by Słren Johansen Department of Economics, University of...

31
1 T HE COINTEGRATED VECTOR AUTOREGRESSIVE MODEL WITH AN APPLICATION TO THE ANALYSIS OF SEA LEVEL AND TEMPERATURE by Slren Johansen Department of Economics, University of Copenhagen and CREATES, Aarhus University

Transcript of by Słren Johansen Department of Economics, University of...

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THE COINTEGRATED VECTOR AUTOREGRESSIVE MODELWITH AN APPLICATION TO THE ANALYSIS OF

SEA LEVEL AND TEMPERATURE

by Søren Johansen

Department of Economics, University of Copenhagenand

CREATES, Aarhus University

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CONTENT OF LECTURES

1. AN EXAMPLE: SEA LEVEL AND TEMPERATURE 1881-19952. A SIMPLE EXAMPLE AND SOME LITERATURE3. THE VECTOR AUTOREGRESSIVE MODEL AND ITS SOLUTION4. HYPOTHESES OF INTEREST IN THE I(1) MODEL5. THE STATISTICAL ANALYSIS6. THE ANALYSIS OF THE DATA7. ASYMPTOTIC ANALYSIS8. CONCLUSION

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1. AN EXAMPLE: SEA LEVEL AND TEMPERATURE 1881-1995Data in levels and differences

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Sea Level

1881 1890 1899 1908 1917 1926 1935 1944 1953 1962 1971 1980 1989 1998­75

­50

­25

0

25

50

75

100

125

Change of Sea Level

1882 1891 1900 1909 1918 1927 1936 1945 1954 1963 1972 1981 1990 1999­15

­10

­5

0

5

10

15

Temperature

1881 1890 1899 1908 1917 1926 1935 1944 1953 1962 1971 1980 1989 1998­0.4

­0.2

0.0

0.2

0.4

0.6

Change of Temperature

1882 1891 1900 1909 1918 1927 1936 1945 1954 1963 1972 1981 1990 1999­0.3

­0.2

­0.1

­0.0

0.1

0.2

0.3

1.Data : 1881:01 to 1995:01, Hansen, J. et al J. Geophys. Res. Atmos. 106 23947(2001)

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2. A SIMPLE EXAMPLE AND SOME LITERATURE

The mathematical formulation of cointegration by a simple example.Take two stochastic processes which have a stochastic trend (random walk)

X1t = a

tXi=1

"1i + "2t � I(1)

X2t = btXi=1

"1i + "3t � I(1)

bX1t � aX2t = b"2t � a"3t � I(0)

1. X(t) is nonstationary and �Xt is stationary: Xt is I(1)2. �0Xt is stationary with � = (b;�a)03. The common stochastic trend is

Pti=1 "1i

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The cointegrated vector autoregressive model

�Xt = ��0Xt�1 +

k�1Xi=1

�i�Xt�i + "t

Literature1. Johansen S. (1996) `Likelihood-Based Inference in Cointegrated Vector Autoregres-sive Models.' Oxford University Press, Oxford. www.math.ku.dk/~sjo

2. Juselius, K. (2006) `The Cointegrated VAR Model: Methodology and Applications.'Oxford University Press, Oxford. www.econ.ku.dk/~katarina.juselius

3. Dennis, J., Johansen, S. and Juselius, K. (2006), `CATS for RATS: Manual to Coin-tegration Analysis of Time Series.' Estima, Illinois.See alsoJohansen, S. (2006) Cointegration: a survey. In: T.C. Mills and K. Patterson (eds.)Palgrave Handbook of Econometrics: Volume 1, Econometric Theory, Basingstoke,Palgrave Macmillan.

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3. THE VECTOR AUTOREGRESSIVE MODEL AND ITS SOLUTIONError correction formulation of the vector autoregressive model

�Xt = ��0Xt�1 +

k�1Xi=1

�i�Xt�i + "t

�(z) = (1� z)Ip � ��0z �k�1Xi=1

(1� z)zi�i

det �(1) = det��0 = 0 =) z = 1 is a root of det �(z) = 0

Question: If the VAR has unit roots and the other roots larger than one, what is themoving average representation and what are the properties of the process?

I(1) condition : det(�(z)) = 0 =) z = 1 or jzj > 1 and

� = Ip �k�1Xi=1

�i; det(�0?��?) 6= 0

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�Xt = ��0Xt�1 +k�1Xi=1

�i�Xt�i + "t

�(z) = (1� z)Ip � ��0z �k�1Xi=1

(1� z)zi�i

THEOREM If det(�(z)) = 0 =) z = 1 or jzj > 1 and det(�0?��?) 6= 0; then

�(z)�1 = C1

1� z +1Xi=0

C�i zi

Xt = CtXi=1

"i +1Xi=0

C�i "t�i + A; �0A = 0

C = �?(�0?��?)

