2.4 — Goodness of Fit and Bias
ECON 480 • Econometrics • Fall 2022
Dr. Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/metricsF22
metricsF22.classes.ryansafner.com
\[\begin{align*} \color{#0047AB}{Y_i} &= \color{#047806}{\hat{Y}_i} + \color{#D7250E}{\hat{u}_i} \\ \color{#0047AB}{\text{Observed}_i} &= \color{#047806}{\text{Model}_i} + \color{#D7250E}{\text{Error}_i} \\ \end{align*}\]
“All models are wrong. But some are useful. — George Box”
How well does a line fit data? How tightly clustered around the line are the data points?
Quantify how much variation in \(\color{#0047AB}{Y_i}\) is “explained” by the model, \(\color{#047806}{\hat{Y}_i}\)
\[\underbrace{\color{#0047AB}{Y_i}}_{\color{#0047AB}{Observed}}=\underbrace{\color{#047806}{\hat{Y}_i}}_{\color{#047806}{Model}}+\underbrace{\color{#D7250E}{\hat{u}_i}}_{\color{#D7250E}{Error}}\]
\[R^2 = \frac{var(\color{#047806}{\hat{Y}_i})}{var(\color{#0047AB}{Y_i})}\]
\[R^2 = \frac{\color{#047806}{SSM}}{\color{#0047AB}{SST}}\]
\[\color{#047806}{SSM} = \sum^n_{i=1}(\hat{Y_i}-\bar{Y})^2\]
\[\color{#0047AB}{SST} = \sum^n_{i=1}(Y_i-\bar{Y})^2\]
\[R^2=1-\left(\frac{\color{#D7250E}{SSR}}{\color{#0047AB}{SST}}\right)\]
\[R^2=(r_{X,Y})^2\]
\[SST = \sum^n_{i=1}(Y_i-\bar{Y})^2\]
\[\color{#047806}{SSM} = \sum^n_{i=1}(\hat{Y_i}-\bar{Y})^2\]
\[\color{#D7250E}{SSR} = \sum^n_{i=1}(\hat{u_i})^2\]
\[R^2 = \frac{SSM}{SST} = \frac{\color{purple}{C}}{\color{red}{A}+\color{purple}{C}}\]
# make a function to calc. sum of sq. devs
sum_sq <- function(x){sum((x - mean(x))^2)}
# find total sum of squares
SST <- school_reg %>%
augment() %>%
summarize(SST = sum_sq(testscr))
# find explained sum of squares
SSM <- school_reg %>%
augment() %>%
summarize(SSM = sum_sq(.fitted))
# look at them and divide to get R^2
tribble(
~SSM, ~SST, ~R_sq,
SSM, SST, SSM/SST
) %>%
knitr::kable()
SSM | SST | R_sq |
---|---|---|
7794.11 | 152109.6 | 0.0512401 |
\[R^2 = \frac{SSM}{SST} = \frac{\color{purple}{C}}{\color{red}{A}+\color{purple}{C}}=0.05\]
R
Ibroom
’s augment()
command makes a lot of new regression-based values like:
.fitted
: predicted values \((\hat{Y_i})\).resid
: residuals \((\hat{u_i})\)testscr | str | .fitted | .resid | .hat | .sigma | .cooksd | .std.resid |
---|---|---|---|---|---|---|---|
690.80 | 17.88991 | 658.1474 | 32.6526000 | 0.0044244 | 18.53408 | 0.0068925 | 1.7612148 |
661.20 | 21.52466 | 649.8608 | 11.3391671 | 0.0047485 | 18.59490 | 0.0008927 | 0.6117112 |
643.60 | 18.69723 | 656.3069 | -12.7068869 | 0.0029742 | 18.59279 | 0.0006996 | -0.6848850 |
647.70 | 17.35714 | 659.3620 | -11.6619808 | 0.0058575 | 18.59441 | 0.0011673 | -0.6294767 |
640.85 | 18.67133 | 656.3659 | -15.5159250 | 0.0030072 | 18.58766 | 0.0010548 | -0.8363024 |
605.55 | 21.40625 | 650.1308 | -44.5807574 | 0.0044603 | 18.47411 | 0.0129531 | -2.4046387 |
606.75 | 19.50000 | 654.4767 | -47.7266907 | 0.0023941 | 18.45548 | 0.0079356 | -2.5716597 |
609.00 | 20.89412 | 651.2984 | -42.2983704 | 0.0034291 | 18.48716 | 0.0089463 | -2.2803484 |
612.50 | 19.94737 | 653.4568 | -40.9567760 | 0.0024438 | 18.49453 | 0.0059658 | -2.2069310 |
612.65 | 20.80556 | 651.5003 | -38.8502504 | 0.0032862 | 18.50537 | 0.0072306 | -2.0943066 |
615.75 | 21.23809 | 650.5142 | -34.7641689 | 0.0040831 | 18.52485 | 0.0072052 | -1.8747872 |
616.30 | 21.00000 | 651.0570 | -34.7569905 | 0.0036136 | 18.52492 | 0.0063680 | -1.8739584 |
616.30 | 20.60000 | 651.9689 | -35.6689130 | 0.0029950 | 18.52080 | 0.0055515 | -1.9225289 |
616.30 | 20.00822 | 653.3181 | -37.0180659 | 0.0024712 | 18.51448 | 0.0049285 | -1.9947233 |
616.45 | 18.02778 | 657.8331 | -41.3830610 | 0.0041152 | 18.49207 | 0.0102908 | -2.2317715 |
617.35 | 20.25196 | 652.7624 | -35.4123888 | 0.0026303 | 18.52202 | 0.0048022 | -1.9083535 |
618.05 | 16.97787 | 660.2267 | -42.1766170 | 0.0071084 | 18.48740 | 0.0185757 | -2.2779936 |
618.30 | 16.50980 | 661.2938 | -42.9937160 | 0.0089166 | 18.48263 | 0.0243010 | -2.3242432 |
619.80 | 22.70402 | 647.1721 | -27.3721439 | 0.0086398 | 18.55446 | 0.0095387 | -1.4795333 |
620.30 | 19.91111 | 653.5394 | -33.2394468 | 0.0024298 | 18.53171 | 0.0039069 | -1.7910748 |
620.50 | 18.33333 | 657.1365 | -36.6364656 | 0.0035203 | 18.51621 | 0.0068913 | -1.9751997 |
621.40 | 22.61905 | 647.3659 | -25.9658367 | 0.0082974 | 18.55936 | 0.0082379 | -1.4032766 |
621.75 | 19.44828 | 654.5946 | -32.8446103 | 0.0024056 | 18.53340 | 0.0037763 | -1.7697780 |
622.05 | 25.05263 | 641.8178 | -19.7677069 | 0.0219144 | 18.57747 | 0.0129635 | -1.0757207 |
622.60 | 20.67544 | 651.7969 | -29.1969420 | 0.0030953 | 18.54804 | 0.0038451 | -1.5737735 |
623.10 | 18.68235 | 656.3408 | -33.2407957 | 0.0029931 | 18.53166 | 0.0048183 | -1.7916534 |
623.20 | 22.84553 | 646.8495 | -23.6495125 | 0.0092313 | 18.56681 | 0.0076172 | -1.2786973 |
623.45 | 19.26667 | 655.0086 | -31.5586344 | 0.0024741 | 18.53877 | 0.0035862 | -1.7005437 |
623.60 | 19.25000 | 655.0466 | -31.4466672 | 0.0024826 | 18.53923 | 0.0035731 | -1.6945175 |
624.15 | 20.54545 | 652.0933 | -27.9432316 | 0.0029272 | 18.55269 | 0.0033295 | -1.5060690 |
624.55 | 20.60697 | 651.9530 | -27.4029716 | 0.0030039 | 18.55462 | 0.0032865 | -1.4770072 |
624.95 | 21.07268 | 650.8913 | -25.9412664 | 0.0037489 | 18.55965 | 0.0036811 | -1.3987447 |
625.30 | 21.53581 | 649.8354 | -24.5354367 | 0.0047766 | 18.56421 | 0.0042044 | -1.3236257 |
625.85 | 19.90400 | 653.5556 | -27.7056741 | 0.0024273 | 18.55357 | 0.0027114 | -1.4928911 |
626.10 | 21.19407 | 650.6145 | -24.5145628 | 0.0039906 | 18.56430 | 0.0035010 | -1.3219776 |
626.80 | 21.86535 | 649.0841 | -22.2840870 | 0.0056821 | 18.57102 | 0.0041331 | -1.2027182 |
626.90 | 18.32965 | 657.1449 | -30.2448510 | 0.0035267 | 18.54397 | 0.0047052 | -1.6306107 |
627.10 | 16.22857 | 661.9349 | -34.8349462 | 0.0101436 | 18.52405 | 0.0181933 | -1.8843463 |
627.25 | 19.17857 | 655.2095 | -27.9594856 | 0.0025232 | 18.55265 | 0.0028710 | -1.5066399 |
627.30 | 20.27737 | 652.7044 | -25.4044560 | 0.0026515 | 18.56148 | 0.0024914 | -1.3690462 |
