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8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Nonlinear Regression Functions
(SW Ch. 6)
Everything so far has been linear in theXs
The approximation that the regression function islinear might be good for some variables, but not for
others.
The multiple regression framework can be extended tohandle regression functions that are nonlinear in one
or moreX.
6-
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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The TestScore! STRrelation looks approximately
linear"
6-#
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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$ut theTestScore! average district income relation
looks like it is nonlinear.
6-%
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&f a relation between YandXis nonlinear'
The effect on Yof a change inXdepends on the valueofX! that is, the marginal effect ofXis not constant
( linear regression is mis-specified ! the functionalform is wrong
The estimator of the effect on YofXis biased ! itneednt even be right on average.
The solution to this is to estimate a regressionfunction that is nonlinear inX
6-)
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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The General Nonlinear Population Regression Function
Yi*f+Xi,X#i,",Xki ui, i* ,", n
Assumptions
. E+uiXi,X#i,",Xki * / +same0 implies thatfis the
conditional expectation of Ygiven theXs.#. +Xi,",Xki,Yi are i.i.d. +same.
%. 1enough2 moments exist +same idea0 the precise
statement depends on specificf.
). 3o perfect multicollinearity +same idea0 the precise
statement depends on the specificf.
6-4
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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6-6
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Nonlinear Functions of a Single Independent Variale
(SW Section 6.!)
5ell look at two complementary approaches'
. olynomials inX
The population regression function is approximated
by a 7uadratic, cubic, or higher-degree polynomial#. 8ogarithmic transformations
Yand9orXis transformed by taking its logarithm
this gives a 1percentages2 interpretation that makessense in many applications
6-:
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Example' the TestScore!Incomerelation
Incomei* average district income in the ithdistrict
+thousdand dollars per capita
@uadratic specification'
TestScorei* / Incomei #+Incomei# ui
Aubic specification'
TestScorei* / Incomei #+Incomei#
%+Incomei% ui
6-B
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Estimation of the quaratic specification in ST!T!
generate avginc2 = avginc*avginc; Create a new regressorreg testscr avginc avginc2, r;
Regression with robust standard errors Number of obs = 420 F( 2, 4!" = 42#$%2 &rob ' F = 0$0000 Rs)uared = 0$%%2 Root +- = 2$!24
. Robust testscr . Coef$ td$ -rr$ t &'.t. /%1 Conf$ nterva35 avginc . 6$#%0% $2#04 4$6 0$000 6$6240 4$6!!! avginc2 . $04260#% $004!#06 #$#% 0$000 $0%!0% $062 7cons . 0!$60! 2$0!%4 20$2 0$000 0$%!# 6$00%
The t-statistic onIncome#is -?.?4, so the hypothesis of
linearity is re;ected against the 7uadratic alternative at the
C significance level. 6-/
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Interpreting the estimate regression function'
+a lot the predicted values
TestScore* 6/:.% %.?4Incomei! /./)#%+Incomei#
+#.B +/.#: +/.//)?
6-
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Interpreting the estimate regression function'
+a Aompute 1effects2 for different values ofX
TestScore* 6/:.% %.?4Incomei! /./)#%+Incomei
#
+#.B +/.#: +/.//)?
redicted change in TestScorefor a change in income to
D6,/// from D4,/// per capita'
TestScore* 6/:.% %.?46 ! /./)#%6#
! +6/:.% %.?44 ! /./)#%4#
* %.)
6-#
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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TestScore* 6/:.% %.?4Incomei! /./)#%+Incomei#
redicted 1effects2 for different values ofX
Ahange inIncome+thD per capita TestScore
from 4 to 6 %.)from #4 to #6 .:from )4 to )6 /./
The 1effect2 of a change in income is greater at low than
high income levels +perhaps, a declining marginal benefit
of an increase in school budgets"aution# 5hat about a change from 64 to 66
Font extrapolate outside the range of the data.
6-%
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Estimation of the cu$ic specification in ST!T!
gen avginc6 = avginc*avginc2; Create the cubic regressorreg testscr avginc avginc2 avginc6, r;
Regression with robust standard errors Number of obs = 420 F( 6, 4" = 2!0$# &rob ' F = 0$0000 Rs)uared = 0$%%#4 Root +- = 2$!0!
. Robust
testscr . Coef$ td$ -rr$ t &'.t. /%1 Conf$ nterva35 avginc . %$0#!! $!0!6%0% !$0 0$000 6$2#2% $4004 avginc2 . $0%#0%2 $02#%6! 6$6 0$00 $%2! $06##6 avginc6 . $000#%% $00064! $# 0$04 6$2!e0 $006!!
