2. analyse univariater stationÄrer prozesse · web viewintroduction to modern time series analysis...

2.ANALYSE UNIVARIATER STATIONÄRER PROZESSE

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

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 Germany, 1961 – 1989

Figure 1.5:Annual Growth Rates of Real Gross Domestic Product (agr) of the Federal Republic of Germany, 1960 – 1989

Figure 1.6:‘Smooth Component‘ and actual values of the Real Gross Domestic Product of the Federal Republic of Germany, 1961 – 2011

Figure 1.7:Example of a Random Walk where only the steps +1 and –1 are possible

Figure 1.8:Exchange Rate Swiss Franc U.S. Dollar, Monthly data, January 1974 to December 2011

2 Univariate Stationary Processes

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

Figure 2.2: AR(1) process with α= 0.5

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

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

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

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

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

Figure 2.8: Estimated partial autocorrelation functions

Table 2.1: Characteristics of the Autocorrelation and the PartialAutocorrelation Functions of AR and MA Processes

Autocorrelation Function

Partial AutocorrelationFunction

MA(q)

breaks off with q

does not break off

AR(p)

does not break off

breaks off with p

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

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 ofEconomicExperts

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 Republic of Germany, 1970 – 1989 (in percent)

Table 3.1 Test for Granger Causality (I): Direct Granger Procedure1/65 – 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.

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

k

Figure 3.2a:Cross-correlations between the residuals of the univariate models of GDP and the quantity of money M1

k

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

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 Test1/65 – 4/89, 100 Observations

Y

x

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

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 Procedure1/65 – 4/89, 100 Observations

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

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 degrees of freedom of the Q statistic.

Table 3.5: Models Estimated with the Hsiao Procedure1/65 – 4/89, 100 Observations

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

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 degrees of freedom of the Q statistic.

Table 3.6: Test for Granger Causality:Direct Granger Procedure with Three Variables1/65 – 4/89, 100 Observations

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

Figure 4.2: Cumulative impulse response functions

Figure 4.3: Impulse response functions

Figure 4.4: Cumulative impulse response functions

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 1/65 – 4/89, 100 Observations

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 theoriginal residuals of nonstationary processes

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

Figure 5.6 Actual and estimated values and residuals of the models with linear deterministic and stochastic trends

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)

18.673 (16.03)

linear trend

0.993 (99.60)

0.191 (9.55)

1.191(59.48)

0.990

0.477

0.973

Durbin-Watson

2.085

0.247

0.247

Figure 5.7:Density of the estimated autocorrelation coefficient and thet 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 Tests1/1983 – 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 constant term. The numbers in parentheses are the p values. The number of lags, k, has been determined with the Hannan-Quinn criterion.

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

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 Inflation1/1969 – 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 (––––)

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

6 Cointegration

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 onCointegration 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

Table 6.2: Critical Values of the Cointegration Testin 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

Table 6.3: Results of the Johansen Cointegration Test

Model

Hypotheses

Eigenvalues

Trace Test

max Test

z1, z3

z1, z6

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

Table 6.4: Results of the Johansen Cointegration Test

Model

Hypotheses

Eigenvalues

Trace Test

max Test

z1, z3, z6VECM(0)

z1, z3, z6VECM(1)

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 Testsfor 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

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 testsfor 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

Table 7.4: Ordered p values for tests for no cointegration

p(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

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

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

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 (continued)