628.25 | 22.98614 | 646.5290 | -18.2789658 | 0.0098456 | 18.58147 | 0.0048593 | -0.9886255 |
628.40 | 20.44444 | 652.3235 | -23.9235136 | 0.0028120 | 18.56620 | 0.0023440 | -1.2893420 |
628.55 | 19.82085 | 653.7452 | -25.1951733 | 0.0024027 | 18.56217 | 0.0022195 | -1.3575986 |
628.65 | 23.20522 | 646.0295 | -17.3794680 | 0.0108552 | 18.58354 | 0.0048531 | -0.9404554 |
628.75 | 19.26697 | 655.0080 | -26.2579595 | 0.0024740 | 18.55863 | 0.0024826 | -1.4149156 |
629.80 | 23.30189 | 645.8091 | -16.0090672 | 0.0113210 | 18.58652 | 0.0042987 | -0.8665030 |
630.35 | 21.18829 | 650.6277 | -20.2777471 | 0.0039786 | 18.57661 | 0.0023882 | -1.0934956 |
630.40 | 20.87180 | 651.3493 | -20.9492352 | 0.0033921 | 18.57483 | 0.0021706 | -1.1293737 |
630.55 | 19.01749 | 655.5767 | -25.0267237 | 0.0026397 | 18.56271 | 0.0024071 | -1.3486822 |
630.55 | 21.91938 | 648.9610 | -18.4109799 | 0.0058443 | 18.58124 | 0.0029028 | -0.9937597 |
631.05 | 20.10124 | 653.1060 | -22.0559340 | 0.0025226 | 18.57177 | 0.0017862 | -1.1885175 |
631.40 | 21.47651 | 649.9706 | -18.5706002 | 0.0046291 | 18.58089 | 0.0023335 | -1.0017633 |
631.85 | 20.06579 | 653.1868 | -21.3368264 | 0.0025016 | 18.57379 | 0.0016576 | -1.1497552 |
631.90 | 20.37510 | 652.4816 | -20.5816166 | 0.0027409 | 18.57584 | 0.0016907 | -1.1091931 |
631.95 | 22.44648 | 647.7593 | -15.8092651 | 0.0076317 | 18.58699 | 0.0028050 | -0.8540965 |
632.00 | 22.89524 | 646.7362 | -14.7361970 | 0.0094455 | 18.58910 | 0.0030274 | -0.7968524 |
632.20 | 20.49797 | 652.2015 | -20.0014969 | 0.0028713 | 18.57736 | 0.0016732 | -1.0779995 |
632.25 | 20.00000 | 653.3368 | -21.0867866 | 0.0024672 | 18.57448 | 0.0015966 | -1.1362620 |
632.45 | 22.25658 | 648.1922 | -15.7422648 | 0.0069451 | 18.58714 | 0.0025275 | -0.8501827 |
632.85 | 21.56436 | 649.7703 | -16.9203622 | 0.0048493 | 18.58468 | 0.0020303 | -0.9128447 |
632.95 | 19.47737 | 654.5283 | -21.5782678 | 0.0023987 | 18.57313 | 0.0016253 | -1.1627056 |
633.05 | 17.67002 | 658.6487 | -25.5987041 | 0.0049700 | 18.56074 | 0.0047638 | -1.3811205 |
633.15 | 21.94756 | 648.8967 | -15.7466959 | 0.0059305 | 18.58715 | 0.0021551 | -0.8499879 |
633.65 | 21.78339 | 649.2710 | -15.6209661 | 0.0054434 | 18.58741 | 0.0019447 | -0.8429946 |
633.90 | 19.14000 | 655.2974 | -21.3973987 | 0.0025479 | 18.57362 | 0.0016981 | -1.1530460 |
634.00 | 18.11050 | 657.6445 | -23.6444923 | 0.0039418 | 18.56702 | 0.0032168 | -1.2750268 |
634.05 | 20.68242 | 651.7810 | -17.7309406 | 0.0031050 | 18.58290 | 0.0014225 | -0.9557378 |
634.10 | 22.62361 | 647.3555 | -13.2554886 | 0.0083155 | 18.59181 | 0.0021516 | -0.7163754 |
634.10 | 21.78650 | 649.2639 | -15.1639357 | 0.0054522 | 18.58833 | 0.0018356 | -0.8183344 |
634.15 | 18.58293 | 656.5674 | -22.4174066 | 0.0031267 | 18.57071 | 0.0022899 | -1.2083620 |
634.20 | 21.54545 | 649.8134 | -15.6134965 | 0.0048011 | 18.58744 | 0.0017114 | -0.8423196 |
634.40 | 21.15289 | 650.7084 | -16.3083914 | 0.0039064 | 18.58602 | 0.0015165 | -0.8794127 |
634.55 | 16.63333 | 661.0121 | -26.4621537 | 0.0084110 | 18.55766 | 0.0086750 | -1.4301811 |
634.70 | 21.14438 | 650.7278 | -16.0278017 | 0.0038893 | 18.58660 | 0.0014583 | -0.8642748 |
634.90 | 19.78182 | 653.8342 | -18.9341744 | 0.0023943 | 18.58006 | 0.0012491 | -1.0202312 |
634.95 | 18.98373 | 655.6537 | -20.7037398 | 0.0026685 | 18.57551 | 0.0016654 | -1.1157341 |
635.05 | 17.66767 | 658.6541 | -23.6040700 | 0.0049762 | 18.56711 | 0.0040554 | -1.2735085 |
635.20 | 17.75499 | 658.4550 | -23.2550243 | 0.0047515 | 18.56818 | 0.0037570 | -1.2545348 |
635.45 | 15.27273 | 664.1141 | -28.6640505 | 0.0151024 | 18.54939 | 0.0185256 | -1.5544390 |
635.60 | 14.00000 | 667.0156 | -31.4156607 | 0.0235965 | 18.53797 | 0.0353765 | -1.7110520 |
635.60 | 20.59613 | 651.9777 | -16.3777393 | 0.0029900 | 18.58589 | 0.0011685 | -0.8827463 |
635.75 | 16.31169 | 661.7454 | -25.9954321 | 0.0097700 | 18.55920 | 0.0097510 | -1.4059202 |
635.95 | 21.12796 | 650.7652 | -14.8152370 | 0.0038565 | 18.58903 | 0.0012354 | -0.7988759 |
636.10 | 17.48801 | 659.0636 | -22.9636613 | 0.0054704 | 18.56903 | 0.0042238 | -1.2392643 |
636.50 | 17.88679 | 658.1545 | -21.6544931 | 0.0044317 | 18.57285 | 0.0030364 | -1.1680036 |
636.60 | 19.30676 | 654.9172 | -18.3172679 | 0.0024552 | 18.58154 | 0.0011989 | -0.9870205 |
636.70 | 20.89231 | 651.3025 | -14.6025459 | 0.0034261 | 18.58944 | 0.0010653 | -0.7872370 |
636.90 | 21.28684 | 650.4030 | -13.5030170 | 0.0041886 | 18.59143 | 0.0011153 | -0.7282390 |
636.95 | 20.19560 | 652.8909 | -15.9408471 | 0.0025865 | 18.58681 | 0.0009568 | -0.8590243 |
637.00 | 24.95000 | 642.0517 | -5.0517338 | 0.0211806 | 18.60155 | 0.0008170 | -0.2748026 |
637.10 | 18.13043 | 657.5990 | -20.4990630 | 0.0039014 | 18.57602 | 0.0023929 | -1.1053874 |
637.35 | 20.00000 | 653.3368 | -15.9868110 | 0.0024672 | 18.58671 | 0.0009177 | -0.8614497 |
637.65 | 18.72951 | 656.2333 | -18.5832373 | 0.0029343 | 18.58090 | 0.0014762 | -1.0015927 |
637.95 | 18.25000 | 657.3265 | -19.3764999 | 0.0036702 | 18.57893 | 0.0020103 | -1.0447333 |
637.95 | 18.99257 | 655.6335 | -17.6835240 | 0.0026608 | 18.58301 | 0.0012114 | -0.9529696 |
638.00 | 19.88764 | 653.5929 | -15.5929415 | 0.0024217 | 18.58752 | 0.0008569 | -0.8402069 |
638.20 | 19.37895 | 654.7527 | -16.5526534 | 0.0024265 | 18.58552 | 0.0009675 | -0.8919220 |
638.30 | 20.46259 | 652.2822 | -13.9821316 | 0.0028317 | 18.59059 | 0.0008063 | -0.7535652 |
638.30 | 22.29157 | 648.1124 | -9.8123916 | 0.0070680 | 18.59698 | 0.0009996 | -0.5299645 |
638.35 | 20.70474 | 651.7301 | -13.3801421 | 0.0031363 | 18.59165 | 0.0008183 | -0.7212312 |
638.55 | 19.06005 | 655.4797 | -16.9296980 | 0.0026056 | 18.58470 | 0.0010872 | -0.9123205 |
638.70 | 20.23247 | 652.8068 | -14.1067884 | 0.0026147 | 18.59037 | 0.0007575 | -0.7602008 |
639.25 | 19.69012 | 654.0432 | -14.7932476 | 0.0023826 | 18.58909 | 0.0007587 | -0.7971007 |
