Chapter 23
Model selection criteria
23.1 Introduction
In some contexts the econometrician chooses between alternative models based on a formal hy-
pothesis test. For example, one might choose amore general modelover amore restricted one if
therestrictioninquestioncanbe formulatedasatestablenullhypothesis,andthe nullisrejected
onanappropriate test.
Inothercontextsonesometimesseeksacriterionformodelselectionthatsomehowmeasuresthe
balance between goodness of fitor likelihood, on the one hand, andparsimonyonthe other. The
balancingisnecessarybecausethe additionofextravariablesto amodelcannotreducethedegree
of fit or likelihood, andis verylikely to increase itsomewhat even if the additional variables are
nottrulyrelevanttothedata-generatingprocess.
Thebestknownsuchcriterion,forlinearmodels estimatedvialeastsquares, istheadjustedR2,
¯
R
2
1 
SSR=n k
TSS=n 1
where n is the number of observations in the sample, k denotes the number of parameters esti-
mated, and SSR and TSS denote the sum of squared residuals and the total sum of squares for
the dependent variable, respectively. Compared to the ordinary coefficient of determination or
unadjustedR
2
,
R
2
1 
SSR
TSS
the“adjusted” calculationpenalizesthe inclusionofadditionalparameters,other thingsequal.
23.2 Information criteria
Amore general criterion in a similar spirit is Akaike’s (1974) “Information Criterion” (AIC). The
originalformulationofthismeasure is
AIC  2‘
ˆ
2k
(23.1)
where ‘
ˆ
 represents the maximumloglikelihood as afunction of the vector of parameter esti-
mates,
ˆ
, andk(as above)denotes the number of “independentlyadjusted parameters within the
model.” Inthis formulation, with AIC negativelyrelatedto the likelihoodandpositivelyrelatedto
thenumber ofparameters,theresearcher seeksthe minimumAIC.
The AIC canbeconfusing,inthatseveralvariants ofthe calculationare “incirculation.” For exam-
ple,DavidsonandMacKinnon(2004)presentasimplifiedversion,
AIC‘
ˆ
 k
whichisjust 2 timesthe original: inthis case,obviously,onewantsto maximizeAIC.
Inthe caseofmodelsestimatedbyleastsquares,the loglikelihoodcanbe writtenas
‘
ˆ
 