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

X19811981.251981.51981.7519821982.251982.51982.7519831983.251983.51983.7519841984.251984.51984.7519851985.251985.51985.7519861986.251986.51986.7519871987.251987.51987.7519881988.251988.51988.7519891989.251989.51989.7519901990.251990.51990.7519911991.251991.51991.7519921992.251992.51992.7519931993.251993.51993.7519941994.251994.51994.7519951995.251995.51995.7519961996.251996.51996.7519971997.251997.51997.7519981998.251998.51998.7519991999.251999.51999.7520002000.252000.52000.7520012001.252001.52001.7520022002.252002.52002.7520032003.252003.52003.7520042004.252004.52004.7520052005.252005.52005.7520062006.252006.52006.7520072007.252007.52007.7520082008.252008.5239.11412799716192241.1671141261852243.23152862176784245.32420903113001247.46094053898167249.64958741649198251.88618589957301254.15798019752199256.44568135206202258.72758076946201260.98730251297496263.21272309778698265.39536802033598267.528848188425269.60935021617303271.63435236698069273.60116726440896275.50584965727694277.33957224210076279.08840619342578280.73342926756823282.25184843890293283.61636998254193284.79602885091799285.75656937790069286.46178567976278286.87550972258214286.96533088923394286.70591098055695286.08206445049569285.09869837305098283.78434658000197282.19034899839926280.385948593644278.45454153937402276.48577991967113274.57423711694798272.81546524100969271.29695444195403270.097072117729269.28352185760679268.91489937543702269.03625524647873269.68014196391601270.86555725068075272.59990644582194274.87654817371202277.67319393308969280.95175288413299284.663590557357288.75667860513897293.17785640764799297.87382727371931302.79455458919369307.89391243188913313.12882524547564318.45713779141369323.83632320869572329.2162558510542334.54240455276232339.75826687816698344.80977258170998349.65190022109101354.244880298184358.555893905554362.55860436475825366.23360502458195369.57353438557863372.58205531818169375.27369143593802377.67576437684738379.82263394381602381.75580512197001383.52233064807575385.17860491910398386.79344689733102388.44547664695301390.21514038049469392.17814407401499394.399511096781396.92831047905185399.79441746346464403.00201778433501406.53024695107575410.33293183325202414.34589760572567418.48667104377699422.65568308616525426.74099005922898430.62342233323687434.19221049214894437.35054535409802440.02174242139063442.14727510665432443.68940418642978444.62834073499903444.95560403836708444.66781944848384443.77144760528364442.28606397755169440.24731934607235437.70373114983403434.71310493189623431.33995023011198427.65420518162887423.72477276975098419.61599807241464415.38571861936401411.08072043721569406.73782323898195402.382930069022Y19811981.251981.51981.7519821982.251982.51982.7519831983.251983.51983.7519841984.251984.51984.7519851985.251985.51985.7519861986.251986.51986.7519871987.251987.51987.7519881988.251988.51988.7519891989.251989.51989.7519901990.251990.51990.7519911991.251991.51991.7519921992.251992.51992.7519931993.251993.51993.7519941994.251994.51994.7519951995.251995.51995.7519961996.251996.51996.7519971997.251997.51997.7519981998.251998.51998.7519991999.251999.51999.7520002000.252000.52000.7520012001.252001.52001.7520022002.252002.52002.7520032003.252003.52003.7520042004.252004.52004.7520052005.252005.52005.7520062006.252006.52006.7520072007.252007.52007.7520082008.252008.5257.39951449300003249.82180317800001241.547748614232.65074736300082228.53608412499995235.58