639.30 | 20.36254 | 652.5103 | -13.2102177 | 0.0027287 | 18.59195 | 0.0006934 | -0.7119262 |
639.35 | 19.75422 | 653.8971 | -14.5471357 | 0.0023896 | 18.58956 | 0.0007359 | -0.7838423 |
639.50 | 19.37977 | 654.7508 | -15.2508001 | 0.0024263 | 18.58820 | 0.0008212 | -0.8217729 |
639.75 | 22.92351 | 646.6717 | -6.9217452 | 0.0095687 | 18.60012 | 0.0006768 | -0.3743132 |
639.80 | 19.37340 | 654.7653 | -14.9653316 | 0.0024285 | 18.58876 | 0.0007915 | -0.8063916 |
639.85 | 19.15516 | 655.2629 | -15.4128952 | 0.0025380 | 18.58788 | 0.0008776 | -0.8305537 |
639.90 | 21.30000 | 650.3730 | -10.4730131 | 0.0042176 | 18.59613 | 0.0006756 | -0.5648343 |
640.10 | 18.30357 | 657.2043 | -17.1043423 | 0.0035727 | 18.58430 | 0.0015246 | -0.9221791 |
640.15 | 21.07926 | 650.8763 | -10.7262653 | 0.0037615 | 18.59579 | 0.0006315 | -0.5783604 |
640.50 | 18.79121 | 656.0926 | -15.5926000 | 0.0028619 | 18.58751 | 0.0010135 | -0.8403739 |
640.75 | 19.62662 | 654.1880 | -13.4380273 | 0.0023811 | 18.59156 | 0.0006257 | -0.7240772 |
640.90 | 19.59016 | 654.2711 | -13.3711093 | 0.0023826 | 18.59168 | 0.0006199 | -0.7204720 |
641.10 | 20.87187 | 651.3491 | -10.2491101 | 0.0033922 | 18.59644 | 0.0005196 | -0.5525298 |
641.45 | 21.11500 | 650.7948 | -9.3448453 | 0.0038309 | 18.59758 | 0.0004882 | -0.5038918 |
641.45 | 20.08452 | 653.1441 | -11.6941452 | 0.0025125 | 18.59439 | 0.0005001 | -0.6301536 |
641.55 | 19.91049 | 653.5409 | -11.9908687 | 0.0024296 | 18.59394 | 0.0005084 | -0.6461161 |
641.80 | 17.81285 | 658.3231 | -16.5230183 | 0.0046083 | 18.58555 | 0.0018389 | -0.8913003 |
642.20 | 18.13333 | 657.5924 | -15.3924778 | 0.0038956 | 18.58790 | 0.0013471 | -0.8300185 |
642.20 | 19.22221 | 655.1100 | -12.9099823 | 0.0024976 | 18.59246 | 0.0006059 | -0.6956653 |
642.40 | 18.66072 | 656.3901 | -13.9900750 | 0.0030210 | 18.59058 | 0.0008615 | -0.7540649 |
642.75 | 19.60000 | 654.2487 | -11.4987090 | 0.0023820 | 18.59469 | 0.0004583 | -0.6195818 |
643.05 | 19.28384 | 654.9695 | -11.9195015 | 0.0024657 | 18.59405 | 0.0005098 | -0.6422822 |
643.20 | 22.81818 | 646.9119 | -3.7119208 | 0.0091149 | 18.60234 | 0.0001852 | -0.2006868 |
643.25 | 18.80922 | 656.0515 | -12.8015382 | 0.0028417 | 18.59264 | 0.0006783 | -0.6899407 |
643.40 | 21.37363 | 650.2052 | -6.8051436 | 0.0043842 | 18.60024 | 0.0002966 | -0.3670482 |
643.40 | 20.02041 | 653.2903 | -9.8902344 | 0.0024772 | 18.59691 | 0.0003527 | -0.5329382 |
643.50 | 21.49862 | 649.9202 | -6.4202311 | 0.0046835 | 18.60056 | 0.0002822 | -0.3463393 |
643.50 | 15.42857 | 663.7588 | -20.2587667 | 0.0142107 | 18.57638 | 0.0086918 | -1.0981272 |
643.70 | 22.40000 | 647.8652 | -4.1652964 | 0.0074592 | 18.60211 | 0.0001902 | -0.2250108 |
643.70 | 20.12709 | 653.0471 | -9.3470412 | 0.0025389 | 18.59759 | 0.0003229 | -0.5036836 |
644.20 | 19.03798 | 655.5300 | -11.3300673 | 0.0026230 | 18.59494 | 0.0004902 | -0.6105686 |
644.20 | 17.34216 | 659.3962 | -15.1961460 | 0.0059033 | 18.58826 | 0.0019977 | -0.8202586 |
644.40 | 17.01863 | 660.1337 | -15.7337076 | 0.0069648 | 18.58716 | 0.0025321 | -0.8497290 |
644.45 | 20.80000 | 651.5129 | -7.0629295 | 0.0032776 | 18.60001 | 0.0002383 | -0.3807408 |
644.45 | 21.15385 | 650.7062 | -6.2562250 | 0.0039083 | 18.60070 | 0.0002233 | -0.3373606 |
644.50 | 18.45833 | 656.8515 | -12.3514896 | 0.0033128 | 18.59337 | 0.0007368 | -0.6658426 |
644.55 | 19.14082 | 655.2955 | -10.7455612 | 0.0025474 | 18.59577 | 0.0004282 | -0.5790481 |
644.70 | 19.40766 | 654.6872 | -9.9872625 | 0.0024171 | 18.59679 | 0.0003508 | -0.5381504 |
644.95 | 19.56896 | 654.3195 | -9.3694538 | 0.0023844 | 18.59756 | 0.0003046 | -0.5048523 |
645.10 | 21.50120 | 649.9143 | -4.8143634 | 0.0046899 | 18.60173 | 0.0001589 | -0.2597116 |
645.25 | 17.52941 | 658.9693 | -13.7192551 | 0.0053527 | 18.59103 | 0.0014748 | -0.7403339 |
645.55 | 16.43017 | 661.4753 | -15.9253223 | 0.0092534 | 18.58673 | 0.0034625 | -0.8610703 |
645.55 | 19.79654 | 653.8006 | -8.2505891 | 0.0023972 | 18.59883 | 0.0002375 | -0.4445676 |
645.60 | 17.18613 | 659.7519 | -14.1518853 | 0.0063978 | 18.59024 | 0.0018796 | -0.7640815 |
645.75 | 17.61589 | 658.7721 | -13.0220905 | 0.0051142 | 18.59224 | 0.0012689 | -0.7026285 |
645.75 | 20.12537 | 653.0510 | -7.3009626 | 0.0025378 | 18.59979 | 0.0001969 | -0.3934265 |
646.00 | 22.16667 | 648.3972 | -2.3972034 | 0.0066367 | 18.60286 | 0.0000560 | -0.1294442 |
646.20 | 19.96154 | 653.4245 | -7.2244596 | 0.0024497 | 18.59986 | 0.0001861 | -0.3892868 |
646.35 | 19.03945 | 655.5267 | -9.1766772 | 0.0026218 | 18.59779 | 0.0003214 | -0.4945238 |
646.40 | 15.22436 | 664.2243 | -17.8243069 | 0.0153857 | 18.58242 | 0.0073020 | -0.9667435 |
646.50 | 21.14475 | 650.7270 | -4.2269747 | 0.0038900 | 18.60208 | 0.0001014 | -0.2279332 |
646.55 | 19.64390 | 654.1486 | -7.5986387 | 0.0023810 | 18.59950 | 0.0002000 | -0.4094351 |
646.70 | 21.04869 | 650.9460 | -4.2459648 | 0.0037035 | 18.60207 | 0.0000974 | -0.2289358 |
646.90 | 20.17544 | 652.9368 | -6.0367973 | 0.0025718 | 18.60088 | 0.0001364 | -0.3253100 |
646.95 | 21.39130 | 650.1649 | -3.2149290 | 0.0044252 | 18.60256 | 0.0000668 | -0.1734068 |
647.05 | 20.00833 | 653.3178 | -6.2678181 | 0.0024712 | 18.60070 | 0.0001413 | -0.3377422 |
647.25 | 20.29137 | 652.6725 | -5.4225136 | 0.0026635 | 18.60133 | 0.0001140 | -0.2922210 |
647.30 | 17.66667 | 658.6563 | -11.3563529 | 0.0049788 | 18.59488 | 0.0009392 | -0.6127092 |
647.60 | 18.22055 | 657.3936 | -9.7936146 | 0.0037254 | 18.59703 | 0.0005213 | -0.5280623 |
647.60 | 20.27100 | 652.7190 | -5.1189788 | 0.0026461 | 18.60154 | 0.0001010 | -0.2758610 |
648.00 | 20.19895 | 652.8832 | -4.8832278 | 0.0025890 | 18.60169 | 0.0000899 | -0.2631489 |
648.20 | 21.38424 | 650.1810 | -1.9809613 | 0.0044088 | 18.60298 | 0.0000253 | -0.1068482 |
648.25 | 20.97368 | 651.1170 | -2.8669730 | 0.0035663 | 18.60270 | 0.0000428 | -0.1545721 |
648.35 | 20.00000 | 653.3368 | -4.9868110 | 0.0024672 | 18.60163 | 0.0000893 | -0.2687144 |
648.70 | 17.15328 | 659.8268 | -11.1267409 | 0.0065060 | 18.59520 | 0.0011818 | -0.6007822 |