n
2
1log2 logn 
n
2
logSSR
(23.2)
210
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Chapter23. Model selection criteria
211
Substituting(23.2)into (23.1)weget
AICn1log2 lognnlogSSR2k
whichcanalsobe writtenas
AICnlog
SSR
n
2kn1log2
(23.3)
Someauthorssimplifytheformulaforthecaseofmodelsestimatedvialeastsquares.Forinstance,
WilliamGreenewrites
AIC log
SSR
n
2k
n
(23.4)
This variant can be derived from (23.3) by dividing through by n and subtracting the constant
1log2. Thatis,writingAIC
G
for theversiongivenbyGreene,wehave
AIC
G
1
n
AIC 1log2
Finally,Ramanathangivesafurthervariant:
AIC
R
SSR
n
e
2k=n
whichistheexponentialofthe one givenbyGreene.
Gretl began by using the Ramanathan variant, but since version 1.3.1 the program has used the
originalAkaikeformula(23.1),andmore specifically(23.3)for modelsestimatedvialeastsquares.
Althoughthe Akaike criterionis designedto favor parsimony, arguablyit does not go far enough
in thatdirection. For instance,if we have two nestedmodels withk 1 andk parametersrespec-
tively, and if the null hypothesis thatparameter kequals 0 is true, in large samples the AIC will
nonethelesstendtoselectthe lessparsimoniousmodelabout16percentofthetime(seeDavidson
andMacKinnon,2004,chapter15).
An alternative to the AIC which avoids this problemis theSchwarz (1978)“Bayesian information
criterion” (BIC).TheBIC canbewritten(inline withAkaike’sformulationoftheAIC)as
BIC 2‘
ˆ
klogn
The multiplication of k by logn in the BIC means that the penalty for adding extra parameters
grows with the sample size. This ensures that, asymptotically, one will notselect a larger model
overacorrectlyspecifiedparsimoniousmodel.
A further alternative to AIC, which again tends to select more parsimonious models than AIC,
is the Hannan–Quinn criterion or HQC (HannanandQuinn,1979). Written consistently with the
formulationsabove,thisis
HQC 2‘
ˆ
2kloglogn
TheHannan–Quinncalculationisbasedonthelawoftheiteratedlogarithm(note thatthelastterm
isthelogofthelogofthesamplesize). Theauthorsarguethattheirprocedureprovidesa“strongly
consistentestimationprocedure for the order ofanautoregression”, and that“comparedto other
stronglyconsistentproceduresthisprocedurewillunderestimate the orderto alesser degree.”
Gretlreportsthe AIC,BIC andHQC (calculatedas explainedabove)for mostsortsofmodels. The
key point in interpreting these values is to know whether they are calculated such that smaller
valuesarebetter,orsuchthatlargervaluesarebetter. Ingretl,smallervaluesarebetter:onewants
tominimizethe chosencriterion.
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Chapter 24
Time series filters
Inaddition to the usual applicationof lags and differences, gretl provides fractional differencing
and various filters commonlyusedinmacroeconomics for trend-cycle decomposition: notablythe
Hodrick–Prescott filter (HodrickandPrescott1997), the Baxter–King bandpass filter (Baxterand
King,1999)andtheButterworthfilter(Butterworth, 1930).
24.1 Fractional differencing
Theconceptofdifferencingatimeseriesdtimesisprettyobviouswhendisaninteger;itmayseem
oddwhendisfractional. However,thisideahasawell-definedmathematicalcontent: considerthe
function
fz1 z
d
;
wherez anddare realnumbers. BytakingaTaylorseriesexpansionaroundz 0,weseethat
fz1dz
dd1
2
z
2

or,more compactly,
fz1
X1
i1
i
z
i
with
k
Q
k
i1
di 1
k!
 
k 1
dk 1
k
Thesameexpansioncanbe usedwiththe lagoperator,so thatifwe defined
Y
t
1 L
0:5
X
t
this couldbe consideredshorthandfor
Y
t
X
t
0:5X
t 1
0:125X
t 2
0:0625X
t 3

Ingretlthistransformationcanbeaccomplishedbythesyntax
genr Y = fracdiff(X,0.5)
24.2 The Hodrick–Prescott filter
Thisfilter is accessed using the hpfilt()function, which takesas itsfirst argumentthe name of
thevariableto be processed. (Afurther optionalargumentis explainedbelow.)
Atimeseriesy
t
maybedecomposedintoatrendorgrowthcomponentg
t
andacyclicalcomponent
c
t
.
y
t
g
t
c
t
; t 1;2;:::;T
212
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Chapter24. Timeseriesfilters
213
TheHodrick–Prescottfiltereffectssuchadecompositionbyminimizingthe following:
XT
t1
y
t
g
t
2

