1. FITTING MODELS WITH NONSTATIONARY TIME SERIES
1
The poor predictive power of early macroeconomic models, despite excellent sample
period fits, gave rise to two main reactions. One was a resurgence of interest in the use of
univariate time series for forecasting purposes, described in Section 11.7.
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
Model
Fit
Define
Fit
2. 2
The other, of greater appeal to economists who did not wish to give up multivariate
analysis, was to search for ways of constructing models that avoided the fitting of spurious
relationships.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
Model
Fit
Define
Fit
3. 3
We will briefly consider three of them: detrending the variables in a relationship,
differencing the variables in a relationship, and constructing error correction models.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
Model
Fit
Define
Fit
4. 4
As noted in Section 13.2, for models where the variables possess deterministic trends, the
fitting of spurious relationships can be avoided by detrending the variables before use.
This was a common procedure in early econometric analysis with time series data.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
Model
Fit
Define
Fit
5. 5
Alternatively, and equivalently, one may include a time trend as a regressor in the model.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
t
b
X
b
b
Y t
t 3
2
1
ˆ
Model
Fit
Define
Fit
Equivalently,
6. 6
By virtue of the Frisch–Waugh–Lovell theorem, the coefficients obtained with such a
specification are exactly the same as those obtained with a regression using detrended
versions of the variables.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
t
b
X
b
b
Y t
t 3
2
1
ˆ
Model
Fit
Define
Fit
Equivalently,
7. 7
However there are potential problems with this approach.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
t
b
X
b
b
Y t
t 3
2
1
ˆ
Model
Fit
Define
Fit
Equivalently,
8. 8
Most importantly, if the variables are difference-stationary rather than trend-stationary, and
there is evidence that this is the case for many macroeconomic variables, the detrending
procedure is inappropriate and likely to give rise to misleading results.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
t
b
X
b
b
Y t
t 3
2
1
ˆ
Model
Fit
Define
Fit
Equivalently,
9. 9
In particular, if a random walk is regressed on a time trend, the null hypothesis that the
slope coefficient is zero is likely to be rejected more often than it should, given the
significance level.
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
t
b
X
b
b
Y t
t 3
2
1
ˆ
Model
Fit
Define
Fit
Equivalently,
Standard error biased downwards when random walk regressed on a trend.
Risk of Type I error is underestimated.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
10. 10
Although the least squares estimator of b2 is consistent, and thus will tend to zero in large
samples, its standard error is biased downwards. As a consequence, in finite samples
deterministic trends may appear to be detected, even when not present.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
t
b
X
b
b
Y t
t 3
2
1
ˆ
Model
Fit
Define
Fit
Equivalently,
Standard error biased downwards when random walk regressed on a trend.
Risk of Type I error is underestimated.
11. 11
Further, if a series is difference-stationary, the procedure does not make it stationary.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
t
b
X
b
b
Y t
t 3
2
1
ˆ
Model
Fit
Define
Fit
Equivalently,
Detrending does remove the drift in a random walk with drift.
12. 12
In the case of a random walk, extracting a non-existent trend in the mean of the series can
do nothing to alter the trend in its variance. As a consequence, the series remains
nonstationary.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
t
b
X
b
b
Y t
t 3
2
1
ˆ
Model
Fit
Define
Fit
Equivalently,
Detrending does remove the drift in a random walk with drift.
However, it does not affect its variance, which continues to increase.
13. 13
In the case of a random walk with drift, the procedure can remove the drift, but again it does
not remove the trend in the variance.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
t
b
X
b
b
Y t
t 3
2
1
ˆ
Model
Fit
Define
Fit
Equivalently,
Detrending does remove the drift in a random walk with drift.
However, it does not affect its variance, which continues to increase.