648.95 | 22.34977 | 647.9798 | 0.9701931 | 0.0072760 | 18.60317 | 0.0000101 | 0.0524053 |
649.15 | 22.17007 | 648.3894 | 0.7605829 | 0.0066482 | 18.60320 | 0.0000056 | 0.0410702 |
649.30 | 18.18182 | 657.4819 | -8.1818441 | 0.0037997 | 18.59890 | 0.0003712 | -0.4411736 |
649.50 | 18.95714 | 655.7143 | -6.2142988 | 0.0026923 | 18.60074 | 0.0001514 | -0.3348954 |
649.70 | 19.74533 | 653.9174 | -4.2174323 | 0.0023883 | 18.60208 | 0.0000618 | -0.2272475 |
649.85 | 16.42623 | 661.4843 | -11.6343227 | 0.0092702 | 18.59443 | 0.0018514 | -0.6290645 |
650.45 | 16.62540 | 661.0302 | -10.5802839 | 0.0084429 | 18.59596 | 0.0013921 | -0.5718342 |
650.55 | 16.38177 | 661.5857 | -11.0356758 | 0.0094622 | 18.59531 | 0.0017009 | -0.5967536 |
650.60 | 20.07416 | 653.1677 | -2.5677456 | 0.0025064 | 18.60281 | 0.0000241 | -0.1383658 |
650.65 | 17.99544 | 657.9068 | -7.2567670 | 0.0041854 | 18.59982 | 0.0003219 | -0.3913683 |
650.90 | 19.39130 | 654.7245 | -3.8244723 | 0.0024223 | 18.60229 | 0.0000516 | -0.2060772 |
650.90 | 16.42857 | 661.4790 | -10.5789340 | 0.0092602 | 18.59595 | 0.0015290 | -0.5719970 |
651.15 | 16.72949 | 660.7929 | -9.6429060 | 0.0080316 | 18.59719 | 0.0010991 | -0.5210635 |
651.20 | 24.41345 | 643.2750 | 7.9250504 | 0.0175731 | 18.59911 | 0.0016561 | 0.4303121 |
651.35 | 18.26415 | 657.2942 | -5.9442148 | 0.0036441 | 18.60095 | 0.0001878 | -0.3204933 |
651.40 | 18.95504 | 655.7191 | -4.3190663 | 0.0026942 | 18.60203 | 0.0000732 | -0.2327595 |
651.45 | 21.03896 | 650.9682 | 0.4818585 | 0.0036853 | 18.60322 | 0.0000012 | 0.0259808 |
651.80 | 20.74074 | 651.6480 | 0.1520070 | 0.0031883 | 18.60323 | 0.0000001 | 0.0081939 |
651.85 | 18.10000 | 657.6684 | -5.8184459 | 0.0039633 | 18.60104 | 0.0001959 | -0.3137625 |
651.90 | 19.84615 | 653.6875 | -1.7875032 | 0.0024092 | 18.60303 | 0.0000112 | -0.0963169 |
652.00 | 21.60000 | 649.6891 | 2.3109075 | 0.0049416 | 18.60289 | 0.0000386 | 0.1246780 |
652.10 | 22.44242 | 647.7685 | 4.3314405 | 0.0076165 | 18.60201 | 0.0002101 | 0.2340045 |
652.10 | 23.01438 | 646.4646 | 5.6353877 | 0.0099721 | 18.60117 | 0.0004679 | 0.3048118 |
652.30 | 17.74892 | 658.4688 | -6.1688330 | 0.0047668 | 18.60077 | 0.0002652 | -0.3327915 |
652.30 | 18.28664 | 657.2429 | -4.9428697 | 0.0036031 | 18.60165 | 0.0001284 | -0.2664984 |
652.35 | 19.26544 | 655.0114 | -2.6614627 | 0.0024747 | 18.60278 | 0.0000255 | -0.1434135 |
652.40 | 22.66667 | 647.2573 | 5.1427251 | 0.0084881 | 18.60151 | 0.0003307 | 0.2779560 |
652.40 | 19.29412 | 654.9461 | -2.5460401 | 0.0024609 | 18.60281 | 0.0000232 | -0.1371930 |
652.50 | 17.36364 | 659.3472 | -6.8471910 | 0.0058378 | 18.60019 | 0.0004010 | -0.3695860 |
652.85 | 19.82143 | 653.7439 | -0.8939202 | 0.0024028 | 18.60318 | 0.0000028 | -0.0481674 |
653.10 | 20.43378 | 652.3479 | 0.7521214 | 0.0028007 | 18.60320 | 0.0000023 | 0.0405349 |
653.40 | 21.03721 | 650.9721 | 2.4278745 | 0.0036820 | 18.60285 | 0.0000317 | 0.1309058 |
653.50 | 19.92462 | 653.5086 | -0.0086306 | 0.0024348 | 18.60323 | 0.0000000 | -0.0004651 |
653.55 | 19.00986 | 655.5941 | -2.0441346 | 0.0026461 | 18.60296 | 0.0000161 | -0.1101581 |
653.55 | 23.82222 | 644.6229 | 8.9271951 | 0.0140425 | 18.59802 | 0.0016672 | 0.4838576 |
653.70 | 19.36909 | 654.7752 | -1.0751999 | 0.0024300 | 18.60316 | 0.0000041 | -0.0579361 |
653.80 | 19.82857 | 653.7276 | 0.0723767 | 0.0024046 | 18.60323 | 0.0000000 | 0.0038999 |
653.85 | 15.25885 | 664.1457 | -10.2957130 | 0.0151833 | 18.59629 | 0.0024033 | -0.5583550 |
653.95 | 17.16129 | 659.8085 | -5.8585474 | 0.0064795 | 18.60101 | 0.0003263 | -0.3163248 |
654.10 | 21.81333 | 649.2027 | 4.8972418 | 0.0055295 | 18.60168 | 0.0001942 | 0.2642940 |
654.20 | 19.07471 | 655.4463 | -1.2463129 | 0.0025944 | 18.60313 | 0.0000059 | -0.0671619 |
654.20 | 25.78512 | 640.1478 | 14.0521378 | 0.0275595 | 18.59014 | 0.0083342 | 0.7669068 |
654.30 | 18.21261 | 657.4117 | -3.1116961 | 0.0037404 | 18.60261 | 0.0000528 | -0.1677809 |
654.60 | 18.16606 | 657.5178 | -2.9178351 | 0.0038305 | 18.60268 | 0.0000476 | -0.1573352 |
654.85 | 16.97297 | 660.2378 | -5.3878525 | 0.0071258 | 18.60135 | 0.0003039 | -0.2910049 |
654.85 | 21.50087 | 649.9151 | 4.9348930 | 0.0046891 | 18.60166 | 0.0001669 | 0.2662135 |
654.90 | 20.60000 | 651.9689 | 2.9311237 | 0.0029950 | 18.60268 | 0.0000375 | 0.1579855 |
655.05 | 16.99029 | 660.1983 | -5.1483570 | 0.0070644 | 18.60151 | 0.0002750 | -0.2780608 |
655.05 | 20.77954 | 651.5596 | 3.4904750 | 0.0032463 | 18.60245 | 0.0000577 | 0.1881578 |
655.05 | 15.51247 | 663.5675 | -8.5174540 | 0.0137442 | 18.59849 | 0.0014845 | -0.4615797 |
655.20 | 19.88506 | 653.5988 | 1.6011176 | 0.0024209 | 18.60307 | 0.0000090 | 0.0862743 |
655.30 | 21.39882 | 650.1477 | 5.1523100 | 0.0044428 | 18.60151 | 0.0001723 | 0.2779077 |
655.35 | 20.49751 | 652.2026 | 3.1474228 | 0.0028708 | 18.60259 | 0.0000414 | 0.1696333 |
655.35 | 19.36376 | 654.7873 | 0.5626794 | 0.0024320 | 18.60321 | 0.0000011 | 0.0303195 |
655.40 | 17.65957 | 658.6725 | -3.2724836 | 0.0049975 | 18.60254 | 0.0000783 | -0.1765619 |
655.55 | 21.01796 | 651.0160 | 4.5339452 | 0.0036464 | 18.60190 | 0.0001094 | 0.2444563 |
655.70 | 19.05565 | 655.4897 | 0.2102249 | 0.0026090 | 18.60323 | 0.0000002 | 0.0113288 |
655.80 | 22.53846 | 647.5496 | 8.2504072 | 0.0079816 | 18.59881 | 0.0007995 | 0.4458073 |
655.85 | 21.10787 | 650.8111 | 5.0389248 | 0.0038170 | 18.60159 | 0.0001414 | 0.2717065 |
656.40 | 20.05135 | 653.2197 | 3.1803139 | 0.0024936 | 18.60258 | 0.0000367 | 0.1713736 |
656.50 | 14.20176 | 666.5557 | -10.0556550 | 0.0221058 | 18.59657 | 0.0033851 | -0.5472630 |
656.55 | 18.47687 | 656.8092 | -0.2591875 | 0.0032838 | 18.60323 | 0.0000003 | -0.0139720 |
656.65 | 18.63542 | 656.4478 | 0.2022480 | 0.0030545 | 18.60323 | 0.0000002 | 0.0109013 |
656.70 | 20.94595 | 651.1802 | 5.5198006 | 0.0035175 | 18.60126 | 0.0001563 | 0.2975913 |
