T emperature Effect on En erg y Demand
Y o u n g d u k K i m a n d In g a n g N a
K o re a E n e r g y E c o n o m i cs In s t it u t e
We
pr ovide
va rious
estim ates
of
tem pera ture
effec t
for
a cc om modating
se as on ality in ener gy dem an d, pa rtic ula rly na tu ra l g as dem and. We ex ploit
tem pera ture
re spon se a nd m onthly
tempe ra ture dis tribu tion to es tim a te
the
tem pera ture effec t on natur a l ga s dem a nd. Both loca l an d globa l smoothed
tem pera ture res pons es a re e stima ted from em piric al re lationsh ip be tween h our ly
tem pera ture and hour ly ene rg y consu mption data during the sa mple pe riod (1990
1996). M onthly tempe ra ture distribu tion e stima tes ar e obtain ed by k erne l dens ity
es tima tion
from
tem pe ra ture
dis pe rs ion
within
a
m on th .
We
integ ra te
tem pera ture res pon se a nd monthly te mper atu re de nsity ove r a ll the tem pera ture s
in the s am ple pe riod to es tima te tem pera tur e e ffe ct on e ner gy dem and. T hen ,
es tima tes
of
te mper atur e
effec t
a re
c om par ed
betwe en
g loba l
an d
loca l
sm oothing m ethods.
1. Introdu ction
T em pera tur e is an importa nt factor to in flu ence e ner gy dem an d, es pe cia lly
he ating a nd cooling de ma nd. E ven if w e rec og nize the importa nce of te mper atur e
on e ner gy dem and, w e a re not we ll inform ed of h ow mu ch te mper atur e affec ts
en erg y dema nd. Hea ting( cooling) dem and is neg ativ ely( pos itively ) a ss oc iated with
tem pera ture . And ene rg y dem and is nonlin ea rly r elate d to tempe ra ture. But we
ha ve little in for ma tion on how muc h nonlin ear it is. It is importa nt to e stima te
tru e r ela tion ship betwe en ene rgy de ma nd a nd tempe ra ture . Other wise , es tima tes
of en erg y dema nd with te mper atur e effect would be bia se d.
Exis ting res ea rch es 1 us e tem pera ture va ria bles as a linea r form , like HDD,
CDD, a nd tem pe ra ture , to e xpla in s ea sona lity of hea ting a nd coolin g dem and.
We w onde r if the line ar a ddition of HDD or tempe ra tu re to the dem an d equ ation
provide s the tr ue re lations hip be tw een en erg y an d tem pera ture s. Our conje ctur e
is that the r ela tion ship be twee n tem pera tur es and ene rgy dem an d should be
nonline ar . HDD or tem pera tur e a s a linea r for m is not so mu ch s moothin g tha t
HDD or tem per atur e in the reg re ssion m ay cre ate a nother sea sona lity.
HDD or
tem pera ture its elf does not he lp ex plain se as on ality in ener gy dem a nd. T h us, it
is ve ry importa nt to a llow the non linea rity to estim ate the rela tionship betwe en
en erg y de ma nd a nd tem pera ture . Since a ny c los ed form of nonlinea rity of the
tem pera ture effect is not known, non par am etric e stima tions a re to be conside red.
We a tte mpt to e stima te n on pa ra m etric nonline ar tem pe ra ture effec t by usin g
g loba l a nd loca l sm oothing m ethods. In g loba l s moothin g, we take Fourie r
flex ible form for the tem pera tur e effec t. For loc al s moothin g, we u se loca lly
we ighted re gr ess ion .
Ma ny studie s on the r ela tion ship betwe en ene rg y and tem pera tur e use m on th ly
da ta to estim ate ene rg y de ma nd. T empe ra ture re late d va ria bles in the studie s
a re
just
in cluded
in
term s
of
monthly
av er ag e
or
s umm ation.
But
the
a gg reg ate s, a ve ra ge s or sum ma tion, ma y not be appr opr iate for se as on ality.
Respons e of en erg y de ma nd to tem pe ra ture
ma y
be v ery
im media te a nd
nonline ar . If this is true, the n a gg reg ate re lationsh ip betwe en tem per atur e a nd
en erg y dema nd will n egle ct n onlin ea r c ha ra cter istics a nd pr oduc e some bia sed
es tima tes.
More ov er, we wonder if tem pera tur e res pons es a re a ll the sa me ove r time.
M os t r ese ar che s on this topic a ssu me tha t te mper atur e re sponses ar e inva ria nt
over time 2 . In oth er w or ds, day tim e r esponse to 10C is a ss ume d to be tota lly
equ alize d as nighttim e re sponse to 10C. T his a ss umption ma y be too re stric tive.
1 Kim (1997), Bentzen and Engs t ed (1993), Silk a nd Joutz (1997), Branch (1993), Nan
and M urr y (1991), Garba cz (1984)
2 J ung(1995), Yoo and Hwang(1997), Park(1997), and M oon(1995) as s ume t hat
t em perat ur e res pons e function is t ime- invariant.
