Vector Time Series Model Representations and Analysis with XploRe

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Vector Time Series Model Representations and Analysis with XploRe

Julius Mungo

CASE - Center for Applied Statistics and Economics Humboldt-Universit¨ at zu Berlin [email protected]

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Motivation 1-1

Multiple time series analysis approach involves a frame work for analyzing time series systems and the possible cross relationships among its levels.

Modelling such systems entails investigating whether

some variables in the system have a tendency to lead others.

there is feedbacks between the variables, the question of contemporaneous movements, impulses (shocks, innovations) transfer from one time series to another

1.5

Motivation 1-2

The modelling procedure in XploRe uses the quantlet library MulTi to model a system of multiple time series.

how XploRe MulTi is used to empirically investigate and modell various MTS systems.

attention is on Vector Autoregressive (VAR) and the Vector Equilibrium Correction (ECM) models representations and modelling.

Granger & Newbold ( 1986)

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Motivation 1-3

Outline

1. Motivation X

2. XploRe Quantlib MulTi

3. Modelling Time Dependent Factor Loadings from a DSFM for IV String Dynamics 4. Summary

5. References

1.5

XploRe Quantlib MulTi 2-1

MulTiplot.xpl

Generates a MTS plot from a k-dimensional time series data, allowing for the series transformation, with graphics, plots and their properties to be investigated

MulTiplot01.xpl

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XploRe Quantlib MulTi 2-2

MulTifr.xpl

For general analysis of the Full VAR Model; VAR order selection criteria, parameter estimation, Residual Analysis, Structural Analysis and Forecasting.

MulTifr02.xpl

1.5

XploRe Quantlib MulTi 2-3

MulTiira.xpl

For VAR impulse response analysis

1 l i b r a r y

(" m u l t i ")

2

x =

3 M u l T i i r a

( x’ ,4 ," M "|" Y "|" I ")

MulTiira01.xpl

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XploRe Quantlib MulTi 2-4

MulTirr.xpl

For Reduced Rank VAR analysis MulTiss.xpl

General analysis for a Subset VAR Model MulTici.xpl

General analysis for cointegration

1.5

VAR modelling with XpLoRe 3-1

VAR modelling

XploRe specifies a k-dimensional VAR(p) model of the form Y

= υ + A

Y

+ A

Y

+, . . . , +A

Y

+ ε

(1)

Y

= (Y

, . . . , Y

)

are observable vectors of k endogenous variables

υ = (υ

, . . . , υ

)

is a vector of intercept terms, A

are (K × K ) coefficient matrices

ε

is a white noise with covariance matrix Σ

> 0

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Modelling Time Dependent Factor Loadings 4-1

Modelling Time Dependent Factor Loadings from a DSFM for IV String Dynamics

The DSFM is represented by Y

= m

(X

) +

l

X

l =1

β

m

(X

) + 

m

are smooth basis function (l = 0, 1, . . . , L) X

are two dimensional covariables

β

are weights of m

depending on time i

β

= (β

, β

, . . . , β

)

form an observed MTS (Fengler et al (2004))

1.5

Time series plots for the beta coeff. series (1999 - 2003)

MTSplot.xpl

Beta1 coeff. time plot

1999 2000 2001 2002 2003

time

0.511.5

number sold(thousands)

Beta2 coeff. time plot

1999 2000 2001 2002 2003

time

-0.3-0.2-0.100.1

number sold(thousands)

Beta3 coeff. time plot

1999 2000 2001 2002 2003

time

-10-50510

number sold(thousands)*E-2

Modelling Time Dependent Factor Loadings 4-3

Min. Max. Mean Median Stdd. Skewn. Kurt.

beta1 0.45 1.53 1.16 1.30 0.25 -0.820 2.69 beta2 -0.28 0.10 0.00 0.00 0.03 -0.25 6.94 beta3 -0.10 0.13 0.00 0.00 0.03 0.93 5.4