656.80 | 21.08548 | 650.8621 | 5.9379523 | 0.0037735 | 18.60095 | 0.0001941 | 0.3201764 |
656.80 | 18.69288 | 656.3168 | 0.4832807 | 0.0029797 | 18.60322 | 0.0000010 | 0.0260483 |
657.00 | 20.86808 | 651.3577 | 5.6422654 | 0.0033860 | 18.60117 | 0.0001572 | 0.3041737 |
657.00 | 19.82558 | 653.7344 | 3.2655619 | 0.0024038 | 18.60254 | 0.0000373 | 0.1759593 |
657.15 | 19.75000 | 653.9067 | 3.2432858 | 0.0023890 | 18.60255 | 0.0000366 | 0.1747577 |
657.40 | 19.50000 | 654.4767 | 2.9233337 | 0.0023941 | 18.60268 | 0.0000298 | 0.1575181 |
657.50 | 18.39080 | 657.0054 | 0.4945557 | 0.0034223 | 18.60322 | 0.0000012 | 0.0266619 |
657.55 | 18.78676 | 656.1027 | 1.4473171 | 0.0028669 | 18.60310 | 0.0000087 | 0.0780043 |
657.65 | 19.77018 | 653.8607 | 3.7892917 | 0.0023922 | 18.60231 | 0.0000500 | 0.2041784 |
657.75 | 19.33333 | 654.8567 | 2.8933427 | 0.0024438 | 18.60269 | 0.0000298 | 0.1559060 |
657.80 | 21.46392 | 649.9993 | 7.8006508 | 0.0045983 | 18.59929 | 0.0004090 | 0.4207880 |
657.90 | 23.08492 | 646.3038 | 11.5962704 | 0.0102929 | 18.59447 | 0.0020464 | 0.6273309 |
658.00 | 21.06299 | 650.9134 | 7.0866316 | 0.0037305 | 18.59998 | 0.0002734 | 0.3821053 |
658.35 | 18.68687 | 656.3305 | 2.0195013 | 0.0029873 | 18.60297 | 0.0000178 | 0.1088493 |
658.60 | 20.77024 | 651.5808 | 7.0191773 | 0.0032322 | 18.60005 | 0.0002321 | 0.3783736 |
658.80 | 19.30556 | 654.9200 | 3.8800005 | 0.0024557 | 18.60226 | 0.0000538 | 0.2090727 |
659.05 | 20.13280 | 653.0340 | 6.0160275 | 0.0025426 | 18.60089 | 0.0001340 | 0.3241860 |
659.15 | 20.66964 | 651.8101 | 7.3398790 | 0.0030873 | 18.59975 | 0.0002424 | 0.3956326 |
659.35 | 22.28155 | 648.1353 | 11.2146930 | 0.0070326 | 18.59507 | 0.0012991 | 0.6056916 |
659.40 | 20.60027 | 651.9683 | 7.4317411 | 0.0029953 | 18.59966 | 0.0002410 | 0.4005656 |
659.40 | 20.82734 | 651.4506 | 7.9494125 | 0.0033204 | 18.59915 | 0.0003059 | 0.4285376 |
659.80 | 19.22492 | 655.1038 | 4.6961767 | 0.0024961 | 18.60181 | 0.0000801 | 0.2530573 |
659.90 | 17.65477 | 658.6835 | 1.2165628 | 0.0050102 | 18.60314 | 0.0000108 | 0.0656382 |
660.05 | 17.00000 | 660.1762 | -0.1261626 | 0.0070301 | 18.60323 | 0.0000002 | -0.0068139 |
660.10 | 16.49773 | 661.3213 | -1.2213189 | 0.0089672 | 18.60314 | 0.0000197 | -0.0660263 |
660.20 | 19.78261 | 653.8324 | 6.3675526 | 0.0023944 | 18.60061 | 0.0001413 | 0.3431032 |
660.30 | 22.30216 | 648.0883 | 12.2116809 | 0.0071055 | 18.59355 | 0.0015566 | 0.6595619 |
660.75 | 17.73077 | 658.5102 | 2.2398043 | 0.0048128 | 18.60291 | 0.0000353 | 0.1208341 |
660.95 | 20.44836 | 652.3146 | 8.6353404 | 0.0028162 | 18.59841 | 0.0003059 | 0.4653970 |
661.35 | 20.37169 | 652.4894 | 8.8605813 | 0.0027376 | 18.59816 | 0.0003130 | 0.4775174 |
661.45 | 20.16479 | 652.9611 | 8.4889091 | 0.0025643 | 18.59858 | 0.0002690 | 0.4574473 |
661.60 | 21.61538 | 649.6540 | 11.9459398 | 0.0049820 | 18.59399 | 0.0010400 | 0.6445202 |
661.60 | 20.56143 | 652.0568 | 9.5431374 | 0.0029466 | 18.59735 | 0.0003909 | 0.5143558 |
661.85 | 19.95551 | 653.4382 | 8.4117628 | 0.0024472 | 18.59866 | 0.0002520 | 0.4532635 |
661.85 | 21.18387 | 650.6378 | 11.2121864 | 0.0039695 | 18.59510 | 0.0007285 | 0.6046244 |
661.85 | 18.81042 | 656.0488 | 5.8011813 | 0.0028404 | 18.60106 | 0.0001392 | 0.3126553 |
661.90 | 20.57838 | 652.0182 | 9.8818390 | 0.0029676 | 18.59692 | 0.0004222 | 0.5326168 |
661.90 | 18.32461 | 657.1564 | 4.7436649 | 0.0035355 | 18.60178 | 0.0001160 | 0.2557495 |
661.95 | 18.82063 | 656.0255 | 5.9244818 | 0.0028291 | 18.60096 | 0.0001446 | 0.3192988 |
662.40 | 20.81633 | 651.4757 | 10.9243049 | 0.0033030 | 18.59551 | 0.0005747 | 0.5889032 |
662.40 | 20.00000 | 653.3368 | 9.0632378 | 0.0024672 | 18.59793 | 0.0002949 | 0.4883728 |
662.45 | 19.68182 | 654.0622 | 8.3878317 | 0.0023821 | 18.59869 | 0.0002439 | 0.4519592 |
662.50 | 19.39018 | 654.7271 | 7.7729334 | 0.0024227 | 18.59933 | 0.0002130 | 0.4188354 |
662.55 | 20.92732 | 651.2227 | 11.3273708 | 0.0034853 | 18.59493 | 0.0006522 | 0.6106874 |
662.55 | 19.94437 | 653.4636 | 9.0864284 | 0.0024426 | 18.59790 | 0.0002935 | 0.4896164 |
662.65 | 20.79109 | 651.5332 | 11.1167801 | 0.0032639 | 18.59524 | 0.0005880 | 0.5992673 |
662.70 | 19.20354 | 655.1526 | 7.5474470 | 0.0025082 | 18.59955 | 0.0002080 | 0.4067027 |
662.75 | 19.02439 | 655.5610 | 7.1890123 | 0.0026340 | 18.59989 | 0.0001982 | 0.3874125 |
662.90 | 17.62058 | 658.7614 | 4.1386136 | 0.0051016 | 18.60212 | 0.0001278 | 0.2233043 |
663.35 | 20.23715 | 652.7961 | 10.5538547 | 0.0026184 | 18.59603 | 0.0004246 | 0.5687378 |
663.45 | 19.29374 | 654.9469 | 8.5030256 | 0.0024611 | 18.59856 | 0.0002590 | 0.4581843 |
663.50 | 18.82998 | 656.0042 | 7.4957941 | 0.0028190 | 18.59960 | 0.0002307 | 0.4039823 |
663.85 | 20.33949 | 652.5628 | 11.2871675 | 0.0027068 | 18.59500 | 0.0005021 | 0.6082824 |
663.85 | 19.22900 | 655.0945 | 8.7554483 | 0.0024938 | 18.59828 | 0.0002782 | 0.4717938 |
663.90 | 17.89130 | 658.1442 | 5.7558152 | 0.0044211 | 18.60109 | 0.0002140 | 0.3104565 |
664.00 | 19.51881 | 654.4338 | 9.5661931 | 0.0023908 | 18.59732 | 0.0003184 | 0.5154548 |
664.00 | 19.08451 | 655.4239 | 8.5760649 | 0.0025870 | 18.59848 | 0.0002770 | 0.4621492 |
664.15 | 19.93548 | 653.4839 | 10.6661536 | 0.0024390 | 18.59588 | 0.0004038 | 0.5747378 |
664.15 | 18.87326 | 655.9055 | 8.2444791 | 0.0027734 | 18.59884 | 0.0002745 | 0.4443222 |
664.30 | 20.14178 | 653.0135 | 11.2864431 | 0.0025486 | 18.59500 | 0.0004726 | 0.6081951 |
664.40 | 23.55637 | 645.2289 | 19.1710835 | 0.0126069 | 18.57923 | 0.0068826 | 1.0383250 |
664.45 | 21.46479 | 649.9973 | 14.4526014 | 0.0046004 | 18.58970 | 0.0014045 | 0.7796128 |
664.70 | 19.19101 | 655.1811 | 9.5188868 | 0.0025156 | 18.59738 | 0.0003318 | 0.5129379 |
664.75 | 20.13080 | 653.0386 | 11.7114129 | 0.0025413 | 18.59437 | 0.0005074 | 0.6310932 |
664.95 | 25.80000 | 640.1139 | 24.8360509 | 0.0276816 | 18.56230 | 0.0261561 | 1.3555326 |
664.95 | 18.77774 | 656.1233 | 8.8267082 | 0.0028772 | 18.59820 | 0.0003265 | 0.4757252 |