We can r ele as e the as sum ption by a llowing te mper atur e re sponse to be differ ent
over a da y. Ev en if morning te mper atur e is the sa me as noon tem pera ture , the
re sponse s betwee n the two time s ma y be differe nt in the form of non linea rity.
Using the tem pe ra ture of a ce rta in tim e of a da y or da ily av era g e te mper atur e
ma y bring out the b ias in the estim ation of the te mper atur e e ffe ct. It is
su gg ested tha t en erg y res pon se to tem pe ra ture depe nd on the time of a da y. W e
ca n use hour ly temper a tu re a nd e ner gy dem an d, which ha ve us eful inform ation
on th e te mper atur e effec t. Us ing the hourly da ta, w e ca n in ves tiga te th e
differ ent pa tte rns of tem pe ra ture - ene rg y r ela tions hip a cr os s a da y by a llowing
differ ent n on linea r for ms over a da y.
Most of stu dies 3 on e ner gy dem an d u se tota l ag gr eg ate da ta ra ther th an per
ca pita cons umption. E spec ially , in ele ctric ity de ma nd res ea rc hes, tota l a g gre ga te
da ta ar e usu ally us ed. W hen dem an d is r apidly g rowing , tem pera tur e effect from
tota l ag gr ega te data ma y le ad to proble m of mis lea ding sea sona lity. Actu ally,
tem pera ture e ffect is not a n a ggr eg ate bu t indiv idua l e ffe ct. If ener gy de ma nd is
sta ble ov er tim e, tota l a gg reg a te
or per
ca pita da ta
doe s not ma tter
in
es tima ting tempe ra ture effec t. Howe ver , if the de ma nd is g rowing ove r time, like
na tura l g as in Kor ea, the n te mper atur e e ffect u sing tota l a g gre ga te da ta ma y
yie ld big forec as t er rors a s w ell a s estim ation err or s. T hus , we inve stiga te th e
re lations hip be tw een en erg y consum ption per ca pita an d tem pera ture .
Section 2 te lls the es tima tion models for te mper atur e res pon se with two
me thods ; global sm oothing w ith Four ier flex ible for m an d loca l sm oothin g with
loca lly we ighte d re gr ess ion. S ection 3 giv es estim ation a nd r es ult. Section 4
g ives conc lusion and implic ation.
2. Methodolog y and Data
We sta rt with the idea tha t th e rela tionship betwe en tem pera ture a nd ene rg y
dem an d is nonlinea r. It is n ot known yet wha t kind of for m th e r ela tions hip
3 J ung et al (1995), Yoo and Hwang(1997), Park (1997) us e t otal aggregate data t o
es tim ate t em perat ure effect in elect ricit y, and Moon et al (1995) in gas dema nd.
ta kes. Giv en the inform a tion, we a ttempt to mea su re tem pera tur e effec t on
en erg y dem an d by es tima tin g tem pera tur e re spon se a nd tem pera ture distrib ution .
For te mper atur e r esponse es tim a tion, we take two methods for n on linea rity; on e
me thod is g lobal smoothing an d the othe r, local sm oothing . For tem pera tur e
distr ibution, w e use ker nel es tima tion a s a nonpar am etric meth od. T he n, w e us e
g as dem and a nd tem pera tur e to e stima te va rious monthly tem pera ture e ffe cts.
Ga s dema nd and tempe ra ture a re obser ve d by thre e h ou rs.
1 ) M e t h o do lo g y
We a ttempt to mea su re the monthly te mper atur e effe ct on ga s dem an d with
thr ee steps . Fir st, we e stima te the nonlinea r r elations hip be twee n te mper atur e
a nd g as
de ma nd. Sec on d, we
use
ke rne l es tima tion
to es tima te m onthly
tem pera ture dis tribu tion. T hir d, we in tegr ate the te mper atur e re sponse ove r the
monthly tem pera tur e dis tribution to estim ate m onthly tem per atur e e ffect.
T wo me th ods ar e ex ploited a s es tima tion of tempe ra ture r es pons es. Global a nd
loca l sm oothing a re us ed to es tima te the nonline ar r elations hip betw een ga s
dem an d a nd tem pera tur e. F or Global s moothin g me thod, we ta ke the Fou rier
flex ible func tion al form , be cau se a ny c los ed form of the tem per atur e re sponse is
not kn ow n. Simply, we ta ke the following form for reg re ssion,
(1)
whe re y an d t repr ese nt g as dem a nd a nd te mper atur e, r espe ctive ly. In th is
me thod, we fa ce a pr oblem to choos e the num ber of coefficie nts. We c hoose th e
num ber of coefficie nts a s low a s pos sible pr ovided the y a dequa tely desc ribe the
re lations hip.
