Chapter 03 - Multiple Regression Analysis#
import stata_setup
stata_setup.config("C:/Program Files/Stata18/", "se", splash=False)
Problem 3.1.#
\(bwght = \beta_0 + \beta_1cigs + \beta_2faminc + u\)
iii. bgwht on cig & family income
%%stata
use bwght.dta, clear
corr(cigs faminc)
reg bwght cigs
display "bwght= " %5.3f _b[_cons] "+ " %5.3f _b[cigs] "cigs; N=" _N ", Rsq=" %5.4f e(r2)
reg bwght cigs faminc
display "bwght= " %5.3f _b[_cons] "+ " %5.3f _b[cigs] "cigs + " %5.3f _b[faminc] "faminc; N=" _N ", Rsq=" %5.4f e(r2)
. use bwght.dta, clear
. corr(cigs faminc)
(obs=1,388)
| cigs faminc
-------------+------------------
cigs | 1.0000
faminc | -0.1730 1.0000
. reg bwght cigs
Source | SS df MS Number of obs = 1,388
-------------+---------------------------------- F(1, 1386) = 32.24
Model | 13060.4194 1 13060.4194 Prob > F = 0.0000
Residual | 561551.3 1,386 405.159668 R-squared = 0.0227
-------------+---------------------------------- Adj R-squared = 0.0220
Total | 574611.72 1,387 414.283864 Root MSE = 20.129
------------------------------------------------------------------------------
bwght | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
cigs |
-.5137721 .0904909 -5.68 0.000 -.6912861 -.3362581
_cons | 119.7719 .5723407 209.27 0.000 118.6492 120.8946
------------------------------------------------------------------------------
. display "bwght= " %5.3f _b[_cons] "+ " %5.3f _b[cigs] "cigs; N=" _N ", Rsq="
> %5.4f e(r2)
bwght= 119.772+ -0.514cigs; N=1388, Rsq=0.0227
. reg bwght cigs faminc
Source | SS df MS Number of obs = 1,388
-------------+---------------------------------- F(2, 1385) = 21.27
Model | 17126.2088 2 8563.10442 Prob > F = 0.0000
Residual | 557485.511 1,385 402.516614 R-squared = 0.0298
-------------+---------------------------------- Adj R-squared = 0.0284
Total | 574611.72 1,387 414.283864 Root MSE =
20.063
------------------------------------------------------------------------------
bwght | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
cigs | -.4634075 .0915768 -5.06 0.000 -.6430518 -.2837633
faminc | .0927647 .0291879 3.18 0.002 .0355075 .1500219
_cons | 116.9741 1.048984 111.51 0.000 114.9164 119.0319
------------------------------------------------------------------------------
. display "bwght= " %5.3f _b[_cons] "+ " %5.3f _b[cigs] "cigs + " %5.3f _b[fami
> nc] "faminc; N=" _N ", Rsq=" %5.4f e(r2)
bwght= 116.974+ -0.463cigs + 0.093faminc; N=1388, Rsq=0.0298
.
Problem3.2. House price:#
\(price = f(sqrft, bdrms)\)
i. reg house price on area & no. of bedrooms. Report result in equation form
%%stata
u hprice1.dta, clear
reg price sqrft bdrms
display "price= " %5.3f _b[_cons] " + " %5.3f _b[sqrft] "sqrft + " %5.3f _b[bdrms] "bdrms; N=" _N ", Rsq=" %5.4f e(r2)
. u hprice1.dta, clear
. reg price sqrft bdrms
Source | SS df MS Number of obs = 88
-------------+---------------------------------- F(2, 85) = 72.96
Model | 580009.152 2 290004.576 Prob > F = 0.0000
Residual | 337845.354 85 3974.65122 R-squared = 0.6319
-------------+---------------------------------- Adj R-squared = 0.6233
Total | 917854.506 87 10550.0518 Root MSE = 63.045
------------------------------------------------------------------------------
price | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
sqrft | .1284362 .0138245 9.29 0.000 .1009495 .1559229
bdrms | 15.19819 9.483517 1.60 0.113 -3.657582 34.05396
_cons | -19.315 31.04662 -0.62 0.536 -81.04399 42.414
------------------------------------------------------------------------------
. display "price= " %5.3f _b[_cons] " + " %5.3f _b[sqrft] "sqrft + " %5.3f _b[b
> drms] "bdrms; N=" _N ", Rsq=" %5.4f e(r2)
price= -19.315 + 0.128sqrft + 15.198bdrms; N=88, Rsq=0.6319
.
ii. The increase in price due to one more bedroms, holding size constant is
%%stata
display _b[bdrms] " (thousand dollars)"
15.198191 (thousand dollars)
The increase in price due to an additional room that has an area of 140 sqrft
%%stata
display _b[bdrms] + _b[sqrft]*140 " (thousand dollars)"
33.17926 (thousand dollars)
iv. The price change explained by change in bdrms & sqrft is about
%%stata
display %5.4f e(r2)*100 "%"
63.1918%
V The predict selling price for the house is
%%stata
display "price= " _b[_cons] + _b[sqrft]*2438 + _b[bdrms]*4
margins, at(bdrms=4 sqrft=2438)
. display "price= " _b[_cons] + _b[sqrft]*2438 + _b[bdrms]*4
price= 354.60525
. margins, at(bdrms=4 sqrft=2438)
Adjusted predictions Number of obs = 88
Model VCE: OLS
Expression: Linear prediction, predict()
At: sqrft = 2438
bdrms = 4
------------------------------------------------------------------------------
| Delta-method
| Margin std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
_cons | 354.6052 8.41493 42.14 0.000 337.8741 371.3364
------------------------------------------------------------------------------
.
