3. metodología box-jenkinsj -v2 (1)

7/21/2019 3. Metodología Box-jenkinsj -V2 (1)

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MUESTREO Y SERIES DE TIEMPO

3. METODOLOGÍA BOX-JENKINS

Zaret Ortego Za!"raoDe#arta!eto $e Ige%er&a

I$'(tr%a)*a+')ta$ $e Ige%er&aU%,er(%$a$ $e At%o'%aMao $e /012



INTRODUI4N A LA METODOLOGÍABOX-JENKINS-------------------------------------------------------------------

/



Pro+e(o e(to+5(t%+o• U #ro+e(o e(to+5(t%+o e( 'a

+o)e++%6 o 7a!%)%a $e ,ar%a")e(a)eator%a( 8Xt9 +o t : T;9

or$ea$a( (eg< e) ('"&$%+e t'e e geera) (e ('e)e%$et%=+ar +o e) t%e!#o. Portato9 #ara +a$a %(tate tte$re!o( 'a ,ar%a")e a)eator%a$%(t%ta re#re(eta$a #or Xt9 +o)o 'e ' #ro+e(o e(to+5(t%+o#'e$e %ter#retar(e +o!o 'a('+e(%6 $e ,ar%a")e( a)eator%a(+'a( +ara+ter&(t%+a( #'e$e,ar%ar a )o )argo $e) t%e!#o. Pore>e!#)o9 (% o"(er,a!o( (6)o 'o(#o+o( ,a)ore( $e t9 te$r&a!o(

'a %!age (%!%)ar a )a $e )a=g'ra (%g'%ete?



E7o'e #ara!@tr%+o

• o e(to( !@to$o( (e #rete$e '(ar )a %7or!a+%6 !'e(tra) #ara

%$et%=+ar e) #ro+e(o geera$or $e )o( $ato(9 )o 'e #er!%t%r5 rea)%ar )a(

e(t%!a+%oe(.

• Se '(ar5 )a !eto$o)og&a Bo-JeC%(9 e( $e+%r9 )a e(tr'+t'ra $e

a'to+orre)a+%6 $e )a (er%e #ara %$et%=+ar e) #ro+e(o e(to+5(t%+o

geera$or $e )o( $ato(.



Pro+e(o e(to+5(t%+o e(ta+%oar%o

U #ro+e(o e(to+5(t%+o (e $eo!%a $@"%)!ete e(ta+%oar%o o e(ta+%oar%o $e

(eg'$o or$e o e(ta+%oar%o e +o,ar%aa (% (' !e$%a (' ,ar%aa (o

+o(tate( e e) t%e!#o e) ,a)or $e )a +o,ar%aa etre $o( #er&o$o(

+'a)e('%era $e#e$e (o)a!ete $e )a $%(ta+%a te!#ora) etre )o( $o(

#er&o$o(9 e( $e+%r9

• La 7'+%6 $e !e$%a EYtFHtH

• La 7'+%6 ,ar%aa arYtF/t/ 0

• La 7'+%6 +o,ar%aa Yt9 YtCFt9tCF-CFCF C 1

• La 7'+%6 $e +orre)a+%6 Y t9 YtCFt9tCF-CFCFC C 0 /

1La +o,ar%aa etre Yt9 YtC9 (e#ara$a #or C %ter,a)o( $e t%e!#o9 e( ))a!a$a )a a'to+o,ar%aa +o

reago C

/'a$o C09 01 2



TIPOS DE NO ESTAIONARIEDAD Y

SUS AUSAS



Pro+e(o e(to+5(t%+o Ga'((%ao

U #ro+e(o e(to+5(t%+o e( ))a!a$o ga'((%ao o or!a)9 (% (' 7'+%6 $e

$%(tr%"'+%6 +o>'ta e( )a or!a) !')t%,ar%a$a

Yt19 Yt/9Q9Yt Nμ,ΣF



Pro+e(o e(to+5(t%+o e(ta+%oar%oU #ro+e(o ga'((%ao e(ta+%oar%o e +o,ar%aa #'e$e (er e(t'$%a$o

'(a$o )o( $o( #r%!ero( !o!eto(. E #art%+')ar e%(te $o( 7'+%oe( 'e

o( #er!%te aa)%ar e) +o!#orta!%eto $e) #ro+e(o9

• La 7'+%6 $e a'to+orre)a+%6• La 7'+%6 $e a'to+orre)a+%6 #ar+%a)



*'+%6 $e a'to+orre)a+%6 A*F

Ya ,%!o( 'e9

Yt9 YtLCFt9tCF-CFCFC C 0

Pro#%e$a$e( $e )a A*?

1. 01

/. C V 1 W -1 V C V 13. C -C Por e(ta #ro#%e$a$ $e (%!etr&a9 e )a #r5+t%+a (6)o (e e(t'$%a C9

C0



E>e!#)o. o(tr'a )a A* #ara )o( $ato( $e )a Ser%e 19 Ser%e / Ser%e 3. ['@

$%7ere+%a( ota\

par(mfrow=c(1,3))

plot(y)

# El gráfico de ρk vs k se llama Correlograma

acf(y,ylim=c(-1,1))

# ráfica la f!"ci" de a!tocovaria"$a C vs% k

acf(y, type = &covaria"ce&, ,ylim=c(-1,1))

10



*'+%6 $e a'to+orre)a+%6 A*F

• S% e e) +orre)ogra!a )o( ,a)ore( $e+ae r5#%$a!ete a +ero9 (e $%+e 'e

e) #ro+e(o Yt e( e(ta+%oar%o9 e( $e+%r?

