Transcript Chapter 13 fitting models with nonstationary time series (EC220)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 13)
Slideshow: fitting models with nonstationary time series 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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Detrending FITTING MODELS WITH NONSTATIONARY TIME SERIES Model Fit Define Fit
Y t
1
2
X t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t Y
ˆ~
t
b
1
b
2 ~
X t
u t
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.
1
Detrending FITTING MODELS WITH NONSTATIONARY TIME SERIES Model Fit Define Fit
Y t
1
2
X t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t Y
ˆ~
t
b
1
b
2 ~
X t
u t
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.
2
Detrending FITTING MODELS WITH NONSTATIONARY TIME SERIES Model Fit Define Fit
Y t
1
2
X t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t Y
ˆ~
t
b
1
b
2 ~
X t
u t
We will briefly consider three of them: detrending the variables in a relationship, differencing the variables in a relationship, and constructing error correction models.
3
Detrending FITTING MODELS WITH NONSTATIONARY TIME SERIES Model Fit Define Fit
Y t
1
2
X t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t Y
ˆ~
t
b
1
b
2 ~
X t
u t
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.
4
Detrending FITTING MODELS WITH NONSTATIONARY TIME SERIES Model Fit Define Fit Equivalently,
Y t
1
2
X t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t
u t Y
ˆ~
t Y
ˆ
t
b
1
b
2 ~
X t
b
1
b
2
X t
b
3
t
Alternatively, and equivalently, one may include a time trend as a regressor in the model. 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.
5
Detrending FITTING MODELS WITH NONSTATIONARY TIME SERIES Model Fit Define Fit Equivalently,
Y t
1
2
X t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t
u t Y
ˆ~
t Y
ˆ
t
b
1
b
2 ~
X t
b
1
b
2
X t
b
3
t
However there are potential problems with this approach.
6
Detrending FITTING MODELS WITH NONSTATIONARY TIME SERIES Model Fit Define Fit Equivalently,
Y t
1
2
X t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t
u t Y
ˆ~
t Y
ˆ
t
b
1
b
2 ~
X t
b
1
b
2
X t
b
3
t
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.
7
FITTING MODELS WITH NONSTATIONARY TIME SERIES Detrending Model Fit Define Fit Equivalently,
Y t
1
2
X t
u t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t Y
ˆ~
t Y
ˆ
t
b
1
b
2 ~
X t
b
1
b
2
X t
b
3
t
Standard error biased downwards when random walk regressed on a trend.
Risk of Type I error underestimated.
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.
8
FITTING MODELS WITH NONSTATIONARY TIME SERIES Detrending Model Fit Define Fit Equivalently,
Y t
1
2
X t
u t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t Y
ˆ~
t Y
ˆ
t
b
1
b
2 ~
X t
b
1
b
2
X t
b
3
t
Standard error biased downwards when random walk regressed on a trend.
Risk of Type I error underestimated.
Although the least squares estimator of
2 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.
9
FITTING MODELS WITH NONSTATIONARY TIME SERIES Detrending Model Fit Define Fit Equivalently,
Y t
1
2
X t
u t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t Y
ˆ~
t Y
ˆ
t
b
1
b
2 ~
X t
b
1
b
2
X t
b
3
t
Detrending does not make a random walk stationary.
Further, if a series is difference-stationary, the procedure does not make it stationary.
10
FITTING MODELS WITH NONSTATIONARY TIME SERIES Detrending Model Fit Define Fit Equivalently,
Y t
1
2
X t
u t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t Y
ˆ~
t Y
ˆ
t
b
1
b
2 ~
X t
b
1
b
2
X t
b
3
t
Detrending does remove the drift in a random walk with drift.
However, it does not affect its variance, which continues to increase.
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.
11
FITTING MODELS WITH NONSTATIONARY TIME SERIES Detrending Model Fit Define Fit Equivalently,
Y t
1
2
X t
u t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t Y
ˆ~
t Y
ˆ
t
b
1
b
2 ~
X t
b
1
b
2
X t
b
3
t
Detrending does remove the drift in a random walk with drift.
However, it does not affect its variance, which continues to increase.
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.
