Transcript Chapter 13 cointegration (EC220)
Christopher Dougherty
EC220 - Introduction to econometrics (chapter 13)
Slideshow: cointegration Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 13). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/139/ Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/
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C t P
2
Y t P v t
COINTEGRATION
log
C t P
log
2
log
Y t P
u t
9.5
0.18
0.16
9 8.5
0.14
0.12
0.1
0.08
8 7.5
DPI PCE
difference 0.06
0.04
0.02
7 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 0 In general, a linear combination of two or more time series will be nonstationary if one or more of them is nonstationary, and the degree of integration of the combination will be equal to that of the most highly integrated individual series.
1
C t P
2
Y t P v t
COINTEGRATION
log
C t P
log
2
log
Y t P
u t
9.5
0.18
0.16
9 8.5
0.14
0.12
0.1
0.08
8 7.5
DPI PCE
difference 0.06
0.04
0.02
7 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 0 Hence, for example, a linear combination of an I(1) series and an I(0) series will be I(1), that of two I(1) series will also be I(1), and that of an I(1) series and an I(2) series will be I(2).
2
C t P
2
Y t P v t
COINTEGRATION
log
C t P
log
2
log
Y t P
u t
9.5
0.18
0.16
9 8.5
0.14
0.12
0.1
0.08
8 7.5
DPI PCE
difference 0.06
0.04
0.02
7 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 0 However, if there is a long-run relationship between the time series, the outcome may be different.
3
C t P
2
Y t P v t
COINTEGRATION
log
C t P
log
2
log
Y t P
u t
9.5
9 8.5
8 7.5
DPI PCE
difference 0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
7 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 0 Consider, for example, Friedman’s Permanent Income Hypothesis and the consumption function C
t P
=
2
Y t P v t
where C
t P
and Y
t P
are permanent consumption and income, respectively, and v
t
is a multiplicative disturbance term.
4
C t P
2
Y t P v t
COINTEGRATION
log
C t P
log
2
log
Y t P
u t
9.5
9 8.5
8 7.5
DPI PCE
difference 0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
7 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 0 In logarithms, the relationship becomes log C
t P
logarithm of v
t .
= log
2 + log Y
t P
+ log v
t
where u
t
is the
5
C t P
2
Y t P v t
COINTEGRATION
log
C t P
log
2
log
Y t P
u t
9.5
0.18
0.16
9 8.5
0.14
0.12
0.1
0.08
8 7.5
DPI PCE
difference 0.06
0.04
0.02
7 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 0 If the theory is correct, in the long run, ignoring short-run dynamics and the differences between the permanent and actual measures of the variables, consumption and income will grow at the same rate and the mean of the difference between their logarithms will be log
2 .
6
C t P
2
Y t P v t
COINTEGRATION
log
C t P
log
2
log
Y t P
u t
9.5
0.18
0.16
9 8.5
0.14
0.12
0.1
0.08
8 7.5
DPI PCE
difference 0.06
0.04
0.02
7 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 0 The figure shows plots of the logarithms of aggregate disposable personal income, DPI, and aggregate personal consumer expenditure, PCE, (left scale), and their difference (right scale) for the United States for the period 1959 –2003.
7
C t P
2
Y t P v t
COINTEGRATION
log
C t P
log
2
log
Y t P
u t
9.5
0.18
0.16
9 8.5
0.14
0.12
0.1
0.08
8 7.5
DPI PCE
difference 0.06
0.04
0.02
7 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 0 It can be seen that the gap between the two has been fairly stable, increasing a little in the first part of the period and declining a little thereafter. Thus, although the series for DPI and PCE are nonstationary, they appear to be wandering together.
8
C t P
2
Y t P v t
COINTEGRATION
log
C t P
log
2
log
Y t P
u t
9.5
0.18
0.16
9 8.5
0.14
0.12
0.1
0.08
8 7.5
DPI PCE
difference 0.06
0.04
0.02
7 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 0 For this to be possible, ut must be a stationary process, for if it were not, the two series could drift apart indefinitely, violating the theoretical relationship.