vi. Find uhat for a house with price $300
%%stata
reg price sqrft bdrms
predict pricehat
predict uh, residual
list price pricehat uh if price==300
. reg price sqrft bdrms
Source | SS df MS Number of obs = 88
-------------+---------------------------------- F(2, 85) = 72.96
Model | 580009.152 2 290004.576 Prob > F = 0.0000
Residual | 337845.354 85 3974.65122 R-squared = 0.6319
-------------+---------------------------------- Adj R-squared = 0.6233
Total | 917854.506 87 10550.0518 Root MSE = 63.045
------------------------------------------------------------------------------
price | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
sqrft | .1284362 .0138245 9.29 0.000 .1009495 .1559229
bdrms | 15.19819 9.483517 1.60 0.113 -3.657582 34.05396
_cons | -19.315 31.04662 -0.62 0.536 -81.04399 42.414
------------------------------------------------------------------------------
. predict pricehat
(option xb assumed; fitted values)
. predict uh, residual
. list price pricehat uh if price==300
+------------------------------+
| price pricehat uh |
|------------------------------|
1. | 300 354.6053 -54.60525 |
12. | 300 394.9769 -94.97694 |
+------------------------------+
.
Problem 3.3.#
i. Reg salary on sales & market value, in constant elasticity
%%stata
use ceosal2.dta, clear
*g lsales = ln(sales)
*g lsalary = ln(salary)
*g lmktval = ln(mktval)
reg lsalary lsales lmktval
. use ceosal2.dta, clear
. *g lsales = ln(sales)
. *g lsalary = ln(salary)
. *g lmktval = ln(mktval)
. reg lsalary lsales lmktval
Source | SS df MS Number of obs = 177
-------------+---------------------------------- F(2, 174) = 37.13
Model | 19.3365617 2 9.66828083 Prob > F = 0.0000
Residual | 45.3096514 174 .260400295 R-squared = 0.2991
-------------+---------------------------------- Adj R-squared = 0.2911
Total | 64.6462131 176 .367308029 Root MSE = .51029
------------------------------------------------------------------------------
lsalary | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
lsales | .1621283 .0396703 4.09 0.000 .0838315 .2404252
lmktval | .106708 .050124 2.13 0.035 .0077787 .2056372
_cons | 4.620917 .2544083 18.16 0.000 4.118794 5.123041
------------------------------------------------------------------------------
.
ii. add profit to the model
%%stata
reg lsalary lsales lmktval profit
Source | SS df MS Number of obs = 177
-------------+---------------------------------- F(3, 173) = 24.64
Model | 19.3509799 3 6.45032663 Prob > F = 0.0000
Residual | 45.2952332 173 .261822157 R-squared = 0.2993
-------------+---------------------------------- Adj R-squared = 0.2872
Total | 64.6462131 176 .367308029 Root MSE = .51169
------------------------------------------------------------------------------
lsalary | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
lsales | .1613683 .0399101 4.04 0.000 .0825949 .2401416
lmktval | .0975286 .0636886 1.53 0.128 -.0281782 .2232354
profits | .0000357 .000152 0.23 0.815 -.0002643 .0003356
_cons | 4.686924 .3797294 12.34 0.000 3.937425 5.436423
------------------------------------------------------------------------------
iii. add CEO tenure to the model
%%stata
reg lsalary lsales lmktval profit ceoten
Source | SS df MS Number of obs = 177
-------------+---------------------------------- F(4, 172) = 20.08
Model | 20.5768102 4 5.14420254 Prob > F = 0.0000
Residual | 44.0694029 172 .256217459 R-squared = 0.3183
-------------+---------------------------------- Adj R-squared = 0.3024
Total | 64.6462131 176 .367308029 Root MSE = .50618
------------------------------------------------------------------------------
lsalary | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
lsales | .1622339 .0394826 4.11 0.000 .0843012 .2401667
lmktval | .1017598 .063033 1.61 0.108 -.022658 .2261775
profits | .0000291 .0001504 0.19 0.847 -.0002677 .0003258
ceoten | .0116847 .005342 2.19 0.030 .0011403 .022229
_cons | 4.55778 .3802548 11.99 0.000 3.807213 5.308347
------------------------------------------------------------------------------
iv.
