Testing for unit roots in autoregressive-moving average models of unknown order: Critical comments

Statistical theory relating to the first order autoregressive unit root process where the autoregressive parameter is equal to one (unstable process) and greater than one (explosive process) dates backs to the writings of Mann and Wald (1943), Rubin (1950), Anderson (1959), White (1958, 1959), and Rao (1961). However, Maddala and Kim (1998) together with Banerjee et. all (1993) and Phillips (1995) give the impression that the seminal work of Nelson and Plosser (1982) marked a paradigm shift in the macroeconomics literature in the 1980s. In fact, in the 1960s and 1970s, conventional practice in time series analysis was to work with data that were differenced a sufficient number of times to render them stationary. This practice was based on informal diagnostics rather than formal statistical tests and Nelson and Plosser (1982) replaced the practice with formal Dickey-Fuller (DF) tests for unit roots (Stock 1994).

The literature on unit roots remains controversial because many test results of unit roots rest on a razor’s edge. Although the literature is extensive and often studies failed to reject the hypothesis, many also failed to reject the hypothesis  at 95 percent, or even 99 percent, confidence interval levels. An early critique by Maddala (1992), Harvey (1997), Maddala and Kim (1998), Phillips (2003) and a recent critique by Moosa (2011), and Luitel and Mahar (2015a, 2015b, 2016) highlight further the continuing controversy in the unit root literature.

On the topic of testing for unit roots, the 1984 paper by Said and Dickey was the 20th most influential paper out of the top 100 highly influential papers published by Biometrika since 1936 (Titterington, 2013). Since the unit root tests have made inroads into other disciplines beyond economics, for example, research work into climate change (Kaufmann et al. 2013), it is time for a critical review of this method.

We have two objectives in this paper: First, we reexamine the reliability of unit root findings in the study by Said and Dickey (1984). We can reaffirm the internal validity of their research design. Second, we found new results not included in that study and explain why the new results cast doubt on the usefulness of the ADF test to determine a unit root in a time series.