Download - ARIMA Modeling
-
7/26/2019 ARIMA Modeling
1/26
ARIMA Modeling:B-J Procedure
9811216905
-
7/26/2019 ARIMA Modeling
2/26
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. $or e%,ytmi%ht #e a
series o dail& stock returns, !here "ossi#le e'"lanator&
varia#les could #e macroeconomic indicators that areavaila#le monthl&.
-
7/26/2019 ARIMA Modeling
3/26
Moving average processes:
(et ut#e a !hite noise "rocess !ith )ero mean * constant
variance 2. +hen
"t= , ut ./ut-/ .0 ut-0 % % %
.+ ut-+
is a -Aq/ model.
-
7/26/2019 ARIMA Modeling
4/26
Autoregressive processes:
An A model is one !here the current value o a varia#ley, de"ends u"on onl& the values that the varia#le tookin "revious "eriods "lus an error term. An A model oorderp, denoted as Ap/, can #e e'"ressed as
Yt = u 1/ "t-/ 10 "t-0 22%% 1p"t-p ut
!here utis a !hite noise distur#ance term. +hee'"ression o A"/ model can #e !ritten morecom"actl& usin% si%ma notation
-
7/26/2019 ARIMA Modeling
5/26
ARMA (p&+# processes
o#tained #& com#inin% the Ap/ * -Aq/ models.
tates that the current value o some seriesy de"ends
linearl& on its o!n "revious values "lus a com#ination ocurrent and "revious values o a !hite noise error term.
+he model could #e !ritten
Yt = u 1/ "t-/ 10 "t-0 22%% 1p "t-p
./ ut-/ .0 ut-0 2%% .+ ut-+ ut
s a A-A ", 3/m model
-
7/26/2019 ARIMA Modeling
6/26
An AR process as:
4 a %eometricall& deca&in% AC$
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$.
-
7/26/2019 ARIMA Modeling
7/26
Process of ARIMAModeling
dentication ta%e
stimation ta%e
:ia%nostic Checkin% $orecastin%
-
7/26/2019 ARIMA Modeling
8/26
Identi3cation
$irst identi& the order o inte%ration on the#asis o visual ins"ection o time series "lot,correlo%ram and ormal unit root testin%.
:etrend and :iferentiatethe series i re3uired to
o#tain a stationar& series. denti& the undrla&in% A "/ and -A 3/
"rocesses in the stationar& series #ased on the#ehaviour o the correlo%ram.
n order to achieve "arsimon& kee" the num#ero A and -A la% orders, " and 3 as small as"ossi#le; "rovided the model still satisactoril&orecast the series.
t is #etter to used mi'ed A-A ", 3/ model
rather than "ure A "/ or -A 3/ model.
-
7/26/2019 ARIMA Modeling
9/26
)stimation
involves estimation of parameters of model
s"ecied in ste" 1.
+his can #e done usin% least s3uares or
another techni3ue, kno!n as -a' likelihood,de"endin% on the model.
-
7/26/2019 ARIMA Modeling
10/26
Model checking
determinin% !hether the model s"ecied and
estimated is ade3uate.u%%ested methods are? should have least
values.
-
7/26/2019 ARIMA Modeling
11/26
-
7/26/2019 ARIMA Modeling
12/26
-
7/26/2019 ARIMA Modeling
13/26
-
7/26/2019 ARIMA Modeling
14/26
Negative MA Coeff
-
7/26/2019 ARIMA Modeling
15/26
-
7/26/2019 ARIMA Modeling
16/26
-
7/26/2019 ARIMA Modeling
17/26
Negative MA Coeff
-
7/26/2019 ARIMA Modeling
18/26
-
7/26/2019 ARIMA Modeling
19/26
-
7/26/2019 ARIMA Modeling
20/26
-
7/26/2019 ARIMA Modeling
21/26
)stimation 5tage
A "ure A model can #e estimatedusin% ( estimator #ut i the modelalso includes -A terms -( estimator
is used.
-
7/26/2019 ARIMA Modeling
22/26
6iagnostic 'ec7ing
-
7/26/2019 ARIMA Modeling
23/26
8o* good te model 3ts te
data9 8o* relia$le *ill $e teprediction $ased on it9
se some indicators o %oodness o tsuch as 2 and ABA.
-
7/26/2019 ARIMA Modeling
24/26
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.
-
7/26/2019 ARIMA Modeling
25/26
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.
.
-
7/26/2019 ARIMA Modeling
26/26
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.