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MODELOS ARCH APLICADOS
MODELOS ARCH APLICADOS
Dr. Luís Miguel Galindo
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“I have heard it said that too much academic research is focused on finding very precise answers to irrelevant questions”
Carol Alexander (2001)
“I have heard it said that too much academic research is focused on finding very precise answers to irrelevant questions”
Carol Alexander (2001)
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“In finance theory the concept of the correct price is determinated by the nature of the modeler. The British, being practical and empirical, might say that the market is right and their model is wrong. The French –rationalist and theoreticians- might say that their model is right and the market is wrong. However, the Americans, pragmatic and diplomatic as they are, would most likely say that both the market and the model are wrong…
“In finance theory the concept of the correct price is determinated by the nature of the modeler. The British, being practical and empirical, might say that the market is right and their model is wrong. The French –rationalist and theoreticians- might say that their model is right and the market is wrong. However, the Americans, pragmatic and diplomatic as they are, would most likely say that both the market and the model are wrong…
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• Volatilidad y correlación son parámetros del proceso estocástico utilizados para modelar variaciones en los precios de activos financieros
1. Volatilidad anual:
A = Factor de anualización (el número de ganancias al año)
A = 250 ó 252
VOLATILIDAD Y CORRELACIÓN:
Dr. Galindo
)%100( AVA
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Comparar volatilidades
La volatilidad anual es
VOLATILIDAD Y CORRELACIÓN:
Dr. Galindo
252
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Conceptos básicos:
(1.1)
(1.2)
(1.3)
VOLATILIDAD Y CORRELACIÓN:
Dr. Galindo
222 XEXE
YX YXEYX ,cov
YvXv
YXYXcorr
,cov,
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MCO:
(2) β = v
v = volatilidad relativa Y (variable dependiente)
VOLATILIDAD Y CORRELACIÓN:
Dr. Galindo
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La volatilidad se mide con la varianza:
Mejor desviación estándar a varianza (unidades de medida)
VOLATILIDAD Y CORRELACIÓN:
Dr. Galindo
T
trT22
1
1
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La volatilidad no es el riesgo porque la solo mide la desviación pero no la forma de la distribución
La volatilidad genera procesos de memoria larga
La volatilidad de diversos activos no se mueve junta
VOLATILIDAD Y CORRELACIÓN:
Dr. Galindo
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Volatilidad: • La volatilidad y la correlación no es observada
directamente en el mercado como los precios
• Volatilidad implícita: el pronostico de la volatilidad que iguala el precio de mercado con el precio del modelo de una opción
• Volatilidad estadística: es una serie de tiempo y depende del modelo especifico
VOLATILIDAD Y CORRELACIÓN:
Dr. Galindo
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Modelos de volatilidad constante y variable:
• Una serie estacionaria tiene una varianza condicional constante
• Una volatilidad variable en el tiempo se describe por una volatilidad condicional
• Una distribución condicional determina la ganancia en un momento particular en el tiempo
VOLATILIDAD Y CORRELACIÓN:
Dr. Galindo
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• La volatilidad condicional en el tiempo t es la raíz cuadrada de la varianza condicional en el tiempo t
Los valores actuales en vez de los valores esperados del pasado se utilizan para estimar la volatilidad condicional
VOLATILIDAD Y CORRELACIÓN:
Dr. Galindo
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AR(1):
(3.1)
et es ruido blanco
MARCO GENERAL: ARCH
Dr. Galindo
1110
1
ttt eyy
2vartete
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= la varianza condicional de
Ello se debe a que un yt-1 fijo implica que la única variación de et es
MARCO GENERAL: ARCH
Dr. Galindo
2e
1t
ty
yE
2e
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(3.2)
(3.3)
MARCO GENERAL
Dr. Galindo
1101
tt
