BAYESIAN MULTIPERIOD FORECASTING FOR INDONESIAN INFLATION USING ARMA MODEL.

International

lournal

of

Applied Mathernatics and Stalistics,

Irrt.

l.,Appl.

Math.

Stat.;

Vol.

55;

Issue

No.

2;

Year 2016, ISSN

0973'7371 (Prinl), I55N 0973'7545

(Online)

Copyright

(O

2016

by CESER PUBLiCATIONS

Bayesian

Multiperiod

Forecasting

for

lndonesia lnflation

Using ARMA

Model

ZulAmry

Department of Mathematics,

State University of Medan, lndonesia

e m a i I : er::rv n:lV_7. r(i; _{8! IA1.|,ePry:l}

ABSTRACT

'l'his paper pr-esenl a Eayestan approach ta fincl the Bayesran forecasting ttsing ARMA model

t.tnclr:r Jeffrey's prlor i)ssLtnlpttotl wtltt qua{lratic krss fi-;rtcllon applied to morthly national

rpflatiop

of

incionesia from ..jartuary 2a0A

b

Decentber 2a14. The forecastlttg ntodel be

ol)taine(l based lhe rnarginal r:onditional poslerior predictive density To conclude whether

the moclel

is

arlequate lsrised theLjung-Box Qsfafisflc, whiletolookattheaccttracyof the

/nodel ls used lhe Root Mean Square error (RMSE)

Key

word:

ARMA model, Bayes theorem, national inflation

Mathernatical Subject Classification: 62C10

1, INTRODUCTION

The ciassical forecasting have been depeloved by Box and Jenkins (1976). There are three steps are

accomplished in the process of fitting the ARMA (p,q) model to a time series: identification of the model,

estimation of the pararneters, ancJ application to forecast. ln the classical approach,

the

parameters

are considereri fixed but unknown, whereas

ihe

Bayesian approach considers the parameters as

ran6om variables, wliich are de scribe d by their probability density functicn, Several of works relating to

Bayesian forecasting in the ARIVA rriodelare Liu (1995), Anrry and Baharum (2015)Fan & Yao (2008),

Kleibergen & Hoek (1996), and Uturbey (2006). This paper focuses to find the Bayesian forecasting model for ARMA modei using Jeffrey's prror with quadratic loss function.

2. MATERIALS AND METHODS

Bayes theorem calculates the posterior distrlbution as proportional to the product of a prior distribution anrj the likelihoocl function, the prior distribution is a probability rrodel describirig the knowledge about

the parameters before observing the currently by the available data. Forecasting in

the

Bayesian

approach is based on the construction of the marginal conditional posterior predictive distribution. Main

idea of Bayesian forecasting is the preclictive distribution r:f the future given the fast data follows directly frorn the joint probabilrstic nrodel, predictive drstribution is derived from the sampling predrctive density

www. ceser. tn/ceserp www. eeserp. com/cp-;ou r

lnternstionol Journol of Applied Mothemqlics ond Stotislics

weighted by posterior distribution (Bijak, 2010)' The method

in this paper

is

study

of

literature by

applyingtheBayesiananalysisunderJeffrey,sassumption,whereasthenraterialsareson,]etheories

inmalhematrcsancJstatisticssuchastheARMAmodel'tsayestheorem,integralion,distributronof stat]stlcs

and

inflation

daia

frr:m January 2000

to

December 2014. The forecasting

model will be

appliedtcforecastthedatafromJanuaryZal|toDecemberZ}llbasedondatafromJanuary2000

to December 2013'

3"1. Likelihood function

The k-step-ahead point forecast of

/

n * I

3. RESULTS

is defined bY:

l(k)=li(r',,,

i.5,, I

Whefe S, =(u,, ).:,.,., i,,,,u r)

The A.RMA (P,q) model is defined bY '

''

=io''

*ia'"'

*"

(32)

where{elissequenceolitclnrrrmalrandomvarialrleswither*NQ,r"1),r>0andunknown,0iand

01 are parsmeters" Residuals are written

as:

, = r,, - i,,, r,

,-f.o

r,

.

(3 3)

0, where r = min(0, P+1*q)' one

kk,W,

.' ,(n 0t, 02,

"''

0q) and

r

(3 1)

(3 4)

(3 5)

By conditioning the first p observations and letting ep = cp-1

='

'=

er =

may approximate by Box & Jenkins [4]' the likelihood function for V= based

:,

is:

ilt-i' I)-l

t\Y r:!;)*t

'

"'t7'

Fufihermoretheequation(34)canbeexpressedas:

r.lv',

'i.s, )

-' r

l

ll.-, ir', I jl \ ,

i '",

where Br = (Yt, _{Yt ',}

By letting W=lJUr

and V=Urxo,

n.

i/

and

D, are maximum likelihood estimator of r'l

| ,l

rl 'i'1

,,

I,f,\

i

-lQ,.r',-,

.rll

-

ir,.

