The Local to Unity Dynamic Tobit Model

TL;DR

This paper develops a new approach to analyze highly persistent time series with nonlinear censoring, showing that OLS estimators remain consistent and proposing tests for unit roots in such models.

Contribution

It introduces a dynamic Tobit model with a local to unity root, demonstrating weak convergence to a constrained non-standard process and enabling valid inference despite censoring.

Findings

OLS estimators are consistent despite censoring.

The proposed test correctly sizes when data has a unit root.

Conventional ADF test over-rejects under the model.

Abstract

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.