TX 1
t2
g
t1
g
t
 g
t
g
t 1
2
:
The firsttermaboveis the sumofsquaredcyclicalcomponentsc
t
y
t
g
t
.Thesecondtermis a
multiple  ofthe sumofsquares ofthe trendcomponent’sseconddifferences. This secondterm
penalizesvariationsinthegrowthrateofthetrendcomponent: thelargerthevalueof,thehigher
is thepenaltyandhence thesmoother thetrendseries.
Note thatthe hpfilt functionin gretlproduces the cyclicalcomponent, c
t
,ofthe original series.
Ifyouwantthesmoothedtrendyoucansubtractthe cyclefromthe original:
genr ct = hpfilt(yt)
genr gt = yt - ct
HodrickandPrescott(1997)suggestthatavalueof1600isreasonableforquarterlydata.The
default value ingretlis 100 times the square of the data frequency(which, ofcourse,yields 1600
for quarterlydata). Thevaluecanbe adjustedusinganoptionalsecondargumenttohpfilt(),as
in
genr ct = hpfilt(yt, 1300)
24.3 The Baxter and King filter
Thisfilter is accessed usingthe bkfilt() function, whichagaintakes the nameofthe variable to
be processed as its first argument. The operation of the filter canbe controlled viathree further
optionalargument.
Considerthe spectralrepresentationofatime seriesy
t
:
y
t
Z
e
i!
dZ!
To extract the component of y
t
that lies between the frequencies !
and
! one could apply a
bandpassfilter:
c
t
Z
F
!e
i!
dZ!
where F
!  1 for !
< j!j <
! and 0 elsewhere. This would imply, in the time domain,
applying to the series a filter with an infinite number of coefficients, which is undesirable. The
BaxterandKingbandpassfilterappliestoy
t
afinitepolynomialinthelagoperator AL:
c
t
ALy
t
whereA(L)isdefinedas
AL
Xk
i k
a
i
L
i
Thecoefficientsa
i
arechosensuchthatF!Ae
i!
Ae
i!
isthebestapproximationtoF
!
for agivenk. Clearly, thehigher kthe better theapproximationis,butsince 2kobservationshave
to be discarded, a compromise is usually sought. Moreover, the filter has also other appealing
theoreticalproperties,amongwhichthe propertythatA1 0, so aseries with asingle unitroot
is madestationarybyapplicationofthe filter.
In practice, the filter is normally used with monthly or quarterly data to extract the “business
cycle” component,namelythecomponentbetween6 and36quarters. Usualchoicesfor kare 8or
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Chapter24. Timeseriesfilters
214
12 (maybe higher for monthly series). The defaultvaluesfor the frequencybounds are 8 and 32,
and the defaultvalue for the approximationorder, k, is 8. Youcanadjustthese values using the
fullformofbkfilt(),whichis
bkfilt(seriesname,f1, f2,k)
wheref1 andf2 representthelowerandupperfrequencyboundsrespectively.
24.4 The Butterworth filter
The Butterworth filter (Butterworth1930) is an approximation to an “ideal” square-wave filter.
The idealfilter divides the spectrumofatimeseries into apass-band(frequenciesless thansome
chosen!
?
foralow-passfilter,orfrequenciesgreaterthan!
?
forhigh-pass)andastop-band;the
gainis1forthepass-bandand0forthestop-band. Theidealfilterisunattainableinpracticesince
it would require an infinite number ofcoefficients, but the Butterworth filter offers a remarkably
goodapproximation. ThisfilterisderivedandpersuasivelyadvocatedbyPollock (2000).
For datay,thefilteredsequence xisgivenby
xy  ÖQM Q
0
ÖQ
1
Q
0
y
(24.1)
where
Öf2I
T
L
T
L
1
T
g
T 2
and M f2I
T
L
T
L
1
T
g
T
I
T
denotes the identity matrix of order T; L
T
 e
1
;e
2
;:::;e
T 1
;0 is the finite-sample matrix
versionof the lagoperator; andQ isdefinedsuchthatpre-multiplicationofa T-vector of databy
Q
0
oforderT 2T produces the seconddifferences ofthe data. Thematrixproduct
Q
0
ÖQf2I
T
L
T
L
1
T
g
T
is aToeplitzmatrix.
Thebehaviorofthe Butterworthfilterisgovernedbytwoparameters: the frequencycutoff!?and
aninteger order,n,whichdeterminesthe numberofcoefficientsused. Thethatappearsin(24.1)
is tan!?=2  2n. Highervalues ofn produce abetter approximationto theidealfilterinprinciple
(i.e.asharpercutbetweenthepass-bandandthestop-band)butthere isadownside: withagreater
numberofcoefficientsnumericalinstabilitymaybeanissue,andthe influenceoftheinitialvalues
inthe samplemaybeexaggerated.
Ingretl the Butterworthfilter is implemented by the bwfilt() function,
1
which takesthree argu-
ments: the series to filter, the order n and the frequency cutoff, !
?
,expressed in degrees. The
cutoffvalue mustbe greater than 0and lessthan180. Thisfunctionoperates as alow-pass filter;
for thehigh-passvariant,subtractthe filteredseriesfromthe original,asin
series bwcycle = y - bwfilt(y, 8, 67)
Pollock recommendsthattheparametersofthe Butterworthfilterbetunedto thedata: oneshould
examine the periodogram of the series in question(possibly after removalofa polynomialtrend)
in search of a “dead spot” of low power between the frequencies one wishes to exclude and the
