multiple regression: categorical independent variables and interaction effects · 2018-11-16 ·...

interactionmodels

Categoricalindependentvariables

Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables

With multiplecategoryvariables

With continuousvariables

Multiple regression:Categorical independent variables

and interaction effects

Johan A. ElkinkSchool of Politics & International Relations

University College Dublin

19 November 2018

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



1 Categorical independent variables

Dummy variables

Multiple categories

2 Interaction models

With dummy variables

With multiple category variables

With continuous variables

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Outline


Dummy variables

Multiple categories





interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Introduction

So far, we have discussed regressions where both thedependent and the independent variables were continuous, orof interval/ratio measurement level.

In particular in the social sciences, variables are oftenqualitative or categorical in nature.

When an independent variable is categorical in nature, theestimation remains the same, but the interpretation changes.

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Dummy variables

A dummy variable is a binary variable that can only havevalues 0 or 1.

In regression analysis, a dummy variable can be added as anindependent variable without any problems. If a categoricalvariable is coded differently, you cannot add it to the model.

respnr gender female1 Male 02 Female 13 Male 04 Male 05 Female 16 Female 17 Female 1

In SPSS: RECODE gender

("Male" = 0) ("Female" =

1) INTO female.

In Stata: recode gender (1 =

0) (2 = 1), gen(female)

In R: female <-

car::recode(gender,

"’male’=0; ’female’=1;

else=NA")

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Regression with dummy variables

Model 1: yi = β1, i.e. a model without any independentvariables.

Here you would simply obtain: β1 = y .

(This also shows that regression is close to estimating meansand the t-test is also the same as for comparing means.)

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables




Model 2: yi = β1 + β2di , where D is a dummy variable. Herethere are two scenarios:

di = 0:yi = β1 + β2 · 0 = β1

and we just estimate the mean of Y for the group where D = 0.

di = 1:yi = β1 + β2 · 1 = β1 + β2

and that sum is the estimated mean of Y for the group whereD = 1.

The estimate β2 is therefore the difference in means for thetwo groups.

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables




Model 3: yi = β1 + β2di + β3xi , where D is a dummy variableand X is continuous. Here there are two scenarios:

di = 0:yi = β1 + β2 · 0 + β3xi = β1 + β3xi

and we have an intercept β1 and a slope coefficient β3 forthe group where D = 0.

di = 1:

yi = β1 + β2 · 1 + β3xi = (β1 + β2) + β3xi

and we have an intercept β1 + β2 and a slope coefficient β3for the group where D = 1.

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Dummy variables and interpretation

So, dummy variables test whether the intercept (means)differ—do not interpret the respective coefficient as “if Xincreases by 1 unit, Y increases by ...”

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Dummy variables and t-tests

yi = β1 + β2di + β3xi

In a regression, the t-test for a coefficient tests whether, giventhe other variables in the model, the slope of a line is differentfrom zero, with zero being no effect of X on Y .

H0 : β3 = 0, so under the null, the slope of the line is zero.

In a regression with a dummy variable, the t-test for thatcoefficient tests whether, given the other variables in themodel, the mean of the two groups differ.

H0 : β2 = 0, so under the null, the two groups have the sameintercept.

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Example: degree and earnings

degree 0.504∗∗∗ 0.340∗∗∗

(0.054) (0.058)

ability 0.018∗∗∗

(0.003)

intercept 2.662∗∗∗ 1.754∗∗∗

(0.028) (0.140)

N 540 540R2 0.139 0.204Adjusted R2 0.138 0.201Residual Std. Error 0.552 0.531F -Statistic 87.020∗∗∗ 68.882∗∗∗

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables




●

●

●

●

●●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

● ●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

● ●

●

●

30 40 50 60

12

34

5

Impact of a degree on future earnings

ability

log(

earn

ings

)

●

●

DegreeNo degree

log(earningsi ) = β1 + β2degreei + β3abilityi

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Outline


Dummy variables

Multiple categories





interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Multiple categories

Instead of just two categories, a categorical variables can havemultiple categories, such as party preference or religiousdenomination. To add these to the regression, we split them upin multiple dummy variables.

