Chapter 6 Modeling Issues
6.1 Import Data
taxprep=read.table("TXTData/TaxPrep.txt", sep ="\t", quote = "",header=TRUE)
#taxprep=read.table(choose.files(), header=TRUE, sep="\t")Data for this study are from the Statistics of Income (SOI) Panel of Individual Returns, a part of the Ernst and Young/University of Michigan Tax Research Database. The SOI Panel represents a simple random sample of unaudited individual income tax returns filed for tax years 1979-1990. The data are compiled from a stratified probability sample of unaudited individual income tax returns, Forms 1040, 1040A and 1040EZ, filed by U.S. taxpayers. The estimates that are obtained from these data are intended to represent all returns filed for the income tax years under review. All returns processed are subjected to sampling except tentative and amended returns.
| Variable | Description |
|---|---|
| MS | is an indicator variable of the taxpayer’s marital status. It is coded one if the taxpayer is married and zero otherwise. |
| HH | is an indicator variable, one if the taxpayer is a head of household and zero otherwise. |
| DEPEND | is the number of dependents claimed by the taxpayer. |
| AGE | is the presence of an indicator for age 65 or over. |
| F1040A | is an indicator variable of the taxpayer’s filing type. It is coded one if the taxpayer uses Form 1040A and zero otherwise. |
| F1040EZ | is an indicator variable of the taxpayer’s filing type. It is coded one if the taxpayer uses Form 1040EZ and zero otherwise. |
| TPI | is the sum of all positive income line items on the return. |
| TXRT | is a marginal tax rate. It is computed on TPI less exemptions and the standard deduction. |
| MR | is an exogenous marginal tax rate. It is computed on TPI less exemptions and the standard deduction. |
| EMP | is an indicator variable, one if Schedule C or F is present and zero otherwise. Self-employed taxpayers have greater need for professional assistance to reduce the reporting risks of doing business. |
| PREP | is a variable indicating the presence of a paid preparer. |
| TAX | is the tax liability on the return. |
| SUBJECT | Subject identifier, 1-258. |
| TIME | Time identifier, 1-5. |
| LNTAX | is the natural logarithm of the tax liability on the return. |
| LNTPI | is the natural logarithm of the sum of all positive income line items on the return. |
6.2 Example 7.2: Income Tax Payments (Page 248)
To illustrate the performance of the fixed-effects estimators and omitted-variable tests, we examine data on determinants of income tax payments introduced in Section 3.2. Specifically, we begin with the error-components model with K = 8 coefficients estimated using generalized least squares.
6.2.1 TABLE 7.1: Fixed effects estimators
taxprep$YEAR<-taxprep$TIME+1981
taxprep$SUBFACTOR<-factor(taxprep$SUBJECT)
library(nlme)
taxprepfx<-lm(LNTAX~MS+HH+AGE+EMP+PREP+LNTPI+DEPEND+MR+SUBFACTOR-1, data=taxprep)
summary(taxprepfx)
Call:
lm(formula = LNTAX ~ MS + HH + AGE + EMP + PREP + LNTPI + DEPEND +
MR + SUBFACTOR - 1, data = taxprep)
Residuals:
Min 1Q Median 3Q Max
-7.4350 -0.3315 -0.0078 0.4586 6.9348
Coefficients:
Estimate Std. Error t value Pr(>|t|)
MS 0.072328 0.255221 0.283 0.776933
HH -0.706799 0.326079 -2.168 0.030421 *
AGE 0.001840 0.322918 0.006 0.995456
EMP -0.244247 0.247434 -0.987 0.323817
PREP -0.029685 0.163207 -0.182 0.855707
LNTPI 0.716755 0.077101 9.296 < 2e-16 ***
DEPEND -0.069021 0.082707 -0.835 0.404184
MR 0.121920 0.008998 13.550 < 2e-16 ***
SUBFACTOR1 -1.941454 0.912856 -2.127 0.033676 *
SUBFACTOR2 -2.076922 0.921470 -2.254 0.024412 *
SUBFACTOR3 -3.762761 0.867812 -4.336 1.59e-05 ***
SUBFACTOR4 -2.390221 0.936929 -2.551 0.010882 *
SUBFACTOR5 -2.383235 0.913485 -2.609 0.009214 **
SUBFACTOR6 -3.442848 0.972091 -3.542 0.000415 ***
SUBFACTOR7 -2.396985 1.026946 -2.334 0.019784 *
SUBFACTOR8 -3.901584 0.984147 -3.964 7.87e-05 ***
SUBFACTOR9 -1.792381 0.935780 -1.915 0.055721 .
SUBFACTOR10 -1.733623 0.887827 -1.953 0.051132 .