Another method for temper a tu re r esponse is loc ally w eigh te d s mooth ing of g a s
dem an d on te mper atur e. T he bas ic idea is to cr ea te a n ew va ria ble that, for
ea ch ga s de ma nd, contains the corre sponding sm oothed v alu e. T h e sm oothe s
va lue s a re ob ta ine d by r unn ing a r eg res sion of ga s dema nd on te mper atur e
us ing only the data a nd a sm a ll a mount of the da ta nea r the poin t. In this
loca lly w eigh ted s ca tte rplot sm oothing , the reg re ssion is weig hted s o tha t the
ce ntra l point ge ts h ighe st w eig ht a nd point far ther a wa y rec eiv e less . T h e
es tima ted re gre ss ion is then to predic t th e sm oothed v alu e of g as dema nd for
ea ch point of g as de ma nd da ta. T he pr oc edure is r epea ted to ob ta in the
re ma ining smoothed va lues , wh ich me ans a se par ate we ighte d r eg res sion is
es tima ted
for
eve ry
point
in
the
data . T his
loca lly
sm oothing is a desir ab le sm oother beca us e of its loc ality
w eigh te d
sca tterplot
it tends to follow th e
da ta . Polynomia l s moothing methods ar e g lob al in tha t wha t ha ppen on the
ex trem e left of a s ca tte rplot ca n affec t the fitted v alu es on the ex trem e rig ht.
Ar e thos e sm oothing m ethods bette r than other s? It is still ope n ques tion. T he
me thods w e u se to estim ate the tem pe ra ture r esponse a re chos en in a s ens e
tha t the y follow the da ta w ell. T he e stima ted re sponses from those s moothin g
me thods ar e ex pe cted to r eflec t the n onlin ea r re lationship from da ta. Since our
g oa l is to mea sur e the monthly tem pera ture e ffe ct, we just nee d to e stima te
a dequa te tem per atur e r espons es in a proce ss to obta in a ppropria te m onthly
tem pera ture e ffe cts.
In
the
sec ond
s tep, we
estim ate
m on thly
tem pe ra ture
distribu tion.
The
es tima ted monthly distr ibution tells how tem pera ture s a re disper se d in a month.
And inte gr ation of tem pera tur e r esponse ove r m on th ly tem per atur e dis tribution
g ives us monthly ag gr eg ates of hourly tem pera tur e r esponse s for all months in
the sa mple per iod. An adv an ta g e of m onthly te mper atu re distrib ution for monthly
tem pera ture e ffe ct is tha t it is conve nient to for ec ast tem pe ra ture effe ct. In other
words , without pre dicting future tem pera ture , we c an for eca st th e tem per atur e
effec t by us ing estim ate d tem pera ture dis tribution.
Her e, we es tima te monthly tem per atur e distr ibution ra th er tha n oth er we ekly ,
qua rte rly, and ye ar ly distribu tions . Wh ile temper a tu re da ta show twin pea ks in a
ye ar , tempe ra ture s in a m on th s eem to ha ve a u nimodal distr ibution. One pe ak
is b etter tha n twin pe aks to m ea sur e its distr ibution. T hus , m on thly distr ibution
ta kes an a dva ntag e in fore ca sting tem pera ture e ffe ct.
For the m onthly tem pera tur e dis tribution, we us e kern el e stima tion. We us e
the most popula r c hoic e of ke rne l, th e g aus sia n ke rne l. Gene ra lly, se lection of
the k erne l is le ss im por tant tha n se lection of the ba ndwidth ove r wh ich
obse rva tions ar e a ve ra ge d. T he mos t comm on me thod of se lecting a n optima l
ba ndwidth is the me th od of c ross - va lidation. It is known tha t this me thod is
robus t and a sy mptotic ally optim al. In this pa pe r, ther efor e, the cr oss- va lidation
cr iterion is applied to monthly tem pera tur e da ta.
In
th e fina l ste p, we
tem pera ture
r esponse
es tima te
func tion
the te mper atur e e ffe ct by
to
trunc ate d
tem pe ra ture
integ ra tion
distr ibution
of
ove r
tem pera ture doma in. T he tempe ra ture effec t is re pres ented as follows,
whe re T E r epre sen ts te mper atur e e ffect, T R( t) m ea ns th e te mper atur e r es pons e
func tion, a nd g (t) is th e monthly tempe ra ture distribu tion. T he te mper atur e
effec t provide s the informa tion of the res pons e of g as dem and to te mper atur e
within a m onth .
2 ) D a t a D e s c rip t io n
For es tima tion of te mper atur e e ffe ct, we use time s er ies of tem per atur e a nd
g as dem and fr om 1990 to 1996. T e mper atur e h as bee n obse rve d in Seoul e ver y
thr ee hours , while ga s dem and is an a gg reg a te of g as c ons umption flowing from
KOGAS to city ga s c om pan ies ob ser ved at 8 time s of a da y.