,

\!

t=,,,]l

I I l I I I I

(-i _=l

I

.-'

L;,

j

where

r, = ,

i" -. ,,

lnternalionol Journcl of Applied Mnlhepnotics

snd

Stotislics

arrd r'i

,

ihe iikelihooci function in eqL;alron (3 5) can be expressed as.

"l

| 't'

r

,

.t

\

-'t

r ,'t' 1t'1"

iJ 6)

'_t]

3.t. P*stericr

distributicn

3:.rsud ihe likeiiiroorl funclion in equatic;tt {l}.6). the "..}effrey's ptior is

/{/,,itr t''fi

By appiying the Bayes theorem lo equatian {3.6) and (3.7), the posterior

cf r"

arid r

'

is

(.3 7')

l,r i I :r') i,

-' r

t

".t7,

( i.ti)

3.3. Conditionai posterior predictiv*

Eased on

,,. , :r,

)-p,, wiih,'

.r{rr

r

'l

will be gotten that

I t L' li '|' ., , 1.:'r,,']

", r,,,, , i, )

t

,

1l ii rxp;re ssr:c

in

'

. it lrtili i;econtes:

..:

ji t, .\' .tlJ r .' \.1:r I r ,. _{. \. , Y0 ,'} , -i

Bascrj ihe equation (3.9), the conditional predictive density of

Yn'

t

is:

{3 e)

(3.1 _{1 )}

: r. ) ,. tt tt tl ) = 'l' h,.,

105

Basucl un the equation (3.9) anci (3.12), the conciitional posterior preeltctive density of Y,,.r' is:

/,

l',,,

l5.l' '1'' 7

t)

'= _'(5' '

'''

"s, )7r _{"', 't,} s:''v'

r

t

)

,.\.tt

l.

,

""''''

rltt

'l

'

o

^

'.'

1,a.

il

(3.13)

r

(\'/'

i

''

t

llr

'

')''

r

''l' + ';r^

-,lt

ll lnlernationol Journol of Applied Molhemotics

snd

Stolistics

' i r .. iril

" r .\7ri ,i';-' -:'Y'JJ. '.\,-t*\'l/t ' , ]',i

- ri

,rt,t,

!,1,;, ,

*v''

Rv'

:v'' 8,,.^ _,

.\', ]

(g'tZ)

where ll ..,t'],,,t. _totl ,1,,0

,

ancj f

y''8,,,

,)'

-'l't

RY

whereZ=W+R

3.4. Marginal conditional posterior predictive

The marginal conditional posterior predictive density of Y,*r is obtained by integrating equation (3.13) with respect

to

il

and

f

7, _{that is:}

intk-l-1tt

I

t;

ll

l

(3.14)

The marginal conditional posterior predictive density

of

Yn*x is a univariate student's t-distribution on

(n+k-1*p) degrees of freedom with mean

,-(.t

a;,,,2

'8,,',. )'' (1t!-,

z

r"')

3.5 Peint forecast

For quadratic loss function, the point forecast of Y,r*r, is the posterir:r mean of the marginal conditional

posterior predictive, that is:

rir -.

.r")

fi

B,; _-,,

,z

'r1.,,,,

,l'

\rt;-,

,z

't']

(3 15)

4. APPLICATICIN

'lhe _{fore casting nroclel rs applied for infiation mclnthly data based on Consumen} Price lndex (CPl), which

lrave coiiecteci for the periocl of January 2000 to December 2014, where the data period of January 2000 - December 2013 are used to forecast the period of January-December 2014.

