Overte pomocí dostupných dat, ze osoby zijící ve svazku manzelském vydelávají více nez jejich svobodné protejsky s jinak srovnatelnými charakteristikami. Navrhnete vhodný funkcní tvar modelu, za kontrolní promenné volte vzdelání, pracovní zkusenosti, indikátor bydliste v urbánních oblastech a pohlaví.
getwd()## [1] "C:/Users/Lukas/Desktop/Ekonometrie/AKM/cviceni 2017/cviceni 6"wage = read.csv("wage.csv")
head(wage) ## wage educ exper tenure nonwhite female married urban
## 1 3.10 11 2 0 0 1 0 1
## 2 3.24 12 22 2 0 1 1 1
## 3 3.00 11 2 0 0 0 0 0
## 4 6.00 8 44 28 0 0 1 1
## 5 5.30 12 7 2 0 0 1 0
## 6 8.75 16 9 8 0 0 1 1\(\log{wage}=\beta_0+\beta_1educ+\beta_2exper+\beta_3exper^2+\beta_4female+\beta_5married+\beta_6urban+\epsilon\)
model = lm(log(wage) ~ educ + exper + I(exper^2) + female + married + urban, data=wage)
summary(model) ##
## Call:
## lm(formula = log(wage) ~ educ + exper + I(exper^2) + female +
## married + urban, data = wage)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.83272 -0.27126 -0.01534 0.24222 1.27378
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.3354092 0.1012208 3.314 0.000985 ***
## educ 0.0762198 0.0070494 10.812 < 2e-16 ***
## exper 0.0360291 0.0051518 6.993 8.30e-12 ***
## I(exper^2) -0.0006364 0.0001110 -5.734 1.67e-08 ***
## female -0.3319148 0.0360529 -9.206 < 2e-16 ***
## married 0.0811993 0.0415425 1.955 0.051165 .
## urban 0.1773454 0.0408536 4.341 1.71e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.406 on 519 degrees of freedom
## Multiple R-squared: 0.4232, Adjusted R-squared: 0.4166
## F-statistic: 63.48 on 6 and 519 DF, p-value: < 2.2e-16library(lmtest) ## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericbptest(model)##
## studentized Breusch-Pagan test
##
## data: model
## BP = 13.322, df = 6, p-value = 0.0382library(sandwich) ## Warning: package 'sandwich' was built under R version 3.2.5coeftest(model, vcov =vcovHC(model, vcov="HC1"))##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.33540920 0.10906411 3.0753 0.002213 **
## educ 0.07621980 0.00805251 9.4654 < 2.2e-16 ***
## exper 0.03602915 0.00502497 7.1700 2.595e-12 ***
## I(exper^2) -0.00063638 0.00010470 -6.0782 2.356e-09 ***
## female -0.33191477 0.03597483 -9.2263 < 2.2e-16 ***
## married 0.08119934 0.04144442 1.9592 0.050620 .
## urban 0.17734539 0.04133441 4.2905 2.127e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Vidíme, ze s kazdým rokem vzdelání se dle naseho modelu zvysuje mzda v prumeru pribilzne o 7.6 %. Dále vidíme, ze s rostoucími zkusenostmi se skutecne mzda nejprve zvysuje a pak zacíná klesat (okolo 30 let). Zena má v prumeru za jinak stejných okolností o 33 % nizsí mzdu nez muz: opet jde pouze o priblizný údaj, presný získáme jako
\(100[\exp{(b_4)}-1]=100[\exp{(-0.3319)}-1]=-28.24\)
Osoba zijící v manzelském svazku má za jinak stejných okolností v prumeru o 8 % vyssí mzdu nez osoba svobodná a osoba zijící ve meste o 17,7 % vyssí nez osoba ve meste nezijící.