respnr party ff fg lab sf1 Fianna Fail 1 0 0 02 Sinn Fein 0 0 0 13 Labour 0 0 1 04 Sinn Fein 0 0 0 15 Fianna Fail 1 0 0 06 Fianna Fail 1 0 0 07 Fine Gael 0 1 0 08 Fine Gael 0 1 0 09 Labour 0 0 1 0

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Multiple categories

respnr party ff fg lab sf1 Fianna Fail 1 0 0 02 Sinn Fein 0 0 0 13 Labour 0 0 1 04 Sinn Fein 0 0 0 15 Fianna Fail 1 0 0 06 Fianna Fail 1 0 0 07 Fine Gael 0 1 0 08 Fine Gael 0 1 0 09 Labour 0 0 1 0

Note that in a regression always one category has to be leftout, and all the other results are relative to this referencecategory, e.g.:

Yi = β1 + β2fgi + β3labi + β4sfi ,

such that all coefficients show the difference relative to FiannaFail voters.

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Example: race and earnings

ethblack −0.239∗∗ −0.198∗∗

(0.098) (0.094)

ethhisp −0.155 0.022(0.105) (0.103)

schoolingFather 0.054∗∗∗

(0.008)

intercept 2.821∗∗∗ 2.164∗∗∗

(0.027) (0.095)

N 540 540R2 0.014 0.101Adjusted R2 0.010 0.096Residual Std. Error 0.591 0.565F -Statistic 3.800∗∗ 20.011∗∗∗

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables




●

●

●

●

●●

●

●●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

● ●

●

●

0 5 10 15 20

12

34

5

Impact of race on future earnings

schooling father

log(

earn

ings

)

●

●

●

WhiteBlackHispanic

log(earningsi ) = β1+β2ethblacki+β3ethhispi+β4schoolingFatheri

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables




ethwhite 2.821∗∗∗

(0.027)

ethblack −0.239∗∗ 2.582∗∗∗

(0.098) (0.095)

ethhisp −0.155 2.666∗∗∗

(0.105) (0.101)

intercept 2.821∗∗∗

(0.027)

N 540 540R2 0.014 0.957Adjusted R2 0.010 0.957Residual Std. Error 0.591 0.591F -Statistic 3.800∗∗ 4,026.080∗∗∗

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Outline


Dummy variables

Multiple categories





interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Interactions

So far, we have only been adding variables in an additivemodel.

Imagine, however, that the relation between X and Y woulddepend on the group—e.g. the effect of ability on income isgreater for those with a degree than those without a degree.

We call this an interaction effect, we have to interact thevariable X with D, for example:

yi = β1 + β2xi + β3di + β4xidi .

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Interaction with dummy variables

Model 4: yi = β1 + β2xi + β3di + β4xidi , where D is a dummyvariable and X is continuous. Here there are two scenarios:

di = 0:

yi = β1 + β2xi + β3 · 0 + β4xi · 0 = β1 + β2xi

and we have an intercept β1 and a slope coefficient β2 for thegroup where D = 0.

di = 1:

yi = β1 + β2xi + β3 · 1 + β4xi · 1 = (β1 + β3) + (β2 + β4)xi

and we have an intercept β1 + β3 and a slope coefficientβ2 + β4 for the group where D = 1.

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Including component variables

Note that this also shows the importance of including thecomponent variables that make up the interaction. E.g.:

yi = β1 + β2di + β3xidi ,

where we exclude the variable X by itself, we would have:

di = 0:yi = β1 + β2 · 0 + β3xi · 0 = β1

and we have an intercept β1 and a slope coefficient 0 (!) forthe group where D = 0.

di = 1:

yi = β1 + β2 · 1 + β3xi · 1 = (β1 + β2) + β3xi

and we have an intercept β1 + β2 and a slope coefficient β3 forthe group where D = 1.

So we arbitrarily fix one slope to zero.

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Including component variables

Or similarly:yi = β1 + β2xi + β3xidi ,

where we exclude the dummy variable D by itself:

di = 0:yi = β1 + β2xi + β3xi · 0 = β1 + β2xi

and we have an intercept β1 and a slope coefficient β2 for thegroup where D = 0.

di = 1:

yi = β1 + β2xi + β3xi · 1 = β1 + (β2 + β3)xi

and we have an intercept β1 and a slope coefficient β2 + β3 forthe group where D = 1.