Basecl

on

the plot

of

series, ACF, PACF

in

Figure 1, 2 and

3

can be concluded that the collection of data is stationary ancl based on the value of d ,

d

and AIC in Table 1,

a

suitable model for the data

lnlernotionol Journsl of Applied Molhemotics

snd

Stolisiics

plot of time series dala

I

$

$

5

I

.t

irJ,l.l

tt

''tt

I

Fiqure 2: plot of autocorrelation function

[image:5.612.99.516.70.767.2]

l

ll,il

tl

I I

Figure 3; plot of partial autocorrelation funclton

Table

1.

value of 'l'

,

e

and A/C

Model a () AIC

ARMA(710)

ARMA(0,7)

0,7Vp5

'Aiisi-

424.53 0.2696

ARMA(1, / -0.2738 0.5250 423.84

By using ihe Bayesian forecasting model in equation (3.15), the results of point forecast are presented in Table 2 as follows:

l[11

0 2ri30'178

I

0.t398343 0.0697564

0.0361444 0.0183071

0.2870595

0.15'15332 0.2759849

94llll]

0"8568429

lnternqtionol Journol of Applied Molhemolics

snd

Siotislics

Tesiing the result

of

point forecast on the forecasting model

to

conclude whether is adequate' investigation to autocorrelations of residuais by using the Ljung-Box Q statistic (Wei, 1994) is:

U:=n{ n I

1/,, ,\

X'\t\ - 11 -qtt (4 1)

l{

Q>X

,,ttt

7,,4)

theaclequacyof

the

model

isreiectatthelevel

o

wherenisthesamplestze'

p,

is lhe autocorrelaticn of residuals at lag k

and

K is

the number of lags being tesi' By supporting the calculation

in lhe

Table

3 it

can be obtained

thar

0 "" !t)

'\|te,

whereas /'"'"(rl\--

l(t

9t90

^n'

i"/

" 7 '',,, ,, g , _{, uo can be concluded that the results} of point forecast are adequate

Table 3. Computation of Ljung-Box Q statistic

one measure to cjetermine the accuracy of a forecasting model is Root Mean Square

(RMSE)'

(Assis et

all

2010), that is:

l? i\15l: {4.2)

,,\. / r

I /t.\tt

-\l-;

where ESS = the

colum '1 _through

5quarus

of'

('

error sum of square,

and

n

-

the number of observations. By calculation in Table 4' 5 contain the period

(/

), observation (

"'),

result of forecast

1t

). residual

('''

), and

) is obtained RMSH

'

0 336964

Number of lags: k _{Arrtocorrelation}

r)' I

-t-'t;

I

0. 1 0304 1

-0.1s0 i

0.022500 [image:6.612.77.532.122.721.2]

[image:7.612.121.512.80.411.2]

Nnfernsfionol Journsl of ,Applied Mathemsiie s

snd

Slqlisiics

Table. 4 . Computation

of

RMSE

169

1.07 10.2630176 0.8069824 0.65'12205939

ltotoA

j

oi:ge:+:

0i241657 0.0144397955

izr

I

cog

_00697564 0.0102436 0.00010493'13

a7z'tt

-

I

c.oiot4+a

0.02

l i

-0.0561444 0.003'1s21937

173

io16

0.0183071 0.1 41 6929 0.0200768779

174 A43

01939937

LI

175

{)

93

i

0 532460U

i

'176

a

47

0 2870595

0.2360063

o.a3is4oo o 1

sisaos

0.05569B9736

-'o

ti3e3325G

b o::+o722bb

a\ a1 ,l { f 1f r'J ,

\J1t : tJ t:)iil,).)z

I

I

otszgaor

oa3346rzz65

01184668

f

A14034s82"7

I

177

r'r B

aie

180

FSS = 1.362533 RMSE = 0 336964

5. CONCLUSION

This paper focus on

ihe

stuciy

of

mathematics

in

the Bayesian forecasting

of

A,RMA model under

lhe Jeffrey's prior assumption with quadratic loss function. By observing the behavior of ACF graph

shaped the damped sine wave in Figure 2 and based on the smallesi AIC value in Table 1 can be

concluded thert the time series daia {cr monthly national inflation of lndonesian starting on January

200Cuntil December20l3wasslationaryasuitablemodelforthedataisARMA{0,1)with

0=0

1696

\,4athematically the poinl fcirecast in equation (3.15) is obtained based the posterior predictive mean

r:f

the

marginal conditional posterior predictive

density. Finally, based

an

investigation to

autocorrelations of residuals by usinE the Ljung-Box Q statistic, can be conclude thai the forecasting

model is adequate with accuracy measure RMSE=0.336964.

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Bayesian Multiperiod Forecasting

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(1995).

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S.

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(1995).

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and

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20A6 Sbckltolnt, Sweden

),

1-7

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Nama Jurnal International Journal of Applied Mathematics and Statistics

097 3-!377,(Pri nt) 0973-7545 on li ne

Vol. 55, No. 2

2016

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c.

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t u rna I ll mia h I nternasional/ I nternasiona I Bereputasi

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lnflation

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