_{i}-\bar{\hat{y}})$, gives the
covariance between the actual and predicted values, which measures how changes
in the actual values are associated with changes in the predicted values. The
denominator normalizes this covariance by the product of the standard
deviations of both the actual and predicted values, ensuring the result is
scaled to a value between $-1$ and 1.

#### Benchmark model

The benchmark model used in this study represents a simple and effective
model, often referred to as the naive model. It relies on the assumption that
the best predictor of the future value of a time series is its most recent
observed value. In the context of this study, the benchmark model is a linear
regression that predicts the monthly inflation rate by using the inflation
rate of the previous month. This model serves as a comparison and is expected
to be outperformed by Lag-Llama and MIDAS regression. Equation 9 shows the
simple structure of our benchmark model.

$\hat{y}_{t}=\beta y_{t-1}$ (9)

where $\hat{y}_{t}$ is the predicted inflation rate for the current month,
$y_{t-1}$ is the observed inflation rate of the previous month, and $\beta$ is
the OLS coefficient of our regression. This benchmark model’s performance was
be evaluated using the same metrics as the other models to provide a clear
comparison.

##  3 Results

In this section, we present the findings of our comparative study between the
MIDAS and Lag-Llama model on the task of nowcasting Eurozone HICP. Initially,
we assess the significance of individual predictors within the MIDAS model, as
detailed in the regression results table, which clarifies how each factor
contributes to the model’s predictive capabilities. Subsequently, we have
compared the predictive accuracy of MIDAS and Lag-Llama, emphasizing their
performance over a critical two-year evaluation period.

###  3.1 Results of the MIDAS Model

In this section, we present the findings from the MIDAS model, focusing on its
predictive accuracy for nowcasting the HICP in the Euro area. The MIDAS
model’s predictions were generated under optimal conditions to ensure a fair
and accurate comparison with the Lag-Llama model.

####  3.1.1 Significance of Predictors

Table 2 shows the estimated coefficients obtained from the regression model.
Each row in the table corresponds to a different predictor variable, with the
following columns providing detailed statistical metrics:

  * •

coef: The name of the predictor variable.

  * •

Estimate: The estimated effect of the predictor on the dependent variable,
holding all other predictors constant.

  * •

std err: The standard error of the estimate, which measures the average amount
that the estimate will differ from the actual value.

  * •

t: The ratio of the estimate to its standard error. Higher absolute values of
the t-value indicate greater statistical significance.

  * •

P$>|t|$: The p-value associated with the hypothesis test that the coefficient
is equal to zero (no effect). A small p-value (typically less than 0.05)
indicates strong evidence against the null hypothesis, thus suggesting a
significant effect.

  * •

[0.025 0.975]: The 95% confidence interval for the coefficient estimate. If
this interval does not include zero, the effect is considered statistically
significant at the 5% level.

Table 2: Regression Results of the MIDAS model | coef | std err | t | P$>|t|$ | 0.025 | 0.975
---|---|---|---|---|---|---
Constant | 0.092 | 0.040 | 2.320 | 0.022 | 0.014 | 0.171
OIL | 0.821 | 0.273 | 3.003 | 0.003 | 0.280 | 1.362
LIBOR_3M | -0.005 | 0.104 | -0.046 | 0.963 | -0.210 | 0.201
INTEREST_SPREAD | 0.108 | 0.050 | 2.190 | 0.030 | 0.010 | 0.206
USD/EUR exchange rate | 2.224 | 1.293 | 1.720 | 0.088 | -0.334 | 4.781
Eurostoxx50_index | -2.087 | 0.916 | -2.279 | 0.024 | -3.899 | -0.275
EUR/CNY exchange rate | -3.621 | 1.809 | -2.002 | 0.047 | -7.200 | -0.042
German_bund | -0.008 | 0.012 | -0.717 | 0.475 | -0.031 | 0.015
HICP t-1 | 0.935 | 0.025 | 37.147 | 0.000 | 0.885 | 0.985