7cons . 00$0! %$0202 !$ 0$000 %0$04 0$0#
The cubic term is statistically significant at the 4C, but
not C, level6-)
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Testing the null hypothesis of linearity, against the
alternative that the population regression is 7uadratic
and9or cubic, that is, it is a polynomial of degree up to %'
%/' popn coefficients onIncome#andIncome%* /
%' at least one of these coefficients is nonGero.
test avginc2 avginc6; -8ecute the test command after running the regression
( " avginc2 = 0$0( 2" avginc6 = 0$0
F( 2, 4" = 6!$
&rob ' F = 0$0000
The hypothesis that the population regression is linear is
re;ected at the C significance level against the
alternative that it is a polynomial of degree up to %.6-4
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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o&r use moel selection criteria +ma'$e later
!. &ogarithmic functions of Yand'orX
ln+X * the natural logarithm ofX
8ogarithmic transforms permit modeling relationsin 1percentage2 terms +like elasticities, rather than
linearly.
%ere(s )h'' ln+xx ! ln+x * ln x
x
+ x
x
+calculus'ln+ , x
x x=
Numericall''
ln+./ * .//BB4 ./0 ln+./ * ./B4% ./ +sort of
6-:
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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*inear+log case, continue
Yi* / ln+Xi ui
for small X,
9
Y
X X
3ow //X
X
* percentage change inX, so a 1%
increase in X (multiplying X by ".") is associated with
a ." "change in Y.
6-#/
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Example- TestScore .s/ ln0Income1
Kirst defining the new regressor, ln+Income
The model is now linear in ln+Income, so the linear-logmodel can be estimated by =8>'
TestScore* 44:.? %6.)#ln+Incomei
+%.? +.)/
so a C increase inIncomeis associated with an
increase in TestScoreof /.%6 points on the test.
>tandard errors, confidence intervals,R#! all theusual tools of regression apply here.
How does this compare to the cubic model
6-#
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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TestScore* 44:.? %6.)#ln+Incomei
6-##
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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II. &oglinear population regression function
ln+Yi * / Xi ui +b
3ow changeX' ln+Y Y * / +X X +a
>ubtract +a ! +b' ln+Y Y ! ln+Y * X
soY
Y
X
or 9Y Y
X
+small X
6-#%
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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*og+linear case, continue
ln+Yi * / Xi ui
for small X, 9Y Y
X
3ow // YY * percentage change in Y, so a change in
X by one unit (X* ")is associated with a " "%change in Y(Y increases by a factor of "+ ").
Note' 5hat are the units of uiand the >EL
ofractional +proportional deviations
ofor example, SER* .# means"
6-#)
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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III. &oglog population regression function
ln+Yi * / ln+Xi ui +b
3ow changeX' ln+Y Y * / ln+X X +a
>ubtract' ln+Y Y ! ln+Y * Iln+X X ! ln+XJ
soY
Y
X
X
or 9
9
Y Y
X X
+small X
6-#4
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*og+log case, continue
ln+Yi * / ln+Xi ui
for small X,
9
9
Y Y
X X
3ow //
Y
Y
* percentage change in Y, and //
X
X
*
percentage change inX, so a 1% change in X is
associated with a 1% change in Y.
In the log-log specification 1has the interpretationof an elasticity.
6-#6
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Example- ln0 TestScore1 .s/ ln0 Income1
Kirst defining a new dependent variable, ln+TestScore,andthe new regressor, ln+Income
The model is now a linear regression of ln+TestScoreagainst ln+Income, which can be estimated by =8>'
ln+ TestScore * 6.%%6 /./44)ln+Incomei +/.//6 +/.//#
(n C increase inIncomeis associated with an
increase of ./44)C in TestScore+factor of ./44)
How does this compare to the log-linear model
6-#:
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Neither specification seems to fit as )ell as the cu$ic or linear+log6-#?
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Summar$% &ogarithmic transformations
Three cases, differing in whether Yand9orXistransformed by taking logarithms.
(fter creating the new variable+s ln+Y and9or ln+X,the regression is linear in the new variables and the
coefficients can be estimated by =8>.
Hypothesis tests and confidence intervals are nowstandard.