t yyyE
1101
ttt
t yyye
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(3.4)
Si la varianza condicional de et es homocedastica:
MARCO GENERAL
Dr. Galindo
1
2
1var
t
t
t
ty
eEyy
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(4.1)
La varianza pronosticada de yt no depende de los valores pasados de et o
ARCH relaja este supuesto
MARCO GENERAL
Dr. Galindo
2
1var e
t
ty
y
2te
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La varianza incondicional es:
(4.2)
MODELO GENERAL: ARCH
Dr. Galindo
2221
2
121
110
eyy
tt
ttt
evaryvar
eyvaryvar
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Despejando:
(4.3)
(4.4) varianza no
condicional
MODELO GENERAL: ARCH
Dr. Galindo
22211 ey
)1( 21
22
ey
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Suponiendo a la heterocedasticidad como función de otra variable:
(4.5)
Como xt-1 es exógena:
(4.6)
La varianza depende de
MODELO GENERAL: ARCH
Dr. Galindo
1 ttt xey
221
1var et
t
t xxy
21tx
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Engle (1992):
(4.7)
(4.8)
MODELO GENERAL: ARCH
Dr. Galindo
ttt uy
2
2110
2
tt
tt
h
y
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Normalización: var(ut) = 1
La varianza condicional de yt depende de sus valores rezagados al cuadrado
MODELO GENERAL: ARCH
Dr. Galindo
22
1varvarvar ttttt
t
t uuyy
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Modelo simple:
(4.9)
Para que el modelo ARCH implique el término de error:Media condicional
(4.10)
MODELO GENERAL: ARCH
Dr. Galindo
22110
2 ... qtqtt yy
tt
t xxyE
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Varianza condicional:
(4.11) Como :
(4.12)
MODELO GENERAL: ARCH
Dr. Galindo
2var tt
tx
y
ttt exy
ttt
ttt xyx
yEye
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es una función de
ut es ruido blanco
(4.13)
ARCH(1):
(4.14)
MODELO ARCH GENERAL
Dr. Galindo
2t 22
1 ,..., qtt ee
ttt ue
22110
2 ... qtqtt ee
2110
2 tt e
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AR(1):
(5.1)
Media condicional:
(5.2)
Varianza condicional (yt-1 es conocida en el tiempo t):
(5.3)
VOLATILIDAD Y CORRELACIÓN
Dr. Galindo
ttt eyy 1
11 tttt yeEyEyE
21 ttt evyvyv
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Media incondicional:
(5.4)
Varianza incondicional:
(5.5)
VOLATILIDAD Y CORRELACIÓN
Dr. Galindo
0tyE
)1( 2
2
tyv
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• Los modelos de volatilidad condicional supone distribución normal y por tanto esta determinado por la media y la varianza
• Correlación incondicional:
(6.1)
VOLATILIDAD Y CORRELACIÓN
Dr. Galindo
tt
tttt
vvcor
11
2121
,cov,
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La correlación condicional permite que la distribución conjunta sea diferente en cada punto en el tiempo
VOLATILIDAD Y CORRELACIÓN
Dr. Galindo
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El precio se determina como un movimiento browniano:
(7)
Rt = tasa de interés del activo sin riesgoZt = Proceso Wiener
Proceso Wiener: dZt es independiente y normalmente distribuida con media cero y varianza dt
VOLATILIDAD IMPLICITA Y CONDICIONAL
Dr. Galindo
dZtRdtts
tds )(
)(
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Volatilidad y correlación histórica:
1.Varianza incondicional:
(8.1)
2 ganancias al cuadrado
2. Correlación incondicional:
(8.2)
MODELOS MA
Dr. Galindo
n
iit
t n1
22ˆ
22
21
1 21ˆ
itit
n
i itit
t
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Exponentially Weighted Moving Averages (EWMA):
EWMA pone más peso en información reciente y por tanto considera el orden de la dinámica de las ganancias
(9.1)
MODELOS MA
Dr. Galindo
12
13
221
...1
...
n
ntn
ttt xxxx
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0 < < 1 Un valor de mayor se le pone mas peso a las observaciones pasadas y por tanto la serie se hace mas suave
Como 0 < < 1 n 0 con n
Converge a:
MODELOS MA
Dr. Galindo
)1(
1
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MA infinito se puede escribir como:
(9.2)
Varianza:
(9.3)
Correlación:
(9.4)
MODELOS MA
Dr. Galindo
it
i x11
212 1ˆ iti
t
ititi
t 211
12 1
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Estimación recursiva:
(9.5)
(9.6)
= determina la intensidad de la reacción de la volatilidad a los eventos de mercado
MODELOS MA
Dr. Galindo
21
21
2 ˆ1ˆ tt