%%stata
corr lmktval profit
(obs=177)
| lmktval profits
-------------+------------------
lmktval | 1.0000
profits | 0.7769 1.0000
Problem 3.4#
i. Find min, max & mean for atndrte, priGPA & ACT
%%stata
use attend.dta, clear
sum atndrte priGPA ACT
. use attend.dta, clear
. sum atndrte priGPA ACT
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
atndrte | 680 81.70956 17.04699 6.25 100
priGPA | 680 2.586775 .5447141 .857 3.93
ACT | 680 22.51029 3.490768 13 32
.
ii. Estimate the model. atndrte = f(priGPA, ACT )
%%stata
reg atndrte priGPA ACT
display "atndrte = " %5.2f _b[_cons] " + " %5.2f _b[priGPA] "priGPA + " %5.2f _b[ACT] "ACT; N=" _N ", Rsq=" %5.4f e(r2)
. reg atndrte priGPA ACT
Source | SS df MS Number of obs = 680
-------------+---------------------------------- F(2, 677) = 138.65
Model | 57336.7612 2 28668.3806 Prob > F = 0.0000
Residual | 139980.564 677 206.765974 R-squared = 0.2906
-------------+---------------------------------- Adj R-squared = 0.2885
Total | 197317.325 679 290.59989 Root MSE = 14.379
------------------------------------------------------------------------------
atndrte | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
priGPA | 17.26059 1.083103 15.94 0.000 15.13395 19.38724
ACT | -1.716553 .169012 -10.16 0.000 -2.048404 -1.384702
_cons | 75.7004 3.884108 19.49 0.000 68.07406 83.32675
------------------------------------------------------------------------------
. display "atndrte = " %5.2f _b[_cons] " + " %5.2f _b[priGPA] "priGPA + " %5.2f
> _b[ACT] "ACT; N=" _N ", Rsq=" %5.4f e(r2)
atndrte = 75.70 + 17.26priGPA + -1.72ACT; N=680, Rsq=0.2906
.
iii. Discuss the estimated slope coeficients.
iv. Predict atndrte at priGPA=3.65 & ACT=20. What do you make of the result?
%%stata
margins, at(priGPA=3.65 ACT=20)
sum atndrte priGPA ACT if atndrte>= 104
. margins, at(priGPA=3.65 ACT=20)
Adjusted predictions Number of obs = 680
Model VCE: OLS
Expression: Linear prediction, predict()
At: priGPA = 3.65
ACT = 20
------------------------------------------------------------------------------
| Delta-method
| Margin std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
_cons | 104.3705 1.4683 71.08 0.000 101.4875 107.2535
------------------------------------------------------------------------------
. sum atndrte priGPA ACT if atndrte>= 104
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
atndrte | 0
priGPA | 0
ACT | 0
.
v. predict difference in atndrte if student A has priGPA=3.1 & ACT=21 & student B has priGPA=2.1 & ACT=26
%%stata
margins, at(priGPA=3.1 ACT=21)
margins, at( priGPA=2.1 ACT=26)
display "Predicted difference in atndrte is " 93.16 - 67.32
. margins, at(priGPA=3.1 ACT=21)
Adjusted predictions Number of obs = 680
Model VCE: OLS
Expression: Linear prediction, predict()
At: priGPA = 3.1
ACT = 21
------------------------------------------------------------------------------
| Delta-method
| Margin std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
_cons | 93.16062 .8823921 105.58 0.000 91.42807 94.89318
------------------------------------------------------------------------------
. margins, at( priGPA=2.1 ACT=26)
Adjusted predictions Number of obs = 680
Model VCE: OLS
Expression: Linear prediction, predict()
At: priGPA = 2.1
ACT = 26
------------------------------------------------------------------------------
| Delta-method
| Margin std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
_cons | 67.31727 1.072343 62.78 0.000 65.21175 69.42279
------------------------------------------------------------------------------
. display "Predicted difference in atndrte is " 93.16 - 67.32
Predicted difference in atndrte is 25.84
.
Problem 3.5: Example3.2 Wage equation#
%%stata
u wage1.dta, clear
reg educ exper tenure
predict r1, r
reg lwage r1
reg lwage educ exper tenure
. u wage1.dta, clear
. reg educ exper tenure
Source | SS df MS Number of obs = 526
-------------+---------------------------------- F(2, 523) = 29.49
Model | 407.946311 2 203.973156 Prob > F = 0.0000
Residual | 3617.48335 523 6.91679416 R-squared = 0.1013
-------------+---------------------------------- Adj R-squared = 0.0979
Total | 4025.42966 525 7.66748506 Root MSE = 2.63
------------------------------------------------------------------------------
educ | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
exper | -.0737851 .0097609 -7.56 0.000 -.0929604 -.0546098
tenure | .0476795 .0183371 2.60 0.010 .011656 .0837031
_cons | 13.57496 .1843245 73.65 0.000 13.21286 13.93707
------------------------------------------------------------------------------
. predict r1, r
. reg lwage r1
Source | SS df MS Number of obs = 526
-------------+---------------------------------- F(1, 524) = 136.41
Model | 30.6376773 1 30.6376773 Prob > F = 0.0000
Residual | 117.692074 524 .224603195 R-squared = 0.2066
-------------+---------------------------------- Adj R-squared = 0.2050
Total | 148.329751 525 .28253286 Root MSE = .47392
------------------------------------------------------------------------------
lwage | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
r1 | .092029 .0078796 11.68 0.000 .0765495 .1075085
_cons | 1.623268 .020664 78.56 0.000 1.582674 1.663863
------------------------------------------------------------------------------
. reg lwage educ exper tenure
Source | SS df MS Number of obs = 526
-------------+---------------------------------- F(3, 522) = 80.39
Model | 46.8741776 3 15.6247259 Prob > F = 0.0000
Residual | 101.455574 522 .194359337 R-squared = 0.3160
-------------+---------------------------------- Adj R-squared = 0.3121
Total | 148.329751 525 .28253286 Root MSE = .44086
------------------------------------------------------------------------------
lwage | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
educ | .092029 .0073299 12.56 0.000 .0776292 .1064288
exper | .0041211 .0017233 2.39 0.017 .0007357 .0075065
tenure | .0220672 .0030936 7.13 0.000 .0159897 .0281448
_cons | .2843595 .1041904 2.73 0.007 .0796756 .4890435
------------------------------------------------------------------------------
.