We u se ga s c ons umption pe r c ons ume r a s ga s de ma nd, beca us e total g a s
dem an d ha s be en g rowin g too fas t to yield appropria te s ea sona lity in the
es tima tion. Actua lly, tem pe ra ture effect is a n individu al e ffe ct, so that w e ca n
ea sily
obtain
th e r egu lar ity on the
rela tionship
betwee n g as
dema nd a nd
tem pera ture by us ing per ca pita consu mption of g as. T he se ries of num ber of
cons ume rs is obta ined fr om city ga s c om pan ies. E ven if we use per ca pita
cons umption in es tima ting te mper atur e r esponse , per ca pita c ons umption m ay b e
g rowing or dec re asing sy stem atica lly. In this cas e, th e tr end in per c apita g a s
cons umption should be c ons ider ed for appropria te tem pera ture res pons e. As a
pre limina ry
tes t for
tr end
in pe r
capita
g as
cons umption, we
ch eck
the
sc atter plot of per ca pita ga s c on sum ption an d tem pe ra ture for eac h y ea r of th e
sa mple per iod. Figur e 1 sh ow s the sc atter plots . In the fig ure 1, we a re not sur e
tha t the rela tionship betwee n te mper atur e a nd ga s de ma nd s hifts up or down
over the s am ple per iod, a nd that the rela tionship is cha ng ing ove r the sa mple
per iod.
Als o, we a rg ue tha t, in a ce rta in tim e, tempe ra ture re spon se ma y be differe nt
from th at in a nother . It is pos sible th at r esponse of g as dem and to tem pera tur e
in the m orning is differ ent from tha t in the night. T he diffe ren ces in patter n as
we ll a s siz e of the r esponse ma y appea r. Fig ure 2 display s sca tter plots of pe r
ca pita consum ption an d tem pe ra ture by 8 diffe ren t time s of a da y dur ing the
sa mple pe riod. Fr om this fig ur e, we k now tha t th e res pons e is differe nt betwee n
da y a nd n ight in pa ttern an d siz e. Es pecia lly, fig ure 2 displa ys sim ilar pa tte rns
betw een 3 a nd 24, betwe en 6 a nd 21, an d a mong 9, 12, 15 a nd 18 o' clock. Since
the 8 r ela tions hips a r e not the sa me, we cla ssify the 8 r esponse s in to 3
su bgr ou ps ; the firs t gr ou p consis ts of the re sponse s a t 3 an d 24; th e s econd
g roup, at 6 an d 21; an d third gr oup, at 9, 12, 15, a nd 18.
In the a bove, we a rgu e that us ing differe nt for ms of r es pons es a mong ye ar s
does not help estim ate tem pera ture r esponse fun ction a nd tha t us ing differe nt
forms of r espons e a mong c erta in tim es of a da y leads to a n appropria te
tem pera ture res pons e function es tima tion. We te st if the differ ent re sponse s
a mong y ea rs or th e differe nt re sponses am ong tim es of a da y fit the da ta w ell.
We use loc ally w eig hted re gre ss ion 4 to estim ate te mper atur e r esponse in the
thr ee ca ses ; we us e, firs t, the as sum ption of a sing le te mper atur e res pon se
func tion; se cond, 7 differ ent y ea rly tem pe ra ture r esponse func tion s; and third, 3
differ ent tem pera ture re spon se fun ctions a mong th ree times of a da y. An d we
ca lcula te sum of squ ar ed e rror s for th ree ca se s. T he s ma ller su m of s quar ed
er rors me ans that the corr esponding cla ss ification e xpla in the da ta b etter. T he
4 35% of dat a are us ed at a point of dat a for t he locally weight ed regres s ion in t he
t hree cas es.
ta ble 1 s hows the sum of squ ar ed e rr or s for thr ee ca ses .
T ab le 1. Sum of s qua red err ors
Cla ssific ation
Sum of squa re d e rror s
T otal
41.1685
Differe nt y ear s
Differe nt tim es of a day
36.8851
19.0488
In T able 1, w e can see that the clas s ification of different res ponses
among
three times
of
a
day
explains
t he
dat a better
than
other
clas s ifications .
3. Es tim ation and Res u lt
1 ) G l o b a l S m o o t h in g M e t h o d
A. T e mper atu re Re sponse
T em pera tur e Re sponse F unction is define d as the r ea ction of cons ume rs to th e
ch ang e of te mper atur e.
For the tem pe ra ture re sponse es tima tion, w e us e th e
Four ier fun ctional form whic h is us eful to tra nsform th e data from tim e doma in
to fre quenc y dom ain .
(3)
wher e ln y is the loga rithm of ga s cons umption per ca pita , and t is the
tem pera ture .
In equa tion ( 3), it is har d to know how m an y ha r monics to fit.
