FITTING MODELS WITH NONSTATIONARY TIME SERIES 1 Detrending 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.

FITTING MODELS WITH NONSTATIONARY TIME SERIES 2 Detrending We will briefly consider three of them: detrending the variables in a relationship, differencing them, and constructing error-correction models.

FITTING MODELS WITH NONSTATIONARY TIME SERIES 3 Detrending 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.

FITTING MODELS WITH NONSTATIONARY TIME SERIES 4 Detrending Problems if X t is a random walk 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.

FITTING MODELS WITH NONSTATIONARY TIME SERIES 5 Detrending Problems if X t is a random walk H 0 :  2 = 0 rejected more often than it should be (standard error underestimated) In particular, if a random walk X t is regressed on a time trend as in the equation at the top, the null hypothesis H 0 :  2 = 0 is likely to be rejected more often than it should, given the significance level.

FITTING MODELS WITH NONSTATIONARY TIME SERIES 6 Detrending Problems if X t is a random walk H 0 :  2 = 0 rejected more often than it should be (standard error underestimated) Although the least squares estimator of  2 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.

FITTING MODELS WITH NONSTATIONARY TIME SERIES 7 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. Detrending Problems if X t is a random walk H 0 :  2 = 0 rejected more often than it should be (standard error underestimated) Trend in variance not removed

FITTING MODELS WITH NONSTATIONARY TIME SERIES 8 Detrending Problems if X t is a random walk with drift H 0 :  2 = 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.

FITTING MODELS WITH NONSTATIONARY TIME SERIES 9 Detrending Problems if X t is a random walk with drift H 0 :  2 = 0 rejected more often than it should be (standard error underestimated) Trend in variance not removed Thus if X t 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.

FITTING MODELS WITH NONSTATIONARY TIME SERIES 10 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. Differencing

FITTING MODELS WITH NONSTATIONARY TIME SERIES 11 Of course differencing overcompensated for the autocorrelation, but if  was near 1, the resulting weak negative autocorrelation was held to be relatively innocuous. Differencing

FITTING MODELS WITH NONSTATIONARY TIME SERIES 12 Unknown to practitioners of the time, the procedure is an effective antidote to spurious regressions. 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. Differencing

FITTING MODELS WITH NONSTATIONARY TIME SERIES 13 A major shortcoming of differencing is that it precludes the investigation of a long-run relationship. Differencing Problem Short-run dynamics only

FITTING MODELS WITH NONSTATIONARY TIME SERIES 14 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. Differencing Problem Short-run dynamics only Equilibrium:

FITTING MODELS WITH NONSTATIONARY TIME SERIES 15 An error-correction model is an ingenious way of resolving this problem by combining a long-run cointegrating relationship with short-run dynamics. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 16 Suppose that the relationship between two I(1) variables Y t and X t is characterized by an ADL(1,1) model. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 17 In equilibrium we would have the relationship shown. This is the cointegrating relationship. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 18 The ADL(1,1) relationship may be rewritten to incorporate this relationship. First we subtract Y t–1 from both sides. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 19 Then we add  3 X t–1 to the right side and subtract it again. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 20 We then rearrange the equation as shown. We will look at the rearrangement, term by term. First, the intercept  1. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 21 Next, the term involving Y t–1. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 22 Now the next two terms. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 23 Finally, the last two terms. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 24 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 Y t–1 and the value predicted by the cointegrating relationship. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 25 The latter term is denoted the error-correction mechanism. The effect of this term is to reduce the discrepancy between Y t and its cointegrating level. The size of the adjustment is proportional to the discrepancy. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 26 The point of rearranging the ADL(1,1) model in this way is that, although Y t and X t 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. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 27 Of course, the  parameters are not known and the cointegrating term is unobservable. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 28 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. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 29 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. Error-correction models ADL(1,1)

FITTING MODELS WITH NONSTATIONARY TIME SERIES 30 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. ============================================================ Dependent Variable: DLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-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 ============================================================

Dependent Variable: DLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-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 31 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.

============================================================ Dependent Variable: DLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-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 32 ZFOOD(–1), the lagged residual from the cointegrating regression, is the cointegration term.

============================================================ Dependent Variable: DLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-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 33 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.

============================================================ Dependent Variable: DLGFOOD Method: Least Squares Sample(adjusted): Included observations: 44 after adjusting endpoints ============================================================ Variable CoefficientStd. Errort-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 The coefficient of the cointegrating term indicates that about 15 percent of the disequilibrium divergence tends to be eliminated in one year.

Copyright Christopher Dougherty 2002–2006. This slideshow may be freely copied for personal use