The results indicate significant relationships between several predictors and
the dependent variable. For instance, the coefficient for OIL is positive and
significant, indicating a strong relationship with the dependent variable
($\textit{coef}=0.821$, $P=0.003$). Contrary to previous findings (Monteforte
and Moretti,, 2012), we find no significant effect of short-term interest
rates, represented by 3-month euro LIBOR rates, on inflation
($\textit{coef}=-0.005$, $P=0.963$). The same can be said about the relation
between long-term interest rates, represented by the prices of German Bunds,
and inflation ($\textit{coef}=-0.008$, $P=0.475$). Another significant
variable is the relatively large negative coefficient for Eurostoxx50_index
($\textit{coef}=-2.087$, $P=0.024$) and EUR/CNY exchange rate
($\textit{coef}=-3.621$, $P=0.047$). The interest rate spread
(INTEREST_SPREAD) is significant at the 5% significance level
($\textit{coef}=0.108$, $P=0.030$). The USD/EUR exchange rate shows a positive
yet marginally insignificant relationship ($\textit{coef}=2.224$, $P=0.088$).
As expected, the variable $HICP_{t-1}$, which corresponds to the previous
month’s inflation, shows a strong positive correlation with the dependent
variable ($\textit{coef}=0.935$, $P<0.001$).

####  3.1.2 Evaluation Metrics

The MIDAS model’s predictions are compared against actual HICP values in
Figure 5. The figure highlights the model’s ability to follow the general
trend of inflation but also reveals some deviations from the actual values.
The performance metrics for the MIDAS model over the 24-month evaluation
period and compared to the benchmark model are summarized in Table 3:

  * •

MAE: The Mean Absolute Error for the MIDAS model is 0.23, indicating the
average absolute deviation of predictions from actual values.

  * •

MAPE: The Mean Absolute Percentage Error is 0.50, reflecting the average
percentage error.

  * •

MSE: The Mean Squared Error is 0.11, showing the average of the squared
differences between predictions and actual values.

  * •

Correlation: The Pearson correlation coefficient is 0.77, suggesting a strong
linear relationship between predicted and actual values.

  * •

R-squared: The R-squared value is 0.77, indicating that 77% of the variance in
the dependent variable is explained by the model.

  * •

Adjusted R-squared: The Adjusted R-squared value is 0.72, which adjusts the
R-squared value based on the number of predictors in the model.

Overall, the MIDAS model shows relatively good performance in inflation
nowcasting, with all the metrics except the Correlation coefficient being
lower than 95% (or higher than 105% where appropriate) compared to the metrics
of the Benchmark model (these numbers are highlighted in bold in Table 3).

Table 3: Comparison of MIDAS Regression to the Benchmark Metric | MIDAS | Benchmark
---|---|---
MAE | 0.23 | 0.26
MAPE | 0.50 | 0.54
MSE | 0.11 | 0.13
Correlation | 0.88 | 0.86
R-squared | 0.77 | 0.73
Adjusted R-squared | 0.72 | 0.68
Figure 5: Predictions of the MIDAS regression

###  3.2 Results of the Lag-Llama Model

In this section, the performance of the Lag-Llama model, a neural network
based on an LSTM architecture, in nowcasting the HICP for the Euro area is
discussed. This model was evaluated over the same 24-month period from June
2019 to June 2021.

####  3.2.1 Evaluation Metrics

The Lag-Llama model’s predictions are illustrated in Figure 6. This figure
includes confidence intervals, providing a probabilistic view of the model’s
predictions. The visual comparison with actual HICP values suggests that the
Lag-Llama model captures the overall inflation trend more closely than the
MIDAS model. The performance metrics for the Lag-Llama model compared to the
benchmark model are presented in Table 4

  * •

MAE: The Mean Absolute Error for Lag-Llama is 0.21, showing a smaller average
deviation from actual values than the MIDAS model.

  * •

MAPE: The Mean Absolute Percentage Error is 0.40, lower than that of the MIDAS
model, indicating better percentage accuracy.

  * •

MSE: The Mean Squared Error is 0.07, significantly lower than the MIDAS model,
highlighting its effectiveness in minimizing large errors.