frequencies one wishes to retain. If !? is placedin such adead spot then the job of separation
canbe done witha relatively smalln, hence avoiding numerical problems. By wayofillustration,
considertheperiodogramforquarterlyobservationsonnewcarssalesinthe US,
2
1975:1to1990:4
(theupperpanelinFigure24.1).
Aseasonal patternis clearly visible inthe periodogram, centered atan angle of90
or 4 periods.
Ifweset!
?
68
(or thereabouts)we shouldbe ableto excise the seasonalityquitecleanlyusing
1ThecodeforthisfilterisbasedonD.S.G.Pollock’sprogramsIDEOLOGandDETREND.ThePascalsourcecodefor
the former is available from http://www.le.ac.uk/users/dsgp1and the Csources for the latterwerekindly made
availabletousbytheauthor.
2
ThisisthevariableQNCfromtheRamanathandatafiledata9-7.
Chapter24. Timeseriesfilters
215
0
50000
100000
150000
200000
250000
300000
0
20
40
60
80
100
120
140
160
180
64.0
10.7
5.8
4.0
3.0
2.5
2.1
degrees
periods
1600
1800
2000
2200
2400
2600
2800
3000
3200
3400
1976
1978
1980
1982
1984
1986
1988
1990
QNC (original data)
QNC (smoothed)
0
0.2
0.4
0.6
0.8
1
0
π/4
π/2
3π/4
π
Figure24.1: TheButterworthfilterapplied
n8. The resultisshown in the lower panel of the Figure,alongwiththe frequency response or
gainplotfor thechosenfilter. Note the smoothandreasonablysteepdrop-off ingaincenteredon
thenominalcutoffof68
3=8.
The apparatus thatsupports this sort ofanalysis inthe gretlGUI canbe found under the Variable
menu inthe main window: the items Periodogram and Filter. Inthe periodogram dialog box you
have the option of expressing the frequency axis in degrees, which is helpful when selecting a
Butterworthfilter;andintheButterworthfilterdialogyouhavetheoptionofplottingthefrequency
response aswellasthe smoothedseriesand/or theresidualorcycle.
Chapter 25
Univariate time series models
25.1 Introduction
Timeseries modelsarediscussedinthischapterandthe nexttwo. Herewe concentrate onARIMA
models, unit root tests, andGARCH. The followingchapter deals withVARs, and chapter27 with
cointegrationanderror correction.
25.2 ARIMA models
Representationandsyntax
The arma command performs estimationof AutoRegressive, Integrated, Moving Average (ARIMA)
models. Thesearemodels thatcanbe writteninthe form
Ly
t
L
t
(25.1)
where L, and L are polynomials inthe lag operator, L, definedsuch that L
n
x
t
x
t n
,and
t
is a white noise process. The exact content of y
t
, of the AR polynomial , and of the MA
polynomial,willbe explainedinthe following.
Meanterms
The processy
t
as written in equation(25.1) has, withoutfurther qualifications, meanzero. If the
modelis to be applied to realdata, it is necessary to include some termto handle the possibility
that y
t
has non-zero mean. There are two possible ways to represent processes with nonzero
mean: oneisto define
t
astheunconditional meanofy
t
,namelythecentralvalue ofitsmarginal
distribution. Therefore,the series ˜y
t
y
t
t
hasmean0, andthe model(25.1)applies to ˜y
t
. In
practice, assumingthat
t
isalinearfunctionofsome observable variablesx
t
,the modelbecomes
Ly
t
x
t
 L
t
(25.2)
This is sometimes known as a “regression model with ARMA errors”; its structure may be more
apparentifwe representitusingtwo equations:
y
t
x
t
u
t
Lu
t
L
t
The modeljustpresentedisalso sometimesknownas “ARMAX” (ARMA + eXogenousvariables). It
seems to us, however, that this label is more appropriately applied to a different model: another
way to include a mean termin (25.1) is to base the representation on the conditional meanofy
t
,
thatis the centralvalue ofthe distributionof y
t
givenits ownpast. Assuming,again,thatthiscan
be representedas alinear combination ofsome observable variables z
t
,the modelwould expand
to
Ly
t
z
t
L
t
(25.3)
The formulation (25.3) has the advantage that   can be immediately interpreted as the vector of
marginal effects of the z
t
variables on the conditional mean of y
t
. And by adding lags of z
t
to
216
Chapter25. Univariatetimeseriesmodels
217
this specification one can estimate Transfer Function models (which generalize ARMA by adding
theeffectsofexogenousvariabledistributedacrosstime).
Gretlprovidesawaytoestimatebothforms. Modelswrittenasin(25.2)areestimatedbymaximum
likelihood;modelswrittenasin(25.3)areestimatedbyconditionalmaximumlikelihood. (Formore
onthese optionsseethe sectionon“Estimation” below.)
In the special case when x
t
 z
t
 1 (that is, the models include a constant but no exogenous
variables)the twospecifications discussedabovereduce to
Ly
t
L
t
(25.4)
and
Ly
t
L
t
(25.5)
respectively. These formulationsareessentiallyequivalent,butiftheyrepresentoneandthe same
process andare,fairlyobviously,notnumericallyidentical;rather