14. Increasing variance has adverse consequences for estimation and inference.
14
In either case the problem of spurious regressions is not resolved, with adverse
consequences for estimation and inference. For this reason, detrending is now not usually
considered to be an appropriate procedure.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
t
t
t X
X
X ˆ
~
t
c
c
Yt 2
1
ˆ
t
t
t u
X
Y
2
1
t
d
d
Xt 2
1
ˆ
t
t
t Y
Y
Y ˆ
~
t
t X
b
b
Y
~
ˆ
~
2
1
t
b
X
b
b
Y t
t 3
2
1
ˆ
Model
Fit
Define
Fit
Equivalently,
15. 15
In early time series studies, if the disturbance term in a model was believed to be subject to
severe positive AR(1) autocorrelation. a common rough-and-ready remedy was to regress
the model in differences rather than levels.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
y
t
t u
u
1
AR(1) auto–
correlation
Difference t
t
t
t u
X
Y
1
2 1
Differencing
t
t
t u
X
Y
2
1
Model
16. 16
Of course, differencing overcompensated for the autocorrelation, but in the case of strong
positive autocorrelation with r near to 1, (r – 1) would be a small negative quantity and the
resulting weak negative autocorrelation was held to be relatively innocuous.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
y
t
t u
u
1
AR(1) auto–
correlation
Difference t
t
t
t u
X
Y
1
2 1
Differencing
t
t
t u
X
Y
2
1
Model
17. 17
Unknown to practitioners of the time, the procedure is also an effective antidote to spurious
regressions, and was advocated as such by Granger and Newbold.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
y
t
t u
u
1
AR(1) auto–
correlation
Difference t
t
t
t u
X
Y
1
2 1
Differencing
t
t
t u
X
Y
2
1
Model
18. 18
If both Yt and Xt are unrelated I(1) processes, they are stationary in the differenced model
and the absence of any relationship will be revealed.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
y
t
t u
u
1
AR(1) auto–
correlation
Difference t
t
t
t u
X
Y
1
2 1
Differencing
t
t
t u
X
Y
2
1
Model
19. 19
A major shortcoming of differencing is that it precludes the investigation of a long-run
relationship.
y
t
t u
u
1
AR(1) auto–
correlation
Difference t
t
t
t u
X
Y
1
2 1
Procedure does not allow determination of long-run relationship.
In equilibrium 0
t
X
Y
Model becomes 0
0
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Differencing
t
t
t u
X
Y
2
1
Model
20. 20
In equilibrium DY = DX = 0, and, if one substitutes these values into the differenced model,
one obtains, not an equilibrium relationship, but an equation in which both sides are zero.
y
t
t u
u
1
AR(1) auto–
correlation
Difference t
t
t
t u
X
Y
1
2 1
Procedure does not allow determination of long-run relationship.
In equilibrium 0
t
X
Y
Model becomes 0
0
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Differencing
t
t
t u
X
Y
2
1
Model
21. 21
We have seen that a long-run relationship between two or more nonstationary variables is
given by a cointegrating relationship, if it exists.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model
In equilibrium .
4
3
2
1 X
X
Y
Y
t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
22. 22
On its own, a cointegrating relationship sheds no light on short-run dynamics, but its very
existence indicates that there must be some short-term forces that are responsible for
keeping the relationship intact.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model
In equilibrium .
4
3
2
1 X
X
Y
Y
t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
23. 23
This implies that it should be possible to construct a more comprehensive model that
combines short-run and long-run dynamics.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model
In equilibrium .
4
3
2
1 X
X
Y
Y
t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
24. 24
A standard means of accomplishing this is to make use of an error correction model of the
kind discussed in Section 11.4. It will be seen that it is particularly appropriate in the
context of models involving nonstationary processes.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model
In equilibrium .
4
3
2
1 X
X
Y
Y
t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
25. 25
It will be convenient to rehearse the theory. Suppose that the relationship between two I(1)
variables Yt and Xt is characterized by the ADL(1,1) model. In equilibrium, we have the
relationship shown.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model
In equilibrium .
4
3
2
1 X
X
Y
Y
t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
26. 26
Hence we obtain equilibrium Y in terms of equilibrium X.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model
Hence
In equilibrium .
