vagaries of the euro-an introduction to arima modeling
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Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Vagaries of the EURO: An Introduction toARIMA Modeling
Econometrıa II
Juan P. Armijo - Nelson Jaque - Andres Medina
Universidad de Santiago de Chile
June 30, 2015
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Items
Datos
Identificacion
Metodologıa Box-Jenkins
Bibliografıa
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Datos Utilizados
I Esta presentacion esta basada en el articulo Vagaries of the Euro:An Introduction to ARIMA Modeling de los autores GuillaumeWeisang y Yukika Awazu.
I Base de Datos OCDEhttp://stats.oecd.org/wbos/default.aspx
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
TC EURO/USD 1994-2008 (MM)
0.7
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1.1
1994 1996 1998 2000 2002 2004 2006 2008
Figure : Tipo de Cambio Euro/USD Enero 1994 - Octubre 2008
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Box-Plot
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Figure : Dispersion y Box-plot para Datos del Tipo de Cambio
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Descomposicion Serie TC
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0.075
1994 1996 1998 2000 2002 2004 2006 2008
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1994 1996 1998 2000 2002 2004 2006 2008
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1994 1996 1998 2000 2002 2004 2006 2008
Figure : Descomposicion en Aleatoridad, Tendencia y Estacionalidad
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
ACF & FACP Datos
0.00
0.25
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0.0 0.5 1.0 1.5lag
acf
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acf
Figure : Autocorrelacion y Autocorrelacion Parcial para Datos de Tipo deCambio
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Test Estacionariedad
Table : Augmented Dickey-Fuller Test
Estadıstico DF Lag p-value-1.654 12 0.7207
Table : Phillips-Perron Unit Root Test
Estadıstico DF Lag p-value-0.803 4 0.9594
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Test Estacionariedad
Table : Box-Ljung Test
Estadıstico Lag p-value1485.974 12 < 2.2e-16
Table : KPSS Test for Level Stationarity
Estadıstico Lag p-value1.0989 2 0.01
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
TC EURO/USD 1994-2008 (MM) Diferenciados
−0.050
−0.025
0.000
0.025
1994 1996 1998 2000 2002 2004 2006 2008
Figure : TC Diferencidos
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
ACF & FACP
0.00
0.25
0.50
0.75
1.00
0.0 0.5 1.0 1.5lag
acf
−0.2
−0.1
0.0
0.1
0.2
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acf
Figure : ACF y FACP TC Diferenciados
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Test Estacionariedad
Table : Augmented Dickey-Fuller Test
Estadıstico DF Lag p-value-2.7168 12 0.2772
Table : Phillips-Perron Unit Root Test
Estadıstico DF Lag p-value-8.9166 4 0.01
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Test Estacionariedad
Table : Box-Ljung Test
Estadıstico Lag p-value29.4232 12 0.003408
Table : KPSS Test for Level Stationarity
Estatıstico Lag p-value0.2955 2 0.1
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Ajuste
Table : Ajuste ARMA(2,0)
Coeff Estimacion σse t Sigφ1 0.3759 0.0764 5.090 0.000 (**)φ2 -0.1829 0.0766 -2.387 0.018 (**)
Intercep -0.0015 0.0020 0.750 0.530
Table : Estadıstica Ajuste ARMA(2,0)
AIC BIC RMSE Ljung-Box DF Sig-812.27 -797.8506 0.020 7.635023 16 0.9589967
Reemplazando los valores de los coeffcientes, se obtiene :
Yt = −7.93 · 10−4 + 1.392 · Yt−1 − 0.577 · Yt−2 + 0.185 · Yt−3
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Boostrap
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Figure : Iteraciones 1.000, Remuestreo 99
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Boostrap
φ1^
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ARMA(2,0)BoostrapIC 95%
φ2^
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ARMA(2,0)BoostrapIC 95%
Figure : Histograma para φ1 y φ2 mediante Boostrap
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Boostrap
Table : Intervalos de Confianza 95% para Ajuste ARMA(2,0) y Boosptrap
Coeff Estimacion σse Lo Upφ1 0.3758 0.0765 0.2287 0.5285φ2 -0.1828 0.0766 -0.3312 -0.0307