Table 1: Summary statistics for Beta coeff. series

beta1 beta2 beta3 beta1 1 -0.64 -0.05

beta2 1 -0.00

beta3 1

Table 2: Contemp. correlation

Betasummary.xpl

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Modelling Time Dependent Factor Loadings 4-4

Betadensity.xpl Kernel density (Epanechnikov, h = 0.0815) and boxplot for levels

Distribution: Beta1

0.5 1X 1.5

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X

0.5 1 1.5

Y

Distribution: Beta2

-0.2 -0.1 0 0.1

X

0510

Y 00.51

X

-0.2 -0.1 0 0.1

Y

Distribution: Beta3

-10 -5 0X*E-2 5 10

051015

Y 00.51

X

-10 -5 0 5 10

Y*E-2

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Modelling Time Dependent Factor Loadings 4-5

0.5 1 1.5 2

X

0.511.52

Y

-0.3 -0.2 -0.1 0 0.1

X

-0.2-0.100.1

Y

-10 -5 0 5 10 15

X*E-2

-10-50510

Y*E-2

Figure 1: Q-Q plots of the normal against the emprical quanttiles for the Beta series

1.5

Modelling Time Dependent Factor Loadings 4-6

preanalysBetas.xpl

Sample autocorrelation function (acf)

0 5 10 15 20 25 30

lag

00.51

acf

Sample partial autocorrelation function (pacf)

5 10 15 20 25 30

lag

00.51

pacf

Sample autocorrelation function (acf)

0 5 10 15lag 20 25 30

00.51

acf

Sample partial autocorrelation function (pacf)

5 10 15lag 20 25 30

00.20.40.60.8

pacf

Sample autocorrelation function (acf)

0 5 10 15 20 25 30

lag

00.51

acf

Sample partial autocorrelation function (pacf)

5 10 15 20 25 30

lag

00.5

pacf

Figure 2: ACF and PACF of levels

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Modelling Time Dependent Factor Loadings 4-7

Testing β

levels for random walk

Coeff. Test Deterministic lags testvalue asymptotic crit. values

term (David & Mackinnon, (1993))

1% 5% 10%

Beta1 ADF constant 1 -2.41 -3.44 -2.86 -2.57

2 -2.24 -3.44 -2.86 -2.57

3 -2.32 -3.44 -2.86 -2.57

7 -2.05 -3.44 -2.86 -2.57

Beta2 ADF constant 1 -6.01 -3.44 -2.86 -2.57

2 -5.01 -3.44 -2.86 -2.57

3 -4.58 -3.44 -2.86 -2.57

4 -4.21 -3.44 -2.86 -2.57

5 -4.03 -3.44 -2.86 -2.57

7 -3.62 -3.44 -2.86 -2.57

Beta3 ADF constant 1 -3.49 -3.44 -2.86 -2.57

2 -3.32 -3.44 -2.86 -2.57

7 -2.87 -3.44 -2.86 -2.57

8 -2.85 -3.44 -2.86 -2.57

Table 3: ADF-Test of unitroot for levels series

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Modelling Time Dependent Factor Loadings 4-8

coeff. lag test statistic crit. values

(Kwiaskowski, (1992))

1% 5% 10%

Beta1 1 const 16.38 0.347 0.463 0.739

2 10.98 0.347 0.463 0.739

3 8.28 0.347 0.463 0.739

7 4.22 0.347 0.463 0.739

Beta2 1 const 29.95 0.347 0.463 0.739

2 15.13 0.347 0.463 0.739

3 11.61 0.347 0.463 0.739

4 9.46 0.347 0.463 0.739

5 8.00 0.347 0.463 0.739

7 6.16 0.347 0.463 0.739

Beta3 1 const 9.61 0.347 0.463 0.739

2 6.47 0.347 0.463 0.739

7 2.52 0.347 0.463 0.739

8 2.25 0.347 0.463 0.739

Table 4: KPSS-Test of stationarity for levels series

unitrootest.xpl

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Modelling Time Dependent Factor Loadings 4-9

Modelling Beta series Results:

At 1% significant level, unit root exist for Beta1 and beta3 at all lags considered

Beta2 indicates of some kind of misspecification

Beta3 do not reject at 1% level, unit-root null hypothesis.

