Introduction to Modern Time Series Analysis · Source: Kirchgässner, Gebhard / Wolters, Jürgen /...

Source: Kirchgässner, Gebhard / Wolters, Jürgen / Hassler, Uwe Introduction to Modern Time Series Analysis, © Springer-Verlag Berlin Heidelberg 2013 ISBN 978-3-642-33435-1 / ISBN 978-3-642-33436-8 (eBook)

Introduction to Modern Time Series Analysis

Gebhard Kirchgässner, Jürgen Wolters and Uwe Hassler

Second Edition

Springer 2013

Teaching Material

The following figures and tables are from the above book. They are provided to help instructors and

students and may be used for teaching purposes as long as a reference to the book is given in class.


1 Introduction and Basics

Figure 1.1: Real Gross Domestic Product of the Federal Republic

of Germany in billions of Euro, 1960 – 2011

Figure 1.2: Quarterly Changes of the Real Gross Domestic Product (ΔGDP)

of the Federal Republic of Germany, 1960 – 1989

100

200

300

400

500

600

700

1960 1970 1980 1990 2000 2010year

bn Euro

-40

-30

-20

-10

0

10

20

30

1965 1970 1975 1980 1985

bn Euro

year


Figure 1.3: Quarterly Growth Rates of the Real Gross Domestic Product (qgr)

of the Federal Republic of Germany, 1960 – 1989

Figure 1.4: Annual Changes of the Real Gross Domestic Product (Δ4GDP) of the Federal Republic of Ger-

many, 1961 – 1989

-20

-15

-10

-5

0

5

10

15

1965 1970 1975 1980 1985

percent

year

-10

-5

0

5

10

15

20

1965 1970 1975 1980 1985

bn Euro

year


Figure 1.5: Annual Growth Rates of Real Gross Domestic Product (agr) of the Federal Republic of Ger-

many, 1960 – 1989

Figure 1.6: ‘Smooth Component‘ and actual values of the Real Gross Domestic Product of the Federal Repub-

lic of Germany, 1961 – 2011

-4

-2

0

2

4

6

8

10

1965 1970 1975 1980 1985year

percent

100

200

300

400

500

600

700

1970 1980 1990 2000 2010

bn Euro

year


-50

-40

-30

-20

-10

0

10

20

30

40

50

20 40 60 80 100

Figure 1.7: Example of a Random Walk where only the steps +1

and –1 are possible


CHF/USD

0.5

1

1.5

2

2.5

3

3.5

1974 1978 1982 1986 1990 1994 1998 2002 2006 2010

-15

-10

-5

0

5

10

15

20

1974 1978 1982 1986 1990 1994 1998 2002 2006 2010

b) Continuous Returns CHF/USD

a) Exchange Rate CHF/USD 1974 – 2011

2011

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

5 10 15 20

c) Estimated Autocorrelations

ˆ

year

year

percent

Figure 1.8: Exchange Rate Swiss Franc U.S. Dollar,

Monthly data, January 1974 to December 2011


2 Univariate Stationary Processes

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

20 15 10 5

t

ˆ

-7.5

-5

-2.5

0

2.5

5

7.5

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

c) Estimated autocorrelation function

with confidence intervals

b) Theoretical autocorrelation function

a) Realisation

xt

Figure 2.1: AR(1) process with α = 0.9


t

-4

-2

0

2

4



ˆ


a) Realisation

-0.8

-0.6

-0.4

-0.2

-1

0

0.2

0.4

0.6

0.8

1

5 10 15 20

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

20 15 10 5

xt

Figure 2.2: AR(1) process with α= 0.5


-5

-2.5

0

2.5

5

t

a) Realisation




1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

5 10 15

ˆ

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

10 15 20

20

5

xt

Figure 2.3: AR(1) process with α = -0.9


40

42

44

46

48

50

52

54

56

1971 1973 1975 1977 1979 1981

a) Popularity of the CDU/CSU, 1971 – 1982

) ( ˆ

) ( ˆ

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

c) Estimated autocorrelation function of the

residuals of the estimated AR(1)-process


b) Autocorrelation (__

) and partial ( )

