Download FITTING MODELS WITH NONSTATIONARY TIME SERIES

FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
Fit Xˆ t  a1  a 2 t  ut
~
Use X t  X t  Xˆ t
Early macroeconomic models tended to produce poor forecasts, despite having excellent
sample-period fits. One response was to search for ways of constructing models that
avoided the fitting of spurious relationships.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
Fit Xˆ t  a1  a 2 t  ut
~
Use X t  X t  Xˆ t
We will briefly consider three of them: detrending the variables in a relationship,
differencing them, and constructing error-correction models.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
Fit Xˆ t  a1  a 2 t  ut
~
Use X t  X t  Xˆ t
For models where the variables possess deterministic trends, the fitting of spurious
relationships can be avoided by detrending the variables before use or, equivalently, by
including a time trend as a regressor in the model.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
Fit Xˆ t  a1  a 2 t  ut
~
Use X t  X t  Xˆ t
Problems if Xt is a random walk
X t  X t 1 t
 X 0   1  ...   t
 X2  t 2
t
However, if the variables are difference-stationary rather than trend-stationary — and for
many macroeconomic variables there is evidence that this is the case — the detrending
procedure is inappropriate and likely to give rise to misleading results.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
Fit Xˆ t  a1  a 2 t  ut
~
Use X t  X t  Xˆ t
Problems if Xt is a random walk
X t  X t 1 t
 X 0   1  ...   t
 X2  t 2
t
H0: a2 = 0 rejected more often than it should be
(standard error underestimated)
In particular, if a random walk Xt is regressed on a time trend as in the equation at the top,
the null hypothesis H0: a2 = 0 is likely to be rejected more often than it should, given the
significance level.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
Fit Xˆ t  a1  a 2 t  ut
~
Use X t  X t  Xˆ t
Problems if Xt is a random walk
X t  X t 1 t
 X 0   1  ...   t
 X2  t 2
t
H0: a2 = 0 rejected more often than it should be
(standard error underestimated)
Although the least squares estimator of a2 is consistent, and thus will tend to 0 in large
samples, its standard error is biased downwards. As a consequence, in finite samples
deterministic trends will tend to be detected, even when not present.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
Fit Xˆ t  a1  a 2 t  ut
~
Use X t  X t  Xˆ t
Problems if Xt is a random walk
X t  X t 1 t
 X 0   1  ...   t
 X2  t 2
t
H0: a2 = 0 rejected more often than it should be
(standard error underestimated)
Trend in variance not removed
Further, if a series is difference-stationary, the procedure does not make it stationary. 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, and the series remains nonstationary.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
Fit Xˆ t  a1  a 2 t  ut
~
Use X t  X t  Xˆ t
Problems if Xt is a random walk with drift
X t   1  X t 1 t
 X 0   1t   1  ...   t
 X2  t 2
t
H0: a2 = 0 rejected more often than it should be
(standard error underestimated)
Trend in variance not removed
In the case of a random walk with drift, detrending can remove the drift, but not the trend in
the variance.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Detrending
Fit Xˆ t  a1  a 2 t  ut
~
Use X t  X t  Xˆ t
Problems if Xt is a random walk with drift
X t   1  X t 1 t
 X 0   1t   1  ...   t
 X2  t 2
t
H0: a2 = 0 rejected more often than it should be
(standard error underestimated)
Trend in variance not removed
Thus if Xt is a random walk, with or without drift, the problem of spurious regressions is not
resolved, and for this reason detrending is not usually considered to be an appropriate
procedure.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Differencing
Yt   1   2 X t  ut
Yt   2 X t  ut
ut  ut 1   t
Yt   2 X t  (   1)ut 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.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Differencing
Yt   1   2 X t  ut
Yt   2 X t  ut
ut  ut 1   t
Yt   2 X t  (   1)ut 1   t
Of course differencing overcompensated for the autocorrelation, but if  was near 1, the
resulting weak negative autocorrelation was held to be relatively innocuous.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Differencing
Yt   1   2 X t  ut
Yt   2 X t  ut
ut  ut 1   t
Yt   2 X t  (   1)ut 1   t
Unknown to practitioners of the time, the procedure is an effective antidote to spurious
regressions. 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.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Differencing
Yt   1   2 X t  ut
Yt   2 X t  ut
ut  ut 1   t
Yt   2 X t  (   1)ut 1   t
Problem
Short-run dynamics only
A major shortcoming of differencing is that it precludes the investigation of a long-run
relationship.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Differencing
Yt   1   2 X t  ut
Yt   2 X t  ut
ut  ut 1   t
Yt   2 X t  (   1)ut 1   t
Problem
Short-run dynamics only
Equilibrium:
Y  X  0
In equilibrium Y = X = 0, and if one substitutes these values into the second equation, one
obtains, not an equilibrium relationship, but an equation in which both sides are 0.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
An error-correction model is an ingenious way of resolving this problem by combining a
long-run cointegrating relationship with short-run dynamics.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Suppose that the relationship between two I(1) variables Yt and Xt is characterized by an
ADL(1,1) model.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Y   1   2Y   3 X   4 X
(1   2 )Y   1  (  3   4 ) X
3  4
1
Y 