234483300103241.433117783250.28656347099999264.97793252799869265.53150385199734266.825672532266.54938127200001269.51649812599999269.59548704199869269.34432646900001269.62175278699863265.78988357499998267.34341449300001269.35010310500002272.89515505499776273.53231046999775276.37773213899999278.35138028400002281.67568070200002285.81712906999923292.47365253300001298.19137848499764303.20957585500003318.85410306199969323.88487690599999321.58845080499964312.42920792999894304.83548219099998290.39540530900001286.32861883799899278.05174378599969261.67510156499969258.21906179500002254.23486058499998255.52447141200079247.61130014400001248.91796802000002247.348622879251.13232772000001248.39081379300001247.56450541699999249.59280648000103260.80338647399969275.52154786499869282.851955019288.53896447599863298.39397962199899304.13093442099893307.67513996099734309.36642090399999312.53422928799893306.29908137000001316.787492085326.06222396999999338.43390869599864354.2363520929967353.21288326199999360.77284532900217364.61644668000002369.64393748399999381.830847178383.87259683399998386.95072337699969393.058894436386.53475529199869389.10805591799863388.69903658799899396.70246141699999403.40163517499963400.86036796699869389.71528423000001380.86109830100003373.12296951500002366.65120166399998363.68945088599997355.82509718599999359.71405761099999362.50819401499899377.72753544499687381.05549250399997386.99255914499764398.68729120299969411.09415334699764440.18133198799717452.95979699199899466.39170517899805466.40320186500003473.28152468399969470.86775149499999461.78254293399999452.79804626399869460.692064786465.65154564099993469.49194678199734462.87271710700003455.10828587499998448.43012279899688443.39886381299999429.68370409099964420.36155656999864413.31269567900193401.93359306399793399.43961684999869393.61405059400181384.48650702699734383.18937530599999Z119811981.251981.51981.7519821982.251982.51982.7519831983.251983.51983.7519841984.251984.51984.7519851985.251985.51985.7519861986.251986.51986.7519871987.251987.51987.7519881988.251988.51988.7519891989.251989.51989.7519901990.251990.51990.7519911991.251991.51991.7519921992.251992.51992.7519931993.251993.51993.7519941994.251994.51994.7519951995.251995.51995.7519961996.251996.51996.7519971997.251997.51997.7519981998.251998.51998.7519991999.251999.51999.7520002000.252000.52000.7520012001.252001.52001.7520022002.252002.52002.7520032003.252003.52003.7520042004.252004.52004.7520052005.252005.52005.7520062006.252006.52006.7520072007.252007.52007.7520082008.252008.500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000010000000100000001000000010000000100000001000000010000000100000001000000010000000100000001000000010000000Z219811981.251981.51981.7519821982.251982.51982.7519831983.251983.51983.7519841984.251984.51984.7519851985.251985.51985.7519861986.251986.51986.7519871987.251987.51987.7519881988.251988.51988.7519891989.251989.51989.7519901990.251990.51990.7519911991.251991.51991.7519921992.251992.51992.7519931993.251993.51993.7519941994.251994.51994.7519951995.251995.51995.7519961996.251996.51996.7519971997.251997.51997.7519981998.251998.51998.7519991999.251999.51999.7520002000.252000.52000.7520012001.252001.52001.7520022002.252002.52002.7520032003.252003.52003.7520042004.252004.52004.7520052005.252005.52005.7520062006.252006.52006.7520072007.252007.52007.7520082008.252008.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000001000000010000000100000001000000010000000