autocorrelation functions with

confidence intervals

year

Percent

Figure 2.4: Popularity of the CDU/CSU, 1971 – 1982


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

10 5 15 20

-10

-5

0

5

10

t



ˆ

a) Realisation


xt

Figure 2.5: AR(2) process with α1 = 1.5, α2= -0.56


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

-1

t




-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

ˆ

-5

-2.5

0

2.5

5

a) Realisation

xt

Figure 2.6: AR(2) process with α1 = 1.4 and α2 = -0.85


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

a) Three month money market rate in Frankfurt

1970 – 1998 ) ( ˆ

c) Estimated autocorrelation function of the

residuals of the estimated AR(2)-process


) ( ˆ

b) Estimated autocorrelation (__

) and partial

autocorrelation (

) functions with confidence

intervals

10 5 15 20

5 10 15 20

0

2

4

6

8

10

12

14

16

1970 1975 1980 1985 1990 1995

year

Percent

Figure 2.7: Three month money market rate in Frankfurt, 1970 – 1998


kk

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

k

AR(1) process with = -0.9

AR(2) process with 1= 1.4, 2= -0.85

kk

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

k

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

k

AR(2) process with 1= 1.5, 2= -0.56

kk

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

k

AR(1) process with = 0.9

kk

5 10 15 20

5 10 15 20

5 10 15 20

5 10 15 20

Figure 2.8: Estimated partial autocorrelation functions

Table 2.1: Characteristics of the Autocorrelation and the Partial

Autocorrelation Functions of AR and MA Processes

Autocorrelation Function

Partial Autocorrelation

Function

MA(q) breaks off with q does not break off

AR(p) does not break off breaks off with p


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

Figure 2.9: Theoretical autocorrelation functions of ARMA(1,1) processes


0

1

2

3

4

5

6

7

8

1994 1996 1998 2000 2002

c) Autocorrelation function of the residuals

of the estimated ARMA(1,1)-process


ˆ

a) New York three month money market rate,

1994 – 2003

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

b) Autocorrelation (__

) and partial ( )

autocorrelation functions of the first

differences with confidence intervals

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5 10 15 20

Percent

year

Figure 2.10: Three month money market rate in New York, 1994 – 2003


Table 2.2: Forecasts of the Council of Economic Experts

and of the Economic Research Institutes

Period R2 RMSE MAE ME â1 U

Institutes

1970 – 1995 0.369 1.838 1.346 -0.250* 1.005* 0.572

1970 – 1982 0.429 2.291 1.654 -0.731 1.193* 0.625

1983 – 1995 0.399 1.229 1.038 0.231 1.081 0.457

Council of

Economic

Experts

1970 – 1995 0.502* 1.647* 1.171* -0.256 1.114 0.512*

1970 – 1982 0.599* 2.025* 1.477* -0.723* 1.354 0.552*

1983 – 1995 0.472* 1.150* 0.865* 0.212* 1.036* 0.428*

‘*’ denotes the ‘better’ of the two forecasts.


3 Granger Causality

Figure 3.1: Growth rate of real GDP and the four quarters lagged interest rate spread in the Federal Re-

public of Germany, 1970 – 1989 (in percent)

Table 3.1 Test for Granger Causality (I): Direct Granger Procedure

1/65 – 4/89, 100 Observations

y x k1 k2 F(yx) F(yx) F(y–x)

4ln(GDPr)

4ln(GDPr)

4ln(M1r)

4ln(M1r)

GLR – GSR

GLR – GSR

4

8

4

8

4

8

4

8

4

8

4

8

6.087***

3.561**

3.160*

1.927(*)

5.615***

2.521*

1.918

1.443

3.835**

2.077*

1.489

1.178

0.391

0.001

0.111

0.279

10.099**

15.125***

‘(*)’, ‘*’, ‘**’, or ‘***’ denote that the null hypothesis that no causal relation exists can

be rejected at the 10, 5, 1 or 0.1 percent significance level, respectively.

-6

-4

-2

0

2

4

6

8

1970 1972 1974 1976 1978 1980 1982 1984 1986 1988

Growth Rate of Real GDP

Interest Rate Spread (t-4)

(GLR - GSR)

percent

year


Figure 3.2. (The dotted lines are the approximate 95 percent confidence intervals.)

