Chapter34. Nonparametricmethods
310
Thelocalregressioncanbemaderobustviaanadjustmentbasedontheresiduals, e
i
y
i
ˆy
i
.
Robustnessweights,
k
,aredefinedby
k
Be
k
=6s
wheresisthemedianoftheje
i
jandBisthebisquarefunction,
Bx
(
x
2
2
forjxj<1
0
forjxj1
Thepolynomialregressionisthenre-runusingweight
k
w
k
x
i
atx
k
;y
k
.
Theloess()functioningretltakesuptofiveargumentsasfollows:theyseries,thexseries,the
orderd,thebandwidthq,andabooleanswitchtoturnontherobustadjustment. Thelastthree
argumentsareoptional: iftheyareomittedthedefaultvaluesared1,0:5andnorobust
adjustment. Anexampleofafullcalltoloess()isshownbelow;inthiscaseaquadraticinxis
specified,threequartersofthedatapointswillbeusedineachlocalregression,androbustnessis
turnedon:
series yh = loess(y, x, 2, 0.75, 1)
AnillustrationofloessisprovidedinExample34.1: wegenerateaseriesthathasadeterministic
sinewavecomponentoverlaidwithnoiseuniformlydistributedon 1;1. Loessisthenusedto
retrieveagoodapproximationtothesinefunction.TheresultinggraphisshowninFigure34.1.
Example34.1:Loessscript
nulldata 120
series x = index
scalar n = $nobs
series y = sin(2*pi*x/n) ) + + uniform(-1, 1)
series yh = loess(y, x, , 2, , 0.75, 0)
gnuplot y yh h x x --output=display --with-lines=yh
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0
20
40
60
80
100
120
x
loess fit
Figure34.1:Loess: retrievingasinewave
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Chapter34. Nonparametricmethods
311
34.2 TheNadaraya–Watsonestimator
TheNadaraya–Watsonnonparametric estimator (Nadaraya1964Watson1964) is an estimator
for theconditional l meanofa variableY, availableinasampleof sizen, for r agivenvalueof a
conditioningvariableX,andisdefinedas
mX
P
n
j1
y
j
K
h
X x
j
P
n
j1
K
h
X x
j
whereK
h
istheso-calledkernelfunction, whichisusuallysomesimpletransformofadensity
functionthatdependsonascalarcalledthebandwidth.Theonegretlusesisgivenby
K
h
xexp
x
2
2h
!
for jxj<   and zero otherwise. . The e scalar  is used to prevent t numericalproblems when the
kernelfunctionisevaluatedtoofarawayfromzeroandiscalledthetrimparameter.
Example34.2: Nadaraya–Watsonexample
# Nonparametric c regression example: : husband’s age e on wife’s age
open mroz87.gdt
# initial l value for the e bandwidth
scalar h = $nobs^(-0.2)
# three e increasingly smooth estimates
series m0 = nadarwat(HA, , WA, , h)
series m1 = nadarwat(HA, , WA, , h * 5)
series m2 = nadarwat(HA, , WA, , h * 10)
# produce e the e graph
dataset sortby WA
gnuplot HA m0 0 m1 1 m2 WA --output=display --with-lines=m0,m1,m2
Example34.2producesthegraphshowninFigure34.2(aftersomeslightediting).
Thechoiceofthebandwidthisuptotheuser:largervaluesofhleadtoasmoothermfunction;
smaller values s makethemfunction follow they
i
values moreclosely, , so that t thefunction
appearsmore“jagged”. Infact,ash!1,mx
i
!
¯
Y;onthecontrary,ifh!0,observationsfor
whichx
i
6arenottakenintoaccountatallwhencomputingmX.
Also,thestatisticalpropertiesofmvarywithh: itsvariancecanbeshowntobedecreasingin
h,whileitssquaredbiasisincreasinginh.Itcanbeshownthatchoosingh1=minimizesthe
RMSE,sothatvalueiscustomarilytakenasareferencepoint.
Notethatthekernelfunction hasits tails “trimmed”. . Thescalar , which h controls thelevel at
whichtrimmingoccursissetbydefaultat4h;thissetting,however,maybechangedviatheset
command. Forexample,
set nadarwat_trim m 10
sets   10h. This s mayat times producemoresensible e results s in regions of withsparse
support; however, youshouldbeawarethatinthosesamecases machineprecision(divisionby
numericalzero)mayrenderyourresultsspurious.Thedefaultisrelativelysafe,butexperimenting
withlargervaluesmaybeasensiblestrategyinsomecases.
