3. Metodología Box-jenkinsj -V2 (1)
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Transcript of 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
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INTRODUI4N A LA METODOLOGÍABOX-JENKINS-------------------------------------------------------------------
/
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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 $e )a=g'ra (%g'%ete?
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E7o'e #ara!@tr%+o
• o e(to( !@to$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(.
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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$o C09 01 2
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TIPOS DE NO ESTAIONARIEDAD Y
SUS AUSAS
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Pro+e(o e(to+5(t%+o Ga'((%ao
U #ro+e(o e(to+5(t%+o e( ))a!a$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
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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)
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*'+%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
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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
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*'+%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.
•
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Pro+e(o(e(ta+%oar%o(
Pro+e(o( oe(ta+%oar%o(De+a%!%eto)eto a +eroF
1/
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*'+%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
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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
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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
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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
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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
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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
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METODOLOGÍA BOX-JENKINS? MODELOSARIMA
-------------------------------------------------------------------
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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
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Mo$e)o( )%ea)e( $e (er%e( $e t%e!#o
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
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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$e 'e atNor!a)09 /
a F
•
//
∑∞
=−−− +=+++=
1
2211
j jt jt t t t t aaaaaY θ θ θ ...R
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Re#re(eta+%6 a'torregre(%,a ARF
E e(ta (eg'$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
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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
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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 $er%,ar )o( f a) +oo+er )o( ,%+e,er(a
/2
t t aBY FER θ = 10
0
==∑∞
=
θ θ θ cFE j
j jBB
t t aY B =RFEφ 10
0
==∑∞
=
φ φ φ cF j
j jBB
FEFE BB 1−=φ θ
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Mo$e)o( )%ea)e( $e (er%e( $e t%e!#o
• 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
/
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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
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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
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Pro+e(o( a'torregre(%,o( $e or$e #
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
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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/
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Pro+e(o( a'torregre(%,o( $e or$e #
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
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AR1F Te6r%+o
3
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Pro+e(o( a'torregre(%,o( $e or$e #
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( $e (o ta)e( 'e B1
32
t t t t aY Y Y ++= −− 2211RRR φ φ
0)( = Bφ
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E>er+%+%o. AR/F
1. S%!')e ' AR/F +o
/. S%!')e ' AR/F +o
['@ ota\ 6!o (e +o!#orta )a A*9 PA* ESA*\
E geera)9 +6!o e(#erar&a 'e (e +o!#ortara )a A*9 PA* ESA*$e ' #ro+e(o AR#F\
3
2P.092.1 /1 −== φ φ
3.092.0 /1 =−= φ φ
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IDENTI*IAI4N9 ESTIMAI4N YDIAGN4STIO
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Pa(o /. E(t%!a+%6
De(#'@( $e %$et%=+ar e) !o$e)o o( %tere(a e(t%!ar )o( #ar5!etro(
• Par5!etro( a'torregre(%,o(
• Par5!etro( $e !e$%a( !6,%)e(
• Me$%a $e) #ro+e(o µ
• ar%aa $e) #ro+e(o γ 0
• ar%aa $e )o( errore( a/
[ ]′= pφ φ φ 1
[ ]′= qθ θ θ 1
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Pa(o /. E(t%!a+%6. M@to$o( $e e(t%!a+%6.
• M@to$o $e )o( !o!eto(. Para #ro+e(o( AR#F9 #ro#or+%oa e(t%!a$ore(
$%re+to(. S% e!"argo9 #ara )o( #ro+e(o( MA o ARMA e) #ro+e(o $e
o"te+%6 e( +o!#)%+a$o9 #'e( ta)e( !o$e)o( o (o )%ea)e( e )o(
#ar5!etro(. No (e re+o!%e$a +o!o e(t%!a$ore( =a)e(. Se e!#)ea
+o!o ,a)ore( %%+%a)e( e #ro+e(o( $e e(t%!a+%6 o )%ea)e(.
• M&%!o( +'a$ra$o( or$%ar%o( OLSF. Mo$e)o( +o t@r!%o( $e !e$%a(
!6,%)e( o #'e$e (er trata$o( +o !&%!o( +'a$ra$o( or$%ar%o( )%ea)e(
• M@to$o $e M5%!a ,ero(%!%)%t'$. E(ta (o)'+%6 e( )a !5( e!#)ea$a #or )o(#a'ete( $e +6!#'to.