ρ(k)

k

Figure 3.2a: Cross-correlations between the residuals of the univariate models of GDP and the quantity

of money M1

ρ(k)

k

Figure 3.2b: Cross-correlations between the residuals of the univariate models of GDP and the

interest rate spread


ρ(k)

k

Figure 3.2c: Cross-correlations between the residuals of the univariate models of the quantity of

money M1 and the interest rate differential

Table 3.2: Test for Granger Causality (II): Haugh-Pierce Test


Y x (0) k S(yx) S(yx) S(y<=>x)

4ln(GDPr) 4ln(M1r) 0.179(*) 4 16.547** 7.036 26.771**

8 17.234* 11.005 31.426*

4ln(GDPr) GLR – GSR 0.076 4 6.031 10.218* 16.826(*)

8 11.270 13.718(*) 25.565(*)

4ln(M1r) GLR – GSR 0.383*** 4 11.967* 9.660* 36.295***

8 14.424(*) 11.270 40.362**

‘(*)’, ‘*’, ‘**’, or. ‘***’ denote that the null hypothesis that no causal relation exists

can be rejected at the 10, 5, 1 or 0.1 percent significance level, respectively.

Table 3.3: Optimal Lag Length for the Hsiao Procedure

Akaike Criterion Schwarz Criterion

Relation *

1k *

2k *

1k *

1k *

2k *

1k

4ln(M1r) 4ln(GDPr) 4 1 1 1 1 1

4ln(GDPr) 4ln(M1r) 5 3 8 4 0 4

(GLR – GSR) 4ln(GDPr) 4 2 1 1 2 1

4ln(GDPr) (GLR – GSR) 5 5 5 5 0 5


Table 3.4: Models Estimated with the Hsiao Procedure


Criterion Akaike Criterion Schwarz Criterion

Explanatory Variable Dependent Variable

4ln(GDPr,t) 4ln(M1r,t) 4ln(GDPr,t) 4ln(M1r,t)

Constant term 0.146

(0.67)

1.263***

(3.42)

0.136

(0.62)

1.139***

(3.94)

4ln(GDPr, t-1) 0.751***

(13.59)

-0.195

(1.32)

0.756***

(13.68)

4ln(GDPr,t-2) -0.283

(1.65)

4ln(GDPr,t-3) 0.369*

(2.54)

4ln(M1r,t-1) 0.159***

(4.62)

1.027***

(10.73)

0.159***

(4.61)

0.972***

(10.12)

4ln(M1r,t-2) -0.173

(1.29)

-0.135

(0.99)

4ln(M1r,t-3) 0.185

(1.36) 0.083

(0.61)

4ln(M1r,t-4) -0.478***

(3.53)

-0.265**

(2.72)

4ln(M1r,t-5) 0.340*

(2.50)

4ln(M1r,t-6) -0.188

(1.36)

4ln(M1r,t-7) 0.192

(1.41)

4ln(M1r,t-8) -0.203*

(2.08)

(û1,û2) 0.012 0.077

R 2 0.694 0.750 0.694 0.726

SE 1.340 1.952 1.340 2.041

Q(m) 23.084* 11.226* 23.344* 16.548*

m 11 4 11 8

The numbers in parentheses are the absolute values of the estimated t statistics. ‘(*)’,

‘*’, ‘**’, or ‘***’ denote that the corresponding null hypothesis can be rejected at the

10, 5, 1 or 0.1 percent significance level, respectively. m denotes the number of de-

grees of freedom of the Q statistic.


Table 3.5: Models Estimated with the Hsiao Procedure


Criterion Akaike Criterion Schwarz Criterion

Explanatory Variable Dependent Variable

4ln(GDPr,t) (GLR – GSR)t 4ln(GDPr,t) (GLR – GSR)t

Constant term 0.327 (1.47)

0.404** (2.80)

0.320 (1.43)

0.293** (2.93)

4ln(GDPr, t-1) 0.730***

(12.22)

-0.034

(0.65)

0.733***

(12.27)

4ln(GDPr,t-2) -0.132*

(2.10)

4ln(GDPr,t-3) 0.021

(0.32)

4ln(GDPr,t-4) 0.154* (2.58)

4ln(GDPr,t-5) -0.083(*)

(1.72)

(GLR – GSR)t-1 -0.105

(0.64)

1.128***

(11.91)

-0.103

(0.63)

1.138***

(12.13)

(GLR – GSR)t-2 0.441**

(2.62)

-0.168

(1.27)

0.438*

(2.60)

-0.198

(1.42)

(GLR – GSR)t-3 -0.347**

(2.69)

-0.316*

(2.32)

(GLR – GSR)t-4 0.481***

(3.70)

0.448**

(3.25)

(GLR – GSR)t-5 -0.274**

(2.95)

-0.327***

(3.53)

(û1,û2) 0.053 0.031

R 2 0.684 0.816 0.684 0.798

SE 1.362 0.732 1.362 0.768

Q(m) 16.513 4.824 16.648 7.118

m 11 7 11 7

The numbers in parentheses are the absolute values of the estimated t statistics. ‘(*)’,

‘*’, ‘**’, or ‘***’ denote that the corresponding null hypothesis can be rejected at the

10, 5, 1 or 0.1 percent significance level, respectively. m denotes the number of de-

grees of freedom of the Q statistic.