Nyní otestujeme nulovou hypotézu, ze osoby zijící v manzelském svazku vydelávají více nez jejich svobodné protejsky. Nulovou hypotézu muzeme formulovat následovne:
\(H_0: \beta_5=0\)
Alternativní hypotéza si v tomto prípade zvolíme jako jednostrannou, protoze se domníváme, ze lidé v manzelském svazku vydelávají více:
\(H_0: \beta_5>0\)
V tomto prípade si musíte dát pozor. P-hodnota je konstruována pro oboustrannou hypotézu, takze pro jednostrannou, bude platit:
\(p=0.0506/2=0.0253\)
Jako hladinu významnosti \(\alpha=0.05\) a provnáme:
\(0.0253<0.05\)
Zamítáme nulovou hypotézu na hladine významnosti \(\alpha=0.05\). Tedy na petiprocentní hladine významnosti proto zamítáme nulovou hypotézu o tom, ze manzelský stav nemá na mzdu vliv, ve prospech jeho jednostranné alternativy.
Co kdyz budeme chtít rozlisit vliv manzelství na mzdu muze a na mzdu zeny? Nepomuze nám jen dosdit do predchozí rovnice jednicku pro female a marrried. Tedy pridáme dalsí promennou, která bude predstavovat interakci promenné married a promenné female.
\(\log{wage}=\beta_0+\beta_1educ+\beta_2exper+\beta_3exper^2+\beta_4female+\beta_5married+\beta_6urban+\beta_7 married \times female+\epsilon\)
V takto specifikovaném modelu je referencní kategorií (base group) svobodný muz.
Za jinak stejných okolností má svobodná zena o \(100\beta_4 \%\) vyssí mzdu nez svobodný muz. Zenatý muz má o \(100\beta_5 \%\) vyssí mzdu nez svobodný muz. Vdaná zena má o \(100 (\beta_4+\beta_5+\beta_7) \%\) vyssí mzdu nez svobodný muz.
model2 = lm(log(wage) ~ educ + exper + I(exper^2) + female + married + urban + married:female, data=wage)
summary(model2)##
## Call:
## lm(formula = log(wage) ~ educ + exper + I(exper^2) + female +
## married + urban + married:female, data = wage)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.91761 -0.23330 -0.03243 0.24144 1.19371
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.2351263 0.1018129 2.309 0.0213 *
## educ 0.0760139 0.0069209 10.983 < 2e-16 ***
## exper 0.0351740 0.0050613 6.950 1.11e-11 ***
## I(exper^2) -0.0006239 0.0001090 -5.724 1.76e-08 ***
## female -0.1326814 0.0564849 -2.349 0.0192 *
## married 0.2536952 0.0558202 4.545 6.85e-06 ***
## urban 0.1682179 0.0401583 4.189 3.30e-05 ***
## female:married -0.3278547 0.0724387 -4.526 7.46e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3986 on 518 degrees of freedom
## Multiple R-squared: 0.4452, Adjusted R-squared: 0.4377
## F-statistic: 59.38 on 7 and 518 DF, p-value: < 2.2e-16bptest(model2)##
## studentized Breusch-Pagan test
##
## data: model2
## BP = 13.746, df = 7, p-value = 0.05589coeftest(model2, vcov =vcovHC(model2, vcov="HC1"))##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.23512626 0.11054638 2.1269 0.03390 *
## educ 0.07601387 0.00794546 9.5670 < 2.2e-16 ***
## exper 0.03517397 0.00496021 7.0912 4.381e-12 ***
## I(exper^2) -0.00062392 0.00010337 -6.0356 3.021e-09 ***
## female -0.13268144 0.05789204 -2.2919 0.02231 *
## married 0.25369522 0.05581832 4.5450 6.844e-06 ***
## urban 0.16821790 0.04114749 4.0882 5.039e-05 ***
## female:married -0.32785473 0.07222272 -4.5395 7.018e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Vidíme, ze svobodná zena by za jinak stejných okolností mela v prumeru priblizne o 13 % nizsí mzdu nez svobodný muz.
Zenatý muz by mel priblizne o 25 % vyssí mzdu nez svobodný muz.
