interaction terms by mumtaz hussain

8/2/2019 Interaction Terms by Mumtaz Hussain

1/6

bbb

bbbb

bbbb

=+++

=+++ +

=+++

+D

i 0 1 1 2 2

i 0 1 2 2 3 1 2

0 1 1 2 2 3 1 2

1

Interactions between Two Continuous Variables

Y

Y ( )

Gerneral Rule:

Y ( ) (a)

Now change inY

i i i

i i i i i

X X u

X X X X u

X X X X

Xbb bb

bb bb

bbbb

bbb bb b

bb

=++D++ +D

-

+D-=++D++ +D

-+++

+D-=++D++ +D

-++

0 1 1 1 2 2 3 1 1 2

0 1 1 1 2 2 3 1 1 2

0 1 1 2 2 3 1 2

0 1 1 1 1 2 2 3 1 2 3 1 2

0 1 1

Y ( ) [( ) ] (b)

Subtract

Y Y Y ( ) [( ) ]

[ ( )]

Y Y Y

[

X X X X X X

b a

X X X X X X

X X X X

X X X X X X X

X bb

bbb bb b

bbbb

b b

b b

bb

bb

bbb b

+

+D-=++D++ +D

----

D=D+D

D=D+D

D=D+

D=+

D

+D=+++D+ +D

+D=

2 2 3 1 2

0 1 1 1 1 2 2 3 1 2 3 1 2

0 1 1 2 2 3 1 2

1 1 3 1 2

1 1 3 1 2

1 1 3 2

1 3 2

1

2

0 1 1 2 2 2 3 1 2 2

( )]

Y Y Y

Y

Y

Y ( )

Y

Now change in

Y Y ( ) [ ( )]

Y Y

X X X

X X X X X X X

X X X X

X X X

X X X

X X

XX

X

X X X X X X

bbbb b b

bbb b

bbbb

bbbb b b

+++D+ + D

-

+D-=+++D+ +D

-+++

+D-=+++D+ + D

0 1 1 2 2 2 2 3 1 2 3 1 2

0 1 1 2 2 2 3 1 2 2

0 1 1 2 2 3 1 2

0 1 1 2 2 2 2 3 1 2 3 1 2

(c)

Subtract c

Y Y Y ( ) [ ( )]

[ ( )]

Y Y Y

X X X X X X X

a

X X X X X X

X X X X

X X X X X X X

bbbb

bbbb b b

bbbb

b b

bb

-+++

=+++D+ + D

----

D=D+ D

D=D+

0 1 1 2 2 3 1 2

0 1 1 2 2 2 2 3 1 2 3 1 2

0 1 1 2 2 3 1 2

2 2 3 1 2

2 2 3 1

[ ]

Y

Y ( )

X X X X

X X X X X X X

X X X X

X X X

X X

bbD

=+D

2 3 1

2

Y X

X

Mu mt a z K h e r a n i - Ma r c h 2 0 1 2 1


2/6

Interactions Between Binary and Continuous Variables

?Different Intercept , Same Slope (D-S)

Different Intercept , Different Slope (D-D)

?Same Intercept , Different Slope (S-D)

1 0 1 2Y i i i X D ubbb=+++

1 0 1 2 3Y ( )i i i i i X D X D ubbbb=+++ +

1 0 1 3Y ( )i i i i X X D ubbb=++ +

2


3/6

bbb

bbbb

=+++

=+++ +

1 0 1 2

1 0 1 2 3

Interactions between binary and continuous variable

Y

Including Interaction Term as a Regressor

Y ( )

General Rule:

i i i

i i

i i i i i

D X u

D X

D X D X u

bbbb

bbb b

bbb b

bbbb

bb

=+++

+D=+++D+ +D

-

+D-=+++D+ +D

-+++

+D-=+

0 1 2 3

0 1 2 3

0 1 2 3

0 1 2 3

0 1

Y ( ) (a)

Now change in X

Y Y ( ) [ ( )] (b)

Subtract

Y Y Y ( ) [ ( )]

[ ( )]

Y Y Y

D X D X

D X X D X X

b a

D X X D X X

D X D X

bbb b

bbbb

bb

bb

bb

b bbbb

++D+ +D

----

D=D+D

D=D+

D=+

D

D= =+=+

D

2 2 3 3

0 1 2 3

2 3

2 3

2 3

3 2 3 2 3

Y

Y [ ]

Y

Effect of X depend upon D

Yincriment to the effect of X when D 1 ( 1 )

D X X D X D X

D X D X

X D X

X D

DX

X

Mu mt a z K h e r a n i - Ma r c h 2 0 1 2


4/6

bbb

bbbb

bbbb

bbbb

=+++

=+++ +

=+++

=

==+++

i 0 1 2

i 0 1 2 3

0 1 2 3

0 1 2 3

Interactions between Binary and Continuous Variables

Y

Y ( )