SUBFACTOR11 -2.175789 0.896572 -2.427 0.015405 *
SUBFACTOR12 -2.884418 0.692702 -4.164 3.39e-05 ***
SUBFACTOR13 -2.124878 0.974428 -2.181 0.029437 *
SUBFACTOR14 -2.489158 0.970216 -2.566 0.010442 *
SUBFACTOR15 -0.886070 0.950740 -0.932 0.351566
SUBFACTOR16 -1.903056 0.902355 -2.109 0.035188 *
SUBFACTOR17 -3.103433 0.948772 -3.271 0.001107 **
SUBFACTOR18 -7.007031 0.976968 -7.172 1.41e-12 ***
SUBFACTOR19 -2.441594 0.948031 -2.575 0.010151 *
SUBFACTOR20 -3.898509 1.028651 -3.790 0.000159 ***
SUBFACTOR21 -3.325560 0.930155 -3.575 0.000366 ***
SUBFACTOR22 -2.071372 0.891475 -2.324 0.020346 *
SUBFACTOR23 -2.350709 0.935508 -2.513 0.012132 *
SUBFACTOR24 -2.066505 0.900165 -2.296 0.021895 *
SUBFACTOR25 -5.681510 0.909637 -6.246 6.17e-10 ***
SUBFACTOR26 -4.114085 0.998612 -4.120 4.10e-05 ***
SUBFACTOR27 -1.895995 0.914310 -2.074 0.038358 *
SUBFACTOR28 -6.776403 0.929486 -7.290 6.17e-13 ***
SUBFACTOR29 -1.979414 0.892364 -2.218 0.026762 *
SUBFACTOR30 -2.253438 0.877853 -2.567 0.010400 *
SUBFACTOR31 -3.109170 0.922367 -3.371 0.000777 ***
SUBFACTOR32 -1.644017 0.934853 -1.759 0.078947 .
SUBFACTOR33 -3.595152 0.880644 -4.082 4.80e-05 ***
SUBFACTOR34 -1.282029 0.868010 -1.477 0.139990
SUBFACTOR35 -1.981843 0.981292 -2.020 0.043682 *
SUBFACTOR36 -3.176758 0.956270 -3.322 0.000925 ***
SUBFACTOR37 -2.881841 0.941031 -3.062 0.002253 **
SUBFACTOR38 -2.037214 0.912517 -2.233 0.025796 *
SUBFACTOR39 -2.490816 0.963816 -2.584 0.009894 **
SUBFACTOR40 -2.021895 0.898985 -2.249 0.024719 *
SUBFACTOR41 -2.514656 0.903545 -2.783 0.005483 **
SUBFACTOR42 -3.547532 0.995653 -3.563 0.000384 ***
SUBFACTOR43 -1.665460 0.942714 -1.767 0.077582 .
SUBFACTOR44 -1.652095 0.914844 -1.806 0.071231 .
SUBFACTOR45 -3.561106 0.950161 -3.748 0.000188 ***
SUBFACTOR46 -2.990858 0.952594 -3.140 0.001740 **
SUBFACTOR47 -2.324781 0.961738 -2.417 0.015811 *
SUBFACTOR48 -2.006750 0.754964 -2.658 0.007981 **
SUBFACTOR49 -2.597448 0.926920 -2.802 0.005171 **
SUBFACTOR50 -3.654927 1.016935 -3.594 0.000341 ***
SUBFACTOR51 -2.202546 0.897783 -2.453 0.014320 *
SUBFACTOR52 -2.796828 0.928213 -3.013 0.002649 **
SUBFACTOR53 -2.152217 0.956570 -2.250 0.024665 *
SUBFACTOR54 -2.381863 0.905095 -2.632 0.008626 **
SUBFACTOR55 -1.922384 0.913789 -2.104 0.035644 *
SUBFACTOR56 -1.156258 0.937277 -1.234 0.217622
SUBFACTOR57 -3.639612 0.953761 -3.816 0.000144 ***
SUBFACTOR58 -1.941540 0.915812 -2.120 0.034244 *
SUBFACTOR59 -1.269146 0.931013 -1.363 0.173123
SUBFACTOR60 -3.086963 0.894974 -3.449 0.000585 ***
SUBFACTOR61 -2.158203 0.896433 -2.408 0.016236 *
SUBFACTOR62 -2.767490 0.906454 -3.053 0.002323 **
SUBFACTOR63 -3.067190 0.941421 -3.258 0.001159 **
SUBFACTOR64 -3.209717 0.915086 -3.508 0.000472 ***
SUBFACTOR65 -3.936287 0.949585 -4.145 3.67e-05 ***
SUBFACTOR66 -1.657242 0.893698 -1.854 0.063974 .
SUBFACTOR67 -3.618607 0.975472 -3.710 0.000219 ***
SUBFACTOR68 -3.442074 1.006949 -3.418 0.000655 ***
SUBFACTOR69 -1.863437 0.892102 -2.089 0.036971 *
SUBFACTOR70 -2.025643 0.962731 -2.104 0.035617 *
SUBFACTOR71 -2.070916 0.909826 -2.276 0.023042 *
SUBFACTOR72 -3.560836 0.933093 -3.816 0.000144 ***
SUBFACTOR73 -1.956272 0.883031 -2.215 0.026952 *
SUBFACTOR74 -2.511433 0.942049 -2.666 0.007799 **
SUBFACTOR75 -1.548801 0.915574 -1.692 0.091023 .
SUBFACTOR76 -1.811015 0.925309 -1.957 0.050595 .
SUBFACTOR77 -1.621423 0.904550 -1.793 0.073345 .
SUBFACTOR78 -1.673650 0.905861 -1.848 0.064951 .