Table 3.6: Test for Granger Causality:

Direct Granger Procedure with Three Variables


Y x z k F(yx) F(yx) F(y–x)

4ln(GDPr) 4ln(M1r) GLR – GSR 4 2.747* 3.788** 0.577

8 2.866** 2.362* 0.127

4ln(GDPr) GLR – GSR 4ln(M1r) 4 0.260 2.426(*) 0.247

8 1.430 1.817(*) 0.229

4ln(M1r) GLR – GSR 4ln(GDPr) 4 7.615*** 0.293 7.273***

8 3.432** 1.009 8.150***

‘(*)’, ‘*’, ‘**’, or ‘***’ denote that the null hypothesis that no causal relation exists can

be rejected at the 10, 5, 1 or 0.1 percent significance level, respectively.


4 Vector Autoregressive Processes

Figure 4.1: Impulse response functions

-0.4

0.0

0.4

0.8

1.2

2 4 6 8 10 12 14 16 18 20

Response of X1 to X1

-0.4

0.0

0.4

0.8

1.2

2 4 6 8 10 12 14 16 18 20


-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2 4 6 8 10 12 14 16 18 20


-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2 4 6 8 10 12 14 16 18 20



Figure 4.2: Cumulative impulse response functions

Figure 4.3: Impulse response functions

-4

-3

-2

-1

0

1

2

3

5 10 15 20

Accumulated Response of X1 to X1

-4

-3

-2

-1

0

1

2

3

5 10 15 20


-1

0

1

2

3

4

5

6

5 10 15 20


-1

0

1

2

3

4

5

6

5 10 15 20


-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

5 10 15 20

Response of DLGDPR to DLGDPR

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

5 10 15 20

Response of DLGDPR to DLM1R

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

5 10 15 20

Response of DLGDPR to GLSR

-2

-1

0

1

2

3

5 10 15 20

Response of DLM1R to DLGDPR

-2

-1

0

1

2

3

5 10 15 20

Response of DLM1R to DLM1R

-2

-1

0

1

2

3

5 10 15 20

Response of DLM1R to GLSR

-0.8

-0.4

0.0

0.4

0.8

1.2

5 10 15 20

Response of GLSR to DLGDPR

-0.8

-0.4

0.0

0.4

0.8

1.2

5 10 15 20

Response of GLSR to DLM1R

-0.8

-0.4

0.0

0.4

0.8

1.2

5 10 15 20

Response of GLSR to GLSR


Figure 4.4: Cumulative impulse response functions

-4

-2

0

2

4

6

8

5 10 15 20

Accummulated Responseof DLGDPR to DLGDPR

-4

-2

0

2

4

6

8

5 10 15 20

Accummulated Response of DLGDPR to DLM1R

-4

-2

0

2

4

6

8

5 10 15 20

Accummulated Response of DLGDPR to GLSR

-4

-2

0

2

4

6

8

5 10 15 20

Accummulated Response of DLM1R to DLGDPR

-4

-2

0

2

4

6

8

5 10 15 20

Accummulated Response of DLM1R to DLM1R

-4

-2

0

2

4

6

8

5 10 15 20

Accummulated Response of DLM1R to GLSR

-4

-2

0

2

4

6

8

5 10 15 20

Accummulated Response of GLSR to DLGDPR

-4

-2

0

2

4

6

8

2 4 6 8 10 12 14 16 18 20

Accummulated Response of GLSR to DLM1R

-4

-2

0

2

4

6

8

5 10 15 20

Accummulated Response of GLSR to GLSR


Table 4.1: Variance Decomposition

Forecast horizon x1 x2

Immediate x1 100.000 0.000

x2 32.834 67.166

4 periods x1 77.866 22.134

x2 23.089 76.911

8 periods x1 65.085 34.915

x2 20.957 79.043

20 periods x1 58.527 41.473

x2 19.838 80.162

Infinity x1 58.020 41.980

x2 19.748 80.252

Table 4.2a: Variance Decomposition


Forecast horizon Δ4ln(GDPr) Δ4ln(M1r) GLR – GSR

immediate

Δ4ln(GDPr) 99.231 0.483 0.286

Δ4ln(M1r) 0.000 92.202 7.798

GLR – GSR 0.000 0.000 100.000

1 year

Δ4ln(GDPr) 82.899 12.479 4.622