• E(to (%g%=+a 'e )a( a'to+orre)a+%oe( $%(!%'e a) #a(ar e) t%e!#o.

E(ta #ro#%e$a$ (e +oo+e +o!o ergo$%+%$a$ $e) #ro+e(o e(to+5(t%+o.

•

11



Pro+e(o(e(ta+%oar%o(

Pro+e(o( oe(ta+%oar%o(De+a%!%eto)eto a +eroF

1/



*'+%6 $e a'to+orre)a+%6 #ar+%a) PA*F

• E( )a +orre)a+%6 +o$%+%oa) etre Yt YtLC o"te%$a a) e)%!%ar )a

%^'e+%a $e YtL19YtL/9Q9YtLC9 e( $e+%r9

_CC orrYt 9YtLC YtL19YtL/9Q9YtLC-1F

• Note 'e (e e)%!%a e) e7e+to $e"%$o a )a +orre)a+%6 #ro$'+%$a #or

reago( ater%ore( a C.

13



E>e!#)o. o(tr'a )a PA* #ara )o( $ato( $e )a Ser%e 19 Ser%e / Ser%e 3. ['@$%7ere+%a( ota\

plot(y)

pacf(y,ylim=c(-1,1))

1



A* !'e(tra) ete$%$a ESA*F

• La ESA* e( 'a `erra!%eta #ara %$et%=+ar !o$e)o( ARIMA $e !aera

geera).

• Para %$et%=+ar e) or$e $e) !o$e)o (e '"%+a e) #atr6 tr%ag')ar a )a

$ere+`a9 +o ,@rt%+e e #9F e e) +'a) to$o( )o( e)e!eto( (o o-

(%g%=+at%,o( re#re(eta$o( +o e) (&!"o)o obF

12



E>e!#)o. o(tr'a )a ESA* #ara )o( $ato( $e )a Ser%e 19 Ser%e / Ser%e 3.['@ $%7ere+%a( ota\

re'!ire(*)

eacf(y)

1



Pro+e(o $e r'%$o ")a+o

• U #ro+e(o e(to+5(t%+o 8at; e( ))a!a$o ' #ro+e(o $e r'%$o ")a+o (% e(

'a ('+e(%6 $e ,.a. %+orre)a+%oa$a( $e 'a $%(tr%"'+%6 =>a 9 a$e!5(9

– EatFHa9 $o$e9 geera)!ete9 Ha 0

– o,at9 atCF /a (% C0c o,at9 atCF 0 (% Cd0

– C 1 (% C0c C 0 (% C d 0

– _CC 1 (% C0c _CC 0 (% C d 0

• S% e) #ro+e(o e(to+5(t%+o 8at; t%ee $%(tr%"'+%6 +o>'ta or!a)9 eto+e(

(e $%+e 'e e( r'%$o ")a+o ga'((%ao. E a$e)ate9 (%e!#re ,a!o( a

+o(%$erar 'e 8at; e( r'%$o ")a+o ga'((%ao

1



E>er+%+%o.

• Geere ' #ro+e(o $e r'%$o ")a+o ga'((%ao +o !e$%a 0 ,ar%aa+o(tate %t? r"orm()F

• Gra='e at9 )a A*9 PA* ESA*.

• aga #r'e"a( $e or!a)%$a$. %t? s+apiro%test()F. er%='e

or!a)%$a$ gr5=+a!ete %t? ''lotcar.,+ist(),/o0plot()F

1



METODOLOGÍA BOX-JENKINS? MODELOSARIMA

-------------------------------------------------------------------

1



Mo$e)o( )%ea)e( $e (er%e( $e t%e!#o

• E(te t%#o $e !o$e)o( ('#oe 'e 'a (er%e $e t%e!#o e( geera$a #or

'a agrega+%6 )%ea) $e (o"re(a)to( a)eator%o(.

• Se $e(ea e!#)ear !o$e)o( 'e '(e #ar5!etro( $e ' !o$o

#ar(%!o%o(o.

• D%+`a #ar(%!o%a (e #'e$e )ograr a) re#re(etar e) #ro+e(o )%ea) e

7'+%6 $e #o+o( t@r!%o( a'torregre(%,o( $e !e$%a( !6,%)e(.