SUBFACTOR79 -5.856583 0.899390 -6.512 1.16e-10 ***
SUBFACTOR80 -3.704689 0.898899 -4.121 4.07e-05 ***
SUBFACTOR81 -3.322793 0.931023 -3.569 0.000375 ***
SUBFACTOR82 -1.864121 0.957077 -1.948 0.051721 .
SUBFACTOR83 -5.491182 0.961071 -5.714 1.45e-08 ***
SUBFACTOR84 -2.609013 0.941254 -2.772 0.005675 **
SUBFACTOR85 -5.323047 0.879880 -6.050 2.03e-09 ***
SUBFACTOR86 -2.829677 0.949784 -2.979 0.002957 **
SUBFACTOR87 -3.703492 0.964595 -3.839 0.000131 ***
SUBFACTOR88 -4.818659 1.016989 -4.738 2.46e-06 ***
SUBFACTOR89 -3.394560 0.930317 -3.649 0.000277 ***
SUBFACTOR90 -1.532264 0.896465 -1.709 0.087711 .
SUBFACTOR91 -1.801299 0.882717 -2.041 0.041544 *
SUBFACTOR92 -8.219328 0.888945 -9.246 < 2e-16 ***
SUBFACTOR93 -2.407979 0.912390 -2.639 0.008436 **
SUBFACTOR94 -2.845610 1.017056 -2.798 0.005240 **
SUBFACTOR95 -2.031485 0.958790 -2.119 0.034348 *
SUBFACTOR96 -2.702229 0.952599 -2.837 0.004648 **
SUBFACTOR97 -5.384899 0.905033 -5.950 3.68e-09 ***
SUBFACTOR98 -2.131225 0.924700 -2.305 0.021379 *
SUBFACTOR99 -2.625805 0.947271 -2.772 0.005673 **
SUBFACTOR100 -2.172483 0.972282 -2.234 0.025671 *
SUBFACTOR101 -2.890329 0.983665 -2.938 0.003374 **
SUBFACTOR102 -3.918986 0.870792 -4.500 7.56e-06 ***
SUBFACTOR103 -1.823848 0.910516 -2.003 0.045431 *
SUBFACTOR104 -2.140979 0.879129 -2.435 0.015048 *
SUBFACTOR105 -2.452705 0.902278 -2.718 0.006672 **
SUBFACTOR106 -2.018929 0.899036 -2.246 0.024938 *
SUBFACTOR107 -3.278959 0.904614 -3.625 0.000304 ***
SUBFACTOR108 -3.951069 0.876380 -4.508 7.29e-06 ***
SUBFACTOR109 -2.577744 0.932250 -2.765 0.005793 **
SUBFACTOR110 -3.002542 0.934017 -3.215 0.001347 **
SUBFACTOR111 -1.118914 0.953960 -1.173 0.241103
SUBFACTOR112 -2.769722 0.939232 -2.949 0.003261 **
SUBFACTOR113 -2.308694 0.913965 -2.526 0.011686 *
SUBFACTOR114 -2.596360 0.928304 -2.797 0.005256 **
SUBFACTOR115 -2.524912 0.957479 -2.637 0.008490 **
SUBFACTOR116 -1.070564 0.964510 -1.110 0.267279
SUBFACTOR117 -2.981548 0.914100 -3.262 0.001144 **
SUBFACTOR118 -2.898291 0.895760 -3.236 0.001253 **
SUBFACTOR119 -1.678321 0.927011 -1.810 0.070517 .
SUBFACTOR120 -3.646692 0.991089 -3.679 0.000246 ***
SUBFACTOR121 -2.360121 0.948188 -2.489 0.012965 *
SUBFACTOR122 -4.301704 0.961525 -4.474 8.54e-06 ***
SUBFACTOR123 -2.321742 0.936552 -2.479 0.013334 *
SUBFACTOR124 -1.885206 0.912533 -2.066 0.039089 *
SUBFACTOR125 -2.760263 0.956213 -2.887 0.003975 **
SUBFACTOR126 -4.599824 0.910184 -5.054 5.13e-07 ***
SUBFACTOR127 -1.495260 0.994569 -1.503 0.133038
SUBFACTOR128 -1.587560 0.943411 -1.683 0.092721 .
SUBFACTOR129 -2.249726 0.941817 -2.389 0.017088 *
SUBFACTOR130 -2.513272 0.931073 -2.699 0.007062 **
SUBFACTOR131 -2.914927 0.902195 -3.231 0.001273 **
SUBFACTOR132 -1.912501 0.895668 -2.135 0.032975 *
SUBFACTOR133 -2.844954 0.883279 -3.221 0.001318 **
SUBFACTOR134 -2.486082 0.961257 -2.586 0.009839 **
SUBFACTOR135 -1.782512 0.945921 -1.884 0.059791 .