W hen lar ge
num ber s of coefficie nts ar e to be fitte d, estim ation bec om es les s e fficient.5 A
bes t wa y to do it is tha t it is norma lly de sir able to keep the num ber of fitting
func tions as low a s pos sible pr ov ided they s till a dequ ately de scr ibe the tim e
se rie s.6
After a ce rta in am ount of ex per imen ta tion, w e choose a numb er of coefficien ts
5 Gr anger and Newbold (1986) chapter 5.
6 Gr anger and Newbold (1986) p171.
a s I=2 and k =1 in equa tion (3). F or the r elations hip betwe en e lectricity a nd
tem pera ture , KEE I ( 1995) use d the sa me orde r of c oe ffic ients .
T h e column 1 in
T able 2 s hows the res ult of Fourie r estim ation and Fig ure 3 s hows the
tem pera ture r esponse us ing equa tion ( 3). In this setting , we fou nd tha t the left
en d tail ha s fla t s lope a nd the r ight end tail has upr ising u pw ar d s lope.
T able 2. Es tima tion of Four ier Func tion
Variable
M ode l1
Model2
d1
d2
Cons tant
1.70 (0.06)
1.35 (0.056)
0.60 (0.10)
0.66 (0.11)
T
- 3.57 (0.29)
- 0.53 (0.27)
- 3.84 (0.52)
- 4.94 (0.56)
T2
3.42 (0.28)
0.07 (0.25)
3.76 (0.52)
4.54 (0.57)
Sin(2πt/ n)
- 0.27 (0.02)
- 0.02 (0.02)
- 0.33 (0.04)
- 0.41 (0.04)
Cos (2πt/ n)
0.32 (0.01)
0.25 (0.01)
- 0.01 (0.01)
- 0.03 (0.01)
Sam ple s ize
20456
20456
R2
0.68
0.87
AIC
- 3.40
- 4.28
T his m ay be the effe ct of nig ht consum ption of ga s. T hu s, w e use a differe nt
model spec ifica tion, by u sing two diffe ren t time dum my va ria bles ,
(4)
wher e d1 is the dum my v ar iable wh ich is one if ob ser va tion time s ar e 6 a m or
9 pm an d d 2 is the dumm y va ria bles whic h is one if obse rv ation tim es a re 3 am
or 0 am .
T he re sult of e qua tion (4) is re por ted in the column 2 in T a ble 2 a nd th e
g ra ph of this e stima tion is s how ed in F igur e 4. T he differe nce be tw een Fig ure 3
a nd Fig ur e 4 lie s in the rig ht a nd le ft end of te mper atu re res pon se function.
T he left end of T R function in Fig ur e 3 s how s the ste ady in cre as ing pa ttern.
T he righ t en d of T R function shows the less s te ep s lope c om par ed to F igur e 3.
T he model 2 is better fitting to da ta than the m ode l 1 in ter ms of Ak aike
Informa tion Crite rion a nd adjusted R 2 .
B. T e mper atu re Distr ibution F unction : Non pa ra m etric Es tima tion
We ca n us e n on pa ra me tric es tim a tion techn iques to ca pture a wide va rie ty of
nonline ar ities w ithou t pa rtic ular spe cifica tion of the non linea r r ela tion.
T he
nonpa ra metr ic estim ation appr oa ch ha s a s tr ong point in tha t this approa ch
re quir es a few as sum ptions a bout the na tur e of the n onlin ea rities .
Howev er,
this approa ch is prone to be ine ffe ctive for s am ple siz e a nd ha ve ov er - fitting
problem s.
T he most comm only u sed nonpar am etr ic e stim ators a re sm oothing
es tima tor s, in w hich obse rva tion err or s ca n be r educe d by av era gin g the da ta in
sophis tic ate d wa ys . Ker nel estim ation ma y b e a n exa mple of s moothin g.
7
In ge ner al, s mooth ing estim ator of Xt m ay be ex pres sed as
(5)
and
(6)
whe re th e weig ht {t , T (x )} ar e la rge for thos e Xt ' s n ear x a nd sm all for those
Xt ' s far fr om x .
We ca n de fine th e w eigh t func tion to be use d in equa tion ( 3) as
(7)
(8)
(9)
whe re K is the proba bility de nsity func tion, calle d ke rne l a nd h is the ban dwidth
. If h is v ery s ma ll, th e a ve ra ging will be done with re spect to a ra th er sm all
7 S plines , art ificial neural netw ork and average derivat ive es tim ators are ot her
s m oothing m ethods . See Campbell
(1977) chapt er 12.