Vdaná zena by mela priblizne o 21 % nizsí mzdu nez svobodný muz (-0.13 + 0.25 - 0.33 = -0.21).
Vdaná zena by tudíz mela priblizne o 8 % nizsí mzdu nez svobodná zena.
Zdá se, ze vstupem do manzelského svazku se plat muzu zvýsí a plat zen snízí.
Chceme otestovat, ze dopad manzelství na mzdu muzu a zen se lisí.
Rozdíl mezi logaritmickou mzdou vdané a svobodné zeny je, za jinak stejných okolností, \(\beta_4+\beta_5+\beta_7-\beta_4\). Testujeme, zda je tento rozdíl pro muze stejný jako pro zeny, tedy zda: \((\beta_5) - (\beta_4+\beta_5+\beta_7-\beta_4)=0\).
Coz je:
\(H_0: \beta_7=0\)
\(H_1: \beta_7<0\)
Dále otestujeme, zdali je efekt manzelství na mzdu zeny statisticky významný.
\(H_0: \beta_5=\beta_7=0\)
\(H_1: non H_0\)
regrese_omezena=lm(log(wage) ~ educ + exper + I(exper^2) + female + urban, data=wage)
waldtest(model2,regrese_omezena, vcov = vcovHC) # robustní verze F-testu## Wald test
##
## Model 1: log(wage) ~ educ + exper + I(exper^2) + female + married + urban +
## married:female
## Model 2: log(wage) ~ educ + exper + I(exper^2) + female + urban
## Res.Df Df F Pr(>F)
## 1 518
## 2 520 -2 12.353 5.742e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1library(AER)## Warning: package 'AER' was built under R version 3.2.5## Loading required package: car## Warning: package 'car' was built under R version 3.2.5## Loading required package: survivallibrary(car)data("CollegeDistance")head(CollegeDistance)## gender ethnicity score fcollege mcollege home urban unemp wage distance
## 1 male other 39.15 yes no yes yes 6.2 8.09 0.2
## 2 female other 48.87 no no yes yes 6.2 8.09 0.2
## 3 male other 48.74 no no yes yes 6.2 8.09 0.2
## 4 male afam 40.40 no no yes yes 6.2 8.09 0.2
## 5 female other 40.48 no no no yes 5.6 8.09 0.4
## 6 male other 54.71 no no yes yes 5.6 8.09 0.4
## tuition education income region
## 1 0.88915 12 high other
## 2 0.88915 12 low other
## 3 0.88915 12 low other
## 4 0.88915 12 low other
## 5 0.88915 13 low other
## 6 0.88915 12 low othergender = factor indicating gender.
ethnicity=factor indicating ethnicity (African-American, Hispanic or other).
score=base year composite test score. These are achievement tests given to high school seniors in the sample.
fcollege=factor. Is the father a college graduate?
mcollege=factor. Is the mother a college graduate?
home=factor. Does the family own their home?
urban=factor. Is the school in an urban area?
unemp=county unemployment rate in 1980.
wage=state hourly wage in manufacturing in 1980.
distance=distance from 4-year college (in 10 miles).
tuition=average state 4-year college tuition (in 1000 USD).
education=number of years of education.
income=factor. Is the family income above USD 25,000 per year?
region=factor indicating region (West or other).