SUBFACTOR136 -3.321478 0.959246 -3.463 0.000557 ***
SUBFACTOR137 -1.364910 0.949280 -1.438 0.150786
SUBFACTOR138 -2.180505 0.975733 -2.235 0.025650 *
SUBFACTOR139 -6.851310 0.964615 -7.103 2.29e-12 ***
SUBFACTOR140 -3.264175 0.961476 -3.395 0.000713 ***
SUBFACTOR141 -3.277959 0.925814 -3.541 0.000417 ***
SUBFACTOR142 -2.047689 0.878153 -2.332 0.019904 *
SUBFACTOR143 -3.311763 0.999429 -3.314 0.000953 ***
SUBFACTOR144 -3.224253 0.882052 -3.655 0.000270 ***
SUBFACTOR145 -1.602488 0.945951 -1.694 0.090560 .
SUBFACTOR146 -3.433803 0.919533 -3.734 0.000199 ***
SUBFACTOR147 -1.962344 0.917847 -2.138 0.032754 *
SUBFACTOR148 -5.720274 0.846794 -6.755 2.39e-11 ***
SUBFACTOR149 -2.394029 0.935963 -2.558 0.010676 *
SUBFACTOR150 -2.313197 0.913255 -2.533 0.011460 *
SUBFACTOR151 -2.661345 1.004407 -2.650 0.008181 **
SUBFACTOR152 -2.874865 0.874572 -3.287 0.001046 **
SUBFACTOR153 -2.324181 0.902537 -2.575 0.010159 *
SUBFACTOR154 -2.125162 0.914003 -2.325 0.020261 *
SUBFACTOR155 -3.781776 0.951065 -3.976 7.49e-05 ***
SUBFACTOR156 -3.755601 0.944757 -3.975 7.53e-05 ***
SUBFACTOR157 -4.081932 0.937647 -4.353 1.48e-05 ***
SUBFACTOR158 -6.112004 0.942740 -6.483 1.39e-10 ***
SUBFACTOR159 -3.983963 0.989367 -4.027 6.07e-05 ***
SUBFACTOR160 -2.913340 0.921931 -3.160 0.001624 **
SUBFACTOR161 -2.042601 0.975596 -2.094 0.036532 *
SUBFACTOR162 -3.397019 0.971739 -3.496 0.000493 ***
SUBFACTOR163 -1.617177 0.917376 -1.763 0.078228 .
SUBFACTOR164 -2.630423 0.889036 -2.959 0.003160 **
SUBFACTOR165 -4.185708 0.893828 -4.683 3.21e-06 ***
SUBFACTOR166 -2.434348 0.917949 -2.652 0.008127 **
SUBFACTOR167 -1.390578 0.974019 -1.428 0.153692
SUBFACTOR168 -4.853027 0.957698 -5.067 4.78e-07 ***
SUBFACTOR169 -2.283081 0.923557 -2.472 0.013596 *
SUBFACTOR170 -4.372778 0.959421 -4.558 5.79e-06 ***
SUBFACTOR171 -3.425975 0.902129 -3.798 0.000155 ***
SUBFACTOR172 -2.343538 0.920833 -2.545 0.011073 *
SUBFACTOR173 -1.710324 0.910104 -1.879 0.060492 .
SUBFACTOR174 -2.098796 0.954269 -2.199 0.028074 *
SUBFACTOR175 -2.797872 0.913217 -3.064 0.002243 **
SUBFACTOR176 -5.046590 0.857151 -5.888 5.31e-09 ***
SUBFACTOR177 -2.893347 0.921484 -3.140 0.001739 **
SUBFACTOR178 -1.841189 0.887291 -2.075 0.038229 *
SUBFACTOR179 -4.466157 0.966026 -4.623 4.26e-06 ***
SUBFACTOR180 -3.730520 0.920546 -4.053 5.45e-05 ***
SUBFACTOR181 -2.869046 0.931615 -3.080 0.002128 **
SUBFACTOR182 -2.424206 0.887707 -2.731 0.006425 **
SUBFACTOR183 -5.356722 0.936315 -5.721 1.39e-08 ***
SUBFACTOR184 -3.066164 0.942504 -3.253 0.001178 **
SUBFACTOR185 -5.124591 0.908341 -5.642 2.18e-08 ***
SUBFACTOR186 -3.251203 0.991743 -3.278 0.001080 **
SUBFACTOR187 -1.677176 0.902537 -1.858 0.063414 .
SUBFACTOR188 -3.472789 0.937982 -3.702 0.000225 ***
SUBFACTOR189 -3.762196 0.964176 -3.902 0.000102 ***
SUBFACTOR190 -2.219572 0.810106 -2.740 0.006253 **
SUBFACTOR191 -2.800552 0.908851 -3.081 0.002115 **
SUBFACTOR192 -3.399641 0.949269 -3.581 0.000358 ***
SUBFACTOR193 -2.837433 0.950723 -2.985 0.002908 **
SUBFACTOR194 -3.019642 0.910231 -3.317 0.000940 ***
SUBFACTOR195 -2.440036 0.937527 -2.603 0.009385 **
SUBFACTOR196 -3.858337 0.947051 -4.074 4.98e-05 ***
SUBFACTOR197 -2.864903 0.978925 -2.927 0.003503 **
SUBFACTOR198 -2.397067 0.987467 -2.427 0.015375 *
SUBFACTOR199 -0.967048 0.997075 -0.970 0.332333
SUBFACTOR200 -3.281440 0.937895 -3.499 0.000488 ***
SUBFACTOR201 -2.309235 0.984349 -2.346 0.019169 *
SUBFACTOR202 -1.779309 0.803265 -2.215 0.026973 *
SUBFACTOR203 -2.595728 0.883891 -2.937 0.003391 **
SUBFACTOR204 -1.802010 0.915415 -1.969 0.049278 *
SUBFACTOR205 -2.116093 0.987569 -2.143 0.032370 *
SUBFACTOR206 -1.809473 0.920028 -1.967 0.049481 *
SUBFACTOR207 -1.560251 0.954835 -1.634 0.102555
SUBFACTOR208 -1.883087 0.892272 -2.110 0.035062 *
SUBFACTOR209 -3.478732 0.939333 -3.703 0.000224 ***
SUBFACTOR210 -3.147438 0.962608 -3.270 0.001112 **
SUBFACTOR211 -2.757256 0.910277 -3.029 0.002515 **
SUBFACTOR212 -1.672145 0.935748 -1.787 0.074239 .