Δ4ln(M1r) 8.994 41.336 49.670

GLR – GSR 9.223 0.487 90.289

2 years

Δ4ln(GDPr) 51.948 15.604 32.448

Δ4ln(M1r) 13.896 34.910 51.194

GLR – GSR 16.124 8.998 74.878

5 years

Δ4ln(GDPr) 48.235 16.049 35.716

Δ4ln(M1r) 14.738 35.244 50.018

GLR – GSR 15.719 13.062 71.219

infinity

Δ4ln(GDPr) 48.187 16.132 35.681

Δ4ln(M1r) 14.733 35.258 50.009

GLR – GSR 15.677 13.079 71.244


Table 4.2b: Variance Decomposition 1/65 – 4/89, 100 Observations

Forecast horizon Δ4ln(GDPr) Δ4ln(M1r) GLR – GSR

immediate

Δ4ln(GDPr) 99.231 0.667 0.102

Δ4ln(M1r) 0.000 100.000 0.000

GLR – GSR 0.000 7.798 92.292

1 year

Δ4ln(GDPr) 82.899 15.740 1.361

Δ4ln(M1r) 8.994 60.685 30.321

GLR – GSR 9.223 7.326 83.450

2 years

Δ4ln(GDPr) 51.948 26.995 21.057

Δ4ln(M1r) 13.896 50.669 35.435

GLR – GSR 16.124 11.184 72.692

5 years

Δ4ln(GDPr) 48.235 25.978 25.787

Δ4ln(M1r) 14.738 50.970 34.292

GLR – GSR 15.719 16.065 68.216

infinity

Δ4ln(GDPr) 48.187 26.033 25.780

Δ4ln(M1r) 14.733 50.999 34.269

GLR – GSR 15.677 16.136 68.188


5 Nonstationary Processes

Figure 5.1: Linear and quadratic trend, superimposed

by a pure random process


Figure 5.2: Realisations of AR(1) processes α = 1.03 (------), α= 0.97 (———)


Figure 5.3: Random walk with (-----) and without (––––) drift


Figure 5.4: Scatter diagrams of the first differences against the

original residuals of nonstationary processes

-12

-8

-4

0

4

8

12

-8 -6 -4 -2 0 2 4 6 8

Original residuals

Fir

st d

iffe

ren

ces

of

the

mo

del

wit

h a

lin

ear

tren

d

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

Original residuals

Fir

st d

iffe

rence

s of

the

rand

om

wal

k


Figure 5.5: Scatter diagrams of the residuals of regressions on a time trend against the original residuals

of nonstationary processes

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

Original residuals

Res

idu

als

of

the

mod

el w

ith a

tim

e tr

end

-16

-12

-8

-4

0

4

8

12

16

-8 -6 -4 -2 0 2 4 6 8

Original residuals

Res

idu

als

of

the

model

wit

h a

sto

chas

tic

tren

d


Figure 5.6 Actual and estimated values and residuals of the models with linear deterministic and stochas-

tic trends

-8

-4

0

4

80

20

40

60

80

100

120

10 20 30 40 50 60 70 80 90 100

Actual and estimated values

Residuals

Model with a linear trend

-20

-10

0

10

20

0

50

100

150

10 20 30 40 50 60 70 80 90 100

Residuals

Actual and estimated values

Model with a random walk with drift


Table 5.1: Results of Linear Trend Elimination

(100 Observations)

Model with a

linear trend random walk random walk with drift

Constant term

5.678

(9.79)

19.673

(16.89)

18.67

3

(16.03)

linear trend

0.993

(99.60)

0.191

(9.55)

1.191

(59.4

8)

2R 0.990 0.477 0.973

Durbin-Watson 2.085 0.247 0.247

Density of the estimated coefficient compared with a

normal distribution with the same variance and = 1

Density of the t statistic

0.1

0.2

0.3

0.4

0.5

0.6

-3.47 -1.96 -1.65 0 -2.88

0

5

10

15

20

25

0.85 0.9 0.95 1

Figure 5.7: Density of the estimated autocorrelation coefficient and the

t statistic under the null hypothesis of a random walk.