summary(CollegeDistance)## gender ethnicity score fcollege mcollege
## male :2139 other :3050 Min. :28.95 no :3753 no :4088
## female:2600 afam : 786 1st Qu.:43.92 yes: 986 yes: 651
## hispanic: 903 Median :51.19
## Mean :50.89
## 3rd Qu.:57.77
## Max. :72.81
## home urban unemp wage distance
## no : 852 no :3635 Min. : 1.400 Min. : 6.590 Min. : 0.000
## yes:3887 yes:1104 1st Qu.: 5.900 1st Qu.: 8.850 1st Qu.: 0.400
## Median : 7.100 Median : 9.680 Median : 1.000
## Mean : 7.597 Mean : 9.501 Mean : 1.803
## 3rd Qu.: 8.900 3rd Qu.:10.150 3rd Qu.: 2.500
## Max. :24.900 Max. :12.960 Max. :20.000
## tuition education income region
## Min. :0.2575 Min. :12.00 low :3374 other:3796
## 1st Qu.:0.4850 1st Qu.:12.00 high:1365 west : 943
## Median :0.8245 Median :13.00
## Mean :0.8146 Mean :13.81
## 3rd Qu.:1.1270 3rd Qu.:16.00
## Max. :1.4042 Max. :18.00\(score=\beta_0+\beta_1 gender+\beta_2 home+\beta_3 fcollege+\beta_4 mcollege+\beta_5 urban+\beta_6 distance+\) \(+ \beta_7 educ +\beta_8 tution + \beta_9 income +\epsilon\)
regrese1=lm(score~gender+home+fcollege+mcollege+urban+distance+education+tuition+income,data=CollegeDistance)
summary(regrese1)##
## Call:
## lm(formula = score ~ gender + home + fcollege + mcollege + urban +
## distance + education + tuition + income, data = CollegeDistance)
##
## Residuals:
## Min 1Q Median 3Q Max
## -21.0097 -5.7068 -0.0247 5.4090 24.5268
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20.41512 0.95330 21.415 < 2e-16 ***
## genderfemale -1.18221 0.21924 -5.392 7.30e-08 ***
## homeyes 1.46751 0.28815 5.093 3.67e-07 ***
## fcollegeyes 1.99928 0.31516 6.344 2.45e-10 ***
## mcollegeyes 0.96359 0.35450 2.718 0.00659 **
## urbanyes -1.47477 0.27190 -5.424 6.12e-08 ***
## distance -0.09985 0.05039 -1.981 0.04760 *
## education 1.99963 0.06462 30.945 < 2e-16 ***
## tuition 2.66690 0.32296 8.258 < 2e-16 ***
## incomehigh 0.38590 0.26305 1.467 0.14243
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.491 on 4729 degrees of freedom
## Multiple R-squared: 0.2604, Adjusted R-squared: 0.259
## F-statistic: 185 on 9 and 4729 DF, p-value: < 2.2e-16Nez zacneme intepretovat statistickou významnost parametru, musíme se ujistit, ze jsou splneny dané predpoklad! Víte jaké jsou predpoklady pro t-test? Co se stane, pokud se náhodná slozka \(\epsilon\) nerídí normálním rozdelením?
bptest(regrese1)##
## studentized Breusch-Pagan test
##
## data: regrese1
## BP = 39.115, df = 9, p-value = 1.098e-05Otestujeme normalitu residuí.
install.packages("normtest")library(normtest)## Warning: package 'normtest' was built under R version 3.2.5Jako jeden z nejznámejsích testu na normalitu je Jarque-Bera test.
\(H_0: Normalita\) vs. \(H_1: Non normalita\)
jb.norm.test(regrese1$residuals, nrepl=2000)##
## Jarque-Bera test for normality
##
## data: regrese1$residuals
## JB = 45.913, p-value < 2.2e-16Pro odhad smerodatných chyb, pouzijeme robustní metodu.
coeftest(regrese1,vcovHC)##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20.415118 0.979098 20.8509 < 2.2e-16 ***
## genderfemale -1.182206 0.220081 -5.3717 8.174e-08 ***
## homeyes 1.467506 0.300459 4.8842 1.073e-06 ***
## fcollegeyes 1.999279 0.302055 6.6189 4.020e-11 ***
## mcollegeyes 0.963594 0.353359 2.7270 0.006416 **
## urbanyes -1.474775 0.275047 -5.3619 8.627e-08 ***
## distance -0.099848 0.047427 -2.1053 0.035318 *
## education 1.999632 0.065858 30.3626 < 2.2e-16 ***
## tuition 2.666899 0.326190 8.1759 3.737e-16 ***
## incomehigh 0.385900 0.257680 1.4976 0.134306
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\(score=\beta_0+\beta_1 gender+\beta_2 home+\beta_3 fcollege+\beta_4 mcollege+\beta_5 gender \times mcollege+\beta_6 urban+\beta_7 distance+\) \(+ \beta_8 educ +\beta_9 tution +\epsilon\)
Nejprve si musíme vytvorit nové, interakcní promenné.