General Rule:

Y ( )

Step 1: at ( 0)

E(Y/ , 0)

i i i

i i i i i

i

X D u

X D X D u

X D X D

D

X D X D

bbbb

bb

bbbb

bbbb

bbbb

bbbb

==+++

==+

=

==+++

==+++

==+++

==+++

0 1 2 3

0 1

0 1 2 3

0 1 2 3

0 1 2 3

0 2 1 3

( )

E(Y/ , 0) (0) ( 0)

E(Y/ , 0) (a)

Step 2: at ( 1)

E(Y/ , 1) ( )

E(Y/ , 1) (1) ( 1)

E(Y/ , 1)

E(Y/ , 1) ( ) ( )

X D

X D X X

X D X

D

X D X D X D

X D X X

X D X X

X D X

bbbbbb

bbbbbb

bb

bbb

bbb

b

bb

-

=- =

=+++-+

=+++--

=+

-

=+-

=+-

=

-

=+

0 2 1 3 0 1

0 1 2 3 0 1

2 3

0 2 0

0 2 0

2

1 3

(b)

Step 3: Subtract

E(Y/ , 1) E(Y/ , 0)

( ) ( ) [ ]

Intercept: Subtract

( )

Slope: Subtract

(

b a

X D X D

X X

X X X

X

b a

b a

X b

bbb

bbb

b

-

=+-

=+-

=

1

1 3 1

1 3 1

3

)

( )

X

X X X

X

X



5/6

=- + -

Interactoions Between Binary and Continuous Variables

:

Application to the Student Teacher Ratio (STR) and the percentage of

English Learners (HiEL)

682.2 0.97 5.6 1.28( )

Example

TestScore STR HiEL STR HiEL

=

=

=- +-

=-

2

(11.9) (0.59) (19.5) (0.97)

R 0.305

Low fraction of English Learners ( 0)

682.2 0.97 5.6(0) 1.28( 0)

682.2 0.97

HiEL

TestScore STR STR

TestScore ST +-

=-

=

=- +-

= +- -

=-

0 0

682.2 0.97

High fraction of English Learners ( 1)

682.2 0.97 5.6(1) 1.28( 1)

(682.2 5.6) 0.97 1.28

687.8 2.25

According to these estim

R

TestScore STR

HiEL

TestScore STR STR

TestScore STR STR

TestScore STR

ates, reducing the by 1 unit is predicted

to increase by 0.97 points in districts with low fraction of

but by 2.25 points in districts with high fraction of .

First, the hypot

STR

TestScore

HiEL HiEL

hesis that the two lines are infact the same can be tested by computing

the F-statistic testing the joint hypothesis that the coefficient on ,and the coefficient on the

interaction term

HiEL

STR HiEL

are both zero. This F-statistics is 89.9 which is singinicant at the 1% level.

Second, the hypothesis that tow lines have the same slope can be tested by testing whether the

coefficient on the intera - =-ction term is zero. The t-statistic ( 1.28 0.97 1.32) is less than 1.645 in

absolute vale, so the null hypothesis that the two lines have the same slope cnnot be jrected using a

two-sided test at the

= =

10% significane level.

Third, the hypothesis that the lines have the same intercept can be tested by testing whether the

population coefficients on is zero. The t-statistic is 5.6 19.5 0.29, soHiEL t the hypothesis that

the lines have the same intercept cannot be rejected at the 5% level.


6/6

bbb

bbbb

bb

=+++

=+++ +

===+

1 0 1 1 2 2

1 2

1 0 1 1 2 2 3 1 2

1 1 1 2 0 1 1

Interactions between two binary variables

Y

Including Interaction Term as a RegressorY ( )

Step 1:

E(Y / , 0)

i i i

i i

i i i i i

i i i

D D u

D DD D D D u

D d D D bb

bbbb

bb

bbbb

bbbb

++

===+++

===+

===+++

===+++

2 2 3 1 2

1 1 1 2 0 1 1 2 3 1

1 1 1 2 0 1 1

1 1 1 2 0 1 1 2 3 1

1 1 1 2 0 1 1 2 3 1

( )

E(Y / , 0) (0) ( 0)

E(Y / , 0) (a)

Step 2:

E(Y / , 1) (1) ( 1)

E(Y / , 1) (b)

St

i i

i i

i i

i i

i i

D d D

D d D d d

D d D d

D d D d d

D d D d d

bbbbbb

bbbbbb

bb

-

==- ==

=+++-+

=+++--

=+

1 1 1 2 1 1 1 2

0 1 1 2 3 1 0 1 1

0 1 1 2 3 1 0 1 1

2 3 1

ep 3: Subtract

E(Y / , 1) E(Y / , 0)

[ ]

i i i i

b a

D d D D d D

d d d

d d d

d


interaction terms by mumtaz hussain

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