Figure 5.8: Development of the Swiss, German/European and US

Euromarket interest rates. Monthly data,

January 1983 – December 2002

Table 5.2: Results of the Augmented Dickey-Fuller Tests

1/1983 – 12/2002, 240 Observations

Variable Levels 1. Differences

k Test Statistic k Test Statistic

SER 3 -1.194

(0.678) 2

-7.862

(0.000)

GER/EER 1 -0.957

(0.768) 0

-11.962

(0.000)

UER 1 -0.995

(0.755) 0

-11.220

(0.000)

The tests were performed for levels with as well as for first differences without a con-

stant term. The numbers in parentheses are the p values. The number of lags, k, has been

determined with the Hannan-Quinn criterion.

0

2

4

6

8

10

12

14

1985 1990 1995 2000

UER

GER/EER

SER


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-3.47 -2.88


normal distribution with the same variance and μ = 0.95

Density of the Dickey-Fuller t statistic

0

2

4

6

8

10

12

14

16

0.8 0.85 0.9 0.95 1

Figure 5.9a: Density of the estimated coefficient and of the t statistic for the null hypothesis

of an AR(1) process with ρ = 0.95


0

2

4

6

8

10

12

0.7 0.75 0.8 0.85 0.9 0.95 1

Density of the Dickey-Fuller t statistic


normal distribution with the same variance and μ = 0.90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-2.88 -3.47

Figure 5.9b: Density of the estimated coefficient and of the t statistic

for the null hypothesis of an AR(1) process with ρ = 0.90

Table 5.3: Results of Unit Root and Stationarity Tests for Inflation

1/1969 – 9/1992, 285 Observations

m/k United States United Kingdom France Germany Italy

Phillips-

Perron

6

12

-8.95**

-10.20**

-9.30**

-10.54**

-5.82**

-6.84**

-10.32**

-11.65**

-6.40**

-7.39**

KPSS 6

12

0.81**

0.51*

1.02**

0.65**

1.57**

0.91**

1.26**

0.80**

0.94**

0.56**

ADF

3

6

12

-4.43**

-3.06*

-1.86

-4.48**

-2.97*

-2.27

-2.71(*)

-1.71

-1.29

-4.98**

-3.49**

-1.75

-3.31*

-2.24

-2.39

‚(*)‘, ,*‘ or ,**‘ denote that the corresponding null hypothesis can be rejected at the 10, 5,

or 1 percent significance level, respectively.

Source: U. HASSLER and J. WOLTERS (1995, Tables 3 and 4, p. 39).


Figure 5.10a: German Inflation Rate: Actual values (––––), permanent component according to S.

BEVERIDGE and CH.R. NELSON (-------), permanent component according to R.J. HODRICK

and E.C. PRESCOTT (– - – - –)

Figure 5.10b: German Inflation Rate: cyclical component according to S. BEVERIDGE and CH.R. NELSON (-

-------), cyclical component according to R.J. HODRICK and E.C. PRESCOTT (––––)

0

1

2

3

4

5

6

7

1975 1980 1985 1990 1995

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

1975 1980 1985 1990 1995


Figure 5.11a: Swiss real money balances M2, 1981 – 2008: Actual values (––––)

and permanent component (--------) due to the Hodrick-Prescott filter

Figure 5.11b: Swiss real money balances M2, 1981 – 2006: Actual values

(––––) and permanent component (--------) due to the Hodrick-Prescott filter

200

250

300

350

400

450

500

1981 1986 1991 1996 2001 2006

200

250

300

350

400

450

500

1981 1986 1991 1996 2001 2006


6 Cointegration

0

0 . 0 1

0 . 0 2

0 . 0 3

0 . 0 4

0 . 0 5

- 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0

Density of the t statistic

0

0 .5

1

1 .5

2

2 .5

3

3 .5

4

4 .5

5

0 0 .2 0 .4 0 .6 0 .8 1

Density of the R2

0

1

2

3

4

5

6

7

8

9

1 0

0 0 .1 0 .2 0 .3 0 .4

Density of the Durbin-Watson statistic

Figure 6.1: Densities of the estimated t value, R2, and

the Durbin-Watson statistic


Table 6.1: Critical Values of the Dickey-Fuller Test on

Cointegration in the Static Model

α

k

1 2 3 4

Model with constant term

0.10 -2.57 -3.05 -3.45 -3.81

0.05 -2.86 -3.34 -3.74 -4.10

0.01 -3.43 -3.90 -4.30 -4.65

Model with constant term and time trend

0.10 -3.13 -3.50 -3.83 -4.15

0.05 -3.41 -3.78 -4.12 -4.43

0.01 -3.96 -4.33 -4.67 -4.97

The values for k = 1 are the critical values of the Dickey-Fuller unit root test.