attach(CollegeDistance)## The following object is masked _by_ .GlobalEnv:
##
## wagegender=as.numeric(gender)-1
mcollege=as.numeric(mcollege)-1
gender_mcollege=gender*mcollege
gender_educ=gender*educationregrese2=lm(score~gender+home+fcollege+mcollege+gender:mcollege+urban+distance+education+tuition,data=CollegeDistance)
regrese2=lm(score~gender+home+fcollege+mcollege+gender_mcollege+urban+distance+education+tuition,data=CollegeDistance)
summary(regrese2)##
## Call:
## lm(formula = score ~ gender + home + fcollege + mcollege + gender_mcollege +
## urban + distance + education + tuition, data = CollegeDistance)
##
## Residuals:
## Min 1Q Median 3Q Max
## -21.0922 -5.6806 0.0175 5.4429 24.7888
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20.22040 0.95389 21.198 < 2e-16 ***
## genderfemale -1.06905 0.23581 -4.534 5.94e-06 ***
## homeyes 1.50788 0.28663 5.261 1.50e-07 ***
## fcollegeyes 2.09531 0.30569 6.854 8.08e-12 ***
## mcollegeyes 1.50551 0.48880 3.080 0.00208 **
## gender_mcollege -0.92709 0.63411 -1.462 0.14380
## urbanyes -1.50189 0.27141 -5.534 3.31e-08 ***
## distance -0.10465 0.05033 -2.079 0.03764 *
## education 2.01328 0.06427 31.325 < 2e-16 ***
## tuition 2.67856 0.32278 8.299 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.491 on 4729 degrees of freedom
## Multiple R-squared: 0.2604, Adjusted R-squared: 0.259
## F-statistic: 185 on 9 and 4729 DF, p-value: < 2.2e-16Opet se musíme presvedcit, zdali jsou splneny GM.
bptest(regrese2)##
## studentized Breusch-Pagan test
##
## data: regrese2
## BP = 35.515, df = 9, p-value = 4.833e-05Pouzijeme robustní odhad chyb.
coeftest(regrese2,vcovHC)##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20.220400 0.979765 20.6380 < 2.2e-16 ***
## genderfemale -1.069052 0.236184 -4.5264 6.148e-06 ***
## homeyes 1.507877 0.299119 5.0411 4.802e-07 ***
## fcollegeyes 2.095310 0.293771 7.1325 1.135e-12 ***
## mcollegeyes 1.505508 0.503255 2.9915 0.00279 **
## gender_mcollege -0.927089 0.640095 -1.4484 0.14758
## urbanyes -1.501893 0.274107 -5.4792 4.494e-08 ***
## distance -0.104647 0.047467 -2.2046 0.02753 *
## education 2.013279 0.065470 30.7511 < 2.2e-16 ***
## tuition 2.678556 0.325819 8.2210 2.582e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Jak interpretovat \(\beta_1\) ?
\(E[score|gender=1,home=1,fcollege=1,mcollege=1,gender_mcollege=1,urban=1,distance=0,education=12,tuition=1.2]=?\)
Jakou hodnotu score ocekáváte, pokud matka dívky nemá VS a ostatní promenné jsou stejné jako v predeslé otázce.