Even at 5% or 10%, rejecting unit root will be marginal.

KPSS clearly rejects its null hypothesis of stationarity around a constant

1.5

Modelling Time Dependent Factor Loadings 4-10

coeff. shift suggested test statistic crit. values (Lanne et, al(2001)) function break date

(shift dummy ) 1% 5% 10%

Beta1 2001.11.06 -1.58 -3.48 -2.88 -2.58

Beta2 2001.11.06 -1.05 -3.48 -2.88 -2.58

Beta3 1999.06.10 -3.20 -3.48 -2.88 -2.58

Table 5: Unitroot-Test of stationarity for levels series in the presence of structural break

We specify a stationary model with first differences and consider fitting an VAR model,

X

= (∆Beta1, ∆Beta2, ∆Beta3)

and determine the autoregressive order for the model

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Modelling Time Dependent Factor Loadings 4-11

’Beta’ Time Series Plot

1999 2000 2001 2002 2003

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Y

Figure 3: First difference plot of Beta series

Modelling Time Dependent Factor Loadings 4-12

Order Selection Criteria

Final Prediction Error FPE (n) =

T + kn + 1 T − kn − 1

k

det(Xˆ

ε(n)) Akaike Information Criterion

AIC = ln Xˆ

ε(n)

 +2(the number of freely estimated parameters) T

= ln Xˆ

ε(n)  +2nK²

T Schwarz Information Criterion

SIC = ln Xˆ

ε(n)  +lnT

T nK² Hannan-Quinnn Information Criterion

HQ = ln Xˆ

ε(n)

 +2ln(lnT ) T nK²

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Modelling Time Dependent Factor Loadings 4-13

order ln(FPE) AIC HQ SC

0 -24.35 -24.35 -24.35 -24.35 1 -24.62 -24.62 -24.60 -24.58 2 -24.67 -24.67 -24.64 -24.59 3 -24.69 -24.69 -24.65 -24.56 4 -24.69 -24.69 -24.63 -24.52 5 -24.60 -24.69 -24.61 -24.47 6 -24.71 -24.72 -24.61 -24.45 7 -24.70 -24.71 -24.59 -24.41 8 -24.69 -24.70 -24.57 -24.36 We choose to apply the order 3 as indicated by HQ.

HQ and HC have been justified as consistent, (see, Paulsen(1984) and Tsay(1984))

1.5

Modelling Time Dependent Factor Loadings 4-14

VAR estimates (OLS) with t-values in parenthesis

2 4 ∆Beta1t

∆Beta2_t

∆Beta3t 3

5 =

2

4 0.12(3.24) 0.19(2.30) −0.06(−0.40)

−0.09(6.35) 0.07(−17.17) −0.07(1.03) 0.02(2.25) 0.02(1.42) −0.26(−8.24)

3 5 2

4 ∆Beta1_t−1

∆Beta2_t−1

∆Beta3_t−1 3 5

+ 2

4 −0.09(−2.40) −0.04(−0.79) 0.11(0.64)

−0.03(−1.58) −0.04(7.94) −0.05(−0.66) 0.00(−0.63) 0.00(0.16) −0.06(−1.89)

3 5 2

4 ∆Beta1_t−2

∆Beta2t−2

∆Beta3_t−2 3 5

+ 2

4 −0.02(−0.58) −0.13(−1.53) 0.14(+0.83) 0.00(+0.03) −0.12(−3.42) −0.02(−0.26)

−0.01(−0.53) −0.02(−1.36) −0.05(−1.52) 3 5 2

4 ∆Beta1_t−3

∆Beta2_t−3

∆Beta3_t−3 3 5

+ 2 4 εˆ_1,t

ˆ ε_2,t ˆ ε_3,t

3 5

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Modelling Time Dependent Factor Loadings 4-15

Covariance matrix of residuals

Σ ˆ

=





+1.58 −0.35 −0.07

−0.35 +0.36 −0.01 +0.07 −0.01 +0.06



 Correlation matrix of residuals

Corr (ε ˆ

) =





1.00 −0.46 +0.23 +1.00 −0.12 +1.00



 The correlation matrix indicates that there is some

contemporaneous correlation structure in the residual vector.

