Deterministic and stochastic trends in the time series models: a guide for the applied economist

Abstract

Applied economists working with time series data face a dilemma in selecting between models with deterministic and stochastic trends. While models with deterministic trends are widely used, models with stochastic trends are not so well known. In an influential paper Harvey (1997) strongly advocates a structural time series approach with stochastic trends in place of the widely used autoregressive models based on unit root tests and cointegration techniques. Therefore, it is important to understand their relative merits. This article suggests that both methodologies are useful and they may perform differently in different models. This article provides a few guidelines to the applied economists to understand these alternative methods.

Acknowledgement

I thank two referees of this journal, Gyaneshwar Rao, Rup Singh and Saten Kumar of the University of the South Pacific and Dr Arusha Cooray of the University of Tasmania for help and useful suggestions. All errors are my responsibility.

Notes

¹ Whether cointegration methods can be applied to relationships with breaks in trends where they are captured with shift dummies is addressed more fully by Muscatelli and Hurn (1992, p. 13).

² Harvey has other reservations on the mainstream time series work with autoregressive equations, unit root tests and cointegration techniques. His main reservation is that these tests and techniques are unreliable due to their lack power. Towards the end of his article he says that‘… autoregressions, whether univariate or multivariate, are very limited in scope. The recent emphasis on unit roots, vector autoregressions and cointegration has focused too much attention on tackling uninteresting problems by flawed methods’.

³ Before time series econometrics made an impact, economists and econometricians at the London School of Economics (LSE) were aware that economic theory is silent on the dynamics of the relationships. Some of these academics were Phillips (Phillips curve fame), Sargan, Lipsey, Mizon and Hendry. They were unsatisfied with the then popular partial adjustment model of dynamics and the general to specific approach (GETS) was developed as an alternative. Negative feedbacks were introduced through Phillips's error correction mechanism (ECM) into GETS. GETS takes the view that dynamics is an empirical issue and through various procedures it is possible to reduce a very general and long dynamic structure into parsimonious specifications. Later developments in the cointegration techniques have used GETS approach to estimate the short-run dynamic equations. Hendry and Mizon are the most well known contemporary proponents of GETS.

⁴ According to them the view that all macroeconomic series are difference-stationary, or that all macroeconomic series are trend-stationary or that we simply do not know and do not care are not tenable and a case-by-case analysis is absolutely necessary. However, Diebold and Senhadji apply their approach to determine if unit root tests are decisively capable of determining if US GNP falls into one of these three categories. Those who believe that GNP is trend stationary have argued that either unit root tests like the ADF test do not have adequate power against the null of unit roots or that if a large enough sample is taken, GNP is likely to be trend stationary. Consequently, the view that ‘we really do not know and therefore do not care’ is popular among some economists. Diebold and Senhadji show that this view is untenable and it is necessary to determine whether variables like output are trend or difference stationary. In this respect we take a similar view. However, our view is not based on the power of unit root tests but on the plausibility of the estimated parameters and how well the two alternative assumptions about trend fit the data.

⁵ Estimation of cointegrating equations with GETS is explained in Rao (2007). For the cointegration the Ericsson and MacKinnon (2002) response surface test is used.

⁶ To conserve space these test results are not reported and may be obtained from the author.

⁷ Strictly speaking estimating cointegrating equations with structural breaks and shift dummies is not satisfactory procedure. However, there is a gap in the literature here. There are only two known procedures to estimate cointegrating equations with structural breaks. First, Gregory and Hansen (1992) developed a method where the two step Engle–Granger equation can be estimated with one endogenous structural break. Second, Juselius (1996) developed a procedure to estimate cointegrating equations with a known break date with the Johansen method.

⁸ Needless to say this conclusion based only on a sample of one experience has many caveats and further experiments are necessary to draw conclusions with more confidence.

⁹ A proof of this is beyond the scope of this article.

¹⁰ The decline in the growth rate seems to have started earlier than the 1997 East Asian financial crisis. The Hodrick–Prescott filtered growth rate also showed 1993 as the beginning of the downturn in the growth rate of Singapore. When 1997 or 1998 was selected as plausible break dates the estimated equations were unsatisfactory in that the share of profits was close to zero and insignificant. The reasons why Singaporean economy started slowing down from 1993 are beyond the scope of this article.