The interpretation of differs from case to case. Ahoice of specification should be guided by ;udgment
+which interpretation makes the most sense in your
application, tests, and plotting predicted values6-#B
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Interactions +et,een Independent Variales
(SW Section 6.-)
erhaps a class siGe reduction is more effective in somecircumstances than in others"
erhaps smaller classes help more if there are many
English learners, who need individual attention
That is,TestScore
STR
might depend onPctE*
More generally,
Y
X
might depend onX#
How to model such 1interactions2 betweenXandX#
5e first consider binaryXs, then continuousXs
6-%/
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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(a) Interactions et,een t,o inar$ ariales
Yi* / 2i #2#i ui
2i,2#iare binary
is the effect of changing2*/ to2*. &n this
specification, this effect oesn(t epen on the .alue of
2#.
To allow the effect of changing2to depend on2#,
include the 1interaction term22i2#ias a regressor'
Yi* / 2i #2#i %+2i2#i ui
6-%
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Interpreting the coefficients
Yi* / 2i #2#i %+2i2#i ui
Neneral rule' compare the various cases
E+Yi2i*/,2#i*# * / ## +b
E+Yi2i*,2#i*# * / ## %# +a
subtract +a ! +b'
E+Yi2i*,2#i*# !E+Yi2i*/,2#i*# * %#
The effect of2depends on #+what we wanted
%* increment to the effect of2, when2#*
6-%#
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Example' TestScore, STR, English learners
8et
%iSTR*
if #/
/ if #/
STR
STR
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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() Interactions et,een continuous and inar$
ariales
Yi* / 2i #Xi ui
2iis binary,Xis continuous
(s specified above, the effect on YofX+holding
constant2 *#, which does not depend on2
To allow the effect ofX to depend on2, include the
1interaction term22iXias a regressor'
Yi* / 2i #Xi %+2iXi ui
6-%)
h ff
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Interpreting the coefficients
Yi* / 2i #Xi %+2iXi ui
Neneral rule' compare the various cases
Y * / 2 #X %+2X +b
3ow changeX'
Y Y* / 2 #+XX %I2+XXJ +asubtract +a ! +b'
Y* #X %2X orY
X
* # %2
The effect ofXdepends on2+what we wanted
%* increment to the effect ofX, when2*
Example' TestScore, STR, %iE*+* ifPctE*#/6-%4
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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TestScore* 6?#.# ! /.B:STR 4.6%iE*! .#?+STR%iE* +.B +/.4B +B.4 +/.B:
Testing various hypotheses' The two regression lines have the same slope the
coefficient on STR%iE*is Gero'
t* !.#?9/.B: * !.%#cant re;ect
The two regression lines have the same intercept thecoefficient on%iE*is Gero'
t* !4.69B.4 * /.#B cant re;ect
Example, ct/
TestScore* 6?#.# ! /.B:STR 4.6%iE*! .#?+STR%iE*, +.B +/.4B +B.4 +/.B:
6-%:
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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!ointhypothesis that the two regression lines are the
same population coefficient on%iE** / and
population coefficient on STR%iE** /'
F* ?B.B) +p-value O .// //
5hy do we re;ect the ;oint hypothesis but neither
individual hypothesis
Aonse7uence of high but imperfect multicollinearity'
high correlation between%iE*and STR%iE*
3inar'+continuous interactions- the t)o regression lines
Yi* / 2i #Xi %+2iXi ui
6-%?