SUBFACTOR213 -2.927508 0.941279 -3.110 0.001922 **
SUBFACTOR214 -3.097024 0.950635 -3.258 0.001159 **
SUBFACTOR215 -2.887754 0.940748 -3.070 0.002200 **
SUBFACTOR216 -1.979758 1.017965 -1.945 0.052070 .
SUBFACTOR217 -2.785545 0.954274 -2.919 0.003588 **
SUBFACTOR218 -4.731554 0.954436 -4.957 8.36e-07 ***
SUBFACTOR219 -4.117183 0.992160 -4.150 3.61e-05 ***
SUBFACTOR220 -3.334648 0.952678 -3.500 0.000485 ***
SUBFACTOR221 -3.477129 0.971094 -3.581 0.000359 ***
SUBFACTOR222 -4.151081 0.821387 -5.054 5.13e-07 ***
SUBFACTOR223 -2.232397 0.882094 -2.531 0.011529 *
SUBFACTOR224 -2.616304 0.866332 -3.020 0.002591 **
SUBFACTOR225 -1.940628 0.875393 -2.217 0.026851 *
SUBFACTOR226 -2.011574 0.908631 -2.214 0.027059 *
SUBFACTOR227 -2.430288 0.908411 -2.675 0.007585 **
SUBFACTOR228 -2.102822 0.903471 -2.327 0.020133 *
SUBFACTOR229 -3.447302 0.927645 -3.716 0.000213 ***
SUBFACTOR230 -2.344091 0.912392 -2.569 0.010335 *
SUBFACTOR231 -3.459879 0.935278 -3.699 0.000228 ***
SUBFACTOR232 -5.658765 1.016771 -5.565 3.34e-08 ***
SUBFACTOR233 -4.783141 0.925671 -5.167 2.86e-07 ***
SUBFACTOR234 -3.819151 0.856194 -4.461 9.08e-06 ***
SUBFACTOR235 -2.024762 0.949219 -2.133 0.033155 *
SUBFACTOR236 -2.784329 0.910186 -3.059 0.002278 **
SUBFACTOR237 -3.198397 0.944980 -3.385 0.000740 ***
SUBFACTOR238 -3.142874 0.919383 -3.418 0.000655 ***
SUBFACTOR239 -3.439833 0.940339 -3.658 0.000267 ***
SUBFACTOR240 -2.622761 0.989240 -2.651 0.008142 **
SUBFACTOR241 -3.996097 0.871946 -4.583 5.15e-06 ***
SUBFACTOR242 -5.086598 0.965775 -5.267 1.69e-07 ***
SUBFACTOR243 -2.900497 0.930985 -3.116 0.001887 **
SUBFACTOR244 -1.575051 0.894947 -1.760 0.078717 .
SUBFACTOR245 -2.699959 0.941336 -2.868 0.004213 **
SUBFACTOR246 -3.595091 0.939563 -3.826 0.000138 ***
SUBFACTOR247 -1.807229 0.982754 -1.839 0.066213 .