�1�0?

1. Xt is nonstationary, �Xt is stationary: Xt is called I(1)2. �0Xt is stationary: Xt is cointegrating (r relations)3. The common trends are �0?

Pti=1 "i (p� r trends)

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Example

�X1t = �14(X1t�1 �X2t�1) + "1t

�X2t =1

4(X1t�1 �X2t�1) + "2t

�(X1t �X2t) = �12(X1t�1 �X2t�1) + "1t � "2t =) X1t �X2t stationary (= yt)

�(X1t +X2t) = "1t + "2t =) X1t +X2t nonstationary random walk (= St)

Granger Representation Theorem

X1t =1

2(St + yt)

X2t =1

2(St � yt)

1. Xt is nonstationary, �Xt is stationary2. �0Xt is stationary with cointegration vector � = (1;�1)03.Pt

i=1("1i + "2i) is a common trend

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Two applications of the Granger Representation Theorem1. The role of deterministic terms

�Xt = ��0Xt�1 +k�1Xi=1

�i�Xt�i + � + "t

Xt = C

tXi=1

("i + �) +

1Xi=0

C�i ("t�i + �) + A

Xt = CtXi=1

"i + C�t + stationary process

Thus1. Linear trend in general2. If C� = �?(�0?��?)

�1�0?� = 0 (or �0?� = 0) : only constant term

Other deterministics.

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2. Asymptotics

Xt = CtXi=1

"i + C�t + stationary process

X[Tu]pT

= C

P[Tu]i=1 "ipT

# dCW (u)

+ C�[Tu]pT+stationary processp

T# P0

The process X[Tu] is proportional to T in the direction C�; but orthogonal to this direc-tion it behaves like T 1=2:

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-

6

Tt

Sealevelt

�0 ��������������

sp(�?)XXXXz

��XtbX1;tb

XXXXy �

��������������������������0Xt�������������

�����

��

���0?Pt

i=1 "i

2.The process X 0t = [SeaLevelt; Tt] is pushed along the attractor set by the common

trends and pulled towards the attractor set by the adjustment coef�cients

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4. HYPOTHESES OF INTEREST IN THE I(1) MODEL1. Hypotheses on the rank

Hr : �Xt = ��0Xt�1 + �1�Xt�1 + �Dt + "t; �p�r; �p�r

H0 � : : : � Hr � : : : � Hp

2. Hypotheses on the long run relations �3. Hypotheses on �

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5. THE STATISTICAL ANALYSISTest for misspeci�cation of the VAR

�Xt = �Xt�1 +k�1Xi=1

�i�Xt�i + �Dt + "t

The VAR model assumes

1. Linear conditional mean explained by the past observations and deterministic terms(Check for unmodelled systematic variation, the choice of lag length, choice of infor-mation set (data), possible outliers, nonlinearity, constant parameters)

2. Constant conditional variance(Check for ARCH effects, but also for regime shifts in the variance)

3. Independent Normal errors, mean zero, variance (Check for lack of autocorrelation, distributional form)

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Estimation of the I(1) model

Hr : �Xt = ��0Xt�1 +

k�1Xi=1

�i�Xt�i + �Dt + "t;

where "t i.i.d. Np(0;) and � and � are (p� r):Maximum likelihood is calculated by reduced rank regression of �Xt onXt�1 correctedfor

�Xt�1; : : : ;�Xt�k+1; Dt

(T. W. Anderson, 1951). The estimate of � are the r linear combinations of the datawhich have the largest empirical correlation with the stationary process �Xt: (Canoni-cal variates and canonical correlations)If �i, are the squared canonical correlations, then

L�2=Tmax (Hr) _ jj _rYi=1

(1� �i); � 2 logLR(HrjHp) = �TpX

i=r+1

log(1� �i)

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6. THE ANALYSIS OF THE DATA

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DLEVEL

1883 1890 1897 1904 1911 1918 1925 1932 1939 1946 1953 1960 1967 1974 1981 1988 1995 2002­15

­10

­5

0

5

10

15Actual and Fitted

1883 1890 1897 1904 1911 1918 1925 1932 1939 1946 1953 1960 1967 1974 1981 1988 1995 2002­3