  * •

Correlation: The Pearson correlation coefficient is 0.92, higher than that of
the MIDAS model, indicating a stronger alignment with actual inflation data.

  * •

Adjusted R-squared: The adjusted R-squared value is 0.81, demonstrating the
model’s high explanatory power.

  * •

R-squared: The R-squared value is 0.84, further indicating the model’s
robustness in fitting the data.

The Lag-Llama model benefits from its neural network architecture, which
allows it to process complex patterns in time-series data more effectively.
The inclusion of probabilistic forecasting with confidence intervals also
provides additional insights into the uncertainty of predictions, which is
valuable for economic forecasting. From Table 4, we can see that all the Lag-
Llama metrics are lower than 95%(or higher than 105% where appropriate) from
results of our Benchmark model.

Table 4: Comparison of Lag-Llama to the Benchmark Metric | Lag-Llama | Benchmark
---|---|---
MAE | 0.21 | 0.26
MAPE | 0.40 | 0.54
MSE | 0.07 | 0.13
Correlation | 0.92 | 0.86
R-squared | 0.84 | 0.73
Adjusted R-squared | 0.81 | 0.68
Figure 6: Predictions of the Lag-Llama model

##  4 Discussion

Our research question was whether the pre-trained Lag-Llama model can
outperform MIDAS regression in the task of inflation nowcasting and how both
models compare to a simple benchmark model using previous monthly values of
inflation. Firstly, it is important to stress that all three models have been
evaluated over a 2-year period between June 2019 and June 2021. The
predictions of the MIDAS regression are shown in Figure 5. The predictions of
the Lag-Llama LSTM model are shown in Figure 6, together with the confidence
intervals as this is a probabilistic time-series prediction.

Before starting the comparative analysis, it is important to note that we have
given MIDAS the best-case scenario. As a result, the following choices were
made. First of all, MIDAS was evaluated only on in-sample predictions and not
on out-of-sample predictions, as this approach typically leads to slightly
better prediction performance. Another advantage MIDAS received is that MIDAS
regression is designed for a specialized task, namely accommodating mixed-
frequency data. Lag-Llama, on the other hand, is not designed for mixed-
frequency data inputs and as a result, we decided to down-sample our data in
order for the Lag-Llama model to still capture some of the information in
daily data. The last advantage of MIDAS regression is the inclusion follows
from the choice of data variables. Despite the data choice being based on the
previous literature, two of our variables turned out to be largely
insignificant. This is not a problem for the MIDAS regression as insignificant
variables are omitted from the predictions. However, this is not the case for
Lag-Llama. Inclusion of data with a low level of information will result in
worse predictions for the Lag-Llama model.

The comparative performance of the MIDAS, Lag-Llama, and benchmark models, as
summarized in Table 5, shows overall slightly better performance of Lag-Llama.
Despite evaluating the MIDAS regression under optimal conditions, the Lag-
Llama model still managed to achieve overall better performance metrics,
although the advantage was not uniformly better by the previously defined 5%
margin. One might argue that this definition, where every performance metric
needs to be lower than 95% (or higher than 105% where appropriate) compared to
the MIDAS regression is rather strict. That is indeed the case. However, this
research is on a specific dataset with a specific variable selection,
therefore more research is needed in order to compare the models. The first
performance metric is the MAE. It’s lowest for the Lag-Llama, at 0.21 compared
to 0.23 for MIDAS and 0.26 for the benchmark model, indicating a more accurate
prediction on average. Similarly, the MAPE for Lag-Llama is 0.40, which is
lower than the 0.50 observed for MIDAS and 0.54 for the benchmark model,
suggesting that Lag-Llama offers a tighter approximation of the actual values
in percentage terms. This trend continues with the MSE, where Lag-Llama
achieves a lower score of 0.07 compared to 0.11 by MIDAS and 0.13 by the
benchmark model, highlighting its greater effectiveness in minimizing the
squared differences between predicted and actual values. Furthermore, the
correlation coefficient, measuring the linear correspondence between the
predicted and actual values, is higher for Lag-Llama at 0.92 compared to 0.88
for MIDAS and 0.73 for the benchmark model. This higher correlation
coefficient for Lag-Llama suggests a stronger alignment with the observed data
trends.