X19811981.251981.51981.7519821982.251982.51982.7519831983.251983.51983.7519841984.251984.51984.7519851985.251985.51985.7519861986.251986.51986.7519871987.251987.51987.7519881988.251988.51988.7519891989.251989.51989.7519901990.251990.51990.7519911991.251991.51991.7519921992.251992.51992.7519931993.251993.51993.7519941994.251994.51994.7519951995.251995.51995.7519961996.251996.51996.7519971997.251997.51997.7519981998.251998.51998.7519991999.251999.51999.7520002000.252000.52000.7520012001.252001.52001.7520022002.252002.52002.7520032003.252003.52003.7520042004.252004.52004.7520052005.252005.52005.7520062006.252006.52006.7520072007.25239.11501412204649241.16791022366795243.23223413801901245.3248223609308247.46145858477701249.65000520506098251.88649625825056254.15817349307256.44574529671098258.72750030009303260.98705975115234263.21229740005964265.39473612995732267.52798450141108269.60822717624302271.63294100535717273.59943790170894275.50377278563394277.33711960599669279.08555208773993280.73015207048348282.24813239571705283.61220725391001284.79142183538602285.75153331360599286.45635152382499286.86972729864397286.95927228379901286.69967415701097286.07577703573901285.09252155550303283.77847903924601282.18503076568823280.38146471911102278.45122541593003276.484016085436274.57446332780711272.81817357302697271.30269114998998270.10643594271926269.29716044114008268.93350340734366269.06054994073895269.71087543110895270.90348531383387272.64577343196498274.93106320886199277.73700227536898281.02540185187769284.74748964890205288.8510534682116293.28269640244395297.98882648859654302.91904631570202308.02679729022663313.26849337718699318.60138580634901323.98226689241704329.36024000983514334.67991179877993339.88382763941587344.91687915995601349.73292831640703354.29102206733199358.55710731943338362.503584369789366.10978278433669369.36711166825694372.27808188553098374.85619405744194377.12793681312405379.127097882481380.89495256985708382.47875864128332383.93565355583695385.33585157040301386.76014488839195388.29206285861869390.011447925786391.98866185125195394.27946624246403396.9219357000116399.93011084383699403.29477737005897406.98333227678648410.94719303593405415.12157221954197419.42671075347613423.77257813797695428.06393602488896432.21580153722198436.15875171109099439.84472352238402443.24455672833705446.34998908691381449.1700228528451.72330562684499454.03075252452493456.11888413579567458.02148404629366459.78069400584138461.44168778489097463.04671889883269464.62990863493837Y19811981.251981.51981.7519821982.251982.51982.7519831983.251983.51983.7519841984.251984.51984.7519851985.251985.51985.7519861986.251986.51986.7519871987.251987.51987.7519881988.251988.51988.7519891989.251989.51989.7519901990.251990.51990.7519911991.251991.51991.7519921992.251992.51992.7519931993.251993.51993.7519941994.251994.51994.7519951995.251995.51995.7519961996.251996.51996.7519971997.251997.51997.7519981998.251998.51998.7519991999.251999.51999.7520002000.252000.52000.7520012001.252001.52001.7520022002.252002.52002.7520032003.252003.52003.7520042004.252004.52004.7520052005.252005.52005.7520062006.252006.52006.7520072007.25257.39951449300003249.82180317800001241.547748614232.65074736300082228.53608412499995235.58234483300103241.433117783250.28656347099999264.97793252799869265.53150385199734266.825672532266.54938127200001269.51649812599999269.59548704199869269.34432646900001269.62175278699863265.78988357499998267.34341449300001269.35010310500002272.89515505499776273.53231046999775276.37773213899999278.35138028400002281.67568070200002285.81712906999923292.47365253300001298.19137848499764303.20957585500003318.85410306199969323.88487690599999321.58845080499964312.42920792999894304.83548219099998290.39540530900001286.32861883799899278.05174378599969261.67510156499969258.21906179500002254.23486058499998255.52447141200079247.61130014400001248.91796802000002247.348622879251.13232772000001248.39081379300001247.56450541699999249.59280648000103260.80338647399969275.52154786499869282.851955019288.53896447599863298.39397962199899304.13093442099893307.67513996099734309.36642090399999312.53422928799893306.29908137000001316.787492085326.06222396999999338.43390869599864354.2363520929967353.21288326199999360.77284532900217364.61644668000002369.64393748399999381.830847178383.87259683399998386.95072337699969393.058894436386.53475529199869389.10805591799863388.69903658799899396.70246141699999403.40163517499963400.86036796699869389.71528423000001380.86109830100003373.12296951500002366.65120166399998363.68945088599997355.82509718599999359.71405761099999362.50819401499899377.72753544499687381.05549250399997386.99255914499764398.68729120299969411.09415334699764440.18133198799717452.95979699199899466.39170517899805466.40320186500003473.28152468399969470.86775149499999461.78254293399999452.79804626399869460.692064786465.65154564099993469.49194678199734462.87271710700003455.10828587499998448.43012279899688443.39886381299999429.68370409099964Z119811981.251981.51981.7519821982.251982.51982.7519831983.251983.51983.7519841984.251984.51984.7519851985.251985.51985.7519861986.251986.51986.7519871987.251987.51987.7519881988.251988.51988.7519891989.251989.51989.7519901990.251990.51990.7519911991.251991.51991.7519921992.251992.51992.7519931993.251993.51993.7519941994.251994.51994.7519951995.251995.51995.7519961996.251996.51996.7519971997.251997.51997.7519981998.251998.51998.7519991999.251999.51999.7520002000.252000.52000.7520012001.252001.52001.7520022002.252002.52002.7520032003.252003.52003.7520042004.252004.52004.7520052005.252005.52005.7520062006.252006.52006.7520072007.25000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000100000001000000010000000100000001000000010000000100000001000000019811981.251981.51981.7519821982.251982.51982.7519831983.251983.51983.7519841984.251984.51984.7519851985.251985.51985.7519861986.251986.51986.7519871987.251987.51987.7519881988.251988.51988.7519891989.251989.51989.7519901990.251990.51990.7519911991.251991.51991.7519921992.251992.51992.7519931993.251993.51993.7519941994.251994.51994.7519951995.251995.51995.7519961996.251996.51996.7519971997.251997.51997.7519981998.251998.51998.7519991999.251999.51999.7520002000.252000.52000.7520012001.252001.52001.7520022002.252002.52002.7520032003.252003.52003.7520042004.252004.52004.7520052005.252005.52005.7520062006.252006.52006.7520072007.2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000