/0




a $o( 7or!a( $e re#re(etar ' #ro+e(o )%ea)

1. Re#re(eta+%6 e Me$%a( M6,%)e( MAF

/. Re#re(eta+%6 A'torregre(%,a ARF

/1



Re#re(eta+%6 $e !e$%a( !6,%)e( MAF

Do$e e( e) #ro+e(o +etra$o.E(ta e+'a+%6 o( #er!%te re#re(etar e) #ro+e(o +o!o 'a ('!a #o$era$ao +o!"%a+%6 )%ea) $e ,a)ore( #re(ete( #a(a$o( $e ' #ro+e(o $e r'%$o")a+o 8at;. E( $e+%r9 e) #ro+e(o e( regre(a$o e 7'+%6 $e ('( $e(,%a+%oe(

#a(a$a(.Re+'er$e 'e atNor!a)09 /

a F

•

//

∑∞

=−−− +=+++=

1

2211

j jt jt t t t t aaaaaY θ θ θ ...R



Re#re(eta+%6 a'torregre(%,a ARF

E e(ta (eg'$a re#re(eta+%6 e) #ro+e(o e( regre(a$o e t@r!%o( $e ('(

#ro#%o( ,a)ore( #a(a$o( !5( ' t@r!%o a)eator%o.

/3

t j

jt jt t t t aY aY Y Y +=+++= ∑∞

=−−−

1

2211R...RRR φ φ φ



O#era$or $e reago(

Sea e) o#era$or $e reago( `a+%a atr5( B9 $o$e?

Note 'e +'a$o e) o#era$or $e reago( e( B-1, 'e$a 'e

Deo!%a$o e) o#era$or $e reago( `a+%a a$e)ate

/

jt t j

t t Y Y BY BY −− == c1

jt t j

t t Y Y BY Y B +

−

+

−

== c1

1



O#era$or $e reago(

Otra 7or!a $e e(+r%"%r )o( $o( !o$e)o( ater%ore( e( te%e$o e +'eta e)

o#era$or $e reago( `a+%a atr5(

1. $o$e

/. $o$e

De $o$e9

E(ta re)a+%6 (e #'e$e '(ar #ara $er%,ar )o( f a) +oo+er )o( ,%+e,er(a

/2

t t aBY FER θ = 10

0

==∑∞

=

θ θ θ cFE j

j jBB

t t aY B =RFEφ 10

0

==∑∞

=

φ φ φ cF j

j jBB

FEFE BB 1−=φ θ




• A''e )a( re#re(eta+%oe( e MA AR (o <t%)e(9 e )a #r5+t%+a o o(

(%r,e #ara re#re(etar )a( (er%e( $e t%e!#o #'e(to 'e $e#e$e $e '

<!ero %=%to $e #ar5!etro( $e(+oo+%$o(.

• E( a(& +o!o $e=%!o( )a +)a(e $e !o$e)o( =%to(?

h Pro+e(o( A'torregre(%,o( $e or$e # AR#FF

h Pro+e(o( $e Me$%a( M6,%)e( $e or$e MAFF

h Pro+e(o( !%to( A'torregre(%,o( $e Me$%a( M6,%)e( $e or$e #9

ARMA#9FF

/



Pro+e(o( a'torregre(%,o( $e or$e # AR#FF

• o(%$ere e) +a(o e(#e+%a) $e )a re#re(eta+%6 AR9 e e) +'a) (6)o)o( #r%!ero( # t@r!%o( (o $%7erete( $e +ero

• E) +'a) e( e'%,a)ete a

• Do$e re#re(eta e) +o>'to $e #ar5!etro( =%to(

$e ' #ro+e(o AR#F

/

t pt pt t t aY Y Y Y ++++= −−−R...RRR φ φ φ 2211

t t p aY B =RFEφ

pφ φ φ ,...,, 21



Pro+e(o( a'torregre(%,o( $e or$e #

E>e!#)o. AR1F

• E #art%+')ar e) #ro+e(o a'torregre(%,o $e #r%!er or$e AR1F9 (ee(+r%"e +o!o

• I,ert%"%)%$a$? S%e!#re

• E(ta+%oar%$a$? (%

• E) #ro+e(o AR1F ta!"%@ e( ))a!a$o #ro+e(o $e MarCo,9 #'e(

e(t5 +o!#)eta!ete $eter!%a$o #or (' reago

/

t t t aY Y += −11RR φ

t Y R

1−t Y R

11 <

φ



30




E>e!#)o. AR1F# im!laci" de procesos *(1)

par(mfrow=c(3,2))y1-arima%sim(list(order=c(1,4,4), ar=4%5), "=644)

plot(y1,yla/=&y&,mai"=(e0pressio"(&*(1) &7p+i7& = 8%5&)))

y2-arima%sim(list(order=c(1,4,4), ar=-4%5), "=644)

plot(y2,yla/=&y&,mai"=(e0pressio"(&*(1) &7p+i7& = -%5&)))

acf(y1,mai"=&&)

acf(y2,mai"=&&)

pacf(y1,mai"=&&)

pacf(y2,mai"=&&)

eacf(y1)

eacf(y2)

31



Pro+e(o( a'torregre(%,o( $e

or$e #

E>e!#)o. AR1F

AR(1) φ = +.8

Time

y

0 100 200 300 400 500

- 4

- 2

0

2

4

6

AR(1) φ = -.8

Time

0 100 200 300 400 500

- 4

- 2

0

2

4

0 5 10 15 20 25

0 . 0

0 . 4

0 . 8

Lag

A C F

0 5 10 15 20 25

- 0 . 5

0 . 0

0 . 5

1 . 0

Lag

0 5 10 15 20 25

0 . 0

0 . 4

0 . 8

Lag

P

0 5 10 15 20 25

- 0 . 8

- 0 . 4

0 . 0

Lag

3/




E>er+%+%o. AR1F1. S%!')e ' AR1F +o

/. S%!')e ' AR1F +o

U(e ,a)ore( $e 1 $%7erete( a )o( $e) e>e!#)o ater%or. E(+r%"a e)

