I would like to thank three anonymous referee, co-editor, Nikolay Gospodinov, Pierre Perron, Barbara Rossi, Jim Stock, and participants of the Econometric Seminar at Harvard and Boston University, the Winter Econometric Society Meeting 2008, and the CIREQ conference in Montreal 2009 for comments. I am grateful to Denis Chetverikov, J. B. Doyle, and Ruitian Lang for research assistance. Financial support from the Castle–Krob Career Development Chair is gratefully acknowledged.
One-Dimensional Inference in Autoregressive Models With the Potential Presence of a Unit Root
Article first published online: 10 JAN 2012
© 2012 The Econometric Society
Volume 80, Issue 1, pages 173–212, January 2012
How to Cite
Mikusheva, A. (2012), One-Dimensional Inference in Autoregressive Models With the Potential Presence of a Unit Root. Econometrica, 80: 173–212. doi: 10.3982/ECTA9371
- Issue published online: 10 JAN 2012
- Article first published online: 10 JAN 2012
- Manuscript received June, 2010; final revision received June, 2011.
- Impulse response;
- grid bootstrap;
- uniform inferences
This paper examines the problem of testing and confidence set construction for one-dimensional functions of the coefficients in autoregressive (AR(p)) models with potentially persistent time series. The primary example concerns inference on impulse responses. A new asymptotic framework is suggested and some new theoretical properties of known procedures are demonstrated. I show that the likelihood ratio (LR) and LR± statistics for a linear hypothesis in an AR(p) can be uniformly approximated by a weighted average of local-to-unity and normal distributions. The corresponding weights depend on the weight placed on the largest root in the null hypothesis. The suggested approximation is uniform over the set of all linear hypotheses. The same family of distributions approximates the LR and LR± statistics for tests about impulse responses, and the approximation is uniform over the horizon of the impulse response. I establish the size properties of tests about impulse responses proposed by Inoue and Kilian (2002) and Gospodinov (2004), and theoretically explain some of the empirical findings of Pesavento and Rossi (2007). An adaptation of the grid bootstrap for impulse response functions is suggested and its properties are examined.