­2

­1

0

1

2

3Standardized Residuals

Lag2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

­1.00

­0.75

­0.50

­0.25

0.00

0.25

0.50

0.75

1.00Autocorrelations

­3.2 ­1.6 0.0 1.6 3.20.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45SB­DH: ChiSqr(2) = 0.07 [0.96]K­S = 0.95 [5% C.V. = 0.08]J­B: ChiSqr(2) = 0.29 [0.86]

Histogram

3.Residual analysis: Plot of �SeaLevelt; and �tted value, normalized residuals, auto-correlations function of residuals and histogram

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DTEMP

1883 1890 1897 1904 1911 1918 1925 1932 1939 1946 1953 1960 1967 1974 1981 1988 1995 2002­0.3

­0.2

­0.1

­0.0

0.1

0.2

0.3Actual and Fitted

1883 1890 1897 1904 1911 1918 1925 1932 1939 1946 1953 1960 1967 1974 1981 1988 1995 2002­3

­2

­1

0

1

2

3Standardized Residuals

Lag2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

­1.00

­0.75

­0.50

­0.25

0.00

0.25

0.50

0.75

1.00Autocorrelations

­3.2 ­1.6 0.0 1.6 3.20.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40SB­DH: ChiSqr(2) = 1.47 [0.48]K­S = 0.89 [5% C.V. = 0.08]J­B: ChiSqr(2) = 1.34 [0.51]

Histogram

4.Residual analysis: Plot of �Temperaturet; and �tted value, normalized residuals,autocorrelations function of residuals and histogram

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Roots of the Companion Matrix

­1.0 ­0.5 0.0 0.5 1.0 1.5

­1.0

­0.5

0.0

0.5

1.0Rank(PI)=2

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The two eigenvectors for temperature sea level

1881 1890 1899 1908 1917 1926 1935 1944 1953 1962 1971 1980 1989­0.36

­0.24

­0.12

0.00

0.12

0.24

1881 1890 1899 1908 1917 1926 1935 1944 1953 1962 1971 1980 1989­2.4

­1.2

0.0

1.2

2.4

3.6

5.The �rst canonical variate is Tt�1 � 0:0031(t=�7:37)

ht�1

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Temperature versus Sea Level

Sea Level

Tem

pera

ture

­80 ­40 0 40 80 120

­0.4

­0.2

0.0

0.2

0.4

0.6

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Rank determination for temperature and sea level data.Rank r = 0 is rejected and rank r = 1 is acceptedp� r r EigVal �2 logLR(HrjHp) 95%Fract p-val2 0 0:168 20:76 15:41 0:0051 1 0:003 0:36 3:84 0:54

The �tted ECM model for temperature and sea level data

�ht = 4:15(t=0:86)

(Tt�1 � 0:0031(t=�7:37)

ht�1)� 0:2805(t=�3:11)

�ht�1 + 3:04(t=0:60)

�Tt�1 + 2:22(t=3:55)

�Tt = �0:40(t=�4:26)

(Tt�1 � 0:0031(t=�7:37)

ht�1)� 0:0024(t=�1:40)

�ht�1 � 0:053(t=�0:54)

�Tt�1 � 0:023(t=�1:91)

Note ht weakly and strongly exogenous because the coef�cients 4:15 to (Tt�1� 0:0031(t=�7:37)

ht�1)

and 3:04 to �Tt�1 are insigni�cant.

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A partial model for (Tt; ht) conditional on the forcing (weakly exogenous) variablesWmgg (CO2 and Methan) and Aerosols (Sulphate)

�Ttht

�= � (T + 1h + 2Wmgg + 3Aerosol)t�1 + � � � + "t

Cointegrating relation

�0Xt = Tt � 0:0065

(t=�3:52)ht � 0:768

(t=4:68)Wmggt � 1:478

(t=3:51)Aerosolt

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Temperature against Temperature*

­0.3 ­0.2 ­0.1 ­0.0 0.1 0.2 0.3

­0.4

­0.3

­0.2

­0.1

­0.0

0.1

0.2

0.3

0.4

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1881 1892 1903 1914 1925 1936 1947 1958 1969 1980 1991­2.5

­2.0

­1.5

­1.0

­0.5

0.0

0.5

1.0

1.5

2.0TEMPWMGGSLEVELSTARAEROSOLSTARTEMPSTAR

6.Plot of the components of the �tted Temperature for simpli�ed forcing variables

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7. ASYMPTOTIC ANALYSIS

Asymptotic properties of the process Xt and its product moments for model withoutdeterministic terms.Three basic results