1 
1
::: 
p
Thegretlsyntaxfor estimating(25.4)issimply
arma p q ; y
The AR and MA lag orders, p and q, can be given either as numbers or as pre-defined scalars.
The parameter canbe droppedifnecessarybyappendingthe option--nc(“noconstant”)to the
command. Ifestimation of (25.5) isneeded, the switch --conditional must be appended to the
command, as in
arma p q ; y --conditional
Generalizingthisprincipleto theestimationof(25.2)or (25.3),yougetthat
arma p q ; y const x1 x2
wouldestimatethe followingmodel:
y
t
x
t

1
y
t 1
x
t 1
:::
p
y
t p
x
t p

t

1
t 1
:::
q
t q
whereinthisinstancex
t

0
x
t;1
1
x
t;2
2
. Appendingthe--conditional switch, asin
arma p q ; y const x1 x2 --conditional
wouldestimatethe followingmodel:
y
t
x
t

1
y
t 1
:::
p
y
t p

t

1
t 1
:::
q
t q
Ideally, the issuebroachedabove couldbemade mootbywritingamoregeneralspecificationthat
neststhealternatives;thatis
L
y
t
x
t
z
t
L
t
;
(25.6)
we would like to generalize the arma command so thatthe user could specify,for anyestimation
method, whether certainexogenous variables shouldbe treated asx
t
sorz
t
s,butwe’re notyet at
thatpoint(andneither are mostothersoftware packages).
Chapter25. Univariatetimeseriesmodels
218
Seasonalmodels
Amore flexible lag structure is desirable whenanalyzing time series thatdisplay strong seasonal
patterns. Model(25.1)canbe expandedto
LØL
s
y
t
LÒL
s

t
:
(25.7)
For suchcases,afuller formofthe syntaxisavailable, namely,
arma p q ; P Q ; y
wherep andqrepresentthe non-seasonalARandMAorders,andPandQthe seasonalorders. For
example,
arma 1 1 ; 1 1 ; y
wouldbeusedto estimate thefollowingmodel:
1 L1 ØL
s
y
t
1L1ÒL
s

t
Ify
t
isaquarterlyseries(andtherefores 4),theaboveequationcanbewrittenmoreexplicitlyas
y
t
y
t 1
Øy
t 4
 Øy
t 5