9
C t P
2
Y t P v t
COINTEGRATION
log
C t P
log
2
log
Y t P
u t
9.5
0.18
0.16
9 8.5
0.14
0.12
0.1
0.08
8 7.5
DPI PCE
difference 0.06
0.04
0.02
7 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 0 When two or more nonstationary time series are linked in such a way, they are said to be cointegrated. In this example, the slope coefficient of log Y
t P
is theoretically equal to 1, making it possible to inspect the divergence graphically in the figure.
10
COINTEGRATION
Y t
1
2
X
2
t
...
k X kt
u t u t
Y t
1
2
X
2
t
...
k X kt
More generally, if there exists a relationship between a set of variables Y
t
, X 2t , …, X
kt
, the disturbance term u
t
can be thought of as measuring the deviation between the components of the model.
11
COINTEGRATION
Y t
1
2
X
2
t
...
k X kt
u t u t
Y t
1
2
X
2
t
...
k X kt
In the short run the divergence between the components will fluctuate, but if the model is genuinely correct there will be a limit to the divergence. Hence, although the time series are nonstationary, u
t
will be stationary.
12
COINTEGRATION
Y t
1
2
X
2
t
...
k X kt
u t u t
Y t
1
2
X
2
t
...
k X kt
If there are more than two variables in the model, it is possible that there may be multiple cointegrating relationships, the maximum number in theory being equal to k – 1.
13
C t P
2
Y t P v t
COINTEGRATION
log
C t P
log
2
log
Y t P
u t
9.5
0.18
0.16
9 8.5
0.14
0.12
0.1
0.08
8 7.5
DPI PCE
difference 0.06
0.04
0.02
7 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 0 To test for cointegration, it is necessary to evaluate whether the disturbance term is a stationary process. In the case of the example of consumer expenditure and income, it is sufficient to perform a standard ADF unit root test on the difference between the two series.
14
COINTEGRATION
============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================
Z
log Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ ============================================================
Y
log Augmented Dickey-Fuller test statistic -1.630141 0.7646 Test critical values1% level -4.180911 5% level -3.515523 10% level -3.188259
C t
Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): 1960 2003 ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) -0.125404 0.076928 -1.630141 0.1107 C 0.548443 0.334069 1.641706 0.1083 @TREND(1959) -0.000340 0.000143 -2.370707 0.0225 ============================================================ The results are shown in the table, with the difference between the logarithms being denoted Z. The ADF test statistic is –1.63, which is less than –3.52, the critical value at the 5 percent level under the null hypothesis of nonstationarity.
15
COINTEGRATION
============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================
Z
log Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ ============================================================
Y
log Augmented Dickey-Fuller test statistic -1.630141 0.7646 Test critical values1% level -4.180911 5% level -3.515523 10% level -3.188259
C t
Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): 1960 2003 ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) -0.125404 0.076928 -1.630141 0.1107 C 0.548443 0.334069 1.641706 0.1083 @TREND(1959) -0.000340 0.000143 -2.370707 0.0225 ============================================================ This is a surprising result, for other studies have found the logarithms of consumer expenditure and income to be cointegrated. Part of the problem is the low power of the test against an alternative hypothesis of u
t
being a stationary process with high autocorrelation.
16
COINTEGRATION
============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================
Z
log Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ ============================================================
Y
log Augmented Dickey-Fuller test statistic -1.630141 0.7646 Test critical values1% level -4.180911 5% level -3.515523 10% level -3.188259
C t
Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): 1960 2003 ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) -0.125404 0.076928 -1.630141 0.1107 C 0.548443 0.334069 1.641706 0.1083 @TREND(1959) -0.000340 0.000143 -2.370707 0.0225 ============================================================ The coefficient of the lagged residual is –0.13, suggesting that the process is approximately AR(1) with autocorrelation 0.87, but the standard error is too large for the null hypothesis of nonstationarity to be rejected.