Table 5: Comparison of Lag-Llama to MIDAS Regression Metric | Lag-Llama | MIDAS
---|---|---
MAE | 0.21 | 0.23
MAPE | 0.40% | 0.50%
MSE | 0.07 | 0.11
Correlation | 0.92 | 0.88
R-squared | 0.84 | 0.77
Adjusted R-squared | 0.81 | 0.72

However, comparing with Table 2, large values for the USD/EUR exchange rate
($\textit{coef}=2.224$), Eurostoxx50_index ($\textit{coef}=-2.087$), and
EUR/CNY exchange rate ($\textit{coef}=-3.621$) coefficients can be observed.
There is a number of reasons that could explain these relatively large
numbers. Figure 3 shows a slightly higher correlation, 0.49, between USD/EUR
exchange rate and EUR/CNY exchange rate. However, this value is still
acceptable according to the standards in the field and does not suggest
multicollinearity. The problem with multicollinearity typically starts at
correlation levels of around 0.8 among the independent variables (Shrestha,,
2020). Our VIF coefficients in Table 1 tell a similar story, as problems with
multicollinearity typically emerge at values of 5 or more (Kim,, 2019).
Another potential problem might be parameter proliferation. As we have
previously mentioned, we are using 22 lags for each independent variable. With
seven independent variables, however, only four of them turned out to be
significant at the 5% significance level. This means that we needed to
estimate 88 parameters in total. A rule of thumb in machine learning to avoid
the curse of dimensionality (also called parameter proliferation in
econometrics) is at least 10 observations for each parameter. With our daily
data over 11 years, we should have enough parameters to avoid this problem.

Another result of our MIDAS regression is that exogenous variables were
insignificant by a large margin. Both 3-month LIBOR rates ($P=0.475$)
representing short-term interest rates and data on 10-year German Bund
($P=0.963$) turned out to be absolutely insignificant. This is contrary to the
findings of Monteforte and Moretti, (2012) and Ekström and Lundgren, (2018),
although the exact significance of these variables is unknown. The potential
reason for this is that the data on the credit spread already contains similar
information about market expectations. This is, however, not supported by the
correlation coefficients in Figure 3. Other reasons might be that Germany is
not as representative for the Eurozone as was our assumption or that
information about short-term and long-term interest rates is not as crucial
for inflation nowcasting. Another insignificant variable at the 5%
significance level was our USD/EUR exchange rates ($P=0.088$). One reason for
this is that some of the changes in import/export prices were already captured
by EUR/CNY exchange rates. One possible improvement in this case is the
exclusion of the weekends in our dataset. For the variables where the data is
available only for business days, values were filled with linear
interpolation. This method, however, does not add any new information to the
regression. As mentioned previously, while this does not influence the
performance of MIDAS regression, it has a negative impact on the performance
of the Lag-Llama model.

One notable limitation of using LSTM models in practical applications,
particularly within public institutions, is their nature as ”black boxes”
(Rudin,, 2019). The fact that we do not know the reason why an LSTM decides to
behave in a certain way can cause problems, such as cases of people wrongly
denied parole based on a machine-learning model (Wexler,, 2017) or machine-
learning pollution models stating that dangerous situations are safe
(McGough,, 2018). In regression analysis, each coefficient represents the
impact of a specific predictor variable on the dependent variable, allowing
for straightforward interpretation and validation against economic theory or
prior research. Furthermore, significance levels show us which variables have
a significant effect on our dependent variable and which do not. This
transparency is crucial for policymakers and analysts who need to justify and
communicate their modeling choices and results. On the other hand, LSTMs, due
to their complex architecture involving multiple layers and numerous
parameters, do not provide the same level of insight into how individual
inputs influence the output. This lack of interpretability can be a
significant drawback in contexts where understanding the contribution of each
variable is essential for decision-making, policy formulation, or regulatory
compliance. There is a field of explainable artificial intelligence that aims
at tackling this issue. Consequently, despite their superior predictive
performance, the adoption of LSTM models is often limited by the need for
models that are not only accurate but also transparent and interpretable.