SUBFACTOR248 -3.003435 0.930749 -3.227 0.001291 **
SUBFACTOR249 -4.050990 0.958601 -4.226 2.59e-05 ***
SUBFACTOR250 -3.054127 0.939440 -3.251 0.001187 **
SUBFACTOR251 -2.856217 0.899253 -3.176 0.001537 **
SUBFACTOR252 -2.139357 0.886316 -2.414 0.015963 *
SUBFACTOR253 -1.074312 0.963784 -1.115 0.265249
SUBFACTOR254 -2.605768 1.015898 -2.565 0.010459 *
SUBFACTOR255 -1.831341 0.927795 -1.974 0.048666 *
SUBFACTOR256 -1.873042 0.951552 -1.968 0.049291 *
SUBFACTOR257 -1.409751 0.951357 -1.482 0.138693
SUBFACTOR258 -0.117362 0.884154 -0.133 0.894426
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.373 on 1024 degrees of freedom
Multiple R-squared: 0.9726, Adjusted R-squared: 0.9655
F-statistic: 136.6 on 266 and 1024 DF, p-value: < 2.2e-166.2.2 TABLE 7.1: Random effects estimator
taxpreprdm1<-lme(LNTAX~MS+HH+AGE+EMP+PREP+LNTPI+DEPEND+MR, data=taxprep, random=~1|SUBJECT, method="ML") #using maximum likelihood estimator to estimate beta coefficients
summary(taxpreprdm1)Linear mixed-effects model fit by maximum likelihood
Data: taxprep
AIC BIC logLik
4813.255 4870.041 -2395.627
Random effects:
Formula: ~1 | SUBJECT
(Intercept) Residual
StdDev: 0.9602161 1.368896
Fixed effects: LNTAX ~ MS + HH + AGE + EMP + PREP + LNTPI + DEPEND + MR
Value Std.Error DF t-value p-value
(Intercept) -2.9603371 0.5705536 1024 -5.188534 0.0000
MS 0.0373000 0.1824839 1024 0.204402 0.8381
HH -0.6889876 0.2320057 1024 -2.969702 0.0031
AGE 0.0207431 0.2000035 1024 0.103713 0.9174
EMP -0.5048035 0.1679848 1024 -3.005054 0.0027
PREP -0.0217036 0.1175229 1024 -0.184675 0.8535
LNTPI 0.7604058 0.0699692 1024 10.867728 0.0000
DEPEND -0.1127475 0.0592818 1024 -1.901891 0.0575
MR 0.1153752 0.0073142 1024 15.774213 0.0000
Correlation:
(Intr) MS HH AGE EMP PREP LNTPI DEPEND
MS 0.176
HH 0.030 0.419
AGE -0.043 -0.167 -0.023
EMP -0.116 -0.069 0.024 -0.030
PREP -0.035 -0.045 0.004 -0.115 -0.112
LNTPI -0.948 -0.180 -0.081 -0.043 0.099 -0.016
DEPEND -0.074 -0.604 -0.269 0.224 -0.038 -0.039 -0.068
MR 0.522 -0.020 0.055 0.149 -0.041 -0.051 -0.698 0.102
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-5.83483692 -0.21263981 0.09677632 0.39814646 5.79731648
Number of Observations: 1290
Number of Groups: 258 6.2.3 Hausman’s test
beta1fix<-coefficients(taxprepfx)
beta1fe<-beta1fix[1:8]
cov1fix<-vcov(taxprepfx)
cov1fe<-cov1fix[1:8, 1:8]
beta1re<-coefficients(taxpreprdm1)
beta1re<-t(beta1re[1, 2:9])
cov1re<-vcov(taxpreprdm1)
cov1re<-cov1re[2:9, 2:9]
HSTEST1<-t(beta1fe-beta1re)%*%solve(cov1fe-cov1re)%*%(beta1fe-beta1re)
beta1fe MS HH AGE EMP PREP
0.072327932 -0.706799308 0.001839538 -0.244247153 -0.029685211
LNTPI DEPEND MR
0.716754955 -0.069020879 0.121919964 beta1re 1
MS 0.03730005
HH -0.68898764
AGE 0.02074305
EMP -0.50480349
PREP -0.02170360
LNTPI 0.76040578
DEPEND -0.11274746
MR 0.11537523HSTEST1 1
1 6.0190066.3 Example 7.2: Income Tax Payments (continued) (Page 255)
6.3.1 Table 7.2: Fixed effects estimators with two variable slopes
ACF(taxpreprdm1, maxlag=10) #Obtain ACF of residuals for within-group residual lag ACF
1 0 1.000000000
2 1 -0.004283774
3 2 -0.223519705
4 3 -0.307380297
5 4 -0.355268841# Compared with SAS, lm in R can estimate fixed effects, but can not code AR(1) for within-subject correlation
taxprepfx2<-lm(LNTAX~MS+HH+AGE+EMP+PREP+LNTPI+DEPEND+MR+SUBFACTOR+SUBFACTOR*MR+SUBFACTOR*LNTPI-1, data=taxprep)
# summary(taxprepfx2)6.3.2 Table 7.2: Variable slopes model
taxpreprdm2<-lme(LNTAX~MS+HH+AGE+EMP+PREP+LNTPI+DEPEND+MR, data=taxprep, method="ML",random=~1+LNTPI+MR|SUBJECT, correlation=corAR1(form=~1|SUBJECT),control = lmeControl(opt = "optim"))