X
1  2 1  2
In equilibrium we would have the relationship shown. This is the cointegrating relationship.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2


The ADL(1,1) relationship may be rewritten to incorporate this relationship. First we
subtract Yt–1 from both sides.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2


Then we add 3Xt–1 to the right side and subtract it again.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2


We then rearrange the equation as shown. We will look at the rearrangement, term by term.
First, the intercept 1.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2


Next, the term involving Yt–1.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2


Now the next two terms.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2


Finally, the last two terms.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2




3  4
1
Yt  (  2  1) Yt 1 

X t 1    3 X t   t
1  2 1  2


Hence we obtain a model that 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.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2




3  4
1
Yt  (  2  1) Yt 1 

X t 1    3 X t   t
1  2 1  2


The latter term is denoted the error-correction mechanism. The effect of this term is to
reduce the discrepancy between Yt and its cointegrating level. The size of the adjustment is
proportional to the discrepancy.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2




3  4
1
Yt  (  2  1) Yt 1 

X t 1    3 X t   t
1  2 1  2


The point of rearranging the ADL(1,1) model in this way is that, although Yt and Xt are both
I(1), all of the terms in the regression equation are I(0) and hence the model may be fitted
using least squares in the standard way.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2




3  4
1
Yt  (  2  1) Yt 1 

X t 1    3 X t   t
1  2 1  2


Of course, the  parameters are not known and the cointegrating term is unobservable.
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FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2




3  4
1
Yt  (  2  1) Yt 1 

X t 1    3 X t   t
1  2 1  2


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.
28
FITTING MODELS WITH NONSTATIONARY TIME SERIES
Error-correction models
ADL(1,1)
Yt   1   2Yt 1   3 X t   4 X t 1   t
Yt  Yt 1   1  (  2  1)Yt 1   3 X t   4 X t 1   t
  1  (  2  1)Yt 1   3 X t   3 X t 1   3 X t 1   4 X t 1   t


  4
1
 (  2  1) Yt 1 
 3
X t 1    3 ( X t  X t 1 )   t
1  2 1  2




3  4
1
Yt  (  2  1) Yt 1 

X t 1    3 X t   t
1  2 1  2


It can be demonstrated that the estimators of the coefficients of the fitted equation will have
the same properties asymptotically as if the true values had been used.
29
FITTING MODELS WITH NONSTATIONARY TIME SERIES
============================================================
Dependent Variable: DLGFOOD
Method: Least Squares
Sample(adjusted): 1960 2003
Included observations: 44 after adjusting endpoints
============================================================
Variable
CoefficientStd. Errort-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
============================================================
The EViews output shows the results of fitting an error-correction model for the demand
function for food using the Engle–Granger two-step procedure, on the assumption that the
static logarithmic model is a cointegrating relationship.
30
FITTING MODELS WITH NONSTATIONARY TIME SERIES
============================================================
Dependent Variable: DLGFOOD
Method: Least Squares
Sample(adjusted): 1960
 2003  1

3  4

Y

(


1
)
Y


X

Included observations:
t
2
t44
1 after adjusting endpoints
t 1    3 X t   t
1  2 1  2
============================================================


Variable
CoefficientStd. Errort-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
============================================================
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.
31
FITTING MODELS WITH NONSTATIONARY TIME SERIES
============================================================
Dependent Variable: DLGFOOD
Method: Least Squares
Sample(adjusted): 1960
 2003  1

3  4

Y

(


1
)
Y


X

Included observations:
t
2
t44
1 after adjusting endpoints
t 1    3 X t   t
1  2 1  2
============================================================


Variable
CoefficientStd. Errort-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
============================================================
ZFOOD(–1), the lagged residual from the cointegrating regression, is the cointegration term.
32
FITTING MODELS WITH NONSTATIONARY TIME SERIES
============================================================
Dependent Variable: DLGFOOD
Method: Least Squares
Sample(adjusted): 1960
 2003  1

3  4

Y

(


1
)
Y


X

Included observations:
t
2
t44
1 after adjusting endpoints
t 1    3 X t   t
1  2 1  2
============================================================


Variable
CoefficientStd. Errort-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
============================================================
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.
33
FITTING MODELS WITH NONSTATIONARY TIME SERIES
============================================================
Dependent Variable: DLGFOOD
Method: Least Squares
Sample(adjusted): 1960
 2003  1

3  4

Y

(


1
)
Y


X

Included observations:
t
2
t44
1 after adjusting endpoints
t 1    3 X t   t
1  2 1  2
============================================================


Variable
CoefficientStd. Errort-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
============================================================
The coefficient of the cointegrating term indicates that about 15 percent of the
disequilibrium divergence tends to be eliminated in one year.
34
Copyright Christopher Dougherty 2002–2006. This slideshow may be freely copied for
personal use.
22.08.06