17
COINTEGRATION
============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================
Z
log Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ ============================================================
Y
log Augmented Dickey-Fuller test statistic -1.630141 0.7646 Test critical values1% level -4.180911 5% level -3.515523 10% level -3.188259
C t
Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): 1960 2003 ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) -0.125404 0.076928 -1.630141 0.1107 C 0.548443 0.334069 1.641706 0.1083 @TREND(1959) -0.000340 0.000143 -2.370707 0.0225 ============================================================ It is likely that persistence in the way that consumers behave is responsible for this. As consumers become more savings conscious, as they seem to have done from 1959 to about 1984, the gap between the logarithms widens.
18
COINTEGRATION
============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================
Z
log Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ ============================================================
Y
log Augmented Dickey-Fuller test statistic -1.630141 0.7646 Test critical values1% level -4.180911 5% level -3.515523 10% level -3.188259
C t
Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): 1960 2003 ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) -0.125404 0.076928 -1.630141 0.1107 C 0.548443 0.334069 1.641706 0.1083 @TREND(1959) -0.000340 0.000143 -2.370707 0.0225 ============================================================ As they become less savings conscious, as seems to have been the case since 1984, it narrows.
19
COINTEGRATION
============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================
Z
log Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ ============================================================
Y
log Augmented Dickey-Fuller test statistic -1.630141 0.7646 Test critical values1% level -4.180911 5% level -3.515523 10% level -3.188259
C t
Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): 1960 2003 ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) -0.125404 0.076928 -1.630141 0.1107 C 0.548443 0.334069 1.641706 0.1083 @TREND(1959) -0.000340 0.000143 -2.370707 0.0225 ============================================================ However, these changes evidently have long cycles, and so even over a period as long as 45 years it is difficult to discriminate between the hypothesis that the gap is a random walk and the alternative that it is stationary, with strong autocorrelation.
20
COINTEGRATION
============================================================ Augmented Dickey-Fuller Unit Root Test on Z ============================================================
Z
log Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ ============================================================
Y
log Augmented Dickey-Fuller test statistic -1.630141 0.7646 Test critical values1% level -4.180911 5% level -3.515523 10% level -3.188259
C t
Augmented Dickey-Fuller Test Equation Dependent Variable: D(Z) Sample (adjusted): 1960 2003 ============================================================ Variable CoefficientStd. Errort-Statistic Prob. ============================================================ Z(-1) -0.125404 0.076928 -1.630141 0.1107 C 0.548443 0.334069 1.641706 0.1083 @TREND(1959) -0.000340 0.000143 -2.370707 0.0225 ============================================================ However, a sufficiently long time series would show that the gap is stationary, for it is not possible for it to decrease indefinitely.
21
COINTEGRATION
Y t
1
2
X
2
t
...
k X kt
u t u t
Y t
1
2
X
2
t
...
k X kt Y
ˆ
t
b
1
b
2
X
2
t
...
b k X kt e t
Y t
Y
ˆ
t
Y t
b
1
b
2
X
2
t
...
b k X kt
In the more general, where the cointegrating relationship has to be estimated, the test is an indirect one because it must be performed on the residuals from the regression, rather than on the disturbance term.
22
COINTEGRATION
Y t
1
2
X
2
t
...
k X kt
u t u t
Y t
1
2
X
2
t
...
k X kt Y
ˆ
t
b
1
b
2
X
2
t
...
b k X kt e t
Y t
Y
ˆ
t
Y t
b
1
b
2
X
2
t
...
b k X kt e
0 In view of the fact that the least squares coefficients are chosen so as to minimize the sum of the squares of the residuals, and that the mean of the residuals is automatically zero, the time series for the residuals will tend to appear more stationary than the underlying series for the disturbance term.
23
COINTEGRATION
Y t
1
2
X
2
t
...
k X kt
u t u t
Y t
1
2
X
2
t
...
k X kt Y
ˆ
t
b
1
b
2
X
2
t
...
b k X kt e t
Y t
Y
ˆ
t
Y t
b
1
b
2
X
2
t
...
b k X kt e
0 To allow for this, the critical values for the test statistic are even higher than those for the standard test for nonstationarity of a time series.