ne ighborh ood ar ound ea ch of the Xt ' s . If h is la rge , the a ve ra gin g will be ove r
la rg er ne ighb or hoods of the Xt ' s. T h us, ban dwidth is the s moothin g pa ra mete r
tha t contr ols the deg ree of a ve ra ging a mounts .8
By subs tituting equa tion (7) a nd (8) into eq uation ( 9), now w e g et the ker nel
es tima tor
of m(x ):
(10)
Unde r c er ta in
re gula rity
conditions
on the
s hape
of the
ker nel K
a nd
ma g nitudes an d beh av ior of the weig hts, it is shown tha t conve rg es to m (x )
a sym ptotica lly in se ver al way s. T h is conv erg enc y pr oper ty holds for a wide
cla ss of ke rne ls. In this pape r, we us e the m os t popula r ch oic e of k ern el, the
g aus sia n ke rne l:
(11)
It is note d that c hoosin g the pr ope r ba ndwidth is cr itica l in an y a pplic ation of
Ker nel es tima tion . T he most comm on me thod of se lectin g a n optima l ban dwidth
is the method of c ross - va lida tion. It is known tha t th is me thod is robust a nd
a sym ptotica lly optima l. T he optima l b andw idth is the one whic h minim izes a
we ighted- av era g e s qua re er ror of the k ern el e stima tor.
(12)
(13)
wher e (Xt ) is the n onn ega tive weig ht fu nction that is r equir ed to r educ e
bounda ry e ffe cts.9 Equa tion (13) is simply the ke rne l e stima tor ba sed on the
da ta s et with obse rv ation j de leted, ev alua ted a t the jth obser va tion Xj . T he
8 As the bandwidt h is increas ed, the local averaging is performed over success ively
wider ranges and t he variability of the kernel es tim ator is reduced. In the lim it, t he
kernel est imat or approaches t he s am ple average of {X t }.
9 F or the det ailed CV criteria, s ee Kim and Cox (1996).
func tion CV(h ) is ca lled the cross - va lida tion func tion bec aus e it va lida te s th e
su cce ss of the ker nel es tima tor in fitting {Yt} a cr os s th e T sub- sa mples , ea ch
with one obse rva tion omitted. T he optima l ba ndwidth is the on e tha t min imize s
this func tion. An a sy mptotic optima lity is esta blishe d for a ba ndwidth se lection
ru le w hich ca n b e in terpr eted in ter ms of c ross- va lidation.
Firs t of a ll, w e estim ate m on th ly te mper atu re distr ibution with the ker nel
me thod. T he is sue in es tim ation of tem pe ra ture distrib ution is the s elec tion of
ba ndwidth a nd th e nu mber of points in the e stim ation. T h e cr oss- va lidation
cr iterion is applied to monthly tem pe ra ture da ta. Howev er, this e ffort res ults in
too wide va rie ty of b andw idth be ca use of sa mple size. T hus, we applied CV
cr iterion to ag gr eg ate monthly da ta , so th at we get more sta ble ban dwidth
a round 0.1.
C. T e mper atur e Effect
T em pera tur e effe ct ca n u se all infor ma tion of the hour ly r ela tion betwe en ga s
cons umption per c apita a nd tem pera ture and of monthly tem pera ture distribu tion.
T hus, tem per atur e e ffe ct c an ca ptu re the e ffect of te mper atur e on consum er ' s
beh av ior . T e mper atu re effec t ca n b e e xpr ess a s follows.
Wher e T E mea ns temper a tu re effect a nd g () is the monthly te mper atur e
distr ibution. Riem an in te gr ation me th od is us ed for in te gr ation. It is noted that
the num ber of point in es tima tion is fixe d to be 50.
T he est imated temperature effect is pres ented in F igure 5 and F igure 6.
T empera ture effect in F igure 5 is t he res ult bas ed on equation (3) and
T empera ture effect in F igure 6 is the res ult bas ed on equation (4). Both
of graphs s how the similar trend over tim e. How ever, F igure 5 and
F igure 6 are different in tw o aspect. F irs t, F igure 6 is cons tantly above
F igure 5 in terms of temperat ure effect. S econd, t he s ummer s eas ons in
F igure 6 a re sm oother than thos e in F igure 5. It is noted that w e cannot
evaluate the appropriat enes s of temperat ure effect s in bot h models at this
s tage.