• M@to$o $e M5%!a ,ero(%!%)%t'$ %+o$%+%oa). E(te !@to$o (e re+o!%e$a
#ara (er%e( e(ta+%oa)e(9 Ser%e( +a(% o e(ta+%oar%a(9 (er%e( +orta(
• M@to$o $e !&%!o( +'a$ra$o( +o$%+%oa)e( 2
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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%"%)%$a$
• Lo( at $e"e (er r'%$o ")a+o9 e( $e+%r9
nLo( at $e"e (er %%$
n La ,ar%aa $e )o( errore(9 a/9 $e"e (er +o(tate e e) t%e!#o
n at Nor!a)09a/F
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Pa(o 3? D%ag6(t%+o ,a)%$a+%6 $e ('#'e(to(F. E(ta+%oar%$a$
Se aa)%a )a( ra&+e( $e) #o)%o!%o AR9 '(a$o
• 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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Pa(o 3? D%ag6(t%+o ,a)%$a+%6 $e ('#'e(to(F. I,ert%"%)%$a$
Se aa)%a )a( ra&+e( $e) #o)%o!%o MA9 '(a$o
• 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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Pa(o 3? D%ag6(t%+o ,a)%$a+%6 $e ('#'e(to(F. Lo( at $e"e (er%$e#e$%ete(
• er%=+ar 'e )a A*9 PA* ESA* $e at (ea e(ta$&(t%+a!ete
')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 $e L>'g-Bo
• Do$e e( e) +oe=+%ete $e a'to+orre)a+%6 $e or$e > $e )a
(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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Pa(o 3? D%ag6(t%+o ,a)%$a+%6 $e ('#'e(to(F. /a e( +o(tate
• Gra=+ar at ,(. t ,er%=+ar 'e o `aa ' #atr6 e(#e+&=+o
• U(ar ' +otra(te $e `%#6te(%(
• Do$e >/ e( )a ,ar%aa $e )a ('"!'e(tra > $e )o( re(%$'a)e(9
#ara >19/
• U(ar )a tra(7or!a+%6 $e Bo-o (o"re t . S% )a ,ar%aa e(
+o(tate9 λp1
0
2
2
2
1
2
2
2
10 :.: σ σ σ σ ≠= a H vs H
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Pa(o 3? D%ag6(t%+o ,a)%$a+%6 $e ('#'e(to(F. at Nor!a)09a/F
• Gr5=+a( $e #ro"a"%)%$a$ [-[F
• U(ar ' +otra(te $e `%#6te(%(
• a+er #r'e"a( e(ta$&(t%+a(? S`a#%ro-q%)C9 Jar'e-Bera9Ko)!ogoro,-S!%ro,9 A$er(o-Dar)%g9 ra!er-,o M%(e(
1
)(~:.~:0 a f a H vs Normal a H t at
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Pa(o 3? D%ag6(t%+o ,a)%$a+%6 $e ('#'e(to(F. o!#ara+%6 etre!o$e)o(E%(te $o( #ro+e$%!%eto( #ara +o!#ara+%6 etre !o$e)o(?
• Ba(a$o( e e) a>'(te $e) !o$e)o o )a 7'+%6 $e ,ero(%!%)%t'$
• Ba(a$o( e )a +a#a+%$a$ $e #ro6(t%+o $e) !o$e)o
/
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Pa(o 3? D%ag6(t%+o ,a)%$a+%6 $e ('#'e(to(F. o!#ara+%6 etre!o$e)o(. M@to$o( "a(a$o( e e) a>'(te $e) !o$e)o
• r%ter%o $e %7or!a+%6 $e ACa%Ce AIF. E) !e>or !o$e)o e( a'e)
'e !%%!%+e e) AI T%e$e a (o"ree(t%!ar )a #arte AR $e)!o$e)oF
• r%ter%o $e %7or!a+%6 Bae(%ao BIF. Se e)%ge e) !o$e)o 'e
!%%!%+e e) BI
• r%ter%o $e S+`lart. T%e$e a (e)e++%oar !o$e)o( !5(
#ar(%!o%o(o( 'e e) AI
• r%ter%o $e M5%!a ero(%!%)%t'$
3
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Pa(o . Pro6(t%+o(
• Se $e(ea #roo(t%+ar e) ,a)or Y) "a(a$o e )a %7or!a+%6 Y19 Y/9
Q 9 Y-19 Y
• ) e( ))a!a$o e) <!ero $e #a(o( a$e)ate $e) #ro6(t%+o o e)
`or%ote $e) #ro6(t%+o
• Deote e) #er&o$o a+t'a) #or 9 ('#oga 'e $e(ea!o(
#roo(t%+ar )a (er%e e e) #er&o$o ). Sea )a
re#re(eta+%6 $e) #ro6(t%+o #ara e) #er&o$o ) `e+`o e e)
or%ge
FEk lnY n +
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Pa(o . Pro6(t%+o(
• E) #ro6(t%+o (e geera a) (a+ar e) ,a)or e(#era$o e e) or%ge $e)
!o$e)o e(+r%to e e) #er&o$o ) .
• E) #ro6(t%+o #ara e) #er&o$o )9 $e"e +o(tr'%r(e ('+e(%,a!ete $e )o(
#ro6(t%+o( #ara )o( #er&o$o( 19 /9Q9 )-1
• Para e(te #ro+e$%!%eto?
n S% Y> o `a o+'rr%$o e e) #er&o$o e( ree!#)aa$o #or e) #ro6(t%+o
n S% a>
o `a o+'rr%$o e e) #er&o$o e( ree!#)aa$o #or +ero
n La( a-> 'e a `a o+'rr%$o (e ree!#)aa #or
n A) +o!%eo $e) #ro+e(o $e #ro6(t%+o (e a('!e 'e a->0 #ara ->V0
FEkk jnY Y a n jn jn −−= −−
FEk jnY n +
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Pa(o . Pro6(t%+o(
E>e!#)o? ARIMA19191F
• Para %)'(trar9 +o(%$ere e) #ro6(t%+o #ara ' #ro+e(o ARIMA19191F.