Lastly, we would like to point out the use of vocabulary in this study.
Although the fields of econometrics and machine learning have some
similarities, which can also be seen in the small field of computational
econometrics, there is often different vocabulary used for the same
phenomenon. In our study, we are using certain words from these two different
disciplines interchangeably. In order to avoid confusion, Table 6 summarizes
interchangeable words that are used in this paper.

Table 6: Vocabulary Differences between Econometrics and Machine Learning Econometrics | Machine Learning
---|---
Dependent (endogenous) variable | Target
Independent (exogenous) Variable (Predictor) | Feature
Error Term | Residual
Parameter Proliferation | Curse of Dimensionality
Goodness of Fit | Model Evaluation
In-sample Predictions | Training Predictions
Out-of-sample Predictions | Test or Validation Predictions

##  5 Conclusion

The results show that the Lag-Llama model, based on LSTM neural network
techniques, has achieved better performance across almost all chosen metrics,
including Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE),
Mean Squared Error (MSE), correlation coefficient, R-squared, and Adjusted
R-squared, compared to the MIDAS regression. The one metric that did not turn
out to be under 95% compared to the MIDAS results was the correlation
coefficient between predictions and target data. This is shown in Table 5
where metrics that are lower than 95% (or higher than 105% where it is
appropriate) are highlighted in bold. We have previously defined that in order
to conclude that Lag-Llama has better performance, all the metrics need to be
at least 5% better than the MIDAS regression. Therefore, we conclude that the
Lag-Llama model does not outperform the MIDAS regression. Furthermore, under
our definition, the MIDAS regression has not been able to outperform our
Benchmark model either, as correlation coefficients were within the 5% range.
One conclusion we can draw is that the Lag-Llama model was able to outperform
our benchmark model. However, we can see that the integration of advanced
machine learning techniques can potentially improve accuracy and timeliness of
predictions in economic forecasting. The performance of the Lag-Llama model
highlights the potential of using foundation models and machine learning
architectures in economic forecasting, which traditionally relies on
econometric models.

## Acknowledgements

I would like to thank Oksana Kavatsyuk for critical feedback on this paper.
Furthermore, I am grateful for the feedback from students Jonathan Opitz,
Tristan Timpers, Twan Tromp and Nina Kubicová.

## Disclosure statement

No potential conflict of interest was reported by the author(s).

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##  Appendix A Appendix

| count | mean | std | min | 25% | 50% | 75% | max
---|---|---|---|---|---|---|---|---
OIL | 3988.000000 | 76.102565 | 27.087402 | 19.330000 | 53.292500 | 67.825000 | 106.930000 | 126.650000
LIBOR_3M | 2771.000000 | 0.009151 | 0.318214 | -3.714000 | -0.027467 | 0.005762 | 0.043488 | 10.986014
INTEREST_SPREAD | 3988.000000 | 0.731351 | 0.546117 | -0.437364 | 0.323852 | 0.693396 | 1.079581 | 2.222250
USD/EUR exchange rate | 2763.000000 | 0.828077 | 0.073159 | 0.674400 | 0.759200 | 0.843800 | 0.892500 | 0.962700
Eurostoxx50_index | 2534.000000 | 3153.874929 | 440.893087 | 1995.010000 | 2913.420000 | 3220.080000 | 3478.747500 | 4158.140000
CNY/EUR exchange rate | 2528.000000 | 0.126901 | 0.009542 | 0.103800 | 0.120900 | 0.127500 | 0.133000 | 0.152200
German_bund | 2951.000000 | 0.788072 | 0.990065 | -0.854000 | 0.085000 | 0.503000 | 1.544000 | 3.497000
HICP | 131.000000 | 1.251145 | 0.928967 | -0.600000 | 0.450000 | 1.300000 | 2.000000 | 3.000000
Table 7: Statistical Summary of Financial Variables and HICP