Problem 3.6#
i. IQ on educ
%%stata
use wage2.dta, clear
reg IQ educ
. use wage2.dta, clear
. reg IQ educ
Source | SS df MS Number of obs = 935
-------------+---------------------------------- F(1, 933) = 338.02
Model | 56280.9277 1 56280.9277 Prob > F = 0.0000
Residual | 155346.531 933 166.502177 R-squared = 0.2659
-------------+---------------------------------- Adj R-squared = 0.2652
Total | 211627.459 934 226.581862 Root MSE = 12.904
------------------------------------------------------------------------------
IQ | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
educ | 3.533829 .1922095 18.39 0.000 3.156616 3.911042
_cons | 53.68715 2.622933 20.47 0.000 48.53962 58.83469
------------------------------------------------------------------------------
.
ii. lwage on educ
%%stata
reg lwage educ
Source | SS df MS Number of obs = 935
-------------+---------------------------------- F(1, 933) = 100.70
Model | 16.1377042 1 16.1377042 Prob > F = 0.0000
Residual | 149.518579 933 .160255712 R-squared = 0.0974
-------------+---------------------------------- Adj R-squared = 0.0964
Total | 165.656283 934 .177362188 Root MSE = .40032
------------------------------------------------------------------------------
lwage | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
educ | .0598392 .0059631 10.03 0.000 .0481366 .0715418
_cons | 5.973063 .0813737 73.40 0.000 5.813366 6.132759
------------------------------------------------------------------------------
c. lwage on educ and IQ
%%stata
reg lwage educ IQ
Source | SS df MS Number of obs = 935
-------------+---------------------------------- F(2, 932) = 69.42
Model | 21.4779447 2 10.7389723 Prob > F = 0.0000
Residual | 144.178339 932 .154697788 R-squared = 0.1297
-------------+---------------------------------- Adj R-squared = 0.1278
Total | 165.656283 934 .177362188 Root MSE = .39332
------------------------------------------------------------------------------
lwage | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
educ | .0391199 .0068382 5.72 0.000 .0256998 .05254
IQ | .0058631 .0009979 5.88 0.000 .0039047 .0078215
_cons | 5.658288 .0962408 58.79 0.000 5.469414 5.847162
------------------------------------------------------------------------------
d. Verify
%%stata
di _b[educ]+_b[IQ]*3.534
.05984021
Problem 3.7#
i. Estimate the model,
%%stata
use meap93.dta, clear
reg math10 lexpend lnchprg
display "math10 = " %5.2f _b[_cons] " + " %5.2f _b[lexpend] "lexpend + " %5.2f _b[lnchprg] "lnchprg; N=" _N ", Rsq=" %5.4f e(r2)
. use meap93.dta, clear
. reg math10 lexpend lnchprg
Source | SS df MS Number of obs = 408
-------------+---------------------------------- F(2, 405) = 44.43
Model | 8063.82429 2 4031.91215 Prob > F = 0.0000
Residual | 36753.3562 405 90.7490276 R-squared = 0.1799
-------------+---------------------------------- Adj R-squared = 0.1759
Total | 44817.1805 407 110.115923 Root MSE = 9.5262
------------------------------------------------------------------------------
math10 | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
lexpend | 6.22969 2.972634 2.10 0.037 .3859705 12.07341
lnchprg | -.3045853 .0353574 -8.61 0.000 -.3740923 -.2350783
_cons | -20.36075 25.07288 -0.81 0.417 -69.64998 28.92848
------------------------------------------------------------------------------
. display "math10 = " %5.2f _b[_cons] " + " %5.2f _b[lexpend] "lexpend + " %5.2
> f _b[lnchprg] "lnchprg; N=" _N ", Rsq=" %5.4f e(r2)
math10 = -20.36 + 6.23lexpend + -0.30lnchprg; N=408, Rsq=0.1799
.
ii. Discuss the intercept. iii. Run simple ols, math10 on lexpend, compare the results
%%stata
reg math10 lexpend
display "math10 = " %5.2f _b[_cons] " + " %5.2f _b[lexpend] "lexpend; N=" _N ", Rsq=" %5.4f e(r2)
. reg math10 lexpend
Source | SS df MS Number of obs = 408
-------------+---------------------------------- F(1, 406) = 12.41
Model | 1329.42517 1 1329.42517 Prob > F = 0.0005
Residual | 43487.7553 406 107.112698 R-squared = 0.0297
-------------+---------------------------------- Adj R-squared = 0.0273
Total | 44817.1805 407 110.115923 Root MSE = 10.35
------------------------------------------------------------------------------
math10 | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
lexpend | 11.16439 3.169011 3.52 0.000 4.934677 17.39411
_cons | -69.3411 26.53013 -2.61 0.009 -121.4947 -17.18753
------------------------------------------------------------------------------
. display "math10 = " %5.2f _b[_cons] " + " %5.2f _b[lexpend] "lexpend; N=" _N
> ", Rsq=" %5.4f e(r2)
math10 = -69.34 + 11.16lexpend; N=408, Rsq=0.0297
.
iv. Correlation between lexpend and lnchprg
%%stata
corr lexpend lnchprg
(obs=408)
| lexpend lnchprg
-------------+------------------
lexpend | 1.0000
lnchprg | -0.1927 1.0000
Problem 3.8#
i.