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Chapter34. Nonparametricmethods
312
30
35
40
45
50
55
60
30
35
40
45
50
55
60
HA
WA
m0
m1
m2
Figure34.2:Nadaraya–Watsonexampleforseveralchoicesofthebandwidthparameter
Acommonvariant of theNadaraya–Watsonestimator r is theso-called“leave-one-out”estimator:
thisisavariantoftheestimatorthatdoesnotusethei-thobservationforevaluatingmx
i
. This
makestheestimatormorerobustnumericallyanditsusageisoftenadvisedforinferencepurposes.
Informulae,theleave-one-outestimatoris
mx
i
P
j6i
y
j
K
h
x
i
x
j
P
j6i
K
h
x
i
x
j
Inordertohavegretlcomputetheleave-one-outestimator,justreversethesignofh:ifwechanged
example34.2bysubstituting
scalar h h = $nobs^(-0.2)
with
scalar h h = -($nobs^(-0.2))
therestoftheexamplewouldhavestayedunchanged,theonlydifferencebeingtheusageofthe
leave-one-outestimator.
Althoughcouldbe, in principle, anyvalue, in thetypicalusageofthis estimator you wantto
computemXforequaltooneormorevaluesactuallyobservedinyoursample,thatismx
i
.
IfyouneedapointestimateofmXforsomevalueofwhichisnotpresentamongthevalid
observationsofyourdependentvariable,youmaywantto addsome“fake”observationstoyour
datasetinwhichyismissingandxcontainsthevaluesyouwantmxevaluatedat.Forexample,
thefollowingscriptevaluatesmxatregularintervalsbetween-2.0and2.0:
nulldata 120
set seed d 120496
# first t part t of the e sample: : actual data
smpl 1 100
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Chapter34. Nonparametricmethods
313
x = normal()
y = x^2 2 + + sin(x) + + normal()
# second d part t of the sample: fake x data
smpl 101 1 120
x = (obs-110) / 5
# compute the Nadaraya-Watson n estimate
# with bandwidth equal to 0.4 4 (note e that
# 100^(-0.2) = 0.398)
smpl full
m = nadarwat(y, x, , 0.4)
# show m(x) for the e fake e x values only
smpl 101 1 120
print x x m m -o
andrunningitproduces
x
m
101
-1.8
1.165934
102
-1.6
0.730221
103
-1.4
0.314705
104
-1.2
0.026057
105
-1.0
-0.131999
106
-0.8
-0.215445
107
-0.6
-0.269257
108
-0.4
-0.304451
109
-0.2
-0.306448
110
0.0
-0.238766
111
0.2
-0.038837
112
0.4
0.354660
113
0.6
0.908178
114
0.8
1.485178
115
1.0
2.000003
116
1.2
2.460100
117
1.4
2.905176
118
1.6
3.380874
119
1.8
3.927682
120
2.0
4.538364
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PartIII
Technicaldetails
314
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Chapter35
GretlandT
E
X
35.1 Introduction
T
E
X —initiallydevelopedbyDonaldKnuthofStanfordUniversityandsinceenhancedbyhundreds
ofcontributorsaroundtheworld — isthegoldstandardofscientifictypesetting. . Gretlprovides
varioushooksthatenableyoutopreviewandprinteconometricresultsusingtheT
E
Xengine,and
tosaveoutputinaformsuitableforfurtherprocessingwithT
E
X.
Thischapterexplainsthefinerpointsofgretl’sT
E
X-relatedfunctionality.Thenextsectiondescribes
therelevantmenuitems; section35.3discusseswaysoffine-tuningT
E
Xoutput; andsection35.4
givessomepointersoninstalling(andlearning)T
E
Xifyoudonotalreadyhaveitonyourcomputer.
(Justtobeclear: T
E
Xisnotincludedwiththegretldistribution;itisaseparatepackage,including
severalprogramsandalargenumberofsupportingfiles.)
Beforeproceeding,however,itmaybeusefultosetoutbrieflythestagesofproductionofafinal
documentusingT
E
X.Forthemostpartyoudon’thavetoworryaboutthesedetails,since,inregard
topreviewingatanyrate,gretlhandlesthemforyou. Buthavingsomegraspofwhatisgoingon
behindthescenceswillenableyoutounderstandyouroptionsbetter.
Thefirststep isthecreationofaplaintext“source”file, containing thetextor mathematics to
betypset, interspersedwithmark-upthatdefines howitshouldbeformatted. . Thesecondstep
istorunthesourcethroughaprocessingenginethatdoestheactualformatting. Typicallythisa
programcalledpdflatexthatgeneratesPDFoutput.
1
(Intimesgonebyitwasaprogramcalledlatex
thatgeneratedso-calledDVI(device-independent)output.)
Sogretlcallspdflatextoprocessthesourcefile. OnMSWindowsandMacOSX,gretlexpectsthe
operatingsystemto findthedefaultviewerforPDFoutput. . OnGNU/Linuxyoucanspecifyyour
preferredPDFviewerviathemenuitem“Tools,Preferences,General,”underthe“Programs”tab.