24
COINTEGRATION
Y t
1
2
X
2
t
...
k X kt
u t u t
Y t
1
2
X
2
t
...
k X kt Y
ˆ
t
b
1
b
2
X
2
t
...
b k X kt e t
Y t
Y
ˆ
t
Y t
b
1
b
2
X
2
t
...
b k X kt e
0
Asymptotic Critical Values of the Dickey-Fuller Statistic for a Cointegrating Relationship with Two Variables
Regression equation contains: 5% 1% Constant, but no trend Constant and trend –3.34
–3.78
–3.90
–4.32
Asymptotic critical values for the case where the cointegrating relationship involves two variables are shown in the table. The test assumes that a constant has been included in the cointegrating relationship, and the critical values depend on whether a trend has been included as well.
25
COINTEGRATION
X t
Y t Z t Xt
,
1 1
Z t
1
2 2
Y t
X
, ~
t
Xt
3
Zt N
Z Yt
Zt t
Yt
In the case of a cointegrating relationship, least squares estimators can be shown to be superconsistent.
26
COINTEGRATION
X t
Y t Z t Xt
,
1 1
Z t
1
2 2
Y t
X
, ~
t
Xt
3
Zt N
Z Yt
Zt t
Yt
An important consequence is that OLS may be used to fit a cointegrating relationship, even if it belongs to a system of simultaneous relationships, for any simultaneous equations bias tends to zero asymptotically.
27
COINTEGRATION
X t
Y t Z t Xt
,
1 1
Z t
1
2 2
Y t
X
, ~
t
Xt
3
Zt N
Z Yt
Zt t
Yt
If |
| < 1, Z is stationary, and X and Y are also stationary.
As an example, consider the model shown. Y
t
exogenous, and
Yt
,
Xt
, and
Zt
and X
t
are endogenous variables, Z
t
is are iid N(0,1) disturbance terms. We expect OLS estimators to be inconsistent if used to fit either of the first two equations.
28
COINTEGRATION
X t
Y t Z t Xt
,
1 1
Z t
1
2 2
Y t
X
, ~
t
Xt
3
Zt N
Z Yt
Zt t
Yt
If
= 1, Z is a random walk, and X and Y are also nonstationary.
The OLS estimator of
2
will be superconsistent.
However, if
= 1, Z is nonstationary, and X and Y will also be nonstationary. So, if we fit the second equation, for example, the OLS estimator of
2 will be superconsistent.
29
COINTEGRATION
X
Y t Z t t Xt
1 .
0
0 .
8
X
0 .
5
t
,
2 .
0
Z t Yt
,
1 0 .
4
Y t
~
Zt N
Xt Zt Z t
Yt
If
= 1, Z is a random walk, and X and Y are also nonstationary.
The OLS estimator of
2
will be superconsistent.
This will be illustrated by a simulation. where the first two equations are as shown.
30
16 14 12 10 8 6 4 2 0 0
X
Y t Z t t Xt
COINTEGRATION
1 .
0
0 .
8
X
0 .
5
t
,
2 .
0
Z t Yt
,
1 0 .
4
Y t
~
Zt N
Xt Zt Z t
Yt
0.2
0.4
0.6
T = 200 T = 100
= 0.5
T = 50 T = 25 0.8
1 The distributions in the figure are for the case
= 0.5. Z is stationary, and so are Y and X. You will have no difficulty in demonstrating that plim a 2 OLS = 0.68.
31
16 14 12 10 8 6 4 2 0 0
X
Y t Z t t Xt
COINTEGRATION
1 .
0
0 .
8
X
0 .
5
t
,
2 .
0
Z t Yt
,
1 0 .
4
Y t
~
Zt N
Xt Zt Z t
Yt
T = 200
= 1.0
0.2
T = 100 T = 50 T = 25 0.4
0.6
T = 200 T = 100
= 0.5
T = 50 T = 25 0.8
1 The distributions to the left of the figure are for
= 1, and you can see that in this case the estimator shows signs of being consistent. But is it superconsistent? The variance seems to be decreasing relatively slowly, not fast, especially for small sample sizes.
32
COINTEGRATION
120 100 80 60 40 20 0 0.3
= 1.0
0.4
T = 3,200 T = 1,600 T = 800 T = 400
X t
Y t Z t Xt
1 .
