FITTING MODELS WITH NONSTATIONARY TIME SERIES 1 Detrending Early macroeconomic models tended to produce poor forecasts, despite having excellent sample-period

FITTING MODELS WITH NONSTATIONARY TIME SERIES

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Detrending

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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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Detrending

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We will briefly consider three of them: detrending the variables in a relationship, differencing them, and constructing error-correction models.

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Detrending

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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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Detrending

Problems if Xt is a random walk

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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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Detrending


H0: 2 = 0 rejected more often than it should be(standard error underestimated)

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In particular, if a random walk Xt is regressed on a time trend as in the equation at the top, the null hypothesis H0: 2 = 0 is likely to be rejected more often than it should, given the significance level.

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Detrending



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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.

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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



Trend in variance not removed

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Detrending

Problems if Xt is a random walk with drift



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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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Detrending

Problems if Xt is a random walk with drift



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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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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

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Of course differencing overcompensated for the autocorrelation, but if was near 1, the resulting weak negative autocorrelation was held to be relatively innocuous.

Differencing

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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.

Differencing

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A major shortcoming of differencing is that it precludes the investigation of a long-run relationship.

Differencing

Problem

Short-run dynamics only

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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:

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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

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Suppose that the relationship between two I(1) variables Yt and Xt is characterized by an ADL(1,1) model.

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In equilibrium we would have the relationship shown. This is the cointegrating relationship.

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The ADL(1,1) relationship may be rewritten to incorporate this relationship. First we subtract Yt–1 from both sides.

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Then we add 3Xt–1 to the right side and subtract it again.

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We then rearrange the equation as shown. We will look at the rearrangement, term by term. First, the intercept 1.

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Next, the term involving Yt–1.

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Now the next two terms.

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Finally, the last two terms.

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


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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.


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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.


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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): 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 ============================================================



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

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

22.08.06

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FITTING MODELS WITH NONSTATIONARY TIME SERIES 1 Detrending Early macroeconomic models tended to produce poor forecasts, despite having excellent sample-period