!o$e)o. E( e(ta+%oar%o\ E( %,ert%")e\ ['@ ota\9 +6!o (e

+o!#orta )a A*9 PA* ESA*\

33

01 <φ

01 >φ



AR1F Te6r%+o

3




E>er+%+%o. AR/F• E #art%+')ar e) #ro+e(o a'torregre(%,o $e (eg'$o or$e AR/F9

(e e(+r%"e +o!o

• I,ert%"%)%$a$? S%e!#re

• E(ta+%oar%$a$? (% )a( ra&+e( $e (o ta)e( 'e B1

32

t t t t aY Y Y ++= −− 2211RRR φ φ

0)( = Bφ



3



E>er+%+%o. AR/F

1. S%!')e ' AR/F +o

/. S%!')e ' AR/F +o

['@ ota\ 6!o (e +o!#orta )a A*9 PA* ESA*\

E geera)9 +6!o e(#erar&a 'e (e +o!#ortara )a A*9 PA* ESA*$e ' #ro+e(o AR#F\

3

2P.092.1 /1 −== φ φ

3.092.0 /1 =−= φ φ



AR/F Te6r%+o

3



Pro+e(o( $e !e$%a( !6,%)e( $e or$e

E>er+%+%o. MA1F• E #art%+')ar e) #ro+e(o $e !e$%a( !6,%)e( $e #r%!er or$e MA1F9

(e e(+r%"e +o!o

• I,ert%"%)%$a$? S%

• E(ta+%oar%$a$? S%e!#re

0

11 −+= t t t aaY θ R

11

<θ



E>er+%+%o. MA1F

1. S%!')e ' MA1F +o

/. S%!')e ' MA1F +o


1

01 <θ

01 >θ



MA1F Te6r%+o

/



Pro+e(o( $e !e$%a( !6,%)e( $e or$e

E>er+%+%o. MA/F

• E #art%+')ar e) #ro+e(o $e !e$%a( !6,%)e( $e (eg'$o or$e

MA/F9 (e e(+r%"e +o!o

• I,ert%"%)%$a$? (% )a( ra%+e( $e (o ta)e( 'e B1

• E(ta+%oar%$a$? S%e!#re

3

2211 −− ++= t t t t aaaY θ θ R

0)( = Bθ



E>er+%+%o. MA/F

1. S%!')e ' MA/F +o

/. S%!')e ' MA/F +o


E geera)9 +6!o e(#erar&a 'e (e +o!#ortara )a A*9 PA* ESA*$e ' #ro+e(o MAF\

2P.092.1 /1 −== θ θ

/D.09P2.0 /1 −=−= θ θ



MA/F Te6r%+o

2



Re('!e $e !o$e)o( AR#F MAF

•U #ro+e(o AR#F e(ta+%oar%o e( e'%,a)ete a ' #ro+e(o MA"F

• U #ro+e(o MAF %,ert%")e e( e'%,a)ete a ' #ro+e(o AR"F

• E ' AR#F )a A* $e+ae r5#%$a!ete a +ero9 !%etra( 'e )a

PA* (e +orta $e(#'@( $e) reago #

• E ' AMF )a PA* $e+ae r5#%$a!ete a +ero9 !%etra( 'e )a

A* (e +orta $e(#'@( $e) reago



Pro+e(o !%to( a'torregre(%,o( $e !e$%a( !6,%)e( $e or$e #9

ARMA#9FF

• E )a #r5+t%+a9 +o e) = $e o"teer 'a #ara!etr%a+%6

#ar(%!o%o(a9 e( e+e(ar%o %+)'%r e e) !o$e)o tato t@r!%o(

a'torregre(%,o( +o!o $e !e$%a( !6,%)e( ota+%6 $e RF

• E) +'a) e( e'%,a)ete a

• Do$e re#re(eta e) +o>'to $e

#ar5!etro( =%to( $e ' #ro+e(o ARMA#9F

qt qt t pt pt t aaaY Y Y −−−− ++++++= θ θ φ φ ...R...RR1111

t qt p aBY B FF θ φ =

pφ φ ,...,1 qθ θ 9...91



Pro+e(o !%to( a'torregre(%,o( $e !e$%a( !6,%)e( $e or$e #9

E>er+%+%o. ARMA191F • E #art%+')ar e) #ro+e(o !%to a'torregre(%,o $e !e$%a( !6,%)e(