4
3
2
1 X
X
Y
Y
t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
X
Y
2
4
3
2
1
1
1
Error correction model
27. 27
Hence we infer the cointegrating relationship.
ADL(1,1) model
Hence
In equilibrium
Cointegrating
relationship
.
4
3
2
1 X
X
Y
Y
t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
X
Y
2
4
3
2
1
1
1
t
t X
Y
2
4
3
2
1
1
1
Error correction model
FITTING MODELS WITH NONSTATIONARY TIME SERIES
28. 28
The ADL(1,1) relationship may be rewritten to incorporate this relationship by subtracting
Yt–1 from both sides, subtracting b3Xt–1 from the right side and adding it back again, and
rearranging.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
X
X
X
Y
X
X
X
X
Y
X
X
Y
Y
Y
)
(
1
1
1
1
1
1
3
1
2
4
3
2
1
1
2
1
4
1
3
1
3
3
1
2
1
1
4
3
1
2
1
1
Cointegrating
relationship
t
t X
Y
2
4
3
2
1
1
1
29. 29
Hence we obtain the error correction model shown.
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
1
FITTING MODELS WITH NONSTATIONARY TIME SERIES
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
X
X
X
Y
X
X
X
X
Y
X
X
Y
Y
Y
)
(
1
1
1
1
1
1
3
1
2
4
3
2
1
1
2
1
4
1
3
1
3
3
1
2
1
1
4
3
1
2
1
1
ADL(1,1) model t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
Cointegrating
relationship
t
t X
Y
2
4
3
2
1
1
1
30. 30
The model states that the change in Y in any period will be governed by the change in X and
the discrepancy between Yt–1 and the value predicted by the cointegrating relationship.
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
Cointegrating
relationship
t
t X
Y
2
4
3
2
1
1
1
31. 31
The latter term is denoted the error correction mechanism, the effect of the term being to
reduce the discrepancy between Yt and its cointegrating level and its size being
proportional to the discrepancy.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
Cointegrating
relationship
t
t X
Y
2
4
3
2
1
1
1
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
32. 32
The feature that makes the error correction model particularly attractive when working with
nonstationary time series is the fact that all of its elements are stationary.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
Cointegrating
relationship
t
t X
Y
2
4
3
2
1
1
1
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
33. 33
If Y and X are I(1), it follows that DYt, DXt, and the error correction term are I(0), the latter by
virtue of being just the lagged disturbance term in the cointegrating relationship.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
Cointegrating
relationship
t
t X
Y
2
4
3
2
1
1
1
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
34. 34
Hence the model may be fitted using least squares in the standard way.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
Cointegrating
relationship
t
t X
Y
2
4
3
2
1
1
1
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
35. 35
Of course, the b parameters are not known and the cointegrating term is unobservable.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
Cointegrating
relationship
t
t X
Y
2
4
3
2
1
1
1
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
36. 36
One way of overcoming this problem, known as the Engle–Granger two-step procedure, is
to use the values of the parameters estimated in the cointegrating regression to compute
the cointegrating term.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
Cointegrating
relationship
t
t X
Y
2
4
3
2
1
1
1
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
37. 37
Engle and Granger demonstrated that, asymptotically, the estimators of the coefficients of
the cointegrating term will have the same properties as if the true values had been used.
As a consequence, the residuals from the cointegrating regression can be used for it.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
ADL(1,1) model t
t
t
t
t X
X
Y
Y
1
4
3
1
2
1
Error correction model
Cointegrating
relationship
t
t X
Y
2
4
3
2
1
1
1
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
38. 38
As an example, we will look at the EViews output showing the results of fitting an error-
correction model for the demand function for food using the Engle–Granger two-step
procedure. It assumes that the static logarithmic model is a cointegrating relationship.
============================================================
Dependent Variable: DLGFOOD
Method: Least Squares
Sample(adjusted): 1960 2003
Included observations: 44 after adjusting endpoints
============================================================
Variable Coefficient Std. Error t-Statistic Prob.