12
FITTING MODELS WITH NONSTATIONARY TIME SERIES Detrending Model Fit Define Fit Equivalently,
Y t
1
2
X t
u t Y
ˆ
X t t
c
1
d
1
c
2
t d
2
t
~
Y t
~
X t
Y X t t
Y
ˆ
t t Y
ˆ~
t Y
ˆ
t
b
1
b
2 ~
X t
b
1
b
2
X t
b
3
t
Increasing variance has adverse consequences for estimation and inference.
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.
13
FITTING MODELS WITH NONSTATIONARY TIME SERIES Differencing Model AR(1) auto – correlation
Y t
1
2
X t u t
u t
1
y
u t
Difference
Y t
2
X t
1
u t
1
t
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.
14
FITTING MODELS WITH NONSTATIONARY TIME SERIES Differencing Model AR(1) auto – correlation
Y t
1
2
X t u t
u t
1
y
u t
Difference
Y t
2
X t
1
u t
1
t
Of course, differencing overcompensated for the autocorrelation, but in the case of strong positive autocorrelation with
near to 1, (
– 1) would be a small negative quantity and the resulting weak negative autocorrelation was held to be relatively innocuous.
15
FITTING MODELS WITH NONSTATIONARY TIME SERIES Differencing Model AR(1) auto – correlation
Y t
1
2
X t u t
u t
1
y
u t
Difference
Y t
2
X t
1
u t
1
t
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. If both Y
t
and X
t
are unrelated I(1) processes, they are stationary in the differenced model and the absence of any relationship will be revealed.
16
FITTING MODELS WITH NONSTATIONARY TIME SERIES Differencing Model AR(1) auto – correlation
Y t
1
2
X t u t
u t
1
y
u t
Difference
Y t
2
X t
1
u t
1
t
Procedure does not allow determination of long-run relationship In equilibrium Model becomes
Y
X t
0
0
0 A major shortcoming of differencing is that it precludes the investigation of a long-run relationship. In equilibrium
Y =
X = 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.
17
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model In equilibrium
Y t
1
2
Y t
1
3
X t
4
X t
1
t Y
1
2
Y
3
X
4
X
.
We have seen that a long-run relationship between two or more nonstationary variables is given by a cointegrating relationship, if it exists.
18
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model In equilibrium
Y t
1
2
Y t
1
3
X t
4
X t
1
t Y
1
2
Y
3
X
4
X
.
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, and thus that it should be possible to construct a more comprehensive model that combines short-run and long-run dynamics.
19
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model In equilibrium
Y t
1
2
Y t
1
3
X t
4
X t
1
t Y
1
2
Y
3
X
4
X
.
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.
20
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model In equilibrium
Y t
1
2
Y t
1
3
X t
4
X t
1
t Y
1
2
Y
3
X
4
X
.
It will be convenient to rehearse the theory. Suppose that the relationship between two I(1) variables Y
t
and X
t
relationship shown.
is characterized by the ADL(1,1) model. In equilibrium, we have the
21
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model In equilibrium Hence
Y t
1
2
Y t
1
3
X t
4
X t
1
t Y
1
2
Y
3
X
4
X
.
Y
1
1
2
1 3
2 4
X
Hence we obtain equilibrium Y in terms of equilibrium X.
22
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model In equilibrium Hence Cointegrating relationship
Y t
1
2
Y t
1
3
X t
4
X t
1
t Y
1
2
Y
3
X
4
X
.
Y
1
1
2
1 3
2 4
X Y t
1
1
2
1 3
2 4
X t
Hence we infer the cointegrating relationship.
23
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model Cointegrating relationship
Y t Y t
1 1
1
2 2
Y t
1
1 3
2 3 4
X X t t
4
X t
1
t Y t
Y t
1
1
(
2
1 )
Y t
1
3
X t
1
(
2
1 )
Y t
1
3
X t
4
X t
1
t
3
X t
1
3
X t
1
4
X t
1
t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3 (
X t
X t
1 )
t
The ADL(1,1) relationship may be rewritten to incorporate this relationship by subtracting
Y t
–1 from both sides, subtracting
3
X t
–1 from the right side and adding it back again, and rearranging.
