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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: 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.

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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 Model Fit Define Fit Detrending

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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. Model Fit Define Fit Detrending

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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. Model Fit Define Fit Detrending

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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. Model Fit Define Fit Detrending

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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. Model Fit Define Fit Equivalently, Detrending

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FITTING MODELS WITH NONSTATIONARY TIME SERIES 6 However there are potential problems with this approach. Model Fit Define Fit Equivalently, Detrending

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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. Model Fit Define Fit Equivalently, Detrending

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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. Model Fit Define Fit Equivalently, Detrending Standard error biased downwards when random walk regressed on a trend. Risk of Type I error underestimated.

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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. Standard error biased downwards when random walk regressed on a trend. Risk of Type I error underestimated. Model Fit Define Fit Equivalently, Detrending

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FITTING MODELS WITH NONSTATIONARY TIME SERIES 10 Further, if a series is difference-stationary, the procedure does not make it stationary. Detrending does not make a random walk stationary. Model Fit Define Fit Equivalently, Detrending

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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. 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.

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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. 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

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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 Model Fit Define Fit Increasing variance has adverse consequences for estimation and inference. Equivalently,

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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. Model AR(1) auto– correlation Difference Differencing

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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. Model AR(1) auto– correlation Difference Differencing

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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 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. Model AR(1) auto– correlation Difference Differencing

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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 Model AR(1) auto– correlation Difference Procedure does not allow determination of long-run relationship In equilibrium Model becomes

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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 Error correction model

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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 Error correction model

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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 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

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FITTING MODELS WITH NONSTATIONARY TIME SERIES 21 It will be convenient to rehearse the theory. Suppose that the relationship between two I(1) variables Y t and X t is characterized by the ADL(1,1) model. In equilibrium, we have the relationship shown. ADL(1,1) model In equilibrium Error correction model

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FITTING MODELS WITH NONSTATIONARY TIME SERIES 22 Hence we obtain equilibrium Y in terms of equilibrium X. ADL(1,1) model Hence In equilibrium Error correction model

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FITTING MODELS WITH NONSTATIONARY TIME SERIES 23 Hence we infer the cointegrating relationship. ADL(1,1) model Hence In equilibrium Cointegrating relationship Error correction model

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FITTING MODELS WITH NONSTATIONARY TIME SERIES 24 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. ADL(1,1) model Cointegrating relationship Error correction model

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FITTING MODELS WITH NONSTATIONARY TIME SERIES 25 Hence we obtain the error correction model shown. ADL(1,1) model Cointegrating relationship 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 Y t–1 and the value predicted by the cointegrating relationship. ADL(1,1) model Cointegrating relationship Error correction model

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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 Y t and its cointegrating level and its size being proportional to the discrepancy. ADL(1,1) model Cointegrating relationship Error correction model

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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), 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. ADL(1,1) model Cointegrating relationship Error correction model

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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 Error correction model

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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 Error correction model

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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 Error correction model

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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 Error correction model

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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): Included observations: 44 after adjusting endpoints ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ ZFOOD(-1) DLGDPI DLPRFOOD ============================================================ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat ============================================================

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Dependent Variable: DLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ ZFOOD(-1) DLGDPI DLPRFOOD ============================================================ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat ============================================================ 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.

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============================================================ Dependent Variable: DLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ ZFOOD(-1) DLGDPI DLPRFOOD ============================================================ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat ============================================================ FITTING MODELS WITH NONSTATIONARY TIME SERIES 35 ZFOOD(–1), the lagged residual from the cointegrating regression, is the cointegrating term.

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============================================================ Dependent Variable: DLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ ZFOOD(-1) DLGDPI DLPRFOOD ============================================================ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat ============================================================ 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.

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============================================================ Dependent Variable: DLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ ZFOOD(-1) DLGDPI DLPRFOOD ============================================================ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criter Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat ============================================================ 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.

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Copyright Christopher Dougherty 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 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 or the University of London International Programmes distance learning course 20 Elements of Econometrics

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