0
0 .
8
X
0 .
5
t
,
2 .
0
Z Yt
,
t
1 0 .
4
Y t
~
Zt N
Xt Zt Z t
Yt
T = 200 0.5
0.6
0.7
The explanation is that the superconsistency becomes apparent only for very large sample sizes. Comparing the distributions for T = 1,600 and T = 3,200, the sample size has doubled and so has the height.
33
COINTEGRATION
============================================================ Dependent Variable: LGFOOD Method: Least Squares Sample: 1959 2003 Included observations: 45 ============================================================ Variable Coefficient Std. Error t-Statistic Prob.
============================================================ C 2.236158 0.388193 5.760428 0.0000
LGDPI 0.500184 0.008793 56.88557 0.0000
LGPRFOOD -0.074681 0.072864 -1.024941 0.3113
============================================================ R-squared 0.992009 Mean dependent var 6.021331
Adjusted R-squared 0.991628 S.D. dependent var 0.222787
S.E. of regression 0.020384 Akaike info criter-4.883747
Sum squared resid 0.017452 Schwarz criterion -4.763303
Log likelihood 112.8843 F-statistic 2606.860
Durbin-Watson stat 0.478540 Prob(F-statistic) 0.000000
============================================================ We will now consider an empirical example, a logarithmic regression of expenditure on food on DPI and the relative price of food using the Demand Functions data set.
34
COINTEGRATION
0.06
0.05
0.04
0.03
0.02
0.01
0 -0.01
1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 -0.02
-0.03
-0.04
The residuals are shown in the figure. The pattern is mixed and it is not possible to say whether it looks stationary or nonstationary.
35
COINTEGRATION
Augmented Dickey-Fuller Unit Root Test on ELGFOOD ============================================================ Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic -1.926585 0.3175 Test critical values1% level -3.588509 5% level -2.929734 10% level -2.603064 ============================================================
Asymptotic Critical Values of the Dickey-Fuller Statistic for a Cointegrating Relationship with Two Variables
Regression equation contains: 5% 1% Constant, but no trend Constant and trend ============================================================ –3.34
–3.78
Variable CoefficientStd. Errort-Statistic Prob. ============================================================ ELGFOOD(-1) -0.207705 0.107810 -1.926585 0.0608 C 0.000791 0.002035 0.388824 0.6994 ============================================================ –3.90
–4.32
The Engle –Granger statistic is –1.93, not significant even at the 5 percent level.
The failure to reject the null hypothesis of nonstationarity suggests that the variables are not cointegrated.
36
COINTEGRATION
Augmented Dickey-Fuller Unit Root Test on ELGFOOD ============================================================ Lag Length: 0 (Automatic based on SIC, MAXLAG=9) ============================================================ t-Statistic Prob.* ============================================================ Augmented Dickey-Fuller test statistic -1.926585 0.3175 Test critical values1% level -3.588509 5% level -2.929734 10% level -2.603064 ============================================================
Asymptotic Critical Values of the Dickey-Fuller Statistic for a Cointegrating Relationship with Two Variables
Regression equation contains: 5% 1% Constant, but no trend Constant and trend ============================================================ –3.34
–3.78
Variable CoefficientStd. Errort-Statistic Prob. ============================================================ ELGFOOD(-1) -0.207705 0.107810 -1.926585 0.0608 C 0.000791 0.002035 0.388824 0.6994 ============================================================ –3.90
–4.32
Nevertheless, the coefficient of the lagged residuals is –0.21, suggesting an AR(1) process with
equal to about 0.8.
37
COINTEGRATION
0.06
0.05
0.04
0.03
0.02
0.01
0 -0.01
1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 -0.02
-0.03
-0.04
Thus the failure of the test to reject the null hypothesis of nonstationarity may merely reflect its low power against the alternative hypothesis that the disturbance term is a highly autocorrelated stationary process. Consequently, it is possible that the variables are in fact cointegrated.
38
Copyright Christopher Dougherty 2011.
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The content of this slideshow comes from Section 13.5 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/ .
Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lse .
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