Christopher Dougherty EC220 - Introduction to econometrics (chapter 13) Slideshow: fitting models with nonstationary time series Original citation: Dougherty,

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

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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.

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Model

Fit

Define

Fit

Detrending


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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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.

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Detrending


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.

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Detrending


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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Equivalently,

Detrending


6

However there are potential problems with this approach.

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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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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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Standard error biased downwards when random walk regressed on a trend.

Risk of Type I error underestimated.


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.

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Standard error biased downwards when random walk regressed on a trend.

Risk of Type I error underestimated.

Model

Fit

Define

Fit

Equivalently,

Detrending


10

Further, if a series is difference-stationary, the procedure does not make it stationary.

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Detrending does not make a random walk stationary.

Model

Fit

Define

Fit

Equivalently,

Detrending


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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Equivalently,

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Detrending does remove the drift in a random walk with drift.

However, it does not affect its variance, which continues to increase.


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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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


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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Fit

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Increasing variance has adverse consequences for estimation and inference.

Equivalently,


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.

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Model

AR(1) auto–correlation

Difference

Differencing


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.

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Difference

Differencing


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.

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Difference

Differencing


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

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Difference tttt uXY 12 1

Procedure does not allow determination of long-run relationship

In equilibrium 0 tXY

Model becomes 00


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

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Error correction model


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


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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.


ADL(1,1) model


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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


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22

Hence we obtain equilibrium Y in terms of equilibrium X.

ADL(1,1) model

Hence


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23

Hence we infer the cointegrating relationship.

ADL(1,1) model

Hence

In equilibrium

Cointegrating relationship

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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


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25

Hence we obtain the error correction model shown.

ADL(1,1) model


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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


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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


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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


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29

Hence the model may be fitted using least squares in the standard way.

ADL(1,1) model


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30

Of course, the parameters are not known and the cointegrating term is unobservable.

ADL(1,1) model


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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


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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


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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 ============================================================



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.

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35

ZFOOD(–1), the lagged residual from the cointegrating regression, is the cointegrating term.

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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.

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37

The coefficient of the cointegrating term indicates that about 15 percent of the disequilibrium divergence tends to be eliminated in one year.

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Copyright Christopher Dougherty 2011.

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used as a resource for teaching an econometrics course. There is no need to

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The content of this slideshow comes from Section 13.6 of C. Dougherty,

Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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Christopher Dougherty EC220 - Introduction to econometrics (chapter 13) Slideshow: fitting models with nonstationary time series Original citation: Dougherty,