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)

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)

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)

-20

-15

-10

-5

0

5

10

15

19651970197519801985

percent

year

0.257

0.006

43.559

(0.00)

0.843

(0.97)

42.715

(0.00)

-10

-5

0

5

10

15

20

19651970197519801985

bn Euro

year

0.843

(0.97)

0.205

0.009

34.276

(0.00)

-4

-2

0

2

4

6

8

10

19651970197519801985

year

percent

1.267

(0.91)

33.010

(0.00)

r2

£

100

200

300

400

500

600

700

19701980199020002010

bn Euro

year

0.448

0.187

0.008

116.587

(0.00)

31.087

(0.00)

-50

-40

-30

-20

-10

0

10

20

30

40

50

20

40

60

80

100

1.204

(0.92)

85.500

(0.00)

29.883

(0.00)

-50

-40

-30

-20

-10

0

10

20

30

40

50

20

40

60

80

100

0.384

0.177

0.008

98.883

(0.00)

29.172

(0.00)

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

20

11

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

5

10

15

20

c)

Estimated Autocorrelations

r(t)

ˆ

t

year

year

percent

1.173

(0.93)

69.711

(0.00)

27.999

(0.00)

0

2

4

6

8

10

12

14

1990 1995 2000 2005

Australia

0

2

4

6

8

10

12

14

1990 1995 2000 2005

Canada

0

2

4

6

8

10

12

14

1990 1995 2000 2005

Denmark

0.5

1

1.5

2

2.5

3

3.5

1974

1978

1982

1986

1990

1994

1998

2002

2006

2010

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

-15

-10

-5

0

5

10

15

20

1974

1978

1982

1986

1990

1994

1998

2002

2006

2010

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

5

10

15

20

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

5

10

15

20

ˆ

CHF/USD

��

��

b) Continuous Returns CHF/USD

a) Exchange Rate CHF/USD 1974 – 2011 2011

��

c) Estimated Autocorrelations

�

�

year

year

percent

0

2

4

6

8

10

12

14

1990 1995 2000 2005

Germany

0

2

4

6

8

10

12

14

1990 1995 2000 2005

Japan

0

2

4

6

8

10

12

14

1990 1995 2000 2005

New Zealand

0

2

4

6

8

10

12

14

1990 1995 2000 2005

Norway

0

2

4

6

8

10

12

14

1990 1995 2000 2005

Sweden

0

2

4

6

8

10

12

14

1990 1995 2000 2005

Switzerland

0

2

4

6

8

10

12

14

1990 1995 2000 2005

United Kingdom

0

2

4

6

8

10

12

14

1990 1995 2000 2005

United States

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

199019921994199619982000200220042006

j

ˆ

b

-

0.4

-

0.2

0

0.2

0.4

0.6

0.8

1

20

15

10

5

t

t

r(t)

ˆ

t

-

7.5

-

5

-

2.5

0

2.5

5

7.5

r(t)