So we fix the value of Y to be identical for the two groupsat the arbitrary point of X = 0.

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Interaction models and t-tests

yi = β1 + β2xi + β3di + β4xidi

So we can think of the following t-tests:

H0 : β2 = 0, so under the null, the slope of the line is zero, forthe group where D = 0.

H0 : β3 = 0, so under the null, the two groups have the sameintercept.

In a regression with an interaction with a dummy variable, thet-test for that coefficient tests whether, given the othervariables in the model, the slope for the two groups differ.

H0 : β4 = 0, so under the null, the two groups have the sameslope between X and Y .

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables




degree 0.340∗∗∗ 0.345(0.058) (0.428)

ability 0.018∗∗∗ 0.018∗∗∗

(0.003) (0.003)

degree × ability −0.0001(0.007)

intercept 1.754∗∗∗ 1.753∗∗∗

(0.140) (0.153)


Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables




●

●

●

●

●●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

● ●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

● ●

●

●

30 40 50 60

12

34

5


ability

log(

earn

ings

)

●

●

DegreeNo degree

log(earningsi ) = β1 +β2degreei +β3abilityi +β4degreei · abilityi

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Example: public sector and earnings

publicSector −0.141∗∗∗ 0.445(0.053) (0.300)

ability 0.026∗∗∗ 0.029∗∗∗

(0.003) (0.003)

publicSector × ability −0.011∗∗

(0.006)

intercept 1.496∗∗∗ 1.329∗∗∗

(0.135) (0.159)


Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Example: public sector and earnings

●

●

●

●

●●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

● ●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

● ●

●

●

30 40 50 60

12

34

5


ability

log(

earn

ings

)

●

●

Public sectorPrivate sector

log(earningsi ) = β1+β2publicSectori+β3abilityi+β4publicSectori ·abilityi

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Outline


Dummy variables

Multiple categories





interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables




log(earningsi ) =β1 + β2blacki + β3hispi + β4abilityi

+ β5blacki · abilityi + β6hispi · abilityi ,

Whites: log(earningsi ) = β1 + β4abilityiBlacks: log(earningsi ) = (β1 + β2) + (β4 + β5)abilityiHispanics: log(earningsi ) = (β1 + β3) + (β4 + β6)abilityi

So β2 and β3 are differences in intercepts, relative to whites; β5and β6 are differences in slopes, relative to whites and t-teststest whether intercepts or slopes, respectively, differ.

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables




ethblack −0.198∗∗ −0.065(0.094) (0.395)

ethhisp 0.022 0.525∗∗

(0.103) (0.229)schoolingFather 0.054∗∗∗ 0.062∗∗∗

(0.008) (0.008)ethblack × schoolingFather −0.011

(0.034)ethhisp × schoolingFather −0.054∗∗

(0.022)intercept 2.164∗∗∗ 2.067∗∗∗

(0.095) (0.104)


Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables




●

●

●

●

●●

●

●●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

● ●

●

●

0 5 10 15 20

12

34

5

Impact of race on future earnings

schooling father

log(

earn

ings

)

●

●

●

WhiteBlackHispanic

interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Outline


Dummy variables

Multiple categories





interactionmodels


Dummyvariables

Multiplecategories

Interactionmodels

With dummyvariables



Interactions between continuous variables

It is possible to interact two continuous variables. Here youexpect the effects of X on Y to gradually change as some thirdvariable Z changes.

yi = β1 + β2xi + β3zi + β4xizi ,

so when we take X as the key independent variable, we have:

Intercept: β1 + β3ziSlope: β2 + β4zi

Both intercept and slope change with Z . These types of modelsare typically somewhat difficult to interpret and there is nostatistical difference between whether the slope between X andY varies for different values of Z , or the slope between Z andY varies for different values of X . It requires a strong theoryon causal relations to be able to make sense of the results.

multiple regression: categorical independent variables and interaction effects · 2018-11-16 ·...

Documents