ARMA191F9 (e e(+r%"e +o!o

• I,ert%"%)%$a$? S% )a( ra%+e( $e (o ta)e( 'e B1

• E(ta+%oar%$a$? S% )a( ra%+e( $e (o ta)e( 'e B1

1111 −− +=− t t t t aaY Y θ φ RR

0)( = Bφ

0)( = Bθ



20



21



E>er+%+%o. ARMA191F

1. S%!')e ' ARMA191F +o

/. S%!')e ' ARMA191F +o

3. S%!')e ' ARMA191F +o

. S%!')e ' ARMA191F +o


E geera)9 +6!o e(#erar&a 'e (e +o!#ortara )a A*9 PA* ESA*

$e ' #ro+e(o ARMA#9F\

2/

0,0 11 <> φ θ

0,0 11 >< φ θ

0,0 11 << φ θ

0,0 11 >> φ θ



Re('!e $e !o$e)o( AR#F9 MAF ARMA#9F

E(ta+%oar%o e %,ert%")e

A*(e +orta

e q

PA*De+ae a +ero

PA*(e

+ortae p

A*De+ae a +ero

A*(e

+ortae q

PA*(e +orta

e p

!o$e)o !%to ARMAF

!o$e)o $e!e$%a(!6,%)e(

MAF

!o$e)oa'to-

regre(%,oARF

D6$e

(e +orta)a A*\

N<!ero $ereago(

#er&o$o(Fa %+)'%r

D6$e

(e +orta)a PA*\

N<!ero $ereago(

#er&o$o(Fa %+)'%r23



Mo$e)o( e(to+5(t%+o( )%ea)e( o e(ta+%oar%o( `o!og@eo( ARIMAF• M'+`a( (er%e( $e t%e!#o (e +o!#orta +o!o (% o t',%era !e$%a

+o(tate9 ta)e( (er%e( (o ))a!a$a( o e(ta+%oar%a( e !e$%a.

• E(ta( (er%e( #'e$e +o,ert%r(e e e(ta+%oar%a( a tra,@( $e )a

$%7ere+%a+%6

2



Mo$e)o( e(to+5(t%+o( )%ea)e( oe(ta+%oar%o( `o!og@eo( ARIMAF

22



Mo$e)o( e(to+5(t%+o( )%ea)e( o e(ta+%oar%o( `o!og@eo( ARIMAF

• A) <!ero $e ,e+e( 'e e( e+e(ar%o $%7ere+%ar 'a (er%e #ara +o,ert%r)a

e e(ta+%oar%a )a $eo!%are!o( d $%re!o( 'e )a (er%e( e( o

e(ta+%oar%a ̀ o!og@ea

• E) o#era$or $%7ere+%a (e $e=e +o!o?

• E( #o(%")e e#re(ar e t@r!%o( $e) o#era$or $e reago( B

• A(&9 'a $%7ere+%a+%6 $e or$e ('#er%or (e #'e$e e#re(ar +o!o

9

2

1−−=∇ t t t Y Y Y

B−=∇ 1

( ) 22 1 B−=∇ ( ) 33 1 B−=∇

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ARIMA#9$9F9 'e (e e(+r%"e +o!o?

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e R2



E re('!eQE(ta+%oar%o\

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e) a$e+'a$o

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No tra(7or!a-+%oe(!5(

+o!#)e>a(

S&

D%ag6(t%+o( Pro6(t%+o(

0



IDENTI*IAI4N9 ESTIMAI4N YDIAGN4STIO

-------------------------------------------------------------------

1



/



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tra,@( $e) a5)%(%( $e $ato( `%(t6r%+o(.

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#ara %$et%=+ar (at%(7a+tor%a!ete ' !o$e)o.

3



Pa(o 1. I$et%=+a+%6

o(%$ere e) !o$e)o ARIMA#9$9F9

La %$et%=+a+%6 +o(%(te e #ro+e$%!%eto( #ara $eter!%ar?

1. La tra(7or!a+%6 #ote+%a) $e j 'e e(ta"%)%a )a ,ar%aa $e Yt

/. Deter!%ar e) or$e $e $

3. Deter!%ar e) or$e $e #

. Deter!%ar )a e%(te+%a o o $e f0

t qt d

p aBY B FEFE θ θ φ λ +=∇ 0



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• De"e!o( $eter!%ar 'a #ote+%a j ta) 'e (ea

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Bo-oF

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• $o$e e( )a !e$%a $e )a (er%e tra(7or!a$a '(a$o j

2

FE FEλ t Y Var

=

≠−

=

0

01

λ

λ λ

λ

λ

λ

siY Ln

siY

Y

t

t

t

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( )∑=

−=n

t t Y S

1

2

λ λ µ λ kFE

λ µ ˆ



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2

2

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1

50

0

150

11

12

t

t

t

t

t

t

t

Y

Y

Y

Y Ln

Y

Y

Y

→=

→=

→=

→=

→−=

→−=

→−=

λ

λ

λ

λ

λ

λ

λ

.

FE

.



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('#er%or `a(ta 'e (e )ogre ' +o!#orta!%eto e(ta+%oar%o.

