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Transcript of Christopher Dougherty EC220 - Introduction to econometrics (chapter 13) Slideshow: fitting models...
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/
http://learningresources.lse.ac.uk/
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.
ttt XXX ˆ~
tccYt 21ˆ
ttt uXY 21
tddX t 21ˆ
ttt YYY ˆ~
tt XbbY ~~̂21
Model
Fit
Define
Fit
Detrending
FITTING MODELS WITH NONSTATIONARY TIME SERIES
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.
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ttt uXY 21
tddX t 21ˆ
ttt YYY ˆ~
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Model
Fit
Define
Fit
Detrending
FITTING MODELS WITH NONSTATIONARY TIME SERIES
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.
ttt XXX ˆ~
tccYt 21ˆ
ttt uXY 21
tddX t 21ˆ
ttt YYY ˆ~
tt XbbY ~~̂21
Model
Fit
Define
Fit
Detrending
FITTING MODELS WITH NONSTATIONARY TIME SERIES
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.
ttt XXX ˆ~
tccYt 21ˆ
ttt uXY 21
tddX t 21ˆ
ttt YYY ˆ~
tt XbbY ~~̂21
Model
Fit
Define
Fit
Detrending
FITTING MODELS WITH NONSTATIONARY TIME SERIES
5
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.
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ttt uXY 21
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ttt YYY ˆ~
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tbXbbY tt 321ˆ
Model
Fit
Define
Fit
Equivalently,
Detrending
FITTING MODELS WITH NONSTATIONARY TIME SERIES
6
However there are potential problems with this approach.
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ttt uXY 21
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ttt YYY ˆ~
tt XbbY ~~̂21
tbXbbY tt 321ˆ
Model
Fit
Define
Fit
Equivalently,
Detrending
FITTING MODELS WITH NONSTATIONARY TIME SERIES
7
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.
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ttt YYY ˆ~
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Model
Fit
Define
Fit
Equivalently,
Detrending
FITTING MODELS WITH NONSTATIONARY TIME SERIES
8
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.
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ttt YYY ˆ~
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tbXbbY tt 321ˆ
Model
Fit
Define
Fit
Equivalently,
Detrending
Standard error biased downwards when random walk regressed on a trend.
Risk of Type I error underestimated.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
9
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.
ttt XXX ˆ~
tccYt 21ˆ
ttt uXY 21
tddX t 21ˆ
ttt YYY ˆ~
tt XbbY ~~̂21
tbXbbY tt 321ˆ
Standard error biased downwards when random walk regressed on a trend.
Risk of Type I error underestimated.
Model
Fit
Define
Fit
Equivalently,
Detrending
FITTING MODELS WITH NONSTATIONARY TIME SERIES
10
Further, if a series is difference-stationary, the procedure does not make it stationary.
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ttt uXY 21
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ttt YYY ˆ~
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tbXbbY tt 321ˆ
Detrending does not make a random walk stationary.
Model
Fit
Define
Fit
Equivalently,
Detrending
FITTING MODELS WITH NONSTATIONARY TIME SERIES
11
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.
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ttt uXY 21
tddX t 21ˆ
ttt YYY ˆ~
tt XbbY ~~̂21
tbXbbY tt 321ˆ
Model
Fit
Define
Fit
Equivalently,
Detrending
Detrending does remove the drift in a random walk with drift.
However, it does not affect its variance, which continues to increase.
FITTING MODELS WITH NONSTATIONARY TIME SERIES
12
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.
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tccYt 21ˆ
ttt uXY 21
tddX t 21ˆ
ttt YYY ˆ~
tt XbbY ~~̂21
tbXbbY tt 321ˆ
Detrending does remove the drift in a random walk with drift.
However, it does not affect its variance, which continues to increase.
Model
Fit
Define
Fit
Equivalently,
Detrending
FITTING MODELS WITH NONSTATIONARY TIME SERIES
13
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.
Detrending
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ttt uXY 21 Model
Fit
tddX t 21ˆ
Define ttt YYY ˆ~
Fit tt XbbY ~~̂21
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Increasing variance has adverse consequences for estimation and inference.
Equivalently,
FITTING MODELS WITH NONSTATIONARY TIME SERIES
14
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.
ytt uu 1
ttt uXY 21
tttt uXY 12 1
Model
AR(1) auto–correlation
Difference
Differencing
FITTING MODELS WITH NONSTATIONARY TIME SERIES
15
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.
ytt uu 1
ttt uXY 21
tttt uXY 12 1
Model
AR(1) auto–correlation
Difference
Differencing
FITTING MODELS WITH NONSTATIONARY TIME SERIES
16
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 Yt and Xt are unrelated I(1) processes, they are stationary in the differenced model and the absence of any relationship will be revealed.
ytt uu 1
ttt uXY 21
tttt uXY 12 1
Model
AR(1) auto–correlation
Difference
Differencing
FITTING MODELS WITH NONSTATIONARY TIME SERIES
17
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.
Differencing
ytt uu 1
ttt uXY 21 Model
AR(1) auto–correlation
Difference tttt uXY 12 1
Procedure does not allow determination of long-run relationship
In equilibrium 0 tXY
Model becomes 00
FITTING MODELS WITH NONSTATIONARY TIME SERIES
18
We have seen that a long-run relationship between two or more nonstationary variables is given by a cointegrating relationship, if it exists.