Source: J.G. MACKINNON (1991, Table 1, p. 275).


a) Logarithm of the per capita real quantity of money M1

b) Logarithm of the per capita real GNP

c) Long-run interest rate

Figure 6.2: Data for the Federal Republic of Germany, 1961 − 1989

4.8

5.2

5.6

6.0

6.4

1965 1970 1975 1980 1985

ln(M )t

year

5.2

5.6

6.0

6.4

6.8

1965 1970 1975 1980 1985

ln(Y )t

year

4

6

8

10

12

1965 1970 1975 1980 1985

percent

year


Table 6.2: Critical Values of the Cointegration Test

in the Error Correction Model

α

k

2 3 4

Model with constant term

0.10 -2.89 -3.19 -3.42

0.05 -3.19 -3.48 -3.74

0.01 -3.78 -4.06 -4.46

Model with constant term and time trend

0.10 -3.39 -3.62 -3.82

0.05 -3.69 -3.91 -4.12

0.01 -4.27 -4.51 -4.72

Source: A. BANERJEE, J.J. DOLADO and R. MESTRE (1998, Table 1, pp. 276f.)

Figure 6.3: German three month money market rate in Frankfurt

3

4

5

6

7

8

9

10

1987 1989 1991 1993 1995 1997

percent

year


Table 6.3: Results of the Johansen Cointegration Test

Model Hypotheses Eigenval-

ues

Trace Test max Test

z1, z3

r 0

r 1

0.257

0.006

43.559(0.00)

0.843(0.97)

42.715(0.00)

0.843(0.97)

z1, z6

r 0

r 1

0.205

0.009

34.276(0.00)

1.267(0.91)

33.010(0.00)

1.267(0.91)

The numbers in parentheses are the p values of the corresponding statistics.

Table 6.4: Results of the Johansen Cointegration Test

Model Hypotheses Eigenval-

ues

Trace Test max Test

z1, z3, z6

VECM(0)

r 0

r 1

r 2

0.448

0.187

0.008

116.587(0.00)

31.087(0.00)

1.204(0.92)

85.500(0.00)

29.883(0.00)

1.204(0.92)

z1, z3, z6

VECM(1)

r 0

r 1

r 2

0.384

0.177

0.008

98.883(0.00)

29.172(0.00)

1.173(0.93)

69.711(0.00)

27.999(0.00)

1.173(0.93)

The numbers in parentheses are the p values of the corresponding statistics.


7 Nonstationary Panel Data

Figure 7.1: 10 year government bond yields, January 1990 – December 2006

Table 7.1: p values of the Augmented Dickey-Fuller Tests

for 10 year government bonds

p p p

Australia

Canada

Denmark

Germany

0.1503

0.3168

0.3392

0.4502

Japan

New Zealand

Norway

Sweden

0.3254

0.0600

0.2677

0.2060

Switzerland

United Kingdom

United States

0.4564

0.3910

0.2298

02468

101214

1990 1995 2000 2005

Australia

0

2

4

6

8

10

12

14

1990 1995 2000 2005

Canada

02468

101214

1990 1995 2000 2005

Denmark

02468

101214

1990 1995 2000 2005

Germany

02468

101214

1990 1995 2000 2005

Japan

02468

101214

1990 1995 2000 2005

New Zealand

02468

101214

1990 1995 2000 2005

Norway

02468

101214

1990 1995 2000 2005

Sweden

02468

101214

1990 1995 2000 2005

Switzerland

02468

101214

1990 1995 2000 2005

United Kingdom

02468

101214

1990 1995 2000 2005

United States


Figure 7.2: Cross-sectional average of the U.S. spreads

Table 7.2: CADF Test statistics for spreads against the U.S.