t

t 1
Ò
t 4
Ò
t 5
Suchamodelis knownas a“multiplicativeseasonalARMA model”.
Gaps inthelag structure
The standardwayto specifyanARMAmodelingretlis viathe AR and MA orders,pandqrespec-
tively. In thiscase alllags from1 to the given orderare included. Insome cases one maywishto
includeonlycertainspecificAR and/orMAlags. Thiscanbedone ineither oftwo ways.
 One canconstruct amatrixcontaining the desired lags (positive integer values) and supply
thename ofthismatrixinplace ofp or q.
 Onecangive acomma-separatedlistoflags,enclosedinbraces,inplace ofp or q.
Thefollowingcodeillustrates theseoptions:
matrix pvec = {1,4}
arma pvec 1 ; y
arma {1,4} 1 ; y
Bothformsabove specifyanARMA modelinwhichARlags 1and4areused(butnot2 and3).
Thisfacilityisavailableonlyfor thenon-seasonalcomponentoftheARMAspecification.
Differencingand ARIMA
The above discussion presupposes that the time series y
t
has already been subjected to all the
transformationsdeemednecessaryfor ensuringstationarity(seealso section25.3). Differencingis
the mostcommonof these transformations, and gretl provides amechanism to include this step
into thearma command: the syntax
arma p d q ; y
wouldestimateanARMAp;q modelonÑdy
t
.Itisfunctionallyequivalentto
Chapter25. Univariatetimeseriesmodels
219
series tmp = y
loop i=1..d
tmp = diff(tmp)
endloop
arma p q ; tmp
exceptwithregardtoforecastingafter estimation(see below).
Whentheseriesy
t
isdifferencedbefore performingtheanalysisthemodelisknownasARIMA(“I”
for Integrated);forthisreason,gretlprovidesthe arima commandasanaliasfor arma.
Seasonaldifferencingishandledsimilarly,withthesyntax
arma p d q ; P D Q ; y
whereD istheorder for seasonaldifferencing. Thus,thecommand
arma 1 0 0 ; 1 1 1 ; y
wouldproducethe same parameter estimatesas
genr dsy = sdiff(y)
arma 1 0 ; 1 1 ; dsy
wherewe use the sdiff functionto createaseasonaldifference(e.g.for quarterlydata,y
t
y
t 4
).
InspecifyinganARIMAmodelwithexogenousregressorswefaceachoicewhichrelatesbacktothe
discussionofthe variantmodels(25.2)and(25.3)above. Ifwechoosemodel(25.2), the “regression
modelwithARMAerrors”,howshouldthisbe extendedtothe caseofARIMA?Theissueiswhether
or not the differencing that is applied to the dependent variable should also be applied to the
regressors. Consider the simplestcase,ARIMA withnon-seasonaldifferencingoforder 1. We may
estimate either
L1 Ly
t
X
t
L
t
(25.8)
or
L
1 Ly
t
X
t
L
t
(25.9)
ThefirstoftheseformulationscanbedescribedasaregressionmodelwithARIMAerrors,whilethe
secondpreservesthelevelsoftheX variables. Asofgretlversion1.8.6,the defaultmodelis(25.8),
in which differencing is applied to both y
t
and X
t
. However, when using the default estimation
method(nativeexact ML, see below),the option--y-diff-onlymaybe given, inwhich case gretl
estimates(25.9).
1
Estimation
The defaultestimation method for ARMA modelsis exact maximum likelihoodestimation(under
the assumption that the error term is normally distributed), using the Kalman filter in conjunc-
tionwiththe BFGSmaximization algorithm. The gradientof the log-likelihoodwithrespectto the
parameter estimates isapproximatednumerically. This methodproducesresults that are directly
comparable with many other software packages. The constant, and anyexogenous variables, are
treated as in equation (25.2). The covariance matrix for the parameters is computed using anu-
mericalapproximationtothe Hessianatconvergence.
The alternative method, invoked with the --conditional switch, is conditional maximumlikeli-
hood(CML),alsoknownas“conditionalsumofsquares”(seeHamilton,1994,p.132). Thismethod
wasexemplifiedinthescript12.3,andonlyabriefdescriptionwillbegivenhere. Givenasampleof
size T,the CMLmethodminimizesthesumofsquaredone-step-aheadpredictionerrorsgenerated
1
Priortogretl1.8.6,thedefaultmodelwas(25.9). Wechangedthisforthesakeofconsistencywithothersoftware.
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