%%stata
use discrim.dta, clear
mean prpblck income
d prpblck income
. use discrim.dta, clear
. mean prpblck income
Mean estimation Number of obs = 409
--------------------------------------------------------------
| Mean Std. err. [95% conf. interval]
-------------+------------------------------------------------
prpblck | .1134864 .0090199 .0957551 .1312177
income | 47053.78 651.6738 45772.73 48334.84
--------------------------------------------------------------
. d prpblck income
Variable Storage Display Value
name type format label Variable label
-------------------------------------------------------------------------------
prpblck float %9.0g proportion black, zipcode
income float %9.0g median family income, zipcode
.
ii. estimate effect of prpblck income on price of soda
%%stata
reg psoda prpblck income
di "psoda = " %5.2f _b[_cons] " + " %5.2f _b[prpblck] "prpblck + " %5.4f _b[income] "income; N=" _N " Rsq=" %5.4f e(r2)
. reg psoda prpblck income
Source | SS df MS Number of obs = 401
-------------+---------------------------------- F(2, 398) = 13.66
Model | .202552215 2 .101276107 Prob > F = 0.0000
Residual | 2.95146493 398 .007415741 R-squared = 0.0642
-------------+---------------------------------- Adj R-squared = 0.0595
Total | 3.15401715 400 .007885043 Root MSE = .08611
------------------------------------------------------------------------------
psoda | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
prpblck | .1149882 .0260006 4.42 0.000 .0638724 .1661039
income | 1.60e-06 3.62e-07 4.43 0.000 8.91e-07 2.31e-06
_cons | .9563196 .018992 50.35 0.000 .9189824 .9936568
------------------------------------------------------------------------------
. di "psoda = " %5.2f _b[_cons] " + " %5.2f _b[prpblck] "prpblck + " %5.4f _b[i
> ncome] "income; N=" _N " Rsq=" %5.4f e(r2)
psoda = 0.96 + 0.11prpblck + 0.0000income; N=410 Rsq=0.0642
.
iii. compare.
%%stata
reg psoda prpblck
di "psoda = " %5.2f _b[_cons] " + " %5.2f _b[prpblck] "prpblck; N=" _N " Rsq=" %5.4f e(r2)
test prpblck = 0.115
. reg psoda prpblck
Source | SS df MS Number of obs = 401
-------------+---------------------------------- F(1, 399) = 7.34
Model | .057010466 1 .057010466 Prob > F = 0.0070
Residual | 3.09700668 399 .007761922 R-squared = 0.0181
-------------+---------------------------------- Adj R-squared = 0.0156
Total | 3.15401715 400 .007885043 Root MSE = .0881
------------------------------------------------------------------------------
psoda | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
prpblck | .0649269 .023957 2.71 0.007 .0178292 .1120245
_cons | 1.037399 .0051905 199.87 0.000 1.027195 1.047603
------------------------------------------------------------------------------
. di "psoda = " %5.2f _b[_cons] " + " %5.2f _b[prpblck] "prpblck; N=" _N " Rsq=
> " %5.4f e(r2)
psoda = 1.04 + 0.06prpblck; N=410 Rsq=0.0181
. test prpblck = 0.115
( 1) prpblck = .115
F( 1, 399) = 4.37
Prob > F = 0.0372
.
iv. Model with a constant price elasticity w.r.t income. Estimate the %age change in psoda, if prpblck increases by 0.2,
%%stata
reg lpsoda prpblck lincome
di "lpsoda = " %5.2f _b[_cons] " + " %5.2f _b[prpblck] "prpblck + " %5.4f _b[lincome] "lincome; N=" _N " Rsq=" %5.4f e(r2)
di _b[prpblck] * .2 *100
. reg lpsoda prpblck lincome
Source | SS df MS Number of obs = 401
-------------+---------------------------------- F(2, 398) = 14.54
Model | .196020672 2 .098010336 Prob > F = 0.0000
Residual | 2.68272938 398 .006740526 R-squared = 0.0681
-------------+---------------------------------- Adj R-squared = 0.0634
Total | 2.87875005 400 .007196875 Root MSE = .0821
------------------------------------------------------------------------------
lpsoda | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
prpblck | .1215803 .0257457 4.72 0.000 .0709657 .1721948
lincome | .0765114 .0165969 4.61 0.000 .0438829 .1091399
_cons | -.793768 .1794337 -4.42 0.000 -1.146524 -.4410117
------------------------------------------------------------------------------
. di "lpsoda = " %5.2f _b[_cons] " + " %5.2f _b[prpblck] "prpblck + " %5.4f _b[
> lincome] "lincome; N=" _N " Rsq=" %5.4f e(r2)
lpsoda = -0.79 + 0.12prpblck + 0.0765lincome; N=410 Rsq=0.0681
. di _b[prpblck] * .2 *100
2.4316051
.
v. add var prppov, what happens to coef. of prpblck?