-

0.4

-

0.2

0

0.2

0.4

0.6

0.8

1

5

10

15

20

c) Estimated autoc

orrelation function

with confidence intervals

b)

Theoretical autocorrelation function

a)

Realisation

x

t

i

b

*

i

b

i,ec

b

%

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

b) Theoretical autocorrelation function

5

10

15

20

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


a) Realisation

xt

2000

3000

4000

5000

6000

7000

2 January 1996

19 May 1999

22 July 1998

8 October 1998

28 October 1997

31 July 1997

-.12

-.08

-.04

.00

.04

.08

2 January 1996

19 May 1999

0

50

100

150

-0.075-0.050-0.0250.0000.0250.050

ˆ

()

rt

t

0.0

0.4

0.8

02468101214161820

t

t

t

-

4

-

2

0

2

4

c) Est

imated autocorrelation function


r(t)

r(t)

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

x

t

0.0

0.4

0.8

02468101214161820

.0

.1

.2

.3

.4

.5

-5-4-3-2-1012345

Standard

normal distribution

Modified t-distribution

t

-4

-2

0

2

4

20

15

10

5

-1

1

0.8

0.6

0.4

0.2

0

-0.2

ˆ

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-0.4

-0.6

-0.8

-1

c) Estimated autocorrelation function�with confidence intervals

0.8


a) Realisation

1

xt

5

10

15

20

-

5

-

2.5

0

2.5

5

t

t

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

r(t)

ˆ

r(t)

-

1

-

0.8

-

0.6

-

0.4

-

0.2

0

0.2

0.4

0.6

0.8

1

10

15

20

20

5

x

t

-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

xt

10

15

20

-5

-2.5

0

2.5

5

t

a) Realisation



5

ˆ

40

42

44

46

48

50

52

54

56

1971

1973

1975

1977

1979

1981

a) Popularity of the CDU/CSU, 1971

–

1982

t

t

r

)

(

ˆ

t

r

)

(

ˆ

t

-

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 function

s

with

confidence intervals

year

Percent

20

15

-0.6

10

0

5

-0.4

1

-0.2

0.8

0.6

0.8

0.4

0.4

0.2

0.2

0

0.6

-0.2

-0.4

-0.6

5

-0.8

10

-1

1

-0.8

ˆ

15

Percent

(

20

)

b)Autocorrelation (__) and partial ( )

autocorrelation functions with confidence intervals

c)Estimated autocorrelation function of the�residuals of the estimated AR(1)-process


ˆ

(

)

-1

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

year

1981

1979

1977

1975

1973

1971

56

54

52

50

48

46

44

42

40

-

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

t

t



r(t)

ˆ

r(t)

a) Realisation

b)


x

t

-10

-5

0

5

10

-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

5

10

15

20

t


xt

ˆ

a) Realisation


-

1

-

0.8

-

0.6

-

0.4

-

0.2

0

0.2

0.4

0.6

0.8

1

5

10

15

20

-

1

r(t)

t

t

b)


c)

Estimated autocorrelation function

with confidence

intervals

-

0.8

-

0.6

-

0.4

-

0.2

0

0.2

0.4

0.6

0.8

1

5

10

15

20

t

r(t)

ˆ

-

5

-

2.5

0

2.5

5

a)

Realisation

x

t

-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

t

b)Theoretical autocorrelation function

c)Estimated autocorrelation function�with confidence intervalssnfidence intervass with α= 0. ��

xt

ˆ

-5

-2.5

0

2.5

5

a)Realisation

-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

197019751980198519901995

year

Percent

0

2

4

6

8

10

12

14

16

197019751980198519901995

b)Estimated autocorrelation (__) and partial autocorrelation (�) functions with confidence intervals