2 ) L o c a l S m o o t h in g M e t h o d
A. T e mper atu re Re sponse Fun ction: Loca l S moothin g Me th od
Another method of es tima tion for tem pe ra ture r es pons e is loc al sm oothin g
me thod, which provide s a rela tionship betwe en tw o va ria bles us ing loca l da ta
a round one point. W e work w ith low ess (robus t loc ally w eig hted re gre ss ion,
Cleve land 1979). T he b asic ide a of lowe ss is to c rea te a new va ria ble th at, for
ea ch y i , conta ins the c or re sponding sm oothed va lue. T he smoothed v alu es ar e
obtain ed b y r unn ing a re gr ess ion of y i on x i usin g on ly da ta ( x i , y i ) a nd a sm all
a mount of the da ta nea r the point. In lowes s, the r eg res sion is weig hted so that
the ce ntra l point (x i , y i ) gets the hig hes t we ight a nd points far ther a wa y rec eive
les s. T he e stima ted r egr ess ion is then to pr edic t the s moothed va lue of ga s
dem an d for e ac h poin t of ga s de ma nd data . T he pr oc edur e is r epea ted to obta in
the re ma ining sm oothed va lues , which me ans a se par ate weig hted reg re ssion is
es tima ted for ev ery point in the data . T his loc ally we ighted
sca tterplot s moothin g is a desir able s moothe r beca us e of its loca lity
it te nds
to follow th e data . In oth er words , a ttra ction of the locality is that the pre dicted
follow s the da ta' s cur va ture . Polynom ial s moothin g m ethods a re g lob al in tha t
wha t ha ppe n on the e xtre me le ft of a sca tterplot ca n a ffe ct the fitted v alue s on
the extr eme rig ht. Lowess me thod is a s follow s. Let y i an d x i be the tw o
va riable s a nd a ss ume the data is orde red so tha t x i< x i + 1 for i=1,,N- 1. For e ac h
y i , a s moothe d va lue y i s is calc ula te d. T he su bset use d in c alc ulation of y i s a re
indic es i_ = ma x( 1, i- k) throug h i+ = min( i+k,N) , wher e k=N*bandw idth. T h e
we ights for ea ch of the obser va tion s betwe en j=i_ , , i+ ar e
ma x (x + - x i , x i - x _ ). T he sm oothed v alu e y i s
pre diction.
wher e
= 1.0001
is th en the weig hted r eg res sion
We trie d to e stima te a tem pe ra ture r esponse by u sing the wh ole da ta. For
this, we as sum e th at the for m of the temper a tu re re sponse is inv ar ian t over
time . In oth er words, r ega rdle ss of sea son, month, we ek, a nd da y, the res pons e
of ga s dema nd to te mper atur e is a ss ume d to be all the s am e. T he r es pons e of
g as dem an d to 10C in spr ing is the sa me a s in fa ll, or w inter . Ac cording to th is
a ssu mption , we ca n use the whole ye ar da ta of the two va ria bles to es tima te
the tempe ra tu re re spon se. Lowe ss is u sed as es tim ation method, a nd ba ndwidth
is 35%. T h at is, 35% of da ta nea r a point is us ed to r egr es s ga s dem an d on
tem pera ture for e ac h point of data . T he r esult is in figur e 7. Bigg er ban dwidth
lea ds to sm oother tem per atur e r es pons e.
T his fig ur e 7 g ive s us a n e xplan ation of rela tionship be tw een te mper atur e a nd
g as de ma nd. As we e xpec te d, the tempe ra tu re r es pons e of ga s dem an d is
dec rea se d below ar ound 22C ( hea ting de ma nd for ga s), while it ha s a slig htly
upwa rd s lope (cooling de ma nd for g as) above 25C. It implie s tha t h ea tin g
dem an d for ga s is ver y s tr ong bu t coolin g dem an d is we ak. In the sa mple
per iod, most of ga s dem an d res pondin g to temper a tu re is due to hea ting. T h e
fa ct that m inim um point of the te mper atur e r esponse is e stima ted a round 22C
g ives a n intu ition that people us ing na tura l g as beg in to res pond below 20C.
Sinc e sm oothin g m ethod puts the m inim um point to high er tha n r ea lity, the
sta rting
tempe ra ture
to
re spond
to tem pera tur e
is
be low
20C.
An othe r
inter es ting point of the te mper atur e res pons e is a t 0 to 5C, wh ere the slope of
the r esponse cha nge s. Be low 0C people r espond to tem pera ture cha nge le ss tha n
a bov e 0C. It tells tha t indiv idua l c on sum er sta rt to r espond at some point
betw een 0C and 10C.
Als o, Fig ure 8 shows th ree te mper atu re r esponse s for da ytime (9, 12, 15, 18),
nig httime( 3, 24), a nd inter time (6, 21). It shows tha t te mper atu re r es pons es ar e
differ ent be tw een day time and n ighttim e. T he da ytime re spon se to a ce rta in
tem pera ture a ppea rs to be bigg er in both s ize and slope th an nig httime re spon se.
B. T e mper atu re Distr ibution
Monthly tempe ra ture distr ibution is e stima ted. Ker nel density e stima tes ar e
produc ed ov er tem pera tur e within a m on th. M on th ly tempe ra tu re dis tr ibution is
es tima ted to produc e monthly tem per atur e effec t. Here , g aus sia n ker nel dens ity
es tima tes ar e produce d. Ke rne l de nsity es tima tion is ex plaine d in the a bov e
se ction of globa l sm oothing me thod. W e us e the optim al ban dw idth whic h
ST AT A prog ra m rec om me nds. T he ban dwidth is dete rm ined by a s follows ,
Als o, for da ytime , in tertim e, and nighttim e tem pera tur e re sponse s, we e stima te
tem pera ture distribu tions of day time, inter time, and nig httime , re spec tively . T he
thr ee distribu tions a re s imila r in s hape , but differ ent in m ea n.