%%stata
reg lpsoda prpblck lincome prppov
di "lpsoda = " %5.2f _b[_cons] " + " %5.2f _b[prpblck] "prpblck + " %5.3f _b[lincome] "lincome + " %5.3f _b[prppov] "prppov; N=" _N " Rsq=" %5.4f e(r2)
. reg lpsoda prpblck lincome prppov
Source | SS df MS Number of obs = 401
-------------+---------------------------------- F(3, 397) = 12.60
Model | .250340622 3 .083446874 Prob > F = 0.0000
Residual | 2.62840943 397 .006620679 R-squared = 0.0870
-------------+---------------------------------- Adj R-squared = 0.0801
Total | 2.87875005 400 .007196875 Root MSE = .08137
------------------------------------------------------------------------------
lpsoda | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
prpblck | .0728072 .0306756 2.37 0.018 .0125003 .1331141
lincome | .1369553 .0267554 5.12 0.000 .0843552 .1895553
prppov | .38036 .1327903 2.86 0.004 .1192999 .6414201
_cons | -1.463333 .2937111 -4.98 0.000 -2.040756 -.8859092
------------------------------------------------------------------------------
. di "lpsoda = " %5.2f _b[_cons] " + " %5.2f _b[prpblck] "prpblck + " %5.3f _b[
> lincome] "lincome + " %5.3f _b[prppov] "prppov; N=" _N " Rsq=" %5.4f e(r2)
lpsoda = -1.46 + 0.07prpblck + 0.137lincome + 0.380prppov; N=410 Rsq=0.0870
.
vi. Correlation between lincome prppov
%%stata
corr lincome prppov
(obs=409)
| lincome prppov
-------------+------------------
lincome | 1.0000
prppov | -0.8385 1.0000
Problem 3.9#
i. Run OLS & estimate.
%%stata
bcuse charity.dta, clear nodesc
reg gift mailsyear giftlast propresp
di "gift = " %5.2f _b[_cons] " + " %5.2f _b[mailsyear] "mailsyear + " %5.3f _b[giftlast] "giftlast + " %5.3f _b[propresp] "propresp ; N=" _N " Rsq=" %5.4f e(r2)
reg gift mailsyear
di "gift = " %5.2f _b[_cons] " + " %5.2f _b[mailsyear] "mailsyear ; N=" _N " Rsq=" %5.4f e(r2)
. bcuse charity.dta, clear nodesc
. reg gift mailsyear giftlast propresp
Source | SS df MS Number of obs = 4,268
-------------+---------------------------------- F(3, 4264) = 129.26
Model | 80700.7052 3 26900.2351 Prob > F = 0.0000
Residual | 887399.134 4,264 208.114244 R-squared = 0.0834
-------------+---------------------------------- Adj R-squared = 0.0827
Total | 968099.84 4,267 226.880675 Root MSE = 14.426
------------------------------------------------------------------------------
gift | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
mailsyear | 2.166259 .3319271 6.53 0.000 1.515509 2.817009
giftlast | .0059265 .0014324 4.14 0.000 .0031184 .0087347
propresp | 15.35861 .8745394 17.56 0.000 13.64405 17.07316
_cons | -4.551518 .8030336 -5.67 0.000 -6.125882 -2.977155
------------------------------------------------------------------------------
. di "gift = " %5.2f _b[_cons] " + " %5.2f _b[mailsyear] "mailsyear + " %5.3f _
> b[giftlast] "giftlast + " %5.3f _b[propresp] "propresp ; N=" _N " Rsq=" %5.4f
> e(r2)
gift = -4.55 + 2.17mailsyear + 0.006giftlast + 15.359propresp ; N=4268 Rsq=0.0
> 834
. reg gift mailsyear
Source | SS df MS Number of obs = 4,268
-------------+---------------------------------- F(1, 4266) = 59.65
Model | 13349.7251 1 13349.7251 Prob > F = 0.0000
Residual | 954750.114 4,266 223.804528 R-squared = 0.0138
-------------+---------------------------------- Adj R-squared = 0.0136
Total | 968099.84 4,267 226.880675 Root MSE = 14.96
------------------------------------------------------------------------------
gift | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
mailsyear | 2.649546 .3430598 7.72 0.000 1.976971 3.322122
_cons | 2.01408 .7394696 2.72 0.006 .5643347 3.463825
------------------------------------------------------------------------------
. di "gift = " %5.2f _b[_cons] " + " %5.2f _b[mailsyear] "mailsyear ; N=" _N "
> Rsq=" %5.4f e(r2)
gift = 2.01 + 2.65mailsyear ; N=4268 Rsq=0.0138
.
iv Add avggift
%%stata
reg gift mailsyear giftlast propresp avggift
corr mailsyear giftlast avggift
. reg gift mailsyear giftlast propresp avggift
Source | SS df MS Number of obs = 4,268
-------------+---------------------------------- F(4, 4263) = 267.33
Model | 194137.386 4 48534.3466 Prob > F = 0.0000
Residual | 773962.453 4,263 181.553472 R-squared = 0.2005
-------------+---------------------------------- Adj R-squared = 0.1998
Total | 968099.84 4,267 226.880675 Root MSE = 13.474
------------------------------------------------------------------------------
gift | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
mailsyear | 1.201168 .312418 3.84 0.000 .5886665 1.81367
giftlast | -.2608573 .0107565 -24.25 0.000 -.2819456 -.239769
propresp | 16.20464 .8175292 19.82 0.000 14.60186 17.80743
avggift | .5269471 .0210811 25.00 0.000 .4856172 .5682769
_cons | -7.327763 .75822 -9.66 0.000 -8.814269 -5.841257
------------------------------------------------------------------------------
. corr mailsyear giftlast avggift
(obs=4,268)
| mailsy~r giftlast avggift
-------------+---------------------------
mailsyear | 1.0000
giftlast | 0.0063 1.0000
avggift | 0.0213 0.9921 1.0000
.