Percent

year

20

15

10

5

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

10

15

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

20

5

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

f

kk

-

1

-

0.8

-

0.6

-

0.4

-

0.2

0

0.2

0.4

0.6

0.8

1

k

AR(1) pro

cess with

a

=

-

0.9

AR(2) process with

a

1

= 1.4,

a

2

=

-

0.85

f

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

a

1

= 1.5,

a

2

=

-

0.56

f

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

a

= 0.9

f

kk

5

10

15

20

5

10

15

20

5

10

15

20

5

10

15

20

0.6

0.4

-0.4

0.2

0.2

0

-0.2

-0.2

0

-0.4

-0.6

1

-0.8

0.6

-1

0.4

(kk

0.8

AR(1) process with = 0.9

k

(kk

5

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

10

k

-1

0.6

-0.6

0.4

15

-0.4

0.2

20

-0.2

0

-0.6

0

-0.2

-0.8

0.2

-0.4

-0.6

1

-0.8

0.4

20

0.6

15

0.8

10

-0.8

5

10

-1

5

1

20

15

10

5

-1

k

0.8

k

1

AR(1) process with = -0.9

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

0.8

(kk

(kk

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

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5

10

15

20

t

t

t

a = 0.5, b = 0.8

a = -0.5, b = -0.8

a = 0.8, b = -0.5

a = -0.8, b = 0.5

t

r(t)

r(t)

r(t)

r(t)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

5101520

15

10

5

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

20

15

10

5

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

20

-0.6

-0.8

-1

20

15

10

5

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

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

a)New York three month money market rate,

1994 – 2003

year

Percent

b)Autocorrelation (__) and partial ( )

autocorrelation functions of the first differences with confidence intervals

2000

1998

1994

1996

2002

8

7

6

5

4

3

2

1

0

20

15

10

5

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

ˆ

c)Autocorrelation function of the residuals

of the estimated ARMA(1,1)-process


20

15

10

5

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

-6

-4

-2

0

2

4

6

8

1970197219741976197819801982198419861988

Growth Rate of Real GDP

Interest Rate Spread (t-4)

(GLR - GSR)

percent

year

ˆ

ρ(k)

ˆ

r

(0)