λ t Y λ t Y



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• A)terat%,a!ete (e #'e$e rea)%ar +otra(te( o #r'e"a( $e

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a ? E) #ro+e(o e( e(ta+%oar%o

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error e(t5$ar. A(&9 e( <t%) gra=+ar )o( )&!%te(

9

0

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dY ∇

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j jk

j

jk r nS ρ

ρ ρ ρ

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1



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• Se $e"e `a+er e) +otra(te

0 ? f0 0 ,(. a ? f0 ≠ 0• A)terat%,a!ete9 (e %+)'e f0 e e) !o$e)o (e e(t%!a. Se

e)%!%a (% f0 o e( e(ta$&(t%+a!ete (%g%=+at%,o

3

µ φ θ )(0 B p=



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• Par5!etro( a'torregre(%,o(

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• Me$%a $e) #ro+e(o µ

• ar%aa $e) #ro+e(o γ 0

• ar%aa $e )o( errore( a/

[ ]′= pφ φ φ 1

[ ]′= qθ θ θ 1



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• M@to$o $e !&%!o( +'a$ra$o( +o$%+%oa)e( 2



Pa(o 3? D%ag6(t%+o ,a)%$a+%6 $e ('#'e(to(F

Se $e"e ,a)%$ar )o( (%g'%ete( ('#'e(to(?

• E(ta+%oar%$a$

• I,ert%"%)%$a$

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nLo( at $e"e (er %%$

n La ,ar%aa $e )o( errore(9 a/9 $e"e (er +o(tate e e) t%e!#o

n at Nor!a)09a/F



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• S% B1 `a e(ta+%oar%$a$

• S% B1 o `a e(ta+%oar%$a$

So)'+%6? D%7ere+%ar )a (er%e

0)( = B pφ



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• S% B1 `a %,ert%"%)%$a$

• S% B1 o `a %,ert%"%)%$a$

P'e$e (er (&to!a $e 'e (e (o"re$%7ere+%6 )a (er%e

0)( = Bqθ



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')a(• U(ar ' +otra(te $e `%#6te(%( +o>'to

• E(te +otra(te e( ))a!a$o e) +otra(te $e PORTMANTEAU 'e (e

"a(a e e) e(ta$&(t%+o $e L>'g-Bo

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(er%e $e re(%$'a)e(

0:....: 210 ≠∃=== jak H vs H ρ ρ ρ ρ

∑=

−−+=k

j

t j a jnnnQ1

21 )(ˆ)()2( ρ

)(ˆ 2t j a ρ



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#ara >19/

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+o(tate9 λp1

0

2

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2

1

2

2

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10 :.: σ σ σ σ ≠= a H vs H



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1

)(~:.~:0 a f a H vs Normal a H t at



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• Ba(a$o( e )a +a#a+%$a$ $e #ro6(t%+o $e) !o$e)o

/



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'e !%%!%+e e) AI T%e$e a (o"ree(t%!ar )a #arte AR $e)!o$e)oF

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!%%!%+e e) BI

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#ar(%!o%o(o( 'e e) AI

• r%ter%o $e M5%!a ero(%!%)%t'$

3



Pa(o . Pro6(t%+o(

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Q 9 Y-19 Y

• ) e( ))a!a$o e) <!ero $e #a(o( a$e)ate $e) #ro6(t%+o o e)

òr%ote $e) #ro6(t%+o

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#roo(t%+ar )a (er%e e e) #er&o$o ). Sea )a

re#re(eta+%6 $e) #ro6(t%+o #ara e) #er&o$o ) è+ò e e)

or%ge

FEk lnY n +



Pa(o . Pro6(t%+o(

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!o$e)o e(+r%to e e) #er&o$o ) .

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#ro6(t%+o( #ara )o( #er&o$o( 19 /9Q9 )-1

• Para e(te #ro+e$%!%eto?

n S% Y> o à o+'rr%$o e e) #er&o$o e( ree!#)aa$o #or e) #ro6(t%+o

n S% a>

o à o+'rr%$o e e) #er&o$o e( ree!#)aa$o #or +ero

n La( a-> 'e a à o+'rr%$o (e ree!#)aa #or

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FEkk jnY Y a n jn jn −−= −−

FEk jnY n +



Pa(o . Pro6(t%+o(

E>e!#)o? ARIMA19191F

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• o e'%,a)ete!ete9 ,er%=+ariiF

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• Ya 'e 9 a) =a) $e) #er&o$o 9 a (e e(t%!a #or

• Para ) /9 (e o"t%ee ' #ro+e$%!%eto (%!%)ar

lnln aBY B ++ −=∇− FEFE 11 11 θ φ

1121111 −++−+−++ −+−+= lnlnlnlnln aaY Y Y θ φ φ FE

[ ] nnnnn aY Y Y Y E 11111 11 θ φ φ −−+== −+ kkFFk

[ ] 01 =+na E nnn Y Y a k−=

[ ] 11111 −+−++ −+== lnlnnln Y Y lY Y E kkFEFEk φ φ



Pa(o . Pro6(t%+o(

U %ter,a)o $e +o=aa a) 1-αF100s #ara e) ,er$a$ero ,a)or $e Y) e(?

Do$e 9

• Zα / e( e) #er+et%) α /F $e )a $%(tr%"'+%6 or!a) e(t5$ar

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a)eator%o(

n Para !o$e)o( ARMA #9F (e +a)+')a $e

n Para !o$e)o( ARIMA#9$9F (e +a)+')a $e )a e+'a+%6 re+'r(%,a $o$e )o( π j (e o"t%ee $e

∑−

=

±1

0

22

2

l

j

jan Z lY ψ σ α kkFEk N

2ˆaσ

2

ˆ jψ

)()()( B B B q p θ ψ φ =

∑−

=−=

1

0

1

j

i

i j j ψ π ψ

)()()1)(( B B B B q p π θ φ =−



METODOLOGÍA BOX-JENKINS? MODELOSSARIMA

-------------------------------------------------------------------



Itro$'++%6• E e) +a#&t')o ater%or9 (e #re(et6 'a +)a(e $e !o$e)o( <t%)e(

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• A`ora +o(%$erare!o( 'a ete(%6 $e e(ta( %$ea( a )a( (er%e(

$e t%e!#o e(ta+%oa)e(.