Problem 3.10#
i. Range of educ
%%stata
use htv.dta, clear
sum edu motheduc fatheduc
sum edu motheduc fatheduc if educ==12
sum edu motheduc fatheduc if educ>=12
. use htv.dta, clear
. sum edu motheduc fatheduc
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
educ | 1,230 13.0374 2.354346 6 20
motheduc | 1,230 12.17805 2.278067 0 20
fatheduc | 1,230 12.44715 3.263835 0 20
. sum edu motheduc fatheduc if educ==12
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
educ | 512 12 0 12 12
motheduc | 512 11.76367 1.697474 3 18
fatheduc | 512 11.78516 2.626214 0 20
. sum edu motheduc fatheduc if educ>=12
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
educ | 1,044 13.62835 1.991304 12 20
motheduc | 1,044 12.46264 2.1308 3 20
fatheduc | 1,044 12.86782 3.121349 0 20
.
ii. Regress education on parents education
%%stata
reg educ motheduc fatheduc
Source | SS df MS Number of obs = 1,230
-------------+---------------------------------- F(2, 1227) = 203.68
Model | 1697.9676 2 848.9838 Prob > F = 0.0000
Residual | 5114.31207 1,227 4.1681435 R-squared = 0.2493
-------------+---------------------------------- Adj R-squared = 0.2480
Total | 6812.27967 1,229 5.54294522 Root MSE = 2.0416
------------------------------------------------------------------------------
educ | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
motheduc | .3041971 .0319266 9.53 0.000 .2415603 .366834
fatheduc | .1902858 .0222839 8.54 0.000 .1465669 .2340046
_cons | 6.964355 .3198205 21.78 0.000 6.336899 7.59181
------------------------------------------------------------------------------
iii. Add abil to the model
%%stata
reg educ motheduc fatheduc abil
Source | SS df MS Number of obs = 1,230
-------------+---------------------------------- F(3, 1226) = 305.17
Model | 2912.30705 3 970.769018 Prob > F = 0.0000
Residual | 3899.97262 1,226 3.18105434 R-squared = 0.4275
-------------+---------------------------------- Adj R-squared = 0.4261
Total | 6812.27967 1,229 5.54294522 Root MSE = 1.7836
------------------------------------------------------------------------------
educ | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
motheduc | .1891314 .0285062 6.63 0.000 .1332051 .2450578
fatheduc | .1110854 .0198849 5.59 0.000 .0720733 .1500976
abil | .5024829 .025718 19.54 0.000 .4520268 .552939
_cons | 8.44869 .2895407 29.18 0.000 7.88064 9.01674
------------------------------------------------------------------------------
iv. Add abilsq to the model
%%stata
g abilsq= abil^2
reg educ motheduc fatheduc abil abilsq
di %5.3f _b[abil] "+" %5.3f 2*_b[abilsq] "abil = 0"
di "abil = " _b[abil] / (2*_b[abilsq])
. g abilsq= abil^2
. reg educ motheduc fatheduc abil abilsq
Source | SS df MS Number of obs = 1,230
-------------+---------------------------------- F(4, 1225) = 244.91
Model | 3027.03706 4 756.759264 Prob > F = 0.0000
Residual | 3785.24262 1,225 3.08999397 R-squared = 0.4444
-------------+---------------------------------- Adj R-squared = 0.4425
Total | 6812.27967 1,229 5.54294522 Root MSE = 1.7578
------------------------------------------------------------------------------
educ | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
motheduc | .1901261 .0280957 6.77 0.000 .1350051 .2452472
fatheduc | .1089387 .0196014 5.56 0.000 .0704827 .1473946
abil | .4014624 .0302875 13.26 0.000 .3420413 .4608835
abilsq | .050599 .0083039 6.09 0.000 .0343076 .0668905
_cons | 8.240226 .2874099 28.67 0.000 7.676356 8.804097
------------------------------------------------------------------------------
. di %5.3f _b[abil] "+" %5.3f 2*_b[abilsq] "abil = 0"
0.401+0.101abil = 0
. di "abil = " _b[abil] / (2*_b[abilsq])
abil = 3.9670977
.
v.