# I changed the initial code to "control = lmeControl(opt = "optim")", because the initial code has convergence problem.
summary(taxpreprdm2) #ESTIMATES ARE CLOSE TO RESULTS FROM SASLinear mixed-effects model fit by maximum likelihood
Data: taxprep
AIC BIC logLik
4443.141 4530.902 -2204.571
Random effects:
Formula: ~1 + LNTPI + MR | SUBJECT
Structure: General positive-definite, Log-Cholesky parametrization
StdDev Corr
(Intercept) 12.05966691 (Intr) LNTPI
LNTPI 1.27245273 -0.988
MR 0.07050666 0.475 -0.602
Residual 1.14826017
Correlation Structure: AR(1)
Formula: ~1 | SUBJECT
Parameter estimate(s):
Phi
0.1346485
Fixed effects: LNTAX ~ MS + HH + AGE + EMP + PREP + LNTPI + DEPEND + MR
Value Std.Error DF t-value p-value
(Intercept) -14.560716 1.4762035 1024 -9.863624 0.0000
MS -0.613181 0.1607932 1024 -3.813475 0.0001
HH -0.766651 0.1991612 1024 -3.849398 0.0001
AGE -0.372122 0.1711989 1024 -2.173622 0.0300
EMP -0.646505 0.1346603 1024 -4.801007 0.0000
PREP -0.303705 0.0960482 1024 -3.162005 0.0016
LNTPI 2.268717 0.1693620 1024 13.395665 0.0000
DEPEND -0.140338 0.0495257 1024 -2.833637 0.0047
MR 0.006456 0.0102326 1024 0.630904 0.5282
Correlation:
(Intr) MS HH AGE EMP PREP LNTPI DEPEND
MS 0.293
HH 0.070 0.450
AGE -0.011 -0.139 -0.001
EMP -0.009 -0.051 0.016 -0.053
PREP 0.053 -0.019 0.012 -0.118 -0.085
LNTPI -0.990 -0.303 -0.095 -0.021 0.002 -0.071
DEPEND 0.044 -0.549 -0.250 0.235 -0.030 -0.037 -0.094
MR 0.733 0.181 0.098 0.099 0.011 0.027 -0.808 0.128
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-7.2788124 -0.1668237 0.0753182 0.3376614 2.7774163
Number of Observations: 1290
Number of Groups: 258 6.3.3 Hausman’s test
beta2fix<-coefficients(taxprepfx2)
beta2fe<-beta2fix[1:8]
cov2fix<-vcov(taxprepfx2)
cov2fe<-cov2fix[1:8, 1:8]
beta2re<-coefficients(taxpreprdm2)
beta2re<-t(beta2re[1, 2:9])
cov2re<-vcov(taxpreprdm2)
cov2re<-cov2re[2:9, 2:9]
HSTEST2<-t(beta2fe-beta2re)%*%solve(cov2fe-cov2re)%*%(beta2fe-beta2re)
beta2fe MS HH AGE EMP PREP LNTPI
-0.28247941 -2.19247828 -0.54479788 -0.12152994 -0.47339937 0.62023798
DEPEND MR
-0.29578737 0.02681867 beta2re 1
MS -0.613180767
HH -0.766650688
AGE -0.372121718
EMP -0.646504958
PREP -0.303704908
LNTPI 1.680299311
DEPEND -0.140337926
MR -0.007261631HSTEST2 #ESTIMATES ARE DIFFERENT FROM RESULTS FROM SAS, BECAUSE THE FIXED EFFECTS ESTIMATORS DID NOT INCLUDE AR(1) 1
1 27.307126.4 TABLE 7.3 Augmented regressions
6.4.1 Create panel data set with subject averages
msavg<-aggregate(taxprep$MS, list(SUBJECT=taxprep$SUBJECT), mean)
names(msavg)<-c("SUBJECT", "msavg")
hhavg<-aggregate(taxprep$HH, list(SUBJECT=taxprep$SUBJECT), mean)
names(hhavg)<-c("SUBJECT", "hhavg")
ageavg<-aggregate(taxprep$AGE, list(SUBJECT=taxprep$SUBJECT), mean)
names(ageavg)<-c("SUBJECT", "ageavg")
empavg<-aggregate(taxprep$EMP, list(SUBJECT=taxprep$SUBJECT), mean)
names(empavg)<-c("SUBJECT", "empavg")
prepavg<-aggregate(taxprep$PREP, list(SUBJECT=taxprep$SUBJECT), mean)
names(prepavg)<-c("SUBJECT", "prepavg")
dependavg<-aggregate(taxprep$DEPEND, list(SUBJECT=taxprep$SUBJECT), mean)
names(dependavg)<-c("SUBJECT", "dependavg")
lntpiavg<-aggregate(taxprep$LNTPI, list(SUBJECT=taxprep$SUBJECT), mean)
names(lntpiavg)<-c("SUBJECT", "lntpiavg")
mravg<-aggregate(taxprep$MR, list(SUBJECT=taxprep$SUBJECT), mean)
names(mravg)<-c("SUBJECT", "mravg")
avg<-merge(msavg, taxprep, by="SUBJECT", all.y=T, sort=T)
avg<-merge(hhavg, avg, by="SUBJECT", all.y=T, sort=T)
avg<-merge(ageavg, avg, by="SUBJECT", all.y=T, sort=T)
avg<-merge(empavg, avg, by="SUBJECT", all.y=T, sort=T)
avg<-merge(prepavg, avg, by="SUBJECT", all.y=T, sort=T)
avg<-merge(dependavg, avg, by="SUBJECT", all.y=T, sort=T)
avg<-merge(lntpiavg, avg, by="SUBJECT", all.y=T, sort=T)
avg<-merge(mravg, avg, by="SUBJECT", all.y=T, sort=T)6.4.2 Models with averages as omitted variables
#VARIABLE INTERCEPTS AND TWO VARIABLE SLOPES
taxprepaug<-lme(LNTAX~MS+HH+AGE+EMP+PREP+LNTPI+DEPEND+MR+msavg+hhavg+ageavg+empavg+prepavg+dependavg+lntpiavg+mravg, data=avg, method="ML",random=~1+LNTPI+MR|SUBJECT, correlation=corAR1(form=~1|SUBJECT),control = lmeControl(opt = "optim"))
#Again, I change the code to "control = lmeControl(opt = "optim")" due to convergence problem.