*

1

k

*

2

k

*

1

k

ˆ

r

R

-0.4

0.0

0.4

0.8

1.2

2468101214161820

Response of X1 to X1

-0.4

0.0

0.4

0.8

1.2

2468101214161820


-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2468101214161820


-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2468101214161820


-4

-3

-2

-1

0

1

2

3

5101520

Accumulated Response of X1 to X1

-4

-3

-2

-1

0

1

2

3

5101520


-1

0

1

2

3

4

5

6

5101520


-1

0

1

2

3

4

5

6

5101520


-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

5101520

Response of DLGDPR to DLGDPR

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

5101520

Response of DLGDPR to DLM1R

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

5101520

Response of DLGDPR to GLSR

-2

-1

0

1

2

3

5101520

Response of DLM1R to DLGDPR

-2

-1

0

1

2

3

5101520

Response of DLM1R to DLM1R

-2

-1

0

1

2

3

5101520

Response of DLM1R to GLSR

-0.8

-0.4

0.0

0.4

0.8

1.2

5101520

Response of GLSR to DLGDPR

-0.8

-0.4

0.0

0.4

0.8

1.2

5101520

Response of GLSR to DLM1R

-0.8

-0.4

0.0

0.4

0.8

1.2

5101520

Response of GLSR to GLSR

-4

-2

0

2

4

6

8

5101520

Accummulated Response

of DLGDPR to DLGDPR

-4

-2

0

2

4

6

8

5101520


of DLGDPR to DLM1R

-4

-2

0

2

4

6

8

5101520


of DLGDPR to GLSR

-4

-2

0

2

4

6

8

5101520


of DLM1R to DLGDPR

-4

-2

0

2

4

6

8

5101520


of DLM1R to DLM1R

-4

-2

0

2

4

6

8

5101520


of DLM1R to GLSR

-4

-2

0

2

4

6

8

5101520


of GLSR to DLGDPR

-4

-2

0

2

4

6

8

2468101214161820


of GLSR to DLM1R

-4

-2

0

2

4

6

8

5101520


of GLSR to GLSR

0

100

200

300

400

500

255075100

-100

0

100

200

300

400

255075100

0

20

40

60

80

100

120

140

160

255075100

-12

-8

-4

0

4

8

12

-8

-6

-4

-2

0

2

4

6

8

O

r

i

g

i

n

a

l

r

e

s

i

d

u

a

l

s

F

i

r

s

t

d

i

f

f

e

r

e

n

c

e

s

o

f

t

h

e

m

o

d

e

l

w

i

t

h

a

l

i

n

e

a

r

t

r

e

n

d

-8

-6

-4

-2

0

2

4

6

8

-8

-6

-4

-2

0

2

4

6

8

O

r

i

g

i

n

a

l

r

e

s

i

d

u

a

l

s

F

i

r

s

t

d

i

f

f

e

r

e

n

c

e

s

o

f

t

h

e

r

a

n

d

o

m

w

a

l

k

-8

-6

-4

-2

0

2

4

6

8

-8

-6

-4

-2

0

2

4

6

8

O

r

i

g

i

n

a

l

r

e

s

i

d

u

a

l

s

R

e

s

i

d

u

a

l

s

o

f

t

h

e

m

o

d

e

l

w

i

t

h

a

t

i

m

e

t

r

e

n

d

-16

-12

-8

-4

0

4

8

12

16

-8

-6

-4

-2

0

2

4

6

8

O

r

i

g

i

n

a

l

r

e

s

i

d

u

a

l

s

R

e

s

i

d

u

a

l

s

o

f

t

h

e

m

o

d

e

l

w

i

t

h

a

s

t

o

c

h

a

s

t

i

c

t

r

e

n

d

-8

-4

0

4

8

0

20

40

60

80

100

120

102030405060708090100

Actual and estimated values

Residuals

Model with a linear trend

-20

-10

0

10

20

0

50

100

150

102030405060708090100

Residuals

Actual and estimated values

Model with a random walk with drift

2

R

100

200

300

400

500

600

700

196019701980199020002010

year

bn Euro

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

0.850.90.951

0

5

10

15

20

25

0.850.90.951


Density of the estimated coefficient compared with a�normal distribution with the same variance and = 1

0.1

0.2

0.3

0.4

0.5

0.6

-3.47

-1.96

-1.65

0

-2.88

0

2

4

6

8

10

12

14

1985199019952000

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

0.80.850.90.951

0

2

4

6

8

10

12

14

16

0.80.850.90.951

Density of the estimated coefficient compared with a�normal distribution with the same variance and μ = 0.95

Density of the Dickey-Fuller t statistic

-2.88

-3.47

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

2

4

6

8

10

12

0.70.750.80.850.90.951

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

0

2

4

6

8

10

12

0.70.750.80.850.90.951

Density of the Dickey-Fuller t statistic

Density of the estimated coefficient compared with a�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

0

1

2

3

4

5

6

7

19751980198519901995

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

19751980198519901995

-40

-30

-20

-10

0

10

20

30

19651970197519801985

bn Euro

year

0

0.01

0.02

0.03

0.04

0.05

-60

-40

-20

0

20

40

60

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 R

2

0

1

2

3

4

5

6

7

8

9

10

0

0.1

0.2

0.3

0.4

Density

of

the Durbin

-

Watson statistic

0

0.01

0.02

0.03

0.04

0.05

-60-40-200204060


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

00.20.40.60.81

Density of the R2

0

1

2

3

4

5

6

7

8

9

10

00.10.20.30.4

Density of the Durbin-Watson statistic

4.8

5.2

5.6

6.0

6.4

19651970197519801985

ln(M )

t

year

5.2

5.6

6.0

6.4

6.8

19651970197519801985

ln(Y )

t

year

4

6

8

10

12

19651970197519801985

percent

year

3

4

5

6

7

8

9

10

198719891991199319951997

percent

year

r0

=

r1

£

2. analyse univariater stationÄrer prozesse · web viewintroduction to modern time series analysis...

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