tρ ki tρ ki

Australia

Canada

Denmark

Germany

Japan

-2.96

-3.69

-3.92

-5.05

-2.80

0

0

0

0

2

New Zealand

Norway

Sweden

Switzerland

United Kingdom

-2.55

-3.13

-1.70

-3.20

-2.90

0

2

1

0

2

Table 7.3: Ordered p values for Augmented Dickey-Fuller tests

for the spreads with α = 0.1

p(j) jα/10 p(j) jα/10

New Zealand

Switzerland

Australia

United Kingdom

Germany

0.0298

0.0406

0.1044

0.1371

0.1385

0.01

0.02

0.03

0.04

0.05

Norway

Denmark

Sweden

Japan

Canada

0.1545

0.2070

0.2367

0.3152

0.4717

0.06

0.07

0.08

0.09

0.10

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

1990 1992 1994 1996 1998 2000 2002 2004 2006


Table 7.4: Ordered p values for tests for no cointegration

j p(j) j p(j)

Denmark

Sweden

United Kingdom

New Zealand

Germany

1.45

1.95

1.56

1.16

1.13

0.0748

0.0841

0.0882

0.1120

0.1200

Switzerland

Australia

Canada

Norway

Japan

0.96

1.56

1.35

1.33

1.26

0.1474

0.1705

0.2394

0.3324

0.4114

Table 7.5: Estimates of the error correction adjustment coefficient

i

i

OLS SUR OLS SUR

Australia

Canada

Denmark

Germany

Japan

-0.045

(-2.77)

-0.036

(-1.56)

-0.065

(-3.47)

-0.063

(-2.06)

-0.022

(-1.13)

-0.067

(-5.12)

-0.066

(-3.68)

-0.079

(-6.05)

-0.083

(-5.67)

-0.037

(-2.54)

New Zealand

Norway

Sweden

Switzerland

United Kingdom

-0.062

(-2.75)

-0.050

(-2.97)

-0.032

(-3.03)

-0.060

(-2.03)

-0.044

(-2.69)

-0.064

(-3.63)

-0.060

(-4.64)

-0.043

(-4.49)

-0.075

(-4.27)

-0.057

(-4.34)

The numbers in parentheses are the t statistics of the estimated parameters.

Table 7.6: Estimates of the short-run US influence

ib

*

ib ib *

ib

Australia

Canada

Denmark

Germany

Japan

0.224

(2.39)

0.104

(1.19)

0.070

(0.98)

0.127

(2.23)

0.103

(1.82)

0.269

(2.83)

0.151

(1.73)

0.136

(1.88)

0.176

(3.08)

0.123

(2.18)

New Zealand

Norway

Sweden

Switzerland

United Kingdom

0.075

(0.91)

0.031

(0.42)

0.010

(0.12)

0.047

(0.93)

0.049

(0.61)

0.114

(1.38)

0.092

(1.25)

0.049

(0.57)

0.100

(1.96)

0.087

(1.07)

The numbers in parentheses are the t statistics of the estimated parameters.


Table 7.7: SUR error correction estimates

i,ec

i,ec

Australia

Canada

Denmark

Germany

Japan

0.828

0.957

1.338

1.096

0.590

New Zealand

Norway

Sweden

Switzerland

United Kingdom

0.588

1.312

1.638

0.938

1.246


8 Autoregressive Conditional Heteroscedasticity

a) German Stock Market Index: Data

b) German Stock Market Index: Continuous returns

c) German Stock Market Index: Histogram of the continuous returns

Figure 8.1: German Stock Market Index, 2 January 1996 until 19 May 1999, 842 observations

2000

3000

4000

5000

6000

7000

2 January 1996 19 May 1999

22 July 1998

8 October 199828 October 1997

31 July 1997

-.12

-.08

-.04

.00

.04

.08

2 January 1996 19 May 1999

0

50

100

150

-0.075 -0.050 -0.025 0.000 0.025 0.050


d) Estimated autocorrelations of the residuals

e) Estimated autocorrelations of the squared residuals

Figure 8.1: German Stock Market Index, 2 January 1996 until 19 May 1999, 842 observations (contin-

ued)

Figure 8.2:Density functions of a normalised t distribution with 5 degrees of freedom, variance one and

a standard normal distribution

0.0

0.4

0.8

0 2 4 6 8 10 12 14 16 18 20

0.0

0.4

0.8

0 2 4 6 8 10 12 14 16 18 20

.0

.1

.2

.3

.4

.5

-5 -4 -3 -2 -1 0 1 2 3 4 5

Standard

normal distribution

Modified t-distribution

ˆ( )

ˆ( )

Introduction to Modern Time Series Analysis · Source: Kirchgässner, Gebhard / Wolters, Jürgen /...

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