summary(taxprepaug)Linear mixed-effects model fit by maximum likelihood
Data: avg
AIC BIC logLik
4412.59 4541.65 -2181.295
Random effects:
Formula: ~1 + LNTPI + MR | SUBJECT
Structure: General positive-definite, Log-Cholesky parametrization
StdDev Corr
(Intercept) 12.48682715 (Intr) LNTPI
LNTPI 1.28389597 -0.992
MR 0.05703604 0.447 -0.555
Residual 1.10067285
Correlation Structure: AR(1)
Formula: ~1 | SUBJECT
Parameter estimate(s):
Phi
-0.04702359
Fixed effects: LNTAX ~ MS + HH + AGE + EMP + PREP + LNTPI + DEPEND + MR + msavg + hhavg + ageavg + empavg + prepavg + dependavg + lntpiavg + mravg
Value Std.Error DF t-value p-value
(Intercept) -22.909909 2.2231930 1024 -10.304957 0.0000
MS -0.563113 0.2425479 1024 -2.321658 0.0204
HH -1.089503 0.2825216 1024 -3.856353 0.0001
AGE -0.408585 0.2792958 1024 -1.462911 0.1438
EMP -0.395533 0.2102914 1024 -1.880881 0.0603
PREP -0.289016 0.1414320 1024 -2.043495 0.0413
LNTPI 2.374719 0.1680609 1024 14.130110 0.0000
DEPEND -0.174946 0.0719544 1024 -2.431338 0.0152
MR 0.030201 0.0107346 1024 2.813397 0.0050
msavg -0.273782 0.3121956 249 -0.876955 0.3814
hhavg 0.456298 0.3823711 249 1.193338 0.2339
ageavg 0.007476 0.3370271 249 0.022184 0.9823
empavg -0.450047 0.2598717 249 -1.731806 0.0845
prepavg 0.035089 0.1833941 249 0.191333 0.8484
dependavg -0.006988 0.0946377 249 -0.073840 0.9412
lntpiavg 0.962655 0.1930848 249 4.985661 0.0000
mravg -0.109881 0.0158988 249 -6.911257 0.0000
Correlation:
(Intr) MS HH AGE EMP PREP LNTPI DEPEND MR
MS 0.159
HH 0.052 0.271
AGE 0.041 -0.043 -0.013
EMP 0.014 0.013 0.005 0.014
PREP 0.056 -0.028 -0.002 -0.018 -0.049
LNTPI -0.716 -0.228 -0.083 -0.045 0.020 -0.073
DEPEND 0.042 -0.433 -0.185 0.101 -0.030 0.012 -0.058
MR 0.423 0.136 0.075 0.130 -0.001 0.057 -0.675 0.068
msavg 0.167 -0.741 -0.195 0.071 -0.006 0.038 0.023 0.354 0.005
hhavg 0.062 -0.183 -0.743 0.046 -0.005 0.020 0.017 0.144 -0.019
ageavg -0.024 0.053 0.015 -0.816 -0.018 0.025 0.034 -0.081 -0.104
empavg -0.033 0.001 -0.001 -0.018 -0.807 0.045 -0.027 0.032 0.008
prepavg -0.032 0.028 0.010 0.016 0.044 -0.774 0.040 -0.007 -0.030
dependavg 0.057 0.360 0.152 -0.086 0.020 0.003 -0.013 -0.762 -0.005
lntpiavg -0.748 -0.021 -0.002 -0.019 -0.038 -0.016 0.083 -0.007 -0.002
mravg 0.641 0.056 0.002 -0.048 0.020 0.019 -0.114 -0.006 -0.247
msavg hhavg ageavg empavg prepvg dpndvg lntpvg
MS
HH
AGE
EMP
PREP
LNTPI
DEPEND
MR
msavg
hhavg 0.378
ageavg -0.153 -0.057
empavg -0.038 0.005 0.001
prepavg -0.035 -0.024 -0.095 -0.086
dependavg -0.523 -0.243 0.201 -0.036 -0.042
lntpiavg -0.253 -0.117 -0.018 0.068 0.006 -0.102
mravg 0.130 0.094 0.113 -0.032 -0.042 0.124 -0.838
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-7.19930371 -0.17697920 0.07578962 0.32209554 3.33317410
Number of Observations: 1290
Number of Groups: 258 #ESTIMATES ARE DIFFERENT FROM SAS BECAUSE fa0(3) WAS CODED IN SAS
beta3re<-coefficients(taxprepaug)
betarand<-t(beta3re[1, 10:17])
cov3re<-vcov(taxprepaug)
cov3re<-cov3re[10:17, 10:17]
ARTEST <- t(betarand)%*%solve(cov3re)%*%betarand
betarand 1
msavg -0.273781537
hhavg 0.456298030
ageavg 0.007476440
empavg -0.450047310
prepavg 0.035089331
dependavg -0.006988012
lntpiavg 0.962655240
mravg -0.109880536ARTEST 1
1 59.97999