φ1 0.3728 0.0307 0.3127 0.4329
φ2 -0.1787 0.0309 -0.2392 0.1182
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Test Normalidad Residuos
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−3 −2 −1 0 1 2 3theoretical
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Figure : Histograma y ACF Residuos
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Test Normalidad Residuos
Table : Jarque Bera Test
Estadıstico Lag p-value3.1028 2 0.2101
Table : Shapiro-Wilk Normality Test
Estatıstico p-value0.9885 0.1965
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Test Estacionariedad Residuos
Table : Test Box-Ljung y χ21−α,m−p = 18.30704
Metodo Rezago Estadıstico p-valueBox-Ljung test 1 4.462283e-05 0.9946701Box-Ljung test 2 0.0004234177 0.9997883Box-Ljung test 3 0.001836062 0.9999791Box-Ljung test 4 0.1005241 0.9987784Box-Ljung test 5 0.7053888 0.9826716Box-Ljung test 6 0.7255762 0.9939224Box-Ljung test 7 1.091118 0.9932218Box-Ljung test 8 4.010621 0.8561639Box-Ljung test 9 4.093958 0.9051202Box-Ljung test 10 6.384053 0.7820311Box-Ljung test 11 7.062077 0.7940343Box-Ljung test 12 7.161289 0.8467733
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Test Estacionariedad Residuos
0.00
0.25
0.50
0.75
1.00
0.0 0.5 1.0 1.5lag
acf
−0.050
−0.025
0.000
0.025
1994 1996 1998 2000 2002 2004 2006 2008
Figure : Serie de Residuos y Autocorrelaciones
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
¿ Es ARIMA(2,1,0) el Mejor Modelo ?
5 10 15 20
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1520
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MA
(q) θ
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510
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−780
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AR(p)φ
MA
(q) θ
Figure : Simulacion de 400 Modelos ARIMA(p,d ,q) con criterios AIC y BIC
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
¿ Es ARIMA(2,1,0) el Mejor Modelo ?
5 10 15 20
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MA
(q) θ
Figure : Simulacion de 400 Modelos ARIMA(p,d ,q) con criterios AIC y BIC
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
¿ Es ARIMA(2,1,0) el Mejor Modelo ?
5 10 15 20
510
1520
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AR(p)φ
MA
(q) θ
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MA
(q) θ
Figure : Simulacion de 400 Modelos ARIMA(p,d ,q).Criterio HQC
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
¿ Es ARIMA(2,1,0) el Mejor Modelo ?
Table : Simulaciones AR(p) y MA(q)
Orden AIC BIC Ljung-Box p-valueAR(0) -793.0581 -784.8462 3.722258e-05AR(1) -808.6678 -797.3500 4.365791e-01AR(2) -812.2744 -797.8506 9.673933e-01AR(3) -810.2809 -792.7512 9.813285e-01AR(4) -808.5141 -787.8784 9.891284e-01AR(5) -806.6443 -782.9027 9.746223e-01MA(0) -793.0581 -784.8462 3.722258e-05MA(1) -813.1804 -801.8626 9.769652e-01MA(2) -811.2049 -796.7811 9.695577e-01MA(3) -810.0536 -792.5239 9.575400e-01MA(4) -808.1814 -787.5457 9.990861e-01MA(5) -807.0935 -783.3519 9.671207e-01
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
¿ Es ARIMA(2,1,0) el Mejor Modelo ?
Table : Simulaciones ARMA(p, q), Criterio AIC
Orden q = 0 q = 1 q = 2 q = 3 q = 4 q = 5p = 0 -793.03 -813.18 -811.20 -810.05 -808.18 -807.09p = 1 -808.67 -811.20 -809.43 -808.10 -806.41 -808.52p = 2 -812.27 -810.28 -808.49 -812.46 -810.49 -808.33p = 3 -810.28 -808.27 -806.61 -810.48 -808.48 -807.01p = 4 -808.51 -808.61 -810.69 -806.24 -806.98 -810.58p = 5 -806.64 -804.75 -804.62 -804.34 -810.05 -808.75
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
¿ Es ARIMA(2,1,0) el Mejor Modelo ?
5 10 15 20
510
1520
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MA
(q) θ
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MA
(q) θ
Figure : Error Mınimo Cuadratico (RMSE)
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
Datos Identificacion Metodologıa Box-Jenkins Bibliografıa
Bibliografıa
Analysis of Financial Time Series, Ruey S. Tsay.
The Art of R Programming, Norman Matloff.
Time Series: Theory and Methods, Peter J. Brockwell & A.Davis.
Introduction to Scientific Programming and Simulation UsingR, O. Jones, R. Mallardet, A. Robinson.
Vagaries of the EURO: An Introduction to ARIMA Modeling Universidad de Santiago de Chile
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