Jakou hodnotu score ocekávate pro muze, jehoz oba rodice mají VS a ostatní promenné jsou stejné jako v predeslé otázce.
linearHypothesis(regrese2,c("mcollegeyes=0","gender_mcollege=0"),test="F",white.adjust="hc0")## Linear hypothesis test
##
## Hypothesis:
## mcollegeyes = 0
## gender_mcollege = 0
##
## Model 1: restricted model
## Model 2: score ~ gender + home + fcollege + mcollege + gender_mcollege +
## urban + distance + education + tuition
##
## Note: Coefficient covariance matrix supplied.
##
## Res.Df Df F Pr(>F)
## 1 4731
## 2 4729 2 5.0088 0.006714 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\(\textbf{RESENI}\)
Vzdy musíte mít na pameti ceteris paribus efekt. Takze \(\beta_1\) nejde interpretovat samostatne, bez prihlédnutí k \(\beta_5\). Pokud tedy se jedná o zenu, jejíz matka mela VS vzdelání, tak rozdíl ve score oproti muzi, jehoz matka mela také VS vzdelání, je dán \(\beta_1+\beta_5\).
\(score=\beta_0+\beta_1 gender+\beta_2 home+\beta_3 fcollege+\beta_4 mcollege+\beta_5 urban+\beta_6 distance+\) \(+ \beta_7 educ +\beta_8 gender \times educ + \beta_9 tution +\epsilon\)
regrese3=lm(score~gender+home+fcollege+mcollege+urban+distance+education+gender_educ+tuition,data=CollegeDistance)
summary(regrese3)##
## Call:
## lm(formula = score ~ gender + home + fcollege + mcollege + urban +
## distance + education + gender_educ + tuition, data = CollegeDistance)
##
## Residuals:
## Min 1Q Median 3Q Max
## -21.819 -5.684 0.014 5.421 24.065
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.46106 1.31545 13.274 < 2e-16 ***
## genderfemale 4.11153 1.70044 2.418 0.01565 *
## homeyes 1.50843 0.28638 5.267 1.45e-07 ***
## fcollegeyes 2.08954 0.30528 6.845 8.65e-12 ***
## mcollegeyes 1.04362 0.35288 2.957 0.00312 **
## urbanyes -1.51028 0.27121 -5.569 2.71e-08 ***
## distance -0.10276 0.05028 -2.044 0.04106 *
## education 2.21889 0.09231 24.037 < 2e-16 ***
## gender_educ -0.38441 0.12211 -3.148 0.00165 **
## tuition 2.65743 0.32260 8.238 2.25e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.484 on 4729 degrees of freedom
## Multiple R-squared: 0.2617, Adjusted R-squared: 0.2602
## F-statistic: 186.2 on 9 and 4729 DF, p-value: < 2.2e-16bptest(regrese3)##
## studentized Breusch-Pagan test
##
## data: regrese3
## BP = 39.87, df = 9, p-value = 8.021e-06coeftest(regrese3,vcovHC)##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.461062 1.345401 12.9783 < 2.2e-16 ***
## genderfemale 4.111529 1.724341 2.3844 0.017146 *
## homeyes 1.508434 0.299228 5.0411 4.802e-07 ***
## fcollegeyes 2.089544 0.293233 7.1259 1.190e-12 ***
## mcollegeyes 1.043619 0.351844 2.9661 0.003031 **
## urbanyes -1.510277 0.274363 -5.5047 3.894e-08 ***
## distance -0.102755 0.047252 -2.1746 0.029706 *
## education 2.218891 0.092544 23.9767 < 2.2e-16 ***
## gender_educ -0.384408 0.122395 -3.1407 0.001696 **
## tuition 2.657430 0.325590 8.1619 4.191e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\(\textbf{RESENI}\)
Opet ceteris paribus. Nelze nezahrnout vliv vzdelání. Nyní vlastne porovnáváme rozdíl mezi muzem a zenou, kdy navíc kontolujeme vzdelání obou pohlaví. Pokud byste interpretovvali pouze odhad parametru \(\beta_1\), tak by to znamenalo, jaký je rozdíl ve score mezi muzem a zenou, pokud mají vzdelání =0. Vzdy si nejprve dosadte do rovnice 0 a 1 a pak vidíte co máte interpretovat.