%%stata
sum abil if abil<3.967
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
abil | 1,035 1.27629 1.980314 -5.631463 3.955591
Problem 3.11#
i. regress math4 on pctsgle
%%stata
use meapsingle.dta, clear
reg math4 pctsgle
. use meapsingle.dta, clear
(Written by R. )
. reg math4 pctsgle
Source | SS df MS Number of obs = 229
-------------+---------------------------------- F(1, 227) = 138.85
Model | 21625.7284 1 21625.7284 Prob > F = 0.0000
Residual | 35354.2892 227 155.745767 R-squared = 0.3795
-------------+---------------------------------- Adj R-squared = 0.3768
Total | 56980.0176 228 249.912358 Root MSE = 12.48
------------------------------------------------------------------------------
math4 | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
pctsgle | -.8328814 .0706815 -11.78 0.000 -.9721572 -.6936056
_cons | 96.77043 1.596802 60.60 0.000 93.62398 99.91688
------------------------------------------------------------------------------
.
ii. add ‘lmedinc’ and ‘free’ to the model
%%stata
reg math4 pctsgle lmedinc free
Source | SS df MS Number of obs = 229
-------------+---------------------------------- F(3, 225) = 63.85
Model | 26201.7562 3 8733.91873 Prob > F = 0.0000
Residual | 30778.2614 225 136.792273 R-squared = 0.4598
-------------+---------------------------------- Adj R-squared = 0.4526
Total | 56980.0176 228 249.912358 Root MSE = 11.696
------------------------------------------------------------------------------
math4 | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
pctsgle | -.1996453 .1587163 -1.26 0.210 -.5124059 .1131153
lmedinc | 3.560128 5.041704 0.71 0.481 -6.374869 13.49512
free | -.3964185 .070346 -5.64 0.000 -.5350398 -.2577973
_cons | 51.72322 58.47814 0.88 0.377 -63.51166 166.9581
------------------------------------------------------------------------------
iii. correlation between lmedinc and free
%%stata
corr lmedinc free
(obs=229)
| lmedinc free
-------------+------------------
lmedinc | 1.0000
free | -0.7470 1.0000
v. Find VIF
%%stata
reg math4 pctsgle lmedinc free
vif
collin math4 pctsgle lmedinc free // help collin
. reg math4 pctsgle lmedinc free
Source | SS df MS Number of obs = 229
-------------+---------------------------------- F(3, 225) = 63.85
Model | 26201.7562 3 8733.91873 Prob > F = 0.0000
Residual | 30778.2614 225 136.792273 R-squared = 0.4598
-------------+---------------------------------- Adj R-squared = 0.4526
Total | 56980.0176 228 249.912358 Root MSE = 11.696
------------------------------------------------------------------------------
math4 | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
pctsgle | -.1996453 .1587163 -1.26 0.210 -.5124059 .1131153
lmedinc | 3.560128 5.041704 0.71 0.481 -6.374869 13.49512
free | -.3964185 .070346 -5.64 0.000 -.5350398 -.2577973
_cons | 51.72322 58.47814 0.88 0.377 -63.51166 166.9581
------------------------------------------------------------------------------
. vif
Variable | VIF 1/VIF
-------------+----------------------
pctsgle | 5.74 0.174186
lmedinc | 4.12 0.242788
free | 3.19 0.313669
-------------+----------------------
Mean VIF | 4.35
. collin math4 pctsgle lmedinc free // help collin
(obs=229)
Collinearity Diagnostics
SQRT R-
Variable VIF VIF Tolerance Squared
----------------------------------------------------
math4 1.85 1.36 0.5402 0.4598
pctsgle 5.78 2.40 0.1730 0.8270
lmedinc 4.13 2.03 0.2423 0.7577
free 3.64 1.91 0.2749 0.7251
----------------------------------------------------
Mean VIF 3.85
Cond
Eigenval Index
---------------------------------
1 4.3011 1.0000
2 0.6266 2.6199
3 0.0602 8.4558
4 0.0120 18.9037
5 0.0001 217.8816
---------------------------------
Condition Number 217.8816
Eigenvalues & Cond Index computed from scaled raw sscp (w/ intercept)
Det(correlation matrix) 0.0416
.
Problem 3.12#
i.
%%stata
use econmath.dta, clear
sum score actmth acteng if score==100
sum score actmth acteng
. use econmath.dta, clear
(Written by R. )
. sum score actmth acteng if score==100
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
score | 0
actmth | 0
acteng | 0
. sum score actmth acteng
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
score | 856 72.59981 13.40068 19.53 98.44
actmth | 814 23.2113 3.773354 12 36
acteng | 814 22.59459 3.788735 12 34
.
ii.
%%stata
reg score colgpa actmth acteng
Source | SS df MS Number of obs = 814
-------------+---------------------------------- F(3, 810) = 177.94
Model | 57165.5682 3 19055.1894 Prob > F = 0.0000
Residual | 86743.1988 810 107.090369 R-squared = 0.3972
-------------+---------------------------------- Adj R-squared = 0.3950
Total | 143908.767 813 177.009554 Root MSE = 10.348
------------------------------------------------------------------------------
score | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
colgpa | 12.3662 .7150624 17.29 0.000 10.96261 13.76979
actmth | .8833519 .1121984 7.87 0.000 .663118 1.103586
acteng | .051764 .1110631 0.47 0.641 -.1662415 .2697696
_cons | 16.17402 2.800439 5.78 0.000 10.67704 21.67099
------------------------------------------------------------------------------
iii.
%%stata
test actmth
test acteng
. test actmth
( 1) actmth = 0
F( 1, 810) = 61.99
Prob > F = 0.0000
. test acteng
( 1) acteng = 0
F( 1, 810) = 0.22
Prob > F = 0.6413
.
d. R-squared
%%stata
di " R-squared=" %5.4f e(r2)
R-squared=0.3972