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#er&o$o e(ta+%oa)

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a'torregre(%,o( e(ta+%oa)e(

n A*? De+ae r5#%$a!ete a +ero9 #ero )a( a'to+orre)a+%oe( (o

$%7erete( $e +ero (6)o e )o( reago( 'e (o !<)t%#)o( $e (

n PA*? Se +orta $e(#'@( $e) reago P(

t Pst Pst st t aY Y Y Y ++++= −−− λ λ λ ...221



Mo$e)o( e(ta+%oa)e( $e !e$%a( !6,%)e(

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!e$%a( !6,%)e( e(ta+%oa)e(.

n A*? Se +orta $e(#'@( $e) reago [(n PA*? De+ae r5#%$a!ete a +ero9 #ero )a( a'to+orre)a+%oe( (o


Qst Qst st t t aaaaY −−− ++++= γ γ γ ...221



Mo$e)o( e(ta+%oa)e( !%to(

Sea e) !o$e)o e(ta+%oa) !%to +o s o"(er,a+%oe( #or #er&o$o

e(ta+%oa)

E) +'a) 'e$a e#re(a$o e t@r!%o( $e) o#era$or $e reago( +o!o?

E(te e( ' !o$e)o e(ta+%oa) ARMAP9[F(

– A*? De+ae r5#%$a!ete a +ero9 #ero )a( a'to+orre)a+%oe( (o


– PA*? De+ae r5#%$a!ete a +ero9 #ero )a( a'to+orre)a+%oe( (o


Qst Qst t Pst Pst t aaaY Y Y −−−− ++++++= γ γ λ λ ...... 11

t s

Qt s

P aBY B FEFE Γ =Λ



Mo$e)o( e(ta+%oa)e( a'torregre(%,o( $e !e$%a( !6,%)e( %tegra$o(

• De !aera !5( geera)9 )a (er%e e(ta+%oa) 8Y t; #o$r&a (er o

e(ta+%oar%a.

• Sea e) o#era$or $%7ere+%a e(ta+%oa)?

• Por e>e!#)o9 )a #r%!era $%7ere+%a e(ta+%oa) 'e$a $a$a #or?

•

E geera)9 (e #o$r&a re'er%r D $%7ere+%a( e(ta+%oa)e( #ara #ro$'+%r'a (er%e e(ta+%oar%a. A(&9 e) o#era$or $%7ere+%a $e or$e D e(?

• Y )a 7or!a geera) $e ' !o$e)o ARIMAP9D9[F( e(?

)1( s

s

B−=∆

st t t

s

t s X X X B X −−=−=∆ )1(

D s D

s B )1( −=∆

t s

Qt Ds

sP aBY B FEFE Γ =∆Λ



Mo$e)o( e(ta+%oa)e( !')t%#)%+at%,o(. Itro$'++%6A +ot%'a+%6 (e !'e(tra )a( +%7ra( !e('a)e( $e e!#)eo e

!%)e(F #ara >6,ee( a!er%+ao( etre )o( 1 1 ao( $e($e 11

`a(ta 11 To!a$o $e qe%9 /002F



Time

C i r a # ' e ' e # e m ! e % ( e & m

i ! e # )

171 172 173 174 175 176 17

6 0 0

8 0 0

1 0 0 0

1 2 0 0

Note 'e?