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8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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6-)/
( ) I t ti t t ti i l
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(c) Interactions et,een t,o continuous ariales
Yi* / Xi #X#i ui
X,X#are continuous
(s specified, the effect ofXdoesnt depend onX#
(s specified, the effect ofX#doesnt depend onX To allow the effect ofXto depend onX#, include the
1interaction term2XiX#ias a regressor'
Yi* / Xi #X#i %+XiX#i ui
6-)
" ffi i t i ti ti i t ti
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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"oefficients in continuous+continuous interactions
Yi* / Xi #X#i %+XiX#i ui
Neneral rule' compare the various cases
Y* / X #X# %+XX# +b
3ow changeX'
Y Y* / +XX #X# %I+XXX#J +a
subtract +a ! +b'
Y* X %X#X or
Y
X
* # %X#
The effect ofXdepends onX#+what we wanted %* increment to the effect ofXfrom a unit change
inX#
Example' TestScore, STR, PctE*6-)#
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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TestScore* 6?6.% ! .#STR! /.6:PctE* .//#+STRPctE*, +.? +/.4B +/.%: +/./B
The estimated effect of class siGe reduction is nonlinear
because the siGe of the effect itself depends onPctE*'
TestScore
STR
* !.# .//#PctE*
PctE* TestScore
STR
/ !.#
#/C !.#.//##/ * !./Example, ct- h'pothesis tests
TestScore* 6?6.% ! .#STR! /.6:PctE* .//#+STRPctE*, +.? +/.4B +/.%: +/./B
6-)%
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Foes population coefficient on STRPctE** /
t* .//#9./B * ./6 cant re;ect null at 4C level
Foes population coefficient on STR* /
t* !.#9/.4B * !.B/ cant re;ect null at 4C level
Fo the coefficients on bothSTRandSTRPctE** /
F* %.?B +p-value * ./# re;ect null at 4C level+
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Application% Nonlinear 0ffects on 1est Scores
of the Student1eacher Ratio
(SW Section 6.2)
Kocus on two 7uestions'
. (re there nonlinear effects of class siGe reduction ontest scores +Foes a reduction from %4 to %/ have
same effect as a reduction from #/ to 4
#. (re there nonlinear interactions betweenPctE*and
STR +(re small classes more effective when there are
many English learners
6-)4
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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5hat is a good 1base2 specification
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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5hat is a good base specification
6-):
The TestScore Income relation
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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The TestScore!Incomerelation
(n advantage of the logarithmic specification is that it is
better behaved near the ends of the sample, especially large
values of income.6-)?
+ase specification
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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+ase specification
Krom the scatterplots and preceding analysis, here are
plausible starting points for the demographic control
variables'
Fependent variable' TestScore
Independent ariale Functional form
PctE* linear*unchP"T linear
Income ln+Income+or could use cubic
6-)B
4uestion 5'
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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4uestion 5'&nvestigate by considering a polynomial in STR
TestScore* #4#./ 6).%%STR! %.)#STR#
./4BSTR%
+6%.6 +#).?6 +.#4 +./#
! 4.):%iE*! .)#/*unchP"T .:4ln+Income +./% +./#B +.:?
&nterpretation of coefficients on'
%iE**unchP"T ln+Income STR, STR#, STR%
6-4/
Interpreting the regression function ia plots
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Interpreting the regression function ia plots
+preceding regression is labeled +4 in this figure
6-4
Are the higher order terms in "#$ statisticall$
8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc
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Are the higher order terms in"#$statisticall$
significant3
TestScore* #4#./ 6).%%STR! %.)#STR# ./4BSTR%+6%.6 +#).?6 +.#4 +./#
! 4.):%iE*! .)#/*unchP"T .:4ln+Income
+./% +./#B +.:?
+a%/' 7uadratic in STRv.%' cubic in STR
t* ./4B9./# * #.?6 +p* .//4
+b%/' linear in STRv.%' nonlinear9up to cubic in STRF* 6.: +p* .//#
6-4#
4uestion 5#' STR-PctE* interactions
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4uestion 5#' STR-PctE* interactions
+to simplify things, ignore STR#, STR%terms for now
TestScore* 64%.6 ! .4%STR 4.4/%iE*! .4?%iE*STR +B.B +.%) +B.?/ +.4/
! .)*unchP"T #.#ln+Income +./#B +.?/
&nterpretation of coefficients on'
STR%iE* +wrong sign%iE*STR*unchP"T ln+Income
6-4%
Interpreting the regression functions .ia plots'
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Interpreting the regression functions .ia plots'
TestScore* 64%.6 ! .4%STR 4.4/%iE*! .4?%iE*STR
+B.B +.%) +B.?/ +.4/! .)*unchP"T #.#ln+Income +./#B +.?/
4Real,orld5 (4polic$5 or 4economic5) importance ofthe interaction term%
TestScore
STR
* !.4% ! .4?%iE**
.# if
.4% if /
%iE*
%iE*
= =
The difference in the estimated effect of reducing theSTRis substantial0 class siGe reduction is more
effective in districts with more English learners
6-4)
Is the interaction effect statisticall$ significant3
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Is the interaction effect statisticall$ significant3
TestScore* 64%.6 ! .4%STR 4.4/%iE*! .4?%iE*STR
+B.B +.%) +B.?/ +.4/
! .)*unchP"T #.#ln+Income +./#B +.?/
+a%/' coeff. on interaction*/ v.%' nonGero interaction
t* !.: not significant at the /C level
+b%/' both coeffs involving STR* / vs.
%' at least one coefficient is nonGero +STRentersF* 4.B# +p* .//%
Next- specifications )ith pol'nomials interactions