2)(11211
2 ˆ1ˆ ttt
211 t
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Con error existe una mayor volatilidad como reacción a la información de mercado
= determina la persistencia de la volatilidad sin importar lo que sucede en t-1 en el mercado
Con un mayor existe una mayor persistencia
Un alto implica una lata persistencia y una baja reacción de mercado (los parámetros no son independientes)
MODELOS MA
Dr. Galindo
21ˆ t
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Regla de dedo del EWMA:
• La volatilidad en los mercados es = 0.75 (alta volatilidad o poca persistencia) o = 0.98 (alta persistencia y no muy reactivo)
• Para pronósticos:
MODELOS MA
Dr. Galindo
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Valores bajos de para pronósticos de CP
Valores altos de para pronósticos de LP
• EWMA equivale a un I – GARCH sin constante
MODELOS MA
Dr. Galindo
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(10.1)
La volatilidad de los pronósticos es:
(10.2)
Con 2 constante
MODELOS MA
Dr. Galindo
tntntR lnln
11111 ... ntttnt
2 nv nt
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Con A ganancias al año entonces el número de días de ganancias al año (n) es A/n:
(10.3) Volatilidad del día
= un día de volatilidad
MODELOS MA
Dr. Galindo
)(/100 2nnAn
2100 A
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Las series de tiempo muestran volatilidad en clusters Heterocedasticidad condicional autoregresiva
ARCH
Volatilidad implica una fuerte autocorrelación en el cuadrado
MODELOS GARCH
Dr. Galindo
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Detección de la volatilidad en clusters
(10.4)
MODELOS GARCH
Dr. Galindo
T
tt
T
ttt
2
4
2
21
2
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El efecto de leverage:
Hecho: la volatilidad es mayor en un mercado de caída que en alza
Prueba:
(10.5)
Si el estadístico es negativo y el BP es estadísticamente
significativo asimetría
MODELOS GARCH
Dr. Galindo
T
tt
T
tt
T
ttt
2
2
2
4
21
2
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Modelo GARCH incluye:
1.Variable dependiente (ganancias)
2.Primera ecuación de la media condicional
et = ganancia inesperada
Opción: Media autoregresiva condicional: AR(1)
3.Segunda ecuación es la varianza condicional
MODELOS GARCH
Dr. Galindo
tt eR 0
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ARCH():
(11.1)
MODELOS GARCH
Dr. Galindo
210
22110
2
0
00
tt
t
ttt
,NIe
,,,
e...e
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GARCH Simétricos:
(12.1)
GARCH(1,1):
(12.2)
MODELOS GARCH:
Dr. Galindo
00 110
2211
22110
2
q
qtqttt
,,,,,,
ee
00
211
2
w
zew ttt
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Describirlo como:
(12.3)
MODELOS GARCH:
Dr. Galindo
23
222
21
23
22
21
21
21
2
1 ttt
ttt
ttt
eeew
ewewew
ew
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La varianza de L.P. se obtiene igualando en la ecuación (12.2):
(12.4)
MODELOS GARCH:
Dr. Galindo
2t
1
12
w
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IGARCH:
Con y suponiendo que :
(13.1)
Tipo de cambio: media y varianza no estacionariaCon w = 0 IGARCH similar a EWMA
MODELOS GARCH:
Dr. Galindo
1
10
1 21
21
2
ttt ew
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Modelo GARCH de componentes permite una variación de largo plazo en la volatilidad:
GARCH(1,1):
(14.1)
MODELOS GARCH:
Dr. Galindo
22
122
12
21
21
22 1
tt
ttt
e
e
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1. Modelo de GARCH de componentes (cambio de parámetros)
2. EGARCH (asimétrico)
3. N-GARCH (no lineales)
4. t- GARCH
MODELOS GARCH:
Dr. Galindo
![Page 52: MODELOS ARCH APLICADOS Dr. Luís Miguel Galindo. “I have heard it said that too much academic research is focused on finding very precise answers to irrelevant](https://reader035.vdocuments.mx/reader035/viewer/2022062305/5665b4e61a28abb57c949a81/html5/thumbnails/52.jpg)
Regla de dedo de la varianza es la raíz cuadrada del tiempo para diferentes periodicidades (no sirve):
Donde α y β so las estimaciones del GARCH
MODELOS GARCH:
Dr. Galindo
nn
n
112
11
2211
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Pruebas de GARCH:
Los errores estandarizados al cuadrado no tengan autocorrelación:
(15.1)
Ello equivale que βρ:
(15.2)
MODELOS GARCH:
Dr. Galindo
2
22
t
t*t
2nT D