C. Steps III: T em pera tur e E ffects
T em pera tur e effec t is es tima ted a s integ ra tion of te mper atur e r espons es ove r
monthly
tem pera ture
distr ibution.
Riema n
s umm ation
is
us ed
to
es tima te
tem pera ture e ffect. T he es tima ted te mper atu re effec t is in Fig ur e 9 an d Fig ur e
10. Fig ure 9 repr ese nts tem per atur e e ffe ct in the ca se of th e one te mper atur e
re sponse
a nd
m on th ly
te mper atur e
distribu tions .
Fig ure
10
s hows
the
tem pera ture effect with thre e tempe ra ture r espons es (da ytim e, in te rtim e, a nd
nig httime) a nd c or re spon ding monthly tem pe ra ture dis tr ibutions.
We es tima te tempe ra ture e ffects w ith g lobal a nd local s mooth ing. Actua lly, w e
es tima te fou r m onth ly se ries of te mper atur e effe cts. In globa l smoothing method
of te mper atur e r esponse , we estim ate th e tota l tempe ra tu re r espons e an d th e
tem pera ture
re sponse s of da ytime , intertim e, a nd nig httime. Also, in
loc al
sm oothing method, the tota l tem per atur e res pon se a nd th e te mper atu re re sponse s
of da ytime , intertim e, a nd nig httime a re estim ated. T hose fou r te mper atur e
re sponse e stima tes a nd m onthly te mper atu re dis tr ibution es tim ate s ar e inte gra ted
over te mper atur e to yield 4 diffe re nt se rie s of te mper atur e effects .
Figur e 11 displa ys the two tem pe ra ture e ffects fr om loca l sm oothing method;
one fr om total te mper atur e r esponse a nd the othe r fr om thre e tem pera tur e
re sponse s depe nding on times of a day (da ytime , in tertim e, an d nigh ttim e). T he
tem pera ture e ffe ct fr om the tota l te mper atur e res pons e is s lightly big ge r tha n
tha t from the thre e te mper atu re re spon ses . Fig ure 12 shows the two tem pera tur e
effec ts from globa l s mooth ing me thod. It sh ows th at the tempe ra ture effe ct fr om
tota l te mper atur e r espons e is low er than that fr om thr ee te mper atur e r es pons es.
Differ ent me thods y ield diffe ren t s ize of tem pe ra ture e ffe ct betwee n tota l a nd
thr ee temper a tu re re sponses .
Figur e 13 compa re s th e tem per atur e e ffe ct w ith total te mper atur e res pon se
betw een
global a nd loca l sm oothing
me thods . Als o, Fig ure
14 shows
the
tem pera ture e ffect with thre e tempe ra tu re r esponse s dependin g on tim es of a
da y betwee n global a nd loca l smoothing m ethods. T he two fig ure s te ll us tha t
loca l sm oothing e stima tes ar e less fluctua ted th an globa l sm oothing e stima tes.
Loca l smoothing e stima te ma gn ifies deg re e of se as on ality lower than g lob al
sm oothing e stima tes .
4. Concluding Rem arks
In this pa per , we estim ate va rious tem pe ra ture effec ts in na tura l g as dem and.
We ha ve thre e s teps for the estim ation; tempe ra tu re re spon se, tem per atur e
distr ibution,
a nd
the
in tegr ation
of
the
tem pe ra ture
r esponse
ove r
the
tem pera ture distr ibution. T o e stima te the tem pera ture r esponse s, we ex ploit two
nonpa ra metr ic smoothing m ethods; globa l a nd loc al sm oothing m ethods. Als o, w e
es tima te tw o tempe ra ture r esponse s for ea ch sm oothin g method; on e repr ese nts
for one te mper atur e r espons e of a da y, the othe r for thr ee diffe rent te mper atur e
re sponse s w ithin a da y. For tem pera ture distr ibution, we use ke rne l dens ity
es tima tion.
W ith
the
te mper atur e
r esponse s
a nd
the
monthly
te mper atur e
distr ibutions, we obta in four te mper atur e e ffe cts of ga s de ma nd. T he n, we
compa re
the four tem per atur e effec ts. We find tha t the g lob al s moothin g
tem pera ture effects a re muc h more fluc tuate d tha n the loca l smoothing e ffects,
a nd tha t th e te mper atu re effec ts with th ree differ ent tempe ra ture r esponse s ar e
mor e fluc tuate d tha n tha t w ith the one tempe ra ture re sponse .
Future work w ill be to dete rmine wh ich one am ong four differ ent te mper atur e
effec ts
is
better . Howeve r,
th at
work
is
dee ply
as socia te d w ith
de ma nd
es tima tion a nd dem and fore cas ting. One of deter min ants w ill be the ma g nitude
of dem an d forec as t er rors . T h os e thin gs belong to our ne xt w or k.
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