============================================================
ZFOOD(-1) -0.148063 0.105268 -1.406533 0.1671
DLGDPI 0.493715 0.050948 9.690642 0.0000
DLPRFOOD -0.353901 0.115387 -3.067086 0.0038
============================================================
R-squared 0.343031 Mean dependent var 0.018243
Adjusted R-squared 0.310984 S.D. dependent var 0.015405
S.E. of regression 0.012787 Akaike info criter-5.815054
FITTING MODELS WITH NONSTATIONARY TIME SERIES
39. ============================================================
Dependent Variable: DLGFOOD
Method: Least Squares
Sample(adjusted): 1960 2003
Included observations: 44 after adjusting endpoints
============================================================
Variable Coefficient Std. Error t-Statistic Prob.
============================================================
ZFOOD(-1) -0.148063 0.105268 -1.406533 0.1671
DLGDPI 0.493715 0.050948 9.690642 0.0000
DLPRFOOD -0.353901 0.115387 -3.067086 0.0038
============================================================
R-squared 0.343031 Mean dependent var 0.018243
Adjusted R-squared 0.310984 S.D. dependent var 0.015405
S.E. of regression 0.012787 Akaike info criter-5.815054 39
In the output, DLGFOOD, DLGDPI, and DLPRFOOD are the differences in the logarithms of
expenditure on food, disposable personal income, and the relative price of food,
respectively.
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
FITTING MODELS WITH NONSTATIONARY TIME SERIES
40. 40
ZFOOD(–1), the lagged residual from the cointegrating regression, is the cointegrating term.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
============================================================
Dependent Variable: DLGFOOD
Method: Least Squares
Sample(adjusted): 1960 2003
Included observations: 44 after adjusting endpoints
============================================================
Variable Coefficient Std. Error t-Statistic Prob.
============================================================
ZFOOD(-1) -0.148063 0.105268 -1.406533 0.1671
DLGDPI 0.493715 0.050948 9.690642 0.0000
DLPRFOOD -0.353901 0.115387 -3.067086 0.0038
============================================================
R-squared 0.343031 Mean dependent var 0.018243
Adjusted R-squared 0.310984 S.D. dependent var 0.015405
S.E. of regression 0.012787 Akaike info criter-5.815054
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
41. 41
The coefficient of DLGDPI and DLPRFOOD provide estimates of the short-run income and
price elasticities, respectively. As might be expected, they are both quite low.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
============================================================
Dependent Variable: DLGFOOD
Method: Least Squares
Sample(adjusted): 1960 2003
Included observations: 44 after adjusting endpoints
============================================================
Variable Coefficient Std. Error t-Statistic Prob.
============================================================
ZFOOD(-1) -0.148063 0.105268 -1.406533 0.1671
DLGDPI 0.493715 0.050948 9.690642 0.0000
DLPRFOOD -0.353901 0.115387 -3.067086 0.0038
============================================================
R-squared 0.343031 Mean dependent var 0.018243
Adjusted R-squared 0.310984 S.D. dependent var 0.015405
S.E. of regression 0.012787 Akaike info criter-5.815054
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
42. 42
The coefficient of the cointegrating term indicates that about 15 percent of the
disequilibrium divergence tends to be eliminated in one year.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
============================================================
Dependent Variable: DLGFOOD
Method: Least Squares
Sample(adjusted): 1960 2003
Included observations: 44 after adjusting endpoints
============================================================
Variable Coefficient Std. Error t-Statistic Prob.
============================================================
ZFOOD(-1) -0.148063 0.105268 -1.406533 0.1671
DLGDPI 0.493715 0.050948 9.690642 0.0000
DLPRFOOD -0.353901 0.115387 -3.067086 0.0038
============================================================
R-squared 0.343031 Mean dependent var 0.018243
Adjusted R-squared 0.310984 S.D. dependent var 0.015405
S.E. of regression 0.012787 Akaike info criter-5.815054
t
t
t
t
t X
X
Y
Y
3
1
2
4
3
2
1
1
2
1
1
)
1
(