ADL(1,1) model
In equilibrium .4321 XXYY
ttttt XXYY 143121
Error correction model
FITTING MODELS WITH NONSTATIONARY TIME SERIES
19
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.
ADL(1,1) model
In equilibrium .4321 XXYY
ttttt XXYY 143121
Error correction model
FITTING MODELS WITH NONSTATIONARY TIME SERIES
20
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.
Error correction model
ADL(1,1) model
In equilibrium .4321 XXYY
ttttt XXYY 143121
FITTING MODELS WITH NONSTATIONARY TIME SERIES
21
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.
ADL(1,1) model
In equilibrium .4321 XXYY
ttttt XXYY 143121
Error correction model
FITTING MODELS WITH NONSTATIONARY TIME SERIES
22
Hence we obtain equilibrium Y in terms of equilibrium X.
ADL(1,1) model
Hence
In equilibrium .4321 XXYY
ttttt XXYY 143121
XY2
43
2
1
11
Error correction model
FITTING MODELS WITH NONSTATIONARY TIME SERIES
23
Hence we infer the cointegrating relationship.
ADL(1,1) model
Hence
In equilibrium
Cointegrating relationship
.4321 XXYY
ttttt XXYY 143121
XY2
43
2
1
11
tt XY2
43
2
1
11
Error correction model
FITTING MODELS WITH NONSTATIONARY TIME SERIES
24
The ADL(1,1) relationship may be rewritten to incorporate this relationship by subtracting Yt–1 from both sides, subtracting 3Xt–1 from the right side and adding it back again, and rearranging.
ADL(1,1) model
Cointegrating relationship
ttttt XXYY 143121
tt XY2
43
2
1
11
Error correction model
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1431211
FITTING MODELS WITH NONSTATIONARY TIME SERIES
25
Hence we obtain the error correction model shown.
ADL(1,1) model
Cointegrating relationship
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tt XY2
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Error correction model
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
26
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.
ADL(1,1) model
Cointegrating relationship
ttttt XXYY 143121
tt XY2
43
2
1
11
Error correction model
.11
)1( 312
43
2
112 ttttt XXYY
FITTING MODELS WITH NONSTATIONARY TIME SERIES
27
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.
ADL(1,1) model
Cointegrating relationship
ttttt XXYY 143121
tt XY2
43
2
1
11
Error correction model
.11
)1( 312
43
2
112 ttttt XXYY
FITTING MODELS WITH NONSTATIONARY TIME SERIES
28
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), Yt, Xt, and the error correction term are I(0), the latter by virtue of being just the lagged disturbance term in the cointegrating relationship.
ADL(1,1) model
Cointegrating relationship
ttttt XXYY 143121
tt XY2
43
2
1
11
Error correction model
.11
)1( 312
43
2
112 ttttt XXYY
FITTING MODELS WITH NONSTATIONARY TIME SERIES
29
Hence the model may be fitted using least squares in the standard way.
ADL(1,1) model
Cointegrating relationship
ttttt XXYY 143121
tt XY2
43
2
1
11
Error correction model
.11
)1( 312
43
2
112 ttttt XXYY
FITTING MODELS WITH NONSTATIONARY TIME SERIES
30
Of course, the parameters are not known and the cointegrating term is unobservable.
ADL(1,1) model
Cointegrating relationship
ttttt XXYY 143121
tt XY2
43
2
1
11
Error correction model
.11
)1( 312
43
2
112 ttttt XXYY
FITTING MODELS WITH NONSTATIONARY TIME SERIES
31
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.
ADL(1,1) model
Cointegrating relationship
ttttt XXYY 143121
tt XY2
43
2
1
11
Error correction model
.11
)1( 312
43
2
112 ttttt XXYY
FITTING MODELS WITH NONSTATIONARY TIME SERIES
32
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.
ADL(1,1) model
Cointegrating relationship
ttttt XXYY 143121
tt XY2
43
2
1
11
Error correction model
.11
)1( 312
43
2
112 ttttt XXYY
FITTING MODELS WITH NONSTATIONARY TIME SERIES
33
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 Sum squared resid 0.006704 Schwarz criterion -5.693405 Log likelihood 130.9312 Durbin-Watson stat 1.526946 ============================================================
============================================================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 ============================================================
FITTING MODELS WITH NONSTATIONARY TIME SERIES
34
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.
ttttt XXYY
31
2
43
2
112 11
)1(
============================================================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 ============================================================
FITTING MODELS WITH NONSTATIONARY TIME SERIES
35
ZFOOD(–1), the lagged residual from the cointegrating regression, is the cointegrating term.
ttttt XXYY
31
2
43
2
112 11
)1(
============================================================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 ============================================================
FITTING MODELS WITH NONSTATIONARY TIME SERIES
36
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.
ttttt XXYY
31
2
43
2
112 11
)1(
============================================================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 ============================================================
FITTING MODELS WITH NONSTATIONARY TIME SERIES
37
The coefficient of the cointegrating term indicates that about 15 percent of the disequilibrium divergence tends to be eliminated in one year.
ttttt XXYY
31
2
43
2
112 11
)1(
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 13.6 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
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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
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11.07.25