35.2 T
E
X-relatedmenuitems
Themodelwindow
ThefullestT
E
XsupportingretlisfoundintheGUImodel window. . This s has amenuitemtitled
“LaTeX”withsub-items“View”,“Copy”,“Save”and“Equationoptions”(seeFigure35.1).
Thefirstthreesub-itemshavebranchestitled“Tabular”and“Equation”.By“Tabular”wemeanthat
themodelisrepresentedintheformofatable;thisisthefullestandmostexplicitpresentationof
theresults. SeeTable35.1foranexample;thiswaspastedintothemanualafterusingthe“Copy,
Tabular”itemingretl(afewlineswereeditedoutforbrevity).
The“Equation”optionisfairlyself-explanatory—theresultsarewrittenacrossthepageinequation
format,asbelow:
˘
ENROLL0:241105
0:066022
0:223530
0:04597
CATHOL 0:00338200
0:0027196
PUPIL 0:152643
0:040706
WHITE
T51
¯
R
2
0:4462 F3;4714:431 1 ˆ0:038856
1
Expertswillbeawareofsomethingcalled“plainT
E
X”,whichisprocessedusingtheprogramtex.Thegreatmajority
ofT
E
Xusers,however,usetheL
A
T
E
Xmacros,initiallydevelopedbyLeslieLamport.gretldoesnotsupportplainT
E
X.
315
Chapter35. GretlandT
E
X
316
Figure35.1:LAT
E
Xmenuinmodelwindow
Table35.1:ExampleofLAT
E
Xtabularoutput
Model1:OLSestimatesusingthe51observations1–51
Dependentvariable:ENROLL
Variable
Coefficient
Std.Error
t-statistic
p-value
const
0:241105
0:0660225
3:6519
0:0007
CATHOL
0:223530
0:0459701
4:8625
0:0000
PUPIL
0:00338200
0:00271962
1:2436
0:2198
WHITE
0:152643
0:0407064
3:7499
0:0005
Meanofdependentvariable
0:0955686
S.D.ofdependentvariable
0:0522150
Sumofsquaredresiduals
0:0709594
Standarderrorofresiduals(ˆ)
0:0388558
UnadjustedR2
0:479466
Adjusted
¯
R2
0:446241
F3;47
14:4306
Chapter35. GretlandT
E
X
317
(standarderrorsinparentheses)
Thedistinctionbetweenthe“Copy”and“Save”options(forbothtabularandequation)istwofold.
First,“Copy”putstheT
E
Xsourceontheclipboardwhilewith“Save”youarepromptedforthename
ofafileintowhichthesourceshouldbesaved. Second, , with“Copy”thematerialis copiedas a
“fragment”whilewith“Save”itiswrittenasacompletefile. Thepointisthatawell-formedT
E
X
sourcefilemusthaveaheaderthatdefinesthedocumentclass(article,report,bookorwhatever)
andtagsthatsay\begin{document}and\end{document}.Thismaterialisincludedwhenyoudo
“Save”butnotwhenyoudo“Copy”,sinceinthelattercasetheexpectationisthatyouwillpaste
thedataintoanexistingT
E
Xsourcefilethatalreadyhastherelevantapparatusinplace.
Theitemsunder“Equationoptions”shouldbeself-explanatory: whenprintingthemodelinequa-
tionform,doyouwantstandarderrorsort-ratiosdisplayedinparenthesesundertheparameter
estimates?Thedefaultistoshowstandarderrors;ifyouwantt-ratios,selectthatitem.
Otherwindows
SeveralothersortsofoutputwindowsalsohaveT
E
Xpreview,copyandsaveenabled. Inthecaseof
windowshavingagraphicaltoolbar,lookfortheT
E
Xbutton. Figure35.2showsthisicon(second
fromtherightonthetoolbar)alongwiththedialogthatappearswhenyoupressthebutton.
Figure35.2:T
E
Xiconanddialog
Oneaspectofgretl’sT
E
Xsupportthatislikelytobeparticularlyusefulforpublicationpurposesis
theabilitytoproduceatypesetversionofthe“modeltable”(seesection3.4).Anexampleofthisis
showninTable35.2.
35.3 Fine-tuningtypesetoutput
There are three aspects to this: : adjusting g the e appearance of f the e output produced d by gretl in
L
A
T
E
Xpreviewmode; adjustingtheformattingofgretl’stabularoutputformodelswhenusingthe
tabprintcommand;andincorporatinggretl’soutputintoyourownT
E
Xfiles.
