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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.