665.10 | 19.10982 | 655.3662 | 9.7337392 | 0.0025687 | 18.59711 | 0.0003543 | 0.5245295 |
665.20 | 19.70109 | 654.0183 | 11.1817591 | 0.0023834 | 18.59515 | 0.0004336 | 0.6025041 |
665.35 | 18.61594 | 656.4922 | 8.8578021 | 0.0030809 | 18.59816 | 0.0003522 | 0.4774498 |
665.65 | 20.99721 | 651.0633 | 14.5866888 | 0.0036085 | 18.58946 | 0.0011200 | 0.7864541 |
665.90 | 20.00000 | 653.3368 | 12.5632378 | 0.0024672 | 18.59303 | 0.0005667 | 0.6769704 |
665.95 | 20.98325 | 651.0952 | 14.8548507 | 0.0035834 | 18.58895 | 0.0011534 | 0.8009022 |
666.00 | 21.64262 | 649.5919 | 16.4080810 | 0.0050542 | 18.58578 | 0.0019907 | 0.8852986 |
666.05 | 20.02967 | 653.2691 | 12.7809101 | 0.0024820 | 18.59268 | 0.0005901 | 0.6887048 |
666.10 | 19.81140 | 653.7668 | 12.3332116 | 0.0024004 | 18.59340 | 0.0005313 | 0.6645532 |
666.15 | 18.00000 | 657.8964 | 8.2536213 | 0.0041755 | 18.59882 | 0.0004154 | 0.4451279 |
666.15 | 19.35811 | 654.8002 | 11.3498483 | 0.0024341 | 18.59491 | 0.0004563 | 0.6115767 |
666.45 | 20.17912 | 652.9284 | 13.5215262 | 0.0025745 | 18.59142 | 0.0006852 | 0.7286469 |
666.55 | 21.11986 | 650.7837 | 15.7663364 | 0.0038405 | 18.58714 | 0.0013932 | 0.8501548 |
666.60 | 23.38974 | 645.6088 | 20.9911376 | 0.0117551 | 18.57447 | 0.0076807 | 1.1364108 |
666.65 | 22.18182 | 648.3627 | 18.2873646 | 0.0066879 | 18.58152 | 0.0032829 | 0.9875065 |
666.65 | 19.94283 | 653.4671 | 13.1828949 | 0.0024419 | 18.59200 | 0.0006176 | 0.7103516 |
666.70 | 17.78826 | 658.3791 | 8.3208116 | 0.0046686 | 18.59875 | 0.0004725 | 0.4488627 |
666.85 | 14.70588 | 665.4064 | 1.4436151 | 0.0186186 | 18.60310 | 0.0000583 | 0.0784267 |
666.85 | 19.04077 | 655.5237 | 11.3263232 | 0.0026207 | 18.59494 | 0.0004895 | 0.6103662 |
667.15 | 20.89195 | 651.3033 | 15.8467229 | 0.0034255 | 18.58699 | 0.0012543 | 0.8543115 |
667.20 | 19.83851 | 653.7050 | 13.4949908 | 0.0024071 | 18.59146 | 0.0006379 | 0.7271560 |
667.45 | 19.52191 | 654.4267 | 13.0232758 | 0.0023903 | 18.59227 | 0.0005899 | 0.7017325 |
667.45 | 20.68622 | 651.7723 | 15.6776717 | 0.0031103 | 18.58734 | 0.0011140 | 0.8450642 |
667.60 | 18.18182 | 657.4819 | 10.1180826 | 0.0037997 | 18.59661 | 0.0005677 | 0.5455776 |
668.00 | 18.89224 | 655.8623 | 12.1377385 | 0.0027542 | 18.59371 | 0.0005909 | 0.6541365 |
668.10 | 24.88889 | 642.1911 | 25.9089194 | 0.0207503 | 18.55900 | 0.0210363 | 1.4090755 |
668.40 | 18.58064 | 656.5726 | 11.8273796 | 0.0031299 | 18.59419 | 0.0006381 | 0.6375305 |
668.60 | 18.04000 | 657.8052 | 10.7947668 | 0.0040890 | 18.59569 | 0.0006957 | 0.5821497 |
668.65 | 17.73399 | 658.5029 | 10.1471688 | 0.0048046 | 18.59656 | 0.0007234 | 0.5474222 |
668.80 | 21.45455 | 650.0207 | 18.7792872 | 0.0045756 | 18.58038 | 0.0023584 | 1.0129934 |
668.90 | 19.92343 | 653.5114 | 15.3886630 | 0.0024344 | 18.58793 | 0.0008390 | 0.8292048 |
668.95 | 20.33942 | 652.5630 | 16.3870389 | 0.0027068 | 18.58587 | 0.0010584 | 0.8831220 |
669.10 | 22.54608 | 647.5322 | 21.5677538 | 0.0080111 | 18.57299 | 0.0054843 | 1.1654219 |
669.30 | 21.10344 | 650.8211 | 18.4789011 | 0.0038083 | 18.58113 | 0.0018977 | 0.9964061 |
669.30 | 18.19743 | 657.4463 | 11.8537387 | 0.0037695 | 18.59414 | 0.0007729 | 0.6391564 |
669.35 | 20.10768 | 653.0913 | 16.2586817 | 0.0025265 | 18.58614 | 0.0009721 | 0.8761255 |
669.35 | 19.15984 | 655.2522 | 14.0977757 | 0.0025350 | 18.59039 | 0.0007334 | 0.7596848 |
669.80 | 19.54545 | 654.3731 | 15.4269235 | 0.0023870 | 18.58785 | 0.0008266 | 0.8312467 |
669.85 | 20.88889 | 651.3103 | 18.5396862 | 0.0034204 | 18.58099 | 0.0017143 | 0.9994891 |
669.95 | 18.39150 | 657.0039 | 12.9461594 | 0.0034211 | 18.59239 | 0.0008361 | 0.6979379 |
670.00 | 19.17990 | 655.2065 | 14.7935495 | 0.0025224 | 18.58909 | 0.0008035 | 0.7971728 |
670.70 | 19.39771 | 654.7099 | 15.9900520 | 0.0024202 | 18.58671 | 0.0009005 | 0.8616041 |
671.25 | 21.67827 | 649.5106 | 21.7393524 | 0.0051503 | 18.57259 | 0.0035616 | 1.1730041 |
671.30 | 19.28889 | 654.9580 | 16.3420043 | 0.0024634 | 18.58597 | 0.0009574 | 0.8805876 |
671.60 | 20.34927 | 652.5405 | 19.0594660 | 0.0027160 | 18.57974 | 0.0014366 | 1.0271479 |
671.60 | 20.96416 | 651.1387 | 20.4612911 | 0.0035495 | 18.57613 | 0.0021675 | 1.1031558 |
671.65 | 19.46039 | 654.5670 | 17.0830307 | 0.0024026 | 18.58437 | 0.0010203 | 0.9204897 |
671.70 | 19.28572 | 654.9652 | 16.7347320 | 0.0024649 | 18.58513 | 0.0010046 | 0.9017504 |
671.75 | 20.91979 | 651.2398 | 20.5101676 | 0.0034724 | 18.57601 | 0.0021302 | 1.1057482 |
671.90 | 20.90021 | 651.2845 | 20.6155514 | 0.0034393 | 18.57573 | 0.0021315 | 1.1114112 |
671.90 | 20.59575 | 651.9786 | 19.9214224 | 0.0029895 | 18.57756 | 0.0017285 | 1.0737475 |
671.95 | 19.37500 | 654.7617 | 17.1882844 | 0.0024279 | 18.58414 | 0.0010439 | 0.9261729 |
672.05 | 19.95122 | 653.4480 | 18.6019998 | 0.0024454 | 18.58086 | 0.0012315 | 1.0023582 |
672.05 | 18.84973 | 655.9592 | 16.0908748 | 0.0027979 | 18.58649 | 0.0010550 | 0.8672009 |
672.30 | 18.11787 | 657.6277 | 14.6723064 | 0.0039268 | 18.58930 | 0.0012339 | 0.7911967 |
672.35 | 19.18341 | 655.1985 | 17.1515175 | 0.0025202 | 18.58422 | 0.0010791 | 0.9242345 |
672.45 | 22.00000 | 648.7772 | 23.6728422 | 0.0060937 | 18.56686 | 0.0050064 | 1.2779368 |
672.55 | 21.58416 | 649.7252 | 22.8248386 | 0.0049004 | 18.56946 | 0.0037338 | 1.2314197 |
672.70 | 20.38889 | 652.4502 | 20.2497577 | 0.0027545 | 18.57671 | 0.0016448 | 1.0913159 |
673.05 | 16.29310 | 661.7878 | 11.2621850 | 0.0098528 | 18.59498 | 0.0018460 | 0.6091222 |
673.25 | 18.27778 | 657.2631 | 15.9868875 | 0.0036192 | 18.58669 | 0.0013493 | 0.8619517 |
673.30 | 19.37472 | 654.7623 | 18.5376731 | 0.0024280 | 18.58102 | 0.0012143 | 0.9988834 |
673.55 | 18.90909 | 655.8239 | 17.7261356 | 0.0027376 | 18.58291 | 0.0012526 | 0.9553027 |
673.55 | 16.40693 | 661.5283 | 12.0216795 | 0.0093532 | 18.59383 | 0.0019948 | 0.6500360 |
673.90 | 15.59140 | 663.3876 | 10.5124688 | 0.0133138 | 18.59601 | 0.0021887 | 0.5695697 |