• E) !o$e)o e e) #er&o$o )9 'e$a e(+r%to +o!o?
• o e'%,a)ete!ete9 ,er%=+ariiF
• a+%e$o ) 19 to!a$o ,a)or e(#era$o9 'e$a?
• 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
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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
• e( )a ,ar%aa $e )o( errore(
• (e o"t%ee a) e#re(ar e) !o$e)o e )a 7or!a $e +`o'e(
a)eator%o(
n Para !o$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 $e
∑−
=
±1
0
22
2
l
j
jan Z lY ψ σ α kkFEk N
2ˆaσ
2
ˆ jψ
)()()( B B B q p θ ψ φ =
∑−
=−=
1
0
1
j
i
i j j ψ π ψ
)()()1)(( B B B B q p π θ φ =−
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METODOLOGÍA BOX-JENKINS? MODELOSSARIMA
-------------------------------------------------------------------
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Itro$'++%6• E e) +a#&t')o ater%or9 (e #re(et6 'a +)a(e $e !o$e)o( <t%)e(
#ara re#re(etar (er%e( $e t%e!#o e(ta+%oar%a( +o te$e+%a.
• A`ora +o(%$erare!o( 'a ete(%6 $e e(ta( %$ea( a )a( (er%e(
$e t%e!#o e(ta+%oa)e(.
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Mo$e)o( e(ta+%oa)e( a'torregre(%,o(
• Sea e) !o$e)o e(ta+%oa) a'torregre(%,o +o s o"(er,a+%oe( #or
#er&o$o e(ta+%oa)
• E(te e( ' #ro+e(o AR $e or$e P9 )o( λ% (o )o( #ar5!etro(
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(#'@( $e) reago P(
t Pst Pst st t aY Y Y Y ++++= −−− λ λ λ ...221
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Mo$e)o( e(ta+%oa)e( $e !e$%a( !6,%)e(
• Sea e) !o$e)o e(ta+%oa) $e !e$%a( !6,%)e( +o s o"(er,a+%oe(
#or #er&o$o e(ta+%oa)
• E(te e( ' #ro+e(o MA $e or$e [9 )o( λ% (o )o( #ar5!etro( $e
!e$%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
$%7erete( $e +ero (6)o e )o( reago( 'e (o !<)t%#)o( $e (
Qst Qst st t t aaaaY −−− ++++= γ γ γ ...221
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Mo$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
$%7erete( $e +ero (6)o e )o( reago( 'e (o !<)t%#)o( $e (
– PA*? De+ae r5#%$a!ete a +ero9 #ero )a( a'to+orre)a+%oe( (o
$%7erete( $e +ero (6)o e )o( reago( 'e (o !<)t%#)o( $e (
Qst Qst t Pst Pst t aaaY Y Y −−−− ++++++= γ γ λ λ ...... 11
t s
Qt s
P aBY B FEFE Γ =Λ
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Mo$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
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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
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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$'++%6
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Mo$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$9FP9D9[F(
t qt
d
p Ba B ∈=∇ )()( θ φ
t s
Qqt Ds
dsP p BBY BB ∈Γ =∆∇Λ FEFEFEFE θ φ
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Mo$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*
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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$9FP9D9[F(
y-ts(c(<%2;,;%<<6,<%51,5%3<1,%4;,14%2:5,11%434,14%552,14%333,%14,<%;56,<%;42,
5%364,<%52,5%52,%:5,14%;35,11%263,11%:2:,11%31,14%;;6,%3;,<%<<6,<%33,
5%15;,<%:::,5%:5:,%5;:,14%262,12%252,11%;3<,11%6<<,12%:1<,%;3<,5%4:,%254, 5%33:,<%5,%:,14%4<5,14%541,12%63,12%222,12%2:;,13%251,14%3;;,5%<34,%;1:,
5%;3,5%<<2,14%5:,14%:66,11%1<,14%655,14%<:,12%<<4,13%512,14%56<,%24,14%26,
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Identificación de d
0 5 10 15 20 25 30
- 1
. 0
- 0
. 5
0
. 0
0
. 5
1
. 0
A
C
F
0 5 10 1 5 20 2 5
- 1
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- 0
. 5
0
. 0
0
. 5
1
. 0
P
a
r i a
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A
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re$agos estacio"ales
e$ago 1
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Identificación de D
0 5 10 15 20 25 30
- 1 . 0
-
0 . 5
0 . 0
0 . 5
1 . 0
A C F
0 5 10 15 20 25
- 1 . 0
-
0 . 5
0 . 0
0 . 5
1 . 0
P a r i a ! A C F
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i"dica"do '!e la difere"cia
estacio"al p!ede ser
"ecesaria
e$ago 12
e$ago 2:
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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
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