T�1=2X[Tu] = CT�1=2P[Tu]

i=1 "i + T�1=2P1

i=0C�i "t�i

d! CW (u)

T�2PT

t=1Xt�1X0t�1

d!R 10 W (u)W (u)

0du

T�1PT

t=1Xt�1"0t

d!R 10 W (dW )

0

whereW is Brownian motion with variance :Test for rank

�2 logLR(HrjHp) = �TpX

i=r+1

log(1� �i)d! trf

Z 1

0

(dB)B0�Z 1

0

BB0��1 Z 1

0

B(dB)0g

where B is standard Brownian motion. Limit invariant to distribution of i.i.d. (0;) errorsbut depends on the choice of deterministic terms

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4. ASYMPTOTIC DISTRIBUTION OF �Consider the model with no deterministic terms, r = 2; and � identi�ed by

� = (h1p�1+ H1p�m1

'1m1�1

; h2p�1+ H2p�m2

'2m2�1

):

THEOREM If "t i.i.d. (0;); then

T�1=2x[Tu]d! CW = G; and T�1S11 = T�2

TXt=1

xt�1x0t�1

d! C

Z 1

0

WW 0duC 0 = G

T

�'1'2

�d!��11H

01GH1 �12H 0

1GH2�21H

02GH1 �22H 0

2GH2

��1 H 01

R 10 G(dV1)

H 02

R 10 G(dV2)

!;

where �ij = �0j�1�j; and V = �0�1W = (V1; V2)0.

Note that V = �0�1W is independent of CW = �?(�0?�?)

�1�0?W: Hence limit is mixedGaussian. The estimators of the remaining parameters are asymptotically Gaussianand asymptotically independent of ' and �.

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ExampleThe simplest example is a cointegrating regression

x1t = �x2t�1 + "1t; and �x2t = "2t

where "t are i.i.d. Gaussian and f"1tg and f"2tg are independent. Then

T (� � �) = T�1PT

t=1 x2t�1"1t

T�2PT

t=1 x22t�1

d!R 10 W2dW1R 10 W

22 (u)du

� Mixed Gaussian

For given fx2t; t = 1; : : : ; Tg

�jfx2tg � N(�;�21PT

t=1 x22t�1

) and(� � �)�1

(

TXt=1

x22t�1)1=2jfx2tg � N(0; 1)

This implies that the marginal distributions satisfy

� � �qV ar(�)

not Gaussian and� � ��1

(

TXt=1

x22t�1)1=2 is Gaussian

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The mixed Gaussian distribution.A simple simulation of the model

�X1t = �1(X1t�1 � X2t�1) + "1t and �X2t = "2twhere "t are i.i.d. N2(0; diag(�21; �22)). DGP (�1 = �1; = 1; �21 = �22 = 1) : X1t =X2t�1 + "1t; �Xt = "2t. MLE by regression:

�X1t = �1X1t�1 + �2X2t�1 + "1t; �1 = �1; = ��2=�1Asymptotic results for testing using the Wald test

1

�1(

TXt=1

(X1t�1 � X2t�1)2)1=2(�1 � �1)d! N(0; 1); or

(1 + 2)1=2

�1T 1=2(�� �)

j�1j�2(

TXt=1

X22t�1)

1=2( � ) d! N(0; 1); orj�1j�2(

Z 1

0

W2(u)2du)1=2T ( � ) d! N(0; 1);

We need the distribution of �1 and the joint distribution of ; T�2PT

t=1X22t�1 to conduct

inference.

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Estimate of gamma against information per observation

Information per observation

gam

ma

0 100 200 300 4000.85

0.90

0.95

1.00

1.05

1.10

1.15

Estimate of alpha against information per observation

Information per observation

alp

ha

0 100 200 300 400­1.28­1.20­1.12­1.04­0.96­0.88­0.80­0.72

Estimate of alpha against information per observation

Information per observation

alp

ha

3.6 4.8 6.0 7.2 8.4 9.6­1.28­1.20­1.12­1.04­0.96­0.88­0.80­0.72

7.From the model �x1t = �(x1t�1 � x2t�1) + "1t; �xt = "2t; T = 100; SIM = 400

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8. CONCLUSION

When modelling stochastically trending variables, the usual regression methods needto be modi�ed, in order to guarantee valid inference.

The vector autoregressive model allowing for I(1) variables and cointegration has beenanalysed in detail, and the challenge is to apply it to nonstationary climate data, in orderto avoid the fallacies involved in spurious correlation and regression.