674.25 | 18.70694 | 656.2847 | 17.9652882 | 0.0029620 | 18.58236 | 0.0013927 | 0.9683002 |
675.40 | 18.32985 | 657.1444 | 18.2556230 | 0.0035263 | 18.58166 | 0.0017140 | 0.9842273 |
675.70 | 17.90235 | 658.1190 | 17.5809889 | 0.0043954 | 18.58321 | 0.0019849 | 0.9482690 |
676.15 | 18.91157 | 655.8182 | 20.3318165 | 0.0027352 | 18.57650 | 0.0016465 | 1.0957276 |
676.55 | 20.32497 | 652.5959 | 23.9540624 | 0.0026934 | 18.56611 | 0.0022503 | 1.2909116 |
676.60 | 20.02457 | 653.2808 | 23.3192006 | 0.0024794 | 18.56806 | 0.0019623 | 1.2565634 |
676.85 | 24.00000 | 644.2176 | 32.6324221 | 0.0150551 | 18.53342 | 0.0239327 | 1.7695996 |
676.95 | 17.60784 | 658.7904 | 18.1595061 | 0.0051360 | 18.58186 | 0.0024782 | 0.9798370 |
677.25 | 19.34853 | 654.8220 | 22.4279950 | 0.0024378 | 18.57070 | 0.0017845 | 1.2085152 |
677.95 | 19.67846 | 654.0698 | 23.8801872 | 0.0023819 | 18.56635 | 0.0019765 | 1.2867295 |
678.05 | 18.72861 | 656.2353 | 21.8147260 | 0.0029354 | 18.57245 | 0.0020349 | 1.1757630 |
678.40 | 15.88235 | 662.7242 | 15.6757917 | 0.0117990 | 18.58720 | 0.0042998 | 0.8486693 |
678.80 | 20.05491 | 653.2116 | 25.5883913 | 0.0024955 | 18.56088 | 0.0023782 | 1.3788507 |
679.40 | 17.98825 | 657.9232 | 21.4768395 | 0.0042012 | 18.57335 | 0.0028301 | 1.1582872 |
679.50 | 16.96629 | 660.2531 | 19.2469438 | 0.0071496 | 18.57917 | 0.0038911 | 1.0395648 |
679.65 | 19.23937 | 655.0709 | 24.5791567 | 0.0024882 | 18.56416 | 0.0021879 | 1.3244624 |
679.75 | 19.19586 | 655.1701 | 24.5799238 | 0.0025127 | 18.56415 | 0.0022097 | 1.3245200 |
679.80 | 19.59906 | 654.2508 | 25.5492004 | 0.0023821 | 18.56101 | 0.0022627 | 1.3766606 |
680.05 | 20.54348 | 652.0978 | 27.9522879 | 0.0029248 | 18.55266 | 0.0033289 | 1.5065553 |
680.45 | 18.58848 | 656.5548 | 23.8952393 | 0.0031189 | 18.56628 | 0.0025952 | 1.2880163 |
681.30 | 15.60419 | 663.3584 | 17.9415882 | 0.0132448 | 18.58220 | 0.0063414 | 0.9720483 |
681.30 | 15.29304 | 664.0678 | 17.2322354 | 0.0149843 | 18.58379 | 0.0066415 | 0.9344406 |
681.60 | 17.65537 | 658.6821 | 22.9178750 | 0.0050087 | 18.56918 | 0.0038483 | 1.2365064 |
681.90 | 17.57976 | 658.8545 | 23.0455580 | 0.0052126 | 18.56879 | 0.0040514 | 1.2435229 |
682.15 | 22.33333 | 648.0172 | 34.1327919 | 0.0072168 | 18.52744 | 0.0123542 | 1.8436407 |
682.45 | 18.75000 | 656.1865 | 26.2634653 | 0.0029097 | 18.55859 | 0.0029236 | 1.4155214 |
682.55 | 18.10241 | 657.6629 | 24.8870583 | 0.0039584 | 18.56311 | 0.0035789 | 1.3420432 |
682.65 | 20.25641 | 652.7522 | 29.8978048 | 0.0026340 | 18.54538 | 0.0034278 | 1.6111786 |
683.35 | 18.80207 | 656.0678 | 27.2821484 | 0.0028496 | 18.55506 | 0.0030893 | 1.4703811 |
683.40 | 18.77230 | 656.1357 | 27.2643275 | 0.0028835 | 18.55512 | 0.0031221 | 1.4694456 |
684.30 | 20.40521 | 652.4130 | 31.8870645 | 0.0027710 | 18.53740 | 0.0041031 | 1.7184969 |
684.35 | 18.65079 | 656.4127 | 27.9372559 | 0.0030340 | 18.55271 | 0.0034503 | 1.5058276 |
684.80 | 20.70707 | 651.7248 | 33.0752493 | 0.0031397 | 18.53237 | 0.0050056 | 1.7828616 |
684.95 | 22.00000 | 648.7772 | 36.1728422 | 0.0060937 | 18.51819 | 0.0116893 | 1.9527273 |
686.05 | 17.69978 | 658.5809 | 27.4691918 | 0.0048924 | 18.55430 | 0.0053989 | 1.4819807 |
686.70 | 21.48329 | 649.9552 | 36.7447808 | 0.0046457 | 18.51560 | 0.0091689 | 1.9821590 |
687.55 | 16.70103 | 660.8578 | 26.6921837 | 0.0081426 | 18.55688 | 0.0085402 | 1.4424181 |
689.10 | 19.57567 | 654.3042 | 34.7958028 | 0.0023837 | 18.52485 | 0.0041997 | 1.8748943 |
691.05 | 17.25806 | 659.5879 | 31.4621134 | 0.0061658 | 18.53893 | 0.0089489 | 1.6984884 |
691.35 | 17.37526 | 659.3207 | 32.0292749 | 0.0058026 | 18.53661 | 0.0087217 | 1.7287909 |
691.90 | 17.34931 | 659.3798 | 32.5201770 | 0.0058814 | 18.53454 | 0.0091147 | 1.7553571 |
693.95 | 16.26229 | 661.8581 | 32.0919525 | 0.0099910 | 18.53606 | 0.0152039 | 1.7358342 |
694.25 | 17.70045 | 658.5793 | 35.6706910 | 0.0048906 | 18.52064 | 0.0091008 | 1.9244552 |
694.80 | 20.12881 | 653.0431 | 41.7569351 | 0.0025400 | 18.49022 | 0.0064467 | 2.2501556 |
695.20 | 18.26539 | 657.2914 | 37.9085959 | 0.0036418 | 18.51004 | 0.0076347 | 2.0439093 |
695.30 | 14.54214 | 665.7797 | 29.5203181 | 0.0197142 | 18.54585 | 0.0258910 | 1.6046353 |
696.55 | 19.15261 | 655.2687 | 41.2813773 | 0.0025396 | 18.49279 | 0.0062997 | 2.2245288 |
698.20 | 17.36574 | 659.3424 | 38.8575607 | 0.0058314 | 18.50509 | 0.0129014 | 2.0973803 |
698.25 | 15.13898 | 664.4190 | 33.8310304 | 0.0158934 | 18.52812 | 0.0272017 | 1.8353793 |
698.45 | 17.84266 | 658.2551 | 40.1948450 | 0.0045362 | 18.49833 | 0.0107106 | 2.1681499 |
699.10 | 15.40704 | 663.8079 | 35.2921243 | 0.0143320 | 18.52161 | 0.0266093 | 1.9131285 |
700.30 | 18.86534 | 655.9236 | 44.3764533 | 0.0027816 | 18.47552 | 0.0079772 | 2.3916032 |
704.30 | 16.47413 | 661.3751 | 42.9249647 | 0.0090664 | 18.48300 | 0.0246377 | 2.3207019 |
706.75 | 17.86263 | 658.2096 | 48.5404085 | 0.0044886 | 18.45005 | 0.0154546 | 2.6182552 |
645.00 | 21.88586 | 649.0374 | -4.0373950 | 0.0057432 | 18.60218 | 0.0001371 | -0.2179132 |
672.20 | 20.20000 | 652.8808 | 19.3191890 | 0.0025898 | 18.57910 | 0.0014071 | 1.0410789 |
655.75 | 19.03640 | 655.5336 | 0.2163941 | 0.0026242 | 18.60323 | 0.0000002 | 0.0116613 |
R
IIR
as the ratio of variances in model vs. actual# as ratio of variances
school_reg %>%
augment() %>%
summarize(r_sq = var(.fitted)/var(testscr)) # var. of *predicted* testscr over var. of *actual* testscr
r_sq |
---|
0.0512401 |
\[R^2 = \frac{var(\hat{Y})}{var(Y)} = \frac{\color{red}{\frac{1}{n-1}}\sum^n_{i=1}(\hat{Y_i}-\bar{Y})^2}{\color{red}{\frac{1}{n-1}}\sum^n_{i=1}(Y_i-\bar{Y})^2} \rightarrow \frac{SSM}{SST}\]
\[\hat{\sigma_u}=\sqrt{\frac{SSR}{n-2}}=\sqrt{\frac{\sum \hat{u}_i^2}{n-2}}\]
R
SSR | df | SER |
---|---|---|
144315.5 | 418 | 18.58097 |
Call:
lm(formula = testscr ~ str, data = ca_school)
Residuals:
Min 1Q Median 3Q Max
-47.727 -14.251 0.483 12.822 48.540
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 698.9330 9.4675 73.825 < 2e-16 ***