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Coeficiente de autocorrelación del cuadrado de los errores estandarizados
(15.3)
Si no existe autocorrelación en las ganancias estandarizadas al cuadrado del GARCH se considera al modelo bien especificado
MODELOS GARCH:
Dr. Galindo
4
22
*t
r
*nt
*tn
n
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Time-varyiny correlation:
La correlación condicional implica variaciones en los parámetros
Bivariate GARCH:
(16.1)
(16.2)
MODELO GARCH MULTIVARIADO:
Dr. Galindo
2111
21111
21 ttt ew
2122
21222
22 ttt ew
![Page 56: MODELOS ARCH APLICADOS Dr. Luís Miguel Galindo. “I have heard it said that too much academic research is focused on finding very precise answers to irrelevant](https://reader035.vdocuments.mx/reader035/viewer/2022062305/5665b4e61a28abb57c949a81/html5/thumbnails/56.jpg)
(16.3)
e1, e2 son ganancias inesperadas de las dos ecuaciones de las medias condicionales
Problema: no se incluyen en la ecuación de la covarianza la que implica que no se captura la correlación asociada con la mayor volatilidad
MODELO GARCH MULTIVARIADO:
Dr. Galindo
212
211 tt ,
2133
21333
23 t,tt ew
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Coeficiente de correlación variable:
(17.1)
Un beta cambiante es:
(17.2)
Pt = correlación condicional
covarianza condicional
varianza condicional
MODELO GARCH MULTIVARIADO:
Dr. Galindo
tt
ttn
ˆˆ
ˆP̂
21
12
ttxt
xytt vP
2
2
2xy
2xt
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1. Especificar el modelo:
(18.1)
(18.2)
ESTIMACIÓN ARCH-GARCH:
Dr. Galindo
111 ,GARCHAR
21
0 t
ttt
,N
eyy
21
2110
2 ttt e
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2. Especificar la función de máxima-verosimilitud de los errores bajo el supuesto de normalidad
(18.3)
3. El programa maximiza la función para obtener los parámetros y sus error- estándar
ESTIMACIÓN ARCH-GARCH:
Dr. Galindo
2
212
21
21
22
ttt
T yyloglogT
L
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ARCH:
(18.1)
(18.2)
Nota: es un estimador insesgado pero impreciso de , Ding, Granger y Engle (1993) sugieren medir la volatilidad directamente del valor absoluto de las ganancias
MODELOS GENERALES:
Dr. Galindo
tt
tt
n,Ne
e
0~
222
211 ttt eewn
2te 2̂
![Page 61: MODELOS ARCH APLICADOS Dr. Luís Miguel Galindo. “I have heard it said that too much academic research is focused on finding very precise answers to irrelevant](https://reader035.vdocuments.mx/reader035/viewer/2022062305/5665b4e61a28abb57c949a81/html5/thumbnails/61.jpg)
Estimar más robusto a asimetrías o no-normalidad
Efecto Taylor: Las ganancias absolutas tienen memoria más largo que las ganancias al cuadrado
MODELOS GENERALES:
Dr. Galindo
![Page 62: MODELOS ARCH APLICADOS Dr. Luís Miguel Galindo. “I have heard it said that too much academic research is focused on finding very precise answers to irrelevant](https://reader035.vdocuments.mx/reader035/viewer/2022062305/5665b4e61a28abb57c949a81/html5/thumbnails/62.jpg)
Inclusión de Dummys:
(19.1)
(19.2)
MODELOS GENERALES:
Dr. Galindo
ttt
ttt
zne
eD
1
1211 tttt Dehwh
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Opciones:
1. Eliminar “aditive dummys” excluyendo los datos
2. Incluir multiplicative outliers que producen un impacto en la volatilidad
MODELOS GENERALES:
Dr. Galindo
ladootroen
fechaunaenD t 0
1
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1. ARCH(q):
(20.1)
(20.2)
MODELOS GENERALES:
Dr. Galindo
tt e
blancoRuido
t
t
ttt
z
,Dz
zne
10
![Page 65: MODELOS ARCH APLICADOS Dr. Luís Miguel Galindo. “I have heard it said that too much academic research is focused on finding very precise answers to irrelevant](https://reader035.vdocuments.mx/reader035/viewer/2022062305/5665b4e61a28abb57c949a81/html5/thumbnails/65.jpg)
El proceso zt es escalado por ht (la varianza condicional) que es función de los valores pasados del cuadrado de los residuales de las ganancias
(20.3)
Varianza incondicional:
(20.4)
MODELOS GENERALES:
Dr. Galindo
q
jjtjt ewh
1
2
q
j j
w
1
2
1
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2. GARCH
(21.1)
Varianza incondicional:
(21.2)
MODELOS GENERALES:
Dr. Galindo
1 1
2
i
q
jjtjitit ehwh
positivoeshcon
w
t
ji
0
1
11
2
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El GARCH es estacionario:
Como entonces:
(21.3)
MODELOS GENERALES:
Dr. Galindo
1 1
1i
q
jji
ttt zhe
tttt hzEheE 22
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3. Integrated GARCH : IGARCH
Con
La varianza incondicional es infinita
MODELOS GENERALES:
Dr. Galindo
1 1
1i
q
jji
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4. Exponential GARCH : EGARCH
(22.1)
ht depende del signo y del tamaño de et
MODELOS GENERALES:
Dr. Galindo
t
tt
kktkktk
jtjq
t
heu
ee
hlnhln
1
0
2
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5. GJR-GARCH: (Glosten, Jagannathan Runkle, 1993):
(23.1)
MODELOS GENERALES:
Dr. Galindo
00
01
1
11
1 1
21
2
t
tt
i
q
jjtjttjtjitit
esi
esiD
eDehwh
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6. Threshold GARCH : TGARCH :
(24.1)
7. Quadratic GARCH : QGARCH
(25.1)
MODELOS GENERALES:
Dr. Galindo
1 110
i
q
jjtjititiitit eDe
12
1 ttt hewh
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8. FIGARCH
9. GARCH con cambio estructural:
(26.1)
10. Components GARCH : CGARCH:
(27.1)
mt = cambios ocasionales de nivel
MODELOS GENERALES:
Dr. Galindo
11211111 tt!RRt heDwDwh
ttt umv
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(27.2)
MODELOS GENERALES:
Dr. Galindo
1adprobabilidcon1
adprobabilidcon01
t
tttt
q
qmm
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(27.3)
mt = tendencia variable en el tiempo o el componente permanente en volatilidad
MODELOS GENERALES:
Dr. Galindo
1
211
1112
1
tttt
tttttt
hePmwm
mhmemh
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11. Regime Switching GARCH : RS-GARCH
(28.1)
MODELOS GENERALES:
Dr. Galindo
régimen111
21111
t
tttttttt
s
shseswssh
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12. Asymmetric Dynamic covariance (Abc)
(29.1)
(29.2)
(29.3)
MODELOS GENERALES:
Dr. Galindo
t1
h0,N~t
ttt
e
mee
iitiith
ijtijjjtiitijtijt hhPh
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(29.4)
1.VECH: P12= 0
2. BEKK
3. FARCH
MODELOS GENERALES:
Dr. Galindo
jttittijtiijtijt ghhgeeabHbw '11
''11
'1
'
10 1212 P
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1. GARCH(1,1):
(30.1)
La varianza condicional es función de:
• Contante•• Noticias sobre la volatilidad del periodo previo
• La varianza pronosticado del periodo anterior:
MODELOS APLICADOS:
Dr. Galindo
21
21
2 ttt ew
21:ARCH te
21:GARCH t
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2. GARCH(ρ,q):
(30.2)
MODELOS APLICADOS:
Dr. Galindo
q
j iitijtjt ew
1 1
222
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3. GARCH-M:
(30.3)
4. Regresores en la ecuación de varianza:
(30.4)
MODELOS APLICADOS:
Dr. Galindo
tttt exy 2' log
'
1
22
1
2t
iitijt
q
jjt zew
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Opciones
1. ARCH-M:• None
• Std Dev.• Variante• Log(var)
2. GARCH-TARCH
• EGARCH• PARCH
• Component-GARCH
MODELOS APLICADOS:
Dr. Galindo
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3. Error distribution
4. Bollersler y wooldrige
Los residuales no son distribuidos condicionalmente como normales
5. Métodos de optimización
MODELOS APLICADOS:
Dr. Galindo
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6. Variante regressors:
• Incluye una constante• Serie positiva
7. Views: pruebas 8. ARCH model procedures: residuales
MODELOS APLICADOS:
Dr. Galindo
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8. TARCH
(30.5)
Con γ1>0 malas noticias incrementan la volatilidad(leverage effect)
MODELOS APLICADOS:
Dr. Galindo
casootroen00si1
1 1 1
2222
yeI
Ieew
tt
q
j i kktktkitijtjt
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9. EGARCH:
(30.6)
Leverage effect:
El impacto es asimétrica si
MODELOS APLICADOS:
Dr. Galindo
11 1
22 loglogk kt
ktk
q
j i it
itijtjt
eew
0i
0i
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10. PARCH:
(30.7)
MODELOS APLICADOS:
Dr. Galindo
0
11
con
eewi
itiiti
q
jjtjt
0Asimetría
0:Simetría
i
i
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11. CGARCH
(30.8)
Variables transitorias = Impacto en el C.P. en volatilidadVariables permanentes = Impacto en el nivel del L.P. de la volatilidad
Efecto asimétrico TARCHqt = es la volatilidad de L.P.
MODELOS APLICADOS:
Dr. Galindo
2
12
11
21
21
2
tttt
tttt
ewmwm
wwewm
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MODELOS ARCH APLICADOS
MODELOS ARCH APLICADOS
Dr. Luís Miguel Galindo