• Re)a+%oe( $etro $e )o(

#er&o$o( re#re(eta )a

+orre)a+%6 etre Q9 Xt-/9 Xt-

19 Xt9 Xt19 Xt/9Q

• Re)a+%oe( etre #er&o$o(

re#re(eta )a +orre)a+%6

etre Q9 Xt-/(9 Xt-(9 Xt9 Xt(9

Xt/(9Q

Mo$e)o( e(ta+%oa)e( !')t%#)%+at%,o(. Itro$'++%6



Mo$e)o( e(ta+%oa)e( !')t%#)%+at%,o(

• P'e(to 'e )a( o"(er,a+%oe( e e) !o$e)o ARIMAP9D9[F( (o $e#e$%ete(

(6)o e )o( !<)t%#)o( $e )o( reago( e(ta+%oa)e(9 Bo-JeC%( %tro$'>ero

'a e(tr'+t'ra $e +orre)a+%6 etre )a( o"(er,a+%oe( $etro $e ' #er&o$o

e(ta+%oa)9 a) #er!%t%r 'e e) r'%$o 8at; (ea re#re(eta$o #or otro !o$e)o

• Do$e 8ut; e( r'%$o ")a+o φ#BF θBF (o #o)%o!%o( e B $e or$e #

9 re(#e+t%,a!ete. S'(t%t'e$o )a ater%or e+'a+%6 e e) !o$e)o

ARIMAP9D9[F( o"tee!o(

• E) +'a) e( ' !o$e)o e(ta+%oa) !')t%#)%+at%,o $e or$e #9$9FP9D9[F9 'e

(e $eota ARIMA#9$9FP9D9[F(

t qt

d

p Ba B ∈=∇ )()( θ φ

t s

Qqt Ds

dsP p BBY BB ∈Γ =∆∇Λ FEFEFEFE θ φ



Mo$e)o( e(ta+%oa)e( !')t%#)%+at%,o(

E>e!#)o 13. Para e) !o$e)o ARIMA09191F09191F1/

1. E(+r%"a e(te !o$e)o e t@r!%o( $e) o#era$or $e reago(.

/. S%!')e e) !o$e)o +o 0. f0.re'!ire(Com/9C)

s-12

y-sarima%im("=14,period=s,model=list(order=c(4,1,1),ma=4%:),

seaso"al=list(order=c(4,1,1),ma=4%;))

3. Gra='e )a A*

. Gra='e )a PA*



E>e!#)o 1. Lo( (%g'%ete( (o $ato( !e('a)e( $e !%))a( tota)e(

,o)a$a( e Ig)aterra #ara )o( ao( etre 1-10. P'e(to 'e )o(

,%a>e( a@reo( t%ee (' to#e !5( a)to $'rate )o( !e(e( $e ,erao9

(e o"(er,a 'a ,ar%a+%6 e(ta+%oa) !ar+a$a e )a (er%e.

Mo$e)e e(to( $ato( '(a$o ' !o$e)o ARIMA#9$9FP9D9[F(

y-ts(c(<%2;,;%<<6,<%51,5%3<1,%4;,14%2:5,11%434,14%552,14%333,%14,<%;56,<%;42,

5%364,<%52,5%52,%:5,14%;35,11%263,11%:2:,11%31,14%;;6,%3;,<%<<6,<%33,

5%15;,<%:::,5%:5:,%5;:,14%262,12%252,11%;3<,11%6<<,12%:1<,%;3<,5%4:,%254, 5%33:,<%5,%:,14%4<5,14%541,12%63,12%222,12%2:;,13%251,14%3;;,5%<34,%;1:,

5%;3,5%<<2,14%5:,14%:66,11%1<,14%655,14%<:,12%<<4,13%512,14%56<,%24,14%26,

%:1,5%1,11%;4<,5%562,12%63<,1:%<6,13%;;<,13%<31,16%114,12%156,14%;:6,12%1;1,

14%5:4,14%:3;,13%65,13%:42,13%143,1:%33,1:%1:<,1:%46<,1;%23:,12%35,11%6:,

12%<<2))



Identificación de d

0 5 10 15 20 25 30

- 1

. 0

- 0

. 5

0

. 0

0

. 5

1

. 0

A

C

F

0 5 10 1 5 20 2 5

- 1

. 0

- 0

. 5

0

. 0

0

. 5

1

. 0

P

a

r i a

!

A

C

F

e$ago 12

e$ago 2:

>ecae le"tame"te de/ido a "o-estacio"aridad ylas mayores a!toccorelacio"es se da" e" los

re$agos estacio"ales

e$ago 1



Identificación de D

0 5 10 15 20 25 30

- 1 . 0

-

0 . 5

0 . 0

0 . 5

1 . 0

A C F

0 5 10 15 20 25

- 1 . 0

-

0 . 5

0 . 0

0 . 5

1 . 0

P a r i a ! A C F

*?" m!estra a!tocorrelacio"es

gra"des, partic!larme"te e"los re$agos 12 y 2:,

i"dica"do '!e la difere"cia

estacio"al p!ede ser

"ecesaria

e$ago 12

e$ago 2:



ACF y PACF de la serie con d=1 y D=1

0 5 10 15 20 25 30 35

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

Lag

A C F

0 5 10 15 20 25 30 35

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

Lag

Series y.d.D



Re7ere+%a(

• qe%9 q%))%a! q. S. T%!e (er%e( aa)(%(. U%,ar%ate a$ !')t%,ar%ate

!et`o$(. A$$%(o qe()e. Seg'$a e$%+%6. /002.

• P@re9 *re$ O. Itro$'++%6 a )a( (er%e( $e t%e!#o. M@to$o( #ara!@tr%+o(.

U%,er(%$a$ $e Me$e))&. /00.

• Z%,ot9 Er%+ qag9 J%a`'%. Mo$e))%g =a+%a) t%!e (er%e( l%t` (-#)'(.

S#r%ger-er)ag. /a e$%+%6. /002.

• George E. P. Bo Gl%)! M. JeC%(. T%!e (er%e( aa)(%( 7ore+a(t%g a$

+otro). Pret%+e a)). 3v e$%+%6. 1.

• S`'!la9 Ro"ert . Stower 9 Da,%$ S. T%!e (er%e( aa)(%( a$ %t(

a##)%+at%o( l%t` R ea!#)e(. S#r%ger. /a e$%+%6. /00.

• Motgo!er D Jo`(o L *ore+a(t%g a$ T%!e Ser%e( Aa)(%(

3. metodología box-jenkinsj -v2 (1)

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