vagaries of the euro-an introduction to arima modeling

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


Items

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


http://stats.oecd.org/wbos/default.aspx


TC EURO/USD 1994-2008 (MM)

0.7

0.8

0.9

1.0

1.1

1994 1996 1998 2000 2002 2004 2006 2008

Figure : Tipo de Cambio Euro/USD Enero 1994 - Octubre 2008



Box-Plot

0.7

0.8

0.9

1.0

1.1

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

●● ●

● ●

0.7

0.8

0.9

1.0

1.1

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Figure : Dispersion y Box-plot para Datos del Tipo de Cambio



Descomposicion Serie TC

−0.050

−0.025

0.000

0.025

0.050

0.075

1994 1996 1998 2000 2002 2004 2006 2008

0.8

0.9

1.0

1.1

1994 1996 1998 2000 2002 2004 2006 2008

−0.010

−0.005

0.000

0.005

1994 1996 1998 2000 2002 2004 2006 2008

Figure : Descomposicion en Aleatoridad, Tendencia y Estacionalidad



ACF & FACP Datos

0.00

0.25

0.50

0.75

1.00

0.0 0.5 1.0 1.5lag

acf

−0.25

0.00

0.25

0.50

0.75

1.00

0.0 0.5 1.0 1.5lag

acf

Figure : Autocorrelacion y Autocorrelacion Parcial para Datos de Tipo deCambio



Test Estacionariedad

Table : Augmented Dickey-Fuller Test

Estadıstico DF Lag p-value-1.654 12 0.7207

Table : Phillips-Perron Unit Root Test





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



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



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

0.3

0.0 0.5 1.0 1.5lag

acf

Figure : ACF y FACP TC Diferenciados




Table : Augmented Dickey-Fuller Test


Table : Phillips-Perron Unit Root Test





Table : Box-Ljung Test


Table : KPSS Test for Level Stationarity

Estatıstico Lag p-value0.2955 2 0.1



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



Boostrap

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−0.075

−0.050

−0.025

0.000

0.025

0.050

0 25 50 75 100

"sim20"

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sim20

sim30

sim40

sim50

0

5

10

15

20

−0.075 −0.050 −0.025 0.000 0.025 0.050

ind

sim10

sim20

sim30

sim40

sim50

Figure : Iteraciones 1.000, Remuestreo 99



Boostrap

φ1^

0.30 0.35 0.40 0.45

05

1015

ARMA(2,0)BoostrapIC 95%

φ2^

−0.25 −0.20 −0.15

05

1015

ARMA(2,0)BoostrapIC 95%

Figure : Histograma para φ1 y φ2 mediante Boostrap



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



Test Normalidad Residuos

0

5

10

15

−0.050 −0.025 0.000 0.025 0.050

0

4

8

12

16

count

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−0.06

−0.03

0.00

0.03

−3 −2 −1 0 1 2 3theoretical

sam

ple

Figure : Histograma y ACF Residuos



Test Normalidad Residuos

Table : Jarque Bera Test


Table : Shapiro-Wilk Normality Test

Estatıstico p-value0.9885 0.1965



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



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



¿ Es ARIMA(2,1,0) el Mejor Modelo ?

5 10 15 20

510

1520

−805

−800

−795

−790

−785

AR(p)φ

MA

(q) θ

5 10 15 20

510

1520

−780

−760

−740

−720

−700

AR(p)φ

MA

(q) θ

Figure : Simulacion de 400 Modelos ARIMA(p,d ,q) con criterios AIC y BIC




5 10 15 20

510

1520

−805

−800

−795

−790

−785

−806

−80

4

−80

2

−800

−798

−796

−794

−792

−79

0

−788

−786

−784

AR(p)φ

MA

(q) θ

5 10 15 20

510

1520

−780

−760

−740

−720

−700

−790

−780

−770

−760

−750

−740

−730

−720

−710

−700

AR(p)φ

MA

(q) θ

Figure : Simulacion de 400 Modelos ARIMA(p,d ,q) con criterios AIC y BIC




5 10 15 20

510

1520

−800

−790

−780

−770

−760

−750

AR(p)φ

MA

(q) θ

5 10 15 20

510

1520

−800

−790

−780

−770

−760

−750

−800

−795

−790 −785

−780

−775

−770

−765

−760

−755

−750

AR(p)φ

MA

(q) θ

Figure : Simulacion de 400 Modelos ARIMA(p,d ,q).Criterio HQC




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




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




5 10 15 20

510

1520

0.0180

0.0185

0.0190

0.0195

0.0200

0.0205

AR(p)φ

MA

(q) θ

5 10 15 20

510

1520

0.0180

0.0185

0.0190

0.0195

0.0200

0.0205 0.018

0.0185

0.019

0.0195

0.02

AR(p)φ

MA

(q) θ

Figure : Error Mınimo Cuadratico (RMSE)



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

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