Not all elements of the parameter matrices are significantly different from zero.

Especially the coefficients for ∆Beta1

.

1.5

Modelling Time Dependent Factor Loadings 4-16

Model Validation

(i ) Multivariate Portmanteau test for autocorrelation H

: E (ε

ε

) = 0, i = 1, . . . , h

H

: at least one autocovariance (autocorrelation) is non zero Test statistic: (Ljung & Box (1978))

Q

= T

h

X

i =1

1 T − i tr n

C

C

C

C

o

∼ χ

C

= T

T

X

t=i +1

ε

ε

C

and C

are the contemporaneous correlations and autocovariance of residuals respectively

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Modelling Time Dependent Factor Loadings 4-17

(ii ) Testing for ARCH effects

Test for neglected conditional heteroscedasticity (ARCH) based on fitting ARCH(q) model to the estimated residuals.

ˆ

ε

= β

+ β

ε ˆ

+ · · · + β

ε ˆ

+ error

H

: β

= · · · = β

= 0, (no ARCH effects) H

: β

6= 0 or . . . or β

6= 0

1.5

Modelling Time Dependent Factor Loadings 4-18

Lagrange Multiplier (LM) statistic: (see, Engle (1982)) ARCH

= 1

2 ε ˆ

ε ∼ χ ˆ

The R

form, test statistic:

T R

∼ χ

R

is the squared multiple R

value of the regression of ˆ ε

on an intercept and q lagged values of ˆ ε

ARCHtest.xpl

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Modelling Time Dependent Factor Loadings 4-19

(iii ) Testing for Nonnormality

H

: E (µ

)

= 0 & E (µ

)

= 0 H

: E (µ

)

6= 0 & E (µ

)

6= 0 Test statistic: (Jarque and Bera (1987))

JB = T 6 T

T

X

t=1

(ˆ µ

)

!

+ T 24 T

T

X

t=1

(ˆ µ

)

− 3

!

The test displays the χ

-statistics associated with the skewness and kurtosis of the standardized residuals for testing nonnormality.

1.5

Modelling Time Dependent Factor Loadings 4-20

Test Q

JB

MARCH

(3) Test statistic 188.39 15.49 980.39

p-value 0.02 0.00 0.00

Table 6: Diagnostic tests for AR(3) models

The tests hypothesis is rejected for p-values smaller than 0.05.

Results show some autocorrelation in the residuals and the presence of heteroscedastic effects in the conditional variance.

We therefore maintain that there is some ARCH effects in model residuals.

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Summary 5-21

Summary

Testing for ARCH effects reveal neglected conditional

heteroscedasticity. This gives an indication of fitting an ARCH or ARCH type model.

Observing that not all elements of the estimated VAR parameter matrices are significantly different from zero, we could choose a subset VAR model where single elements of the estimated coefficient matrices are restricted to zero.

1.5

References 6-22

References

G.C Reinsel

Elements of Multivariate Time Series Anylysis.

Springer Verlag, New York, 1993.

W. H¨ ardle, Z. Hl´ avka and S. Klinke XploRe Application Guide

Springer-Verlag, Heidelberg, 2000.

H. L¨ utkepohl

Introduction to Multiple Time Series Analysis.

Springer Verlag,1993.

K. Patterson

An Introduction to Applied Econometrics a time series approach.

Macmillan Press Ltd, 2000.

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