As witnessed by the number of journal articles published in this area, unit root tests have been extensively studied. Here, I will review some of these papers that I will introduce sorted by topic rather than by chronology.
The paper Granger and Newbold (1974) can be regarded as the origin of this specific area in Econometrics. These authors emphasized the consequences --already warned on textbooks at that time-- of the presence of autocorrelation in the residuals (the model is misspecified, regardless of the value of the coefficient of determination, R2). As a general recommendation in the context of economic data, the authors proposed also specifying a regression model for the first differences of the series rather than the levels.
The question of deterministic and stochastic trends
The latter idea of differencing the data is more closely related to the subject of unit root tests. In this sense, the papers Dickey and Fuller (1979) and Phillips (1987) showed that, in the presence of a unit root, the test statistics for the significance of deterministic components, such as a linear trend, do not follow the traditional distributions.
Soon around that time, the first test statistics for testing the null of a unit root and the corresponding tabulated critical values were published. Dickey and Fuller (1981), Said and Dickey (1984) and Phillips and Perron (1988). Kwiatkowski, Phillips, Schmidt and Shin (1992) reversed the null hypothesis and designed a statistic to test the null of stationarity (possibly around a linear trend) against a unit root.
Before the publication of these enlightening works, the standard approach to detrend a time series was by means of a deterministic linear trend that is fitted to the data. After the above-mentioned papers, the question whether a deterministic trend or a stochastic trend should be modelled became an issue to be explored as a preliminary stage in applied works in econometrics and time series analysis.
The question extends to seasonality
This issue was naturally extended to the seasonal component observed in most series. Seasonal unit root tests were developed to address the question whether seasonality should be modelled as a deterministic or as a stochastic component. Hylleberg, Engle, Granger and Yoo (1990) and Beaulieu and Miron (1993) extend unit root tests for quarterly and monthly data, respectively. Canova and Hansen (1995) propose a statistic to test for a stationary seasonal component against a stochastic component.
Critics and weaknesses
Test statistic for unit roots did not emerge without criticism, in some cases due to the interdisciplinary nature of Econometrics.
- Some practitioners were puzzled with the interpretation of unit roots. Some claimed that a random walk (a pure stochastic trend model) could not be supported by economic theories, because it involves an everlasting effect of shocks, as opposed to the short-term persistence of shocks that is observed in most macroeconomic data. This critic was concisely captured by the statement that a stochastic trend implies that summer may become winter, which does not fit with well with economic theories and reasoning.
- Unit root tests were found to have low power and could be distorted by the presence of level shifts or structural changes.
The first critic does not seem an issue any more among econometricians, probably because the dangers of ignoring a unit root are more severe than the converse (assume a deterministic trend when there is a unit root). A deterministic trend does not always agree with economic theories and intuition either, especially when it comes to forecasting. Thus, working with the differenced data instead of the level of the series has become the common practice.
The second critic revealed a weakness of unit root tests. Perron (1989) warned that the presence of level shifts or structural breaks may distort the original unit root tests and may result in wrong conclusions. This led to another wave of journal articles addressing this issue, among them, Franses and Vogelsang (1998), Hassler and Rodrigues (2004). The performance of unit root tests has also been investigated under a Markov-switching-regime model, Nelson, Piger and Zivot (2000).
Lag order selection methods
In order to deal with possible serial correlation in the disturbance term, the regression used in the Dickey-Fuller test is extended with lags of the dependent variable. Different methods to choose the appropriate lag order have been proposed in several papers. Some of them are based on the Akaike's and Schwarz's information criteria, others follow a top-down strategy where lags are tested for significance starting from a maximum lag order, others follow a bottom-up approach. Some references studying this issue are: Ng and Perron (1995), Ng and Pierre Perron (2001), Hylleberg (1995) and Burridge and Taylor (2001).
Variations of the Dickey-Fuller test
In order to improve the power of unit root tests, some variations based on the Dickey-Fuller test have been designed. The weighted least squares estimator has been proposed and assessed as an alternative to deal with serial correlation, Park and Fuller (1995). Leybourne (1995) and Leybourne and Taylor (2003) propose applying the Dickey-Fuller test for the original series and for the reverse series and take the maximum of both statistics as the test statistic, for which new critical values were provided.
Bootstrap comes along
Considering the issues mentioned above, bootstrap methods qualify as a potential tool to improve the performance of these tests with real data. Bootstrapping time-dependent data is not straightforward and different attempts and approaches have been designed. Chang and Park (2003) apply the sieve-bootstrap to the augmented Dickey-Fuller test. Burridge and Taylor (2003) describe a bootstrapping method for the HEGY test for seasonal unit roots. Psaradakis (2000) propose a bootstrap test for unit roots in a seasonal autoregressive model. A grid bootstrap is proposed by Hansen (1999) to obtain a confidence interval in a first order autoregressive model.
Response surface analysis
The distribution of unit root test is often non-standard and tabulated values obtained by means of simulations have to be employed. The critical values depend on the model specification (e.g. including a constant or not), the sample size, the periodicity of the data and on possible lags included in the regression. Simulations for the most common combination of these variables are provided in the original papers. Those table that not cover all cases, interpolation needs to be somehow applied and sometimes critical values are not available in the original paper (for example test for seasonal root in series of frequency other than quarterly or monthly).
In the last years, the approach described in MacKinnon (1996) has been applied to obtain approximations to the distribution of the unit root test statistics. The idea involves extensive simulations where several quantiles of the distribution of the statistic are obtained and used to apply response surface regressions where critical values as well as the corresponding p-value are estimated. MacKinnon (1994) uses this idea to obtain critical values for the DF test, Cheung and Lai (1995) do the same exercise for the augmented DF test, Harvey and van Dijk (2006) follow the same idea for the HEGY test, Otero and Smith (2012) apply this technique to the Leybourne (1995) unit root test mentioned above, Díaz-Emparanza (2014) uses response surface regressions to generalize the HEGY test to series of any periodicity.
Perhaps, the topics that I mention below were developed separately in other fields, but they are clearly related to the general setting of unit root tests. Thus, I would say that the following topics can be considered a branch of the studies mentioned so far: periodic integration, Osborn, Chui, Smith and Birchenhall (1988), Boswijk and Franses (1995); fractional integration; near unit root processes, Phillips, Moon and Xiao (2001).
Despite the potential problems of blindly using deterministic trends, the issue of deterministic versus stochastic trends has not been extended to all fields, it seems an issue intrinsic to economic data. For example, in climatology it is still common to fit linear trends as a measure and description of temperatures trends. A discussion is given in this article (López-de-Lacalle, 2012), which is summarized in this previous post of this blog.
Some people became tired of finding the same topic published on journals and felt that researchers were overlooking more interesting areas or research. I heard that a reputable researcher withdrew the subscription to a journal because of the large number of papers that were being published about unit root tests.
For me, the literature on unit root tests has been a great topic that introduced me in the field of time series analysis. I wrote some of my first software programs and conducted simulation experiments in this context. It allowed me to get a hands-on understanding of what a distribution is and what it entails, what a null hypothesis is. This are simple concepts but experimenting with them gave me some views that I couldn't get from textbooks and journal articles.