24
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model Cointegrating relationship
Y t Y t
1 1
1
2 2
Y t
1
1 3
2 3 4
X X t t
4
X t
1
t Y t
Y t
1
1
(
2
1 )
Y t
1
3
X t
1
(
2
1 )
Y t
1
3
X t
4
X t
1
t
3
X t
1
3
X t
1
4
X t
1
t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3 (
X t
X t
1 )
t
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X t
t
.
Hence we obtain the error correction model shown.
25
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model Cointegrating relationship
Y t Y t
1 1
1
2 2
Y t
1
1 3
2 3 4
X X t t
4
X t
1
t
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X t
t
.
The model states that the change in Y in any period will be governed by the change in X and the discrepancy between Y
t
–1 and the value predicted by the cointegrating relationship.
26
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model Cointegrating relationship
Y t Y t
1 1
1
2 2
Y t
1
1 3
2 3 4
X X t t
4
X t
1
t
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X t
t
.
The latter term is denoted the error correction mechanism, the effect of the term being to reduce the discrepancy between Y proportional to the discrepancy.
t
and its cointegrating level and its size being
27
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model Cointegrating relationship
Y t Y t
1 1
1
2 2
Y t
1
1 3
2 3 4
X X t t
4
X t
1
t
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X t
t
.
The feature that makes the error correction model particularly attractive when working with nonstationary time series is the fact that, if Y and X are I(1),
Y t
,
X t
, and the error correction term are I(0), the latter by virtue of being just the lagged disturbance term in the cointegrating relationship.
28
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model Cointegrating relationship
Y t Y t
1 1
1
2 2
Y t
1
1 3
2 3 4
X X t t
4
X t
1
t
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X t
t
.
Hence the model may be fitted using least squares in the standard way.
29
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model Cointegrating relationship
Y t Y t
1 1
1
2 2
Y t
1
1 3
2 3 4
X X t t
4
X t
1
t
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X t
t
.
Of course, the
parameters are not known and the cointegrating term is unobservable.
30
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model Cointegrating relationship
Y t Y t
1 1
1
2 2
Y t
1
1 3
2 3 4
X X t t
4
X t
1
t
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X t
t
.
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.
31
FITTING MODELS WITH NONSTATIONARY TIME SERIES Error correction model ADL(1,1) model Cointegrating relationship
Y t Y t
1 1
1
2 2
Y t
1
1 3
2 3 4
X X t t
4
X t
1
t
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X t
t
.
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.
32
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 Sum squared resid 0.006704 Schwarz criterion -5.693405 Log likelihood 130.9312 Durbin-Watson stat 1.526946 ============================================================ 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.
33
FITTING MODELS WITH NONSTATIONARY TIME SERIES ============================================================ Dependent Variable: DLGFOOD Method: Least Squares
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X
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 Sum squared resid 0.006704 Schwarz criterion -5.693405 Log likelihood 130.9312 Durbin-Watson stat 1.526946 ============================================================
t
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.
34
FITTING MODELS WITH NONSTATIONARY TIME SERIES ============================================================ Dependent Variable: DLGFOOD Method: Least Squares
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X
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 Sum squared resid 0.006704 Schwarz criterion -5.693405 Log likelihood 130.9312 Durbin-Watson stat 1.526946 ============================================================
t
ZFOOD( –1), the lagged residual from the cointegrating regression, is the cointegrating term.
35
FITTING MODELS WITH NONSTATIONARY TIME SERIES ============================================================ Dependent Variable: DLGFOOD Method: Least Squares
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X
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 Sum squared resid 0.006704 Schwarz criterion -5.693405 Log likelihood 130.9312 Durbin-Watson stat 1.526946 ============================================================
t
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.
36
FITTING MODELS WITH NONSTATIONARY TIME SERIES ============================================================ Dependent Variable: DLGFOOD Method: Least Squares
Y t
(
2
1 )
Y t
1
1
1
2
1 3
2 4
X t
1
3
X
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 Sum squared resid 0.006704 Schwarz criterion -5.693405 Log likelihood 130.9312 Durbin-Watson stat 1.526946 ============================================================
t
The coefficient of the cointegrating term indicates that about 15 percent of the disequilibrium divergence tends to be eliminated in one year.
37
Copyright Christopher Dougherty 2011.
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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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