James Duffy

This paper considers highly persistent time series that are subject to nonlinearities in the form of censoring or an occasionally binding constraint, such as are regularly
encountered in macroeconomics. A tractable candidate model for such series is the dynamic
Tobit with a root local to unity. We show that this model generates a process that converges
weakly to a non-standard limiting process, that is constrained (regulated) to be positive.
Surprisingly, despite the presence of censoring, the OLS estimators of the model parameters
are consistent. We show that this allows OLS-based inferences to be drawn on the overall
persistence of the process (as measured by the sum of the autoregressive coefficients), and for
the null of a unit root to be tested in the presence of censoring. Our simulations illustrate
that the conventional ADF test substantially over-rejects when the data is generated by a
dynamic Tobit with a unit root, whereas our proposed test is correctly sized. We provide an
application of our methods to testing for a unit root in the Swiss franc / euro exchange rate,
during a period when this was subject to an occasionally binding lower bound.

We provide new limit theory for functionals of a general class of processes lying at the boundary between stationarity and nonstationarity -- what we term weakly nonstationary processes (WNPs). This includes, as leading examples, fractional processes with d=1/2, and arrays of autoregressive processes with roots drifting slowly towards unity. We first apply the theory to study inference in parametric and nonparametric regression models involving WNPs as covariates. We then use these results to develop a new specification test for parametric regression models. By construction, our specification test statistic has a chi-squared limiting distribution regardless of the form and extent of persistence of the regressor, implying that a practitioner can validly perform the test using a fixed critical value, while remaining agnostic about the mechanism generating the regressor. Simulation exercises confirm that the test controls size across a wide range of data generating processes, and outperforms a comparable test due to Wang and Phillips (2012, Ann. Stat.) against many alternatives.