arima modeling

7/26/2019 ARIMA Modeling

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ARIMA Modeling:B-J Procedure

A k [email protected]

9811216905


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Structural modelsmultivariate in nature, and

Y = F (movements in current or past values of oter(e!planator"# varia$les#%

Univariate time series models

Yt= F (own past values& 'urrent past values of an

error term#

useul !hen a structural model is ina""ro"riate.

!hen other varia#les are not o$serva$leor not

measura$le )!planator"varia#les are measured at a lo*er

fre+uenc" of o$servation tan yt. $or e%,ytmi%ht #e a

series o dail& stock returns, !here "ossi#le e'"lanator&

varia#les could #e macroeconomic indicators that areavaila#le monthl&.


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Moving average processes:

(et ut#e a !hite noise "rocess !ith )ero mean * constant

variance 2. +hen

"t= , ut ./ut-/ .0 ut-0 % % %

.+ ut-+

is a -Aq/ model.


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Autoregressive processes:

An A model is one !here the current value o a varia#ley, de"ends u"on onl& the values that the varia#le tookin "revious "eriods "lus an error term. An A model oorderp, denoted as Ap/, can #e e'"ressed as

Yt = u 1/ "t-/ 10 "t-0 22%% 1p"t-p ut

!here utis a !hite noise distur#ance term. +hee'"ression o A"/ model can #e !ritten morecom"actl& usin% si%ma notation


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ARMA (p&+# processes

o#tained #& com#inin% the Ap/ * -Aq/ models.

tates that the current value o some seriesy de"ends

linearl& on its o!n "revious values "lus a com#ination ocurrent and "revious values o a !hite noise error term.

+he model could #e !ritten

Yt = u 1/ "t-/ 10 "t-0 22%% 1p "t-p

./ ut-/ .0 ut-0 2%% .+ ut-+ ut

s a A-A ", 3/m model


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An AR process as:

4 a %eometricall& deca&in% AC$

4 a num#er o non)ero "oints o AC$ 7 A

order.

A MA process as:

4 num#er o non)ero "oints o AC$ 7 -A order4 a %eometricall& deca&in% AC$.

A com$ination ARMA process as:4 a %eometricall& deca&in% AC$

4 a %eometricall& deca&in% AC$.


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Process of ARIMAModeling

dentication ta%e

stimation ta%e

:ia%nostic Checkin% $orecastin%


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Identi3cation

$irst identi& the order o inte%ration on the#asis o visual ins"ection o time series "lot,correlo%ram and ormal unit root testin%.

:etrend and :iferentiatethe series i re3uired to

o#tain a stationar& series. denti& the undrla&in% A "/ and -A 3/

"rocesses in the stationar& series #ased on the#ehaviour o the correlo%ram.

n order to achieve "arsimon& kee" the num#ero A and -A la% orders, " and 3 as small as"ossi#le; "rovided the model still satisactoril&orecast the series.

t is #etter to used mi'ed A-A ", 3/ model

rather than "ure A "/ or -A 3/ model.


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)stimation

involves estimation of parameters of model

s"ecied in ste" 1.

+his can #e done usin% least s3uares or

another techni3ue, kno!n as -a' likelihood,de"endin% on the model.


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Model checking

determinin% !hether the model s"ecied and

estimated is ade3uate.u%%ested methods are? should have least

values.


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Negative MA Coeff


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Negative MA Coeff


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)stimation 5tage

A "ure A model can #e estimatedusin% ( estimator #ut i the modelalso includes -A terms -( estimator

is used.


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6iagnostic 'ec7ing


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8o* good te model 3ts te

data9 8o* relia$le *ill $e teprediction $ased on it9

se some indicators o %oodness o tsuch as 2 and ABA.


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Is te model sucientl" speci3edor it re+uires including some moreAR or MA terms9

'amine the residuals. +heircorrelo%ram can %ive a clue iresiduals are "ure !hite noise or atleast ree rom serial correlation. ?

and (=statistics o residuals is useulor detectin% serial correlation. se (testor A in residuals.


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6oes te model include someunnecessar" parameters (AR or MAterms# *ic can $e dropped out*itout a signi3cant loss in predictiveecienc" of te model9

tstatistic o individual coeDcient can%ive an initial #ut not conclusive/clue.

.


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Forecasting

n A-A ", 3/ the "rediction e3uationis sim"l& a linear e3uation that reersto "ast values o ori%inal time series

and "ast values o the errors. n an A-A ",d,3/ model !here dE0,the orecastin% can #e made at t!odiferent levels at the level o diferenced stationar& time

series, and

at ori%inal inte%rated time series.

arima modeling

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