PreviewingintheGUI
As regards preview mode, you u can control theappearanceof gretl’s output using a file named
gretlpre.tex,whichshouldbeplacedinyourgretluserdirectory(seetheGretl CommandRef-
erence). Ifsuchafileisfound,itscontentswillbeusedasthe“preamble”totheT
E
Xsource. The
defaultvalueofthepreambleisasfollows:
\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
Chapter35. GretlandT
E
X
318
Table35.2:Exampleofmodeltableoutput
OLSestimates
Dependentvariable:ENROLL
Model1
Model2
Model3
const
0.2907
0.2411
0.08557
(0.07853)
(0.06602)
(0.05794)
CATHOL
0.2216

0.2235

0.2065

(0.04584)
(0.04597)
(0.05160)
PUPIL
0.003035
0.003382
0.001697
(0.002727)
(0.002720)
(0.003025)
WHITE
0.1482

0.1526

(0.04074)
(0.04071)
ADMEXP
0.1551
(0.1342)
n
51
51
51
¯
R2
0.4502
0.4462
0.2956
96.09
95.36
88.69
Standarderrorsinparentheses
*indicatessignificanceatthe10percentlevel
**indicatessignificanceatthe5percentlevel
Chapter35. GretlandT
E
X
319
\usepackage{amsmath}
\usepackage{dcolumn,longtable}
\begin{document}
\thispagestyle{empty}
Notethattheamsmathanddcolumnpackagesarerequired.(Forsomesortsofoutputthelongtable
packageisalsoneeded.) Beyondthatyoucan,forinstance,changethetypesizeorthefontbyal-
teringthedocumentclassdeclarationorincludinganalternativefontpackage.
Inaddition, if you wishto typesetgretloutput inmorethanonelanguage, you canset upper-
languagepreamblefiles.A“localized”preamblefileisidentifiedbyanameoftheformgretlpre_xx.tex,
wherexxisreplacedbythefirsttwo letters ofthecurrentsettingoftheLANGenvironmentvari-
able. Forexample,ifyouarerunningtheprograminPolish,usingLANG=pl_PL,thengretlwilldo
thefollowingwhenwritingthepreambleforaT
E
Xsourcefile.
1. Lookforafilenamedgretlpre_pl.texinthegretluserdirectory.Ifthisisnotfound,then
2. lookforafilenamedgretlpre.texinthegretluserdirectory.Ifthisisnotfound,then
3. usethedefaultpreamble.
Conversely, supposeyouusuallyrungretlinalanguageother thanEnglish,andhaveasuitable
gretlpre.texfileinplaceforyournativelanguage.IfonsomeoccasionsyouwanttoproduceT
E
X
outputinEnglish,thenyoucouldcreateanadditionalfilegretlpre_en.tex: thisfilewillbeused
forthepreamblewhengretlisrunwithalanguagesettingof,say,en_US.
Command-lineoptions
Afterestimatingamodelviaascript—orinteractivelyviathegretlconsoleorusingthecommand-
lineprogramgretlcli—youcanusethecommandstabprint oreqnprintto printthemodelto
fileintabular formator equation n formatrespectively. . Theseoptions s areexplained intheGretl
CommandReference.
Ifyouwishaltertheappearanceofgretl’stabularoutputformodelsinthecontextofthetabprint
command,youcanspecifyacustomrowformatusingthe--formatflag. Theformatstringmust
beenclosedindoublequotesandmustbetiedtotheflagwithanequalssign. Thepatternforthe
formatstring isasfollows. . Therearefourfields,representingthecoefficient, , standarderror, t-
ratioandp-valuerespectively. Thesefieldsshouldbeseparatedbyverticalbars;theymaycontain
aprintf-typespecification for r the e formatting g ofthenumeric valueinquestion, or may beleft
blanktosuppresstheprintingofthatcolumn(subjecttotheconstraintthatyoucan’tleaveallthe
columnsblank).Hereareafewexamples:
--format="%.4f|%.4f|%.4f|%.4f"
--format="%.4f|%.4f|%.3f|"
--format="%.5f|%.4f||%.4f"
--format="%.8g|%.8g||%.4f"
Thefirstofthesespecificationsprintsthevaluesinallcolumnsusing4decimalplaces.Thesecond
suppressesthep-valueandprintsthet-ratioto3places. Thethirdomitsthet-ratio. Thelastone
againomitsthet,andprintsbothcoefficientandstandarderrorto8significantfigures.
Onceyousetacustomformatinthisway,itisrememberedandusedforthedurationofthegretl
session.Toreverttothedefaultformattingyoucanusethespecialvariant--format=default.
Furtherediting
Onceyouhavepastedgretl’sT
E
Xoutputintoyourowndocument,orsavedittofileandopenedit
inaneditor,youcanofcoursemodifythematerialinanywishyouwish. Insomecases,machine-
generated T
E
X is s hard d to o understand, but gretl’s s output is s intended d to o be human-readableand
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