Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc

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-


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The TestScore! STRrelation looks approximately

linear"

6-#


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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-)


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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


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6-6


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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-:


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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


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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-/


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Interpreting the estimate regression function'

+a lot the predicted values

TestScore* 6/:.% %.?4Incomei! /./)#%+Incomei#

+#.B +/.#: +/.//)?

6-


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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-#


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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.

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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-)


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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


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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-:


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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.

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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-#


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TestScore* 44:.? %6.)#ln+Incomei

6-##


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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

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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-#)


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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

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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.

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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

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Neither specification seems to fit as )ell as the cu$ic or linear+log6-#?


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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


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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

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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

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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#*

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Example' TestScore, STR, English learners

8et

%iSTR*

if #/

/ if #/

STR

STR


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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-%)

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Interpreting the coefficients



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


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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:

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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


6-%?


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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


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"oefficients in continuous+continuous interactions

Yi* / Xi #X#i %+XiX#i ui


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-)#


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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-)%


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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+


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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


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5hat is a good 1base2 specification


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5hat is a good base specification

6-):

The TestScore Income relation


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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


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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'


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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


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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$


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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

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Introduction to Econometrics- Stock & Watson -Ch 6 Slides.doc