str -2.2798 0.4798 -4.751 2.78e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 18.58 on 418 degrees of freedom
Multiple R-squared: 0.05124, Adjusted R-squared: 0.04897
F-statistic: 22.58 on 1 and 418 DF, p-value: 2.783e-06
summary()
command in Base R
gives:
Multiple R-squared
Residual standard error
(SER)
r.squared | adj.r.squared | sigma | statistic | p.value | df | logLik | AIC | BIC | deviance | df.residual | nobs |
---|---|---|---|---|---|---|---|---|---|---|---|
0.0512401 | 0.0489703 | 18.58097 | 22.57511 | 2.8e-06 | 1 | -1822.25 | 3650.499 | 3662.62 | 144315.5 | 418 | 420 |
r.squared
is 0.05
\(\implies\) about 5% of variation in testscr
is explained by our modelsigma
(SER) is 18.6
\(\implies\) average test score is about 18.6 points above/below our model’s prediction\[\color{orange}{Y}=\color{teal}{\beta}(\color{purple}{X})\]
where \(\color{orange}{Y}\) is numeric:
\[Y_i = \beta_0+\beta_1 X_i+u_i\]
OLS estimators \((\hat{\beta_0}\) and \(\hat{\beta_1})\) are computed from a finite (specific) sample of data
Our OLS model contains 2 sources of randomness:
Inferential statistics analyzes a sample to make inferences about a much larger (unobservable) population
Population: all possible individuals that match some well-defined criterion of interest
Sample: some portion of the population of interest to represent the whole
Roughly, but probably not exactly
Sampling variability describes the effect of a statistic varying somewhat from sample to sample
If we collect many samples, and each sample is randomly drawn from the population (and then replaced), then the distribution of samples is said to be independently and identically distributed (i.i.d.)
Each sample is independent of each other sample (due to replacement)
Each sample comes from the identical underlying population distribution
Calculating OLS estimators for a sample makes the OLS estimators themselves random variables:
Draw of \(i\) is random \(\implies\) value of each \((X_i,Y_i)\) is random \(\implies\) \(\hat{\beta_0},\hat{\beta_1}\) are random
Taking different samples will create different values of \(\hat{\beta_0},\hat{\beta_1}\)
Therefore, \(\hat{\beta_0},\hat{\beta_1}\) each have a sampling distribution across different samples
If neither of these are true, we have other methods (coming shortly!)
One of the most fundamental principles in all of statistics
Allows for virtually all testing of statistical hypotheses \(\rightarrow\) estimating probabilities of values on a normal distribution
The CLT allows us to approximate the sampling distributions of \(\hat{\beta_0}\) and \(\hat{\beta_1}\) as normal
We care about \(\hat{\beta_1}\) (slope) since it has economic meaning, rarely about \(\hat{\beta_0}\) (intercept)
\[\hat{\beta_1} \sim N(\mathbb{E}[\hat{\beta_1}], \sigma_{\hat{\beta_1}})\]
\[\hat{\beta_1} \sim N(\mathbb{E}[\hat{\beta_1}], \sigma_{\hat{\beta_1}})\]
\(\mathbb{E}[\hat{\beta_1}]\); what is the center of the distribution? (today)
\(\sigma_{\hat{\beta_1}}\); how precise is our estimate? (next class)
In order to talk about \(\mathbb{E}[\hat{\beta_1}]\), we need to talk about population \(u\)
Recall: \(u\) is a random variable, and we can never measure the error term
\[\mathbb{E}[u]=0\]
\[\mathbb{E}[u]=0\]
\[var(u|X)=\sigma^2_{u}\]
\[\mathbb{E}[u]=0\]
\[var(u|X)=\sigma^2_{u}\]
\[cor(u_i,u_j)=0 \quad \forall i \neq j\]
\[\mathbb{E}[u]=0\]
\[var(u|X)=\sigma^2_{u}\]
\[cor(u_i,u_j)=0 \quad \forall i \neq j\]
\[cor(X, u)=0 \text{ or } E[u|X]=0\]
\[\mathbb{E}[u]=0\]
\[var(u|X)=\sigma^2_{u}\]
\[var(u|X)=\sigma^2_{u}\]
Assumption 2 implies that errors are “homoskedastic”: they have the same variance across \(X\)
Often this assumption is violated: errors may be “heteroskedastic”: they do not have the same variance across \(X\)
This is a problem for inference, but we have a simple fix for this (next class)
\[cor(u_i,u_j)=0 \quad \forall i \neq j\]
For simple cross-sectional data, this is rarely an issue
Time-series & panel data nearly always contain serial correlation or autocorrelation between errors
e.g. “this week’s sales look a lot like last week’s sales, which look like…etc”
There are fixes to deal with autocorrelation (coming much later)
\[cor(X, u)=0\]
This is the absolute killer assumption, because it assumes exogeneity
Often called the Zero Conditional Mean assumption:
\[E[u|X]=0\]
“Does knowing \(X_i\) give any useful information about \(u_i\)?”
\[E[\hat{\beta_1}]=\beta_1\]
Does not mean any sample gives us \(\hat{\beta_1}=\beta_1\), only the estimation procedure will, on average, yield the correct value
Random errors above and below the true value cancel out (so that on average, \(E[\hat{u}|X]=0)\)
Calculate the mean height of our sample \((\bar{H})\) to estimate the true mean height of the population \((\mu_H)\)
\(\bar{H}\) is an estimator of \(\mu_H\)
Unbiasedness: does the estimator give us the true parameter on average?
Efficiency: an estimator with a smaller variance is better
\(\mathbf{\hat{\beta_1}}\) is the Best Linear Unbiased Estimator (BLUE) estimator of \(\mathbf{\beta_1}\) when \(X\) is exogenous1
No systematic difference, on average, between sample values of \(\hat{\beta_1}\) and the true population \(\beta_1\):
\[E[\hat{\beta_1}]=\beta_1\]
\[cor(X,u)=0\]
\[E(u|X)=0\]
For any known value of \(X\), the expected value of \(u\) is 0
Knowing the value of \(X\) must tell us nothing about the value of \(u\) (anything else relevant to \(Y\) other than \(X\))
We can then confidently assert causation: \(X \rightarrow Y\)
\[cor(X,u)\neq 0\]
We find \(\hat{\beta_1}>0\)
Does this mean Ice cream sales \(\rightarrow\) Violent crimes?
\[E[\hat{\beta_1}]=\beta_1+cor(X,u)\frac{\sigma_u}{\sigma_X}\]
Is this an accurate reflection of \(education \rightarrow wages\)?
Does \(E[u|education]=0\)?
What would \(E[u|education]>0\) mean?
Is this an accurate reflection of \(taxes \rightarrow consumption\)?
Does \(E[u|tax]=0\)?
What would \(E[u|tax]>0\) mean?